Biomechanical Modelling of the Human Eye

Home , Globe (human eye), Human eye

JOHANNESKEPLER UNIVERSITÄTLINZ Netzwerk für Forschung, Lehre und Praxis

Dissertation

zur Erlangung des akademischen Grades

Doktor der Technischen Wissenschaften

im Doktoratsstudium der technischen Wissenschaften

Angefertigt am Institut für Anwendungsorientierte Wissensverarbeitung (FAW)

Eingereicht von:

Dipl.-Ing. (FH) Michael Buchberger

Betreuung:

Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner

Beurteilung: Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner

Univ.-Doz. Dipl.-Ing. Dr. Thomas Haslwanter

Linz, März 2004

Johannes Kepler Universität A-4040 Linz · Altenbergerstraße 69 · Internet: http://www.uni-linz.ac.at · DVR 0093696

Eidesstattliche Erklärung

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe.

Linz, im März 2004 Michael Buchberger To Bianca and my parents... Abstract

The goal of this work was the development of a biomechanical model of the human eye. An interactive software system was implemented, called „SEE++“ which allows also physicians to obtain a better understanding of the mechanics of eye movements. This software visualizes and simulates pathologies and eye muscle surgeries, based on the biomechanics of the eye. It can be used in preoperative planning, medical training and basic research, and shows how Medical- Informatics can improve the diagnosis and treatment of patients.

The interdisciplinary nature of the project required contributions from very different fields. Anatomical studies, in cooperation with researchers as well as practicing physicians, provided data for defining a mathematical representation of human eye movements. The biomechanical model included a geometrical representation of eye movements, a muscle force prediction model, and a kinematic model that balances muscle forces by using mathematical optimization methods. High-resolution magnetic resonance imaging studies were carried out to visualize eye muscle morphology, and image processing methods used to reconstruct three dimensional approximation models of human eye muscles. Modern software engineering methods provided the basis for an extensible object-oriented software design. Three dimensional interactive visualization and a user interface optimized for medical use were combined into a unique software simulation system for the clinic and for teaching.

The „SEE++“ software system is currently the most advanced biomechanical representation of the human eye, with respect to simulating eye movements and eye muscle surgeries. The extensive possibilities for parametrization of the human eye model allow interactive simulations of pathological cases and surgical corrections, and the predictions correspond well with clinical data. This system is used in various clinical facilities as computer aided decision support for strabismus (squint) surgeries. In medical training and education, it substantially improves the functional understanding of human eye movements.

Keywords: Biomechanical Modelling, Eye Surgery, Strabismus, Eye Motility, Medical Decision Support Systems

v Kurzfassung

Diese Forschungsarbeit hat das Ziel, ein biomechanisches Modell des menschlichen Auges zu entwickeln. Das implementierte Software System, „SEE++“, soll es Medizinern ermöglichen, ein besseres Verständnis der Mechanik der Augenbewegungen zu bekommen. Augenbewegungsstö- rungen und Augenmuskeloperationen werden dabei auf biomechanische Ursachen und Wirkungen zurückgeführt. Der erfolgreiche klinische Einsatz dieses Systems zeigt, wie computerbasierte Me- thoden der Medizin-Informatik die Diagnose und Behandlung von Patienten verbessern können.

Die interdisziplinären Anforderungen dieses Projekts erforderten Beiträge aus stark unterschied- lichen medizinisch-technischen Forschungsbereichen. Anatomische Studien lieferten Grunddaten für die Formulierung eines mathematischen Modells der menschlichen Augebewegungen. Biome- chanische Überlegungen führten zu einer geometrischen Beschreibung von Augenbewegungen, einer Muskelkraftsimulation und eines kinematischen Modells. Augenpositionen wurden mit mathematischen Optimierungsmethoden aus dem Kräftegleichgewicht der Augenmuskulatur berechnet. Um die Morphologie der Augenmuskulatur besser zu verstehen, wurden umfangreiche Studien mit hochauﬂösender Magnetresonanztomographie durchgeführt, und mit Bildverarbeitungsme- thoden die dreidimensionalen Rekonstruktionen berechnet. Der Einsatz von modernen Methoden des objektorientierten Software-Engineering bildete die Grundlage für eine ﬂexible Implementie- rung. Dreidimensionale interaktive Visualisierung und optimiertes Benutzerschnittstellen-Design wurden in einem einzigartigen Software System kombiniert.

Das biomechanische Simulationssystem „SEE++“ ist derzeit das weltweit detaillierteste und mo- dernste Softwaresystem für die Modellierung und Simulation von Augenbewegungsstörungen. Das System ermöglicht durch umfangreiche Möglichkeiten der Parametrisierung die Simulation von pathologischen Fällen und deren operative Korrektur. Die Simulationsergebnisse zeigen nach- weislich eine gute Übereinstimmung mit verfügbaren klinischen Vergleichsdaten. Derzeit wird dieses System für die computerbasierte Entscheidungsunterstützung in verschiedenen klinischen Einrichtungen verwendet. Der Einsatz in der medizinischen Ausbildung verbessert das Verständ- nis über Funktion und Wirkungsweise von menschlichen Augenbewegungen. In der medizinisch- physiologischen Grundlagenforschung ermöglicht das System die Auswertung und Evaluierung von Messergebnissen, und gibt somit einen detaillierteren Einblick in die komplizierte Struktur des menschlichen Auges.

Schlüsselwörter: Biomechanische Modelle, Augenoperation, Strabismus, Augenmotilität, Klini- sche Entscheidungsunterstützung

vi Acknowledgements

Within eight years of research, many people have greatly contributed to this work. First of all, I would like to express my deepest gratitude, respect and admiration to Prim. Prof. Dr. Siegfried Priglinger, head of the ophthalmologic department at the convent hospital of the „Barmherzigen Brüder“ in Linz. He did not only start this project, but also greatly contributed to this work as teacher, mastermind and highly experienced medical expert, always emphasizing improvement in patient care. His exceptional personality and social engagement inspired me beyond my technical work.

I would like to thank Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner, head of the Research Institute for Applied Knowledge Processing (FAW) at the University of Linz for reviewing and supporting this work.

From the ETH-Zurich, I would like to thank Univ.-Doz. Dipl.-Ing. Dr. Thomas Haslwanter for explaining medical details and giving valuable advice and guidance throughout the creation of this thesis. From the medical side, many physicians and researchers have been involved in this research work. For valuable cooperation I would like to thank Dr. Joel Miller from the Smith Kettlewell Eye Research Institute. Furthermore, Prim. Univ.-Prof. Dr. Erich Salomonowitz, Prim. Univ.-Doz. DDr. Armin Ettl and Dr. Jörg Hildebrandt from the hospital St. Pölten supported this work by providing clinical patient data. Additionally, the radiologic department of the Wagner Jauregg hospital in Linz lead by Prim. Dr. Johannes Trenkler with the help of Univ.-Doz. Dr. Franz Fellner and Univ.-Doz. DDr. Dipl.-Ing. Mag. Josef Kramer provided necessary equipment and medical expert knowledge to carry out diﬀerent physiological studies.

For usability testing and evaluation of the software system „SEE++“ I would like to thank Univ.- Prof. Dr. Andrea Langmann from the university in Graz for believing in this work. From the university of Innsbruck I appreciate the help of OA Dr. Eduard Schmid, OA Dr. Ivo Baldissera and OA Dr. Cornelia Stieldorf. From the hospital of the „Barmherzigen Schwestern“ in Ried I would like to thank OA Dr. Robert Hörantner for extensively testing the software and providing valuable feedback.

From the Upper Austrian University of Applied Sciences in Hagenberg I would especially like to thank Univ.-Prof. Dipl.-Ing. Dr. Witold Jacak for initiating and greatly supporting the „SEE- KID“ project. Moreover, thanks go to FH-Prof. Dipl.-Ing. Dr. Herwig Mayr, who also supported the project in its beginnings with his project engineering knowledge.

Many diploma students were involved within this project. First of all, I would like to give my respect to Dipl.-Ing. (FH) Thomas Kaltofen for extensive implementation work and above- average participation within this work. Additionally, Dipl.-Ing. (FH) Martin Wiesmair, Dipl.-Ing. (FH) Franz Pirklbauer, Dipl.-Ing. (FH) Stefan Satzinger and Dipl.-Ing. (FH) Michael Lacher spent their internship and diploma semester in the ﬁeld of the „SEE-KID“ project. I would also like to thank Dipl.-Ing. (FH) Thomas Kern, Dipl.-Ing. (FH) Johannes Dirnberger, Mag. Michael Giretzlehner and Dr. Thomas Luckeneder for proof reading this work.

Last but not least, I would like to thank Dipl.-Ing. Dr. Otmar Höglinger from the Upper Austrian Research GmbH for believing in new technologies and providing a fascinating infrastructure for application-oriented research in the ﬁeld of Medical-Informatics. This work was also supported by grants from the Austrian Ministry of Science (FFF) and the Upper Austrian government.

vii Contents

1 Introduction 1 1.1 Overview ...... 1 1.2 Medical Informatics ...... 3 1.3 Clinical Decision Support ...... 4 1.4 Eye muscle surgery ...... 6 1.5 SEE-KID ...... 7

2 Medical Foundations 9 2.1 Anatomy of the Human Eye ...... 11 2.1.1 Eye Muscle „Pulleys“ ...... 16 2.1.2 Global/Orbital Eye Muscle Layers ...... 18 2.1.3 Anatomical Measurements ...... 19 2.1.4 High Resolution MRI-Imaging of the Orbit ...... 21 2.1.5 Human Dissection of the Orbit ...... 24 2.2 Eye Movement Physiology ...... 26 2.2.1 Actions of the Extraocular Muscles ...... 27 2.2.2 Kinematic Principles of Eye Movements ...... 29 2.2.2.1 Donders’ Law ...... 30 2.2.2.2 Listing’s Law ...... 30 2.2.3 Sensorimotor Control ...... 33 2.2.3.1 Innervation of the Eye Muscles ...... 34 2.2.3.2 Oculomotor Neurons ...... 35 2.2.3.3 Neural Signal Encoding ...... 36 2.2.3.4 The Superior Colliculus ...... 37 2.2.3.5 Brainstem Control of Saccades ...... 38 2.2.3.6 Control of Smooth-Pursuit Movements ...... 40 2.2.3.7 Hering’s Law ...... 40 2.2.3.8 Sherrington’s Law ...... 41 2.3 Measurement Techniques ...... 41 2.3.1 Eye Movement Measurements ...... 42 2.3.1.1 Electro-Oculography ...... 42 2.3.1.2 Infrared-Oculography ...... 43 2.3.1.3 Scleral Search Coils ...... 44 2.3.1.4 Video-Oculography ...... 46 2.3.2 Physiologic Muscle Force Measurements ...... 47 2.3.3 Measurement of Motion in the Orbit ...... 51

viii 3 Strabismus 53 3.1 Overview ...... 54 3.1.1 Visual Acuity ...... 54 3.1.2 Symptoms ...... 54 3.1.3 Treatment ...... 55 3.2 Binocular Vision ...... 55 3.2.1 Projection ...... 57 3.2.2 Diplopia ...... 58 3.3 Ocular Dissociation ...... 58 3.4 Clinical Assessment ...... 61 3.4.1 Corneal Light Reﬂex Tests ...... 62 3.4.2 Cover Tests ...... 64 3.4.3 Subjective Clinical Tests ...... 65 3.4.4 Hess-Lancaster Test ...... 66 3.5 Eye Motility Disorders ...... 69 3.5.1 Concomitant Strabismus ...... 70 3.5.2 Incomitant Strabismus ...... 71 3.5.2.1 Paralytic Strabismus ...... 73 3.5.2.2 Duane’s Syndrome ...... 77 3.5.2.3 Fibrosis Syndrome ...... 80 3.5.2.4 Supranuclear Disorders ...... 81 3.6 Strabismus Surgery ...... 83 3.6.1 Recession Surgery ...... 84 3.6.2 Resection Surgery ...... 86 3.6.3 Transposition Surgery ...... 87 3.6.4 Amount of Surgery ...... 87

4 Biomechanical Modelling 89 4.1 Analytical Models ...... 90 4.2 Structure of Biomechanical Models ...... 91 4.3 History of Modelling of the Human Eye ...... 92 4.4 Ocular Geometry ...... 94 4.4.1 Coordinate Systems ...... 94 4.4.2 Mathematical Description of Eye Rotations ...... 96 4.4.2.1 Rotation Matrices ...... 96 4.4.2.2 Quaternions ...... 98 4.4.2.3 Listing’s Law ...... 102 4.4.2.4 Deﬁnition of Eye Positions ...... 104 4.4.3 Geometrical Abstractions ...... 105 4.4.3.1 Globe ...... 105 4.4.3.2 Muscles ...... 106 4.4.3.3 Evaluation of Muscle Action ...... 118 4.4.4 Passive Geometrical Changes ...... 121 4.5 Muscle Force Prediction ...... 123 4.5.1 Length-Tension Relationship ...... 123 4.5.2 Elastic Force Data ...... 125 4.5.3 Contractile Force Data ...... 126

ix 4.5.4 Total Force Data ...... 128 4.6 Kinematics ...... 130 4.6.1 Orbital Restoring Force ...... 130 4.6.2 Globe Translation ...... 131 4.6.3 Balancing Forces ...... 133 4.6.3.1 Solving for Eye Positions ...... 140 4.6.3.2 Solving for Innervations ...... 141 4.7 Brainstem Simulation ...... 142 4.7.1 Simulation of Binocular Function ...... 143

5 Visualization of Muscle Action 148 5.1 Image Analysis ...... 149 5.1.1 Picture Setup ...... 150 5.1.2 Generation of Polyhedron ...... 152 5.2 Surface Reconstruction ...... 155 5.2.1 Calculation of the Muscle Path ...... 156 5.2.1.1 Analyzing Surface Distribution ...... 159 5.2.2 Approximation of Muscle Surface ...... 161 5.2.2.1 Optimized Rendering ...... 162 5.2.3 Interpolation of Muscle Models ...... 164 5.3 Reconstruction Results ...... 166 5.3.1 Validation ...... 167

6 Software Design and Implementation 169 6.1 Design of the Biomechanical Model ...... 171 6.2 Design of the „SEE++“ Software System ...... 176 6.3 The „SEE++“ Software System ...... 180 6.3.1 „SEE++“ Simulation Task Flow ...... 181 6.3.2 Simulation properties ...... 182 6.3.2.1 Globe Data ...... 183 6.3.2.2 Muscle Data ...... 184 6.3.2.3 Distribution of Innervation ...... 185 6.3.2.4 Gaze Patterns ...... 185 6.4 Evaluation ...... 186 6.4.1 Abducens Palsy ...... 186 6.4.1.1 Simulation of the Pathology ...... 186 6.4.1.2 Simulation of Surgical Correction ...... 188 6.4.2 Superior Oblique Palsy ...... 189 6.4.2.1 Simulation of the Pathology ...... 189 6.4.2.2 Simulation of Surgical Correction ...... 191 6.4.3 Superior Oblique Overaction ...... 191 6.4.3.1 Simulation of the Pathology ...... 192 6.4.3.2 Simulation of Surgical Correction ...... 193 6.4.4 Heavy-Eye Syndrome ...... 195 6.4.4.1 Simulation of the Pathology ...... 196 6.4.4.2 Simulation of Surgical Correction ...... 199

x 7 Conclusion 201 7.1 Goals Achieved ...... 203 7.2 Future Work ...... 204

Literature 205

xi List of Figures

1.1 Example Applications Based on Data from the Visual Human Project r . . . . . 4 1.2 Classiﬁcation of Clinical Decision Support Systems ...... 5 1.3 SEE++ Virtual Eye Muscle Surgery Software ...... 8

2.1 Definition of Anatomical Planes ...... 10 2.2 View of the Bony Orbita of a Right Eye ...... 11 2.3 Anatomy of the Globe ...... 13 2.4 Anatomy of the Extraocular Eye Muscles ...... 15 2.5 Extraocular Muscles of a Right Eye from above ...... 15 2.6 Schematic View of the Extraocular Tissue Architecture ...... 17 2.7 Example of Lateral Rectus Path Influenced by Pulley ...... 18 2.8 Different Fiber Layers of Rectus Muscles ...... 19 2.9 Axial MR Scan, showing Lateral and Medial Rectus of both Eyes ...... 23 2.10 Axial MR Scan using Contrast Agent, showing Lateral and Medial Rectus of both Eyes...... 23 2.11 Comparison of MR Scans in different Gaze Positions ...... 24 2.12 Human Dissection of the Orbit showing Medial Rectus Pulley ...... 25 2.13 Line of Sight, Vertical and Horizontal Axes; Rotations to Other Eye Positions . . 26 2.14 Binocular Fixation of an Object in Space ...... 27 2.15 Rotational Directions for both Eyes ...... 28 2.16 Simple Abstraction of Listing’s Law ...... 31 2.17 Definition of Listing’s Law w.r.t. Primary Position ...... 32 2.18 Recording of Saccadic Eye Movements ...... 34 2.19 Ocular Motor Neurons and Motor Nuclei ...... 35 2.20 Oculomotor Nerve Circuit ...... 36 2.21 Location and Distribution of Oculomotor Nerves ...... 37 2.22 Motor Circuit for Horizontal Saccades ...... 39 2.23 EOG Eye Movement Measurement Technique ...... 42 2.24 Schematic Overview of Purkinje Corneal Reflections ...... 44 2.25 Purkinje Reflections ...... 45 2.26 Electromagnetic Search Coil Eye Position Measurement ...... 45 2.27 Insertion Procedure for a Scleral Search Coil Lens ...... 46 2.28 Video-Oculography Pupil Detection ...... 46 2.29 Chronos VOG System ...... 47 2.30 Different Types of Muscle Force Measurement ...... 48 2.31 Example of measuring Force with Forceps ...... 49 2.32 Muscle Force Transducer for Intraoperative Measurements ...... 50

xii 2.33 Intraconal Tissue Motion around the Optic Nerve ...... 51

3.1 Projection of Objects in Space onto the Retina on an Eye ...... 57 3.2 Different Forms of Binocular Fixation ...... 58 3.3 Example for Inward Squinting ...... 60 3.4 Example for Outward Squinting ...... 60 3.5 Example for Upward Squinting ...... 60 3.6 Example for Downward Squinting ...... 60 3.7 Pseudo-Esotropia due to wide Bridge and Epicanthal Skin Fold ...... 62 3.8 Hirschberg Light Reflex Test Method ...... 63 3.9 Krimsky Light Reflex Test Method ...... 63 3.10 Prism-Cover Test ...... 64 3.11 Cover Tests for Tropias and Phorias ...... 65 3.12 Maddox-Wing Test ...... 66 3.13 Binocular Fixation in the Hess-Lancaster Test ...... 67 3.14 Hess-Lancaster Diagram for Right Eye (Left Eye Fixing) ...... 68 3.15 Interpretation of Hess-Diagram according to Muscle Actions ...... 68 3.16 Hess-Diagram Interpretation ...... 69 3.17 Example of Disturbance of the Binocular Muscular Team ...... 72 3.18 Example of Abnormal Head Posture, Compensating Esotropia ...... 73 3.19 Example of a Right Superior Rectus Palsy ...... 74 3.20 Hess-Lancaster Chart for Right Superior Rectus Palsy ...... 75 3.21 Example of a Right Superior Oblique Palsy ...... 76 3.22 Hess-Lancaster Chart for Right Superior Oblique Palsy ...... 77 3.23 Example for Duane’s Retraction Syndrome Type 3 of a Right Eye ...... 78 3.24 Example for Brown’s Syndrome of a Right Eye ...... 79 3.25 Example for an Abducens Gaze Palsy ...... 82 3.26 Schematic Example of Muscle Recession ...... 84 3.27 Preparation for Medial Rectus Recession ...... 84 3.28 Medial Rectus Recession, continued ...... 85 3.29 Medial Rectus Recession, continued ...... 85 3.30 Schematic example of Muscle Resection ...... 86 3.31 Medial Rectus Resection ...... 86 3.32 Medial Rectus Resection, continued ...... 87

4.1 Halle’s Ophthalmotrope ...... 92 4.2 Ruete’s Ophthalmotrope ...... 93 4.3 Coordinate System of a Left Eye ...... 95 4.4 Gimbal Systems for describing 3D Eye Position ...... 95 4.5 Geometrical Abstraction of the Globe ...... 105 4.6 Geometrical Abstraction of an Eye Muscle ...... 106 4.7 Geometrical „String Model“ ...... 107 4.8 Point of Tangency in the „String Model“ ...... 108 4.9 Tape Model Calculations ...... 110 4.10 Geometrical Tape Model ...... 111 4.11 Comparison of Conventional vs. Pulley Model ...... 112 4.12 Primary Position in Pulley Model ...... 114

xiii 4.13 Tertiary Position in Pulley Model ...... 115 4.14 Muscle Action Circles in Pulley Model ...... 116 4.15 Muscle Direction Vector in Pulley Model ...... 117 4.16 Muscle Rotation Axis and Action Circle Center in Pulley Model ...... 118 4.17 Point of Tangency in Pulley Model ...... 119 4.18 Muscle Path Comparison using Diﬀerent Geometrical Models ...... 119 4.19 Muscle Force Distribution in String and Tape Model ...... 121 4.20 Muscle Force Distribution in the Pulley Model ...... 121 4.21 Elastic Force Data ...... 125 4.22 Contractile Force Data ...... 127 4.23 Total Force Data ...... 128 4.24 Apex Coordinate System for measuring Globe Translation ...... 132 4.25 Torque Error Function for Listing Positions in a Healthy Eye ...... 135 4.26 Torque Error Function for Pathological Eye ...... 136 4.27 Torque Error Minimization in solving for Eye Positions ...... 141 4.28 Squint-Angles Diagram for Binocular Fixation ...... 143 4.29 Simulation Task Flow for the Hess-Lancaster Test ...... 145

5.1 DXF Model Generation Tasks using Marching Cubes ...... 149 5.2 Sorting of Color Index Table ...... 150 5.3 Deﬁnition of 2D Polygon for MRI Segmentation ...... 151 5.4 Threshold Region for MRI Slice ...... 151 5.5 Marching Cube Traversal ...... 152 5.6 Marching Cubes Standard Classes ...... 153 5.7 Surface Reconstruction Tasks ...... 155 5.8 DXF Model with Area Centroids ...... 157 5.9 Approximation of the Muscle Path ...... 158 5.10 Angular Measurement of Surface Points ...... 159 5.11 Example of Analysis of Variance ...... 160 5.12 NURBS Approximation of Muscle Cross-Section ...... 162 5.13 Linear Interpolation of NURBS-generated Cross-Section ...... 163 5.14 Reconstruction of a Left Medial Rectus Muscle ...... 166 5.15 Shaded Reconstruction of a Left Medial Rectus Muscle ...... 167 5.16 Morphology of Reconstructed Medial Rectus Muscle ...... 167 5.17 Shaded Reconstruction of a Left Medial Rectus Muscle with MRI Data ...... 168

6.1 Structure of the „SEE++“ Software System ...... 170 6.2 Model-View-Controller Structure of the „SEE++“ Software System ...... 171 6.3 Abstraction of Muscles ...... 172 6.4 Primary Abstractions for Biomechanical Model ...... 173 6.5 Extensible Software Design for the Biomechanical Model ...... 174 6.6 Optimizer and Related Classes ...... 175 6.7 Structure of the „SEE++“ Package ...... 176 6.8 Structure of the „SeeMedic“ Package ...... 177 6.9 Structure of the „SeeModel“ Package ...... 178 6.10 Structure of the „SeeView“ Package ...... 179 6.11 Muscle Force Vector Diagram ...... 179

xiv 6.12 Simulation Task Flow for using „SEE++“ ...... 181 6.13 Default View of the „SEE++“ Software System ...... 183 6.14 Globe Data Parameters of the „SEE++“ System ...... 183 6.15 Muscle Data Parameters of the „SEE++“ System ...... 184 6.16 Gaze Pattern Dialog of the „SEE++“ System ...... 185 6.17 Muscle Properties for Abducens Palsy Simulation ...... 187 6.18 Hess-Lancaster Test for Abducens Palsy ...... 187 6.19 Simulation of Right Lateral Rectus Resection ...... 188 6.20 Postoperative Hess-Lancaster Simulation for Abducens Palsy ...... 189 6.21 Muscle Properties for Superior Oblique Palsy ...... 190 6.22 Hess-Lancaster Simulation of Superior Oblique Palsy ...... 190 6.23 Postoperative Hess-Lancaster Simulation for Superior Oblique Palsy ...... 191 6.24 Classiﬁcation of Superior Oblique Overaction ...... 192 6.25 Hess-Lancaster Simulation of Superior Oblique Overaction ...... 193 6.26 Simulation of Superior Oblique Surgery ...... 194 6.27 Hess-Lancaster Simulation of Superior Oblique Surgery ...... 194 6.28 Muscle Displacement as Hypothesis for Heavy-Eye Syndrome ...... 195 6.29 Measured Values from Patient with Heavy-Eye Syndrome ...... 196 6.30 Simulation Results for Resized Globes according to Patient Data ...... 197 6.31 Simulation Attempt using Data suggested by Schroeder and Krzizok ...... 197 6.32 Superior Oblique Muscle Insertion Transposition in Heavy-Eye Simulation . . . . 198 6.33 Hess-Lancaster Simulation Results of the Heavy-Eye Syndrome ...... 199 6.34 Muscle Surgery for Heavy-Eye Simulation ...... 199 6.35 Hess-Lancaster Simulation Results of the Heavy-Eye Surgery ...... 200

xv Chapter 1

Introduction

The research work presented in this document describes a successful combination and application of interdisciplinary research in the ﬁelds of computer science and eye muscle strabismus surgery. Basic medical research was incorporated in a mathematical model and subsequently realized as interactive software system for clinical usage. The main goal of this work was application- oriented research in order to realize a software system that directly aﬀects advancement in patient treatment.

The goal of this thesis is to prove that software simulation systems can be of valuable assistance in surgical decision making in the ﬁeld of eye muscle strabismus surgery. The combination of essential diagnostic data of individual surgical cases, its diagnosis and and its possible treatments into a well designed, clinically applicable computer software system is presented. A new way of interactive, virtual eye muscle surgery evaluation an preoperative planning method is proposed, based on a biomechanical model of the human eye in order to predict surgical outcomes on the basis of objective, anatomically related and measurable data. Additionally, physiologically meaningful three-dimensional visualization proves to support detailed evaluation and meaningful interpretation of disorders of the human eye movements. However, it is not intended to replace common clinical diagnostics, nor surgical expertise, instead, the proposed work should be considered as a clinical decision support tool to process diagnostic data into possible choices and amounts of surgery in an objective manner.

1.1 Overview

Due to the extensive medical implication of this work, the ﬁrst two Chapters give an overview of medical foundations within the ﬁeld of ophthalmology with special regard to strabismus. In Chap- ter 2, the basic anatomy of the human eye, its muscles and the orbita are explained. Additionally, anatomical measurements provide quantitative data based on available literature. Magnetic resonance imaging studies and human dissection were performed within the presented research work that provide detailed data on the anatomy and physiology of the extraocular eye muscles. Results from magnetic resonance imaging (MRI) studies explained in Sec. 2.1.4 were used to visualize three dimensional muscle morphology (cf. Ch. 5) which resulted in the most detailed model-based visualization of human eye muscles currently available.

1 CHAPTER 1. INTRODUCTION 2

Understanding eye movement physiology is very important in order to build a biomechanical model. Therefore, Chapter 2 contains valuable information on kinematic principles of ocular movements as well as detailed information on sensorimotor control and neural signal encoding. Especially important for biomechanical modelling are kinematic and sensory principles that have been identiﬁed as central laws that constrain ocular movements. All these principles need to be included in a biomechanical model that gives anatomically related predictions.

In order to relate predictions of a simulation model to clinical data, diagnostic methods are needed to quantify eye motility disorders. These methods are presented in Sec. 2.3 where measurement results serve as starting point for the simulation of pathological situations. Additionally, physiologic muscle force measurements that were carried out by a partner institution (Smith Kettlewell Eye Research Institute, San Francisco) are introduced in Sec. 2.3.2. These measurements were included in the biomechanical model in order to simulate muscle forces.

In Chapter 3 of this thesis, the field of strabismus is introduced. Since the main goal of the biomechanical model and the software system that are presented in this thesis is the application in the field of strabismus, it is essential to understand function and basic medical diagnosis and treatment in the field of strabismus. Therefore, Chapter 3 gives an overview of the physiology of binocular vision and ocular dissociation in Sec. 3.2 and Sec. 3.3. In Sec. 3.4 the clinical assessment of eye motility disorders is described. Without the use of simulation model, these mostly subjective measurement methods are currently the only diagnostic basis. In Sec. 3.5, major categories and examples of important eye motility disorders are explained, whereas Sec. 3.6 describes the most important treatment methods within surgical interventions.

In Chapter 2 as well as in Chapter 3 of this thesis, illustrations of eye muscle anatomy and examples of major eye muscle disorders were already produced with the biomechanical simulation system „SEE++“.

In Chapter 4, the major part of the presented research work is described. In Sec. 4.2, the structure of a biomechanical model is described which can be split into geometrical, muscle force and kinematic sub models. In this work, a unique mathematical formulation of ocular geometry using quaternions is described in Sec. 4.4. Furthermore, the mathematical implementation of the muscle force prediction model is explained in Sec. 4.5. Another major achievement that accounts for the stability of simulation predictions is the formulation of the kinematic model, explained in Sec. 4.6, that connects geometry and muscle forces in order to solve a standard non- linear minimization problem. Currently, this is the only biomechanical eye model that strictly dissociates geometry from muscle force and kinematics. Through the application of standard non-linear numerical minimization methods, the accuracy of simulation predictions with respect to clinical measurements is unique for this type of medical application.

In Chapter 5 of this thesis, a new approach of three dimensional reconstruction of shape and morphology of the extraocular muscles is presented. Reconstruction results were incorporated in the software simulation system „SEE++“ in order to visualize anatomically related data. Image analysis and image processing methods were used to generate interpolated muscle models that were connected to the muscle force simulation of the biomechanical model.

Finally, Chapter 6 gives an overview of the software system design for the biomechanical model as well as for the software system „SEE++“ that has been implemented. Modern methods in object oriented software engineering were used to build a generic extensible and robust software system CHAPTER 1. INTRODUCTION 3 that provides ﬂexibility for further advancements. An additional feature is that the biomechanical model can be seen as autonomous part within the software simulation system „SEE++“. While the software system provides user interface, most advanced three dimensional interactive visualization and representation of biomechanical parameters as anatomically meaningful properties, the biomechanical model implements and encapsulates the mathematical formulation for geometric, muscle force and kinematic models. Therefore, in Sec. 6.1, main parts of the software design of the biomechanical model are explained and in Sec. 6.2, main aspects of the design model of the „SEE++“ software system are described.

Additionally, in Chapter 6 the user interface and simulation parameters of the „SEE++“ software system are explained in Sec. 6.3. This Chapter then concludes with case studies showing diﬀerent eye motility disorders that were simulated with the „SEE++“ system. Currently, „SEE++“ is the only simulation system that is able to also simulate complex eye motility disorders as shown in the „Heavy-Eye“ example in Sec. 6.4.4.

1.2 Medical Informatics

The rapid development of computer systems increasingly enables the use of software systems within the medical field. Efficient computers for image processing and 3D-graphics in combination with specialized systems offer a practical supplement e.g. in medical diagnostics. In building such systems, interdisciplinary research in the fields of e.g. medicine, mathematics, physics and informatics is inevitable. Generally, research activities in the described field can be assigned to the field of Medical Informatics, which is defined as the application of computers, communications and information technology and systems to all fields of medicine - medical care, medical education and medical research [BM00a]. Therefore, this field combines medical science with several technologies and disciplines in the information and computer sciences and provides methodologies by which these can contribute to better use of the medical knowledge base and ultimately to better medical care [BM00a].

Medical Informatics range from computer-based patient records to image processing and from primary care practices to hospitals of health care. Processing information plays a vital role in health care since the ﬁeld of medical knowledge is growing every day, and there needs to be a way to properly formalize, store, process, and access that knowledge. Diseases and their diagnosis are being explored more and more so health care professionals have to take advantage of the information. The number of administrative and legal requirements dealing with processing information is on the rise. Information technology, if utilized correctly, can improve health care tremendously which is why health care professionals have to be educated about health and medical informatics.

The presented research work focuses on simulation software systems providing medical decision support by using methods and technologies for the ﬁelds of mathematics, physics and software engineering. A substantial criterion for the application of such systems in practice is the reliability of medically relevant data or results, as well as the scope of interpretation applied to such data. In the application of virtual reality in connection with surgical interventions, the success of an operation is substantially inﬂuenced by data obtained from such a system. Detailed graphical visualization enables the surgeon to preoperatively simulate a disease, and afterwards by means of CHAPTER 1. INTRODUCTION 4

(a) Segmentation (b) Rendering

Figure 1.1: Example Applications Based on Data from the Visual Human Project r , from [Ack02] interactive „virtual surgery“, plan, check and possibly even correct a surgical procedure in order to achieve the best result. Continuous endeavor of current research is associated with the so called „Visual Human Project r “ (Fig. 1.1), an anatomically detailed, three-dimensional representation of human bodies, in order to use these data in diﬀerent ﬁelds of application (e.g. Visible Human Explorer, Cross Sectional Anatomy, Voxelman r , Body Navigator, etc.) [Ack02]. Acquisition of transverse computed tomography (CT), magnetic resonance (MR) and cryosection images of representative male and female cadavers has been completed. The male was sectioned at one millimeter intervals, the female at one-third of a millimeter intervals. The long-term goal of the Visible Human Project r is to produce a system of knowledge structures that will transparently link visual knowledge forms to symbolic knowledge formats such as the names of body parts. Another goal of this work is to represent the function of the human body as realistically as possible by trying to apply well-known relationships from the mechanics to the anatomy of humans. Complex mathematical models of skeletons, muscles, joints and their graphical, three-dimensional visualization form the basis of an interactive system. By means of systematic studying of such systems, new insight can be derived, integrated into the model and subsequently be used to extend research.

1.3 Clinical Decision Support

In order to improve a patients prognosis, clinicians continuously make decisions on what diagnostic procedures they should perform and on what therapeutic actions they should take. Clinical decision support tools may provide predictions on e.g. the diagnosis of a disease and potential promising treatment suggestions on the basis of clinical information about a patient. However, clinical information can be gathered in many different ways by using existing information technology infrastructure. Clinical decision support tools often make use of different sources of information (Fig. 1.2 [BK03a]) e.g. medical information systems provide electronic medical patient records or protocol-based systems offer standard algorithms that define one precise manner in which certain classes of patients should be evaluated or treated. Language coding and classification systems complement these systems by realizing natural, intuitive interfaces. Com- munication systems in health care (e.g. workflow management tools and electronic transmission CHAPTER 1. INTRODUCTION 5 of diagnostic findings) properly arrange selective distribution of medically relevant information.

Figure 1.2: Classiﬁcation of Clinical Decision Support Systems

Intelligent clinical decision support systems overlap the described technologies for the purpose of combining medical and administrational data in order to build up clinical useful knowledge within certain disciplines. Methods and models that operate on this data and are specifically designed for investigations in certain medical fields enable the formulation of predictive conclusions in order to evaluate diagnosis and treatment options. By means of systematic analysis of patient data, such systems may support clinicians in their decision-making processes. The development of predictive decision support tools concentrates on five main steps [BM00a]:

1. Analysis: In this step, the clinical problem and its speciﬁc characteristics must be examined. This process must be described clearly, including the potential role of the decision support tool.

2. Outcome: The desired clinical outcome that clinicians consider central for decision making must be indicated. In most cases, these outcomes are deﬁned in clinically related groups of diseases or diagnoses.

3. Predictors: This determines the clinical characteristics that might be used as predictors of the clinical outcome. Predictors directly inﬂuence the way a patient is treated and thus transitively determine the treatment outcome.

4. Quantification: This step defines a relation between the predictors and the clinical outcome. The quantification may be provided by expert clinicians who possess the clinical knowledge and is captured within a mathematical description (e.g. statistical model).

5. Presentation: The predictive decision support tool needs to be presented to the users in an applicable way. Often, this includes the design and implementation of an interactive software system, incorporating predictors and quantiﬁcation relationships.

The research work presented here describes the analysis, design, implementation and application of a clinical decision support tool in the ﬁeld of eye muscle strabismus surgery. CHAPTER 1. INTRODUCTION 6

1.4 Eye muscle surgery

Surgery of the eye muscles is surgery to weaken, strengthen, or reposition any of the eye muscles that move the eyeball (the extraocular muscles). The purpose of eye muscle surgery is generally to align the pair of eyes so that they gaze in the same direction and move together as a team, either to improve or maintain binocular vision, or to improve the orientation of both eyes. To achieve binocular vision, corresponding images need to be projected onto corresponding areas of the retinae of both eyes. In addition, eye muscle surgery can improve eye alignment in people with other e.g. neurologically caused eye disorders (Nystagmus, Duane Syndrome, etc.).

Depth perception (stereopsis) develops around the age of three months. In case of misaligned eyes, successful development of binocular vision and the ability to perceive three-dimensionally is constricted. Surgery should not be postponed past the age of four since immediate sensory adaptations occur in the early stage of growth.

The six extraocular muscles attach via tendons to the sclera (the white, opaque, outer protective covering of the eyeball) at different places just behind an imaginary equator encircling the top, bottom, left, and right of one eye. The other end of each of these muscles attaches to a part of the orbit (the eye socket in the skull). These muscles enable the eyes to move up, down, to one side or the other, or any angle in between. Modification of one eye muscle also affects control signals of the brain to the other eye. Normally both eyes move together, receive the same image on corresponding locations on both retinas, and the brain fuses these images into one three-dimensional image. The exception is in strabismus which is a disorder where one or both eyes deviate out of alignment, most often outwardly (exotropia) or toward the nose (esotropia), sometimes upward (hypertropia) or downward (hypotropia). The brain now receives two different images, and either suppresses one or the person sees double (diplopia). This deviation can, in most cases, be adjusted by weakening or strengthening the appropriate muscles to move the eyes toward the center. For example, if an eye deviates upward, the muscle at the bottom of the eye could be strengthened. Both eyes are controlled by twelve eye muscles (six muscles per eye) including combined bilateral brain signals that form a highly complex mechanical system. In a nutshell, the human mechanical eye system seems to work very hard and complicated in order to give a simple impression.

One main problem in treating eye alignment disorders is, that there is no clinical theory available that exactly defines surgical treatment consequences based on diagnostic measurements. Up to now, clinicians use their own practical experience and raw statistical data to derive dose-response relationships for certain pathological situations [SS00]. This results in repeated surgery, especially in case of complicated, combined eye alignment disorders. This leads to the establishment of different „treatment philosophies“ without verification of underlying correlation to mechanical and/or neurological properties. Another issue affects incorporation of patient specific data in order to adjust accuracy of existing statistical approaches. Moreover, the actions of eye muscles vary as a function of eye position, thus, effect of eye muscle surgery must be evaluated within a specific field of gaze. Realigning both eyes with respect to only one eye position is most often not sufficient to achieve a satisfying result. Many clinical approaches have been undertaken to define certain guidelines for eye muscle surgery, however, all of these assumptions are based on different ideas of the mechanical behavior of the human eye system (e.g. [Pri81]). Clinicians and teachers still legitimate decisions based upon outdated assumptions of the action of the extraocular muscles. Since the problem of building up fundamental strategies for clinical use, in CHAPTER 1. INTRODUCTION 7 this case, lies in the lack of definitions for elementary concepts describing the modes of operation of the human extraocular system, new approaches had to be found.

1.5 SEE-KID

The research work SEE-KID (Software Engineering Environment for Knowledge based Inter- active Eye motility Diagnostics), described in this work, tries to connect aspects of biomechanical modelling with methods of modern software engineering [BM00b]. This work is carried out at the Upper Austrian Research Center at the department for Medical-Informatics in close cooperation with the Upper Austrian University of Applied Sciences in Hagenberg. With international partners (e.g. ETH-Zürich and Smith Kettlewell Eye Research Institute in San Francisco), this project started in 1995 and was supported by funding of the Austrian Ministry for Science and Technology (FFF) within the years 2000 and 2002. Additionally, partners from Austrian hospitals and research centers in Linz, St. Pölten, Innsbruck and Graz greatly contributed to this work as evaluation partners.

Originally, this research project was initiated by Prim. Prof. Dr. Siegfried Priglinger, head of the department of ophthalmology at the convent hospital of the „Barmherzigen Brüder“ in Linz, Austria. This department has specialized in correcting eye motility disorders, particularly in infants, by e.g. recession or resection of certain eye muscles. Most of these surgeries must be performed at an early age. In order to avoid a permanent misalignment and a sensory adaptation resulting from it, children must be operated according to individual strategies (e.g. ﬁbrosis syndrome) as soon as possible. Prerequisite for such surgeries is an early diagnosis and a conservative treatment plan that includes e.g. masking (covering the better eye to stimulate the recovery of the pathological eye).

For the success of an eye muscle surgery, an understanding of the disease mechanism and the anatomically functioning mechanisms is necessary, in order to avoid wrong or multiple surgical treatments.

Such model-supported eye muscle operations have been performed at the hospital of the „Barm- herzigen Brüder“ in Linz, Austria, since 1978. Also, new operation techniques and treatment options have been developed during this time. Particularly complicated surgeries must be planned in detail and suitable operation steps must be selected. At present, surgical procedures can be evaluated and improved so far only directly at the patient. In complicated eye motility disorders, even an experienced surgeon will depend on documented empirical values, which often lead to multiple treatments until the result is satisfying. The result of this research work is a software system (SEE++, Fig. 1.3), which enables physicians to simulate eye motility disorders on the basis of measurements from the patient and to perform all possible surgical treatments interactively. Using a 3D representation of the anatomy of the human eye, the surgeon can model disorders as deviations from a non-pathological „healthy“ eye. Thus the surgeon can determine the optimal treatment for the patient and plan its proceeding in detail.

The simulated outcome of a virtual surgery is displayed interactively in the 3D visualization, as well as through measuring parameters and diagrams familiar to clinicians. In addition, reference points and measured values are displayed to the surgeon, enabling better orientation while oper- ating. Moreover, the SEE++ system permits to exchange the model base interactively, therefore, CHAPTER 1. INTRODUCTION 8

Figure 1.3: SEE++ Virtual Eye Muscle Surgery Software model predictions can be compared by applying different modelling strategies on the fly. An- other benefit of this system is that it is capable of simulating binocular highly complex diseases including neurologically caused pathologies (e.g. nuclear, inter- or supranuclear lesions). Chapter 2

Medical Foundations

Understanding visual experience has long challenged the best of human minds, from the Ancient Greek’s interest in optics to the study of visual perception by contemporary psychologists and neuroscientists. Today’s scientiﬁc study of perception seeks to understand the nature of our experience in terms of the underlying mechanisms by which it occurs. Understanding the anatomy and functional behavior of the human eye is causally related to explaining the process of visual perception.

Ongoing new discoveries implying extensive clinical consequences clearly point out, that the basic anatomy and physiology of the human eye is still not explored to the full extent. Especially, new findings concerning the structure and behavior of eye muscles suggest that the human eye ranks among the most complex human organs. New radiologic inventions (e.g. MRI, CT, PET and hy- brid technologies [BWM+01]) extend the abilities for diagnostic and scientific exploration. This also explains why today’s fundamental medical research needs interdisciplinary approaches com- prising collaborations of clinicians, technical engineers, mathematicians and physicists. Without the invention and incorporation of new technologies, medical research would not emerge that effectively.

This chapter will give a basic overview of the human eye’s anatomy which will be required in order to fully understand principles and properties of the oculomotor plant that will be described throughout this thesis. Additionally, physiologic and neurologic principles of oculomotor control will be explained providing deeper insight into important aspect of the visual system. Moreover, common measurement techniques are presented, giving new insight into the function of the human visual system. Especially in research, these methods are essential for the discovery and validation of various properties of the oculomotor system.

9 CHAPTER 2. MEDICAL FOUNDATIONS 10

Medical professionals and researchers in the medical-technical fields often refer to sections of the body in terms of anatomical planes (flat surfaces). These planes are imaginary lines - vertical or horizontal - drawn through an upright body. The terms are used to describe a specific body part. Fig. 2.1 gives an overview of these anatomical naming conventions.

Figure 2.1: Deﬁnition of Anatomical Planes

In Tab. 2.1 general anatomical terms and their meanings are listed. The naming for the anatomical planes often diﬀer between coronal plane or frontal plane, sagittal plane or lateral plane and axial plane or transverse Plane.

Anatomical Terms Direction Medial Toward the midline of the body Lateral Away from the midline of the body Proximal Toward a reference point (extremity) Distal Away from a reference point (extremity) Inferior Lower or below Superior Upper or above Cephalad or Cranial Head Caudal or Caudad Tail, tail end Anterior or Ventral Toward the front Posterior or Dorsal Toward the back

Table 2.1: Terms of Medical Directions CHAPTER 2. MEDICAL FOUNDATIONS 11

2.1 Anatomy of the Human Eye

The eyes, the „sense of perception“ , rank among the most important sensory organs of the human organism. They supply us with a constantly updated picture of the environment. The following explanations refer to a right eye. The eyeball or globe (lat. bulbus oculi, brieﬂy bulbus, øapprox. 24 mm, approximately spherical [PD01]) lies protected in the orbita, recessed in the head.

Fig. 2.2 shows the bony orbita of a right eye. The human orbita is shaped pyramidal and is approx. 40-50 mm deep. The back end of the orbita (orbital apex) is situated right next to the optic canal which acts as passage for the optical nerve and the ophthalmic arteries. Except for the solid orbital boundary, orbital bones are extremely thin.

(a) Orbital bones (b) Orbital apex with nerves and arteries

Figure 2.2: View of the Bony Orbita of a Right Eye [KJCS99]

Referring to Fig. 2.2(a), the following bone structures can be identiﬁed:

(1) Os lacrimale - the lacrimal bone, an irregularly rectangular thin plate, forming part of the medial orbital wall of the orbit behind the frontal process of the maxilla (the upper jaw bone). (2) Os ethmoidale - the ethmoid bone, an irregularly shaped bone lying between the orbital plates of the frontal and anterior to the sphenoid bone (3). (3) Os sphenoidale - a bone of most irregular shape occupying the base of the skull. (4) Os zygomaticum - the zygomatic bone, a quadrilateral bone which forms the prominence of the cheek. (5) Os frontale - the frontal bone, a large single bone forming the forehead and the upper margin and roof of the orbit on either side. (6) Os maxillare - the maxillary bone or the maxilla, an irregularly shaped bone that with its fellow forms the upper jaw.

In Fig. 2.2(b), a schematic view of the orbital fissures, consisting of a inferior and superior part. The inferior orbital fissure is a cleft between the greater wing of the sphenoid and the orbital plate CHAPTER 2. MEDICAL FOUNDATIONS 12 of the maxilla, through which pass the maxillary division and the orbital branch of the trigeminal nerve (face nerve) and the infraorbital vessels. The superior orbital fissure is located between the greater and the lesser wings of the sphenoid establishing a channel of communication between the middle cranial fossa and the orbit, through which pass the oculomotor and trochlear nerves, the ophthalmic division of the trigeminal nerve, the abducens nerve, and the ophthalmic veins. For this work, the area to concentrate on will be the superior orbital fissure, since it provides a passage for all nerves that control the visual organ.

According to Fig. 2.2(b), the following nerves and vessels can be distinguished:

(1) Vena ophthalmica superior - the superior ophthalmic vein

(2) Nervus lacrimalis - the lacrimal nerve, a branch of the ophthalmic nerve supplying sensory ﬁbres to the lateral part of the upper eyelid, conjunctiva, and lacrimal gland (the gland that secretes tears).

(3) Nervus frontalis - the frontal nerve, a branch of the ophthalmic nerve which divides within the orbit into the supratrochlear and the supraorbital nerves.

(4) Nervus trochlearis - the trochlear nerve that controls the superior oblique eye muscle.

(5) Nervus oculomotoris - the oculomotor nerve is responsible for motor innervation of the upper eyelid muscle, extraocular muscle and pupillary muscle.

(6) Nervus abducens - the abducent nerve, innervates the lateral rectus eye muscle.

(7) Nervus nasociliaris - the nasociliary nerve is a branch of the ophthalmic nerve in the superior orbital ﬁssure, passing through the orbit, giving rise to the communicating branch to the ciliary ganglion, the long ciliary nerves, the posterior and anterior ethmoidal nerves, and terminating as the infratrochlear and nasal branches, which supply the mucous membrane of the nose, the skin of the tip of the nose, and the conjunctiva.

(8) Vena ophthalmica inferior - the inferior ophthalmic vein

(9) Nervus opticus - the optic nerve which is carrying all impulses for the sense of sight.

(10) Arteria ophthalmica - the ophthalmic artery originating from the internal carotid artery and distributing to the eye, orbit and adjacent facial structures.

The orbita is covered by the periost (or periorbita), a membrane of ﬁbrous connective tissue which closely invests all bones except at the articular surfaces. From there, connective tissue and septa stabilize and cover intraorbital structures (e.g. globe, muscles and vessels). This anatomical structure is also known as Tenon’s capsule.

The globe or bulbus (Fig. 2.3) is built bulb ﬂat-like, composed of three layers of skin [SS98]:

sclera (leather skin) - outer eye skin,

choroidea (vein skin) - middle eye skin, the middle layer of the globe, between retina and sclera CHAPTER 2. MEDICAL FOUNDATIONS 13

retina (retina) - internal eye skin.

Figure 2.3: Anatomy of the Globe [KJCS99]

Like a high sophisticated camera, the eye has multiple discrete parts which must function together properly to produce a clear vision. To illustrate this, the path of light as it travels through the eye is discussed, and the various ocular structures are identiﬁed.

Cornea: The first surface encountered by a ray of light is the tear film. The eye’s surface must be kept moist at all times. To achieve this, glands in and near of the eyelids produce both tears and a special oil which mix together and coat the eye. This tear film coats the cornea which normally is the crystal clear window to the eye. Behind the cornea, the anterior chamber is situated, which is filled with aqueous fluid. The aqueous is usually clear like water and is responsible for maintaining the pressure of the eye.

Iris: Inside the anterior chamber is the iris. This is the part of the eye which is responsible for the eye-color perceived from an outside viewer. It acts like the diaphragm of a camera, dilating and constricting the pupil to allow more or less light into the eye.

Lens: The next structure encounter is the crystalline lens. The lens is responsible for focusing light onto the retina. It changes shape slightly to allow adaption of focus between objects that are near and those that are far. During the process of aging, the lens becomes less ﬂexible and able to „accommodate“ or change focus.

Vitreous: This is a jelly-like substance that fills the body of the eye. It is normally clear and in early life, it is firmly attached to the retina behind it. With age, the vitreous becomes more water-like and may detach from the retina. Often, little clumps or strands of the jelly form and cast shadows which are perceived as „floaters“.

Retina: Finally, light reaches the retina, a thin tissue lining the innermost wall of the eye. The retina acts much like the ﬁlm in a camera. The retina responds to light rays hitting it and converts them to electrical signals carried by the optic nerve to the brain. The outlying parts of the retina are responsible for peripheral vision while the center area, called the CHAPTER 2. MEDICAL FOUNDATIONS 14

macula, is used for ﬁne central vision and color vision. The very center of the macula is called the fovea. It has a very high concentration of special cells (cones) which make it the only part of the retina capable of 20/20 vision1.

Retinal Layers: Like film, the retina is composed of several layers with different roles. The first layer encountered by light is called the nerve fiber layer. Here, the nerve cells travel from all the parts of the retina to the optic nerve. Under this layer, most of the retinal blood vessels are located. They are responsible for supplying the inner parts of the retina. The outermost layer is the the photoreceptor layer. The photoreceptor layer, composed of cones for fine and color vision, and rods for vision in dim light, consists of the cells that actually convert light into nerve impulses. There are approximately 120 million rods and 6 million cones in a human retina. Most of the cones are located in the macula. The photoreceptor cells lie on top of a layer of cells called the retinal pigment epithelium or RPE. The RPE is responsible for keeping the photoreceptors healthy and functioning well. Under the RPE is the retina’s second set of blood vessels which are in a layer called the choroid. The RPE, fed by the blood vessels of the choroid, supply the photoreceptors.

Optic Nerve: The optic nerve is the structure which takes the information from the retina as electrical signals and delivers it to the brain where this information is interpreted as a visual image. The optic nerve consists of a bundle of about one million nerve ﬁbers. The position in the back of the eye where the nerve enters the globe is corresponds to the „blind spot“ since there are no rods or cones in these location. Normally, a person does not notice this blind spot since rapid movements of the eye and processing in the brain compensate for this absent information.

The globe is tightly suspended within the orbita, surrounded by six extraocular muscles which are responsible for the movement of the globe. The four straight eye muscles (musculi recti) and the upper diagonal (superior oblique) eye muscle originate in the posterior part of the orbita. Only the lower diagonal (inferior oblique) muscle originates from the orbital plate of the maxilla (see page 11). All eye muscles insert at the leather skin of the globe. Fig. 2.4 shows the muscle origins in the posterior orbita and their insertions. Each eye muscle aﬀects the eyeball in three components, whereby the muscle path determines the main direction of pull. The main eﬀect of each muscle can be derived from its designation [BKP+03].

Perpendicular arranged to each other, the musculi recti originate in the anulus of Zinn, a point at the posterior end of the orbita. Their tendons unite to a circular plate (Zinns’ ring) and their insertions lie before the equatorial plane of the globe [Gue86]. In contrast, the musculi obliqui insert behind the globe equator and pull diagonally forward. The m. obl. sup. is the longest of all eye muscles. Starting at its insertion, it runs above the globe towards the nasal frontal bone, pulls through a cartilaginous hole (the trochlea) and runs from there directly to its origin close at Zinns’ ring. Obl. inf. originates at the nasal edge of the bony orbita, runs below the globe, crosses the m. rect. inf. and inserts within the rear range of the eyeball. Within the crossing area, m. obl. inf. and m. rect. inf. are connected by ligamentum Lockwood [Gue86]. Each eye muscle consists, apart from the purely muscular portion, also of a tendon which connects the muscle at the origin on one side, and at the point of insertion on the other side. The overall

1’20/20 vision’ is a term used to describe normal distance vision. The ’20’ is a distance of 20 feet, which is a standard testing distance for eyesight, used by Optometrists and Doctors. CHAPTER 2. MEDICAL FOUNDATIONS 15

Figure 2.4: Anatomy of the Extraocular Eye Muscles [KJCS99] length (muscle and tendon) of eye muscles is very diﬀerent. The largest diﬀerences occur in tendon lengths [Kau95]. The m. obl. inf. has the shortest tendon (0-2 mm) and the m.obl.sup. the longest (25-30 mm). The actual muscle length lies between 30 mm (mm obliqui) up to 39 mm (m. rect. inf.). Due to the insertion lying before or behind the equator of the globe, each muscle partially contacts the eyeball’s surface. At the point of tangency, the muscle loses contact to the globe and pulls toward its origin. With each movement of the globe, the relative position of a muscles insertion changes with respect to the orbita.

Figure 2.5: Extraocular Muscles of a Right Eye from above [uS98]

In referring to Fig. 2.4 and Fig. 2.5, the names and primary directions of action for each muscle is identiﬁed in Tab. 2.2.

Until 1994, eye muscles were assumed to be string like structures that run from the origin straight to the insertion. At the point where a eye muscle touches the globe (point of tangency), the muscles were expected to move freely. If the muscles could move freely between insertion and CHAPTER 2. MEDICAL FOUNDATIONS 16

Fig.Nr. Muscle name Primary action(s) (1) musculus rectus superior or superior rectus muscle upward (upper straight eye muscle) (2) musculus rectus lateralis or lateral rectus muscle sideways outward (outside straight eye muscle) (3) musculus rectus inferior or inferior rectus muscle downward (lower straight eye muscle) (4) musculus rectus medialis or medial rectus muscle sideways inward (internal straight eye muscle) (5) musculus levator palpebrae or levator muscle raises the upper eyelid (upper eyelid muscle) (6) musculus obliquus superior or oblique superior muscle downward and inside (upper diagonal eye muscle) (7) musculus obliquus inferior or oblique inferior muscle upward and outside (lower diagonal eye muscle)

Table 2.2: Eye Muscle Names and Primary Actions origin during an eye movement (shortest path hypothesis), a shift of the muscle path on the globe surface would occur, especially in extreme gaze positions. Thus, the muscle path and thus the direction of pull would change considerably according to the actual eye position (loss of main direction of pull).

2.1.1 Eye Muscle „Pulleys“

In order to prevent an eye muscle from slipping away while the globe rotates, connective tissue surround the globe and stabilizes the muscles within the area of the point of tangency. These stabilizers are called Pulleys [BM00b][DMP+95]. Early radiographic studies in monkeys and computed tomography studies in human subjects suggested that the bellies of contracted recti extraocular muscles have paths that are very stable relative to the orbit despite changes in gaze. Miller and Demer [DMP+95] used magnetic resonance imaging (MRI) and manual 3D image reconstruction to demonstrate the extreme stability of the recti extraocular muscle paths throughout the normal range of ocular rotations. The result was, that only the anterior parts of the tendons moved relative to the orbit, as they must, because they are attached to the globe. Histochemical and Immunohistochemical investigations within this study confirmed the findings of the MRI analysis. Fresh orbital specimens were obtained at autopsy from adult cadavers, dissection and globe enucleation was performed. All specimens were fixed in 10% neutral buffered formalin, dehydrated and embedded in paraffin. The resulting 7 to 10–µm sections were mounted on glass slides and chemically colored distinguishing collagen, elastin, cartilage, smooth muscle, striated muscle and tendon. Smooth muscle is generally involuntary and differs from striated muscle in the much higher actin/myosin ratio, the absence of conspicuous sarcomeres and the ability to contract to a much smaller fraction of its resting length. CHAPTER 2. MEDICAL FOUNDATIONS 17

Figure 2.6: Schematic View of the Extraocular Tissue Architecture from [DMP+95] IR = inferior rectus; LPS = levator palpebrae superioris; LR = lateral rectus; MR = medial rectus; SO = superior oblique; SR = superior rectus.

In Fig. 2.6, a schematic representation of the orbital tissues with respect to eye muscle pulleys is shown. The horizontal section shows the lateral and medial rectus muscles (LR, MR) suspended by ﬁbroelastic sleeves consisting of collagen and elastin at the posterior part of Tenon’s capsule, approx. 10 mm posterior to the muscle insertion. The sleeves itself are coupled to the orbit by musculoﬁbroelastic septae, extending to the periorbita and to adjacent muscle sleeves. The coronal sections of Fig. 2.6 are represented at the level of the superior rectus tendon (SR tndn), and the superior oblique tendon (SO tndn). CHAPTER 2. MEDICAL FOUNDATIONS 18

Figure 2.7: Example of Lateral Rectus Path Inﬂuenced by Pulley

In Fig. 2.7, a left eye elevated by 35 ◦ is shown, focusing the left lateral rectus muscle. The figure shows how the muscle path of the left lateral rectus is inflected by the pulley indicated in red. Since the lateral rectus muscle, in its primary function, pulls the eye outward, away from the nose (see Tab. 2.2), this main action must be retained, also when the globe rotates to other positions. Through the inflection of the muscle path at the location of the pulley, the main function of the lateral rectus muscle is preserved and the globe would be rotated laterally in case contraction of the muscle occurs. Comparing the posterior and anterior part of the muscle with respect to the pulley location, the anterior part moves with the rotation of the globe, while the posterior part of the muscle stays stable relative to the point of origin.

2.1.2 Global/Orbital Eye Muscle Layers

Recent anatomic studies of whole orbits confirmed the existence of pulleys (e.g. [Mil89]). Tech- nical improvements have made it possible to reconstruct orbital histology even more detailed and without the need to remove the structures from the orbital bones. This leaves the normal spatial relationships as is, thus providing a better insight in the actions of the different muscles. Former studies on mammals (e.g. [MGGN75]) already suggested that the extraocular muscles consist of two different layers.

Referring to Fig. 2.8, the global layer is continuous from the origin at the annulus of Zinn to the tendinous scleral insertion on the globe, in contrast to the orbital layer, obviously terminating posterior to the insertion. Recent studies have shown that the position of the rectus muscle pulleys change slightly as a function of gaze [DOP00], thus modifying the mechanical properties of muscle action. Pulley positions were identified by the sharp inflections of the muscle path in different gaze positions. A study carried out by Demer et.al [KCD02] showed that for each rectus muscle, a discrete inflection moved significantly posteriorly during contraction and anteriorly during relaxation of the muscle. These changes in path inflections have been interpreted as antereoposterior shifts in the rectus muscles pulleys. This so called Active-Pulley hypothesis CHAPTER 2. MEDICAL FOUNDATIONS 19

Figure 2.8: Diﬀerent Fiber Layers of Rectus Muscles from [DOP00] is quantitatively supported by recent studies (e.g. [KCD02][DOP00][CJD00]). The resulting consequence is that human rectus muscle pulleys move to shift the ocular rotation axis in order to attain commutative behavior of the oculomotor system.

2.1.3 Anatomical Measurements

Around 1869 by A.W. Volkmann [Vol69], a statistic analysis was carried out with several patients. The results of this study were defined as „standard eye“, an average eye of humans. Volkmann’s specifications referred to the median lines of eye muscle tendinous insertions (i.e. the approximated center line of a muscle volume), and accuracy was limited due to the seamless transition of muscle tendon into neighboring tissue. In 1965, Nakagawa used hand dissected slices of cadavers in order to measure each eye muscles individual cross-section [Nak65]. These clinical results en- dured as basis for many modelling approaches until studies from Mühlendyck and Miller revised the data [MR84][MKM84]. With the discovery of Pulleys, new geometrical measurement using MRI and histologic approaches were defines [CJD00]. CHAPTER 2. MEDICAL FOUNDATIONS 20

Globe Radius: 12.43 mm Cornea Radius: 5.50 mm MR LR SR IR SO IO Origin x (mm) -17.00 -13.00 -16.00 -16.00 -15.27 -11.10 y (mm) -30.00 -34.00 -31.78 -31.70 8.24 11.34 z (mm) 0.60 0.60 3.80 -2.40 12.25 -15.46 Insertion x (mm) -9.15 10.44 0.00 0.00 -0.07 6.73 y (mm) 8.42 6.75 7.33 7.65 -8.05 -10.46 z (mm) 0.00 0.00 10.05 -9.80 9.48 0.00 Length (mm) 39.96 49.65 44.05 44.80 22.28 34.03 L0 (mm) 31.60 36.20 33.90 35.00 32.00 29.10 Tendon Length (mm) 3.80 8.40 5.40 4.80 26.50 1.50 Tendon Width (mm) 10.30 9.20 10.60 9.80 11.00 9.80

Table 2.3: Muscle Parameters measured by Volkmann from [Gue86]

The geometrical data for muscle length from Tab. 2.3 also includes the tendon of each muscle, whereas the muscle length (L0) represents only the contractile part of the muscle without its tendon. These values are defined on the basis of relaxed denervated muscles (muscles which are not innervated). All coordinates in this table are defined with respect to an oculocentric, head-fixed coordinate system originating in the center of the globe.

Data from Tab. 2.3 data was reﬁned by Miller et.al. due to the discovery of pulleys and new measurement results for further studies [Mil89][CJD00][KCD02]. CHAPTER 2. MEDICAL FOUNDATIONS 21

Globe Radius: 11.99 mm Cornea Radius: 5.50 mm MR LR SR IR SO IO Origin x (mm) -17.00 -13.00 -15.00 -17.00 -18.00 -13.00 y (mm) -30.00 -34.00 -31.76 -31.76 -31.50 10.00 z (mm) 1.00 -1.00 3.60 -2.40 5.00 -15.46 Insertion x (mm) -9.65 10.08 2.76 1.76 2.90 8.00 y (mm) 8.84 6.50 6.46 6.85 -8.00 -9.18 z (mm) 0.00 0.00 10.25 -10.22 8.82 0.00 Pulley x (mm) -14.00 12.00 -5.16 -5.16 -15.27 -13.00 y (mm) -5.00 -8.00 -10.78 -8.78 11.00 10.00 z (mm) 0.14 0.33 10.00 -12.00 11.75 -15.46 Length (mm) 39.96 49.65 44.05 44.80 22.28 34.03 L0 (mm) 31.92 37.50 33.82 35.60 34.15 30.55 Tendon Length (mm) 4.91 7.71 5.40 4.80 31.90 1.33 Tendon Width (mm) 10.50 9.30 10.80 10.00 10.80 8.60

Table 2.4: Revised Muscle Parameters by Miller et.al. [MR84]

In the revised muscle data from from Tab. 2.4, muscle insertions as well as the pulley positions were measured in three-dimensional space. This means that assuming the globe as geometrical abstraction using a spherical object, each muscle’s insertion does not lie exactly on the globe, due to the fact that in reality, the globe is shaped ellipsoid. This leads to a compromise in choosing the globe radius as the smallest distance between coordinate system origin and muscle insertion of all six muscles. The substantial mathematical diﬀerence of these data consists of the fact that now each muscle rotates its own „virtual globe“, in turn aﬀecting lever arm and force behavior.

2.1.4 High Resolution MRI-Imaging of the Orbit

Imaging techniques have become an indispensable diagnostic tool in the field of ophthalmology. In most hospitals, computed tomography is still the method of choice for orbital imaging because of its low costs and excellent depiction of bony details. However, resolution of MRI scanners has improved during the past years providing sufficient accuracy to show detailed tiny anatomical structures. In contrast to CT, MRI allows for multiplanar imaging without the need for repositioning the patient. Whereas CT uses ionizing radiation, which is known to be harmful to health, MRI utilizes the nuclear magnetic resonance (MR) effect.

Nuclei with a net magnetic moment, such as hydrogen ions (protons) which occur in living matter, line up in parallel in a strong magnetic field and change to a higher energy level when a radio frequency (RF) impulse is applied at right angles to the static magnetic field. The CHAPTER 2. MEDICAL FOUNDATIONS 22 strength of this static magnetic field in MR scanners ranges between 0.5 to 2 Tesla, whereby one Tesla corresponds to a unit of magnetic flux. The tesla (symbolized T) is the standard unit of magnetic flux density. Reduced to base units in the International System of Units, 1 T represents one kilogram per second squared per ampere (kg/s2A). In practice, the tesla is a large unit, and is used primarily in industrial electromagnetics. Once the RF impulse is turned off, the nuclei relax to the original energy level and release a RF signal that can be detected by a coil (RF receiver). Proton density of the examined tissue plays an important role when applying MRI investigations to in vivo subjects. The nuclear relaxation process depends on the mobility of the protons in the substance of interest.

These properties are distinguished by T1 (spin-lattice relaxation time) and T2 (spin-spin relaxation time) values. T1 is influenced by molecular rotation time and are lowest in the tissues with intermediate-sized molecules such as fat. T1 values are dependent upon magnetic field strength (increasing field strength gives increasing T1). T1 values at 1.5 T are around 800 − 1000 ms for brain tissue while it is considerably higher (3000 ms) for water and lower (300 ms) for fat. After the RF signal is applied, energy of the spin system in the examined tissue will be redistributed and the measurable signal will decay away as the spins. Long T2 relaxation times only occur in molecules which are tumbling rapidly in solution. More immobile species and solids have rapid spin-spin relaxation and often produce MR signals which decay away before they can even be detected in MR imaging. There are several distinct mechanisms occurring at the molecular level which can contribute to relaxation, but dipole-dipole interactions are the major cause of T2 relaxation. Additionally, extrinsic factors that are set on the MR scanner influence image acquisition. Time of repetition (TR) and time of echo (TE) are the most important values. TR is the time between the emission of RF pulses, and TE is the time between the excitation of an RF pulse and the reception of the measured signal. Thus, the relative contribution of any of the tissue specific factors can be varied by choosing different pulse sequences in order to achieve a weighting of a specific parameter. Hence, in MR imaging, tissues with higher T1 values appear dark on T1 weighted images and tissues with higher T2 values appear bright on T2 weighted images. Cortical structures like bones and fast flowing blood inside arteries and larger veins give no signal on MRI scans. Protons of fast flowing blood that had been excited by an RF pulse, have left the imaging slice before their signal can be detected.

Following a study of orbital MR imaging [Ett99], MR signal intensities, with respect to ocular and orbital tissues, can be identiﬁed from Tab. 2.5.

Tissue type T1w T2w cornea, sclera L L aqueous, vitreous L H normal clear lens M L extraocular muscles M M orbital fat H M connective tissue septa M M nerves M M

Table 2.5: MR Signal Intensities of Ocular and Orbital Tissues on T1-weighted (T1w) and T2-weighted (T2w) images from [Ett99]. H=high signal, M=medium signal, L=low signal. CHAPTER 2. MEDICAL FOUNDATIONS 23

Figure 2.9: Axial MR Scan, showing Lateral and Medial Rectus of both Eyes, from [FPBK03]. T1 weighted scan, 1.5 mm slicing, TR: 602 TE: 16

Figure 2.10: Axial MR Scan, showing Lateral and Medial Rectus of both Eyes, from [FPBK03]. T1 weighted scan, 1.5 mm slicing, TR: 602 TE: 16

Within the SEE-KID research work, a MRI study has been carried out in order to visualize the extraocular eye muscles by using newest MRI scanner equipment and special head coil receptors [FPBK03]. In this study, three human subjects were scanned using a 1.5 Tesla Siemens Magnetom Symphony scanner. Coronal and axial scans with T1 and T2 weighting were taken, using a 8 channel head phase array as receptor coil. Image series with approx. 19 images and a slice thickness of 1.5 mm, 512 × 416 matrix were taken while the subject was ﬁxating reference points CHAPTER 2. MEDICAL FOUNDATIONS 24

(see Fig. 2.9 and Fig. 2.10). Both eyes were put in local anesthesia using proprietary eye drops. Additionally, one subject was injected Gadolinium Magnivist 0.2ml/kg radiopaque material. All subjects were captured focusing diﬀerent stable gaze positions that were indicated by dots in the scanner tube.

(a) Subject fixating straight (b) Subject fixating to the right (c) Subject fixating to the right ahead bottom

Figure 2.11: Comparison of MR Scans in diﬀerent Gaze Positions [FPBK03]

In Fig. 2.11 diﬀerent gaze positions are compared with respect to the medial rectus muscles of both eyes. In Fig. 2.11(a), the subject was looking straight ahead, showing the muscles in a rather low innervated state. Compared to Fig. 2.11(b) and Fig. 2.11(c), where the subject is looking to the right and to the right bottom respectively, deformation of the physiologic cross-sectional area (PCSA) can be noticed. In repeating this described process for every slice in each image series, eye muscle morphology was reconstructed and graphically visualized.

The goal of this study was to show morphology of eye muscles, especially the rectus muscles while fixating different reference points. These data was analyzed and reconstructed as 3D dynamic representation of extraocular eye muscles in different gaze positions for later use in the SEE-KID research project [BK03b].

High resolution MRI enables exact delineation of space occupying orbital processes in relation to surrounding anatomical structures, thus facilitating planning of surgical procedures, which will be essential for computer-assisted surgery and treatment planning [Ett99].

2.1.5 Human Dissection of the Orbit

Since the discovery of pulleys, existing surgical procedures need to be revised and checked for mechanical consequences that can arise when surgical modification of pulleys and surrounding tissue occurs. Treatment of strabismus probably affects the action of orbital extraocular muscle layers on their pulleys, can cause unintended effects, especially when altering relationships between muscle insertion and pulley positions and should be better understood and perhaps considered in surgical planning [DOP00].

Human dissection was performed in order to better understand the functional implications of the newly discovered anatomical structures within eye muscles (Pulleys, global and orbital layers, etc.) [FAP03]. Compared to striated skeletal muscles, the six extraocular eye muscles show specific differences concerning structure, distribution of muscle fibers and neural sensitivity to CHAPTER 2. MEDICAL FOUNDATIONS 25 innervation. Thus, human eye muscles take up a exceptional position among human anatomy with regard to human skeletal muscles.

Figure 2.12: Human Dissection of the Orbit showing Medial Rectus Pulley [FAP03]

All six extraocular muscles consist of 2 distinct muscle portions, a consistently very thin orbital muscle layer with a high mitochondrial density in the muscle fibers, encircled by a thick C-shaped global layer. This global muscle layer consists of muscle fibers with a variable mitochondrial density, additionally, average physiologic cross-sectional areas are larger compared to the orbital muscle layer (see Sec. 2.1.2). Concerning muscle fiber types, human extraocular muscles consist of single innervated twitch fibres and multiple innervated non-twitch fibres which are usually only found in skeleton musculature. In Fig. 2.12, the medial rectus muscle of a right eye was dissected, showing the muscle stretched to indicate the connective tissue structure, encircling parts of the muscle in order to inflect the muscle path and therefore directing muscle force, compared to the anatomic origin, towards a much more anteriorly located point. The Pulley structure can be located posterior to the globe equator and is comprised of a dense collagen matrix with alternating bands of collagen fibers precisely arranged at right angles to one another. This three-dimensional organization most likely gives high tensile strength to the pulley. Connective tissue and smooth muscle bundles suspend the pulley from the periorbita. Smooth muscle is distributed in small, discrete bundles attached deeply into the dense pulley tissue. Fine structural observations confirm the existence and substantial structure of a pulley system in association with the medial rectus extraocular muscle. The presence of pulleys must be considered in models of the oculomotor plant. However, the nature of the connective tissue-smooth muscle suspending the pulley system to the orbit supports the suggestion that the pulley position, and thus the directional force of the eye muscles, may be adjustable [PPBD96]. CHAPTER 2. MEDICAL FOUNDATIONS 26

2.2 Eye Movement Physiology

The movement of the globe approximately corresponds to a rotation of an object in the three- dimensional space around a certain axis. The globe center can be regarded as rotation center. The line of sight is a vector from the globe center through the center of the pupil. Perpendicular to this vector, the vertical and the horizontal axes are deﬁned, whereby the intersection of these three axis lies in the globe center Fig. 2.13. Eye positions can be classiﬁed by their rotational properties [Kau95]:

Figure 2.13: Line of Sight, Vertical and Horizontal Axes; Rotations to Other Eye Positions

Primary position: The eye looks straight forward, with the head ﬁxed and upright (Fig. 2.13 (a)). It is accepted that in this position all muscles exhibit minimum force. From primary position, all other eye positions can be reached with as small an energy expenditure as possible.

Secondary position: From the primary position, a rotation around the horizontal or vertical axis (Fig. 2.13(b)) is performed. The eye looks to the left or right or upward or down.

Tertiary position: From the primary position a rotation around the horizontal and vertical axis (Fig. 2.13(c)) is performed. The eye looks e.g. to the left and down. The combination around two axis can be represented also by a rotation axis, which lies in the plane spanned by the horizontal and vertical coordinate axes (Fig. 2.13(d)).

Both eyes can be moved only in binocular community with one another, i.e. the movement of only one eye is usually not possible [Gue86]. Eye muscles are able to reposition the eye accurately and very fast, moreover they can hold a certain position without exhaustion. The rotation of an eye around a certain axis results in a certain eye position and thus also realigns the line of sight to a new gaze direction. The gaze direction designates the orientation of the eye, whereby an eye position always includes the fixation of a target object. Each muscle is defined through its origin, point of tangency and insertion. The muscle paths from the origin to the insertion at the globe are however additional held by connective tissues, retaining movements, so called „Pulleys“. For spatial perception, both eyes must exactly fixate an object of interest. The image of this object is then projected onto the foveae of both eyes, which provides best visual acuity. A cube fixated with both eyes is projected upside down (like in a camera) and skewed (due to different distances of both eyes) onto the retina. In the brain, both images will be merged in order to perceive one upright three dimensional cube (fusion) (Fig. 2.14(a)).

As an example, a patient suffering from inward squinting (esotropia) of the right eye. According to this, the right eye fails to fixate the cube and misaligns compared to the left eye (Fig. 2.14(b)). Different images are now projected onto the foveae of both eyes, and the brain fails to merge this CHAPTER 2. MEDICAL FOUNDATIONS 27

(a) (b)

Figure 2.14: Binocular Fixation of an Object in Space [Tho01] information. The result is, that disturbing (uncrossed) double images are perceived. Especially in infants, the brain will compensate on these errors by trying to eliminate those images that originate from the squinting eye. This process leads to a fatal outcome: the squinting eye will lose visual acuity and will suﬀer from functional deﬁciency.

2.2.1 Actions of the Extraocular Muscles

Basically, the eye can rotate in three dimensions around fixed axes of a head-fixed coordinate system. Each extraocular muscle rotates the globe in specific directions, also dependent of the current position of the eye. Movement of the eye nasally is adduction; temporal movement is abduction. Elevation and depression of the eye are termed sursumduction and deorsumduction, respectively. Torsional eye movements rotate the eye around its visual axis, whereby incycloduc- tion or incyclotorsion (intorsion) is nasal rotation of the vertical meridian and excycloduction or excyclotorsion (extorsion) is temporal rotation of the vertical meridian.

Referring to Fig. 2.15, the coordinate system used for torsional rotations is diﬀerent in both eyes. Referring to the ﬁgure above, directions of rotation of one eye around the respective axes can be described as follows:

Rotation around Z-axis positive angle = adduction (towards the nose) negative angle = abduction (away from the nose)

Rotation around X-axis positive angle = elevation or sursumduction (upward) negative angle = depression or deorsumduction (downward) CHAPTER 2. MEDICAL FOUNDATIONS 28

Rotation around Y-axis positive angle = intorsion (inward rolling) negative angle = extorsion (outward rolling)

Figure 2.15: Rotational Directions for both Eyes [BKP+03]

As shown in Fig. 2.15, coordinate axes for ab-/adduction and in-/extorsion point in different directions for both eyes. This provides a uniform definition of the rotation direction by using consistent designations (e.g. abduction = away from the nose). In both eyes, positive torsion is defined to rotate the eye inward around the line of sight. If the two axes would not show different signs, then intorsion would roll one eye inward and one outward. The axes for elevation and depression remain the same in both eyes, since up and down rotations specify the same directions for both eyes. Additionally, in case of binocular eye movement, abduction of one eye results in adduction of the other eye, thus eye positions are mirrored for ab-/adduction and in-/extorsion movements.

The primary muscle that moves an eye in a given direction is known as the agonist. A muscle in the same eye that moves the eye in the same direction as the agonist is known as the synergist, while a muscle in the same eye that moves the eye in the opposite direction of the agonist is the antagonist. For example, in abduction of the right eye, the right lateral rectus muscle is the agonist, the right superior and inferior oblique muscles are partly synergists, and the right medial rectus is the antagonists. A muscle that moves the opposite eye in the same direction as the agonist is the contralateral synergist, whereas a muscle in the opposite eye, that moves the eye in the opposite direction as the agonist is called contralateral antagonist.

Yoke muscles are the primary muscles in each eye that accomplish a binocular eye movement (e.g. for right gaze, the right lateral rectus and left medial rectus muscles). Each extraocular muscle has a yoke muscle in the opposite eye to accomplish movement of both eyes in the same direction. The primary action of an extraocular muscle is the direction of rotation of the eye when that muscle contracts. This term also indicates the gaze position in which the eﬀects of a muscle most easily are demonstrated. Knowledge of primary actions is important as squinting often increases in the ﬁeld of action of a weak eye muscle. CHAPTER 2. MEDICAL FOUNDATIONS 29

Medial and lateral rectus muscles have only horizontal actions. The medial rectus muscle is the primary adductor of the eye, and the lateral rectus muscle is the primary abductor of the eye.

Superior and inferior rectus muscles are the primary vertical movers of the eye. The superior rectus acts as the primary elevator and the inferior rectus acts as the primary depressor of the eye. This vertical action is greatest with the eye in the abducted position.

The direction of pull of the vertical recti muscles forms a 23 ◦ angle relative to the visual axis in the primary position, giving rise to secondary and tertiary functions. The secondary action of vertical rectus muscles is torsion. The superior rectus is an incyclotorter, and the inferior rectus is an excyclotorter. The tertiary action of both muscles is adduction.

Superior and inferior oblique muscles are the primary muscles of torsion. The superior oblique creates incyclotorsion, and the inferior oblique creates excyclotorsion. As the direction of pull for both muscles forms a 51 ◦ angle (relative to the visual axis in the primary position), secondary and tertiary actions occur.

The secondary action of the oblique muscles is vertical, and it is best demonstrated when the eye is adducted with the superior oblique acting as a depressor and the inferior oblique acting as an elevator of the eye. The tertiary action for each muscle is abduction.

Within the orbita, the eye is surrounded by ﬂexible fat pads, which, besides rotation, permit globe translation up to a certain limit. Usually, this translation is negligibly small, however in some pathological situations involving co-contraction of muscles (e.g. Duane syndrome), additional information about globe translation provides an important clue for the estimation and correction of eye motility disorders. A forward movement of the globe along the Y-axis is denoted as Protrusion, a backward motion is deﬁned as retraction.

2.2.2 Kinematic Principles of Eye Movements

The main task of the eye movement system is to move the eye quickly from the current gaze position to a new location, in order to ﬁxate an object of interest. Binocular eye movements are either conjugate (versions) or disconjugate (vergences). Versions are movements of both eyes in the same direction (e.g. right gaze in which both eyes move to the right). Dextroversion is movement of both eyes to the right, and levoversion is movement of both eyes to the left. Sur- sumversion and deorsumversion are elevation and depression of both eyes, respectively. Vergence movements are movements where both eyes move in opposite horizontal directions to permit the acquisition of a near or far object. When a subject is Looking at a near object, both eyes move together slightly to maintain binocular vision of it, as the object recedes away from the subject, then the two eyes diverge again.

Vision is blurred during an eye movement, so the length of time that the eye is moving must be minimized. In order to minimize the time during which no clear image is captured on the fovea, eye movements that move the fovea from one object/point to another are very rapid. These saccadic eye movements are among the fastest movements the body can make. The eyes can rotate at over 500 deg/sec, and subjects make well over one hundred thousand of these saccades daily.

Stabilization movements assure that the image of an object or region in the center of the ﬁeld-of- CHAPTER 2. MEDICAL FOUNDATIONS 30 view is kept over the fovea. Sophisticated mechanisms exist to accomplish this goal in the face of eye, head, body, and object motion. These eye movements are often grouped into four categories:

The vestibular-ocular reﬂex (VOR) rotates the eyes to compensate for head rotation and translation. Rotational and linear acceleration are detected by the semicircular canals and otolith organs in the inner ear. The resultant signals are used to command compensating eye movements.

The Optokinetic reflex stabilizes the retinal image caused by large-field motion. Retinal slip induced by field motion is used to initiate eye movements at the appropriate rate to cancel out image motion.

Smooth-pursuit eye movements are similar to the optokinetic reﬂex, but allow arbitrarily sized targets to be stabilized instead of large-ﬁeld motion. A moving target is required for smooth eye movements so that the eyes can move smoothly across a stationary object.

Vergence eye movements counter-rotate the eyes to maintain the images of an object at a given depth to be maintained at corresponding locations on the two retinae.

In the generation of eye movements, the brain follows certain consistent patterns of behavior (laws). However, it is important to realize, that the brain only adheres to these laws only when it suits a given purpose [FHMT97].

2.2.2.1 Donders’ Law

Donders’ law states, that the 3D angular position of the eye is always the same for any particular gaze position [Don48]. If the eye adopts a particular torsional orientation when looking forward at one instance (e.g. after a rotation from the right), it will adopt the same orientation at all other instances (e.g. after a rotation from the left) [FHMT97]. Hence, torsional eye position is not arbitrary, but uniquely determined by the gaze position.

2.2.2.2 Listing’s Law

Listing’s Law can be considered as an extension to Donders’ law, in that it specifies a specific torsional value for any particular eye position. It states that, with the head stationary, upright and the eyes fixating a distant object, all rotation axes of the eye lie in the same plane. Listing’s law is considered one of the most important principles in eye movement physiology, by which the brain couples redundant degrees of freedom, in this case by relating the two dimensional space of target directions relative to the fovea to a 2D manifold of 3D eye positions [FHMT97].

When the eye leaves one target object and fixates upon another, it rotates about an axis perpendicular to a plane cutting both, the former and the present lines of sight. Applying this principle in a broader sense, one could say that Listing’s law is probably the consequence of the facts that the primary position is the average of all eye positions during the day, that most eye movements are directed radially from or to the primary position and that rotations about a single axis are easier to perform than rotation about two axes or about an axis that changes during the CHAPTER 2. MEDICAL FOUNDATIONS 31 movement. In considering both, horizontal and vertical eye movements, in which case the eye is treated as if it were a pointer (defined by the line of sight). These two dimensional descriptions of eye rotations specify the orientation of this pointer in space. Since the eye can also make torsional movements around the line of sight, a full description of the rotational behavior also requires measurement of torsion. Accounting for all three degrees of freedom of eye rotations leads to some unique neurological consequences regarding oculomotor organization. Although this subject can be explained simply, it still causes a lot of confusion. Some of the difficulty arise from the need to define the kinematics of eye rotations quantitatively, setting up coordinate systems based on rigorously determined sets of spherical angles. Although this is useful for empirical verification, it is a poor way to introduce this subject. Rather than describing eye rotations with respect to a horizontal and vertical angular coordinate system, this explanation will begin intuitively.

(a) (b) (c)

Figure 2.16: Simple Abstraction of Listing’s Law [Nak83]

A simple physical „ball and membrane“ model of the eye is used [Nak83]. The model consists of a spherical globe attached to a very tightly stretched elastic membrane, carefully secured to the end of a cylinder, so that the membrane is equally taut in all directions Fig. 2.16. Because of the elastic quality of this membrane, there is a natural resting place for the globe. The stalk that is attached to this globe represents the line of sight. Additionally, a cross is mounted on top of this stalk to reveal the amount of twist (torsion) of the globe Fig. 2.16(a). The position of rest is defined as the primary position, and the direction of the line of sight (the direction the stalk points to) corresponds to the primary gaze direction. The plane spanned by the membrane of this model in resting position shall correspond to Listing’s plane for the eye. It is the fronto-parallel plane passing through the center of the globe. In Fig. 2.16(b), the globe is rotated with full degrees of rotational freedom. It can be rotated horizontally and vertically by displacing the stalk and torsionally by twisting the stalk between the fingers. The most important case is shown in Fig. 2.16(c) where the globe is moved by pushing the stalk with a smooth rod. Using this technique, the globe can be moved to any desired gaze position and the model has sufficient rotational freedom. Compared to Fig. 2.16(b), only two degrees of freedom are available to position the globe, since the twist of the globe around its visual axis (the stalk) is no longer under external control because it cannot be twisted between the fingers. Its orientation is dictated by the elastic membrane, which ensures that it corresponds to the position having the CHAPTER 2. MEDICAL FOUNDATIONS 32 lowest potential energy. Thus, each gaze position is associated with exactly one orientation of the globe.

What is essential to remember is that the behavior of this particular ball and membrane model is isomorphic with Donders and Listing’s laws. This phenomenon is also described as „pseudoto- stion“, since the eye does not actively rotate around its visual axis, instead, this torsional rotation is intrinsic to the behavior of the mechanical system of the human eye. The orientation of the eye can be predicted by assuming that the eye has made a geodesic (shortest path) rotation from the primary position to any other ﬁxation position. The axis of this shortest path rotation is perpendicular to the intended gaze position and thus lies in Listing’s plane. For the case of the rubber membrane model, it means that the axis of the shortest path rotation lies in the plane spanned by the membrane. The behavior of the model shows a radial or axial symmetry, such that there is no net torsion of the eye with respect to an axial reference position (the primary position).

Eye movements from tertiary positions to other tertiary positions do occur, but are less frequent and do not occur predominantly in down-, up-, right- or left-gaze. During eye movement from one tertiary to another tertiary position, Listing’s law is fulfilled only if the rotation takes place about an axis that is tilted to Listing’s plane by half the angle between the momentary tertiary position and primary position [Hel63]. Hence, an axis that can change during the movement. Using end points of rotation vectors to plot 3D eye positions, all positions will lie closely scattered along a plane (Listing’s Plane). Recordings of eye movements in humans and monkeys show that the standard deviation of rotation vectors describing eye positions from this plane is only about 0.5- 1.0 deg. The best plane to fit these data is called „displacement plane“ [Has95]. The orientation of this displacement plane also depends on the reference position used to describe eye positions. In shifting the reference position, the plane describing exactly the same eye positions will only shift half the angle in the same direction (see Fig. 2.17). In Fig. 2.17(a), the displacement plane is perpendicular to the reference position, whereas in Fig. 2.17(b), the reference position has shifted 2α deg. forward. Since the displacement plane tilts only α degree in Fig. 2.17(b), it is now tilted α deg. backward, with respect to a head fixed coordinate system.

(a) Reference position perpendic- (b) Reference position shifted by (c) Reference position shifted by ular to displacement plane 2α deg. 4α deg.

Figure 2.17: Deﬁnition of Listing’s Law w.r.t. Primary Position from [Has95]

For every data-set of arbitrary eye positions, there is one reference position such that the corresponding displacement plane is exactly perpendicular to the reference position. Only in this case, this position of the eye is termed primary position and the corresponding displacement plane is termed Listing’s Plane [Has95]. In Literature it is often preferred to use the term Listing’s plane in a broader sense, referring to any plane of rotation vectors. Applying this approach, all planes CHAPTER 2. MEDICAL FOUNDATIONS 33 in Fig. 2.17 would be termed Listing’s plane.

2.2.3 Sensorimotor Control

The visual system can be seen as a combination of a high-acuity detector, the eye, with a high- precision movement system, the oculomotor system [Car00]. Information processing during visual perception is strongly inﬂuenced by constraints arising from the human oculomotor system. The oculomotor system is characterized by the interaction between peripheral reﬂexes and central motor commands of visual origin.

Generally, eye movements fall into three broad classes:

Gaze-stabilization movements - shift the lines-of-sight of the two eyes to precisely compensate for self-motion, stabilizing the visual world on the retina. Gaze-stabilization movements are accomplished by two partially independent brain systems. The vestibulo-ocular system employs the inertial velocity sensors attached to the skull (the semi-circular canals) to determine how quickly and in what direction the head is moving and then rotates the eyes an equal and opposite amount to keep the visual world stable on the retina. The optokinetic system extracts information from the visual signals of the retina to determine how quickly and in what direction to rotate the eyes to stabilize the visual world.

Gaze-aligning movements - point a portion of the retina specialized for high resolution (the fovea) at objects of interest in the visual world. Gaze-aligning movements are also divided into two broad classes, saccades and smooth pursuit movements. Saccadic eye movements rapidly shift the lines-of-sight of the two eyes, with regard to the head, from one place in the visual world to another at rotational velocities up to 500−1000 deg /s. Saccades are the most common eye movements and occur rapidly (e.g. 3 saccades per second). In contrast, smooth pursuit eye movements rotate the eyes at a velocity and in a direction identical to those of a moving visual target, stabilizing that moving image on the retina.

Gaze-shifting movements - or vergence movements, operate to shift the lines-of-sight of the two eyes with regard to each other so that both eyes can remain ﬁxated on a visual stimuli at diﬀerent distances from the head (see Fig. 2.14).

The neural control of gaze in natural conditions requires the interaction between different strategies of oculomotor control. In particular, some eye movements are controlled by visual feedback (e.g. smooth pursuit) whereas others are controlled in open loop with respect to vision (saccades). Indeed, visual tracking of moving targets requires the combination of smooth pursuit eye movements with catch-up saccades. Only in a small part of the visual field, i.e. within the central 2 deg of the field of vision (the fovea), visual acuity permits the analysis of fine structured objects. In tracking saccades while a subject is looking at a picture (Fig. 2.18(a)), it can be shown, how the oculomotor control system directs the sense of perception through a visual stimulus (Fig. 2.18(b)). The recorded saccadic eye movements from Fig. 2.18 clearly show how the picture is sketched by discrete point to point eye movements.

Consequently, the brain needs to sample information of complicated scenes by many fixations, e.g. intervals, during which the eyes perform only microscopic movements on different parts of CHAPTER 2. MEDICAL FOUNDATIONS 34 an image (see Fig. 2.18). These fixations are separated by rapid eye movements called saccades. During reading, mean fixation durations on words ranges roughly from 200 ms to 300 ms, depending on lexical difficulties of the words. These fixations are interrupted by saccades with a mean duration between 20 ms and 40 ms. The brain continuously transforms sensory information into meaningful perceptions. The brain also accomplishes visual transformation task, such as encoding sequences of line points into curves, curves into shapes, and shapes into recognizable images. Though these processes are mostly visual, they also make use of more general brain functions, such as information chunking, learning and memory, that are employed by other brain regions.

(a) (b)

Figure 2.18: Recording of Saccadic Eye Movements from [KSJ00]

2.2.3.1 Innervation of the Eye Muscles

All eye movements are rotations, accomplished by three antagonistic pairs of muscles (see Sec. 2.2.1). These six muscles are controlled by three brainstem nuclei (Fig. 2.19(b)). These nuclei contain the cell bodies for all of the motor neurons (Fig. 2.19) which innervate the oculomotor muscles and thus serve as a final common path through which all eye movement control must be accomplished (Fig. 2.21). Each eye muscle is innervated by approximately 1000 motoneurons. The single motoneurons branch out in the eye muscles and innervate approximately 4 to 40 muscle fibers respectively. A motor unit is the aggregation of those muscle fibers, which are connected to one and the same motoneuron (see Fig. 2.19(b)).

The brain uses two diﬀerent possibilities to increase the tractive force of a muscle:

1. It recruits motor units, which were inactive before and

2. it concentrates the activity of those motor units, which have been active before, but which have not been working to full capacity. CHAPTER 2. MEDICAL FOUNDATIONS 35

(a) Innervation of a Muscle by a Motoneuron [uS98] (b) The Ocular Motor Nuclei [KSJ00]

Figure 2.19: Ocular Motor Neurons and Motor Nuclei

2.2.3.2 Oculomotor Neurons

The eye muscles are innervated by three cerebral nerves. The nervus oculomotoris (III. cerebral nerve) innervates the medial rectus muscle, the superior rectus muscle, the inferior rectus muscle and the inferior oblique muscle (moreover the levator palpebrae muscle as well as - with its parasympathetic part - the ciliaris muscle and the sphincter pupillae muscle) (see Fig. 2.20). The nervus trochlearis (IV. cerebral nerve) innervates the superior oblique muscle, the nervus abducens (VI. cerebral nerve) innervates the lateral rectus muscle.

The cell bodies of the motoneurons lie together in groups and form the so-called nuclei in the brainstem (Fig. 2.19(b)). The center of both oculomotor nerves lies in the midbrain (Fig. 2.19(b)). The alignment of those parts of the nuclei, which are associated with the speciﬁc eye muscles, is very complicated and most of the details have been discovered in the last few years. The cell bodies of the medial rectus muscle, the inferior rectus muscle and the inferior oblique muscle lie ipsilateral (e.g. for the right eye on the right nucleus-side). Only the cell bodies of the superior rectus muscle lie contralateral (e.g. for the right eye on the left nucleus-side). The ﬁbers of the nerve of the superior rectus muscle cross in the area of the nucleus oculomotoris to the other side. The cell bodies of the levator palpebrae muscle lie close to the middle line, both ipsilateral and contralateral. The two nuclei of the nervus trochlearis also lie in the midbrain, just below the nucleus oculomotoris. The motoneurons of the nervus trochlearis originate contralateral and cross behind the aqueduct, below the quadrigeminal bodies, to the other side. The nuclei of the nervus abducens lie in the bridge and are ipsilateral connected with the respective lateral rectus muscle (see Fig. 2.20). CHAPTER 2. MEDICAL FOUNDATIONS 36

(a) (b)

Figure 2.20: Oculomotor Nerve Circuit [KSJ00]

2.2.3.3 Neural Signal Encoding

Engineering models of the eye and its muscles indicate that motor neurons must generate two classes of muscle forces to accomplish any eye rotation. A pulsatile burst of force that regulates the velocity of an eye movement and a long-lasting increment or decrement in maintained force that, after the movement is complete, holds the eye stationary by resisting the elasticity of the muscles which would slowly draw the eye back to a straight-ahead position [Rob64]. Physiological experiments have demonstrated that all motor neurons participate in the generation of both of these two types of forces.

These two forces, in turn, appear to be generated by separable neural circuits. In the 1960s it was suggested that changes to the long-lasting force required after each eye rotation could be computed from the pulse-, or velocity-signal by the mathematical operation of integration. In the 1980s the lesion of a discrete brain area, the nucleus prepositus hypoglossi (Fig. 2.22), was shown to eliminate from the motor neurons the long-lasting force change required for leftward and rightward movements without affecting eye velocity during these movements. This, in turn, suggested that most or all eye movements are specified as velocity commands and that brainstem circuits involving the nucleus prepositus hypoglossi compute, by integration, the long-lasting force required by a particular velocity command. More recently, a similar circuit has been identified that appears to generate the holding force required for upwards, downwards, and torsional CHAPTER 2. MEDICAL FOUNDATIONS 37

Figure 2.21: Location and Distribution of Oculomotor Nerves [Tho01] movements.

2.2.3.4 The Superior Colliculus

The superior colliculus is a major visumotor region, integrating visual and motor information into oculomotor signals for the brainstem (Fig. 2.20(b)). It is a multilayered structure in the midbrain and can be divided into two functional regions:

1. the superﬁcial layers

2. the intermediate and deep layers.

The three superficial layers of the superior colliculus receive both direct input from the retina and a projection from striate cortex for the entire contralateral visual hemifield. Neurons in the superficial layers respond to visual stimuli.

In the two intermediate and deep layers cell activity is primarily related to oculomotor actions. The movement-related cells receive visual information from prestriate, middle temporal, and parietal cortices and motor information from the frontal eye ﬁeld. These layers also contain representations of the body surface and of the locations of sound in space.

Individual movement-related neurons in the superior colliculus discharge before saccades of specific amplitudes and directions, just as individual vision-related neurons in the superior colliculus respond to stimuli at specific distances and direction from the fovea. The movement-related neurons form a map of potential eye movements that is in register with visual and auditory receptive maps, so that the neurons that control eye movements to a certain target are found in the same CHAPTER 2. MEDICAL FOUNDATIONS 38 region as the cells excited by the sounds and image of that target. The region of the visual field that contains the targets for the saccades controlled by a given movement-related neuron in the superior colliculus is called the movement field of that neuron. Electrical stimulation of the intermediate layers of the superior colliculus evoke saccades into the movement fields of the neurons at the site of the stimulating electrode. The actual eye movement is encoded by the entire ensemble of these broadly tuned cells.

Activity in the superficial and intermediate layers of the superior colliculus can occur independently. Thus, sensory activity in the superficial layers need not lead to motor output, and motor output can occur without sensory activity in the superficial layers. In fact, the neurons in the superficial layers do not have a large, direct projection to the intermediate layers. Lesions of a small part of the colliculus affect the latency, accuracy, and velocity of saccades, whereas lesions of the entire colliculus render e.g. a monkey unable to make any contralateral saccades, although with time, the ability to make contralateral saccades is recovered.

The saccade-related activity of the superior colliculus neurons is shaped by inputs from the posterior parietal cortex, the frontal eye fields, and the substantia nigra pars reticulata (Fig. 2.19(b)). The posterior parietal cortex is involved in the visual guidance of saccades by shaping the visual inputs to the superior colliculus. The posterior parietal cortex contains neurons that are modu- lated by visual attention, i.e., by how behaviorally relevant a visual stimulus is. They respond more effectively when the stimulus is the target for an eye movement. The frontal eye fields form an executive center that can selectively activate superior colliculus neurons, playing a role in the selection and production of voluntary saccades. The activity of frontal eye fields neurons reflects the selection of the visual target for a saccadic eye movement when several potential goals for movements are available. The frontal eye fields is also involved in suppressing reflexive saccades and generating voluntary, non-visual saccades. The complementary executive control exerted on saccade generation by the frontal eye fields and the superior colliculus is revealed by the effect of selective and combined ablation.

Lesions of the superior colliculus prevent the generation of reflexive saccades, whereas the generation of voluntary saccades is disrupted by frontal eye fields lesions. Although saccades can still be produced after the ablation of either the superior colliculus or the frontal eye fields, a combined ablation of these two structures results in the complete abolition of saccadic eye movements.

2.2.3.5 Brainstem Control of Saccades

The saccadic system, in order to achieve a precise gaze-shift, must supply the brainstem circuits with a command that controls the amplitude and direction of a movement. Considerable research now focusses on how this signal is generated. Current evidence indicates that this command can originate in either of two brain structures. The superior colliculus of the midbrain or the frontal eye ﬁelds of the neocortex (Fig. 2.19(b)), both containing laminar sheets of neurons that code all possible saccadic amplitudes and directions in a topographic map-like organization [HM03]. Activation of neurons at a particular location in these maps is associated with a particular saccade and activation of neurons, adjacent to that location associated with saccades having adjacent coordinates. Lesion experiments indicate that either of these structures can be removed without permanently preventing the generation of saccades.

How these signals that topographically encode the amplitude and direction of a saccade are CHAPTER 2. MEDICAL FOUNDATIONS 39

Figure 2.22: Motor Circuit for Horizontal Saccades [KSJ00]. Excitatory neurons are orange and inhibitory neurons are gray. The dotted line represents the midline of the brainstem. translated into a form appropriate for the control of the oculomotor brainstem is not known. One group of theories proposes that these signals govern a brainstem feedback loop which accelerates the eye to a high velocity and keeps the eye in motion until the desired eye movement is complete [Rob75a]. Other theories place this feedback loop outside the brainstem, or generate saccadic commands without the explicit use of a feedback loop. In any case, it seems clear that the superior colliculus and frontal eye ﬁelds are important sources for these signals because if both of these structures are removed, no further saccades are possible [STC80]. The superior colliculus and frontal eye ﬁelds, in turn, receive input from many areas within the visual processing streams including the visual cortex, as well as the basal ganglia and brain structures involved in audition and somatosensation. These areas are presumed to participate in the processes that must precede the decision to make a saccade, processes like „attention“.

As an example, in Fig. 2.22, the motor circuit for horizontal saccades in the brainstem is shown. CHAPTER 2. MEDICAL FOUNDATIONS 40

The eye velocity component of the motor signal arises from excitatory burst neurons in the paramedian pontine reticular formation (PPRF) that synapse on motor neurons and interneurons in the abducens nucleus. The abducens motor neurons, project to the lateral rectus muscles while the interneurons of the abducens nucleus project to the contralateral medial rectus muscle via ﬁbers that cross the midline and ascend in the medial longitudinal fasciculus (MLF). Excitatory burst neurons also drive ipsilateral inhibitory burst neurons that inhibit contralateral abducens and excitatory burst neurons. The medial vestibular nucleus also inhibits contralateral abducens neurons. Omnipause neurons inhibit excitatory burst neurons and abducens neurons, preventing unwanted eye movements. The eye position component of the motor signal arises from a „neural integrator“ comprised of neurons distributed throughout the medial vestibular nuclei and nucleus prepositus hypoglossi on both sides of the brain stem. These neurons receive velocity signals from excitatory burst neurons and integrate its velocity signal into a position signal. The position signal is then transmitted to the ipsilateral abducens neurons.

2.2.3.6 Control of Smooth-Pursuit Movements

In the smooth pursuit system, signals carrying information about target motion are extracted by motion processing areas in visual cortex and then passed to the dorso-lateral pontine nucleus of the brainstem. There, neurons have been identiﬁed which code either the direction and velocity of pursuit eye movements, the direction and velocity of visual target motion, or both. These signals proceed to the cerebellum where neurons have been shown to speciﬁcally encode the velocity of pursuit eye movements [SK84]. These neurons, in turn, make connections with cells known to be upstream of the nucleus prepositus hypoglossi (Fig. 2.22), the integrator of the oculomotor system for horizontal saccades). As in the saccadic system, the brainstem integrator appears to compute the long-term holding force from this signal and then to pass the sum of these signals to the motor neurons.

All eye movement control signals must pass through the ocular motoneurons which serve as a ﬁnal common path. In all cases these neurons carry signals associated both with the instantaneous velocity of the eye and the holding force required at the end of the movement. Eye movement systems must provide control signals of this type, presumably by ﬁrst specifying a velocity command from which changes in holding force can be computed. In the case of saccades, this command is produced by brain structures that topographically map all permissible saccades in amplitude and direction coordinates. In the case of pursuit, the brain appears to extract target motion and to use this signal as the oculomotor control input. Together these systems allow humans to redirect the line-of-sight to stimuli of interest and to stabilize moving objects on the retina for maximum acuity.

2.2.3.7 Hering’s Law

Neural processes of binocular vision involve the control of both eyes at the same time. One very fundamental function of the visual system is to provide information on the location of objects in space. The left and the right eyes, positioned about 6 cm apart, each capturing an image of the environment from a slightly diﬀerent point of view, and comparison by the brain of these two retinal images yields information about distance. As long as the diﬀerence between the left and right eye image (the binocular disparity) is not too large, the mechanism of stereopsis CHAPTER 2. MEDICAL FOUNDATIONS 41

(binocular vision) both interprets the disparity as depth and, by means of sensory fusion, causes the desperate images to be perceived as single impression. Thus, the two eyes must be well aligned for normal stereopsis to function properly. This implies that both eyes accurately ﬁxate a single point in space.

According to Hering’s law of equal innervations [Her68], the two eyes are not independently controlled by the brain. Hering observed that many eye movements are conjugate from birth, even when one eye is covered. Hering argued that a common innervation must be sent to the eyes to achieve this conjugacy, implying that the two eyes are controlled as one unit. The means, that changes of innervation are programmed in one of two modes:

1. for a conjugate eye movement, in which both eyes rotate through the same angle and in the same direction, or

2. for a disjunctive eye movement, in which both eyes rotate through equal angles but in opposite directions.

In positioning both eyes with respect to conjugate eye movements, the brain accepts one eye as the leading or fixating eye, from where motor commands are received. The other eye will be considered as the following eye, and will receive innervations based on those from the fixing eye. By the Hering law, yoke muscles receive equal and simultaneous innervation. The magnitude of this innervation is determined by the fixating eye. Since the magnitude of innervation to yoke muscles is determined by the fixating eye, the angle of deviation between eyes (strabismus) may vary depending on which eye is fixating. The primary deviation is misalignment with the normal eye fixating. If the paretic eye fixates, the ensuing secondary deviation is typically larger than the primary deviation.

2.2.3.8 Sherrington’s Law

Rotation of the eyes can be derived from a change in the distribution of active muscle tension between antagonist muscles, although their sum remains constant. From a stable position (e.g. primary position), the change of muscle force distribution occurs in phases in order to begin, carry out, and stop the saccadic or smooth pursuit movement. The distribution is then stabilized at the new eye position.

Sherrington’s law of reciprocal innervation states that increase of innervation and contraction of a muscle is associated with a reciprocal decrease in innervation of the antagonist muscle [DE73].

2.3 Measurement Techniques

Measurement techniques in the field of oculomotor physiology can be divided into measurements of eye movements (positions) and into measurements of the underlying anatomy and physiology (e.g. neural activity, motion and muscle force). These measurements can be accomplished by the use of different measurement techniques, ranging from minimal invasive methods (e.g. clinical CHAPTER 2. MEDICAL FOUNDATIONS 42 observations) to methods that need access to the human anatomy. However, in order to understand the fundamental workflow of the oculomotor system, all measurement results need to be considered.

2.3.1 Eye Movement Measurements

Eye movements not only dictate where a subject looks or what a subject sees, they are indispensable for diagnosing ophthalmologic, neurological, and otologic problems. Measurements of eye movements are used to study how our cortex processes information, what the cerebellum is doing, and how learning takes place in the brainstem. And they are increasingly used for technical applications, like monitoring the alertness of drivers or controlling the auto-focus of cameras. Measurement techniques are comprised of mostly non invasive techniques that monitor moment of the eyes in two or three dimensions.

2.3.1.1 Electro-Oculography

The electro-oculogram (EOG) is a measurement of bipotentials under two diﬀerent light intensities (light adapted – non-light adapted eye) produced by changes in two constant eye positions. It is used as a clinical test for retinal dysfunction. The changes of bipotentials in electro-oculography is also used under constant light intensity for the measurement of diﬀerent eye positions. In the normal eye, there is a steady potential of approximately 6 to 10 mV between the cornea and the retina known as the corneal-retinal potential. The corneal-retinal potential can be measured by placing a single surface electrode directly lateral to each of the orbits of the eye and a reference electrode to the bridge of the nose.

(a) Application of EOG measurement elec- (b) EOG recording: The subject was in- trodes structed to look to the far left, hold his gaze, look to the far right, hold his gaze and re- peat.

Figure 2.23: EOG Eye Movement Measurement Technique [FFG+02]

By placing electrodes superior and inferior to the orbit of each eye, and a reference electrode lateral to the eye of interest, vertical eye movements can also be measured (see Fig. 2.23(a)). When a test subject is gazing straight ahead, the corneal-retinal dipole is symmetric between the two electrodes, and measured EOG output is zero. As the subject gazes to the left, the cornea becomes closer to the left lateral electrode, therefore causing the EOG output to become more CHAPTER 2. MEDICAL FOUNDATIONS 43 positive (see Fig. 2.23(b)). The inverse of this is true when the subject looks in the right direction. When measuring the EOG output, there is a fairly linear relationship between the horizontal angle of gaze and the EOG output. This relationship remains true up to approximately thirty degrees of arc. Through careful calibration, horizontal eye movements can be measured with a resolution of approximately one degree of arc. The EOG technique is preferred for recording eye movements in sleep and dream research.

One speciﬁc type of EOG measurements is the electronystagmograhy (ENG). ENG measurements are used to measure a condition called nystagmus. Measurement of nystagmus, a characteristic pattern of eye movement, is invaluable in the diagnosis of various vestibular and balance dysfunctions.

The EOG is not a very stable signal and measurements can vary as a result of varying ambient light conditions. By having a patient carry out eye movements of constant amplitude in the dark and then in the light, any change in the biopotential would reﬂect a change in the corneal-retinal potential. In normal eyes, this potential decreases during dark adaptation, and increases during light adaptation. However, the signal can change when there is no eye movement. It is prone to drift and giving spurious signals, the state of the contact between the electrodes and the skin produces and other source of variability. There have been reports that the velocity of the eye as it moves may itself contribute an extra component to the EOG. It is not a reliable method for quantitative measurement, particularly of medium and large saccades. However, it is a cheap, easy and non-invasive method of recording large eye movements, and is still frequently used by clinicians.

2.3.1.2 Infrared-Oculography

If a fixed light source is directed at the eye, the amount of light reflected back to a fixed detector will vary with the eye’s position. This principle enables the determination of the current eye position. Infrared light is used as this is „invisible“ to the eye, and doesn’t serve as a distraction to the subject. As infrared detectors are not influenced to any great extent by other light sources, the ambient lighting level does not affect measurements. Spatial resolution reaches up to 0.1 ◦, and temporal resolutions of 1ms can be achieved. It is better for measuring horizontal than vertical eye movements. Blinks can be a problem, as not only do the lids cover the surface of the eye, but the eye retracts slightly, altering the amount of light reflected for a short time after the blink. The corneal reflection of the light source is measured relative to the location of the pupil center. Corneal reflections are known as the Purkinje reflections, or Purkinje images [Cra94].

Due to the construction of the eye, four Purkinje reflections are formed (see Fig. 2.24). Video- based eye trackers typically locate the first Purkinje image. With appropriate calibration procedures, these eye trackers are capable of measuring a viewer’s point of regard (POR) on a suitably positioned (perpendicularly planar) surface on which calibration points are displayed. Two points of reference on the eye are needed to separate eye movements from head movements. The positional difference between the pupil center and corneal reflection changes with pure eye rotation, but remains relatively constant with minor head movements.

Since the infra-red light source is usually placed at some ﬁxed position relative to the eye, the Purkinje image is relatively stable while the eyeball, and hence the pupil, rotates in its orbit (see Fig. 2.25). So-called generation-V eye trackers also measure the fourth Purkinje image, CHAPTER 2. MEDICAL FOUNDATIONS 44

Figure 2.24: Schematic Overview of Purkinje Corneal Reflections, from [Cra94] PR, Purkinje reflections: 1, reflection from front surface of the cornea; 2, reflection from rear surface of the cornea; 3, reflection from front surface of the lens; 4, reflection from rear surface of the lensùalmost the same size and formed in the same plane as the first Purkinje image, but due to change in index of refraction at rear of lens, intensity is less than 1% of that of the first Purkinje image; IL, incoming light; A, aqueous humor; C, cornea; S, sclera; V, vitreous humor; I, iris; L, lens; CR, center of rotation; EA, eye axis (line of sight); a ≈ 6mm; b ≈ 12.5mm; c ≈ 13.5mm; d ≈ 24mm; r ≈ 7.8mm from [Cra94] however, due to the anatomical structure of the eye, this reflection gives a very weak signal. By measuring the first and fourth Purkinje reflections, these dual-Purkinje image (DPI) eye trackers separate translational and rotational eye movements. Both reflections move together through exactly the same distance upon eye translation but the images move through different distances, thus changing their separation, upon eye rotation. Unfortunately, although the DPI eye tracker is quite precise, head stabilization may be required.

2.3.1.3 Scleral Search Coils

One of the most precise eye movement measurement methods involves attaching a mechanical or optical reference object mounted on a contact lens which is then worn directly on the eye. This technique evolved to the use of a modern contact lens to which a mounting stalk is attached (see Fig. 2.26 and Fig. 2.27). The contact lens is necessarily large, extending over the cornea and sclera (the lens is subject to slippage if the lens only covers the cornea). Various mechanical or optical devices have been placed on the stalk attached to the lens: reﬂecting phosphors, line diagrams, and wire coils have been the most popular implements in magneto-optical conﬁgurations.

The principle method employs a wire coil, which is then measured moving through an electromagnetic field. When a search coil (see Fig. 2.26(a) is put into an oscillating magnetic field, a voltage is induced in the coil [Has95]. Three orthogonal magnetic fields are emitted by an electromagnetic field frame (see Fig. 2.26(b)). Usually, one contact lens has 2 search coils mounted, one coil for the measurement along each axis of a two-dimensional coordinate system. The voltages induced by the electromagnetic field frame can directly be related to elements of the rotation matrix that describes the current eye position relative to a reference position where the coils line CHAPTER 2. MEDICAL FOUNDATIONS 45

Figure 2.25: Purkinje Corneal Reﬂections I. and VI. marked with crosses, from [Kan87] up with the magnetic ﬁelds. The coils commonly used for recording 3D eye positions are dual search coils (produced by Skalar Instruments, Delft, The Netherlands) which are oriented in such a way that they are approximately parallel to the axes spanned by the electromagnetic frame.

(a) Scleral Search Coil Lens (b) Electromagnetic Frame for Search Coil Measurements

Figure 2.26: Electromagnetic Search Coil Eye Position Measurement, from [Ska03]

Problems, like the determination of oﬀsets which are frequently superimposed on the induced voltages have been investigated in detail by Hess et al [HVOS+92]. The determination of angular rotations for a measured eye position also involves consideration of „false torsion“ (see Sec. 2.2.2.2) and appropriate correction (cf. [Has95]).

Insertion of the contact lens is shown in Fig. 2.27. Although the scleral search coil is the most precise eye movement measurement method (accurate to about 5-10 arc-seconds over a limited range of about 5 ◦), it is also the most intrusive method. Insertion of the lens requires care and practice. Wearing of the lens causes discomfort and therefore is not ideal for clinical application. CHAPTER 2. MEDICAL FOUNDATIONS 46

Figure 2.27: Insertion Procedure for a Scleral Search Coil Lens, from [Ska03]

2.3.1.4 Video-Oculography

With the development of video and image analysis technology, various methods of automatically extracting the eye position from images of the eye have been developed. Tracking the relative movements of these images gives an eye position signal. 3D video-oculography (3D VOG) systems commonly work on the following principle: first, the center of the pupil is found. This is achieved by thresholding the gray-level signal of the image of the eye, finding the pupil, and fitting an ellipse to its outline (see Fig. 2.28). The center of this ellipse determines the horizontal and vertical eye position. Then, the light-dark distribution of the iris is measured along a circle around the center of the pupil. Cross-correlating this iris signature with a reference pattern gives the torsional eye position (cf. [MHCS94] and [Has95]). Up to now, no algorithm exists that can distinguish between a translation of the camera with respect to the head, and a shift of the pupil by a rotation of the eye. However, looking at Fig. 2.28, it is obvious that images of the eye contain more information than just the center of the pupil. Corneal reflections, also used in infrared oculography, patterns of the iris and the shape of the upper and lower eye-lid may give additional signals to improve the stability of VOG.

Figure 2.28: Video-Oculography Pupil Detection, [SMI]

However, image based methods tend to have temporal resolutions lower than that achieved with IR techniques. One reason why existing video-oculography (VOG) systems have not filled this need is the difficulty of measuring the rotation of the eye around the line of sight. Techniques for tracking the horizontal and vertical position of the pupil are straightforward, and a number of different algorithms give acceptable results. But measuring the rotation of the eye around the line of sight is much more difficult, since it requires not only the detection of the pupil, but also relies on the resolution of fine details in the structure of the iris (cf. [Has95], [MHCS96] and [SH03]). Even small displacements between head and camera can cause large measurement artifacts (e.g. CHAPTER 2. MEDICAL FOUNDATIONS 47 a camera displacement of only 1 mm shifts the center of the pupil by the same amount as a 5 ◦-rotation of the eye). Currently, no published algorithms exist for the automated selection of suitable landmarks on the iris, which is necessary for automated measurements of ocular torsion. Such algorithms would dramatically improve the applicability of video-oculography, also to other scientific and medical applications. However, current system accomplish the 3D measurement of eye positions, provided that the head is upright and not moving rapidly (see Fig. 2.29).

Figure 2.29: Chronos VOG System, from [Ska03]

The VOG eye tracker shown in Fig. 2.29 consists of a head unit, which is individually adjustable, and carries the CMOS cameras for recording eye-in-head images. Additionally, this head unit carries the system unit, which accommodates the custom-designed architecture for the online, real-time acquisition and pre-processing of image and signal data. This is designed around a standard Windows PC with a PCI plug-in board. The head unit is connected to the system unit through high-speed digital data links. These provide the necessary data channels for the transfer of high bandwidth image and signal data from the head unit to the system unit, and the command sequences from the system unit to the head unit. The image of the eye is reflected by the dichroic mirror to the optical lens and projected onto the image sensor. An infrared pass filter is fitted in front of the image sensing area in order to exclude sporadic incident light from the environment. These optical elements and the cameras are arranged on the head unit to facilitate maximal field-of-view for the test subject. A field-of-view approaching ±90 ◦ horizontal and +40/ − 60 ◦ vertical is attained.

Nevertheless, there is an increasing requirement in both the clinical and research ﬁelds for non- invasive precise measurement of three-dimensional eye movement by using 3D VOG systems.

2.3.2 Physiologic Muscle Force Measurements

During the investigation of muscle force, two types of contraction measurements can be differentiated: isotonic contraction and isometric contraction. An isotonic contraction is the measurement of the muscle length and length change due to an activation with constant load (Fig. 2.30(A)). In the case of isometric contraction, the strength and the change of force of the muscle is measured with the length held constantly (Fig. 2.30(B)). The contraction mechanism takes place on molecular level and is described by the so-called filament sliding theory. Actin and myosin filaments are connected over archings and realize activation-steered shortening, and thus development of CHAPTER 2. MEDICAL FOUNDATIONS 48 muscle force.

Figure 2.30: Diﬀerent Types of Muscle Force Measurement, from [BKP+03]

Generally, static and dynamic characteristics of a muscle can be differentiated. Static force behavior is also called force-length behavior, whereby isometric force measurement is accomplished as a starting point and, in dependence of the adjusted length, measurement of the produced force is carried out. Dynamic characteristics of a muscle refers to contraction speed and is analyzed by using isotonic measurements. In the representation of the static characteristics of a muscle, active and passive muscle force can be differentiated. Active muscle force results from activation (innervation) of a muscle initiated by the brain, while passive forces represent the flexible stretch characteristics of a muscle, which works opposite to the active force. By applying repeated isometric force measurements with differently adjusted lengths, a force length curve of the muscle behavior results. In relating these data to the activation potential of a muscle, a three-dimensional force-length-activation function can be defined. This function can be divided again into its active and passive forces, receiving an active and a passive force curve, which correspond to the total force curve of a muscle. The passive force curve describes the flexible forces a muscle exerts, if it is stretched or contracted accordingly. If a detached muscle is sufficiently stretched, then, at a certain length, the muscle stops to behave like a non-linear spring, but will get stiff very fast, allowing no further flexibility. This is called the „leash region“ and becomes apparent in a drastic rise of the passive strength with increased stretch length. On the other hand, if a muscle gets shorter and loses its stretch, then it will get slack and can not exert force anymore. This is called the „slack region“ of the muscle force function.

Meaningful force measurements of the extraocular muscles are extremely diﬃcult to obtain. How- ever, there are diﬃculties deriving the physiologic behavior of extraocular muscles from such measurements. A clinically useful device to measure force was introduced by Scott et al in 1972 (see Fig. 2.31) [SCO72].

Collins described instrumented duction forceps that carried strain gauges to measure force, and an ultrasonic microphone to measure distance, in conjunction with an ultrasonic sound source [Col78]. The Collins forceps allowed convenient length-tension measurements on an isolated, disinserted muscle or an intact eye.Others followed with variations of this approach (e.g. [SCW+84]). CHAPTER 2. MEDICAL FOUNDATIONS 49

Figure 2.31: Example of measuring Force with Forceps

A common clinical diagnostic test is the forced duction, sometimes called passive duction test. This test is used to identify the cause of the lack of rotation of a muscle from two possible broad causes. In weakness of the muscle or paresis of the nerve supplying the muscle, the eye does not rotate as it should because, ultimately, the muscle is not contracting properly. In mechanical causes of lack of rotation of the eye, the muscle receives input instructing it to contract and rotate the eye, but mechanical forces are preventing the eye from moving properly. In the exam chair, it is performed by topically anesthetizing the eye with drops, and then further anesthesia may be given by soaking an applicator with topical anesthetic and applying it to the intended point of contact with the eye. Classically, a forceps is used to perform the test, by grabbing the eye and mechanically rotating it while the patient looks in the direction of gaze being tested, and seeing if the examiner can rotate the eye for the patient. Free rotation implies that the eye is not mechanically resisted, which means the rotation problem is a paresis of the nerve supplying the muscle the impulse to contract. If there is mechanical resistance, often not only does the eye not fully rotate in the direction being tested, but the examiner may feel the resistance. Some examiners prefer to do the test by pushing the globe in the intended direction with an applicator, rather than grabbing the eye with a forceps, but the overall concept is the same (see Fig. 2.31).

According to Miller [MD96], there exist four problems with all isometric measurements:

Relationship of innervation, length, and tension - is not physiologically meaningful during isometric measurements, since the eye to be measured is held at a ﬁxed position and moved from there. Normally, when innervation increases, muscle force increases, and the muscle shortens. When innervation decreases, muscle force also decreases, and the muscle lengthens. With forced duction, innervation is held constant, thus, muscle force increases as the muscle is lengthened, and decreases as it is shortened, resulting in the inverse behavior compared to the normal relationship. Isometric measurements also disturb the normal relationship between length and tension, though not so badly as forced ductions. Neither forced duction measurements nor isometric measurements can directly predict normal behavior of the extraocular muscles. The force produced in a suddenly stretched muscle tends to decay by stress relaxation or yield. Similarly, a sudden increase in load results in an abrupt initial stretch, followed by a gradual increase in stretch, called creep. Seeking to avoid yield and creep by pulling and relaxing the muscle quickly does not help, since CHAPTER 2. MEDICAL FOUNDATIONS 50

increasing stretch velocity increases viscosity, spuriously increasing measured forces during the pull and decreasing them during the release. Viscous effects can be minimized by stretching slowly, but this increases yield and creep. Methods that require the measured muscle to be disinserted - makes it nearly impossible to measure force at primary position length. Once the cut end of the muscle tendon is attached to the measuring device, attempts to determine primary position length by abut- ting it to its severed insertion, while holding the eye in primary position and preventing globe translation, do not inspire confidence. The need to „unwrap“ the muscle from the globe to take the actual measurement provides another opportunity for error. Rotating the globe to measure stiffness - may be done in an intact eye [CCSJ81] or with some muscles disinserted. Here the worry is that translation as well as rotational forces are applied by the duction forceps. There is no way to avoid translating the eye when pulling at one point on its surface. Translation stretches some muscles and relaxes others, distorting force measurements in complex ways. The degree of elastic coupling to surrounding structures - with measurements on disinserted muscles is seldom clear and definitely not physiologic.

Figure 2.32: Muscle Force Transducer for Intraoperative Measurements [MD96]

To address these problems, Miller et al developed a method that is related to the method of Collins [COS75], in that it leaves the eye free to rotate, preserving the physiologic relationship between muscle tension and length (see Fig. 2.32). The muscle is not disinserted, and its path length (e.g. degree of stretch in a given eye position) is not significantly altered. This device lies flat on the muscle tendon, causing little modification of ocular mechanics. Since the measured muscle remains attached to the globe, primary position muscle length is easily established. Problems of globe translation do not arise, and musculo-fascial couplings are physiologic. However, signals are slightly distorted as the transducer rotates under the lids. The method is most useful with awake patients, but even with anesthetized patients it has the advantage that primary position forces can be determined. CHAPTER 2. MEDICAL FOUNDATIONS 51

2.3.3 Measurement of Motion in the Orbit

It has been shown, that the measurement of the motion of orbital tissue can improve the diagnosis and management of orbital disorders and also give more detailed insight to the kinematics of orbital movements [DMP+95][CJD00][DOP00]. It is easy to measure the motion of the eyes as a function of gaze, since the eyes are easily observed visually and their motion is readily accessible for inspection. However, the motion of the orbital tissues behind the globe is effectively hidden by the orbit, the eye and the eyelids, so that it is very difficult to apply measurements in vivo. Attempting to measure motion objectively by introducing instruments or devices into the orbit is not yet an option. The risk of damaging the optic nerve and consequently causing blindness is simply too large. Additionally, such instruments may influence the very motion they are supposed to measure. As an alternative, Abràmoff suggested to measure the motion first by using MR imaging and then using image analysis techniques to measure the motion objectively [Abr01]. A suitable image device constructs an image, or a series of images from some regionally varying physical properties. In this case, an image can be considered as an ordered set of vectors, where each vector represents the magnitude of one or more of the measured physical properties. Reconstructing optical flow vector fields in these images gives insight into the motion of the captured region.

Cine MRI time sequences were obtained using T1-weighted volumes on a 1.5 T MR scanner and a head-coil (TE 6.9, TR 12, matrix 256x256x46) with an acquisition time of 15 sec. Two dimensional image sequences were extracted from these volumes on a transversal axis intersecting both, horizontal rectus muscles and the optic nerve (cf. Sec. 2.1.4).

Figure 2.33: Intraconal Tissue Motion around the Optic Nerve, from [Abr01] A: ﬂow ﬁeld displayed over a static MRI of the left orbit. B: Schematic view of motion as expressed by B.

Because the true motion field in the orbit is unknown, simulations and measurements of controlled motion of an object (i.e. a sirloin steak) was used to compare the computed flow fields with the known motion fields. A sirloin steak was mounted in a transparent box fitted with an angle ruler. A sirloin steak consists of bands of several millimeters width of alternating muscle and fat tissue that approximates the alternating fat-muscle-fibrous tissue structure of orbital soft tissue. This object was rotated 5 degrees per captured frame, and a sequence of 21 frames was obtained by the MR scanner. Pre-filtering using Gaussian smoothing and nonlinear diffusion was performed on all images. CHAPTER 2. MEDICAL FOUNDATIONS 52

Optical flow algorithms were used to extract the spatio-temporal patterns of image or signal intensity to estimate the optical flow field. In a second stage, the resulting system of equations is solved to estimate the actual optical flow. The computed flow fields were displayed using a mapping technique that shows all flow vectors as colored pixels plotted over the original MR image (see Fig. 2.33). Thus, a multi-modality image is obtained that combines functional (motion) and anatomical information in a single image. Chapter 3

Strabismus

Squint (strabismus) is the name given to usually persistent or regularly occurring misalignment of the eyes. Strabismus is a visual defect in which the eyes are misaligned and point in diﬀerent directions. One eye may look straight ahead, while the other eye turns inward, outward, upward or downward. Strabismus is a common condition among children. About 4 percent of all children suﬀer from strabismus. It can also occur later in life. It occurs in males and females and may run in families. However, many people with strabismus have no relatives with this problem.

People that have strabismus suﬀer not only from the frequently disﬁguring externally visible abnormality, the visual impairment associated with squint is an even greater burden. Squint is not just a blemish but often a severe visual impairment. The earlier a child develops a squint and the later it can be treated, the worse the visual impairment will be. By the time a child reaches school age, the prospects of successful treatment decline dramatically. Babies and small children with strabismus should be treated at the earliest possible moment.

This chapter will give an overview of some important fundamentals in strabismus along with its various pathological classifications. Along with representative examples, basic clinical diagnosis and treatment will be covered, including the effects of surgery, evaluation of eye motility disorders or determination of the amount of surgery. However, more detailed information on strabismus pathology and surgery is well documented and can be found in various literature (e.g. [Kau95], [DE73], [RSS01], [Kan87] etc.). The goal of this chapter is to give insight into current clinical practice in the field of strabismus diagnosis and treatment in order to better understand the application of a new, computer-aided basis for this medical subject.

53 CHAPTER 3. STRABISMUS 54

3.1 Overview

In order to be able to correctly perceive the environment, both eyes must look in the same direction. This causes almost identical images to be generated within each eye. These two images are then fused together in the brain to form a single three-dimensional visual impression. If a squint is present, the difference between the two images caused by the misalignment is too great and the brain is unable to converge and to fuse them. The result is irritating double vision. The juvenile brain is able to respond to double images by simply suppressing the image arriving from the deviant eye. This process generally has calamitous consequences as vision in the unused eye gradually becomes weak (amblyopic). Amblyopia is the term used to describe weak vision in an otherwise organically healthy eye. In the absence of treatment almost 90% of all children who suffer from a squint develop amblyopia on one side. If this squint-related visual weakness is not detected and treated within early time, it will remain a lifelong affliction. The child will then never learn to see with both eyes or even have three-dimensional vision and will be at greater risk of accidents and restricted in everyday life. Prompt treatment can almost always prevent or cure amblyopia and sometimes also produce good spatial vision.

3.1.1 Visual Acuity

Babies are able to perceive their environment through their eyes quite soon after birth - but only indistinctly. Among all human senses, the visual sense develops within shortest time, but visual acuity still has to be developed through constant exercise. Only a limited period in growth is available for this purpose. By the time school age is reached, the eyes’ learning program is virtually complete. The old adage that „what you don’t learn as a child, you’ll never learn as an adult“ applies to eyesight, too. In the ﬁrst weeks of life a child is still unable to exactly coordinate the movements of the two eyes. Brief misalignments during this time are no cause for concern. They may also occur occasionally again in the course of the coming months. The ability to gaze also has to be learned, but if one eye constantly deviates from the direction of the other, there is no time to loose. The ophthalmologist can diagnose the problem even in infancy and must initiate the treatment at the right time.

3.1.2 Symptoms

Children with conspicuous squint have the best prospects because they are taken to the ophthalmologist within short time by their parents on account of the „blemish“. Unfortunately, there exists a number of barely visible or invisible deviations. They are only detected when one eye is already amblyopic - such as during the eye test when starting school, when it is generally too late for an entirely successful treatment. For this reason alone 4% of all people suﬀer from serious one-sided visual deﬁciency. It is therefore very important to know and to heed all characteristics that might indicate an impending or existing squint: sensitivity to light, with tears, squeezing one eye shut, bad mood or irritability, chronic blepharitis, head held to one side and clumsy motion are alarm signals. Each sign is a valid reason in its own right to obtain an ophthalmologist’s opinion immediately. CHAPTER 3. STRABISMUS 55

3.1.3 Treatment

The primary prerequisite of treatment is to establish an optimum visual acuity for both eyes. Only then, a squint surgery can yield successful and enduring results. Eyes with total loss of ﬁxation (e.g. blindness) often re-establish squinting after surgical treatment and cannot hold long-lasting parallel alignment.

Eyeglasses Many squinting children in Europe are farsighted. Exceeding near ﬁxation can cause squinting with these children due to the convergence impulse. Eyeglasses, determined under full relaxation, using anesthetic, relaxing eye drops, can minimize or even heal a disorder.

Occlusion treatment Occlusion treatment, in which a light restraining plaster is applied over the squinting and normal eyes in a speciﬁc rhythm as instructed by the ophthalmologist, serves to prevent as well as combat amblyopia. The plaster covering on the normal eye is intended to have the eﬀect of exercising the squinting eye. Changing over the plaster alternately prevents weak vision in the normal eye caused by the occlusion. The main prerequisite for the success of amblyopia treatment is strict adherence to the treatment/exercise phases for the squinting eye and the normal eye that have been precisely determined by the ophthalmologist in every single case. If glasses, occlusion and eye-drops do not result in an improvement in visual acuity in older pre-school children and younger school children with amblyopia, a training program prescribed by the ophthalmologist can occasionally provide further help. The amblyopia check-ups and treatment must generally be continued over a period of years into the growth phase, in addition to glasses and even after a successful operation. The skin plaster can, after improval of vision, often be replaced by an occlusion using blurring spectacle sheets.

Strabismus surgery and subsequent treatment Half of the children with a squint need correction of the faulty alignment by means of an operation on the extraocular eye muscles. Sometimes, in a mechanical restriction of eye movements, operative positional correction is a prerequisite for all other measures. As a general rule, the operation is only carried out when the child wears glasses reliably, can see more or less equally well in both eyes and can be adequately examined (normally shortly before starting school). In order to improve preoperative diagnosis, the wearing of prisms, also for longer time, is useful to determine the true squint angle. The operation does not eliminate the weak vision, neither does it produce an immediate improvement in spatial vision. Both generally require further ophthalmic treatment. The operation does not eliminate the need for glasses, because they are the only means of correcting refractive errors. The type of misalignment and the result of the preliminary treatment determine whether a single operation is suﬃcient. Strabismus operations are carried out on the children under general anaesthetic by the ophthalmologist.

3.2 Binocular Vision

In vergence disconjugate movement, both eyes move synchronously and symmetrically in opposite directions, where convergence movements describe the ability to move both eyes inward (to the nose) and divergence deﬁnes the movement of both eyes outward (away from the nose). In CHAPTER 3. STRABISMUS 56 convergence, voluntary and in voluntary reﬂexes occur, in order to fuse the images of both eyes to a single stereoscopic picture.

Tonic convergence - describes the tonus of the extraocular muscles (especially the medial recti muscles) when awake.

Proximal convergence - is the convergence induced by the knowledge of the proximity of an object and occurs in optical instruments even though they are set for inﬁnity.

Fusional convergence - is a reflex that ensures the bilateral projection of corresponding areas onto the retinae of both eyes. This reflex occurs without a change in optical refraction and is stimulated by the perception of double images (diplopia). Thus, fusional convergence generates compensatory eye movements in order to overcome disparity of the retinal images. The amplitude of fusion denotes the maximum magnitude of eye movement that resides from fusional convergence. Fusional amplitudes may be corrected through the application of prisms and can be measured with a synoptophore. The usual fusional amplitude of convergence that is measured to the breaking point of diplopia for fixation of far targets is approximately 30 prism dioptres and for near fixation approximately more than 35 prism dioptres, whereas one dioptry corresponds to approximately 0.5 degrees. Generally, fusional convergence supports to control exophoria (latent divergent strabismus), whereas fusional divergence helps to compensate for esophoria (latent convergent strabismus).

Accommodative convergence - is convergence induced by accommodation. For each dioptre is associated with a near linear increasing relationship to the angle of accommodative convergence and results in the so called accommodative convergence/accomodation (AC/A) ratio. Normally 3 to 5 prism dioptres per dioptre of accommodation. Anomalies in the AC/A ratio most often indicate a cause for strabismus. A high AC/A ratio due to accommodation for the ﬁxation of near objects may induce excessive convergence and result in esotropia (inward squinting). Thus, a low AC/A ratio may lead to divergence that is causally related to exotropia (outward squinting) while a subject ﬁxates a near object.

Voluntary convergence - is convergence that can be produced at will.

Binocular vision develops within the ﬁrst years of life and is additionally accomplished by the ability to three-dimensionally perceive an image due to stereopsis. Three essential factors are required for the successful development of binocular vision and stereopsis.

• Clear and undisturbed refraction in both eyes,

• the ability of diﬀerent areas in the brain to fuse slightly diﬀerent images from both eyes, and

• the ability of exact coordination of eye movements in all possible gaze positions within the physiologic ﬁeld of gaze.

Thus, the ability of fusion strongly depends on the relationship between both retinae. CHAPTER 3. STRABISMUS 57

3.2.1 Projection

Projection is defined as the interpretation of the position of an object in space on the basis of corresponding areas that are stimulated on the retina (see Fig. 3.1). Following Fig. 3.1(a), an object (F’) and an additional element (T’) are stimulating the right fovea (F) and the right temporal area of the retina (T), respectively, the brain will perceive the red object (F’) as straight ahead, whereas the black object (T’) will be located in the nasal visual field of the right eye. This situation corresponds to the normal projection of images onto the retina of an eye. According to this definition, nasally located elements will project onto temporal areas on the retina, whereas elements that are located in the upper part of the physiologic field of gaze are projected onto lower areas onto the retina, and vice versa.

(a) Right Eye (b) Both Eyes

Figure 3.1: Projection of Objects in Space onto the Retina on an Eye, adapted from [Kan87]

When both eyes are kept open, the red object (F’) in Fig. 3.1(b) stimulates the retinae of both eyes (F), and the black object (T’) stimulates temporal areas of the retina in the right eye (T) as well as nasal areas of the retina in the left eye (N). Accordingly, the right eye projects the object into temporal areas of the visual ﬁeld, and the left eye projects into nasal areas of the visual ﬁeld. Since the stimulated retinal elements both represent the same object, both retinal points will project to the same location in space and no double images will occur.

According to the implications of retinal projection, a horopter (see Fig. 3.1(b)) represents a virtual spherical arc that captures all points in space that correspond to diplopia free binocular vision. Points that are ﬁxated beyond or before this horopter will be accompanied by double images (diplopia) and are the foundation for physiological double vision.

The line of sight or visual axis, corresponds to the line that is virtually drawn from the ﬁxation point in space to the retinal point on the fovea and additionally almost intersects the center of the pupil. From Fig. 3.1(b) it is easy to see that both red lines of sight intersect at the desired ﬁxation point (F’) in order project a single, sharp image onto the foveae. The retinal areas (F) in both eyes are denoted as corresponding retinal areas. CHAPTER 3. STRABISMUS 58

(a) (b) (c)

Figure 3.2: Diﬀerent Forms of Binocular Fixation, from [Kan87]

3.2.2 Diplopia

In strabismus, a dissociation of the lines of sight of both eye occurs which can be latent (phoria) or manifest (tropia). Concerning manifest forms of strabismus there can be two diﬀerent problems in the alignment of the visual axes, namely confusion and diplopia. In Fig. 3.2(b), a convergent manifest strabismus of the right eye is shown, so that the fovea of the right eye is stimulated by a black triangle and the red object is projected onto the left fovea. Confusion results in the overlapping of these diﬀerent images into the same vertical axis in space.

After that, diplopia results from the stimulation of an excentric retinal area in the fovea of one eye. In Fig. 3.2(c), the red object in space does not stimulate corresponding retinal areas, instead, a nasally shifted position on the retina is stimulated in the left eye, whereas the object is on the fovea of the right eye. In case of convergent strabismus, uncrossed double images occur (see Fig. 3.2(c)), while in divergent strabismus diplopia is perceived with crossed double images. Mainly in young children, a mechanism of alternating suppression leads to a temporary phenomenon in binocular vision, where the misaligned image is actively masked out by the brain. When the fixating eye is covered, the squinting eye takes up the fixation and suppression stops immediately. But in not alternating (monocular) strabismus, amblyopia can be seen as a result of continuous monocular suppression of the image of the affected eye.

3.3 Ocular Dissociation

The term orthophoria defines the ideal condition of ocular balance wherein the oculomotor system is in perfect equilibrium so that both eyes retain their normal positional relationship (i.e. both eyes remain directed upon a fixation point or remain parallel). In such a condition, the positions of both eyes are identical and a correct posture of the eyes is maintained without effort for all directions of gaze and all physiological distances of a fixation point. However, this ideal equilibrium is rare for distant and seldom existent, especially when fixating near objects. Usually, both eyes are maintained on the fixation point only under stress with the aid of corrective CHAPTER 3. STRABISMUS 59 fusion reflexes, originating from the brain. This can be classified as a tendency for deviation of the eyes from parallelism, which can be prevented by the mechanism of binocular fusion. This results in a typical deviation of both eyes, when they are dissociated (e.g. one eye is covered), and binocular vision is mutually prevented on purpose. This „common“ deviation is called heterophoria and better represents the practical reality compared to the idealized case of orthophoria. However, functional heterophoria is clinically referred to as a significant, latent deviation of the eyes, compared to the unrealized ideal of orthophoria or the physiologically normal deviations of heterophoria.

Additionally, the time-incidence of the deviation is denoted as follows:

Latent strabismus (Heterophoria) - is referred to as „hidden“ strabismus, which only occurs when binocular vision is prevented by dissociation of the eyes, usually achieved by covering and subsequent uncovering of one eye.

Manifest strabismus (constant strabismus, Heterotropia) - is deﬁned as deviation that occurs all the time, also without dissociation of the eyes.

Variations in the characteristics of deviation of movements of the eye are also distinguished using the following terms:

Concomitant strabismus - occurs when the deviation remains the same, or approximately the same, in all position of the eyes and additionally is unaltered no matter which eye is used for fixation of a target object. Thus, both eyes move together in coordination, and the visual axes, although abnormally directed, retain the same abnormal relationship to each other throughout the complete physiologic field of gaze. It is now well known that so-called concomitant deviations are in fact variable by nature (cf. [RSS01]), and that these variations can be measured. Therefore the term concomitant is too unspecific, if it has to signify that the angle of deviation is invariable. Presently, this definition is used if there is no limitation on duction movements, in spite of an oculomotor disorder, and that there is an accompanying sensorial disorder to a greater or lesser degree.

Incomitant strabismus - means that the deviation alters with changing the position of the eyes and varies depending on whether the non-pathological or pathological eye is used for ﬁxation. This usually originates in paretic or paralytic, sometimes spastic problems. In paralytic strabismus, when the eyes are turned away from a paralyzed muscle, which is thus not contracted, eye positions may be relatively normal. However, when the eyes move in the direction of action of the paralyzed muscle, movement becomes limited or even absent and deviation grows with the angle of excitation.

According to the direction, time-incidence and degree of deviation, heterophoria and heterotropia are described by using the following terminology [DE73]:

Esophoria and Esotropia (latent and manifest convergent strabismus) - is apparent, when the deviating eye turns inward, towards the nose (see Fig. 3.3). CHAPTER 3. STRABISMUS 60

Figure 3.3: Example for Inward Squinting

Exophoria and Exotropia (latent and manifest divergent strabismus) - designates a outward (away from the nose) deviating eye (see Fig. 3.4).

Figure 3.4: Example for Outward Squinting

Hyperphoria and Hypertropia (latent and manifest vertical strabismus) - occurs when the eye is deviating upward. Vertical upward deviating strabismus is also often termed strabismus sursumvergence (see Fig. 3.5).

Figure 3.5: Example for Upward Squinting

Hypophoria and Hypotropia (latent and manifest vertical strabismus) - occurs when the eye is deviating downward. Vertical downward deviating strabismus is also often termed strabismus deorsumvergence (see Fig. 3.6).

Figure 3.6: Example for Downward Squinting

In order to denote the aﬀected pathological eye within this terminology, the following terms are used commonly:

Monocular strabismus - can be denoted as right or left uniocular or monocular strabismus CHAPTER 3. STRABISMUS 61

that always affects one eye so that the other eye is preferred as the leading fixating eye. The affected eye will then follow the fixating eye and deviate according to the actual pathology. Binocular or alternating strabismus - sometimes affects one eye and sometimes the other, so that fixation can be assumed and maintained by each eye in turn, and when both eyes are open, each is able to hold fixation freely, without any obvious preference for fixing with either eye. Sometimes, however, it is legitimate to retain the term when there is a preference to take up fixation with one eye that is distinctly dominant, despite the absence of an equal level of vision and the ability of each eye to retain fixation of the non-dominating eye after covering the other eye.

It has to be noted that the mechanism controlling movements of the eyes is variable and struc- turally determined, but is non-obligatory, non-rigid and physiological in nature although the oculomotor control system conforms to definite laws within the limits of which it must remain (see Sec. 2.2.2). It is obvious, however, that a mechanism of such complexity and precise control must possess the weakness of vulnerability and it is not surprising that disruptions frequently occur from structural and functional causes, both in its peripheral and central parts. Abnormal- ities in ocular motility are therefore frequent and, indeed, constitute one of the most common of ocular disabilities. The essential factor determining the efficiency of the oculomotor control system is the early (in childhood) development of the ability of fixation and the fusional reflexes accomplished by the brain.

3.4 Clinical Assessment

In evaluation of ocular alignment a ﬁrst decision about the information that is required must be taken. Measurements can give information on the eye alignment during everyday binocular viewing, or the maximal deviation of the visual axes under conditions of disrupted binocular vision, or both of these. Subjective methods are useful for cooperative, communicative older patients, but objective methods must be used in younger patients or those less cooperative. Finally, some testing methods are useful only under research laboratory conditions.

Most laboratory tests are objective but depend on the measurement instrument that is used. The absolute position of the eye in space may be determined by measurement of the quantity of light reﬂected by the cornea from a deviated eye. The electro-oculogram (see Sec. 2.3.1.1) is generated by alterations of eye positions and electrodes capture the imbalance of electrical potentials. Insulated wire placed in a silicone rubber (eye coil) generates response to a magnetic ﬁeld (see Sec. 2.3.1.3) and can be used to exactly determine eye position.

Clinical investigations imply the analysis of atypical head positions that may indicate restrictive or paralytic strabismus, the presence of a null point in a patient who has nystagmus, or alphabet pattern strabismus. Usually, a patient places the head in a position that provides comfortable single binocular vision for the straight ahead view, but occasionally the head is placed to separate diplopic images maximally. The examiner must diﬀerentiate between head turns, tilts, and vertical head positions and attempt to quantify these.

In patients who have vertical strabismus, lid asymmetry is often found. If hypotropia is present and the non fixing eye is lower than the fixing eye, then the lid position is lower in this non fixing CHAPTER 3. STRABISMUS 62

Figure 3.7: Pseudo-Esotropia due to wide Bridge and Epicanthal Skin Fold, from [Kan87] eye. This is termed pseudoptosis, if the lid regains normal position when the previously hypotropic eye ﬁxates in primary position. Epicanthal skin folds that extend over the nasal sclera in a small child may simulate esotropia (see Fig. 3.7). Vertical displacement of an orbit may simulate vertical strabismus, and an abnormal increase in the interorbital distance may simulate exotropia. The examiner can declare a patient to be strabismic only after the appropriate alignment testing has been performed.

Objective clinical methods to determine and measure deviations of the visual axes include corneal light reﬂex tests, cover tests, and haploscopic tests. These tests do not require any response by the patient and thus are independent of the patient’s ability to interpret the testing environment.

3.4.1 Corneal Light Reﬂex Tests

Corneal light reflex tests, the oldest testing methods, are suitable for all patients. The angle kappa (i.e. the angle formed between two imaginary lines: the visual axis and the pupillary axis) has to be taken into account, and the fixation of one eye is required. The Hirschberg method relies on a pupil size of 4mm and assumes each millimeter of light displacement across the cornea is equivalent to approximately 7 ◦ of deviance. A light reflection at the pupillary border signifies a 15 ◦ deviation (see Fig. 3.8), at the mid-iris a deviation of 30 ◦ and at the limbus a deviation of 45 ◦. The patient shown in Fig. 3.8 has a left esotropia, thus, the corneal light reflex of the left eye is displaced temporally with respect to the pupillary border of the left eye, while the reflex is centered in the pupil of the right eye. CHAPTER 3. STRABISMUS 63

Figure 3.8: Hirschberg Light Reﬂex Test Method, from [Kan87]

The test should be performed with the light centered in each eye’s pupil to detect the presence of secondary deviations. The disadvantages of this test are the estimations necessary to measure the eye deviation and the inability to control accommodation when testing at near fixation, as the measuring light serves as the fixation target. Distance testing is difficult because the dimness of the target light is reflected in the corneas.

The Krimsky test quantifies the light reflex displacement using appropriately held prisms. The original description suggested to place the prism before the aligned, fixating eye (see Fig. 3.9), but most users today find it easier to hold the prism before the deviating eye. The strength of a base-out prism over the fixing right eye to center the pupillary light reflex in the esotropic left eye is defined as the amount of left esotropia (see Fig. 3.9).

Figure 3.9: Krimsky Light Reﬂex Test Method, from [Kan87]

Prisms must be appropriately handled to yield accurate measurement of strabismus. They deflect light toward their base, but the patient views the light as deflected toward the prism apex. The prism diopter is defined as the strength of prism necessary to deflect a light beam 1cm at 1m distance. Glass prisms are calibrated when positioned with the back surface perpendicular to the visual axes. Plastic prisms, whether loose or in bar form must be held with the rear surface in the frontal plane, to approximate closely the position of minimal deviation of light through the prism. Prisms cannot be stacked base to base as the sum prism strength is much greater than the sum of each individual prism strength, but they may be stacked with bases 90 ◦ apart. CHAPTER 3. STRABISMUS 64

Large deviations are best neutralized when the prisms are divided between the two eyes. For measurements when patients view in eccentric gaze positions and for those in the head tilt test, care is required to ensure the prisms are held in the frontal plane.

3.4.2 Cover Tests

These objective tests detect and measure horizontal and vertical strabismus, but they cannot measure torsional deviations and detect only some and not all torsional deviations. All cover tests demand the ability of each eye to look at a ﬁxation target at near and distance, and to move to take up ﬁxation upon that target.

Figure 3.10: Prism-Cover Test, from [Kan87]

The monocular cover test detects constant visual axis deviations. The examiner observes the uncovered eye for movement as its fellow eye is covered with a paddle, the hand or the thumb. A nasal movement implies exotropia, temporal movement esotropia, upward movement hypotropia, and downward movement hypertropia of the uncovered eye (see Fig. 3.11). Each eye is covered in turn. An accommodation-controlling ﬁxation target is presented to the patient, who ideally describes the target. Small toys are suitable for young children, but bright white lights are too be avoided as the patient cannot accommodate on the contours of a light. Tropias established by the cover test may be measured using the simultaneous prism and cover test. A prism of appropriate strength held in the appropriate direction is introduced before one eye as its fellow is covered (see Fig. 3.10). Prism strength is increased until eye movement ceases and the prism strength corresponds to the size of the strabismus. The test is then repeated with the prism before the other eye.

The uncover test requires observation of the covered eye as the cover is removed. If that eye deviated under cover it may regain ﬁxation or may remain deviated. The former implies the CHAPTER 3. STRABISMUS 65 presence of a phoria, a latent deviation held in check by sensory fusion, or an intermittent tropia; the latter implies a tropia with ﬁxation preference for the fellow eye.

Figure 3.11: Cover Tests for Tropias and Phorias, from [DE93]. (a) For exotropia, covering the right eye drives inward movement of the left eye to take up fixation; uncovering the right eye shows recovery of fixation by the right eye and leftward movement of both eyes; covering the left eye discloses no shift of the preferred right eye. (b) For esotropia, covering the right eye drives outward movement of the left eye to take up fixation; uncovering the right eye shows recovery of fixation by the right eye and rightward movement of both eyes; covering the left eye discloses no shift of the preferred right eye. (c) For hypertropia, covering the right eye drives downward movement of the left eye to take up fixation; uncovering the right eye shows recovery of fixation by the right eye and upward movement of both eyes; covering the left eye shows no shift of the preferred right eye. (d) For exophoria, the left eye deviates outward behind a cover and returns to primary position when the cover is removed. An immediate inward movement denotes a phoria, a delayed inward movement denotes an intermittent exotropia.

Phorias may be detected more directly using the alternate cover test, in which each eye is occluded alternately to dissociate the visual axes maximally. Care must be taken to permit time for each eye to reside behind the cover (the cover must not be „fanned“ before the eyes). Appropriately held prisms enable quantification of the phoria (see Fig. 3.10). Some patients have poorly defined end points and a range over which eye movements shift from one direction to the opposite as prism strength is increased, thus, the strabismus measurement may be estimated as the midpoint between clearly defined movements in each direction.

If the cause of strabismus is paralytic or restrictive, patients may have greater cover test measurements when the paretic or restricted eye fixes in a given gaze position (secondary deviation) than when the unaffected eye fixates (primary deviation). This phenomenon arises from Hering’s Law (see Sec. 2.2.3.7), which demands equal innervation to yoke muscles, thus, the yoke of a paralyzed or restricted muscle receives excess innervation when the pathologic eye is fixing.

Strabismus should be detected and measured in primary position at distance and near ﬁxation, and in gaze up, down, right, and left 30 ◦ from primary position. These nine diagnostic gaze positions include the above plus up and right, up and left, down and right, and down and left and are useful to measure cyclovertical muscle palsies. For patients who have oblique muscle dysfunction, measurements are taken with the head tilted right and left at distance ﬁxation.

3.4.3 Subjective Clinical Tests

Subjective clinical methods include diplopia tests and haploscopic tests, which require cooperation, intelligence, and the ability of the patient to communicate the sensory percept to the CHAPTER 3. STRABISMUS 66 examiner.

Figure 3.12: Maddox-Wing Test, from [YD03]

The Maddox-Rod test dissociates both eyes for near and distance ﬁxation and measures heterophoria (see Fig. 3.12). A red Maddox rod in a trial frame may be used to evaluate subjective ocular torsion. The grooves must be aligned with the mark on the rim, as they tend to rotate within. The red glass requires the patient to alert the examiner when the red light viewed behind a red ﬁlter before the right eye and a white light viewed with the left eye are superimposed or displaced one from the other. Fusion is disrupted by the red glass and thus horizontal and vertical phorias are uncovered and measured. The gaze position of maximal image separation is a clue to the identity of paretic or restricted muscles. This is a useful bedside test, but accommodation is not controlled.

The Maddox rod consists of closely aligned, powerful glass or plastic cylinders. When illuminated, these cylinders project a line upon the patient’s retina perpendicular to the groove orientation. The line is aligned horizontally to detect and measure horizontal phorias (accommodation cannot be controlled with this test). Torsion may be detected and quantiﬁed using the Maddox rod which is placed in a trial frame scaled in degrees. It is common to place two Maddox rods of diﬀering color in each trial frame cell and permit the patient to rotate each to his or her perception of the horizontal. The torsional position of each eye may be read directly in degrees from the angular scale used for cylinder axes.

3.4.4 Hess-Lancaster Test

The Hess-Lancaster test is a test for binocular alignment using separated images for both eyes. Generally, there are many similar testing methods (e.g. Less-Screen, Helmholtz-Test, etc.) that use the same testing principle as the Hess-Lancaster test. The also exists the so called Hess test, which uses a diﬀerent coordinate system for measurement, but the basic interpretation of the results is identical.

During the clinical Hess-Lancaster test the following steps are carried out:

1. The patient wears red-green-glasses with the red ﬁlter in front of the e.g. right eye (ﬁxing eye) initially (see Fig. 3.13).

2. The patient gets a green light pointer, the examiner gets a red light pointer. CHAPTER 3. STRABISMUS 67

3. The examiner projects a red light spot onto the so-called Hess screen and asks the patient to bring the green light spot (following eye) over the red light spot. Under normal conditions both light spots overlay in all nine main gaze directions (see Fig. 3.13(a)).

4. Now the red ﬁlter is put in front of the other eye and the previous steps are repeated with the other eye, which is now the ﬁxing eye.

If the patient fixates with the „normal“ unaffected left eye, the fixation can be reached with normal innervation. However, if the right medial rectus muscle is palsied, the patient’s green light pointer will point into a direction, which does not correspond to the real location of the fixation point in space (see Fig. 3.13(c)). After the test is finished, the relative positions are connected with straight lines.

(a) (b) (c)

Figure 3.13: Binocular Fixation in the Hess-Lancaster Test, from [Kan87]

If e.g. the patient suffers from a palsy of the lateral rectus muscle on the right eye and also fixates with the right eye (red filter), the „normal“ left medial rectus muscle receives an excessive innervation (Hering’s law). As a result, the patient’s green light pointer will point to a position, which lies far beyond the correct one (see Fig. 3.13(b)). The results of the Hess-Lancaster test are in general two diagrams (left eye and right eye fixing) with the corresponding gaze positions, which in turn show the deviation and the squint angle (see Fig. 3.14).

In the Hess-Lancaster diagram from Fig. 3.14, the blue points represent those gaze positions, which the patient should ﬁxate (intended gaze positions) and the red points represent those gaze positions, which the patient was able to reach with the following eye. The diﬀerence between the blue and the red points shows the respective deviation of binocular coordination. At the same time next to each red point (following gaze position) the torsion of the following eye is shown (as text) in order to get the torsional position of the following eye. The following list gives an overview of the interpretation of Hess or Hess-Lancaster diagrams for a right medial rectus muscle palsy: CHAPTER 3. STRABISMUS 68

Figure 3.14: Hess-Lancaster Diagram for Right Eye (Left Eye Fixing)

1. The two diagrams from Fig. 3.15 (left eye and right eye ﬁxing) are compared.

2. The smaller diagram shows the eye with the palsied muscle.

3. The larger diagram corresponds to the (a) eye with the overacting muscle.

4. The smaller diagram shows the biggest restriction in the main functional direction of the palsied muscle (in Fig. 3.15(b), the right medial rectus muscle).

5. The larger diagram shows the biggest expansion in the main functional direction of the synergistic muscle (in Fig. 3.15(a), (b) the left lateral rectus muscle). Figure 3.15: Interpretation of Hess-Diagram according to Muscle Actions, from [Kan87] CHAPTER 3. STRABISMUS 69

Changes in the Hess diagrams are a prognostic help. A palsy of the right superior rectus muscle for example will show a restriction of the aﬀected muscle and an overfunction of the synergist (left inferior oblique muscle) (see Fig. 3.16(a)). As a result of this signiﬁcant incomitant reaction shown in both diagrams, the diagnosis can be made directly out of the diagrams. However, when the palsied muscle has recovered, both diagrams show approximately normal values again. If the palsy is not corrected, the shapes of both diagrams change and a secondary contractor of the ipsilateral antagonist (right inferior rectus muscle) develops, which can be seen as an overfunction in the diagram. This in turn can lead to a secondary (inhibition) palsy of the left superior oblique muscle, which is represented in the diagram through reduced activity (see Fig. 3.16(b)) and which gives the false impression that the left superior oblique muscle is the real cause for the pathological Figure 3.16: Hess-Diagram situation. By-and-by the two diagrams get even more concomitant Interpretation, from up to the point were it is impossible to determine which muscle [Kan87] was the primary palsied muscle (see Fig. 3.16(c)).

3.5 Eye Motility Disorders

Pathological anomalies in ocular motility are mainly differentiated in two fundamental types. The reasons for motility disorders can therefore be a failure in the development of the fixation reflexes or a disruption of fixation reflexes from structural or functional causes. Each of these categoric disorders will produce different symptoms and therefore demand also different therapy.

Anomalies in the conjugate fixation reflex make it impossible to fixate objects, since both eyes cannot move in binocular community. This will result in heterophoria (latent strabismus) when the fixation can be established with stress, and in concomitant, manifest strabismus when fixation is not possible at all. Similarly, the near fixation reflex may be disrupted, resulting in an anomaly in convergence. Some structural or neuro-muscular lesion may prohibit the development of adequate movements, already from birth on, so that congenital incomitant strabismus is developed. On the other hand, if a lesion occurs in the peripheral (infranuclear) neuro-muscular mechanism after the reflexes have been established, an acquired incomitant strabismus, resolvable in terms of individual eye movements results due to some pathological accident. A lesion in the central (supranuclear) mechanism results in binocular deviation, resolvable in terms of associated binocular eye movements.

Thus, a supranuclear lesion indicates disorders of movement caused by the destruction or functional impairment of brain structures above the level of the motor neurons, such as the motor cortex (see Sec. 2.2.3). Furthermore, infranuclear lesions refer to disruptions below a nucleus of a nerve (e.g. eye muscle palsies), whereas intranuclear lesions aﬀect connections between other oculomotor nuclei (palsies that aﬀect interneuronal connections, e.g. the medial longitudinal fasciculus, see Sec. 2.2.3.5).

The fundamental fact about the development of pathological dissociation is, that if the normal CHAPTER 3. STRABISMUS 70

fixation reflexes are not attained and to some extent fixed by usage in the early months and years of life and certainly before the age of five years, while the nervous system is still flexible enough for adaptations, these reflexes will never develop.

3.5.1 Concomitant Strabismus

Concomitant strabismus can be classified as congenital, sometimes also acquired, functional eye motility disorder, caused by common predisposition that lead to a pathological development of eye movement. Functional eye motility disorders are influenced by many different factors, whereby three groups of factors can be distinguished and are particularly important [DE73]:

Static anatomical conditions - that produce a faulty position of the eyes relative to each other, excessive or abnormal innervations - particularly important in a disharmony of the relationship between accommodation and convergence, and deﬁcient development of the ability to fuse - due to a unilateral sensory failure or a lack of central organization.

It is important to notice that it is extremely rare that strabismus is caused by one of these factors alone. However, generally, all these factors depend on some obstruction to the development of the binocular ﬁxation reﬂexes which coordinate eye movements.

Generally, the following reasons can be responsible for the development of concomitant strabismus:

Hyperopia (long sightedness) - is a very common cause for Esophoria, since a young child must accommodate in order to get a sharp image, when ﬁxating far objects. Unfortu- nately, accommodation also implies convergence of both eyes and can subsequently lead to strabismus, when not cured within certain time.

Fusional dysfunction - that can be congenital or acquired (e.g. caused by scarlet fever or measles). Additionally, concussion or accidents can cause acquired concomitant strabismus.

Unbalanced refraction in both eyes - can hinder fusion due to diﬀerently sized retinal images that occur when ﬁxating objects.

Developmental disorders of eye muscles - can additionally contribute to mechanically caused concomitant strabismus.

In the great majority of the cases, good fusional reﬂexes can compensate for a mechanical misalignment, and the result is latent strabismus. The common occurrence of divergent strabismus, wherein, except in most young children, the eyes revert to their divergent resting position after the development of impaired vision or blindness in one eye. Generally, children tend to convergent resting positions, whereas adults tend to divergent resting positions of the eyes. This illustrates the tendency of anatomical conditions to make their inﬂuence felt in the absence of fusion. CHAPTER 3. STRABISMUS 71

Innervational, particularly accommodational influences similarly tend to the development of concomitant strabismus, but permanent manifest strabismus does not occur if the binocular fixation reflexes have not yet been developed and the fusional vergence is good. If the fusional ability is too weak, concomitant strabismus develops in childhood, before fusion has become stabilized.

A deﬁciency of fusion can also be responsible for the development of concomitant strabismus, especially when complex neural control mechanisms are labile or unstable. Exhaustion or excitability and an unstable neuropathic constitution (i.e. a sudden shock or a psychological trauma) leads to the disruption of the visual system, and strabismus occurs.

Heredity is most frequently an evidence for concomitant strabismus, as relatives of a family in the same generation show a higher incidence. However, concomitant strabismus does not take the same form throughout a family, instead it is varying considerably in being uniocular or alternating. Concerning the incidence and type of amblyopia, variations have even been seen in monozygotic twins.

3.5.2 Incomitant Strabismus

Incomitant Strabismus can be seen as a dissociation of the ocular movements, wherein the deviation is irregular, varying in an uncoordinated manner, in different directions of gaze. This type of strabismus results of defects in the final motor path of the binocular reflexes. The essential difference between concomitant and incomitant strabismus is, that in concomitant strabismus, the central organization of binocular vision functions inadequately or not at all, whereas in incomitant strabismus, the brain is usually able to command movements, but the motor apparatus is restricted in carrying out these instructions with adequacy. However, this distinction is variable, since in many forms of incomitant strabismus in early life, disruptions of the sensory stimuli are established, similar to those typically occurring in concomitant strabismus. Conversely, certain forms of concomitant strabismus (e.g. accommodative strabismus or hyperopia) do not show any disruptions of the sensory stimuli, despite the failure to maintain binocular single vision on near fixation. In incomitant strabismus, lesions occur in the lower motor neuron level (the nuclei, nerves or muscles), and consequently deviations are not resolvable in terms of eye movements, but in terms of individual muscles. Herein also lies the distinction from conjugate deviations due to lesions above the motor neuron level, affecting the supranuclear mechanism which concerns itself with the control of movements and does not take the contraction of individual muscles into account. Concomitant strabismus, therefore, gives relatively constant deviations in all directions of gaze, whereas incomitant strabismus may become evident only when the eye is turned into the field of action of the affected muscle where deviation always grows proportional to the gaze position.

Incomitant strabismus may be either paretic or spastic in nature, but most cases show clinically a simple under- or over-action of a muscle, although this is mostly associated with changes in activity of other muscles, so that the resulting deviation may be very complicated and may eventually rapidly assume entirely diﬀerent characteristics.

One way to simplify diagnostics is to divide the symptoms into four distinctive categories [DE73]:

Overaction of the ipsilateral antagonist which, if maintained for longer time, may result in CHAPTER 3. STRABISMUS 72

permanent contractual changes.

Overaction of the contralateral synergist - which is due to the demand of equal innervation (Hering’s law, see Sec. 2.2.3.7) which requires an equal distribution of innervational impulses between both eyes. The overaction of the contralateral synergist (yoke muscle) becomes apparent when an attempt is made to move the paretic muscle in the other eye, and may also result in contractual changes.

Underaction of the antagonist of the contralateral synergist - becomes noticeable through Sherrington’s law (see Sec. 2.2.3.8) of reciprocal innervation. This is due to an compensatory overaction of the contralateral synergist and results in an inhibitory underaction of its antagonist, predetermined by the law of reciprocal innervation.

Overaction of the ipsilateral synergist(s) - describes a compensatory reaction that is carried out in an attempt to accomplish the deﬁcient movement. This type of overaction is seldom noticeable, since ipsilateral synergists always have very limited compensatory actions with respect to an aﬀected muscle.

Figure 3.17: Example of Disturbance of the Binocular Muscular Team

In Fig. 3.17, one example of a disturbance of the binocular muscular team is illustrated. A palsy of the right lateral rectus muscle results in an overaction of the right medial rectus (the ipsilateral antagonist). Additionally, this pathological situation produces a compensatory overaction of the left medial rectus muscle (the contralateral synergist) and an inhibitory underaction of the left lateral rectus muscle (the ipsilateral antagonist of the contralateral synergist).

If the paretic eye is habitually used for fixation, the secondary deviation (deviation when fixing with a pathologic eye) produced by the contralateral synergist becomes accustomed. Conversely, if the non-paretic eye is constantly used for fixation, the primary deviation (deviation when fixing CHAPTER 3. STRABISMUS 73 with a normal, healthy eye) will function against the recovery of the affected, palsied muscle of the paretic eye, by facilitating the overaction of the ipsilateral antagonist.

It is also possible that a pathological situation (e.g. described in Fig. 3.17) disappears so that the originally paretic strabismus changes its characteristics and subsequently shows a purely spastic deviation. However, if the muscle palsy disappears, it leaves permanent changes of the contracture or stretching in the various muscles involved. Thus, the original incomitant strabismus may become virtually concomitant.

3.5.2.1 Paralytic Strabismus

Paralytic strabismus means, that one of the muscles attached to the globe is paralyzed and the eye affected may turn in, out, up or down depending on the muscle involved i.e. the eye movement is restricted in the direction of the action of the paralyzed muscle. The reasons for paralytic squint can originate from a breakdown of motor control before or after the binocular fixation reflexes have been developed. It may also be caused by certain nerve palsies, which in turn may be caused by peripheral diseases of the cranial nerves (e.g. meningitis, encephalitis etc.). Clinical progress and treatment of paretic strabismus therefore differs considerably with respect to its different causes. The following objective signs of paralytic strabismus can be mentioned:

• Abnormal deviation of the eyes,

• deviation of movements, and

• the adoption of compensating postural attitudes (abnormal head posture).

A common symptom for paralytic strabismus is binocular diplopia (double images) that disappear when either eye is closed. This is caused by a misalignment of the eyes, which can be secondary to nerve or muscle related disorders. Diplopia can also cause a reduction of reading, driving, and vocational skills.

Figure 3.18: Example of Abnormal Head Posture compensating Esotropia, from [Kau95] CHAPTER 3. STRABISMUS 74

Especially abnormal head postures are a noticeable sign for paralytic strabismus. Patients with early manifest strabismus show abnormal head posture in that the head is tilted to the shoulder so that ﬁxation in excessive adduction of the leading eye can be accomplished at the same time. The tilt of the head occurs mostly in the direction where the leading eye is situated (see Fig. 3.18). The head is turned towards the ﬁeld of action of the paralyzed muscle, whereas in paralysis of any of the recti muscles the chin is elevated in paralysis of the elevators of the eye - superior recti and inferior obliques. It is depressed in paralysis of the depressors of the eye - inferior recti and superior oblique. The head is tilted towards the normal side in paralysis of the superior oblique. It is tilted towards the side of the paralyzed muscle in paralysis of the superior and inferior recti, and the inferior oblique muscle.

In the following, some examples of individual muscle palsies are given along with the respective clinical signs. In contrast to neurogenic palsies that occur due to lesions in the neural oculomotor pathway, muscle palsies are restricted to limitations of the function of speciﬁc muscles.

Palsy of the Right Superior Rectus Muscle

In primary position, the right superior rectus muscle is mainly an elevator so that the primary deviation of the affected eye is downwards (see Fig. 3.19(e)), due to an overaction of the right inferior rectus muscle (the ipsilateral antagonist). The secondary deviation of the unaffected eye during fixation with the affected eye is upwards (see Fig. 3.19(f)), due to overaction of the left inferior oblique and the left superior rectus (the contralateral synergists). Usually, the secondary deviation is greater than the primary, which complies to the definition of incomitant deviation. However, if contraction develops in the ipsilateral antagonist (the right inferior rectus muscle), the primary deviation increases to become approximately equal to the secondary deviation, assuming concomitant features. Since the right superior rectus muscle is, in its secondary and tertiary functions also adductor and intorter, the loss of the adducting influence is of little significance as long as the right medial rectus is intact, but there is noticeable extorsion due to an overaction of the right inferior rectus and the right inferior oblique muscles.

(a) (b) (c)

(e) Primary Deviation

(d) (g)

(f) Secondary Deviation

(h) (i) (j)

Figure 3.19: Example of a Right Superior Rectus Palsy CHAPTER 3. STRABISMUS 75

(a) Primary Deviation, Left Eye Fixing (b) Secondary Deviation, Right Eye Fix- ing

Figure 3.20: Hess-Lancaster Chart for Right Superior Rectus Palsy

Ocular movements show limitations of the palsied eye in an upward and outward direction (see Fig. 3.19(a)) so that there is an increase of deviation when looking up and to the right using the unaffected eye. Using the right eye to fixate, the secondary deviation results in an upshoot of the unaffected left eye, due to an overaction the left inferior oblique (the contralateral synergist) which is also apparent in right lateral gaze (see Fig. 3.19(d)). A downshoot of the affected right eye occurs due to an overaction of the right inferior rectus muscle (the ipsilateral antagonist), or otherwise, a defective downward movement of the unaffected left eye, when fixating with the right eye, due to a weakness of the left superior oblique (the antagonist of the contralateral synergist) which is evident particularly on looking down and to the right (see Fig. 3.19(h)). The deviation is also evident on upward movement from primary position (see Fig. 3.19(b)), but there is seldom any defect on downward movement (see Fig. 3.19(i)). The vertical ocular movements are relatively normal in abduction since the right inferior oblique compensates for the weak right superior rectus muscle in these positions (see Fig. 3.19(c), Fig. 3.19(g)), although there may be a slight overaction of the ipsilateral synergist when looking to the left and down (see Fig. 3.19(j)). There is an increase in extorsion in the affected eye during right gaze, due to overaction of the right inferior oblique, also in left gaze, due to an overaction of the right inferior rectus muscle.

The compensatory (abnormal) head posture in a right superior rectus palsy is usually a rotation of the head into the field of action of the affected muscle. In this case, the head would be turned up and to the right, and also tilted to the side of the affected eye.

The Hess-charts in Fig. 3.20 show a shrinkage, away from the direction of action of the right superior rectus muscle with enlargement towards the direction of the right inferior rectus and the left inferior oblique accompanied by a slight shrinkage away from the action of the left superior oblique. When fixating with the left, unaffected eye, the resulting primary deviation in Fig. 3.20(a) shows the overaction of the ipsilateral antagonist in the right eye through a CHAPTER 3. STRABISMUS 76 downward movement of the diagram towards the direction of action of the right inferior rectus muscle. When fixating with the affected, right eye, the secondary deviation in Fig. 3.20(b) shows the overaction of the left superior rectus muscle, which results in a movement of the diagram towards the direction of action of the left superior rectus muscle.

Palsy of the Right Superior Oblique Muscle

In primary position, the right superior oblique muscle is primarily a depressor, so that the primary deviation of the affected right eye in this particular situation is an upward movement, due to an overaction of the right inferior oblique muscle (see Fig. 3.21(e)). The secondary deviation of the unaffected eye during fixation with the affected right eye is downward movement due to overaction of the left inferior rectus muscle (see Fig. 3.21(f)).

(a) (b) (c)

(e) Primary Deviation

(d) (g)

(f) Secondary Deviation

(h) (i) (j)

Figure 3.21: Example of a Right Superior Oblique Palsy

Since the right superior oblique muscle is also an abductor and intorter, the loss of the abducting inﬂuence is of little signiﬁcance, as long as there is an intact right lateral rectus muscle, but there is some extorsion due to overactions of the right inferior oblique and the right inferior rectus muscles, which can easily be seen in Fig. 3.21(f) and Fig. 3.21(e) when looking at the dark (blue) cross in the pupil.

Ocular movements show limitations of the palsied eye when looking downwards and inwards, since this is the direction of action of the affected muscle (see Fig. 3.21(j)). This situation is ac- centuated by an upshoot of the affected right eye, due to an overaction of the right inferior oblique muscle, which is evident in all positions of left gaze (Fig. 3.21(c), Fig. 3.21(g) and Fig. 3.21(j)). Particularly, when looking up and to the left, a downshoot of the unaffected eye, due to overaction of the left inferior rectus muscle can be noticed. Restricted downward movement of the affected right eye is also evident on downward movement (see Fig. 3.21(i)), and to some extent in upward movement from primary position (see Fig. 3.21(b)). The ocular movements are rather normal on looking to the palsied side (see Fig. 3.21(a), Fig. 3.21(d) and Fig. 3.21(h)).

The compensatory (abnormal) head posture is usually a rotation of the face to the left and CHAPTER 3. STRABISMUS 77

(a) Primary Deviation, Left Eye Fixing (b) Secondary Deviation, Right Eye Fix- ing

Figure 3.22: Hess-Lancaster Chart for Right Superior Oblique Palsy downwards, accompanied by a tilting of the head to the left. Both, the turning of the face, as well as the tilting of the head are oriented in the direction of the unaﬀected eye.

The Hess-Lancaster charts shown in Fig. 3.22 show a shrinkage away from the direction of action of the right superior oblique muscle with a slight enlargement towards the direction of action of the right inferior oblique (see Fig. 3.22(a)), and an enlargement in the direction of action of the left inferior rectus muscle with a shrinkage away from the direction of action of the left superior rectus muscle (see Fig. 3.22(b)).

3.5.2.2 Duane’s Syndrome

Duane’s syndrome is a congenital ocular motility disorder characterized by limited abduction and/or limited adduction. The palpebral fissure narrows (the globe retracts) on attempted adduction. Upward or downward deviation may occur with attempted adduction due to a leash effect. Often associated with this condition is a „tether“ phenomenon consisting of overelevation, overdepression, or both, in adduction as the retracted globe escapes from its horizontal rectus restrictions. It is a condition of aberrant innervation that results in co-contraction of the medial and lateral recti in the affected eye. Thus, Duane’s syndrome can be considered to be a congenital miswiring of the medial and the lateral rectus muscles such that globe retraction occurs on adduction.

Duane’s syndrome is often clinically subdivided into three types (1-3). Diﬀerent clinical types may be present within the same family suggesting that the same genetic defect may produce a range of clinical presentations.

Duane’s syndrome type 1: The ability to move the aﬀected eye outward towards the ear (abduction) is limited, but the ability to move the aﬀected eye inward towards the nose CHAPTER 3. STRABISMUS 78

(a) Duane’s Retraction Syndrome Type 3 (b) Hess-Lancaster Chart for Duane’s Retraction Syn- drome

Figure 3.23: Example for Duane’s Retraction Syndrome Type 3 of a Right Eye

(adduction) is normal or nearly so. The eye opening (palpebral ﬁssure) narrows and the eyeball retracts into the orbit when looking inward towards the nose (adduction). When looking outward towards the ear (abduction) the reverse occurs.

Duane’s syndrome type 2: The ability to move the affected eye inward towards the nose (adduction) is limited, whereas the ability to move the eye outward (abduction) is normal or only slightly limited. The eye opening (palpebral fissure) narrows and the eyeball retracts into the globe when the affected eye attempts to look inward towards the nose (adduction).

Duane’s syndrome type 3: The ability to move the affected eye both inward towards the nose (adduction) and outward towards the ear (abduction) is limited. The eye opening (palpebral fissure) narrows and the eyeball retracts when the affected eye attempts to look inward towards the nose (adduction).

Each of these three types can be further classified into three subgroups designated A, B, and C to describe the eyes when looking straight (in primary position). In subgroup A the affected eye is turned inward towards the nose (esotropia). In subgroup B the affected eye is turned outward towards the ear (exotropia), and in subgroup C the eyes are in a straight primary position.

In Fig. 3.23, Duane’s Retraction Syndrome Type 3 for a right eye is illustrated. Due to a co- contraction of the right medial and right lateral rectus muscles, the aﬀected right eye will show retraction as indicated in Fig. 3.23(a) when looking to the right. The Hess-Lancaster chart for the right eye in Fig. 3.23(b) shows the typical restriction in ab/adduction of the right eye, due to co-innervation of both recti muscles. This results in a shrinkage of the diagram for both sides, away from the ﬁelds of action of the right lateral rectus and the right medial rectus muscles. CHAPTER 3. STRABISMUS 79

Brown’s Syndrome

This ocular motility disorder, characterized by an inability to elevate the adducted eye actively or passively, was ﬁrst described by Brown. It has since become recognized that there is a variety of causes, that the condition may be congenital or acquired, and that the defect can be permanent, transient, or intermittent.

Brown’s syndrome is characterized by a deficiency of elevation in the adducting position. Im- proved elevation is usually apparent in the midline, with normal or near-normal elevation in abduction (see Fig. 3.24). There is occasional widening of the palpebral fissure on attempted elevation in adduction. With lateral gaze in the opposite direction, the affected eye may depress in adduction, although no overdepression simulating overaction of the superior oblique muscle occurs on duction testing. Exodeviation (V pattern) often occurs as the eyes are moved upward in the midline. Many patients are orthophoric and experience diplopia in the primary position, although with time hypotropia may develop with a compensatory head posture turn towards the opposite eye. In some cases, there is discomfort on attempted elevation in adduction, the patient may feel or even hear a click under the same circumstances, and there may be a palpable mass or tenderness in the trochlear region. A positive forced duction test is the hallmark of Brown’s syndrome.

(a) Primary Deviation, Left Eye ﬁxing (b) Secondary Deviation, Right Eye Fix- ing

Figure 3.24: Example for Brown’s Syndrome of a Right Eye

From the Hess-Lancaster charts of Brown’s syndrome in Fig. 3.24, the characteristic V-Pattern of the primary and secondary deviations can be recognized. The Hess chart for the primary deviations (fixing with the unaffected eye) in Fig. 3.24(a) shows a deficiency in elevation when the right eye is in adduction and near normal elevation in abduction. When fixing with the affected eye (see Fig. 3.24(b)), improved elevation of the unaffected eye is evident when the affected right eye looks in adduction.

The anatomical cause of the syndrome is a tight superior oblique tendon. Acquired Brown CHAPTER 3. STRABISMUS 80 syndrome has been attributed to a variety of causes, including superior oblique surgery, scleral buckling bands, trauma, focal metastasis to the superior oblique, and following sinus surgery and inflammation in the trochlear region. An identical motility pattern, as seen in Brown’s syndrome, can be acquired by patients with juvenile or adult rheumatoid arthritis. It appears that this form of Brown’s syndrome shares similar characteristics to inflammatory disorders that affect the tendons of the fingers.

3.5.2.3 Fibrosis Syndrome

This syndrome is characterized by replacement of normal muscle tissue by fibrous tissue in varying degrees. The various clinical presentations depend on the number of muscles affected, the degree of fibrosis, and whether the involvement is unilateral or bilateral. The condition is congenital, with males and females equally affected.

Fibrosis Syndrome is characterized encompassing

• ﬁbrosis of the extraocular muscles,

• ﬁbrosis of Tenon’s capsule,

• adhesions between muscles, Tenon’s capsule, and globe,

• inelasticity and fragility of the conjunctiva,

• absence of elevation or depression of the eyes,

• little or no horizontal movement,

• eyes ﬁxed 20 to 30 degrees below the horizontal,

• chin elevation and

• the condition being present at birth.

Congenital ﬁbrosis of the inferior rectus is probably a variant of the general ﬁbrosis syndrome. The inferior rectus alone or together with a levator may be involved, with little or no involvement of the other extraocular muscles. The condition may be unilateral or bilateral and is commonly asymmetric. Because patients cannot typically elevate their eyes even to the midline, they adapt a compensating head posture with their chin up to maintain binocular vision.

Another variant of the general fibrosis syndrome is strabismus fixus, in which the eyes are in a markedly fixed position of esotropia or exotropia. The eyes are so firmly fixed that they cannot be actively or passively moved in a horizontal direction, although vertical movement is usually possible.

Another possible variation of the general ﬁbrosis syndrome is the vertical retraction syndrome. In this condition, horizontal movements are normal, but elevation and depression are reduced, while the eye is abducted. In addition to the vertical limitation, there is retraction of each eye during attempted depression, with the eye in the abducted position. CHAPTER 3. STRABISMUS 81

Congenital fibrosis of the extraocular muscles (CFEOM) describes a group of rare congenital (present at birth) eye movement disorders that result from the dysfunction of all or part of the oculomotor nerve (cranial nerve III) and/or the muscles this cranial nerve innervates. Patients affected with CFEOM are typically born with ophthalmoplegia (an inability to move the eyes in certain directions) and ptosis (droopy eyelids). In addition, the eyes are usually fixed in an abnormal position.

3.5.2.4 Supranuclear Disorders

A supranuclear gaze palsy is an inability to look in a particular direction as a result of cerebral impairment. There is a loss of the voluntary aspect of eye movements, but, as the brainstem is still intact, all the reﬂex conjugate eye movements are normal.

The type of gaze problem is dependent upon the lesion - thus a right hemisphere lesion, particularly the frontal lobes, leads to a contralateral gaze palsy, i.e. an inability to look away from the lesion.

As an example for a supranuclear gaze palsy, the abducens gaze palsy will be described in more detail.

Abducens Gaze Palsy

Cranial nerve VI, also known as the abducens nerve, innervates the ipsilateral lateral rectus (LR), which functions to abduct the ipsilateral eye.

Patients usually present with horizontal diplopia and an esotropia in primary gaze. The deviation, as would be expected, is noted to be greater when the patient ﬁxates with the paretic eye. Patients also may present with an abnormal head posture to maintain binocularity and binocular fusion and to minimize diplopia.

It is rare to ﬁnd true congenital sixth nerve palsy. A typical workup of a sixth nerve palsy involves excluding paresis of other cranial nerves (including VII and VIII), a check of ocular muscle motility and evaluating pupillary responsiveness. Only the ipsilateral lateral rectus that is solely innervated by the involved peripheral sixth cranial nerve is aﬀected, therefore, only deviations in the horizontal plane are produced. In isolated cases of peripheral nerve lesions, no vertical or torsional deviations are present.

Central nervous system lesions of the abducens nerve tract are localized easily secondary to the typical ﬁndings associated with each kind of lesion. Damage to the sixth nerve nucleus results in an ipsilateral gaze palsy. The lack of contralateral adduction defects (see Fig. 3.25) makes it easy to diﬀerentiate nuclear from a fascicular or non-nuclear lesion (see Sec. 3.5.2.1).

Abducens palsy frequently is seen as a postviral syndrome in younger patients. CHAPTER 3. STRABISMUS 82

(a) Primary Deviation, Left Eye ﬁxing (b) Secondary Deviation, Right Eye Fixing

Figure 3.25: Example for an Abducens Gaze Palsy

Nystagmus

The most primitive fault is a failure in fixation, due to anomalies in central vision, in which case innate attempts to fixate cause rapid jumps known as nystagmus. Nystagmus is generally described as an involuntary movement of the eyes, which reduces vision. The movement is usually side to side (but can be up and down or circular motion) and can be either jerk or pendular. Normal (physiological) nystagmus occurs for example when a passenger of a train watches as telegraph poles pass the window. The eyes will travel one way, and then jump in the opposite direction to begin watching the next pole. There are over 40 different types of nystagmus but the main division is between congenital and acquired Nystagmus.

Congenital nystagmus - is thought to be present at birth, but is usually not apparent until the age of ﬁve months.

Acquired nystagmus - occurs later than 6 months of age, and can be caused by a stroke, disease such as multiple sclerosis, or even a heavy strike to the head. Patients with nystagmus may suﬀer from the perception of a moving world, known as oscillopsia. The number of people that suﬀer from congenital nystagmus is much less than those with acquired nystagmus. This is thought to be because an infant can adapt to the perception of motion better that an adult that previously had normal vision.

It is not clear whether congenital nystagmus is actually present at birth, or whether it occurs early in the child’s vision development. It is therefore also referred to as early-onset or infantile nystagmus. Many children with nystagmus have no other vision or brain problems. This is known CHAPTER 3. STRABISMUS 83 as idiopathic, which means of unknown cause. However, nystagmus is often a symptom of other conditions such as albinism, aniridia, cataracts, cone dysfunction and many others. Nystagmus can be present with cerebral palsy, Down’s Syndrome and many motor system diseases. A type of congenital nystagmus is latent nystagmus, and is only present when one eye is covered. This is not usually noticed until the ﬁrst visit to the physician.

3.6 Strabismus Surgery

The goal of surgery of the extraocular muscles is to balance an ocular misalignment in order to restore binocular alignment. Generally, it is also desirable to reestablish single binocular vision. Generally, the attainment of peripheral fusion with fusional vergence amplitudes suﬃcient to maintain alignment of the eyes, comfortable single binocular vision to enable the patient to perform visual tasks without asthenopia and improved esthetic appearance are the most common objectives.

Ideal preoperative evaluation of the strabismus surgical patient includes quantiﬁcation of the misalignment in primary positions at distance and at near, as well as in the nine diagnostic gaze positions (see Sec. 3.4). In most patients, the maximal deviation under conditions of complete dissociation of the visual axes is the deviation for which surgery is to be designed. As indicated, measurements are taken both with and without the appropriate optical correction. Finally, duction and version testing and, when appropriate, forced duction testing may be performed.

All surgical methods are accomplished through the modification of the eye muscles since this is the most effective way to influence the mechanical properties of the oculomotor system. Basically, there are three different options to modify the mechanical functions of eye movements, when considering extraocular muscle surgery:

• Weakening surgery (recession), in order to reduce the traction of a muscle,

• strengthening surgery (resection), in order to raise traction of a muscle,

• transposition surgery, in order to inﬂuence the pulling direction of a speciﬁc muscle.

Strabismus surgery demands careful planning. Barring unusual circumstances or unusual anatomy, the surgical plan is prepared before anesthetic is given. It is helpful to keep a description of the surgical plan, for ready reference before and during surgery. The exact location and incision technique depends upon the muscles to be operated, pre-existing scarring, and the patient’s previous surgical history. Absorbable sutures generally are utilized when vascular healing of tissues occurs, such as in typical recession and resection techniques. These sutures generally absorb within 7-10 days if covered with conjunctiva, slightly longer if exposed. Permanent sutures are utilized if avascular tissue such as the superior oblique tendon is harnessed, as in the superior oblique tuck or silicone band-lengthening procedure. CHAPTER 3. STRABISMUS 84

3.6.1 Recession Surgery

Recession surgery aims a transposition of the muscle insertion posterior along the muscles primary direction of action. This surgical technique can be applied to all six extraocular muscles. In this procedure (see Fig. 3.26), the surgeon must detach the muscle from the eye and reattach it further back on the eye, thereby reducing the relative strength of the muscle.

Figure 3.26: Schematic Example of Muscle Recession, from [Kau95]

As an example for the weakening of a extraocular muscle, a medial rectus recession is described in detail (cf. [YD03]).

To perform a medial rectus recession (Fig. 3.27), the surgeon grasps the eye at the conjunctiva- Tenon’s capsule junction with a 0.3mm forceps and rotates the eye into elevation and abduction (Fig. 3.27(a)). The surgeon then elevates the conjunctiva at the base of the fornix, and incises the conjunctiva 8mm from the limbus (Fig. 3.27(b)). At this point, all visible conjunctival vessels are prepared lightly to ensure good visibility during localization of the tendon. The surgeon grasps the fascia within the conjunctival incision with gentle pressure against the sclera, and elevates it from the globe (Fig. 3.27(c)). The scissors are then used to incise Tenon’s capsule at this point, and expose the sclera (Fig. 3.27(d)). Visualization of the sclera is maintained using the posterior forceps.

(a) (b) (c) (d) (e)

Figure 3.27: Preparation for Medial Rectus Recession, from [YD03]

A self-retaining muscle hook is then passed behind the medial rectus muscle with no posterior movement of the hook further than the site of the incision itself (Fig. 3.27(e)). When the medial rectus is on the hook, the surgeon conﬁrms that the entire tendon is engaged. The posterior arm of the 0.3mm forceps may be used as a probe to locate the superior pole of the muscle (Fig. 3.28(a)). The pole is secured using the forceps, and the muscle hook is withdrawn to the inferior aspect of the tendon (Fig. 3.28(b)). Then, the hook is passed beyond the superior pole CHAPTER 3. STRABISMUS 85 held by the forceps (Fig. 3.28(c)). This way, it is certain that the full length of the tendon is secured completely.

A small hook is introduced between the insertion of the tendon and Tenon’s capsule anterior and used to dissect carefully the fascia from the surface of the tendon, as it is moved posteriorly along its long axis (Fig. 3.28(d)). Afterwards, the scissors are used to incise the superior aspect of Tenon’s fascia (Fig. 3.28(e)).

(a) (b) (c) (d) (e)

Figure 3.28: Medial Rectus Recession, continued, from [YD03]

Gentle traction, away from the operated muscle, allows the surgeon to control the hook beneath the tendon and accomplish suture passage at the insertion. The surgeon may then control the forceps to position optimally and to control the globe. A lock to the edge of the tendon by applying a suture (Fig. 3.29(a)) is performed. A scissors is used for removal of the tendon (Fig. 3.29(b)). Visualization of the tendon may be preserved if a dry cotton pledget is passed between the tendon and globe.

The sutures are then drawn in the direction in which they were passed, to avoid breaking the sutures out of the sclera. The receded muscle is demonstrated where an attempt is made to preserve the original orientation of the tendon to the globe at its normal width (Fig. 3.29(c)). After the tendon has been secured to the globe, a hook is introduced above the superior pole of the earlier insertion and the fascia is rotated inferiorly (Fig. 3.29(d)) over the operative site. The incision is closed with a single plain suture (Fig. 3.29(e)).

(a) (b) (c) (d) (e)

Figure 3.29: Medial Rectus Recession, continued, from [YD03]

The principles of recession for the other recti muscles are basically the same as described for the medial rectus. However, recession of the lateral rectus and vertical recti includes visualization and preservation of the neighboring oblique muscles before the procedure is performed. CHAPTER 3. STRABISMUS 86

3.6.2 Resection Surgery

Resection of a muscle is performed in order to strengthen its force. In a muscle resection procedure shown in Fig. 3.30, the surgeon must detach the muscle from the eye (Fig. 3.30(a)), excise a portion of the distal end, and reattach the muscle to the eye on the same position (Fig. 3.30(b)). The shortening of the muscle provides greater pull in the ﬁeld of action that the muscle functions.

(a) Resection of Detached Muscle (b) Reinsertion of Muscle

Figure 3.30: Schematic Example of Muscle Resection, from [Kau95]

The preparation procedure in resection surgery is related to the recession surgery in that the conjunctival vessels are prepared and Tenon’s fascia is entered. A large muscle hook is passed beneath the tendon, going no further posteriorly than the insertion itself. The Tenon’s capsule beneath the olive tip of the large muscle hook is incised (Fig. 3.31(a)) in order to visualize the sclera at both poles of the tendon. Two small hooks are passed along the long axis of the tendon so that the muscular fascia can be exposed for incision (Fig. 3.31(b)). A second large muscle hook is passed beneath the tendon, and traction is applied between the two muscle hooks, with the insertion and both hook tips kept parallel. The anterior arm of the caliper is placed on the midportion of the anterior hook, and the posterior portion delineates the site for needle passage (Fig. 3.31(c)). A marking pen may be used if it is not desirable to remeasure during suture passage.

(a) (b) (c) (d) (e)

Figure 3.31: Medial Rectus Resection, from [YD03]

The same technique of suture passage is employed as described earlier for recession. The needle is passed tangentially to the tendon and globe, and woven through the tendon from its midportion to the superior pole and then locked upon itself. The needle at the opposite end of the suture is then passed from the midportion to the inferior pole and once again locked, and the tendon CHAPTER 3. STRABISMUS 87 secured at the point of desired resection (Fig. 3.31(d)). The tendon is cut ahead of the clamp (Fig. 3.31(e)), and the resection of the tendon is completed at the original insertion.

(a) (b) (c)

Figure 3.32: Medial Rectus Resection, continued, from [YD03]

Once the surgeon is satisﬁed with the location and depth of the scleral pass, the ﬁrst suture throw is placed before the muscle is drawn forward and knotted (Fig. 3.32(a) & Fig. 3.32(b)). The conjunctiva can be closed with two interrupted plain sutures (Fig. 3.32(c)).

3.6.3 Transposition Surgery

Transposition surgery has the goal to modify that direction of action of the extraocular muscles in order to correct anomalies of ocular alignment. The vertical transposition surgery of the horizontal recti muscles is applied to correct symptomatic „A“- and „V“-patterns, that arise from eso- and exotropia, in absence of signiﬁcant overaction of the oblique muscles. The eﬀects of muscle transposition are not completely predictable, and only a few of the many transpositions with therapeutic potential are attempted by clinicians (cf. [MDR93]).

An „A“-symptomatic esotropia is treated with a bilateral recession of both horizontal medial recti muscles in combination with an upward transposition.

An „A“-symptomatic exotropia is treated with a bilateral recession of both horizontal lateral recti muscles in combination with a downward transposition.

A „V“-symptomatic esotropia is treated with a bilateral recession of both horizontal medial recti muscles in combination with a downward transposition.

A „V“-symptomatic exotropia is treated with a bilateral recession of both horizontal medial recti muscles in combination with an upward transposition.

In applying these combined surgical methods, careful preoperative planning is necessary, especially the consideration of eﬀects that modify lever-arm and unreel-strain of a muscle gives information about the possible results or complications that may arise and need to be avoided.

3.6.4 Amount of Surgery

The amount of surgery is the main criteria that is to be decided preoperatively on the basis of diagnosis and measurements. Since the oculomotor systems forms a relatively complex mechanical CHAPTER 3. STRABISMUS 88 system, these decisions are not trivial. However, it is now generally accepted, that the choice and amount of surgery are matters of importance on which the end result closely depends (cf. [RSS01]). From the purely geometric point of view, a 1mm rectus muscle recession, combined with a 1mm advancement of the ipsilateral antagonist, should result in a 4.7 ◦ change of angle for a globe of 24.5mm diameter, 5 ◦ for a globe of 23mm diameter, using the simple spherical circumference. In reality, however, the result is reduced by the passive forces of the other muscles and anatomical influences like pulleys and other movement retaining components. In single- muscle surgery, the effect is further reduced by the fact that the counter-tension of the antagonistic muscle remains unchanged. The coefficient of „surgical effectiveness“, or the ratio between the theoretical geometric effect and the actual result, is close to 1 only when the counter-tension of the antagonistic muscle is adjusted by an equivalent amount in the opposite direction (in terms of tension and not of distance). The effect also depends on the muscle that is operated on and additionally on factors like age, the manner in which surgery is carried out, the eye that is operated on (affected or unaffected), the binocular status and type of the strabismus, concomitant or incomitant features and a lot of different other relations. Finally, there is no linear relationship between the amount of surgery and the effect of a surgical procedure. All these factors have led to a wide range of figures being proposed, giving little credibility to establishing the amount of surgery [RSS01].

„Although for some authors the precise amount of surgery is unimportant, or purely indicative, this attitude is currently unacceptable [RSS01].“

From the analysis of their results, some authors have shown that, in spite of variables, it is possible to measure the amount of surgery to be performed. Postoperative statistical analysis of results together with known, and in particular mechanical parameters, have, at least for some, provided the basis for a mean surgical effectiveness chart. Figures and formulae proposed for calculating the amount of surgery necessary to obtain a certain surgical effect are only valid from 2 to 7 or 1 8 mm, and are only applicable from the age of 2 2 to 3 years. However, diagrams and formulae provide average values, applicable only when patient data is in this certain field of tolerance.

Thus, the preoperative process of measurement, diagnosis and planning deals with a certain amount of tolerance that is intrinsic to every part of this process, since it is unlikely possible to carry out a surgical procedure with an accuracy of e.g. 1 mm. Additionally, the terminology for pathologies in the field of strabismus is different in each language and also not clearly separable, since in some pathological cases, the exact cause is still unknown. Dealing with preoperative planning and the determination of the amount for eye muscle surgery is somehow the key fact in this thesis, which proposes an entirely different way to describe ocular motility defects. Com- puter programs that realize and calculate results of formulae that have been based on statistical result are already used, but do not sufficiently solve the problem. Instead, this thesis proposes biomechanical modelling of pathologies in terms of simulation parameters that are confined to an underlying model of geometry, muscle force and kinematics of the human eye. Essentially supported by an interactive and clinically applicable computer simulation program, eye motility disorders can be expressed in their impact on a „virtual“ patient that represents all measured values from the real patient that should be treated. Chapter 4

Biomechanical Modelling

Different strategies in modelling diagnosis and treatment of strabismus in the field of medicine have been proposed over the last years (cf. [C72¨ ], [Jam22], [SS00]). These models had the primary goal to suggest or predict possible outcomes of a treatment procedure based on measurements that were acquired from patients. Since many authors proposed new standard values based upon clinical results, these values differ considerably. It is probable, that these authors did not take the same amounts of surgery, or even comparable or representative types of strabismus into account. The problem becomes evident when considering natural variations in the anatomy and physiology of different patients, which makes it nearly impossible to drive a model that is based on clinical results to exactly the same results, when predicting surgery for comparable patients. This also expresses the non-linearity of the relationship between surgical dose and postoperative response. However, a model-driven approach, able to coordinate broad ranges of laboratory research and clinical experience, can accelerate progress of diagnosis and treatment of strabismus.

This chapter proposes a new approach of predicting surgical results for strabismus surgeries in that the the human oculomotor system is described by a biomechanical model that incorpo- rates anatomically related parameters in order to enable the simulation of pathological situations through the modiﬁcation of these parameters. Biomechanics is a branch of mechanics and mechanics again is a branch of physics. Therefore, biomechanics deals with the mechanical laws and rules of biological structures and the interaction of these biological structures. Since such models already existed before (cf. [MR84]), but were used rarely in clinical application, this thesis proposes a diﬀerent mathematical approach in combination with an interactive, easy to use software simulation system that enables interactive eye motility simulation and preoperative planning of surgery.

89 CHAPTER 4. BIOMECHANICAL MODELLING 90

4.1 Analytical Models

Analytical models are mathematical models that have a closed form solution, i.e. the solution to the equations used to describe changes in a system can be expressed as a mathematical analytic function. Scientific practice involves the construction, validation and application of scientific models, thus, an analytical approach implies the invention of hypotheses and theories and the subsequent effort of validation within the scientific model space. Reapplication of model results into the real world may support or reject the proposed theories. Further more, models provide an environment for interactive engagement. Evidence from science education research shows that significant gains in knowledge and understanding are achieved within interactive activities. Thus, it is important that the environment created around a model provides an interactive experience in order to be valuable for practical use. Working with models can enhance systems thinking abilities by allowing sensitivity studies to assess how changes in key system variables alter the system’s behavior. Such sensitivity studies can help to identify leverage points of a system to either help to affect a desired change with a minimum effort, or to help estimate the risks or benefits associated with proposed or accidental changes in a system. Concerning medical application, especially in the field of strabismus surgery, there exist different model types that aim clinical improvement on the basis of analytical models.

Medical expert systems model the relationships between symptoms and diseases by using explicit domain knowledge that is embedded into a model. Such systems use inference strategies to apply knowledge to the available data (which is often noisy and incomplete) through heuristic reasoning. As with all expert systems, the design of medical diagnostic systems raises essential questions:

• How to find a suitable representation of the observed data? • How to define a suitable representation of the knowledge of the medical domain the system should work in? • How to obtain measurements that could validate a diagnostic hypothesis? • How to handle inaccuracy and uncertainty of the observed data? Early systems adopted a sequential data-driven strategy to perform heuristic classification. Knowledge was encapsulated in production rules in the form of „if ... then“ statements [CHG00]. Rules are a simple and modular mechanism, but their formalism is restrictive, and a huge amount of rules are needed to deal with real-life problems.

The application of expert systems can be seen as an effective way to distribute extant professional expertise, which is specific and also very limited to the domain of appliance. Conversely, expert systems don’t seem to be an adequate model to prove hypothesis or theories due to their limited generality. Empirical models are such types of models that generalize empirical knowledge in terms of surgical „dose-response“ tabular relationship (cf. [MD99]). Whether developed informally or with computerized databases and statistical techniques, empirical generalizations sum- marize experience and are therefore models of observations. Empirical generalizations are probably the basis of professional competence in most fields. However, because these models are so closely related to experience, they prevent fundamental insight into the causes of observed patterns. CHAPTER 4. BIOMECHANICAL MODELLING 91

Homeomorphic models are organized by following the human structure in that these models are built of comparable physiologic parts found in the domain of interest. All interactions with a homeomorphic model reﬂect underlying physiological processes. Thus, a homeomorphic model can treat arbitrary new situations only so long as they can be expressed using the model’s terms.

Biomechanical models are homeomorphic models that try to simulate physiologic functions of the human body by using parts, properties and parameters that are comparable to the human example. Moreover, biomechanical models also include constrains and relationships between the model parts and therefore deﬁne the essential functions that can be evaluated and modiﬁed.

In the following sections, this work presents the formulation of a biomechanical model of the oculomotor system, based on geometric, force and kinematic principles that have been identiﬁed by studying human subjects (cf. [BKPH03]).

4.2 Structure of Biomechanical Models

Biomechanics, as the name implies, is a branch of mechanics that examines forces acting upon biological structures and the eﬀects induced by these forces. Thus, for successfully developing biomechanical models, a good understanding of three diﬀerent areas is required: Biological structures, mechanical analysis and an understanding of movements [Mil96].

Initially important for creating a biomechanical model is to extract the relevant anatomic structures of the oculomuscular system (see Sec. 2.1.4) and form proper abstractions that can be modelled using mathematical methods. Therefore, abstract representations of all six extraocular eye muscles and the globe need to be deﬁned and used as parts of the overall biomechanical model. A muscle itself is composed of several other abstract representations. These are for example the insertion and origin of a muscle which are decomposed again.

However, representing muscles and the globe does not suffice in order to form a complete biomechanical model. A description of the geometrical interaction between all abstract representations is also required. This part of a biomechanical model is defined as the geometrical model. It is desirable to form the geometrical model as autonomous, exchangeable part that interacts with other biomechanical components in order to preserve system variability. A well formed biomechanical model consist of exchangeable components that provide flexibility and compatibility for later modifications.

Apart from a geometrical part of the system model, there is also need for a model describing muscle forces and the inﬂuence of these forces on the ocular geometry. Therefore, a biomechanical model also requires a muscle force model which simulates and predicts the transformation of innervations into muscle force and subsequently supplies this muscle force to the ocular geometry, which resolves forces in terms of eye rotations.

Finally, a biomechanical model also needs to consider kinematics of ocular movements. Adopted to a kinematic model of the human eye, forward and inverse kinematics can be identiﬁed as CHAPTER 4. BIOMECHANICAL MODELLING 92 predicting eye position on the basis of muscle innervations and predicting muscle innervation on the basis of eye positions.

Thus, a biomechanical model consists of several abstract representations and sub-models. Ab- stract representations refer to diﬀerent anatomical parts of the human eye, whereas sub-models imply the mathematical descriptions of geometrical properties, muscle forces and kinematics [BKPH03].

4.3 History of Modelling of the Human Eye

One of the ﬁrst modelling attempts to get a better understanding about the oculomotor system was the so called „ophthalmotrope“, which has been designed and constructed by C.G.T. Ruete in 1845 [ST90]. An ophthalmotrope is a mechanical model of the eye, usually built out of copper, and is used to gain better understanding about eye rotations around diﬀerent axes in 3D-space (see Fig. 4.1).

Figure 4.1: Halle’s Ophthalmotrope, from [ST90]

In 1848, F.C. Donders made an interesting discovery. If the eye ﬁxates an object somewhere in space, the position of this object also determines the gaze position of the eye. But the position of the object in space does not specify the amount of torsional rotation, nor is this amount of torsion arbitrary. In fact, the amount of torsion is clearly speciﬁed through the gaze position of the eye itself.

When the German edition of Donders’ work appeared, it drew the attention of H. von Helmholtz, who noticed that Donders had found the existence of pseudotorsion and proposed to call Donders’ discovery „Donders’ Law“ [Hel63]. Pseudotorsion is a problem as it is a measured value of torsion which does not really exist, but is evoked by the measuring procedure. As Donders did not give a reason for pseudotorsion, it was von Helmholtz, who gave an explanation for its occurrence. Pseudotorsion is caused by the fact that, in tertiary positions of gaze, the vertical meridian CHAPTER 4. BIOMECHANICAL MODELLING 93 through the eye does not coincide with a vertical line in space, nor does the horizontal meridian coincide with a horizontal line in space. „The reason for this discrepancy is that ’horizontal’ and ’vertical’ are deﬁned according to the coordinate system used“ [ST90]. Thus, a solution had to be found to eliminate the problem of pseudotorsion.

Since pseudotorsion is caused by an unsuitably deﬁned coordinate system, J. B. Listing had the idea to use polar coordinate systems to overcome this problem. Listing also noticed that in this coordinate system all tertiary positions of gaze can be reached by a single rotation around one particular axis. Therefore, when Ruete read about Listing’s discovery in 1853, he suggested to call it „Listing’s Law“. He also noticed that his ﬁrst ophthalmotrope was not correct, as it violated Listing’s law, so he developed a new version of the ophthalmotrope, which complied with Listing’s law and already included parts for simulating muscles (see Fig. 4.2).

Figure 4.2: Ruete’s Ophthalmotrope, from [ST90]

As Donders’ law and Listing’s law were published, a consistent sequence of rotations for describing gaze rotations had to be found. As a consequence, in 1854 A. Fick introduced the so called Fick- sequence of rotations, where the position of the eye is characterized by rotations around the vertical, the horizontal and then the torsional axis. Later, in 1863, von Helmholtz suggested a different sequence by exchanging the first two axes. Therefore, the Helmholtz-sequence describes first a rotation around the horizontal, then around the vertical and finally around the torsional axis [Has95]. Such gimbal systems are a convenient way to describe the sequence of rotations.

In 1869 A. W. Volkmann provided the basis for all future developments in the ﬁeld of extraocular mechanical research, especially for the development of improved models based upon ophthal- motropes. Volkmann made a statistical analysis of the characteristics of the eyes of numerous patients and published the data of an average human eye [Vol69].

Using Volkmann’s eye data, Krewson [Kre50] published the first geometrical model of the human CHAPTER 4. BIOMECHANICAL MODELLING 94 eye called „string model“. The „string model“ defines muscles as strings which take the shortest path from the insertion to the origin without considering any orbital connective tissues or muscle forces. A second attempt for building a geometrical model using Volkmann’s data was made by Robinson [Rob75b]. Robinson named his model „tape model“ and tried to include some sort of fixing structure to reduce muscle side-slip during tertiary positions of gaze. Although this was a first step in the right direction, the muscles in the „tape model“ still showed too much side-slip compared to actual observations of the oculomuscular apparatus. Later, Kusel and Haase [KH77] tried to improve the „tape model“ and introduced the restricted „tape model“, but the predictions of their model were still very close to the original „tape model“.

Robinson was the first who noticed that a geometrical model alone is not sufficient to successfully build a realistic model of the human eye. So he also took muscle forces into account and built the first biomechanical model. Following Robinson’s observation, Günther [Gue86] extended the restricted „tape model“ of Kusel and Haase into a biomechanical model. Although the predictions of these models were better than those of the previous ones, they still were not perfect.

The latest discovery in the ﬁeld of extraocular research was the analysis of muscle pulley structures in 1995. Based on this new research results, Miller and Demer [MD99] started another attempt to build a biomechanical model of the human eye. Their so called orbit model represents the latest development in the ﬁeld of biomechanical eye motility research.

4.4 Ocular Geometry

The mathematical derivation for the modelling of ocular geometry is comprised of the geometrical definition of eye rotations and the geometrical representation of the extraocular muscles that cause these rotations. In order to describe the transformation of eye muscle action into respective angular rotations, the lever-arm of each eye muscle must be defined. This is accomplished by the geometric interpretation of muscle action, that assumes that each muscle, in its static behavior, rotates the eye around a specific axis in space. Additionally, geometrical constrains like the sequence of rotation and conformance to Listing’s law needs to be taken into account (see Sec. 2.2.1 and Sec. 2.2.2.2).

4.4.1 Coordinate Systems

In order to describe or measure 3D eye positions, coordinate systems need to be deﬁned, allowing the exact deﬁnition of the orientation the eye. Let {X,Y,Z} denote a cartesian coordinate system of a left eye (see Fig. 4.3), so that the positive X-axis points to the right, the positive Z-axis points upward and the negative Y-axis coincides with the line of sight through the center of the pupil when the eye is in primary position.

For 3D orientation of the eye, Euler’s theorem can by applied, which states that for every two orientations of an object, the object can always move from one position to the other by a single rotation around a fixed axis. This combined rotation is usually decomposed into three consecutive rotations around well defined, hierarchically nested axes. The sequence of rotation plays an important role, since the execution of rotations specifying the same angles but in different order, leads to a different final orientation of the rotated object. One important distinction within CHAPTER 4. BIOMECHANICAL MODELLING 95

Figure 4.3: Coordinate System of a Left Eye this definition is to consider active and passive rotation behavior. Active rotations are defined as rotations where the coordinate system is constant and not affected by preceding rotations, whereas passive rotations change the coordinate system, reorienting the coordinate axes with each rotation that is applied. Thus, in passive rotations, each rotational modification also changes the coordinate axes around which successive rotations will be performed. In adopting these properties to eye rotations, active rotations correspond to rotations in a head-fixed coordinate system, and passive rotations are performed in an eye-fixed coordinate system.

(a) Fick Gimbal (b) Helmholtz Gimbal

Figure 4.4: Gimbal Systems for describing 3D Eye Position, adapted from [Has95]

A combination of a horizontal and a vertical rotation of the eye is a well defined sequence, uniquely characterizing the direction of the line of sight. However, this does not completely determine the 3D eye position, since the rotation around the line of sight is still unspecified. A third rotation is needed to completely determine the orientation of the eye. Systems that use such a combination of three rotations for the description of eye positions generally use passive rotations, or rotations of the coordinate system. Due to non-commutativity of rotations in 3D CHAPTER 4. BIOMECHANICAL MODELLING 96 space, it is necessary to use a uniform order of rotations to describe eye positions and to denote this rotation order along with the definition of eye positions. Such rotations of the coordinate system can effectively be demonstrated by using gimbal systems, in which the hierarchy of passive rotations is automatically implemented as intrinsic system property. A coordinate system was defined by Fick [Fic54], whereby rotations have to be executed first around the vertical axis, then around the horizontal and around the torsional axes (see Fig. 4.4(a)). In contrast, Fig. 4.4(b) shows a rotation sequence defined by Helmholtz, first around the horizontal, afterwards around the vertical and torsional axes [Hel63]. Whereas the Fick gimbal system rotates three angular components {α, β, γ} around the coordinate axes {Z,X,Y } respectively, the Helmholtz gimbal will rotate around the axes {X,Z,Y }. Both systems of rotation describe a gimballed suspension of the globe with consideration of the respective rotation sequence, where rotations are executed from the outside to the inside. From the representation of the two systems in Fig. 4.4 it is easy to recognize that some specified angles of an eye position lead to different orientations of the globe and thus to different eye positions when comparing both systems.

4.4.2 Mathematical Description of Eye Rotations

Based on the coordinate system for one eye, depicted in Fig. 4.3, the angular rotations around the main coordinate axis are described by the Fick sequence of rotations (cf. Sec. 4.4.1). This ensures that each eye position is expressed by three angular rotations {α, β, γ} around the coordinate axes {Z,X,Y } respectively. There are many diﬀerent ways to represent rotations in three- dimensional space. In this work, matrices and quaternions will be used to deﬁne rotations and eye positions for the derivation of the ocular geometry. Eye rotations in three-dimensional space consist of rotations as well as translations. The discussion here will be restricted to the rotational components of the total eye movement. Translations of the eye will be treated later in this document.

First, let a vector SC~ (sx, sy, sz) represent a line originating from the center of the eye ﬁxed coordinate system. Next, it will be described how this vector can be rotated according to a given eye position.

4.4.2.1 Rotation Matrices

In order to specify a rotation matrix that describes an eye position in Fick rotational order, the standard deﬁnition of rotations around the three principle axes are used. The rotation matrices used in this derivation use passive rotational order.

Generally, elements of a rotation matrix can be indexed in the following row-based form,   R11 R12 R13 R =  R21 R22 R23  (4.1) R R R 31 32 33 . CHAPTER 4. BIOMECHANICAL MODELLING 97

A rotation of γ radians around the Y-Axis is deﬁned as,  cos(γ) 0 − sin(γ)  RY (γ) =  0 1 0  (4.2) sin(γ) 0 cos(γ) , a rotation of β radians around the X-Axis is deﬁned as,  1 0 0  RX (β) =  0 cos(β) sin(β)  (4.3) 0 − sin(β) cos(β) ,

ﬁnally, a rotation of α radians around the Z-Axis is deﬁned as,  cos(α) sin(α) 0  RZ (α) =  − sin(α) cos(α) 0  (4.4) 0 0 1 .

Thus, a generalized rotation matrix that can rotate the eye around each of the principle coordinate axis in the Fick rotational sequence (Z-X-Y) can be deﬁned using,

0 T T T RF ick(α, β, γ) = RY RX RZ , (4.5) which conforms to an active rotation (i.e. a rotation of the object), where T denotes matrix inversion. If rotated in the object space (eye ﬁxed coordinate system), the rotation sequence needs to be inverted to, RF ick(α, β, γ) = RZ RX RY , (4.6) and solved for the matrix product, this results in cos(α) cos(γ) + sin(α) sin(β) sin(γ) sin(α) cos(β) sin(α) sin(β) cos(γ) − cos(α) sin(γ) ! RF ick(α, β, γ) = sin(α) − cos(γ) + cos(α) sin(β) sin(γ) cos(α) cos(β) cos(α) sin(β) cos(γ) + sin(α) sin(γ) cos(β) sin(γ) − sin(β) cos(β) cos(γ) .

Please note that the sequence of rotations from Eqn. 4.5 around the principle axes in world coordinate space (head-ﬁxed) can be rewritten to perform rotation in the object’s coordinate space (eye-ﬁxed) by inverting Eqn. 4.5 to Eqn. 4.6.

Given the rotation matrix from Eqn. 4.6, the vector S~ can be reoriented to S~0 according to the angular Fick coordinates (α, β, γ) using,

~0 ~ S = RF ick(α, β, γ) S. (4.7) Thus, an eye position can be described by reorienting the geometry of the eye according to Eqn. 4.7. The eye movement can be described by linearly interpolating the rotational angles (α, β, γ) between two positions, which then describe the shortest path rotation between these two positions. Moreover, the product of multiple Fick rotation matrices (CF ick) describes the ﬁnal eye position that is reached by subsequent execution of each rotation. The ﬁnal eye position of a rotation (α, β, γ) from primary position, followed by a rotation (α0, β0, γ0) can be described by, 0 0 0 CF ick = RF ick(α , β , γ ) RF ick(α, β, γ). (4.8) CHAPTER 4. BIOMECHANICAL MODELLING 98

However, this construction assumes that the eye initially always starts its movement from primary position (i.e. the position where the line of sight is equal to the Y-Axis of the eye ﬁxed coordinate system). If the shortest path rotation from primary position to a position that is composed of two or more Fick rotations of the form (4.6) needs to be known, Fick angular coordinates can be resolved from the compound rotation matrix of the form (4.8).

In order to regain angular coordinates from a Fick rotation matrix of the form (4.1), the following entities can be used: cos(β) sin(α) tan(α) = = R12 , cos(α) cos(β) R22 sin(β) = −R32, (4.9) cos(β) sin(γ) tan(γ) = = R31 . cos(β) cos(γ) R33

The presented derivation enables the deﬁnition of eye position by using rotation matrices of the form (4.6) and to determine eye position from given rotation matrices using Eqn. 4.9. When specifying all entities of the ocular geometry, a new eye position can be described by transforming each geometric entity using Eqn. 4.7. However, rotation matrices are complicated to handle, especially when deﬁning rotation around arbitrary axis.

Using the presented method of Euler angles to represent a set of three rotations specified in successive order may suffer from a problem known as „gimbal-lock“. The problem is that no matter what the order in which the three rotations are carried out, there will always be a value for one angle of rotation that yields infinite values of the other two angles. For example, Euler angles are normally represented as yaw, pitch, and roll, in that order. Given that order of rotation, it is easy to show that when the pitch angle is headed straight up or straight down, the values of yaw and roll are undefined. In a physical gimbal system, this is known as gimbal-lock and refers to the situation in that two or more principle axis of a coordinate system align, resulting in a loss of rotational degree of freedom.

The notion of axis-angle deﬁnition for rotation represents a more convenient way with respect to deﬁning eye positions and eye movements, and shall be introduced by using quaternion algebra.

4.4.2.2 Quaternions

Each rotation may be parameterized by a unit vector along the axis of rotation, and the angle of rotation. Rotations collectively form a three-dimensional space which can be pictured as a solid 3-sphere with diametrically opposite points of its surface identiﬁed. Rotations may also be parameterized by Euler angles via various combinations of rotations around the X, Y, Z axes (see Sec. 4.4.2.1). Unfortunately, there are 12 such parameterizations possible, and the use of Euler angles can give rise to gimbal-lock problems (see Sec. 4.4.1). Using an axis and angle representation in specifying an arbitrary axis and an angle (positive if in a counterclockwise direction), this is an eﬃcient way to avoid gimbal-lock. Additionally, axis-angle representations are a more intuitive and practical oriented method of representing rotations of objects. Quaternion algebra is a powerful mathematical tool that combines vector notation with rotational operations.

Basically, quaternions are hypercomplex numbers, just as a single complex number, z = x + iy, CHAPTER 4. BIOMECHANICAL MODELLING 99 can be used to specify a point or vector in a two dimensional space, a single quaternion,

q = a + bI + cJ + dK, (4.10) can be used to specify a point in a four dimensional space, and a quaternion with a = 0 can be used to describe vectors in Euclidean 3-space. A quaternion can be rewritten as,

q = (s, V~ ), (4.11) where s = a, and V~ = bI +cJ +dK. In this notation, s is called the scalar part of the quaternion, and V~ is the vector part. Additionally,

Scal(q) = s, (4.12) V ect(q) = V,~ (4.13) extract scalar and vector parts from a quaternion so that Scal(q) results in a real number and V ect(q) gives the original cartesian three-dimensional vector.

This deﬁnition can be extended to include cartesian vector spaces: Let {E~1, E~2, E~3} be a set of base vectors in a cartesian vector space. Then, a vector V~ = (x1, x2, x3) can be represented as:

V~ = x1E~1 + x2E~2 + x3E~3.

There exists an isomorphism of a three-dimensional vector space into the four-dimensional vector space of quaternions.

Let V~ = a1E~1 + a2E~2 + a3E~3, and w = a0 + a1I + a2J + a3K. w could also be written as: w = a0 + V~ , where a0 is called the scalar part of the quaternion, and V~ , the vector part. The quantities {I, J, K}, the unit quaternions, stand in relation to quaternions in the same way that {E~1, E~2, E~3} unit vectors relate to vectors. However, {I, J, K} do not combine in exactly the same way as the unit vectors, {E~1, E~2, E~3}.

This leads to the deﬁnition that a vector S~ = (x1, x2, x3) can be represented by a quaternion of the form, q = [0, (x1, x2, x3)], (4.14) where the scalar part is zero and the vector part of the quaternion contains the identical cartesian vector. The general form of a quaternion is denoted as,

qs = [s, (x1, x2, x3)], (4.15)

Given two quaternions according to Eqn. 4.15,

q1 = [a0, (a1, a2, a3)],

q2 = [b0, (b1, b2, b3)], CHAPTER 4. BIOMECHANICAL MODELLING 100 the following algebraic operations can be deﬁned:

q1 + q2 = [a0 + b0, (a1 + b1, a2 + b2, a3 + b3)], (4.16)

q1 − q2 = [a0 − b0, (a1 − b1, a2 − b2, a3 − b3)]. (4.17)

According to Eqn. 4.11, q1 and q2 can also be written as,

q1 = [s1, V~1], (4.18)

q2 = [s2, V~2], where s1 and s2 are the scalar parts and V~1 and V~2 are the vector parts of the quaternions q1 and q2 respectively. Additionally, a „pure“ quaternion is a quaternion whose sca1ar part is zero,

qp = [0, V~ ]. (4.19)

Let q1 and q2 be in the form of (4.18), then, multiplication of two quaternions can be accomplished by,

q1q2 = [s1s2 − V~1 · V~2, (s1V~2 + s2V~1 + V~1 × V~2)], (4.20) where V~1 × V~2 stands for the vector-cross-product, and V~1 · V~2 stands for the vector-scalar-product.

Two „pure“ quaternions in the form (4.19) can be multiplied using,

qp1qp2 = [−V~1 · V~2, (V~1 × V~2)]. (4.21)

Multiplication of a quaternion in the form (4.15) or (4.19) by a scalar number n is deﬁned as,

qn = nq = [ns, nV~ ] = [nx0, (nx1, nx2, nx3)]. (4.22)

In relation to vector operations in cartesian three-dimensional space, the following entities can be identiﬁed when using quaternion algebra:

The conjugate of a quaternion q in the form of (4.11) inverts the vector part in that,

q0 = [s, −V~ ], (4.23) the absolute value, or magnitude of a quaternion the form of (4.11) can be calculated using, q |q| = s2 + V~ · V,~ (4.24) and the norm n of a quaternion is denoted as,

n(q) = (s2 + V~ · V~ ) = |q|2. (4.25) CHAPTER 4. BIOMECHANICAL MODELLING 101

A very important operation with quaternions, particularly with regard to representing rotations, is the inverse function, 1 1 q−1 = ( )[s, −V~ ] = 1/q = q0, (4.26) |q|2 n(q) whereby the division of two quaternions can be expressed by,

q1 −1 −1 q3 = = q1(q2 ) = q1q2 . (4.27) q2

In order to rotate a vector by a quaternion, the vector is multiplied on the right by the quaternion, and on the left by the inverse of the quaternion.

From (4.19) it follows that a „pure“ quaternion vp can be rotated using,

0 −1 vp = Rot(vp, q) = q vpq. (4.28)

0 0 The result vp will always be a quaternion with zero scalar component, [0, V ect((vp))]. This guarantees that: Rot(vp1, q)Rot(vp2, q) = Rot(vp1vp2, q) (4.29) which implies that dot and cross products are preserved. This eﬀect is embedded in the quaternion product.

By using the inverse of a quaternion q = [s, V~ ] as q−1[s, −V~ ]/ |q|2, the eﬀects of magnitude are divided out so that any scalar multiple of a quaternion gives the same amount of rotation.

When the magnitude of q = 1, these unit-quaternions lie on a sphere of radius 1, q = [w, (x, y, z)] such that w2 + x2 + y2 + z2 = 1. These unit quaternions carry the amount of rotation in w, as cos(θ/2), while the vector part (x, y, z) points along the axis of rotation with magnitude sin(θ/2) and, the axis of rotation is that line in space which remains unmoved during the rotation.

In referring to Eqn. 4.24, the magnitude of,

q = [cos(θ/2), sin(θ/2)(~u)] = 1, (4.30) since cos2 + sin2 = 1 and where ~u denotes the unit-vector of the axis of rotation.

This sphere of unit quaternions spans a 4-dimensional vector space (S4) and forms a sub-group of the 3-dimensional vector space (S3). This spherical metric of S3 is the same as the angular metric of S3.

Thus, a quaternion for rotating is stored as,

qr = [cos(θ/2), sin(θ/2)(~u)], (4.31) where θ refers to the radian angle of rotation around the unit-vector of the axis of rotation ~u.A short (axis-angle) notation of a rotation quaternion can therefor be denoted as:

qr = [θ, (~u)]. (4.32) CHAPTER 4. BIOMECHANICAL MODELLING 102

Rotation quaternions of the form (4.31) can be combined by quaternion multiplication, using Eqn. 4.29.

An eye position can therefore be represented by a unit-quaternion that describes a rotation from primary position around a speciﬁed rotation axis ~u with an appropriate angle θ using Eqn. 4.31.

In order to represent eye movement, interpolation of rotations is conveniently performed using quaternion algebra. Let qa and qb denote two distinct quaternions that describe the start and end position of an eye movement respectively. Then the rotation quaternion, −1 t q¯ = qa(qa qb) , (4.33) where the quaternion power operator is deﬁned as,

qt = [st, V~ ]. (4.34)

This will describe any desired intermediate rotation that lies on the shortest path between the two eye positions. The time parameter t can be introduced into the angle so that the adjustment of q varies uniformly over the great arc between qa and qb. Eqn. 4.7 can now be rewritten using a rotation quaternion q in the form (4.28) that represents the current eye position and S~0 = V ect(Rot(S,~ q)), (4.35) represents the rotation of a vector S~ into a new position S~0.

Finally, a quaternion can be transformed into a rotation matrix. Let q be a rotation quaternion in the form of (4.15), then, the corresponding rotation matrix is deﬁned by,  2 2  1 − 2x2 − 2x3 2x1x2 − 2x3S 2x1x3 + 2x2S 2 2 R =  2x1x2 + 2x3S 1 − 2x1 − 2x3 2x2x3 − 2x1S  . (4.36) 2 2 2x1x3 − 2x2S 2x2x3 + 2x1S 1 − 2x1 − 2x2

Further information on quaternions, derivations and proofs can be found in [PW82] or [WP03].

4.4.2.3 Listing’s Law

In order to generate 3D eye positions based on 2D reference positions, torsional rotation angles need to be calculated in order to fulﬁll Listing’s law. Referring to Sec. 2.2.2.2, Listing’s law can be derived geometrically. Based on the coordinate system deﬁnition from Sec. 2.2.1 and Sec. 4.4.1, an eye position can be represented by three successive rotations around the coordinate system axes.

Let {α, β, γ} denote three angular rotations around the coordinate axes X, Z and Y respectively, whereas α represents ab-/adduction, β elevation or depression and γ torsional eye movements. This sequence of rotation corresponds to the deﬁnition of the Fick rotation order from Sec. 4.4.1. Since the Y-Axis represents the line of sight, torsional rotation in accordance with Listing’s law needs to be determined as a function of the angles α and β only. CHAPTER 4. BIOMECHANICAL MODELLING 103

The torsional rotation of a Point P (x, y, z) around the line of sight results in a new reference position P 0(x0, y0, z0) and can be described using Eqn. 4.37, x0 = x cos(γ) − z sin(γ), (4.37) y0 = y, z0 = z cos(γ) + x sin(γ).

Additionally, P 0 is rotated around the horizonal X-Axis into position P 00(x00, y00, z00) in order to represent elevation or depression movements using Eqn. 4.38, x00 = x0, (4.38) y00 = y0 cos(β) + z sin(β), z00 = z0 cos(β) + y0 sin(β), and finally ab-/adduction movement is represented by a rotation of P 00 around the Z-Axis using Eqn. 4.39, defining the final position of P 000(x000, y000, z000), x000 = x00 cos(α) + y00 sin(α), (4.39) y000 = y00 cos(α) − x00 sin(α), z000 = z00.

Substitution of Eqn. 4.37 and Eqn. 4.38 in Eqn. 4.39 gives Eqn. 4.40, a combined rotation around all three axes of an eye ﬁxed coordinate system, obeying the Fick sequence of rotation, x000 = y cos(γ) sin(α) − (z cos(β) + x sin(α) sin(β)) sin(γ) + (4.40) cos(α)(x cos(γ) + y sin(β) sin(γ)), y000 = y cos(α) cos(β) − x cos(β) sin(α) + z sin(β), z000 = z cos(β) cos(γ) + cos(α)(−y cos(γ) sin(β) + x sin(γ)) + sin(α)(x cos(γ) sin(β) + y sin(γ)).

According to Listing’s law, Eqn. 4.40 deﬁnes a rotation axis in 3D-space, so that any eye position deﬁned through the angles {α, β, γ} can be reached by a single rotation around this axis. Since this axis stays constant during the rotational movement, it can be found by setting P 000 = P in Eqn. 4.40 and subsequently solving for x, y and z. Thus, all points of the rotation axis satisfy Eqn. 4.41, y cos(γ) sin(α) − z cos(β) sin(γ) + y cos(α) sin(β) sin(γ) x = , (4.41) 1 − cos(α) cos(γ) + sin(α) sin(β) sin(γ) x cos(β) sin(α) − z sin(β) y = , −1 + cos(α) cos(β) cos(α)(−y cos(γ) sin(β) + x sin(γ)) + sin(α)(x cos(γ) sin(β) + y sin(γ))) z = − . (4.42) −1 + cos(β) cos(γ)

Since Listing’s law also states, that all rotation axes of the eye lie in the fronto-parallel plane (Listing’s plane), this constraint needs to be added by using the equation for the X − Z plane, y = 0, (4.43) CHAPTER 4. BIOMECHANICAL MODELLING 104 constraining all coordinates of the rotation axes from Eqn. 4.41. By replacing Eqn. 4.43 into Eqn. 4.41, the system of equations can be solved by forward substitution of Eqn. 4.41 in Eqn. 4.42, resulting in Eqn. 4.44, z cos(β) sin(γ)(cos(γ) sin(α) sin(β) + cos(α) sin(γ)) z = − . (4.44) (−1 + cos(β) cos(γ))(−1 + cos(α) cos(γ) − sin(α) sin(β) sin(γ))

Solving Eqn. 4.44 for γ eliminates the last free variable z and expresses the torsional angle of rotation as a function of α and β, ensuring that each eye position is assigned a unique torsional value according to Eqn. 4.45, cos(α) + cos(β) ⊗ (α, β) = γ = cos−1 . (4.45) 1 + cos(α) cos(β)

4.4.2.4 Deﬁnition of Eye Positions

In describing eye positions with the methods presented in Sec. 4.4.2.1 and Sec. 4.4.2.2, there are basically two possibilities:

3D eye positions are used to fully specify all three degrees of freedom for a given eye position in space. This corresponds to the deﬁnition of three angular coordinates in Fick rotational order that specify add-/abduction, elevation/depression and torsion respectively. 2D eye positions are used when specifying ab-/adduction and elevation/depression angles, except the torsional rotation angle is calculated as suggested by Listing’s law and therefore constrains eye rotation axes to lie in Listing’s plane (see Sec. 2.2.2.2).

Eye positions that fulﬁll Listing’s law can therefore be deﬁned by using Eqn. 4.6 in the following way: RListing(α, β) = RF ick(α, β, ⊗(α, β)), (4.46) where γ is substituted by Eqn. 4.45.

Using quaternion notation, 3D eye positions can be defined by considering passive rotation history of every coordinate system axis. In Fick rotation sequence, first, the X-axis of the coordinate system is rotated around the Z-axis by α degrees using a quaternion of the form (4.32). Let XA~ (1, 0, 0), YA~ (0, 1, 0) and ZA~ (0, 0, 1) denote the base vectors of the eye fixed coordinate system. Then, the rotation quaternion,

qrz = [α, (ZA~ )], deﬁnes a rotation of α degrees around the base axis ZA~ , and,

0 XA~ = V ect(Rot(XA,~ qrx)), reorients the original X-axis to a new vector XA~ 0. Accordingly, the y-Axis of the eye ﬁxed coordinate system is rotated around the Z- and X-axis respectively by using,

0 qry = [β, (XA~ )]qrz, 0 YA~ = V ect(Rot(Y~ A, qry)), CHAPTER 4. BIOMECHANICAL MODELLING 105 resulting in a new Y-Axis YA~ 0. The rotation quaternions that rotate around each of the pre- oriented eye ﬁxed axes can now be deﬁned as,

0 qry = [γ, (YA~ )], 0 qrx = [β, (XA~ )], (4.47)

qrz, where γ is given by Eqn. 4.45.

The complete rotation quaternion for deﬁning eye positions according to the Fick sequence of rotation is given as qListing = qryqrxqrz. (4.48)

4.4.3 Geometrical Abstractions

In the derivation of the ocular geometry, geometrical abstractions will serve as modelling elements that mathematically deﬁne constrains and conditions that represent its human counter- part. Additionally certain equations can be identiﬁed as interfaces to other elements of the overall biomechanical model.

4.4.3.1 Globe

The ﬁrst essential geometrical abstraction is the modelling of the globe. The geometrical object of a sphere is used to represent the human eyeball and is deﬁned by using the spherical equation, x2 + y2 + z2 = r2, (4.49) where x, y, z are coordinates that lie on the sphere with radius r.

Figure 4.5: Geometrical Abstraction of the Globe

The illustration in Fig. 4.5 shows that the globe is placed in the center of the eye ﬁxed coordinate system which is identical with the rotation center of the sphere. The line of sight originates from CHAPTER 4. BIOMECHANICAL MODELLING 106 the center of rotation (C) and intersects the center of the pupil. In primary position, the line of sight coincides with the Y-axis of the head-ﬁxed coordinate system. The orientation of the globe (i.e. the eye position) is changed according to the rotation of the line of sight using Eqn. 4.35.

Although, this geometrical abstraction assumes that the globe is spherical, further derivations will show, that the „natrual“ ellipsoid shape of the human eyeball will be considered when deﬁning the lever-arm of each muscle. Thus, the spherical approximation of the globe was chosen for easier visualization and can be replaced by a more realistic (i.e. more complex) geometrical structure, without aﬀecting the models behavior.

4.4.3.2 Muscles

In order to geometrically deﬁne the action of the extraocular muscles, speciﬁc landmark-points that describe the muscle path and the direction of pull are chosen for each muscle. Derivations will be given starting from older, historic modelling approaches up to the geometrical model used in this thesis, the so called „Pulley“ model.

Figure 4.6: Geometrical Abstraction of an Eye Muscle

An extraocular muscle can geometrically be approximated by a straight line. Muscle force and deformation due to contraction or relaxation are not of primary interest in the geometrical model. These properties will be added by additional models of muscle force and dynamic 3D-visualization. The geometrical model itself is only responsible for predicting how a muscle can transfer its force into a rotation axis that aﬀects eye position. Following Fig. 4.6, the requirements for a speciﬁc geometrical model of the extraocular muscles are the calculation of the following entities (cf. [BKPH03] and [Buc02]):

The origin of each muscle represents the point where a muscle originates in the posterior area of the orbit (i.e. the anulus of Zinn). The point of tangency of each extraocular eye muscle is defined as that point where the muscle first contacts the globe. This special point is intrinsic to the geometrical definition, since CHAPTER 4. BIOMECHANICAL MODELLING 107

it changes as a function of gaze and is defined using different mathematical formulations in different geometrical models.

The insertion point is that point on the globe where the tendon of the muscle ﬁnally inserts on the globe.

The muscle action circle is deﬁned as that circle that lies on the globe and is deﬁned by the muscle’s plane of action.

The arc of contact of a muscle is deﬁned as the circular path between the insertion and the point of tangency. This path is always part of the muscle action circle.

The muscle rotation axis is deﬁned as a vector that represents an axis around which a speciﬁc muscle rotates the eye. Thus axis is perpendicular to the muscle action circle or the muscle’s plane of action.

String Model

The simplest geometrical representation of extraocular muscle action is the „string model“ (see Fig. 4.7). This model is also known as the „shortest path hypothesis“ (cf. [Kre50]), which means that muscle strings are assumed to be always tight and therefore take the shortest possible connection between the insertion and the origin (see Fig. 4.7(a)).

(a) Muscle Path in Primary Position (b) Muscle Path in Secondary Position

Figure 4.7: Geometrical „String Model“, from [BKPH03]

The deﬁnition of the „string model“ can be reduced to specifying the point of tangency as a function of gaze position (see Fig. 4.8). Let I~ be the vector from the center of the coordinate system C to the insertion point of a muscle from Sec. 2.1.3 and I~0 the vector to the insertion point in the actual eye position, such that,

I~0 = V ect(Rot(I,~ q)), (4.50) where the rotation quaternion q in the form (4.32) speciﬁes the current eye position. Let O~ denote the origin of an eye muscle, according to Sec. 2.1.3 and R~ denote a vector with length

~ ~ r = R that is perpendicular to the vector O. Then, the right triangle O − R − C gives l as the CHAPTER 4. BIOMECHANICAL MODELLING 108 shortest path from origin to insertion by using,

r 2 2 ~ ~ l = O − R . (4.51)

From the deﬁnition of the right triangle, the angle between O~ and R~ is deﬁned by,

δ = cos−1(l/r). (4.52)

The rotation axis RM~ of the muscle can be deﬁned as vector that is perpendicular to the muscle action plane spanned by the vectors O~ and I~0 using,

~ ~0 ~ O × I RM = . (4.53) ~ ~0 O × I

Finally, the vector QN~ , which points along the vector O~ with length r needs to be rotated around the muscle rotation axis RM~ by the angle δ. The vector QN~ is denoted as, ~ ~ O QN = r. (4.54) ~ O Using a rotation quaternion of the form (4.32),

rq = [δ, (RM~ )], (4.55) the vector QN~ can be rotated into the position of the point of tangency T by,

T~ = V ect(Rot(QN,~ rq)). (4.56)

Figure 4.8: Point of Tangency in the „String Model“

In secondary gaze positions (see Fig. 4.7(b)), the shortest path hypothesis in this model leads to a displacement of the point of tangency which is known as „side-slip“. In this model, the muscle CHAPTER 4. BIOMECHANICAL MODELLING 109 action circle is always a great circle on the globe and its center coincides with the center of the globe.

However, the predictions of the „string model“ do not correspond to clinical expectations. Par- ticularly in secondary and tertiary gaze positions, this model gives inadequate results. Only in primary position, predictions for muscle path and muscle match clinical expectations. Fig. 4.7(a) shows a left eye with the lateral rectus muscle in primary position, whereas Fig. 4.7(b) shows an adduction of about 34 ◦ (secondary gaze position). It is easy to recognize that the path of the lateral rectus muscle shifts upwards with a change in gaze position (the lateral rectus muscle „shifts its muscle path with the line of sight“), which does not correspond to clinical observations. This shift results in a drastic change of the muscle path and thus in an abnormal rotation of the eye (i.e. the lateral rectus now mainly elevates the eye!). The lateral rectus muscle cannot sustain its mainly abducting eﬀect on the globe, it „slips“ away as the eye turns towards the nose.

Tape Model

Based on the „string model“, Robinson ([Rob75a]) developed the so called „tape model“ which tries to reduce the side-slip of the muscles in secondary and tertiary positions. Therefore, Robinson limited the side-slip of the point of tangency with an empirical value depending on the extent of the rotation of the eye. This side-slip retaining component is introduced in the „tape model“ via a reduction of the maximum side-slip angle that is calculated with respect to the „string model“.

The maximum side-slip angle is determined by measuring the angular displacement of two perpendicular vectors with respect to the current eye position in a head-ﬁxed coordinate system (see Fig. 4.9). Let MP~ be the vector perpendicular to I~ and O~ , and let TP~ be the „tangential“ vector with respect to the arc of contact of the muscle such that, ~ ~ ~ I × O MP = , (4.57) ~ ~ I × O ~ ~ ~ MP × I TP = . (4.58) ~ ~ MP × I

These two base vectors are deﬁned with respect to I, the insertion point in primary position. Now, these two base vectors are also deﬁned for the current gaze position, using the vector I~0 from Eqn. 4.50,

I~0 × O~ ~ 0 MP = , (4.59) ~0 ~ I × O ~ ~0 ~ 0 MP × I TP = . (4.60) ~ ~0 MP × I

The maximum angle of deviance is measured between the vectors MP~ and MP~ 0. However, since the vector MP~ is deﬁned in primary position, it needs to be transformed in the head-ﬁxed coordinate system in order to measure the true angle of deviance. Let q be the quaternion that CHAPTER 4. BIOMECHANICAL MODELLING 110

Figure 4.9: Tape Model Calculations deﬁnes the current eye position in the form (4.32). Then, q−1 will represent a backward rotation into the head-ﬁxed coordinate space, and,

−1 MP~ h = V ect(Rot(MP~ , q )), (4.61) −1 TP~ h = V ect(Rot(T~ P , q )) will transform the vectors MP~ and TP~ into head-ﬁxed vectors MP~ h and MP~ h respectively. Now, the maximum angle of deviation vmax of the muscle action circles between the primary position and the current gaze position with respect to the „string model“ is deﬁned as,

~ 0 −1 ~ 0 vmax = sgn(MP · TP~ h) cos (MP · MP~ h), (4.62) where the first part of this equation specifies the angular direction in terms of the sign of the deviation angle. The actual deviation angle e with respect to the vector MP~ h can be calculated using, −1 e = cos (MP~ h · O~ ). (4.63) Finally, the reduced side-slip angle v can be defined as a fraction of the maximum deviation angle vmax with respect to the actual excitation angle e using,

v = vmax |cos(e)| . (4.64)

The angle v now deﬁnes the the actual reduced displacement angle for the „tape model“ that is applied by using a rotation around the axis that is deﬁned by the vector O~ based on the point of tangency from the „string model“. Thus, the point of tangency T from the „string model“ (4.56) in primary position is rotated using the following rotation quaternion qt of the form (4.32),

qt = [v, (O~ )], (4.65) resulting in the deﬁnition of the point of tangency TT in the „tape model“,

TT~ = V ect(Rot(T~ , qt)). (4.66) CHAPTER 4. BIOMECHANICAL MODELLING 111

Because of the modification to the movement of the point of tangency, the center of the muscle action circle of a specific muscle does no longer coincide with the globe center in secondary and tertiary positions. Instead, the center of a specific muscle action circle is now defined through a muscle’s point of tangency vector TT~ and origin vector O~ , just like the muscle rotation axis. As in the „string model“, the vector perpendicular to the plane spanned through these three points represents the muscle rotation axis RM~ , such that,

~ ~ ~ TT × O RM = . (4.67) ~ ~ TT × O

The center of the muscle action circle can now be calculated as the intersection point of the rotation axis with the muscle action plane. Since the muscle rotation axis is perpendicular to the muscle action plane, and the muscle action plane contains the point of tangency and origin of a muscle, the center of the muscle action circle lies in the extension of the muscle rotation axis. Thus, the center of the muscle action circle can be calculated using,

C~ − TT~ C~c = (C~ − RM~ ) . (4.68) RM~ − C~

(a) Muscle Path in Primary Position (b) Muscle Path in Secondary Position

Figure 4.10: Geometrical Tape Model, from [BKPH03]

The limitation of the movements of the point of tangency results in a reduced muscle side-slip in secondary and tertiary positions. Fig. 4.10 shows this effect by comparing an eye in primary position (Fig. 4.10(a)) and after an adduction of 34 ◦ (Fig. 4.10(b)). Although the side-slip is reduced in the „tape model“, the model predictions still differ from anatomical findings. In the following part of this thesis, a new geometrical pulley model will be described which tries to overcome this problem and also takes pulleys into account. CHAPTER 4. BIOMECHANICAL MODELLING 112

Pulley Model

Results of investigations of muscle pulley structures provided the reason why all the conventional models like the string or „tape model“ where not exact and the predictions of these models did not compare to clinical findings (see Sec. 2.1.1). But pulleys did not only influence existing models. They also considerably affected existing surgery techniques. Before the discovery of pulleys most surgeons tended to remove portions of the orbital connective tissues covering the eye muscles, however, it has been noticed that if too much of the connective tissue is removed, unpredictable and unwanted results occurred after surgery. With the discovery of pulleys, the reason for this behavior became clear. When surgeons damaged or even removed pulley structures, they de- stroyed the functional origin of a muscle and therefore destabilized the muscle and sometimes even the whole oculomuscular system (see [CJD00]).

In Fig. 4.11, the difference between conventional models and the pulley model is illustrated. While conventional models assume that the muscle tendon is coupled tightly to the globe (see Fig. 4.11(A)), the pulley model introduces an new anatomical structure that lets the muscle slide through a fascial pulley which is elastically coupled to the orbital wall (see Fig. 4.11(B)). The functional consequence is obvious in comparing both models when the eye is elevated. While conventional models tend to define a constant axis of rotation with respect to different gaze positions (compare Fig. 4.11(A) and Fig. 4.11(B)), the pulley model adapts the axis of rotation such that the primary direction of action of the muscle is preserved with respect to the current gaze position (compare Fig. 4.11(C) and Fig. 4.11(D)). This means, that the axis of rotation of a muscle is expressed according to an eye-fixed coordinate system in the pulley model, since the coordinate system axes are moved as a function of gaze position.

Figure 4.11: Comparison of Conventional vs. Pulley Model, from [MD99] CHAPTER 4. BIOMECHANICAL MODELLING 113

For the mathematical derivation of the pulley model, a new geometrical pulley point is introduced, which also serves as functional origin of a muscle. In contrast to other models, the pulley model redirects a muscle’s main direction of pull towards the functional pulley point instead of using the anatomical origin (o) as force directing reference position. This results in a stabilization of the muscle path in the posterior area of the orbit as well as an adaption of the muscle rotation axis with respect to the current eye position.

For the calculation of the point of tangency, the muscle action circle must be defined. In primary position, the muscle action circle of each muscle is a great circle. Therefore, in primary position, the center of a muscle action circle coincides with C as shown in Fig. 4.12. In secondary and tertiary positions, the muscle action circle of a specific muscle is no longer a great circle and for this reason, the center has to be calculated in a different way.

Thus, to calculate the center of a muscle action circle in all different eye positions, the first step is the definition of three vectors {SX,~ SY,~ SZ~ } based on the insertion in primary position I, the pulley location P and the center of the globe C.

According to Fig. 4.12, let SZ~ denote the unit vector from C to I, ~ ~ I SZ = , (4.69) ~ I

SY~ the vector perpendicular to the plane spanned by the three points C, P and I,

~ ~ ~ P × I SY = , (4.70) ~ ~ P × I and SX~ be the cross product of SY~ and SZ~ , SX~ = SY~ × SZ.~ (4.71)

In order to calculate the center of a muscle action circle in a secondary or tertiary position the eye and, consequently, the vectors {SX,~ SY,~ SZ~ } have to be rotated out of the primary position into a secondary or tertiary position. Since the orientation of the eye is deﬁned through a rotation quaternion q of the form (4.32), q is also used for reorienting three vectors {SX,~ SY,~ SZ~ }, such that, TX~ = V ect(Rot(SX,~ q)), (4.72) TY~ = V ect(Rot(SY~ , q)), TZ~ = V ect(Rot(SZ,~ q)).

Thus, the resulting vectors {TX,~ TY,~ TZ~ } deﬁne the three base-vectors {SX,~ SY,~ SZ~ } in the current eye position (see Fig. 4.13). Additionally, the insertion point I0 is deﬁned as the rotated insertion point in primary position I according to (4.50).

From Fig. 4.13, it can be seen that the rotated primary position muscle action circle that is deﬁned by the pulley model is no longer the shortest possible path between the insertion and the origin CHAPTER 4. BIOMECHANICAL MODELLING 114

Figure 4.12: Primary Position in Pulley Model, from [BKPH03] on the surface of the bulbus. Therefore, three new vectors {GX,~ GY~ , GZ~ } are calculated which form the basis of a new muscle action circle representing the shortest path between the insertion and the pulley. These vectors are calculated with the same equations shown in (4.69)-(4.71), but with I replaced by I0, using,

~0 ~ I GZ = , (4.73) ~0 I ~ ~0 ~ P × I GY = , ~ ~0 P × I GX~ = GY~ × GZ.~

There are three diﬀerent muscle action circles which can be determined with a set of given equations. These circles are the primary position muscle action circle, the rotated primary position muscle action circle in tertiary and secondary positions (zero side-slip) and the shortest path muscle action circle in tertiary and secondary positions (full side-slip).

The two diﬀerent muscle action circles in secondary and tertiary positions are important, because the „correct“ muscle action circle of a speciﬁc muscle in secondary and tertiary positions lies somewhere between these two (see Fig. 4.14). In order to determine the actual muscle action circle, the full side-slip angle Ψ is calculated, which lies between the rotated primary position muscle action circle and the shortest path muscle action circle measured at the insertion point. The angle between the two muscle action circles is calculated using, ! −GX~ · TY~ Ψ = tan−1 . (4.74) −GX~ · −TX~ CHAPTER 4. BIOMECHANICAL MODELLING 115

Figure 4.13: Tertiary Position in Pulley Model, from [BKPH03]

However, as already mentioned before, the „correct“ muscle action circle lies somewhere between the rotated primary position muscle action circle and the shortest path muscle action circle. Therefore, the angle Ψ has to be reduced by an empirical side-slip scaling parameter defined as α. This scaling parameter was fitted to comply to other models (i.e. Orbit [MR84]) and describes the rotational force due to muscle width that tends to rotate the eye around the axis I~0. The different values for α of each muscle have been taken from the EyeLab model (cf. [PWD00]) and are listed in Tab. 4.1. Muscle Side-slip Scalar (α) medial rectus 0.3443 lateral rectus 0.2909 inferior rectus 0.2954 superior rectus 0.2850 inferior oblique 0.0722 superior oblique 0.1240

Table 4.1: Side-slip Scaling Values for Pulley Model

Using Eqn. 4.74 for calculating Ψ and the side-slip scaling parameter α, the side-slip angle θ can be calculated using, ! GX~ · TY~ θ = −α tan−1 . (4.75) GX~ · TX~

With the side-slip angle θ and the vectors TX~ and TY~ , a vector D~ can be deﬁned which describes the direction in which the muscle departs from its insertion (see Fig. 4.15). This vector D~ can CHAPTER 4. BIOMECHANICAL MODELLING 116

Figure 4.14: Muscle Action Circles in Pulley Model, from [Kal02] be deﬁned as a linear combination of TX~ and TY~ such that,

D~ = TY~ sin(θ) − TX~ cos(θ). (4.76)

To determine the center of the actual muscle action circle in secondary and tertiary gaze positions a vector N~ , which is perpendicular to the actual muscle action circle and represents the axis of rotation of the globe is deﬁned. However, calculating N~ is easy, since I0,P and D~ are all lying in the plane of the muscle action circle and therefore can be used for calculating the perpendicular vector of the plane spanned by the vector D~ and by the vector deﬁned through the points P and I0 (see Fig. 4.16). Consequently, the equation for calculating N~ is given by,

~ ~0 ~ D × I P N = . (4.77) ~ ~0 D × I P

Using the vector N~ , the calculation of Cc, the center of the pulley model muscle action circle, is now possible by using, 0 Cc = (I~ N~ )N.~ (4.78)

With the calculation of Cc, the center of the muscle action circle of a specific muscle in a primary, secondary or tertiary position is defined. Moreover, the vector N~ corresponds to the rotation axis of a specific muscle. Therefore, the only aspect remaining is the determination of the point of tangency.

In order to calculate the point of tangency, two vectors {EX,~ EY~ } are deﬁned, which form the basis of the pulley model muscle action circle and are shown in Fig. 4.17. These two vectors are CHAPTER 4. BIOMECHANICAL MODELLING 117

Figure 4.15: Muscle Direction Vector in Pulley Model, from [Kal02]

formed using P , Cc and N~ according to, ~ ~ ~ P − Cc EX = , (4.79) ~ ~ P − Cc ~ ~ ~ N × EX EY = . ~ ~ N × EX

With the help of EX~ and EY~ , the angle β which describes the angle between the vector from 0 Cc to I and the vector from Cc to P , can be deﬁned. The angle β is also shown in Fig. 4.17 and is calculated using, ! I~0 · EY~ β = tan−1 . (4.80) I~0 · EX~

However, in order to determine the point of tangency, the calculation of the angle γ is necessary (see Fig. 4.17). Therefore, the radius of the pulley model muscle action circle is required and the length of the vector from Cc to P . The radius r of the muscle action circle and the length l of the vector from the pulley to the center of the muscle action circle can be calculated using,

~ ~ l = P − Cc , (4.81)

~0 ~ r = I − Cc .

Since the angle between the vector from the point of tangency to Cc and the vector from the point of tangency to the pulley P must be exactly 90 ◦, implied by the deﬁnition of the tangent CHAPTER 4. BIOMECHANICAL MODELLING 118

Figure 4.16: Muscle Rotation Axis and Action Circle Center in Pulley Model, from [Kal02] function, the angle γ can be calculated by, r γ = 90 ◦ − sin−1 . (4.82) l

Thus, the angle between the insertion and the point of tangency at the center of the muscle action circle is (β − γ). With this angle, the point of tangency t can be calculated by rotating 0 the point I around an axis deﬁned through the vector N~ using a rotation quaternion rq of the form (4.32), rq = [β − γ, (N~ )]. (4.83)

0 Next, the rotation quaternion rq is used to rotate the point I in the muscle’s direction of pull by (β − γ) degrees and the resulting quaternion is assigned to the point T ,

0 T = V ect(Rot(I~ , rq)). (4.84)

4.4.3.3 Evaluation of Muscle Action

In order to complete the definition of extraocular geometry, one needs also to consider muscles and their directions of pull. A muscle’s geometric description is based on definition of important reference points, while the muscle’s direction of pull is determined by the definition of a rotation axis, around which the muscle would rotate the eye. The geometric description of a muscle is defined by a muscle path from its origin to its insertion and how this muscle path changes with different gaze positions. The illustration in Fig. 4.18(c) shows the medial rectus muscle defined by muscle origin, pulley, point of tangency and insertion. The pulley stabilizes the muscle path in CHAPTER 4. BIOMECHANICAL MODELLING 119

Figure 4.17: Point of Tangency in Pulley Model, from [BKPH03] the rear orbit. The point of tangency marks that range, at which the muscle path first contacts the globe. Reactions of the muscle path to changes in gaze position are differently represented by different models. Thus, substantial differences exist in the definition of the muscle path between string, tape and pulley models. The „string model“ and the „tape model“ use origin, point of tangency and insertion to define the muscle path, whereas the pulley model additionally consider the pulley point.

(a) „String Model“ Path (b) Tape Model Path (c) Pulley Model Path

Figure 4.18: Muscle Path Comparison using Diﬀerent Geometrical Models from [BKP+03]

In comparing the muscle path representation of these models (see Fig. 4.18), noticeable diﬀerences in the movement of the point of tangency occur. In the „string model“ (Fig. 4.18(a)) as well as in the „tape model“ (Fig. 4.18(b)), the point of tangency and thus also the entire rear muscle path is pulled downward. This also substantially aﬀects the direction of pull in other gaze positions. Using the pulley model (Fig. 4.18(c)), stabilization of the direction of pull of the muscle is reached by introducing a new reference point (pulley), and thus a substantially more realistic result occurs.

To geometrically evaluate the rotational behavior of a muscle according to clinical methods, the rotation axis can be split into its respective components of action. This results in specifying normalized rotational magnitudes of rotational action for ab-/adduction, elevation/depression and in-/extrorsion. These rotational actions are measured within an oblique coordinate system CHAPTER 4. BIOMECHANICAL MODELLING 120

{D~1, D~2, D~3} where D~3 denotes the head-fixed vertical axis, D~1 describes the head-fixed horizontal axis that is rotated around D~3 according to ab-/adduction movements and D~2 corresponds to the line of sight. Thus, the orientation of this new coordinate system depends only on ab- /adduction and elevation/depression movements of the globe. In order to measure muscle action in a certain eye position, the rotation axis of a muscle, defined using string, tape or pulley model needs to be transformed into the new coordinate system.

Let {α, β, γ} denote an eye position according to the Fick rotational sequence and RA~ be a normalized muscle rotation axis according to the string, tape or pulley model (4.53, 4.67 or 4.77). Using the rotation quaternions qrx and qrz from (4.47), the rotation quaternion,

qt = qrxqrz, (4.85) specifies the rotational transformation of coordinates from head-fixed space into the new measurement coordinate system. Therefore, the rotation axis RA~ (x, y, z) can be transformed into an axis RA~ 0(x0, y0, z0) using, 0 RA~ = V ect(Rot(RA,~ qt)), (4.86) where z0 represents the ab-/adducting rotational component, x0 represents the elevating/depressing component and y0 specifies the torsional rotation. Since RA~ 0 is a unit vector, all rotational components lie between −1 and 1.

The muscle force distribution shows this relative rotational components for selected eye muscles along a horizontal view range (see Fig. 4.19) in a certain elevation/depression level. The rotational components are indicated in standardized values between -1 and 1 for ab-/adduction, elevation/depression and in-/extorsion. The illustration in Fig. 4.19 show the force distributions of the lateral rectus muscle in the „string“ and „tape“ models with an elevation of 15 ◦ along the horizontal ﬁeld of vision with up to 60 ◦ of abduction/adduction of a left eye.

Here, the physiologically incorrect prediction of the „string model“ becomes clearly evident at 36 ◦ of adduction. The lateral rectus muscle drastically changes its abducting effect into adduction (see Fig. 4.19(a)). The comparison with the „tape model“ in Fig. 4.19(b) shows better behavior for accurately the same scenario due to retention of the main direction of pull of the lateral rectus. These differences result from the differentiated mathematical modelling of the anatomical structures. While in the „string model“ the muscle path within its contact range with the globe was defined by the shortest path between insertion and origin, the „tape model“ contains an angle-reducing component, which describes the muscle path by the relative motion of the point of tangency as a function of gaze position. This allows the simulation of stabilizing connective tissues, in order to limit the movement of muscles in extreme gaze positions. Graphically, this simulation results in a bent muscle path between insertion and point of tangency, which substantially better corresponds to anatomical conditions. CHAPTER 4. BIOMECHANICAL MODELLING 121

(a) „String Model“ Distribution

(b) Tape Model Distribution

Figure 4.19: Muscle Force Distribution in String and Tape Model from [BKP+03]

Figure 4.20: Muscle Force Distribution in the Pulley Model

Only through the introduction of a model that includes pulleys, more detailed simulations and pathological case studies can be accomplished (see Fig. 4.20). The simulation of pulleys shows an absolute retention of a muscle’s main action in the entire physiological ﬁeld of vision and is thus best suitable for being used as geometrical representation.

4.4.4 Passive Geometrical Changes

Passive changes in ocular geometry occur when some muscles move the globe into a different gaze position. Due to the rotation of the globe, muscle length changes in every eye muscle, since a contracting muscle shortens and an antagonistic muscle lengthens during eye movement. These geometric variations are very essential to the biomechanical model because muscle length CHAPTER 4. BIOMECHANICAL MODELLING 122 and length changes are major parameters that affect muscle force prediction. The mathematical description of muscle length can be split into the sum of the contact arc length and the length of the posterior muscle part. The length of the contact arc can be calculated as the spherical distance along the muscle action circle between the insertion point and the point of tangency. Let T~ and I~0 be the vectors from the center of the muscle action circle to the insertion and origin point. Then, the radius of the muscle action circle is defined as,

~0 rad = I , (4.87) and the angle λ between the two vectors can be calculated using, λ = cos−1(I~0 · T~). (4.88) Using the radius and the angle λ, the spherical distance on the muscle action circle between insertion and point of tangency is, darc = rad · λ. (4.89)

For the pulley model, the length of the posterior part of the muscle is also calculated in two steps. First, the length from the origin point of the muscle to the pulley point is given by the length of the vector PO~ ,

~ l1 = PO . (4.90) Then, for the length from the pulley point to the point of tangency, an adapted formula from (4.51) for the shortest path distance is used, q ~0 ~0 l2 = P~ · P~ − I · I − Tl, (4.91) where P~ is the vector from the center of the globe to the pulley point, I~0 is the vector from the center of the globe to the insertion point and Tl denotes the length of the tendon which is given in Sec. 2.1.3. The subtraction of Tl is important, since the net muscle length is measured without the muscle tendon.

For geometrical models that do not use pulleys, the posterior part of the muscle length is simply calculated using the shortest path distance between origin and point of tangency using,

~ ~ l = O − T − Tl. (4.92)

Finally, the overall muscle length ml can be computed using,

ml = darc + l1 + l2, (4.93) for the pulley model, and, ml = darc + l, (4.94) for any other models that do not use pulleys.

Based upon these calculations, the passive length change with respect to geometrical measurements from Sec. 2.1.3 can be computed. The passive length change dl is always measured in percent, based on the geometrical muscle length in primary position, using,

100 · (ml − L0) dl = . (4.95) L0 CHAPTER 4. BIOMECHANICAL MODELLING 123

4.5 Muscle Force Prediction

Besides ocular geometry, muscle force prediction is another essential part of the overall biomechanical model. The muscle force prediction model has the function of providing forces for each muscle, based on muscle length, tension and actual innervation.

For skeletal muscles, many different modelling approaches can be found in literature. Most models are devoted to muscle force prediction in that they only provide models for estimating the global uniaxial output force of given muscles in defined conditions and experiments (cf. [MWTT98] and [KS98]). Conversely, other models are concerned with the understanding of the contractile mechanism, and describe the chemico-mechanical aspects of the contraction process, but are hardly related to a realistic global output force involving the 3D anatomical and passive properties of a specific muscle. Only few studies attempt to provide a model of muscles including anatomical and mechanical, active and passive properties, allowing for a realistic simulation of its contractile behavior in relation with its deformation and its global output force (cf. [BL99] and [BCL99]).

However, since all of these studies primarily investigated skeletal muscles, results cannot directly be related to eye muscles. Further, since the anatomy of human eye muscles is still not fully explored, models that take ﬁber-types and contraction velocities into account cannot be adapted to produce accurate results due to currently many unknown properties of human eye muscles. Therefore, for current proposes, a static model of eye muscle force prediction using the best available data is preferable. The static muscle force prediction model presented in this thesis is adapted from studies conducted by Miller and Robinson (cf. [MR84] and [Rob75b]), taking length-tension-innervation relationships into account.

4.5.1 Length-Tension Relationship

As mentioned before, muscle force depends on muscle length and tension. Tension is a function of two variables, namely muscle stretch and muscle innervation. The more a muscle is stretched the higher the tension, the second variable is motor command or innervation. High motor commands lead to muscle contraction which also increases tension. Based upon measurements that were carried out by Robinson et al. [ROS69], Collins et al. [CSO69] and Miller (see Sec. 2.3.2), the length-tension relationship was modelled for a given stretch and motor command by a hyperbola. This length-tension relationship was modelled according to the following equation, r ! k k2 F (dl , e ) = λ (dl + e ) + (dl + e )2 + a2 . (4.96) i i i i 2 i i 4 i i

Here k is the asymptotic slope (equal to stiffness for large extensions) and a determines the sharpness of curvature of the transition to the „slack“ state. Miller and Robinson approximated the values of k = 1.8g and a = 6.24g to fit to the measured data. The other parameters in (4.96) are indexed by muscle number. Changes in muscle length dl is expressed as a percentage of the length of the muscle in primary position using Eqn. 4.95. Thus, Eqn. 4.96 can be applied to all muscles with respect to their differing reference lengths (L0). The parameter ei translates the whole curve left or right along the length axis. It thus simulates a change in innervation. It CHAPTER 4. BIOMECHANICAL MODELLING 124 is thought that muscle force is linearly proportional to the average muscle cross sectional area, accordingly, the parameter λi is used to scale the lateral rectus muscle length-tension hyperbola. Therefore, λ is 1 for the lateral rectus muscle and the scaling parameter for each of the other five muscles is taken to be its cross sectional area relative to that of the lateral rectus, shown in Tab. 4.2. Muscle λ Lateral Rectus 1.00 Medial Rectus 1.07 Superior Rectus 0.80 Inferior Rectus 0.97 Superior Oblique 0.41 Inferior Oblique 0.38

Table 4.2: Cross Sectional Muscle scaling Parameter Values

The remaining parameters dli and ei are now used to formulate the muscle force prediction based upon experimental data. The parameter ei slides the force curve along the length axis and therefore inﬂuences the output force when the length of a muscle does not change. This is related to a simulation of changing innervation, thus, a mathematical formulation for innervation can be derived from the parameter ei by using,

ivi(ei) = Fi(0, ei) − F (0, e0), (4.97)

e0 = −21.7264.

The first term in (4.97) describes the total isometric force for zero length change and the second term describes the passive isometric muscle force for zero length change. The net change in force, and therefore also innervation can be determined in subtracting the passive isometric force from the total isometric force. In Eqn. 4.97, e0 is that constant value of e that fits the passive force curve from the force data. Thus, changes in e0 modifies default passive force behavior for all muscles as initial force displacement ratio in units of percent length changes with respect to L0 from (4.95). Since this mathematical derivation of innervation is based upon the parameter e of Eqn. 4.96, the units for iv are arbitrary. However, it can be seen that a change in innervation alters the total force output by the same amount as a numerically equal change of dl.

In using this muscle force prediction model, two fundamental problems arise. When a detached muscle is stretched sufficiently it stops behaving like a non linear spring and becomes stiff very quickly. This so called leash region is not incorporated in the length-tension relationship from Eqn. 4.96. Conversely, if the muscle is shortened sufficiently it will become slack and exert no force. However, the hyperbolic nature of Eqn. 4.96 ensures that the tension for muscles will never become zero. Miller and Robinson addressed these problems by creating three force surfaces corresponding to elastic (passive), contractile (developed) and their sum total force that were modified by hand to fit experimental data. The relationship between elastic force FE and contractile force FC can be defined as,

FT (dl, iv) = FE(dl) + FC (dl, iv), (4.98) where F represents the total force curve and iv deﬁnes innervation according to Eqn. 4.97. CHAPTER 4. BIOMECHANICAL MODELLING 125

4.5.2 Elastic Force Data

Elastic force is that force that a muscle exhibits when it is stretched or compressed due to its material properties. Thus, the elastic force data is that part of Eqn. 4.96 that is only sensitive to length changes dl. The curve consists of static values for passive isometric tension values given at diﬀerent muscle lengths (cf. Sec. 2.3.2).

Figure 4.21: Elastic Force Data

It is easy to see from Fig. 4.21 that the elastic force function is constant throughout the innervation axis (iv), thus only depends on the length change of the muscle (dl axis). To incorporate the leash and slack regions, the dl axis was split into three regions. In the region 17% ≤ dl ≤ 40%, elastic force was simply calculated using Eqn. 4.96,

FE(dl) = F (dl, e0), 17% ≤ dl ≤ 40%. (4.99)

In the region dl < 17%, muscle force was smoothly tapered by hand to zero in order to reﬂect the slack region. Similarly for dl > 40%, force was extrapolated to an arbitrarily large value to model the rapid increase in stiﬀness exhibited in the leash region. From Fig. 4.21 it can be seen that at a value of dl >= 54, force data stays constant at 82g, which simulates rigid behavior of the muscle.

Muscle data for the leash and the slack regions are stored as tables that are used for interpolation. Let,

TSd[x] = (x1, x2, x3, . . . , xn), (4.100) denote the table that stores the dl values of the slack region and,

TSf [y] = (y1, y2, y3, . . . , yn), (4.101) CHAPTER 4. BIOMECHANICAL MODELLING 126 denote the table that stores the force value for each xi. Then, any real value x2 ≤ u ≤ xn−2 can be found by using a cubic interpolation for an index i, such that,

xa ≤ TSd[xi] ≤ xb, 2 ≤ a, b ≤ n − 2. (4.102)

Based on 4 sample points (v1, v2, v3, v4) = (TSf [xi−1],TSf [yi],TSf [yi+1],TSf [yi+2]), the approximated force value FSa can be calculated using,

a0 = v4 − v3 − v1 + v2,

a1 = v1 − v2 − a0,

a2 = v3 − v1, (4.103)

a3 = v2, 0 1 u = (u − xa) (xb − xa) 03 02 0 FSa(u) = u a0 + u a1 + u a2 + a3.

For the leash region of the elastic muscle data, Eqn. 4.100 - Eqn. 4.103 is used with the respective table data that extrapolates force values up to 82g, and according to the cubic interpolation, a function FLa(u) can be deﬁned in the same manner as Eqn. 4.103. Using these interpolation values, the elastic force function can be completed in addition to Eqn. 4.99 as,   FSa(dl), dl < 17%, FE(dl) = FLa(dl), dl > 40%, (4.104)  82, dl ≥ 54%.

In order to simulate muscle dysfunction concerning elastic force, FE is additionally scaled by an elastic force scaling factor es such that,

FE(dl, es) = FE(dl) · es, 0 ≤ es ≤ 2. (4.105)

4.5.3 Contractile Force Data

Contractile muscle force can be deﬁned as a function of innervation iv and length change dl. Based on Eqn. 4.98, the contractile force function FC can also be extracted from Eqn. 4.96 with additional manual changes. Compared to force data analysis of human skeletal muscles, contractile force in eye muscles does not fall oﬀ when muscle length increases above normal. This could be due to the sudden rise of FE at large dl. However, it is obvious, that for large dl values, the total output force is dominated mainly by the elastic force function FE. On the other hand, contractile force FC needs to be zero when innervation iv is zero. Thus, the contractile muscle force function, shown in Fig. 4.22, is calculated in three regions.

In the ﬁrst region, where −20% ≤ dl ≤ 45%, 0g ≤ iv ≤ 100g, the contractile force is calculated using Eqn. 4.96 by,

FC (dl, iv) = F (dl, iv) − FE(dl), −20% ≤ dl ≤ 45%, 0g ≤ iv ≤ 100g. (4.106) CHAPTER 4. BIOMECHANICAL MODELLING 127

Figure 4.22: Contractile Force Data

Here, the first term evaluates the total force using Eqn. 4.96 and the second term uses Eqn. 4.99 to subtract the elastic force to get the contractile output force based on Eqn. 4.98. The second region is defined as −20% ≤ dl ≤ 45%, 0g ≤ iv > 100g and is interpolated in a manually generated force data table. To reflect saturation of innervation, contractile force is restrained at a maximum value of approximately 100g for high iv. Bicubic interpolation is used as table lookup strategy. Let,

TCd[x] = (x1, x2, x3, . . . , xn), (4.107) denote the table that stores the dl values of the interpolation region and,

TCi[y] = (y1, y2, y3, . . . , yn), (4.108) denote the table that stores the innervation values of this region. Then, a two dimensional table of the form,

TCf [x, y] = (z1, z2, z3, . . . , zn), (4.109) can be deﬁned which stores the force values for each combination of xi and yi such that for any real values x4 < u < xn−4 and y4 < v < yn−4, an interpolated force value in the table TCf can be found by using an average of 16 points, surrounding the closest corresponding value. Therefore, two indices xi and yi for Eqn. 4.109 can be found for u and v respectively, such that, x ≤ T [xi] ≤ x , a Cd b (4.110) ya ≤ TCi[yi] ≤ yb, 4 ≤ a < b ≤ n − 4.

Using xi and yi as reference point, a weighted sum of force values of 16 surrounding points can be calculated using, 2 2 X X FCa(u, v) = TCf [xi + m, yi + n] · R(m − dx) · R(dy − n), (4.111) m=−1 n=−1 CHAPTER 4. BIOMECHANICAL MODELLING 128

where dx = TCd[xi] − u and dy = TCd[yi] − v denote the real differences between table values and target coordinates u and v. Finally, R is the cubic weighting function that defines how much each of the 16 points that are used for interpolation contributes to the final value,

1 3 3 3 3 R(x) = 6 · [P (x + 2) − 4P (x + 1) + 6P (x) − 4P (x − 1) ], x, x > 0, (4.112) P (x) = 0, x ≤ 0.

In the remaining section, where dl > 45%, force in the muscle is taken as that value for dl = 45%, which has already been deﬁned. Hence, the deﬁnition for the contractile force function, can be completed from Eqn. 4.106 in adding, FCa(dl, iv), −20% ≤ dl ≤ 45%, 0g ≤ iv > 100g, FC (dl, iv) = (4.113) FC (45, iv), dl > 45%.

In order to simulate muscle dysfunction concerning contractile force, FC is additionally scaled by an contractile force scaling factor cs such that,

FC (dl, iv, cs) = FC (dl, iv) · cs, 0 ≤ cs ≤ 2. (4.114)

4.5.4 Total Force Data

The total force of a muscle can be calculated by summing elastic and contractile force data. This results in a total force function FT that depends on innervation iv and length change dl, using Eqn. 4.105 and Eqn. 4.114 such that,

FT (dl, iv, es, cs, ts) = ts · (FE(dl, es) + FC (dl, iv, cs)), 0 ≤ ts ≤ 2. (4.115)

In Eqn. 4.115, the parameter ts indicates the total force scaling parameter, which provides the simulation of muscle dysfunction that aﬀects all properties of muscle force.

+ =

(a) Elastic Force (b) Contractile Force (c) Total Force

Figure 4.23: Total Force Data

In Fig. 4.23, the addition of elastic and contractile force is shown. The resulting total force curve (Fig. 4.23(c)) shows the dominating elastic leash region (Fig. 4.23(a)) for high values of dl and CHAPTER 4. BIOMECHANICAL MODELLING 129 innervation dependent muscle force (Fig. 4.23(b)) that is generated with rising iv. In using this force prediction model, Eqn. 4.115 provides diﬀerent scaling parameters to modify the behavior of a muscle in pathological situation.

Elastic strength (es) scales the passive force function in order to inﬂuence the elastic properties of a muscle. A reduction of this value, for example, results in a muscle with a reduced elastic force when the muscle stretches.

Contractile strength (cs) scales the active force function and simulates either overaction or a muscle palsy in relation to the cerebral innervation. A value of zero would correspond to a complete muscle palsy (complete loss of the contractile strength).

Muscle strength (λ) scales the total force function (passive and active force). For this value, default values are used which describe the diﬀerent scaling of all muscles in relation to the lateral rectus muscle (see Tab. 4.2). If this value is changed, the muscle is strengthened or weakened relative to the lateral rectus muscle. In order to scale the total force in relation to the current muscle only, one should change the muscle’s total strength.

Total strength (ts) scales the total force function in relation to the current muscle, but only after the total muscle force has already been scaled by the factor λ.

For the simulation of surgeries, the length of a muscle is very important. The length of the muscle and the tendon also changes the way how the simulation interprets the muscle force curves. As an example, a rubber band with a length of 10mm (relaxed length L0 = 10mm) is stretched by 10mm, which results in a path length of 20mm and a relative stretch of 100%. Now, if the band is shortened to 5mm relaxed length, and once again stretched by 10mm, the relative stretch is now 200%. This would lead to a totally different resulting force in the force model and shows, how the modification of the muscle length can influence force behavior of a muscle. Referring to Sec. 4.4.4, the geometrical model offers different parameters to modify muscle length.

Muscle length (L0) deﬁnes the length of a relaxed, denervated muscle without tendon in mm.

Tendon length (Tl) deﬁnes the additional length of the tendon in mm.

To simulate a resection of the lateral rectus muscle by 5mm, the length of the muscle and/or the tendon has to be changed. When the insertion of the lateral rectus muscle is disinserted from the globe, the muscle is shortened by 5mm and afterwards reattached again at the same position. In carrying out this procedure, 1mm of muscle length is lost during the separation and 1mm during the ﬁxation of the length of the muscle tendon. Since the model cannot predict this condition automatically, it is necessary to shorten the muscle by 7mm in this situation. Therefore, the tendon length of the lateral rectus muscle is changed from 8.4 to 1.4mm. If a larger resection is carried out, it is possible that the muscle loses its tendon (Tl = 0) and in such a case, the remaining shortening has to be applied to the muscle length parameter (L0), consequently both parameters have to be changed. CHAPTER 4. BIOMECHANICAL MODELLING 130

4.6 Kinematics

Using the orbital geometry from Sec. 4.4.3 and the muscle force prediction from Sec. 4.5, the kinematics of the biomechanical model can be defined. The goal of the kinematic model is to relate geometry and muscle force prediction in such a way that eye positions and innervations can be derived from given parameterizations of one eye. Thus, the kinematic model is responsible for the correct transformation between muscle simulation and geometrical representation. Starting point for the derivation of the kinematic model is the complete definition of data for one eye. This reference data set is given by a geometrical model and the muscle force prediction model for all 6 eye muscles together with initial values for all parameters. Then, two essential operations that conform to forward and inverse kinematics, need to be defined.

Forward Kinematics deﬁnes the operation that resolves an eye position based upon a given set of innervations for all eye muscles.

Inverse Kinematics deﬁnes the operation that resolves a set of innervations for all eye muscles that are required to drive the eye into a given position.

To solve the kinematic operations, a mathematical deﬁnition of a stable eye position is required. Such a stable eye position is only reached, when all forces that act on the globe are in an equilibrium. To measure if an equilibrium is reached, a force balance equation is deﬁned, that derives a torque imbalance vector based on the current eye position and muscle forces. The torque imbalance vector ~t can therefore be derived using,

6 X ~t = FT i(dli, ivi, esi, csi, tsi) · ~ni, (4.116) i=1 where FT i denotes the total output force of a muscle i based on Eqn. 4.115, and the vector ~ni denotes the unit moment vector from Eqn. 4.77. A stable eye position can therefore be deﬁned as the constraint,

~t ≈ 0. (4.117)

It is however not enough to just balance force of the eye muscles to describe a stable eye position. Additional moments that arise from orbital restoring forces and globe translation that alter forces and rotational behavior must also be considered.

4.6.1 Orbital Restoring Force

In addition to the torque that is exerted by the muscles as the eye moves further away from primary position, it encounters resistance from non-muscular elastic tissues in the orbit such as Tenon’s Capsule which acts to restore the eye to primary position. This resistance can be deﬁned by a passive moment vector P~ (Pα,Pβ,Pγ), where the components of the vector correspond to angular coordinates (α, β, γ) based on the angular coordinates of the current eye position using Eqn. 4.9. In modelling the orbital restoring force, a simple nonlinear spring model of the form,

0 03 K(w, w ) = K1 · w + K2 · w . (4.118) CHAPTER 4. BIOMECHANICAL MODELLING 131 is used, where K1 and K2 denote the spring constants. The torsional restoring force that acts in the opposite direction of γ can therefore be denoted as,

0 3 Pγ = Kt · γ + Kt · γ . (4.119)

0 ◦ In Eqn. 4.119, Kt and Kt are the torsional stiﬀness constants, taken from [MR84], as Kt = 0.5g/ 0 ◦3 and Kt = 324µg/ . Similarly, the orbital restoring force acting in the opposite direction of α and β can also be formulated with respect to Eqn. 4.118, P α α α = K · + K0 · (α2 + β2) · , (4.120) Pβ β β where the spring constants K and K0 were also taken from [MR84], as K = 0.25g/ ◦ and K0 = 81µg/ ◦.

The orbital restoring torque vector P~ needs to be considered when balancing forces using Eqn. 4.116.

4.6.2 Globe Translation

Globe translation is the anteroposterior movement of the globe during eye movements (see 2.2.1). Usually the eye’s center of rotation remains fixed within the head-fixed coordinate system. How- ever, in pathological situations, globe translation can be of great value in diagnosis and treatment, and therefore it is of interest to the biomechanical model. When the globe translates during rotation, the center of rotation is shifted and thus, the axis of rotation is modified. Additionally, globe translation alters the relative position to the muscle origin and therefore also the stretch of each muscle, which in turn modifies output force and force balance in Eqn. 4.116.

In order to measure globe translation, a new coordinate system is defined that describes the orbital cone through the definition of an apex point that lies midway between the origin point of the superior rectus (Osr) and inferior rectus (Oir) muscles (see Fig. 4.24). Let {H~x, H~y, H~z} ~ ~ ~ Osr+Oir denote a head-fixed coordinate system, and let V = 2 , denote the apex point in the posterior region of the orbit. The new apex coordinate system can then be defined by three base vectors {A~x, A~y, A~z},

A~y = V,~

A~x = H~z × A~y

A~z = A~x × A~y, where the base vector A~y represents the vector from the origin to the apex point, the vector A~x is perpendicular to the head-ﬁxed vertical axis and the vector A~y and the vector A~z is perpendicular to A~x and A~y. Then, the rotation quaternions that transform between head-ﬁxed and apex coordinate systems can be formulated using two rotation angles ψ and ω, such that,

−1 ψ = cos (A~y · H~y), −1 ω = cos (A~z · H~z), CHAPTER 4. BIOMECHANICAL MODELLING 132

Figure 4.24: Apex Coordinate System for measuring Globe Translation

where ψ denotes the angle between the head-ﬁxed axis E~y and the apex axis A~y, and ω denotes the angle between the two vertical axes E~z and A~z. The forward transformation to the apex coordinate system can therefore be represented as combined rotation in the form of (4.32), around the head-ﬁxed axes H~x and H~z,

qapex = [ω, (H~x)] · [ψ, (H~z)]. (4.121)

The reverse transformation into the head fixed coordinate system can be formulated using the inverse rotation of (4.121), −1 qhead = qapex . (4.122) To measure globe translation, a relationship between the torque imbalance equation (4.116) and the amount of translation must be defined. This is reached by the introduction of a stiffness vector F~a = (54, 27, 54), that is linearly related to translation values. Thus, the translation vector Gtrans~ can be found by first transforming the torque imbalance vector ~t to the apex coordinate system, and then restrict the length of this vector using the stiffness values F~a, which leads to,

V ect(Rot(~t, qapex)) Gtrans~ = . (4.123) 2F~a

The amount of globe translation gt along the axis A~y for a given eye position can then be deﬁned as,

~ gt = Gtrans , (4.124) CHAPTER 4. BIOMECHANICAL MODELLING 133 and the translation that affects ocular geometry can be represented by a vector T rans~ that is defined with respect to the head-fixed coordinate system using,

T rans~ = V ect(Rot(Gtrans~ , qhead)). (4.125) This translation vector T rans~ can now be used to modify ocular geometry in order to reflect rotational changes. Therefore, the vector −T~ rans is used to translate the origin point O and the pulley point P of each muscle (see Sec. 4.4.3.2) in the opposite direction, prior to calculating the torque imbalance vector ~t, using Eqn. 4.116. Instead of translating the whole ocular geometry, the pulley and the origin point are translated in the opposite direction, which results in the same relative effect. In order to formulate this modification, the calculation of the muscle rotation axis within the ocular geometry is necessary. Therefore, Eqn. 4.53, Eqn. 4.67 and Eqn. 4.77 that define the muscle rotation axes in the string, tape and pulley model respectively, need to be modified, so that each rotation axis is calculated with respect to a displaced pulley and muscle origin such that, Gt(T rans,~ RA~ ) := RA~ → −T~ rans, (4.126) where RA~ denotes any muscle rotation axis from Eqn. 4.53, Eqn. 4.67 or Eqn. 4.77, and the operator → ensures that these rotation axes are calculated with modified pulley and origin data.

4.6.3 Balancing Forces

In order to find stable eye positions, it is necessary to refine the torque imbalance equation (4.116) to additionally consider orbital restoring forces and globe translation, since eye muscle are not the only actors that exert force onto the globe. This leads to the refined torque imbalance equation,

6 X T~ = P~ + FT i(dli, ivi, esi, csi, tsi) · Gt(trans,~ ~rai), (4.127) i=1 where the force of each eye muscle is multiplied by the unit moment vector which is calculated with translated origin and pulley points (Gt), and P~ denotes the orbital restoring force, that modifies the global rotational balance using Eqn. 4.119 and Eqn. 4.120. Each value for the torque imbalance vector T~ in Eqn. 4.127 is mainly determined by six innervations (iv~ ) and six length changes (dl~ ). Each modification to the current eye position results in different length change values and thus also in a different torque imbalance vector. Conversely, changing innervations give different force values and therefore also affect torque balance. It can therefore be seen that different eye positions lead to different dl values by applying Eqn. 4.95 and that different innervation values lead to different values for e, using Eqn. 4.97. Thus, an adequate way to control the value of T~ is to define an innervation vector I~v and an eye position vector E~p such that,

I~v = {iv1, iv2, iv3, iv4, iv5, iv6}, (4.128)

E~p = {ex, ey, ez}, where the values {iv1, iv2, . . . , iv6} contain the innervation values for each eye muscle and the values {ex, ey, ez} contains a rotation vector that describes an eye position based on a rotation quaternion of the form (4.32), 1 E~ = · tan(θ) · U,~ (4.129) p 2 CHAPTER 4. BIOMECHANICAL MODELLING 134 where the orientation vector U~ is parallel to the axis of rotation and the length is given by half the tangens of the rotation angle in radians. In order to transform an eye position E~p into a 6-vector of dl values, passive length changes need to be calculated, based on E~p by applying the method described in Sec. 4.4.4.

A rotation vector E~p of the form (4.129) can be transformed into a quaternion using,

−1 −1 Q(E~p) = [cos(tan (|E~p|)), sin(tan (E~p))], (4.130) and,

R(q) = tan(cos−1(Scal(q)) · V ect(q), (4.131) gives the rotation vector from the quaternion q using Eqn. 4.12.

However, before the minimization can be applied, the degrees of freedom for the innervational dependency of the objective function needs to be reduced, since the muscles are always working in pairs according to Sherrington’s law described in Sec. 2.2.3.8, Sec. 2.2.1 and Sec. 3.5.2. Assuming that the innervation vector Iv has the following form,

~0 Iv = {ivRL, ivRM , ivRS, ivRI , ivOS, ivOI }, (4.132) where agonist muscle innervations correspond to odd, and antagonistic muscle innervations correspond to even vector indices, then, only agonist muscle innervations need to be speciﬁed, since the antagonist muscle innervations can be found by using Sherrington’s law of reciprocal innervations. Thus, the degrees of freedom are reduced from 6 to 3, where antagonist muscle innervations are dictated by innervations speciﬁed for agonist muscles.

~0 Sherrington’s law of reciprocal innervations can be applied over Iv by, (h + w)2 Ive(Ivo) = − w, (4.133) Ivo + w

~0 where Ive and Ivo are 3-vectors of even and odd innervations from Iv respectively, and w = 7.6, h = 14.3 are constants taken from [Rob75b] and ﬁt experimental data. Thus, a new innervation vector I~v can be deﬁned as, ~ Iv = {Ivo(1),Ive(1),Ivo(2),Ive(2),Ivo(3),Ive(3)}, (4.134) resulting in a 6-vector of innervations that is determined by Ivo only.

Thus, a stable eye position depends on the parameter vectors I~v and E~p where the model parameters ix and ex dictate the model function deﬁned in Eqn. 4.127. This can easily be seen when looking at the muscle force prediction model in Sec. 4.5, where Eqn. 4.96 is a function where the parameters are non linearly dependent on the function values.

To quantify how well model parameter values in I~v and E~p approximate a stable eye position, Eqn. 4.127 can be used such that the squared length LT of the vector T~ gives information about kinematic instability, 2 6 ~ ~ ~ X ~ ~ LT (Iv, Ep) = P + FT i(dli, ivi, esi, csi, tsi) · Gt(T rans, RAi) , (4.135) i=1 CHAPTER 4. BIOMECHANICAL MODELLING 135

Figure 4.25: Torque Error Function for Listing Positions in a Healthy Eye

where the dli values are calculated from E~p using the equations from Sec. 4.4.4. When plotting 2D eye positions of the form (4.46), the error function (4.135) can be visualized.

Fig. 4.25 shows the torque error function for eye positions between −30 ◦ and 30 ◦ ab-/adduction and elevation/depression, where the torsional position is implicitly specified using Listing’s torsion from Eqn. 4.45. It can be seen that the minimum of this function also specifies the best approximation to a stable eye position, thus, the desired goal is a minimization of Eqn. 4.135 which can be reached by finding values for I~v and E~p such that,

minLT (I~v, E~p). (4.136)

The non linear manner of the error function (4.135) becomes evident when the ocular geometry is changed to some pathological situation. The curve that is shown in Fig. 4.26 depicts the torque imbalance function when the lateral rectus muscle has a pathological muscle path (i.e. the muscle insertion is transposed towards the superior rectus muscle).

In comparing Fig. 4.25 and Fig. 4.26, it can be seen that the shape of the curve changes drastically with different geometrical or muscle force prediction values. Thus, the procedure for finding the minimum in Eqn. 4.135 needs to be as stable as possible. Additionally, the structure of the mathematical formulation also guarantees that all components of the biomechanical model can independently be exchanged by other models without invalidating the model itself. It is therefore also desirable to use a stable minimization approach in order to provide flexibility with respect to the mathematical model of ocular geometry and muscle force prediction. CHAPTER 4. BIOMECHANICAL MODELLING 136

Figure 4.26: Torque Error Function for Pathological Eye

However, the minimization algorithm that is used to find a stable eye position within the torque imbalance equation is based upon a nonlinear least-squares problem, where Eqn. 4.135 can be used as objective function. The model is known to have the form LT (I~v, E~p) and the goal is to choose values for I~v or E~p that give the best fit in order to minimize LT . The nonlinear least- squares problem can be regarded as a general unconstrained minimization problem. In least squares problems, the objective function f has usually the form, m 1 X f(x) = r (x)2, (4.137) 2 j j=1 where each rj is a residual and x is a column vector of the free model parameters (x1, x2, ··· , xn). Using Eqn. 4.137, a residual vector r can be defined such that,

T r(x) = (r1(x), r2(x), ··· , rm(x)) , (4.138) where T transforms the column vector r(x) into a row-based vector. The ﬁrst derivatives of f(x) is denoted as the Jacobian matrix J of r,

 ∂r1 ∂r2 ··· ∂rm  ∂x1 ∂x1 ∂x1  . . .. .  J(x) =  . . . .  . (4.139) ∂r1 ∂r2 ··· ∂rm ∂xn ∂xn ∂xn

The gradient (∇f(x)) is given by, m X T ∇f(x) = rj(x)∇rj(x) = J(x) r(x). (4.140) j=1 CHAPTER 4. BIOMECHANICAL MODELLING 137

The second derivatives of the objective function can be computed in the form of the Hessian matrix and is described as,

m m 2 X T X 2 ∇ f(x) = ∇rj(x)∇rj(x) + rj(x)∇ rj(x) = (4.141) j=1 j=1 m T X 2 = J(x) J(x) + rj(x)∇ rj(x). j=1

The Jacobian matrix can be used to determine the direction of decent for the objective function, whereas the Hessian matrix gives information whether a current residual vector is a local minimum. The goal of minimization is to calculate the gradient in Eqn. 4.140 and the Hessian in Eqn. 4.141 and use this information to iteratively converge to a minimum of the objective function. Therefore, the first part of the Hessian is already known through Eqn. 4.139 as J(x)T J(x), and the second part of the Hessian needs to be calculated as the summation of the second partial derivatives and the residual values. However, the distinctive feature of least squares problems is that by knowing the Jacobian and therefore being able to compute the first part of the Hessian for free. Moreover, the term J(x)T J(x) in Eqn. 4.141 is often more important than the second summation term in Eqn. 4.141, either because of nearly linear compliance of the model near the 2 solution when ∇ rj is very small or because of small residuals rj. Most algorithms for minimizing non linear least squares using equations of the form (4.137) exploit these structural properties of the Hessian matrix. The simplest of these algorithmic methods is the Gauss-Newton method (cf. [NW99]) which excludes the second order term in Eqn. 4.141 and therefore approximates the Hessian matrix and uses a line search strategy to find a local minimum. The idea of a line search method is to use the direction of the chosen step, but to control the length, by solving a one-dimensional problem of minimizing,

φ(α) = f(αpk + xk), (4.142) where α is the step size and pk is the search direction chosen from the position xk. The property,

0 φ (α) = ∇f(αpk + xk)pk, gives the information that if the gradient can be computed, an eﬃcient one-dimensional search with derivative can be used. The search direction can be evaluated by solving,

T T Jk Jkpk = −Jk rk. (4.143)

Typically, an effective line search only looks towards α > 0 since a reasonable method should guarantee that the search direction is a descent direction, which can be expressed as φ0(α) < 0. It is typically not worth the effort to find an exact minimum of Eqn. 4.142, since the search direction is rarely exactly the right direction, thus it is enough to move closer. However, each iterative step in the minimization can be computed using,

xk+1 = xk + αkpk. (4.144)

Most rules for terminating the iterative solver are based on the residual,

2 rk = ∇ f(xk)pk + ∇f(xk), (4.145) CHAPTER 4. BIOMECHANICAL MODELLING 138

and evaluate the relative change of rk with respect to each iteration in the minimization procedure. Therefore, the iterative approach can be terminated by using,

|rk| ≤ ηk |∇f(xk)| , (4.146) where ηk could be chosen to be min(0.5, |∇f(xk)|), or any other problem dependent value within the interval [0; 1]. Additionally, a maximum number of iterations kmax is often used to limit non-converging situations.

Using the Gauss-Newton approach, if J(xk) is rank-deficient for some k, the coefficient matrix in Eqn. 4.143 will become singular. However, the system in Eqn. 4.143 will still have a solution, in fact, there are infinitely many solution for pk in this case. To overcome this weakness, the line search strategy of the Gauss-Newton method is replaced by a trust-region search strategy. This results in the realization of the Levenberg Marquardt method, which still uses the Hessian approximation in the same manner as the Gauss-Newton method, but improves convergence by using a more sophisticated directional search method. Line search and trust-region methods both generate steps with the help of a quadratic model of the objective function, but they use this model in different ways. The main difference is that line search methods use the model for generating a search direction, and subsequently focus on finding a suitable value for the step length α along this direction, whereas trust-region methods define a region around the current iterate within which they trust the model to be an adequate representation of the objective function. The step within trust-region methods is then evaluated as an approximate minimum within the defined region. In case a step is not acceptable, the size of the region is reduced. In general, the step direction changes whenever the size of the trust-region is altered. However, choosing the size of the trust region is crucial with resect to finding a good approximation for a local minimum. The model function for the trust-region approach can be denoted as, 1 m (p) = f + ∇f T p + pT B p, (4.147) k k k 2 k 2 where fk = f(xk), ∇fk = ∇fk(xk) and Bk = ∇ f(xk). The derivation of Eqn. 4.147 is identical to the first two terms of the Taylor-series expansion of f around xk due to the fact that, 1 f(x + p) = f(x) + ∇f(x)T p + pT ∇2f(x + tp)p, (4.148) 2 can be solved for some t ∈ [0; 1], and is used to study all the points in the immediate vicinity of f(xk) in order to identify a local minimum. In Eqn. 4.147, m(p) approximates the function values around the current iteration step, thus, the goal is to find a minimum for p ≤ ∆k, where ∆k is the trust-region radius. Consequently, to obtain each step, a solution of the subproblem, 1 min m (p) = f + ∇f T p + pT B p, (4.149) n k k k k p∈R 2 is required. Adapted to the current situation, the model function in Eqn. 4.147 can be rewritten as, 1 1 min m (p) = |r |2 + pT J T r + pT J T J p. (4.150) n k k k k k k p∈R 2 2 and the subproblem to be solved according to Eqn. 4.149 is stated as,

1 2 min |Jkp + rk| , where |p| ≤ ∆k. (4.151) p 2 CHAPTER 4. BIOMECHANICAL MODELLING 139

The solution to the subproblem (4.151) can be characterized with respect to comparing the step GN direction with a Gauss-Newton approach. When a Gauss-Newton solution (pk ) of Eqn. 4.143 GN GN lies strictly inside the trust-region (that is, pk < ∆k), then this step pk also solves the subproblem (4.151). Otherwise, there is a λ > 0 such that the solution p = pLM of Eqn. 4.151 satisﬁes |pk| = ∆k (cf. [NW99]) and,

T LM T (Jk Jk + λI)pk = −Jk rk. (4.152)

LM The vector pk is a solution to the trust-region problem (4.151) for some ∆k > 0 if and only if there is a scalar λ ≥ 0 such that,

T LM T (J J + λI)pk = −J r, LM λ(∆k − pk ) = 0.

Analytically, the minimization problem of (4.152) becomes,

T 2 LM T (Jk Jk + λDk)pk = −Jk rk, (4.153) and equivalently solves the linear least squares problem of the form, Jk rk min √ pk + , (4.154) pk λDk 0 where the matrix Dk is a diagonal matrix that is used to scale the step direction p and can change from iteration to iteration as new information about the typical range of values is gathered. The trust-region radius ∆k is adjusted between iterations according to the agreement between predicted and actual reduction in the objective function such that,

f(xk) − f(xk + pk) ρk = . (4.155) mk(0) − mk(pk)

For a step to be accepted, the ratio ρ must exceed a small positive number (typically 0.0001). If this test fails, the trust region is decreased and the step is recalculated. When the ratio is close to one, the trust region for the next iteration is expanded. The solution of Eqn. 4.154 can be found by applying matrix factorization methods (e.g. QR factorization) to solve a linear system of equations. The local convergence behavior of the Levenberg Marquardt method is similar to the Gauss-Newton method. Near a solution with small residuals, the model function (4.151) will give an accurate picture of the objective function (4.137). The trust region will eventually become inactive, and the algorithm will take unit Gauss-Newton steps, giving the rapid linear convergence rate that is provided by the Gauss-Newton method. Generally, with the determination of λ in Eqn. 4.154 the Levenberg Marquardt method changes is preference between a line search and trust-region strategy with respect to the parameter λ. However, the introduction of the trust- region approach for determining λ is just one possibility among many other strategies including the direct modiﬁcation of this value using heuristic approaches (see [NW99]).

The Levenberg Marquardt algorithm for minimizing a n-dimensional non-linear function can be formulated as follows: CHAPTER 4. BIOMECHANICAL MODELLING 140

First initial values for x0 need to be defined and the objective function (4.137) f(x0) is evaluated, the iteration counter k = 0, ∆0 = ∆¯ where ∆¯ is an overall bound on step lengths and λ = λ0. (1) Test for convergence: If the condition defined in Eqn. 4.146 is satisfied, or the maximum number of iteration is exceeded, the algorithm terminates with xk as the solution. (2) Solve trust-region subproblem This step solves the subproblem Eqn. 4.154 using a trust- region algorithm. According to Eqn. 4.154, a new vector pk is determined that gives the new search direction for the current iteration. The multiplier λk is set to zero if the minimum- GN norm Gauss-Newton step pk is smaller than ∆k, otherwise, λk is chosen such that,

∆k = |Dkpk| . (4.156)

(2.1) Evaluate ρk: Using Eqn. 4.155, the ratio ρk gives information on the alternation of the trust-region radius ∆k. (3) Improve step The next iteration point is updated by,

xk+1 = xk + pk. (4.157)

Then, k is increased by 1 and the procedure proceeds by starting again from (1).

A complete implementation and analysis of the Levenberg Marquardt algorithm can be found in [Fin96]. Details information on diﬀerent methods of the trust-region implementation, proofs and analysis of convergence can also be found in [PTVF97] and [NW99].

Recalling the torque imbalance equation (4.135), the Levenberg Marquardt method is used to either ﬁnd a stable eye position for constant innervations, or innervations for a constant eye position.

By using Eqn. 4.135 as objective function for the Levenberg Marquardt minimization method, stable eye positions can be approximated by minimizing the torque imbalance vector in that either I~v or E~p is set constant and Eqn. 4.135 is minimized over the free parameters. Thus, the solution to the minimization problem,

LM min LT (I~v, E~p), (4.158) gives a stable eye position in terms of balanced muscle and orbital forces.

4.6.3.1 Solving for Eye Positions

Eqn. 4.158 can be used to solve the forward kinematics in that a stable eye position is found for a given set of innervations. According to Eqn. 4.135, this results in setting Iv constant and minimize over Ep such that,

LM Epmin(E~p) = min LT (I~v, E~p), I~v constant. (4.159) E~p

The resulting eye position Epmin in the form of (4.129) can then be regarded as one solution that satisﬁes Eqn. 4.117 for a given constant set of 6 innervations I~v. CHAPTER 4. BIOMECHANICAL MODELLING 141

Figure 4.27: Torque Error Minimization in solving for Eye Positions

In Fig. 4.27, the optimization steps for a pathological situation was rendered. Constant innervations I~v that would hold a „normal “ healthy eye in primary position are given. The indicated minimum of this function corresponds to a modiﬁed primary position, due to the diﬀerent mechanical properties of the pathological situation that was modelled according to Fig. 4.26.

However, it must be noticed that each step in the minimization of Eqn. 4.159 modiﬁes the vector E~p in all 3 components, whereas Fig. 4.27 only shows ab-/adduction and elevation/depression axis for visualizing eye positions. Thus, the torque error function will change its shape within this 3-dimensional representation, whenever the torsional eye position (that is ey of E~p) is adjusted by the minimization procedure. This also explains, why the minimization steps that were rendered as vectors in Fig. 4.27 do not align with the shape of the function unless the minimum is reached.

4.6.3.2 Solving for Innervations

Using Eqn. 4.158 to solve for innervations is denoted as solving the inverse kinematics problem. In this case, a given eye position E~p will be constant in Eqn. 4.158, and innervations I~v need to be determined. Here, the odd innervation vector Ivo from Eqn. 4.133 is used in order to reduce the degrees of freedom, and all 6 innervations are regained by applying Eqn. 4.134. This leads to CHAPTER 4. BIOMECHANICAL MODELLING 142 a minimization of Eqn. 4.158 over I~v such that, LM Ivmin(I~v) = min LT (Ivo(I~v), E~p), E~p constant. (4.160) I~v

By using Eqn. 4.133, it is possible to reduce the inverse kinematics problem from 6 to 3 degrees of freedom, thus allowing the optimizer to converge to values for I~v that are confined to analytically formulated laws that are supposed to represent anatomical behavior. However, the goal of this procedure is to find meaningful values in terms of innervations that correspond to anatomical findings in that innervations of antagonist muscles are dictated by agonist muscle innervations.

4.7 Brainstem Simulation

According to Sec. 2.2.3.1, innervations of the oculomotor system originate in the brainstem nuclei which are in turn connected to supranuclear brain structures. In order to build a simple interface to controlling these structures in a simulation model, the mathematical representation of innervations will be used. The presented biomechanical model does not provide an independent simulation of the brainstem or the supranuclear structures. However, it provides a model of the distribution of innervations for each eye, which makes it possible to change the percentage of innervation of the oculomotor nuclei to the respective muscles. If such a distribution of innervation is changed, all innervations in relation to the aﬀected muscle for the aﬀected eye are scaled. The maximum allowable innervation of 250% corresponds to the maximum allowable stimulation which the nucleus shall be able to generate.

If the distribution of innervation is modified, it is possible to simulate either inter- or supranuclear gaze palsies or even retraction syndromes (see Sec. 3.5.2.2). For simulating a nuclear gaze palsy, changes to the distribution of innervation of the nervus abducens to the lateral rectus muscle on the right eye and to the nervus oculomotorius to the medial rectus muscle of the left eye are required. During an internuclear gaze palsy only one eye would be affected and therefore also only one eye would require changes to the distribution of innervation. Since the presented model uses a so-called invisible reference eye (see Sec. 4.7.1), which forms the basis for most of the calculations, and if the distribution of innervation of this reference eye is changed, only the following eye would reflect these changes in the Hess-Lancaster test. This enables for example the simulation of a lesion, which is only visible through the following eye alternately depending whether the left or the right eye is fixing.

The mathematical formulation for simulating brainstem control is straight forward, by supplying a distribution matrix to each eye that scales innervation prior to the application of the muscle force prediction model (see Sec. 4.5). Therefore, each eye is associated with a matrix Dn of the form,

RL RM RS RI OS OI   s11 s12 ··· s16 Abducens Oculo/RM (4.161)  . . .. .  Oculo/RS Dn = . . . . Oculo/RI , 0 ≤ sij ≤ 2.5,   Oculo/OS s61 s62 ··· s66 Oculo/OI where n denotes the corresponding eye (i.e. left, right or reference eye) and the matrix elements sij correspond to the scaling parameter of a oculomotor nerve i with respect to a muscle j. CHAPTER 4. BIOMECHANICAL MODELLING 143

Each row in the Matrix of Eqn. 4.161 has the form of (4.132) and each column is ordered by the respective oculomotor nucleus for each muscle (see Sec. 2.2.3.2). The initial innervation distribution matrix for an eye is chosen as the identity matrix Dn = I. In order to scale speciﬁc ~0 innervations for one eye, an innervation vector Iv of the form (4.132) is multiplied by Dn such that,

~ 0 T I~v = (Iv · Dn) , (4.162) where T denotes matrix transposition.

4.7.1 Simulation of Binocular Function

The forward and inverse kinematics solutions from Sec. 4.6.3.1 and Sec. 4.6.3.2 provide two essential operations to assemble a procedure that aims to simulate behavior of binocular eye movements. The presented system uses a model for simulating the binocular Hess-Lancaster test (see Sec. 3.13). The results of the test can be visualized in the form of Hess-diagrams (left eye and right eye fixing) and in the form of a text-based view (squint-angles diagram). The main difference between the two visualizations is the declaration of the deviations. While the Hess-diagram (see Sec. 3.13) offers some visual impression of the pathological situation and can be calculated for arbitrarily chosen gaze positions, the squint-angles diagram offers a text-based illustration of deviation values in the nine main gaze directions.

(a) Left Eye (b) Right Eye

Figure 4.28: Squint-Angles Diagram for Binocular Fixation

In the squint-angles diagram from Fig. 4.28, the deviations for each gaze direction for the left and the right eye ﬁxing are shown, whereby HD denotes horizontal deviation and VD denotes vertical deviation. In the white ﬁelds of the diagram, the torsional deviation is given as excyclo or incyclo (EX or IN) and the protrusion is denoted in mm (PR for negative protrusion = retraction). All values except for protrusion are given in degrees.

The signs for the horizontal deviations are interpreted as follows:

Positive HD: Towards the nose (adduction) Negative HD: Away from the nose (abduction) CHAPTER 4. BIOMECHANICAL MODELLING 144

The signs for the vertical deviations are defined differently for right eye and left eye fixing:

Right eye ﬁxing

Positive VD: Deviation downwards (depression)

Negative VD: Deviation upwards (elevation)

Left eye ﬁxing

Positive VD: Deviation upwards (elevation)

Negative VD: Deviation downwards (depression)

A gaze pattern is the basis for simulation of the Hess-Lancaster test. At the execution of the simulation, for every eye position in the gaze pattern, a fixation of the particular fixing eye is carried out. The other eye will then, by the description of the simulation, be regarded as following eye and will deviate from the fixing eye, according to some current pathological situation. The positions of fixation specified in the gaze pattern can be assigned to the blue points in a Hess- Lancaster diagram. The red points depict the resulting deviation points of the following eye (see Fig. 3.14).

Fig. 6.12 shows a schematic simulation task flow for the Hess-Lancaster test, which is performed for each gaze position of the fixing eye gaze pattern. The goal is the determination of the deviation of the following eye based on positions of the fixing eye. Thereby, the fixing and following eye can be exchanged, i.e. first the right eye is the fixing eye and the left eye the following and vice versa. This makes it possible to calculate the right resp. left fixation. For simplicity, let FP and FI denote two operations that find eye positions and innervation respectively of a specific eye, according to each eye’s specific parameters, using Eqn. 4.159 and Eqn. 4.160 such that,

FP F ix(I~v) = Epmin(I~v), for the ﬁxing eye,

FP F ol(I~v) = Epmin(I~v), for the following eye, (4.163)

FP Ref (I~v) = Epmin(I~v), for the reference eye, and,

FIF ix(E~p) = Ivmin(E~p), for the ﬁxing eye,

FIF ol(E~p) = Ivmin(E~p), for the following eye, (4.164)

FIRef (E~p) = Ivmin(E~p), for the reference eye.

Thus, if the left ﬁxing eye e.g. looks to an eye position A~(ax, ay, az) denoted in the form of (4.129), then,

B~ = FP F ol(FIF ix(A~)), will describe the position of the right, following eye that results from the innervations that are required to hold the left, ﬁxing eye in position A~. Note that the right eye will turn against the left eye in this case, since the coordinate systems of both eyes are not identical (see Sec. 2.2.1), and the innervations are applied to the right eye without any modiﬁcation. CHAPTER 4. BIOMECHANICAL MODELLING 145

Figure 4.29: Simulation Task Flow for the Hess-Lancaster Test

(A) Starting from a „healthy“ reference eye and a predetermined fixation position a complete 3D eye position is calculated. If a fixation position is specified, only the abduction/adduction and elevation/depression can be chosen freely. The specification of a torsional value is not possible, since a patient does not explicitly rotate the eye around a specified torsional angle on command. As a result, the complete position of the fixing eye has to be calculated by generating innervations for eye positions with listing torsion with the help of the reference eye until the position of the fixing eye matches with these innervations and the desired 2D eye position. This is similar to a minimization problem of the form (4.143) using the objective function,

I~vp = FIRef (P~ ), 2 ~ ~ ~ ~ ~ ~ f(P, Ivp) = (ZAng(FP F ix(P )),XAng(FP F ix(P ))) − (ZAng(P ),XAng(P )) ,

torsion = YAng(f(P,~ I~v)), (4.165)

where I~vp are starting innervations of the reference eye that correspond to the rotation vector P~ of the form (4.129) that describes the 2D eye position in that the torsional rotation CHAPTER 4. BIOMECHANICAL MODELLING 146

is always 0. f is the objective function that describes the length of the angular diﬀerence vector between reference eye and ﬁxing eye, where XAng,YAng and ZAng extract the radian angle from a rotation vector of the form (4.129). A valid 3D eye position is then found, when the innervations of the reference eye drive the following eye to a 2D position that matches P~ , and f(P,~ I~vp) ≈ 0 holds. The value for torsion will then be the torsional rotation that expands the 2D eye position P~ into a 3D eye position vector PF~ ick such that,

PF~ ick = (ZAng(P~ ),XAng(P~ ), torsion), (4.166)

where PF~ ick is a 3D eye position of the form (4.6), and the corresponding rotation quaternion qp can be computed by using Eqn. 4.48,

qp = qryqrxqrz, (4.167)

where qrz, qrx and qry correspond to the quaternions that describe rotation of angles ZAng(PF~ ick),XAng(PF~ ick) and YAng(PF~ ick) respectively around each coordinate axis.

(B) Using the ﬁxing eye parameters and the 3D eye position pq from (A), the innervations for all six eye muscles are calculated to bring the ﬁxing eye in the desired position using Eqn. 4.164,

T IvF~ ix = (FIF ix(R(pq)) · DF ix) , (4.168)

where R transforms the quaternion pq into a rotation vector using Eqn. 4.131, and DF ix scales the output innervation vector according to the innervation distribution matrix for the ﬁxing eye using Eqn. 4.161.

(C) These innervations are now passed into the reference eye in order to calculate its eye position. If the fixing eye is pathological, the eye position calculated with the reference eye differs from the previously calculated eye position. The result of this calculation is now the intended fixation position of the fixing eye F~p ix such that,

T PF~ ix = FP Ref ((IvF~ ix · DRef ) ), (4.169)

where the innervation vector IvF~ ix from (B) is scaled by the brainstem matrix DRef from Eqn. 4.161 with the addition that the columns for the medial rectus and the lateral rectus muscles are exchanged prior to multiplication using,

RM RL RS RI OS OI  D D ··· D  22 21 16 Abducens  D12 D11 ··· D16  Oculo/RM (4.170) D =   Oculo/RS , 0 ≤ D ≤ 2.5. Ref  . . .. .  Oculo/RI ij  . . . .  Oculo/OS Oculo/OI D62 D61 ··· D66

Note that the values D11,D22 and D21,D12 are additionally transposed within the matrix in order to reﬂect the downward order of the rows that correspond to the oculomotor nuclei.

(D) During this step, the intended ﬁxation position of the ﬁxing eye is mirrored in order to get the intended position of the following eye. Since innervations cannot be mirrored (the contralateral synergist of the right lateral rectus muscle is the left medial rectus muscle, which may not act in exactly the same way) this „indirection“ of mirroring the eye position CHAPTER 4. BIOMECHANICAL MODELLING 147

is necessary. Thus, the rotation vector PF~ ix from (C) is converted into mirrored Fick rotational angles PF~ ick such that,

0~ ~ ~ ~ PF ick = (−ZAng(PF ix),XAng(PF ix), −YAng(PF ix)), (4.171)

and the corresponding rotation quaternion qf can be computed by using Eqn. 4.48. The mirroring of an eye position is necessary, since abduction/adduction and torsion of both eyes are diﬀerent for describing the same direction (see Sec. 2.2.1).

(E) Now, innervations are determined, using the reference eye with the mirrored eye position qf so that it will be possible to calculate the position of the following eye. These innervations can be found by applying,

IvF~ ol = (FIRef (R(qf )). (4.172)

(F) The innervations of the reference eye IvF~ ol now determine the position of the following eye. This position is one of the red points in the Hess-diagram and can be found by using,

T P~F ol = FP F ol((IvF~ ol · DF ol) ), (4.173)

where DF ol denotes the innervation distribution matrix for the following eye. Chapter 5

Visualization of Muscle Action

Besides the path of a muscle, which can be calculated using the methods from Sec. 4.4, the most important information about a muscle is its volume and the distribution of its volume. In fact, the distribution of the volume changes with different muscle activations. In order to simulate the deformation process of a given muscle, the initial shape of the muscle at different activations needs to be acquired. To provide this information, a three-dimensional representation of MR images is used for calculating the basic parameters of the muscle. This data, constructed from a set of MR images, each representing a slice of the muscle, defines the surface and thus the volume of the model (see Sec. 2.1.4).

The goal of this chapter is to outline the image processing techniques that were used to reconstruct a three dimensional representation of the extraocular muscles and additionally make the resulting model deformable according to underlying MR data and biomechanical predictions. The basic data was acquired within a study [FPBK03], where MR images were captured in diﬀerent gaze positions in order to analyze the shape and deformation of the extraocular muscles (see Sec. 2.1.4). A 3D polygon model was built, and diﬀerent states of innervation of a muscle can interactively be visualized by interpolating between static reconstructions of MRI data.

The reconstruction procedure consists of two main parts. First, MRI data is segmented by defining selection polygons on each image slice. These selections are used to scan each slice and detect muscle boundaries in order to define a first basic polygon model. The second step involves smoothing and interpolation using spline approximation and data variance analysis in order to render different states of innervation. The resulting representation of a muscle is then incorporated into the biomechanical model and connected to the muscle force prediction model (see Sec. 4.5) in order to visualize model predictions in terms of muscle deformation.

148 CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 149

5.1 Image Analysis

One basic method used for constructing three-dimensional models is the marching cube algorithm. The idea of this algorithm is to generate a continuous surface by analyzing the relationship of neighboring points in space. Each point is classified as either inside or outside the volume. Surfaces are created between points, situated on opposite sides. This process fills an area in 3D-space with a large number of small cubes, determining which corners of the cubes lie inside or outside of the volume. According to a predefined set of rules, this information can be used to generate a continuous surface, which approximates the surface of the original volume with an accuracy inversely proportional to the size of the cubes. To achieve this goal, the marching cube algorithm creates a series of planes, intersecting MRI data. These planes are all aligned in parallel and are evenly spaced. Then, for each plane, the parts that lie inside and outside of the traced object are determined. In this case, this is done by assigning one MR image to each plane. Image pixels of a certain color are defined to be inside, while others are outside. The value of the points between the planes is assumed to be of the same value as the nearest pixel. The result of this step is a volume representation, where each pixel represents a small cube (i.e. a voxel), either (partially) inside or outside of the original object.

MR data from imaging device

picture setup

sort color table

define polygons

set threshold

generate polyhedron

define data volume

use scan-line/marching- cubes algorithms

generate surface

storage

set threshold

DXF file

Figure 5.1: DXF Model Generation Tasks using Marching Cubes, from [Sat03]

The overall process of image analysis is depicted in Fig. 5.1. Starting from raw MRI data, each MR image is modiﬁed by the „picture“ setup procedure, where the color index table is sorted, a segmentation polygon is deﬁned and a threshold for analyzing muscle boundaries is selected. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 150

5.1.1 Picture Setup

The color index table of each MR image ﬁle is used to reorder the color values, whereby individual pixels in the picture refer to the appropriate entry in the color index table. The color index table is encoded in RGB color scheme, but, due to the grayscale MR images, the respective RGB entries are always identical (A grayscale color 50 is expressed as RGB(50,50,50)). In MR images, usually not all of the 256 possible colors are used. Thus, the ﬁrst step in this process is to investigate the picture with respect to the used colors, mark these colors for the subsequent sort procedure and shift these entries to the end of the color index table. Additionally, unused color entries are zeroed after sorting. The sorting procedure then orders all used color entries according to their grayscale values (see Fig. 5.2). The changed color index table is than reapplied to the actual image in that each pixel reference value is exchanged in order to reference the same color as in the unsorted color index table.

Figure 5.2: Sorting of Color Index Table

This ensures, that the appearance of the picture is not aﬀected by this action. The reason for this preprocessing step is that color comparisons can be eﬃciently implemented within a sorted color index table, in that only the color index of the pixels need to be compared, since higher color index values directly correspond to higher grayscale colors. This comparison operation will be used within the next procedure to determine the boundaries of the muscle by thresholding the color values of the image.

The next step in the picture setup procedure is the deﬁnition of 2D polygons to surround the region of interest for each MRI slice.

In Fig. 5.3, the deﬁnition of a image segmentation polygon is shown. The region of interest (the superior rectus muscle in Fig. 5.3(a)) is surround by a manually deﬁned area in order to use a scanline approach to rasterize every point within this area (see Fig. 5.3(b)) until the complete area has been processes, as shown in Fig. 5.3(c). Within this rasterization process, the marching cubes algorithm, together with color threshold analysis is used to identify boundaries of the muscle within the region of interest.

In order to decide whether an image pixel lies inside or outside the muscle area, a threshold region is additionally defined for the color index table of each MRI slice. Fig. 5.4 depicts a threshold region that is defined within the 256 possible grayscale values and classifies an image pixel as part of the muscle when the pixel index value for the color table is within the interval [8; 134]. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 151

(a) Deﬁnition of 2D (b) Scanline Rasteri- (c) Rasterized Poly- Polygon zation gon

Figure 5.3: Deﬁnition of 2D Polygon for MRI Segmentation, from [Pri01]

Figure 5.4: Threshold Region for MRI Slice, from [Pri01]

Thus, each image of size SIZE(I) = m × n can be represented as a table of color index values,

I[i] = {p1, p2, ··· , pm×n}, (5.1) and an index function PI(x, y) can be deﬁned such that,

PI(x, y) = I[x + (y · n)], 0 ≤ x < m, 0 ≤ y < n. (5.2)

Additionally, an image stack I[i] is deﬁned, such that,

I[i] = {PI1, PI2, ··· , PIj} (5.3) where j MR images are collected, each associated with a 2D polygon deﬁnition and a threshold area such that,

P [i] = (p1, p2, ··· , pj),

T [i] = (t1, t2, ··· , tj), where P [i] holds the polygon deﬁnitions as lists of 2D points pi = (P1,P2, ··· ,Pk) with individual lengths ki, and T [i] stored the threshold regions in the form of a list of intervals ti = [ta; tb] where 0 ≤ ta ≤ tb ≤ 255. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 152

5.1.2 Generation of Polyhedron

In the second stage of the image analysis procedure, a polygon mesh is generated by applying the surface reconstruction method of marching cubes. Therefore, a series of MR images is incorporated into a volume data set, based on the deﬁnition from (5.3) such that,

V [x, y, z] = {PIz1(x, y), PIz2(x, y), ··· , PIj(x, y)}, (5.4) where the size of all images is reduced to the size of the largest rectangular boundary of all polygons P [i] for all MRI slices. The scanline algorithm (cf. [Pav94]) is used to generate V according to Fig. 5.3. This algorithm takes a list of the polygon edges P [i] and performs topological sorting so that the inner polygon area can be scanned along a horizontal line that is moved from the top most y-value of the polygon points to the bottom.

Figure 5.5: Marching Cube Traversal, from [BK03b]

Thus, each pixel value within the area of the polygon is visited once. The resulting volume V then has the following properties:

• The data volume is comparable to a cuboid that contains voxels (i.e. 3D pixels) that encompasses all pixels that are deﬁned by each polygon.

• Each image slice contained in V only contains pixels that are situated within the respective polygon area. Pixels that are outside the polygon edges are set inactive.

• All inactive pixels are assigned a color index value that is outside the respective threshold range (i.e. V [xo, yo, zo] ∈/ T [i], where (xo, yo, zo) represents a pixel that is outside a polygon). • All pixel values that are inside the polygon edges are transferred to V without any changes.

To determine the surface of the muscle, the algorithm assigns ﬂags to each possible group of eight neighboring pixels within two image slices to the corners of a cube (see Fig. 5.5). Every cube can be generated using,

Q(x, y, z) = (V [x, y, z],V [x + 1, y, z],V [x + 1, y + 1, z],V [x, y + 1, z], (5.5) V [x, y, z + 1],V [x + 1, y, z + 1],V [x + 1, y + 1, z + 1],V [x, y + 1, z + 1]), 1 ≤ x ≤ m − 2, 1 ≤ y ≤ n − 2, 1 ≤ z ≤ j − 2. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 153

Then, for each cube defined by (5.5), a classification function is defined that can determine if a cube corner is inside or outside the threshold region such that,

T rue, t ≤ c ≤ t , C(c, t) = a b (5.6) F alse, otherwise.

To generate the surface, the algorithm performs the following steps:

1. Determine the state of the corners from the pixel values.

2. Find a transformation T that transforms the actual cube, matching one of the 15 predeﬁned classes (see Fig. 5.6).

3. Build a set of triangles, representing the surface passing through this cube.

4. Transform the triangles using the inverse transformation T −1 from step 2.

5. Add triangles to the set that contains the already constructed surface.

Figure 5.6: Marching Cubes Standard Classes, from [BK03b]

In the first step, the corners of each marching cube are set such that, 0 = F alse when the corner of the cube is outside and 1 = T rue when the corresponding corner of the cube is inside the muscle surface according to Eqn. 5.6. Because the cube has eight corners and each corner CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 154 has to be in one of the two states, 28 = 256 possible cube configurations exist. Each of these configurations defines if and how the surface passes through the cube.

Using the cube corner values that deﬁne the points as inside or outside the muscle surface, it can be shown that by rotation and inversion of the corners, each of the diﬀerent cubes can be mapped to one out of standard classes (cf. [LC87]), as shown in Fig. 5.6.

The implementation of the marching cubes algorithm performs additional optimization to produce a smoother surface. It is not only classifying the corners of a cube as inside or outside, but also determines how far inside or outside each corner is by evaluating the specific color value of each pixel. Once a specific standard configuration is identified according to Fig. 5.6, the extension of the triangle along each edge of the cube can be calculated by linear interpolation of the color values. Let A and B be two corner points of a marching cube, ca and cb their respective color values and T [i] = [ta; tb] the respective threshold interval for a specific slice i. The marching cube standard classes from Fig. 5.6 define triangle corners between cube edges where one cube corner lies inside the muscle area (V [xo, yo, zo] ∈ T [i]) and one cube corner lies outside the muscle area (V [xo, yo, zo] ∈/ T [i]). Then, the distance d of one corner point of the triangle within the cube can be calculated using,

 t−cb 1 − , ca > cb,  ca−cb t−ca d = c −c , ca < cb, (5.7)  b a 0.5, ca = cb, where t is the lower or upper bound of the threshold interval such that,

t , c ≤ a ≤ c , c ≤ c , t = a a b a b (5.8) tb, ca ≤ b ≤ cb, ca ≤ cb.

After performing these steps for all cubes in the area of interest, the result is a set of triangles, describing the surface of the object. It is obvious from the above layout, that for j images of size m × n pixels, the asymptotic runtime of this algorithm is O(j × m × n). It is therefore desirable to keep the images as small as possible.

Pixels with color values near a given threshold are assumed to be nearer to the surface then pixel values far from the threshold. The algorithm is then moving the corner points of the surface parts inside or outside, depending on the ratio of the distances of the corners of the cube. This is especially important when the area of the muscle cross section shrinks rapidly from one MR image to the next.

The resulting list of triangles that is generated by the marching cubes algorithm can be deﬁned as,

L = {TR1, TR2, ··· , TRn}, (5.9)

TRi = {X1,X2,X3}, where L denotes a list of n triangles TRi, each consisting of a list of 3 points X1 to X3. The resulting resolution of the triangle mesh depends on the size of the cubes where image pixels that would lie between two MR images are linearly interpolated. Output data of this process is represented using DXF ﬁles, storing each triangle as single primitive within the ﬁle. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 155

5.2 Surface Reconstruction

Based on the output of the first stage of the reconstruction process, described in Sec. 5.1, the polygon model is analyzed and interpolation parameters are defined. The goal of the surface reconstruction process is to define a mathematical interpolation model based on the data from Sec. 5.1.

While output data from the marching cubes algorithm is interchangeable and easy to generate, it is very insuﬃcient in terms of memory usage and rendering speed. Moreover, this data structure does not contain any connectivity or volume information. Since the last two points are very important, input data needs to be transformed into a more suitable representation. In Fig. 5.7, an overview of the main processing steps is depicted.

DXF file from MR analysis

calculation of gravity line

import DXF

transform DXF format

calculate area centroids

spline interpolation

calculate volume and display surface

interpolation between the two EOM models

Figure 5.7: Surface Reconstruction Tasks, from [Sat03]

The first step is to find connectivity information. For this purpose a simplified boundary representation method is used. Because the surface is built from triangles only, there is no need to store edge information but only triangles and corners. In order to build a usable data structure, the input file is transformed into a list of triangles. Corner points of two or more triangles, which are within a certain limit of spatial proximity, are welded together. This replaces all points that are to be welded with a new point, which is the geometric average of these points. It can be shown that, because of the way the marching cube algorithm works, points are either equal to 1 each other, or have a spatial distance which is higher or equal to 2 σ where σ is the smallest distance between two pixels in the stack of images used. Due to this, as long as the threshold for 1 the welding is smaller than 2 σ, the resulting set of triangles will describe the same (intended) volume as the triangles in the DXF file.

The boundary representation is then transformed into an indexed triangle list. In this form, connectivity information can still be gained from the list, but at higher cost. However, this representation is ideal for current rendering hardware. This list eliminates the references from CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 156 the points to the triangles. Neighboring triangles can still be identiﬁed by searching through the list of triangles and looking for references to the same corner points.

Based on the deﬁnition from Eqn. 5.9, the data structure for the list of triangles is modiﬁed such that,

L = {(p1, q1, r1), (p2, q2, r2), ··· , (pn, qn, rn)}, (5.10)

PTS = {PT 1, PT 2, ··· , PT m},

PT i = (xp, yp, zp), where L denotes a list of n triangles where each list element contains 3 indices p, q, r into a vertex list PTS. The vertex list holds all corner points PT i, where each point contains the coordinates (xp, yp, zp). Because the model can be stored using a small amount of memory, and a linear traversal over the primitives can be used to display them all, this method of storing geometry is widely used in today’s rendering applications. When the input data has been transformed into a data structure that can eﬃciently be rendered, restoration of the volume information is still needed. Before de- ciding on how to do this, the transformation of the volume should be taken into account. Fig. 5.7 gives a brief overview of the generation tasks involved in constructing the surface representation model.

Given two volume models of the same muscle in two different activation states, the actual goal is to generate a new model in an intermediary state, by interpolating between these states. The two surface representations do however not contain any topological information. The positions of the triangles of one surface model are not connected in any way with the position of the triangles on the other surface model. To overcome this flaw, the final model needs to contain topological data as well as volume information. The following information can be identified as vital for interpolation:

• position and orientation of the muscle in space, • length of the muscle, • activation of the muscle, • volume of the muscle.

It is important to distinguish muscle movement from muscle volume deformation in a set of input models. This distinction is especially important since the interpolation of position and volume works in an entirely diﬀerent way. Activation cannot be extracted from the input model itself but has to be supplied from some external source (e.g. from the physician who acquired the MR data). The model does not only contain the current volume of a muscle, but also information on how the volume is distributed along the muscle.

5.2.1 Calculation of the Muscle Path

Based on the data deﬁnition from Eqn. 5.10, the approximation of the muscle path (see Fig. 5.28) can be calculated. This can be done by deﬁning the area centroids on each image slice using all CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 157 triangle corner points that belong to one MR image and were generated by the marching cubes algorithm.

Let P˜(d) be the function that describes the muscle path as approximated curve that connects all area centroids. To get the path, the muscle is divided into a series of slices along the z-axis. Because the images PIi(x, y) used for generating input data are coronal cuts and the image axes x and y are mapped to x and y in 3D space respectively, the muscle always has its longest extents along the z-axis.

Figure 5.8: DXF Model with Area Centroids

A bounding parallelepiped is defined to exactly fit the muscle and to find the longest dimension of extension. The muscle is then cut into equally spaced slices along this dimension. The area centroid of each slice is calculated (see Fig. 5.8) and the resulting set of points is approximated by a series of Hermite splines. An area centroid is calculated such that,

n−1 1 X A = (x y − y y ), (5.11) 2 i i+1 i+1 i i=0 n−1 1 X c = (x + x )(x y − x y ), x nA i i+1 i i+1 i+1 i i=0 n−1 1 X c = (y + y )(x y − x y ), y nA i i+1 i i+1 i+1 i i=0 where A denotes the area deﬁned by the polygon of all n points (xi, yi) on one slice and (cx, cy) denotes the area centroid.

The algorithm is generating splines by using cubic regression on the surface points of each slice, starting with one spline and recursively splitting and adding splines until the whole set of splines satisfies a least mean square threshold. The identification of the slices is easy, as a cut can be performed every δ units, starting at −0.5 ∗ δ, where delta is the resolution of the marching cubes (e.g. δ = 1). This way, each slice contains exactly the points which were defined by one image in the marching cube algorithm. In the ideal case (i.e. the model contains no noise) these points form a ring whose borders are two star shaped polygons around the center of gravity. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 158

The best fitting spline is calculated using normal cubic regression. The control points are defined to be equally spaced along the spline. When Ci, i ∈ {0..n} are the area centroids and ui, i ∈ {0..n} define the corresponding locations along the hermite spline curve,

S(u) = au3 + bu2 + cu + d, (5.12) then the error ,

n X 2 = (S(ui) − Ci) , (5.13) i=0

needs to be minimized. After the regression, the average error n is compared to a previously deﬁned threshold. If the criterion is met, the process stops, otherwise the list of points is split into two halves and regression is performed on each one. This process is repeated recursively until the criterion is met on each small spline segment. The process can be proven to stop because when the amount of points for one spline segment is 4 or less, then the regression produces a spline where is zero. The complete curve is made continuous by setting the start and end points as well as the tangents of two consecutive splines equal. In Fig. 5.28 an example of the calculation of the muscle path is depicted, showing the interpolated curve P (d) with respect to the area centroids.

Figure 5.9: Approximation of the Muscle Path

Thus, the muscle path can be denoted as function of a set of connected Hermite splines,

3 P˜(d): R → R , (5.14) in order to represent position, orientation and length of the muscle, where the parameter d denotes the position along the path.

In fact, most of the defined slices consist of too few points to represent a muscle cross-section. This leads to falsely calculated area centroids, thus resulting in an incorrect muscle path. Hence, CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 159 the slices including too few points have to be identified and filled up with additional points to represent a better approximation of muscle cross-section. The reason for the existence of such muscle slices are artifacts in the MRI data and the arbitrarily chosen resolution when applying the marching cubes algorithm.

5.2.1.1 Analyzing Surface Distribution

The purpose of analyzing the distribution of the surface along the muscle path is to identify cross- sections which include either too few points or false points. Once identiﬁed, the cross-sections are corrected by using points from the cross-section nearest to the ones with not enough points.

By applying an analysis of variance such that each muscle slice is treated as group of values,

Xi = {l1, l2, ··· , ln}, (5.15) where each value li is deﬁned as the angle α that is enclosed by a vector from the area centroid to a surface point and a vector that is parallel to the x-axis of the muscle plane as shown in Fig. 5.10.

Figure 5.10: Angular Measurement of Surface Points

Then, the arithmetic mean of the angular distribution of the points around the area centroid is deﬁned as, Pn l x¯ = i=1 i . (5.16) n − 1

The variation with respect to the arithmetic mean can then be calculated using, Pn (l − x¯)2 s2 = i=1 i , (5.17) n − 1 2 where s is defined as variation of angular displacements li and, √ s v = · 100, (5.18) x¯ CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 160 is the variation coefficient that relates variation s to the arithmetic mean and provides a better comparison between different groups of data.

An example result of the complete identiﬁcation process is shown in Fig. 5.11, where all de- ﬁned cross-sections were analyzed and visualized in a diagram. The x-axis shows the number of arbitrary chosen cross-sections of one muscle and the y-axis shows the value assigned to each cross-section by means of analysis of variance. It can be seen that there are sections with low and high values, but only six of them reach a value which is higher than 80. These six sections with high values in variance are the cross-sections that consist of points which were derived directly from the MR images, all others sections lie between the MR images and were approximated by the marching cubes algorithm, thus not including enough information for further processing.

Hence, the diagram shows which cross-sections have to be edited by adding additional points in order to smoothen the calculation of the gravity line.

120

100

variance 40

0 1 5 9 13172125293337414549535761656973 number of cross-sections

Figure 5.11: Example of Analysis of Variance, from [Sat03]

The following steps are performed in order to smoothen the distribution of surface points around the muscle path:

• Iterate over the values received from the analysis of variance.

• If a value is below a certain threshold, start searching for a slice which has a better variance.

• Find the best variance by searching left and right, take the ﬁrst slice found which has correct points and the shortest distance to the faulty slice.

• Add surface points to the faulty slice in order to produce a better distribution around the area centroid.

Finally, after all slices are corrected, the muscle path P˜(d) is recalculated in order to reﬂect surface modiﬁcations. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 161

5.2.2 Approximation of Muscle Surface

After the complete path is established, a radius function is deﬁned that describes a NURBS (cf. [PT97]) interpolated curve that approximates all surface points around the muscle path within a muscle slice.

Before describing the interpolation itself, it is necessary to determine the input points over which the approximation should take place. All surface points of one cross-section are split according to their angular displacement from Eqn. 5.15 into two parts, those having an angle which is lower than 180 degrees and those whose angle is greater than 180 degrees (see Fig. 5.12). Now, the partitioned points of each cross-section have to be processed in order to contain only unique points. The reason why all points of one slice have to be diﬀerent from each other is that in some cases it is possible that a spline curve is created which has a cusp (visual discontinuity), although the resulting spline curve is always c1 continuous1.

All slices are processed by calculating a NURBS curve for each cross-section, so that this curve ﬁts exactly between the points used as control points. Thus a control point vector,

P c = {P c1, P c2, ··· , P cn}, (5.19) includes all surface points on one cross-section in order to approximate a curve. The NURBS approximation can be denoted as,

n P Ni,p(u)wiP ci i=0 C(u) = n , a ≤ u ≤ b, (5.20) P Ni,p(u)wi i=0 where the points P ci are the control points, wi are the weights that inﬂuence the control points and Ni,p are p-degree B-spline functions deﬁned recursively as,

1 : if u ≤ u < u N (u) = i i+1 i,0 0 : otherwise u − ui ui+p − u Ni,p(u) = Ni,p(u) + Ni+1,p−1(u), (5.21) ui+p ui+p+1 − ui+1 where U denotes a knot-vector,

U = {a, . . . , a, up+1, . . . , um−p−1, b, . . . , b}. (5.22) | {z } | {z } p+1 p+1 containing values for u at which pieces of the curve join continuously. The knot vector usually contains internal and external knots, in the form that the values a and b are repeatedly occurring p + 1 times at the beginning and at the end of the vector. Therefore, any value u for Eqn. 5.20 needs to be within a and b. The spline basis functions Ni,p only depend on the value of p and the values of the knot vector, where p is the order of the curve. Increasing the order p also increases the continuity and smoothness of the curve at the knots, but tends to move the curve away from its control points. Detailed information on NURBS and B-splines can be found in [PT97]. Within

1“c1 continuous” means that the ﬁrst derivative is continuous. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 162 the presented research work, partly based on an MRI study described in Sec. 2.1.4, a software system „Visu“ was implemented which is described in [Pri01], [Lac01] and [Sat03].

In using the NURBS approximation for the muscle surface around the muscle path, two continuous curves can be approximated for the upper and lower half of the muscle cross-section as shown in Fig. 5.12.

Figure 5.12: NURBS Approximation of Muscle Cross-Section, from [Sat03]

The next step is to improve the surface also in the length of the model. This has to be done, because the already calculated shapes of the cross-sections have slightly diﬀerent area centroids which result in a shift of the sections to each other and thus in a „bumpy“ surface.

5.2.2.1 Optimized Rendering

In order to display the muscle surface and to realize a correct mapping of a texture to the surface of the muscle, it is vital that all spline points have a constant distance to each other. This leads to point locations of each cross-section corresponding to points of a consecutive cross-section by lying in the same plane. This property is, however, not realized after the NURBS interpolation. The previously calculated splines have to be processed in order to obtain equally spaced points.

Since for all points of a spline the angle α from Eqn. 5.15 can be calculated, the idea is to have a vector P~ = P (i) that samples spline points along a circular path, centered at the area centroid. According to Fig. 5.13, the area centroid of a muscle slice is denoted as M and the vector P~ can be deﬁned as,

cos(θ · i) P (i) = M~ + , (5.23) sin(θ · i)

360 ◦ where θ = m and m is the number of points to be generated. Consequently, i needs to be within the interval [0; m].

The angle θ · i can be used to find points of the spline curve that fit the criteria β ≤ θ · i < γ, where β and γ define two vectors V~1 and V~2 that point to two consecutive points of the spline curve. The intermediate spline point Xi can then be calculated as the intersection point between CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 163

0 0 P~ and V~12 = V~2 − V~1, where P~ is deﬁned as elongated vector P~ such that,

0 P~ = P~ · (|V~1| + |M~ |). (5.24)

Thus, the intersection point can be found by solving,

0 X~ i = M~ + s · P~ (5.25)

X~ i = V~1 + t · V~12, for the free variables s and t, resulting in,

M~ + s ∗ P~ − V~ t = y y 1y (5.26) V~12y M~ ∗ V~ − M~ ∗ V~ + V~ ∗ V~ − V~ ∗ V~ s = x 12y y 12x 12x 1y 12y 1x . V~12x ∗ P~y − V~12y ∗ P~x

The resulting vectors X~ i (drawn in dashed lines in Figure 5.13) now deﬁne equally spaced, spline approximated muscle surface points.

Figure 5.13: Linear Interpolation of NURBS-generated Cross-Section, from [Sat03] CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 164

By using the same spline interpolation again on the sum of all spline interpolated cross-sections, a homogenous surface without disturbances is obtained.

5.2.3 Interpolation of Muscle Models

Once all of the input models are processed, they are assigned values for their activation according to Sec. 2.1.4. Currently this assignment has to be set manually, as there is no direct way to identify the activation of a muscle based on pure MRI data. Then, the complete set of input models, along with their activation values can be used to interpolate the muscle at any given activation in between.

Both activation and length change whenever interpolation is applied. Due to activation, the length of a muscle changes. Passive length change is imposed on all other muscles when one is activated, because of the mechanical properties of the eye. Eye muscles are always working in pairs, when one muscle contracts, the other muscle is extended and vice versa (see Sec. 2.2.1). Both, activation and length change of a muscle can be obtained from the biomechanical model (see Sec. 4.4.4 and Sec. 4.5). The length change and the activation can be handled as separate problems. First the muscle is interpolated from the input images and the length change is disregarded. Afterwards the muscle length is adapted and the muscle is scaled with the constraint to keep its volume constant. The interpolation is done purely on parameter basis.

The process of interpolating the muscle volume without length changes can be split into the following steps:

1. Identify all input models used for interpolation. 2. Interpolate the muscle path. 3. Interpolate all muscle surface points on each muscle cross-section.

The first step consists of defining weighting factors for the different input models, depending on their activation relative to the a desired activation A. Let wM be the weight associated with an input model M. Interpolation of nearest neighbors is applied, so the process starts by finding two models M and N with activations AM and AN respectively, for which AM ≤ A ≤ AN and there is no model O for which AM < AO < AN . Then a weight is assigned to each of the two neighbors that depends inversely linear on the distance from the desired activation. The weights are defined to be, |AM − A| wM = 1 − (5.27) |AM − AN | |AN − A| wN = 1 − . |AM − AN |

The next step is to interpolate the muscle path. Linear interpolation is used for the path. So if P M (d) is the muscle path of model M, according to Eqn. 5.14, and P N (d) is the path of model N, then the resulting interpolated path P i(d) is can be denoted as,

P i(d) = P M (d) · wM + P N (d) · wN . (5.28) CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 165

Since two muscle paths may not have an equal number of spline segments, splitting of muscle paths will be necessary. For this, a splitting operation is deﬁned, such that it is possible to cut out a new spline from an existing one such that all points of the new spline are also points of the old spline. This conﬁguration can be achieved by recursively splitting larger splines into two halves when the corresponding part of the other spline set also consists of multiple splines (cf. [Lac01]).

The method for interpolating the muscle surface modiﬁes the distribution of the volume along the muscle path. The change of the distribution is given by the input models. Provided that all input models use the same resolution when deﬁning the muscle surface points on each cross- section (i.e. parameter m in see Eqn. 5.23), all surface point on a cross-section can directly be related to surface points within the same cross-section in another model.

From Eqn. 5.25 let,

XM = {XM 1, XM 2, ··· , XM m}, (5.29)

XN = {XN 1, XN 2, ··· , XN m}, be the surface points belonging to one cross-section of the model M and N respectively. Then, for a given activation A, according to Eqn. 5.27, the intermediate muscle surface distribution can be found by deﬁning a weighting factor w such that,

|A − A| w = M , (5.30) AM − AN and use this factor to scale the vector that connects source and destination surface points by using,

XIi = (XN i − XM i) · w, (5.31)

th where XM i and XN i denote the i surface points on a muscle cross-section and XIi gives the interpolated surface point according to the weighting factor w.

When repeating the conversion process described in Eqn. 5.31, a new list of surface points XI results that describes an interpolated distribution of the muscle surface according to a given activation and an interpolated muscle path from Eqn. 5.28. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 166

5.3 Reconstruction Results

In order to reach good matching results, high resolution MR images were taken from a study that was carried out within this research work, described in Sec. 2.1.4. Therefore, DICOM output images from the MR scanner were used directly for the reconstruction of the muscle model. The reconstructed model was rendered using the OpenGL graphics platform. Each of the following screenshots shows one MR image representing one cross-section of a muscle and is overlapped with the model at the corresponding location.

In Fig. 5.14, a medial rectus muscle of a left eye was reconstructed, while the patient was looking in primary position. Underlying MRI data was transparently visualized in order to evaluate accuracy of the algorithm. The wireframe muscle shown in Fig. 5.14 is the result of the NURBS approximation described in Sec. 5.2.2.

Figure 5.14: Reconstruction of a Left Medial Rectus Muscle

Additionally, texturing was applied to the wireframe model in order to improve visual quality. Fig. 5.15 shows a shaded, textured representation of a medial rectus muscle in a left eye.

The screenshots from Fig. 5.16 show a medial rectus muscle in diﬀerent innervation states. Fig. 5.16(a) shows the muscle model with an innervation of 0, thus corresponding to an initial reconstruction when the patient was looking in primary position. The second static reconstruction shown in Fig. 5.16(c), which depicts the muscle model that was reconstructed from MRI data, when the subject was looking in secondary position. An arbitrary innervation of 1 is associated with Fig. 5.16(c). By using the interpolation methods described in this chapter, it is now possible to interpolate muscle surface between the two states shown in Fig. 5.16(a) and Fig. 5.16(c). Therefore, an innervation value of 0.5 is assumed and the resulting interpolated muscle model is shown in Fig. 5.16(b).

In the case depicted in Fig. 5.16(b), an innervation of 0.5 means that the interpolated muscle model shows a state that is exactly in the middle between the primary and the secondary positions. CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 167

Figure 5.15: Shaded Reconstruction of a Left Medial Rectus Muscle

(a) Primary Position (b) Interpolated Position (c) Secondary Position

Figure 5.16: Morphology of Reconstructed Medial Rectus Muscle

5.3.1 Validation

Validation of the resulting reconstructions of MRI data concentrates on the evaluation of reconstruction results with respect to the underlying MRI data. However, validation of the reconstruction process cannot be realized by simply comparing pictures. Instead, the extracted data has to be compared with real-life dimensions, which means to use data measured on within data that is provided by the MR scanner. A possible way is to compare muscle dimension measurements from the extracted data with muscle dimensions data that can be measured in the MR images.

The standard software Adobe Photoshop was used to measure the pixel dimensions of muscles in MR images and real dimensions in mm were calculated from DICOM parameters that are included with each MR image. Since the reconstructed muscle model is deﬁned in a world coordinate system with units in mm, the measured values from underlying MR data can directly be compared.

The measured lengths and heights from eleven coronal MRI cross-sections are listed in Tab. 5.1, whereby the height of the muscle was taken as the longest dimension of the muscle on the current CHAPTER 5. VISUALIZATION OF MUSCLE ACTION 168

Figure 5.17: Shaded Reconstruction of a Left Medial Rectus Muscle with MRI Data

MR image in vertical direction (the axis that develops vertically within the sagittal plane), and the width was taken as the longest extension of the muscle in horizontal direction (the axis that develops horizontally within the coronal plane).

Muscle Image Height Width Length Cross-Section MR Model MR Model MR Model 1 9.4 8.44 5.1 4.94 0.0 0.0 2 9.4 8.48 4.7 4.14 1.8 1.41 3 9.4 9.07 4.3 4.43 3.6 3.19 4 9.8 9.17 4.7 4.46 5.4 5.43 5 9.8 9.51 4.3 4.35 7.2 6.99 6 10.2 9.70 4.3 4.27 9.0 8.77 7 9.8 9.87 4.3 4.12 10.8 10.84 8 9.8 9.83 3.9 4.00 12.6 12.20 9 9.4 9.77 3.5 3.83 14.4 14.40 10 9.4 9.63 3.9 3.78 16.2 16.17 11 9.4 9.59 3.9 3.96 18.0 17.92

Table 5.1: Comparison of Medial Rectus Muscle Dimension Data in Millimeters, measured form MR Images and the Model

The inter-slice distance was given by DICOM parameters as 1.8 mm, thus the length of the muscle that can be displayed according to MRI data is 10 · 1.8mm = 18mm. Chapter 6

Software Design and Implementation

The biomechanical model as well as the eye muscle surface reconstruction described in Ch. 4 and Ch. 5 were implemented and incorporated into an interactive simulation software system „SEE++“. This system enables the simulation of pathological situations as well as the prediction of surgical interventions by means of graphical three-dimensional visualization of both eyes including the extraocular muscles.

This chapter gives a short overview of the software design model that was used to implement the biomechanical model. Additionally, the integration of the biomechanical model into a software system „SEE++“ is described. The software design model is structured in a way such that the biomechanical model is represented as standalone software component that can be integrated into diﬀerent applications. The software system „SEE++“ is therefore one possible application that uses the biomechanical model to simulate eye muscle pathologies and surgeries. Moreover, the biomechanical model contains exchangeable sub-models (i.e. geometry, muscle force prediction and kinematics) that provide ﬂexibility and scalability for future extensions.

The software system described in this chapter was implemented using the C++ programming language and the MFC class library framework. Computer-aided Software-Engineering tools (i.e. Rational Rose, Rational Uniﬁed Process) were used to build a design model using the Uniﬁed Modelling Language (UML) within an integrated round-trip engineering approach (cf. [Buc01]).

169 CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 170

The basic structure of the software system „SEE++“ that was implemented within this research work is depicted in Fig. 6.1. The research work described in this thesis concentrates on results of the development of a biomechanical model for the human eye within the „SEE-KID“ project, which subsequently can be split into the development of a mathematical model and a software system. From Fig. 6.1 it can be seen that the implementation of the biomechanical model is an autonomous part within the „SEE++“ software system. The structure of the software system itself, besides the biomechanical model, consists of a „SEE++“ and a GUI package. The „SEE++“ package acts as pure interface to the underlying biomechanical model, whereas the GUI package contains the application and the user interface.

Figure 6.1: Structure of the „SEE++“ Software System

Because of the strict division of the software system into biomechanical model and application components, the „SEE++“ system provides exchangeable structure such that any other biomechanical model could be incorporated as long as it conforms to the „SEE++“ interface package.

Several tools were used to manage the high complexity of the system design and implementation. For realizing the software design, Rational Rose 2002 was used (cf. [Cor02b]) and implementation is based on Microsoft Visual Studio .NET using the MFC (Microsoft Foundation Classes) framework (cf. [Cor02a]). The graphical three-dimensional visualization within the „SEE++“ software system was implemented using the OpenGL graphics library. The design is described in UML notation and consists mainly of class diagrams which are structured in packages.

However, Fig. 6.2 illustrates that the design of the „SEE++“ system is based on a slightly modiﬁed form of the Model-View-Controller (MVC) concept. The idea behind the MVC concept is to separate the model, the views and the controller to make the model as independent as possible. Moreover, the MVC concept enables the use of multiple views onto one shared model (cf. [GHJV94]).

In the „SEE++“ system, the See++ package is used by the instancer to provide an interface between the GUI package and the biomechanical model. Consequently, the GUI package only uses the classes of the See++ package and the See++ package only uses the instancer to access data and functionality of the biomechanical model (cf. Fig. 6.2). Since the „SEE++“ system uses the MVC concept, all data is stored within the model, thus the views and the controller do not CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 171

Figure 6.2: Model-View-Controller Structure of the „SEE++“ Software System store any data on their own.

One of the main goals of the „SEE++“ system is flexibility and reusability. Therefore, different design patterns have been used. The instancer uses the Singleton design pattern, which means that a global point of access is provided to the other classes and only one instance of the instancer class can exist at the same time. The instancer also uses the Proxy design pattern. Since most of the objects required during runtime are managed by the instancer and returned to the See++ package and the GUI package when needed, the usage of the Proxy design pattern has the consequence that an object is not allocated in memory until it is requested for the first time. The advantage of this procedure is that it reduces the start-up time and memory consumption of the system.

The reason, why the GUI package is placed between the view and the controller section in Fig. 6.2 is that the GUI package implements the user interface parts of the diﬀerent views of the „SEE++“ system, but also handles the mouse and keyboard input, which, according to the MVC concept, is realized by the controller. Nevertheless, the actual handling of the user input and the implementation of the views resides in the See++ package, since the GUI package only delegates these tasks.

6.1 Design of the Biomechanical Model

The structure of the biomechanical model should be easy to extend in order to permit integration of latest research results. As a consequence, the model design should be able to handle model extensions and modiﬁcations to existing parts of the model that do not completely invalidate current model predictions.

In order to satisfy these requirements, the following design guidelines need to be applied: CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 172

• Not every part of the model should be a direct sub-model of the biomechanical model. Instead, sub-models should use other sub-models, so that the biomechanical model uses only those parts of the model which are directly required for calculating model predictions.

• Sub-models which have a lot in common should not duplicate this shared functionality in each model. A better approach is to base these models on an abstract base model which contains all the common functionality, but cannot be instantiated on its own. As a result, the addition of new models is much easier, since only the model-speciﬁc parts have to be implemented in the new model.

• All model data should be stored once in order to avoid redundancies. The biomechanical model should therefore provide an abstract representation of anatomical parts.

The software design for the biomechanical model starts with ﬁnding proper abstractions for the medical relevant entities of the oculomotor structures. First, all muscles are modelled using an abstract base class that unites all common operations and additionally aggregates anatomical subparts like insertion, origin and pulley points (see Fig. 6.3).

Figure 6.3: Abstraction of Muscles

The class diagram shown in Fig. 6.3 treats each geometrical point of a muscle as own class and therefore provides diﬀerent operations when accessing insertion, origin or pulley of a muscle. This for example would mean that, when changing the radius of the bulbus, the muscle insertions and the pulleys are extended, whereas the origin points stays the same.

It is now desirable to incorporate these geometrical abstractions into a design for a complete biomechanical model, consisting of geometrical, muscle force and kinematic model according to Sec. 4.2. One possibility is illustrated in Fig. 6.4, which deﬁnes one superclass „Biomechanical Eye Model“ that references all sub-models for modelling geometry, muscle force and kinematics. Moreover, this superclass also directly accesses the six diﬀerent eye muscle classes and the bulbus class which hold all data for one eye.

The model structure shown in Fig. 6.4 does however lead to some implementation diﬃculties. Since the superclass „Biomechanical Eye Model“ directly references diﬀerent geometrical sub- models like String- or Tape model, common functionality for geometrical calculations cannot be incorporated into this design. When for example, an additional pulley model needs to be added, CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 173

Figure 6.4: Primary Abstractions for Biomechanical Model implementation redundancies will occur in that all basic functionality needs to be duplicated within a new geometrical model. Moreover, external usage of the „Biomechanical Eye Model“ depends on knowing which geometrical models are available and also needs to diﬀerentiate between diﬀerent access modes of these models. Identical considerations hold for the muscle force model and the kinematic model.

A conceptual description of a biomechanical model that is very similar to that shown in Fig. 6.4 is suggested in [MD99] and [Gue86]. However, the previously mentioned ﬂaws are also present in these models making them hard to modify and to extend.

In order to provide the necessary ﬂexibility and extensibility of the design model, new abstractions need to be incorporated into the structure from Fig. 6.4. This implies the introduction of abstract base classes for each of the sub-models for geometry, muscle force and kinematics. This way, superclasses need to access only one interface in order to be able to use diﬀerent sub-models within the same functional responsibilities.

Fig. 6.5 shows an extended version of the model structure, where the biomechanical model now uses only one sub-model, namely the kinematic model. The kinematic model itself uses only the muscle force model. The muscle force model is an abstract base class with currently one derived subclass. Derived models are concrete realizations of an abstract base model, where the abstract base class holds all common functionality. Finally, the muscle force model uses the geometrical model which is also an abstract base model. The geometrical model has three derived models, the String-, Tape- and Pulley model, and uses the models of the six extraocular eye muscles and the bulbus.

The structure for modelling the eye muscles, previously shown in Fig. 6.3 has also been extended in Fig. 6.5. All muscles are now derivations of one abstract base class and the diﬀerent parts of a muscle, like insertion, pulley and origin, are now parts of the abstract base model only, so they need not be added to each individual muscle.

The following properties can be derived from the system design in Fig. 6.5: CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 174

Figure 6.5: Extensible Software Design for the Biomechanical Model

• New sub-models can be added to the biomechanical model more easily. • Only parts aﬀected directly by additions have to be validated for their correctness once again. • Extensions to existing models only inﬂuence parts of the model and not the whole model.

According to the desired structure of a biomechanical model (see Sec. 4.2), the system design model exposes the „Biomechanical Eye Model„ class on the top of the class hierarchy. This class references the kinematic model which is responsible to connect geometry and muscle force model, but the geometrical model is only accessible through the muscle force model.

In the „SEE-KID“ model (cf. Fig. 6.5), the requirements for a speciﬁc geometrical model that are enforced by the abstract base class „Geometrical Model“ are the calculation of the following entities:

• The point of tangency of each extraocular eye muscle, • the rotation axis around which a speciﬁc muscle rotates the eye (muscle rotation axis), • the center of the muscle action circle, • and the length of a speciﬁc muscle.

The listed requirements are speciﬁc to each geometrical model. In the „string model“, for example, the center of the muscle action circle is also the center of the bulbus (cf. Sec. 4.8). The muscle CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 175 action circle is deﬁned by the muscle rotation axis and since the muscle rotation axis changes with nearly every movement of the bulbus, the muscle action circle reacts in the same way.

Apart from containing the bulbus and the muscles, the class „GeometricalModel“ also provides methods for specifying eye positions, performing eye rotations (using Fick or Helmholtz rotation sequences) and manipulating the properties of the contained bulbus and muscles. These functionalities are independent of a speciﬁc geometrical model and are therefore implemented in the class „GeometricalModel“. Thus, the derived models of „GeometricalModel“ do not have to reimplement this functionality.

Similar conditions can be applied to the muscle force model. The abstract base class „Muscle- Model“ defines common functionality to load and store muscle force tables. Additionally, methods for cubic and bicubic data interpolation are provided (cf. Sec. 4.5). Each specific muscle force prediction model then only implements functionality that is specific to a certain model.

For the kinematic model, the calculation of the forward and inverse kinematics are non-linear optimization problems and optimization methods have to be used (see Sec. 4.6.3). The „SEE- KID“ model contains two diﬀerent algorithms for solving kinematics. The Levenberg Marquardt algorithm, which uses gradients, and the Downhill Simplex algorithm, which only needs function evaluations to perform function minimization (cf. Sec. 4.6). Concerning convergence speed, the Levenberg Marquardt algorithm performs much faster since it uses gradients, whereas the downhill simplex method is slower but much more accurate in ﬁnding minimums.

Figure 6.6: Optimizer and Related Classes

In the „SEE-KID“ model, the object-oriented design shown in Fig. 6.6 enables the use of both algorithms (see Fig. 6.6). Therefore, an abstract base class called „Optimizer“ is introduced which provides a common interface for solving the non-linear optimization problems. The classes „LevenbergMarquardt“ and „DownhillSimplex“ are derived from the class „Optimizer“ and implement the respective algorithms. Moreover, the class „Optimizer“ uses the classes „ParamObject“ and „FunctionObject“. The class „ParamObject“ enables the parametrization of the algorithms, whereas the class „FunctionObject“ is an abstract base class used for deﬁning the objective function (cf. Eqn. 4.135). Thus, a derived class of „FunctionObject“ implements the speciﬁc function for the optimization and since the forward and inverse kinematics are the problems to be solved, the kinematic model is derived from „FunctionObject“. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 176

6.2 Design of the „SEE++“ Software System

According to Fig. 6.2, the design model for the „SEE++“ system consists of the GUI and the „SEE++“package. These packages hold functionality that provides the user interface and the interface to the biomechanical model, described in Sec. 6.1. The content of the „SEE++“ package is visualized in Fig. 6.7. The main functionality is contained within the packages „SeeMedic“ and „SeeModel“.

Figure 6.7: Structure of the „SEE++“ Package

The packages that are contained within the „SEE++“ package, according to Fig. 6.7 can be described as follows:

• The SeePoint package provides classes for the diﬀerent kinds of points like insertion points or points of tangency.

• The SeeGeneral package contains the base class SeeObject from which nearly all other classes of the See++ package are derived.

• The SeePatientData package is used for storing and loading data of patients.

• The SeeConstraint package describes limits which are applied to most of the user-controlled values in the GUI package.

• The SeeHelp package implements the context sensitive help.

• The SeeSurgery package provides diﬀerent surgery techniques in connection with classes of the SeeView package. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 177

• The SeeProjection package prepares data returned by the SeeModel package for use in the SeeView package.

The basic process of data flow within the „SEE++“ system starts with loading patient data (SeePatient package) and distributing these data into classes contained in the SeeMedic package. Next, simulations are performed using the biomechanical model through the interface of the SeeModel package, where modifications of patient specific data during simulation are checked for validity by classes that are contained in the SeeConstraint package. Simulation results are graphically visualized by functionality that is assigned to the SeeProjection and SeeView packages. Surgical interventions that define specific methods of how the patient data can be modified in order to simulate or correct pathological situations are realized within the SeeSurgery package.

Figure 6.8: Structure of the „SeeMedic“ Package

The SeeMedic and the SeeModel packages utilize the Adapter design pattern, which means that each of the classes of these packages adapts the interface of another class (cf. [GHJV94]). Thus, the SeeMedic package provides access to the six extraocular eye muscles and their diﬀerent components, although the classes of the SeeMedic package do not store any values (see Fig. 6.8). Instead, these values are retrieved from the class SeeDataModel, which is located in the SeeModel package (see Fig. 6.9). Since the SeeModel package also uses the Adapter design pattern, the SeeDataModel in turn retrieves the data from the biomechanical model through the instancer. The other classes in the SeeModel package are adapters of the diﬀerent models contained in the biomechanical model „SEE-KID“.

The class SeeMedic is the base class for all medical objects and the class SeeOrbita is used as a container for the six extraocular eye muscles and the bulbus. Moreover, for the four straight and CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 178 the two oblique muscles, two diﬀerent abstract base classes called SeeRectus and SeeObliquus have been introduced, both derived from SeeMuscle.

The approach of using the Adapter design pattern in the SeeMedic and the SeeModel packages may seem too much eﬀort. However, the advantage of this design is that the model stays independent from the views, since the views implemented in the SeeView package only use the classes of the SeeMedic package. On the other hand, the GUI package and the views are independent of the biomechanical model, which makes it quite easy to exchange either the biomechanical model or the GUI.

Figure 6.9: Structure of the „SeeModel“ Package

The class SeeModel is the base class for all models in the SeeModel package. The class SeeData- Model provides access to the data stored in the biomechanical model and the class SeeStateModel serializes the internal state of the See++ package and the GUI package between diﬀerent execu- tions of the „SEE++“ software system.

In order to visualize output data of the „SEE++“ software system, three diﬀerent types of diagrams (views) are supported. The muscle force distribution (MFD) diagram, the muscle force vector (MFV) diagram and the Hess diagram. The MFD and the MFV diagram represent diﬀer- ent visualizations of the output data of a geometrical model, whereas the Hess diagram is based on the output data of the biomechanical model (see Sec. 4.7.1). The implementation of these diagrams is not part of the „SEE-KID“ model, as they are implemented in the SeeView package, shown in Fig. 6.10, which is part of the See++ package.

Fig. 6.10 shows the SeeView package which contains the classes of the diﬀerent views that the „SEE++“ software system supports. All views are derived from the abstract base class SeeView and all diagrams are additionally derived from the abstract base class SeeDiagramView. The classes SeeVText and SeeV3D visualize the diﬀerent properties of the muscles and the bulbus, either in a textual or in a three-dimensional form. The three diagrams which are all derived from SeeDiagramView are visualizations of the output data of the biomechanical model.

For the calculation of the MFD diagram, implemented in the class SeeVMFDD, the muscle rotation axis of a specific muscle is used. It is also possible to calculate one MFD diagram for several muscles by adding up the diagrams for each muscle. Thus, the MFD diagram visualizes the force distribution of one or several muscles of one eye in a specific elevation plane (cf. Sec. 4.4.3.3). Each curve of the diagram shown in Fig. 4.19, represents a different rotational component of the visualized muscle(s) in an arbitrary but fixed elevation plane within a specific range.

The MFV diagram and the implementation of this diagram can be found in the class SeeVMFVD. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 179

Figure 6.10: Structure of the „SeeView“ Package

For the calculation of the MFV diagram, shown in Fig. 6.11, arbitrary gaze positions are chosen which are only constrained by anatomical boundaries. These gaze positions are drawn in a 2D- diagram where the x-axis represents adduction and abduction and the y-axis represents elevation and depression in a specific range. Each of the gaze positions is then used for drawing a vector, whereas the length and the direction of the vector are specified by the force distribution of a specific muscle along with its particular pulling direction. The MFV diagram, like the MFD diagram, also supports the visualization of several muscles by adding up the diagrams of each muscle.

Figure 6.11: Muscle Force Vector Diagram

Fig. 6.11 illustrates a MFV diagram of the lateral rectus muscle of a left eye with a set of gaze CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 180 positions within the 30 degree physiologic ﬁeld of vision. These gaze positions are considered as default gaze positions for a „normal“ human subject.

However, MFD and MFV diagrams themselves cannot be gathered through clinical measurements. For this reason the “SEE++„ system supports the Hess diagram which can also be measured clinically (cf. Sec. 3.13).

The Hess diagram is a visualization of combined output data of the kinematic model and is implemented in the class SeeVHess (see Fig. 6.10). The Hess diagram also depends on arbitrary gaze positions which must not exceed anatomic boundaries and are plotted in the diagram as points. These gaze positions are then used as an input for the calculation of the simulation (cf. Sec. 4.7.1). The axes of the Hess diagram are defined according to the MFV diagram, where the x-axis represents ab-/adduction and the y-axis represents elevation/depression within a specific range (see Fig. 3.14). A Hess diagram contains different points which represent the specified and the calculated gaze positions. To improve the visibility of the diagram, points of specified and calculated gaze positions are drawn using different colors and are connected through thin lines.

In addition to the described packages, diﬀerent helper classes for matrix calculations, quaternion algebra and OpenGL are used. For matrix calculations the „NewMat“ framework was used (see [Dav03]) and for quaternion and 3D calculations the „3DMath“ classes (see [Fal99]) were adapted.

6.3 The „SEE++“ Software System

The „SEE++“ software is a new simulation system that aims at the forecast of clinical operation results, as well as the representation of pathological situations in the ﬁeld of strabismus surgery. The system is based on a highly developed mathematical simulation model (biomechanical model), which copies the behavior of the human eye realistically and thus provides an experimental platform for the simulation of pathologies and the evaluation of possible treatments.

„SEE++“ is a biomechanical system for the interactive three-dimensional simulation and visualization of eye motility disorders and their surgical correction.

The „SEE++“ software system oﬀers the following functions:

• Compact, descriptive and thus well understandable knowledge transfer in teaching and training,

• scientiﬁcally oriented procedures for practice,

• fundamental references and numerous examples,

• a basis for individual considerations of diagnostics and operational correction of eye motility disorders.

The „SEE++“ software system aims the following target user group:

• Ophthalmologists, specialists as well as orthoptics can use „SEE++“ to support measurements and to archive pathologies and treatment methods. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 181

• Researchers in the field of ophthalmology, strabismology and neurology, pediatrists as well as researchers in the field of biophysics use „SEE++“ as an extensive scientific tool for the investigation of the mechanics of eye movements.

• „SEE++“ oﬀers a substantial support to teachers through descriptive representation of the fundamentals for understanding eye movements.

• Students have the possibility to interactively deepen their knowledge and to recall and describe previously studied fundamentals of eye movement and strabismus.

6.3.1 „SEE++“ Simulation Task Flow

When using „SEE++“ to model and simulate eye motility disorders, a certain simulation task flow can be defined in order to perform simulation and evaluation of model predictions. In Fig. 6.12, this task flow is illustrated as a sequence of simulation steps that can be performed iteratively within the software system.

Figure 6.12: Simulation Task Flow for using „SEE++“

1. Parametrization with patient data: One main goal of „SEE++“ is to give close-to-reality related predictions of a patient-specific situation. In this first step of the simulation task flow, model values will be based directly on the patient. Parameters can be modified such as globe radius, cornea radius, muscle lengths, insertions, tendons etc. At the same time, general data are entered like name or description.

2. Simulation of a pathology: During the simulation of a pathology, model parameters are changed in such a way that the resulting model predictions correspond to measured values of the patient, as closely as possible. A model prediction is done in „SEE++“ by the simulation of a clinical Hess-Lancaster test, whereby the representation for right or left fixation is used. By determination of the Hess-Lancaster data of the patient, these values can be compared to simulated data to find whether the simulation corresponds to the pathology of the patient. Also the 3D representation of the patient offers an additional support regarding the evaluation of a simulation. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 182

3. Comparison of simulation results with patient data: This comparison refers to the Hess-Lancaster investigation already mentioned, whereby the process of comparing can also serve as a verification of the diagnosis posed before. Thus, in this step it is to determine whether the simulation result agrees sufficiently with the measured patient data. This, at the same time, offers a basis for a later simulated treatment by interactive virtual surgery of the modelled pathology.

4. Simulation of surgery: Here the actual operation is simulated by interactively modifying different model parameters using the mouse within the 3D representation. Points of reference support orientation and the dosage of the surgery performed. Furthermore, different operation techniques are available such as transposition and tangential repositioning of a muscles insertion. Muscle force and innervation parameters are changed manually in the program so that they correspond to a comparable surgical procedure. For example, a muscle resection can be accomplished by changing the value of the parameter for muscle length. The 3D model visualizes these changes immediately after confirmation of entered values.

5. Evaluation of results: According to step 3, a comparison of the simulation results is carried out again. On the basis of the binocular Hess-Lancaster test, the outcome of a surgery can be judged regarding to the correction of a pathological situation, and whether it is still necessary to apply additional changes (simulation trials).

6. Simulation result: The simulation result represents the last condition of all model parameters in the task ﬂow of the simulation of a pathology with „SEE++“. The system enables the user to assign and archive scenarios to a patient. Thereby the results of diﬀerent simulations may be compared and e.g. simulation strategies can be developed. Each scenario stores any step of a treatment of a patient and can later be recalled in textual or graphic ways.

6.3.2 Simulation properties

When starting the program, the default view depicted in Fig. 6.13 appears. All functions of the program are accessible through the main menu in a structured manner. At the same time, a tree representation in the Treeview provides the same functions for direct access. The „SEE++“ system features four different diagram windows, where each diagram can be displayed arbitrarily within each view window. A fixed component of the system is the 3D view, in which the current simulation is illustrated by a „virtual patient“. The „Toolbar for 3D-View“ is used in order to configure this view, i.e. to show or hide muscles, the globe, points of reference, etc.. The „Toolbar for Main Functions“ allows quick and direct access to the most important features of the main menu or the Treeview, as well as it allows to switch the current model. Depending on the current position of the mouse cursor, additional information is displayed in the Status Bar at the bottom of the main window.

Medical data includes all data necessary for modelling and simulating a „virtual patient“. When referring to a patient ﬁle, patient data and interactions can be distinguished. Interactions map simulation and surgeries of the „virtual patient“ through appropriate choice of the simulation parameters. All values of these simulation parameters, which describe a pathological situation or simulate a surgical intervention, are centralized into medical data. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 183

Figure 6.13: Default View of the „SEE++“ Software System

Thus, Medical data includes data for left eye, right eye and a so-called „reference eye“.

6.3.2.1 Globe Data

Every eye simulated by „SEE++“ has its own globe data parameters. In this explanation, the left eye is used as an example. Of course, all details apply similarly to the right eye and the reference eye.

Figure 6.14: Globe Data Parameters of the „SEE++“ System

This dialog shown in Fig. 6.14 displays the current patient’s name if previously entered in the patient’s data. The values to be adjusted here are „Globe Radius“ and „Cornea Radius“, whereby preset default values depend on the geometric model selected. The globe affects the result of CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 184 the simulation fundamentally, since an enlarged or reduced radius leads to different forces of the muscles acting on the globe. When this value is modified, additionally all points of insertion and pulleys (functional origins) of all muscles are adapted automatically. Changing the cornea radius affects only the geometric shape of the globe, but not the result of the simulation.

6.3.2.2 Muscle Data

The muscle data dialog shown in Fig. 6.15 is one of the most important elements of the „SEE++“ system. Here, the muscle force model can be adapted to pathological conditions with regard to particular eye muscles. All data in this dialog inﬂuences the development of force of all or of certain muscles only. Consequently, muscle palsies, overactions or ﬁbroses can be simulated (cf. Sec. 3.5). Again, muscle data is present for all eyes simulated by „SEE++“ (left eye, right eye and reference eye).

Figure 6.15: Muscle Data Parameters of the „SEE++“ System

The muscle data dialog also offers the possibility to manually modify some geometric properties (i.e. origin, pulley and insertion). Modifying these parameters causes alteration of muscle path, and thus a different result of the simulation. All values are defined in primary position, even if the 3D view depicts a different eye position, the muscle dialog, in its geometric values, always refers to primary position.

Modiﬁcation of muscle force parameters applies to the muscle force curves (i.e. elastic, contractile and total force) situated on the right side of the dialog. These curves can be adjusted according to the simulation parameters deﬁned in Sec. 4.5. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 185

6.3.2.3 Distribution of Innervation

The dialog for modifying the distribution of innervation controls the activation potential of the muscles on the basis of stimuli generated by motor nuclei of the cranial nerves (cf. Sec. 2.2.3). Since this distribution is connected to every eye, it represents an abstraction from the actual anatomical structure (cf. Sec. 2.2.3). Diﬀerent nuclear or supranuclear lesions can be modelled by appropriate adaptation of the distribution of innervation for the left and/or the right eye. The distribution of innervation is available for all eyes simulated by „SEE++“ (left eye, right eye and reference eye).

6.3.2.4 Gaze Patterns

Additionally, a gaze pattern can be defined for the left and the right eye, each specifying fixation positions for the simulation of the Hess-Lancaster test (cf. Sec. 3.13). The dialog shown in Fig. 6.16 enables modification and storage of gaze patterns, as well as the possibility to manually enter measured patient values in order to compare simulation predictions with patient measured data.

Figure 6.16: Gaze Pattern Dialog of the „SEE++“ System

Changes concerning medical data are saved as interactions in scenarios that are assigned to a patient. They are saved along with the patient ﬁle as well. By using this scenario concept, simulation tasks can be retrieved at a later time without loosing the order in which they were performed. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 186

6.4 Evaluation

In the following sections, different examples for the simulation and correction of pathological cases are described. The first two examples show common basic cases of pathologies that affect only a single muscle. In further examples, simulation results are also compared to clinical measurements acquired pre- and postoperatively from patients that had surgery.

6.4.1 Abducens Palsy

In the ﬁrst example, an abducens palsy of the right eye will be simulated. Abducens palsy is an incomitant form of squinting, i.e. the squint angle increases in the main functional direction of the muscle concerned - namely the lateral rectus muscle - towards abduction. Clinically, a patient increasingly shows double images (uncrossed) towards abduction, thus in right gaze positions. Possible causes for an abducens palsy are, among other things, traumata of the peripheral nerve (that is the entire nerve without the nucleus) caused for example by a basal skull fracture. As a result, a damage to the nerve from the nucleus (in the brainstem) up to the insertion can take place. In case of a damage directly within the area of the nucleus, there is a possibility that also the interneurons (the connections between the nucleus of the abducens nerve and the conjugated muscle on the other side, the left medial rectus muscle in this case), which are situated in the neighborhood, are hurt. This case is not assumed in this example.

6.4.1.1 Simulation of the Pathology

As a consequence of a lesion to the nerve, the contractile strength of the muscle has to be reduced and furthermore its elastic parts have to be modiﬁed. This is the basis for the simulation. In the „SEE++“ software system, this force reduction is carried out in the muscle data dialog, shown in Fig. 6.17. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 187

Figure 6.17: Muscle Properties for Abducens Palsy Simulation

In the ﬁeld „Total Strength“ the strength of the muscle can be reduced by changing the value from 1(%/100) to 0.5(%/100). When the Hess-diagram calculation is displayed, the modiﬁcations from the muscle data dialog are immediately taken into account (see Fig. 6.18).

Figure 6.18: Hess-Lancaster Test for Abducens Palsy

In the calculated Hess-diagram shown in Fig. 6.18, for the left eye (right eye fixing) the exceeding CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 188 reaction of the following left eye into adduction can be seen. Conversely, in the second Hess- diagram the restriction of the right eye (left eye fixing) in abduction is evident. On the basis of the results of the simulation, which are represented by the Hess-diagrams in Fig. 6.18, the simulation of the abducens palsy can be considered as sufficient.

6.4.1.2 Simulation of Surgical Correction

For the surgical correction of the simulated abducens palsy, a strengthening of the paretic muscle is necessary. The surgery is carried out by shortening the affected lateral rectus muscle (resection). During a resection surgery, the insertion of a muscle is separated from the globe, a piece of the muscle is cut off or folded in and afterwards, the muscle is fixed again at the same position on the globe. For shortening a muscle in „SEE++“, the muscle data dialog is used again. The relaxed, denervated muscle length (L0) of the right lateral rectus muscle is reduced by 4 mm from 37.5 mm to 33.5 mm (see Fig. 6.19).

Figure 6.19: Simulation of Right Lateral Rectus Resection

Now the two Hess-diagrams in Fig. 6.20 show that the target of the surgical correction has been achieved, namely to get the double image-free zone into the primary position without substantially weakening adduction. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 189

Figure 6.20: Postoperative Hess-Lancaster Simulation for Abducens Palsy

However, a complete „healing“ of the palsy by modifying the innervational component in the model is clinically not possible, since a surgical modiﬁcation of innervation is impossible.

6.4.2 Superior Oblique Palsy

In the second example, a palsy of the superior oblique muscle of the right eye will be simulated. This palsy is, similar to the abducens palsy, an incomitant form of squint. The vertical deviation of the palsied eye increases towards adduction and depression, equally, extorsion increases in abduction. The horizontal component is aﬀected in the sense of a convergent deviation. Again, similar to the abducens palsy, a possible cause can be a basal skull fracture, since the trochlear nerve, like the abducens nerve, is vulnerable to traumatic injuries due to its length. Clinically, the patient usually takes an abnormal head posture (tilt to the left) for balancing extorsion and for maintenance of binocular vision. Vertically shifted and outward-tilted double images increase in depression and convergence (in the main functional range of the concerned muscle).

6.4.2.1 Simulation of the Pathology

Similar to the simulation of the abducens palsy (see Sec. 6.4.1), the contractile strength of the muscle has to be reduced and furthermore, its elastic parts have to be changed (see Fig. 6.21). CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 190

Figure 6.21: Muscle Properties for Superior Oblique Palsy

For successfully simulating a superior oblique palsy, the strength of the superior oblique muscle of the right eye has to be reduced. In the „SEE++“ software system, this force reduction is carried out in the muscle data dialog shown in Fig. 6.21.

In the ﬁeld „Total Strength“ the strength of the muscle can be reduced by changing the value from 1(%/100) to 0.3(%/100).

Figure 6.22: Hess-Lancaster Simulation of Superior Oblique Palsy CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 191

In the calculated Hess-diagram shown in Fig. 6.22, the exceeding reaction of the inferior rectus muscle of the left eye (right eye fixing) can be seen. In the second Hess-diagram, the increasing incomitant restriction of the superior oblique muscle of the right eye (left eye fixing) in adduction can be clearly seen. On the basis of the results of this simulation, the simulation of the superior oblique palsy can be considered as sufficient.

6.4.2.2 Simulation of Surgical Correction

For the surgical correction of the simulated superior oblique palsy, a strengthening of the paretic muscle is necessary. The surgery is carried out by shortening the aﬀected superior oblique muscle (resection). According to Sec. 6.4.1, the relaxed, denervated length (L0) of the muscle (superior oblique muscle) is reduced by 5 mm from 34.15 mm to 29.15 mm and the simulation results are shown in Fig. 6.23.

Figure 6.23: Postoperative Hess-Lancaster Simulation for Superior Oblique Palsy

Now, the two Hess-diagrams in Fig. 6.23 show that the target of the surgical correction has been achieved, namely to get the double image-free zone into the primary position. However, as explained in Sec. 6.4.1, a complete „healing“ of the palsy, even in extreme adduction and depression, is clinically not possible, since a surgical modiﬁcation of the innervation is impossible.

6.4.3 Superior Oblique Overaction

Vertically incomitant squinting is characterized as horizontal misalignment of the eyes in which the magnitude of the horizontal deviation diﬀers in upgaze when compared to downgaze. Com- monly, so called A-patterns and V-patterns are seen with respect to the Hess-Lancaster test. These patterns are named using letters of the alphabet whose shapes have visual similarities to the ocular motility patterns that they describe. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 192

(a) Increased Divergence in Downgaze (b) Increased Convergence in Upgaze

Figure 6.24: Classiﬁcation of Superior Oblique Overaction

The term A-pattern designates a vertically incomitant horizontal deviation, shown in Fig. 6.24, in which there is more convergence in midline upgaze (see Fig. 6.24(b)) and less convergence (increased divergence) in midline downgaze (see Fig. 6.24(a)). An A-pattern esotropia is an inward deviation of the visual axes in which there is more inward deviation of the eyes in midline upgaze than in midline downgaze. An A-pattern exotropia is an outward deviation of the visual axes in which there is more divergence of the eyes in midline downgaze than in midline upgaze.

With significant A-patterns, version testing usually reveals superior oblique muscle overaction. The tertiary abduction effect of the superior oblique muscle is believed to produce the A-pattern. The abducting force is greatest in downgaze within the superior oblique’s primary field of action, causing an increased relative divergence of the eyes in downgaze.

6.4.3.1 Simulation of the Pathology

In order to simulate superior oblique overaction, clinically measured patient data was used to compare predictions of the „SEE++“ software system when adjusting simulation parameters. The simulated pathology is shown in Fig. 6.25, where the measured patient data is displayed as green Hess-Lancaster diagram and the simulation results are shown in red.

In order to simulate the pathology, the path of the superior oblique muscle from the trochlea to the insertion was modiﬁed. In the „SEE++“ software system, this can be done by displacing the virtual pulley of the superior oblique muscle away from the nose. Thus, the pulley position of the superior oblique was changed from −15.270/11.000/11.750 to −13.250/11.000/11.750. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 193

Figure 6.25: Hess-Lancaster Simulation of Superior Oblique Overaction

In order to show an overacting superior oblique muscle, the total muscle strength was changed from 1.00(%/100) to 2.10(%/100), additionally the relaxed, denervated muscle length (L0) was reduced from 34.150 mm to 32.150 mm. Conversely, the total muscle strength of the inferior oblique muscle was reduced from 1.00(%/100) to 0.20(%/100) and the muscle length (L0) was changed from 30.55 mm to 34.55 mm. Thus, the inferior oblique muscle was lengthened, whereas the superior oblique muscle was shortened. The result of the simulation can be seen in Fig. 6.25.

6.4.3.2 Simulation of Surgical Correction

In order to correct the pathological situation shown in Fig. 6.25, a weakening of the concerned muscle needs to be carried out. Therefore, the right superior rectus muscle was transposed along its main direction of action, as shown in Fig. 6.26. This leads to a suﬃcient correction within horizontal gaze positions. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 194

(a) Recession and Transposition of the Superior (b) Final Position of the Superior Oblique Muscle Oblique Muscle after Surgery

Figure 6.26: Simulation of Superior Oblique Surgery

The complete surgical treatment is characterized by a recession of the right superior oblique muscle of 8.6 mm followed by a tangential transposition of 4.5 mm (see Fig. 6.26(a)). The resulting position of the superior oblique muscle can be seen in Fig. 6.26(b).

The simulation results of this operation are shown in Fig. 6.27.

Figure 6.27: Hess-Lancaster Simulation of Superior Oblique Surgery

It can be seen that the results of the simulation predictions correspond to postoperative treatment CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 195 prospects in that within the normal physiologic ﬁeld of gaze, binocular function has been restored to near normal conditions.

6.4.4 Heavy-Eye Syndrome

Motility disorders caused by a myopic globe with high axial length were reported in literature long ago. Donder reported in 1864 [Don64] frequent divergence in high myope patients whose elongated globes ﬁnd diﬃculty in orientation. The same phenomenon is described in a text book by Duke Elder (1968) [DE73] in patients with only one myopic eye. A deviation of this type was called the heavy eye syndrome by Ward (1967) (cf. [Kau95]). Such an eye is frequently limited in vertical excursions.

For this reason R. Hugonnier and Magnard (1960) suggested the designation „the nervous syndrome of high myopia“, which, because of the muscular aetiology, was changed to „myopic myosi- tis“ by Hugonnier (1965) [Hug65].

The term heavy eye syndrome is descriptive for hypotrophia. Anatomic studies showed an irregular path of the extraocular muscle from the origin through the orbit to the point of insertion. Kaufmann [Kau95] interpreted the heavy eye syndrome with a dislocation of the medial and lateral rectus muscles into a caudal position. These translocations should cause an overweight of downward rotating force. Furthermore, Kaufmann reports a shift of the superior and inferior rectus muscle in nasal direction with the result of adduction overweight. Detailed information was gathered using MRI analysis of myope patients.

Several publications analyzed the exact muscle path and quantified the translocation of the affected muscle. Krzizok [KKT97] reported a dislocation of the lateral rectus muscle into the temporocaudal quadrant by 3.4 mm based on MRI findings. Furthermore, Krzizok suggested a fixation of the dislocated muscle in the physiological meridian as causal therapy.

In a publication by Schroeder [SKT98], the dislocation of all four rectus muscles is quantiﬁed. Schroeder reported a dislocation of 2.9 mm of the lateral rectus muscle into the lower temporal quadrant, the path of the superior rectus muscle was altered 1.5 mm medially and the path of the inferior rectus muscle was shifted 1.3 mm medially, the medial rectus muscle was dislocated 1.3 mm downwards. Before these MRI studies, aetiology of the heavy eye syndrome was unclear.

(a) Lateral Rectus Muscle with Normal Path (b) Displacement of the Lateral Rectus Muscle into Temporocaudal Quadrant (Base Hypothesis for Hypotrophy in Heavy-Eye Syndrome)

Figure 6.28: Muscle Displacement as Hypothesis for Heavy-Eye Syndrome CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 196

Fig. 6.28 illustrates the eﬀect of dislocation of the pulley in temporocaudal position. The pulling direction of the muscle shown in Fig. 6.28(a) is shifted in the same way. Contraction of the muscle shown in Fig. 6.28(b) causes mainly abduction but also a downward movement. The biomechanical relevance of these dislocations with the result of downward movement of the eye was not analyzed up to now.

Dislocations of rectus muscles are reported in a range from 1.3 mm up to 3.4 mm. The operation is usually performed with permanent sutures or silicon loops. This technique is also known as „guide pulleys“, which aims to establish an artificial fixation and avoid a further side-slip of the muscle. These „guide pulleys“ fixate the muscles to the globe and therefore influence the „natural“ pulleys. MRI analysis was not able to clarify mechanical effects which may be responsible for the heavy eye syndrome.

6.4.4.1 Simulation of the Pathology

The primary aspect for the simulation of the Heavy-Eye Syndrome is the enlargement of the globe of the affected eye. Due to an oversized globe, the rotational effect of all muscles that act on the affected eye is much lower compared to a normal eye. Moreover, muscle tension increases, since the distance from the pulley to the insertion also extents, while muscle length stays constant.

For this simulation, a 48 years old patient with Heavy-Eye Syndrome was measured and compared with predictions of the „SEE++“ biomechanical simulation system. The measured values in Fig. 6.29 are shown in green color.

Figure 6.29: Measured Values from Patient with Heavy-Eye Syndrome

The ﬁrst step in the simulation is the modiﬁcation of the globe radius for both eyes according to ultrasonic globe measurements of the patient. Thus, the globe radius of the left eye was changed to 14.00 mm and the globe radius of the right eye was changed to 16.50 mm. The results of the simulation for these changes can be seen in Fig. 6.30. CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 197

Figure 6.30: Simulation Results for Resized Globes according to Patient Data

The second step is the simulation of the muscle dislocations in order to reproduce the Heavy-Eye Syndrome according to suggestions from Schroeder and Krzizok. By dislocating the pulleys of all four rectus muscles, the typical downward overaction of the aﬀected eye could not be reproduced in the „SEE++“ system, using the suggested approach, as illustrated by the simulation results in Fig. 6.31.

Figure 6.31: Simulation Attempt using Data suggested by Schroeder and Krzizok

In contrast to suggestions found in literature, biomechanical simulation of pulley dislocations using the „SEE++“ software system barely aﬀects horizontal gaze positions in the Heavy-Eye Syndrome. Therefore, the oblique muscles obviously contribute to this pathology due to superior oblique overaction and inferior oblique underaction which in turn inﬂuences horizontal gaze positions.

In order to simulate the Heavy-Eye Syndrome according to measured patient data, modiﬁcation of the oblique muscles was performed as shown in Fig. 6.34, in that the insertion of the superior CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 198 oblique was moved so that both oblique muscle form a steeper angle to the median-sagittal plane (compare Fig. 6.32(a) and Fig. 6.32(b)). Additionally, the oblique superior muscle was strengthened, whereas the inferior oblique muscle was weakened. This results in ampliﬁcation of the downward movement of the globe.

(a) Normal Insertion Location of (b) Transposed Superior Oblique the Superior Oblique Muscle Insertion

Figure 6.32: Superior Oblique Muscle Insertion Transposition in Heavy-Eye Simulation

The complete values for the simulation parameters that lead to the simulation results shown in Fig. 6.33 are denoted as follows:

• Globe radius of the right eye was changed from 11.90 mm to 16.50 mm.

• Globe radius of the left eye was changed from 11.90 mm to 14.00 mm.

• The right lateral rectus pulley was displaced 2.90 mm inferiorly.

• The right medial rectus pulley was displaced 1.30 mm inferiorly.

• The right superior rectus pulley was displaced 1.50 mm medially.

• The right inferior rectus pulley was displaced 1.30 mm medially.

• The right medial rectus contractile strength was changed from 1.00(%/100) to 2.00(%/100).

• The right lateral rectus contractile strength was changed from 1.00(%/100) to 0.50(%/100), the elastic strength was changed from 1.00(%/100) to 0.20(%/100).

• The right superior rectus contractile strength was changed from 1.00(%/100) to 1.10(%/100), the elastic strength was changed from 1.00(%/100) to 0.50(%/100).

• The right inferior rectus total strength was changed from 1.00(%/100) to 1.80(%/100). CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 199

Figure 6.33: Hess-Lancaster Simulation Results of the Heavy-Eye Syndrome

6.4.4.2 Simulation of Surgical Correction

The patient with Heavy-Eye syndrome simulated in Sec. 6.4.4.1 was treated with superior oblique, medial and lateral rectus surgery. The insertion of the superior oblique muscle was transposed to the vertical pole (see Fig. 6.34(b). The medial rectus muscle was recessed by 4 mm and tangentially transposed downward by 4 mm, whereas plication surgery was performed at the lateral rectus muscle by 7 mm followed by a 4 mm upward tangential transposition.

(a) Directions of Muscle Trans- (b) Fixation of Muscle to the positions Vertical Pole (1) Anterior/Posterior Transpo- sition (2) Tangential Transposi- tion

Figure 6.34: Muscle Surgery for Heavy-Eye Simulation

The same surgery was simulated using the „SEE++“ software system. The following modiﬁcations CHAPTER 6. SOFTWARE DESIGN AND IMPLEMENTATION 200 to simulation parameters were performed:

• The right superior oblique muscle was positioned at the vertical pole (see Fig. 6.34(b)), in that the insertion was moved 13.20 mm horizontally posterior and subsequently moved -2.90 mm tangentially (see Fig. 6.34(a)).

• The insertion of the right medial rectus muscle was moved 4.00 mm horizontal posterior and subsequently moved -4.00 mm tangentially (see Fig. 6.34(a)).

• The insertion of the right lateral rectus muscle was moved 5.00 mm tangentially (see Fig. 6.34(a)).

• The right lateral rectus muscle length (L0) was changed from 37.50 mm to 32.00 mm.

The simulation result is shown in the Hess-Lancaster diagram in Fig. 6.35. The results show satisfactory reorientation of the lines of sight in primary position. However, full restoration of ocular motility in case of Heavy-Eye Syndrome is most often not possible due to the disease’s extensive pathological impact.

Figure 6.35: Hess-Lancaster Simulation Results of the Heavy-Eye Surgery

The simulation results from Fig. 6.35 show that the biomechanical model described in this thesis is also capable of simulating complicated pathological situations as well as their surgical treatment. Amounts of surgery that were performed on the patient can be used identically or with only small adjustments within the biomechanical model, which provides proper anatomical conformance in most cases. This way, the „SEE++“ software system provides a good approximation for clinical pre- and postoperative simulation. Chapter 7

Conclusion

The research work described in this thesis has been inspired by the idea that modern technologies can also contribute and even advance medical processes, in this case, the ﬁeld of strabismus surgeries. Besides the technical skills in informatics, software engineering and biomechanics, extensive medical knowledge was needed to create a modern, interactive software system that supports surgeons in complicated medical decision making and explains students fundamentals about the way how the oculomotor system works.

One of the most challenging task within this research work was communication with physicians and medical personnel. Since this research work already started in 1996, almost the first two years were spent with gathering medical knowledge and finding common ways to communicate and talk about medical and technical topics. This common language can be identified as vital for all further work that was accomplished within this research project.

From the technical point of view, a complex software system needed to be designed that provides flexibility and extensibility, since many state of the art research results within the field of extraocular physiology needed to be incorporated successively. Moreover, it was even necessary to carry out medical studies with partner institutions in order to gain deeper insight in specific topics of extraocular muscle function and provide the possibility to study pathological situations. One of these studies is described in Sec. 2.1.4 and Sec. 5, where high resolution MRI studies were carried out in order to visualize 3D reconstructions of the morphology of the extraocular muscles in normal human subjects. However, many other research data was needed to develop a biomechanical model of the human eye. Many other partners provided data about extraocular muscle force measurements, extraocular geometry and eye motility diagnostics. Without these medical basis, a mathematical model would never show any relation in its predictions compared to the human eye.

While developing this model and the software system, it was early realized that the basic understanding of the function of the oculomotor system is still under heavy discussion. After implementing and simulating ocular geometry based on the string and tape models (cf. Sec. 4.4), the pulley hypothesis proposed a very different operation mode of the oculomotor system. Inte- grating pulleys into the biomechanical model also clarified how muscle force predictions can be related to eye positions in a way such that muscle force equilibrium defines a stable eye position. Nevertheless, these new findings were incorporated into other biomechanical models before, but

201 CHAPTER 7. CONCLUSION 202 one of the main achievements of this work was that the problem of ﬁnding stable eye positions was reduced to a common non-linear optimization problem, without invalidating model predictions.

Another main feature of the developed software system „SEE++“ is, that it provides a simple and easy to use interface which controls biomechanical model parameters in way that is familiar to medical personnel and physicians. Moreover, one of the goals was that a physician working with this software system does not need extensive mathematical background knowledge in order to simulate complex pathological eye motility disorder. In contrast, the user is able to identify anatomically related parameters and surgical techniques that closely correspond to clinical experience. Thus, the biomechanical model can be parameterized, using measured patient data without additional modiﬁcation.

Concerning the mathematical background of the biomechanical model, implementation was split into pure modelling of the biomechanics of the oculomotor system and representation of anatomical abstraction that form an interface to the biomechanical model. This way, maximum flexibility on both ends of the software system was reached. The user interface can be modified without the need to adapt the biomechanical model. Conversely, the biomechanical model can be modified in its workflow or its behavior that controls model predictions, without modifying the user interface that controls anatomically related parameters that influence the model.

One of the most important parts of the system is the interactive 3D representation of a „virtual“ patient. To provide anatomically relevant models in three dimensions, extensive work has been carried out in the ﬁeld of image analysis and image processing (cf. Sec. 5). Therefore, knowledge in 3D computer graphics and geometry needed to be incorporated into the software system. Modern methods for user interface design and interactive control of 3D scene rendering were implemented to give the user most intuitive control (cf. [Fal99]).

Numerical mathematics, especially algorithms for solving non-linear optimization problems were used to connect ocular geometry with muscle force simulation and solve forward and inverse kinematics. Based on these formulations, clinical test methods were investigated and checked for suitability with respect to integration into the software system. Especially the Hess-Lancaster test of binocular function (cf. Sec. 3.13) turned out to give valuable information due to its extensive clinical usage and its relation to other similar measurement methods (cf. Sec. 3.4) within clinical assessment of eye motility disorders. Therefore, the clinical Hess-Lancaster test was implemented based on the biomechanical model (cf. Sec. 4.7.1). This abstraction of a clinical measurement technique also solves the crucial problem of connecting both eyes in order to transform innervations of one eye to innervations of the fellow eye. Since the horizontal rectus muscles are mirrored with respect to their locations between both eyes, innervations can be transformed by using a virtual reference eye that transforms innervations based on mirrored eye positions. This also provided a basis for simulating neural control of the oculomotor nuclei in order to also simulate neurological disorders.

Within the last four years, research within this project has been concentrated on refining the biomechanical model by evaluating different pathological cases and comparing model predictions to clinically measured patient data. Extensive coordination between clinical studies and technical realization was performed during this time. Different partner institutions have been engaged within this process. The university hospitals of Graz and Innsbruck have contributed to this work in providing clinical data and studies. The hospital St. Pölten and the Wagner Jauregg hospital in Linz assisted in carrying out anatomical and physiological studies using MRI and CHAPTER 7. CONCLUSION 203 human dissections. The convent hospital of the Barmherzigen Brüder is the primary cooperation partner within this research work and has been the first institution to use the „SEE++“ software system clinically. Currently, the university hospital of Vienna evaluates the system for educational purposes. International cooperations supported this work by providing in depth knowledge about ocular physiology and anatomy. The Smith Kettlewell Eye Research Institute in San Francisco provided muscle force measurement data (cf. Sec. 2.3.2) and the university hospital of the ETH Zurich provided measurement data of eye movements.

Besides the efforts of research and software implementation, the most challenging task was the integration of the software system into the clinical environment. During the last four years, publications and conference presentations in the field of strabismus research have been published and worldwide attention has been gained. In order to convince physicians of the reasonable practical relevance of such a software system, pure mathematical proofs do not suffice. Instead, many case studies have additionally been presented in order to show clinical compliance of simulation predictions. The idea was to propose a new way of planning and simulating eye muscle surgeries and to provide methods to study the functions of the oculomotor system. Currently, a book on eye motility disorders and computer aided simulation and treatment is being published to give deeper insight into possibilities and future prospects in using this new methodology.

7.1 Goals Achieved

Within the medical ﬁeld, the main goals were to advance the clinical treatment of eye motility disorders and to provide computer aided teaching support. Within the clinical integration process of the software system „SEE++“ these goals were achieved by carrying out clinical trials and comparisons of patient data. Thus, clinical application of the „SEE++“ improved patient care by supporting surgeons in diagnosis and preoperative planning. Often, up to three successive operations are performed in order to achieve a successful treatment result. Especially in such cases, the application of computer aided surgical treatment contributes to minimize repeated surgical treatment which results in beneﬁts for the patients and directly reduces treatment costs.

The integration of the biomechanical eye model into the ﬁeld of medical training and education enables students and teachers to interactively explain and study basic functional aspects as well as surgical methods. Educational trial lessons have shown that the presented software system supports students in self-studying the function of the oculomotor system and considerably enhance basic understanding of functional implications of diﬀerent surgical treatment methods.

In basic research, this biomechanical model provides a way of gaining a more detailed understanding in principles and processes that aﬀect oculomotor control. In this case, biomechanical models provides an eﬃcient method for checking hypotheses and verifying experimental data. Moreover, state of the art research results in anatomy and physiology can be incorporated into the biomechanical model and subsequently improve simulation predictions.

The „SEE++“ software system currently implements the worldwide most accurate and up to date biomechanical model of the human eye. Due to a well structured object oriented design, this system provides adequate ﬂexibility for further improvements. Integration of interactive three dimensional visualization methods and a user interface that corresponds to anatomical notions provide an eﬃcient new way for physicians to intuitively handle biomechanical simulations. CHAPTER 7. CONCLUSION 204

7.2 Future Work

During research, many additional topics and ideas for enhancements and future investigations have been discovered. Most of the future work that is planned will concentrate on further improvements of the biomechanical model and the software system „SEE++“. However, additional work will try to use parts of the „SEE++“ software for the development of new measurement and clinical diagnostic methods.

One of the next improvements of the biomechanical model will introduce the active pulley hypothesis (cf. Sec. 2.1.2) in that a muscle consists of two distinct layers, one inserting at the pulley and one on the globe. Additionally, data from Demer et. al. [KCD02] suggests that pulleys are dislocated as a function of eye position or muscle innervation. Integration of active pulleys may lead to even more accurate simulation predictions, especially when simulating pathological situations.

Up to now, a physician that is interested in exploring surgical methods needs to reproduce a pathological situation in terms of model parameter values before simulation of surgery can be accomplished. One major extension to the biomechanical model will deal with the automated generation of pathological simulations based upon measured patient data. Therefore, biomechanical model parameter values need to be ﬁt to actual clinical measurements. Using such functionality would greatly improve clinical usage.

Currently, the extension of the „SEE++“ system into a scalable component architecture, realizing a biomechanical „construction kit“ is evaluated. This construction kit should provide standard types of elements in order to aggregate and combine new biomechanical models based on a graphical interactive way of programming.

In order to assign standard cases to speciﬁc classes of pathologies, patient data will be - additionally to the computation of functional interpretations - stored in a knowledge base. This enables the system to suggest a suitable surgery for a new pathological case that ﬁts into one of already stored pathological classes. This practice of evidence-based medicine means integrating individual clinical expertise with the best available external clinical evidence from systematic research.

One of the most exciting future development will aim improvements of eye position measurement methods (cf. Sec. 2.3), especially within eﬃcient objective measurements using video oculography (cf. Sec. 2.3.1.4). Currently, two techniques exist for the recording of eye positions: scleral search coils (cf. Sec. 2.3.1.3), and video-oculography (VOG). While scleral search coils have a high temporal and spatial resolution, they are too expensive and invasive to be used outside the research laboratory. One part of the future work is the development of new VOG algorithms that compensate for translations of the camera with respect to the head. Additionally, the „SEE++“ software system shall serve as front end for VOG measurements in order to show how eye movements change after surgery on the extraocular muscles. Literature

[Abr01] M.D. Abràmoff. Objective Measurement of Motion in the Orbit. Abràmoff Transatlantic Publishing, New York, 2001. ISBN 0-9713598-0-6. [Ack02] M.J. Ackerman. Visible Human Project: From Data to Knowledge, pages 115–117. Yearbook of Medical Informatics. Schattauer Verlagsgesellschaft mbH, Stuttgart, 2002. [BCL99] I.E. Brown, E.J. Cheng, and G.E. Loeb. Measured and modeled properties of mammalian skeletal muscle. II. The effects of stimulus frequency on force-length and force-velocity relationships. Journal of Muscle Research and Cell Motility, (20):627–643, 1999. [BK03a] M. Buchberger and T. Kaltofen. Computer-based Simulation Systems in the Field of Medical- Informatics. Brown bag symposium talk, Smith Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco CA 94115-1813, feb 2003. URL www.ski.org. [BK03b] M. Buchberger and T. Kaltofen. An Ophthalmologic Diagnostic Tool Using MR Images for Biomechanically-Based Muscle Volume Deformation. In Proc. of SPIE Medical Imaging, volume 4 of 23, pages 60–71. San Diego, 2003. ISBN 0-8194-4833-8. [BKP+03] M. Buchberger, T. Kaltofen, S. Priglinger, J.M. Miller, and M. Wiesmayr. SEE++ User’s Manual. Upper Austrian Research GmbH, Hagenberg, Austria, second edition, 2003. [BKPH03] M. Buchberger, T. Kaltofen, S. Priglinger, and R. Hörantner. Construction and Application of an Object-Oriented Computer Model for Simulating Ocular Positioning Defects. Spektrum Augenhelkd., 17(4):151–157, 2003. [BL99] I.E. Brown and G.E. Loeb. Measured and modeled properties of mammalian skeletal muscle. I. The effects of post-activation potentation on the time course and velocity dependencies of force production. Journal of Muscle Research and Cell Motility, (20):443–456, 1999. [BM00a] J.H. Bemmel and M.A. Musen, editors. Handbook of Medical Informatics. Springer-Verlag, Heidel- berg, Germany, second edition, 2000. ISBN 3-450-63351-0. [BM00b] M. Buchberger and H. Mayr. SEE-KID: Software Engineering Environment for Knowledge-based Interactive Eye Motility Diagnostics. In S. Olsson and H. Pettersson, editors, Proc. International Symposium on Telemedicine, pages 67–79. Gothenburg, Sweden, 2000. [Buc01] M. Buchberger. Einführung in die UML (in German). Lecture notes, Upper Austrian University of Applied Sciences, Hagenberg, 2001. [Buc02] M. Buchberger. A Biomechanical Model for Eye Motility Diagnostics (in German). Spektrum Augen- helkd., 16(4):176–182, 2002. [BWM+01] T. Beyer, C. C. Watson, C. C. Meltzer, D.W. Townsend, and R. Nutt. Der Biograph: Ein kombinierter PET-CT-Tomograph der Premiumklasse für die klinische Onkologie. Electromedica, 69(2):126–132, 2001. [C72¨ ] C. Cüppers. Korrektur der Horizontalabweichung. Arbeitskreis Schielbehandlung, Wiesbaden, (5): 11–19, 1972. [Car00] R.H.S. Carpenter. The neural control of looking. Current Biology, (10):291–293, 2000. [CCSJ81] C.C. Collins, M.R. Carlson, A.B. Scott, and A. Jampolsky. Extraocular muscle forces in normal human subjects. Investigative Ophthalmology and Visual Science, 20(5):652–664, 1981. [CHG00] F. Chabat, D.M. Hansell, and Yang G.Z. Computerized Decision Support in Medical Imaging. IEEE Engineering in Medicine and Biology, (5):89–96, Sept./Oct. 2000.

205 LITERATURE 206

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Michael Buchberger

Address

Bahnhofstr. 63 A-4230 Pregarten Austria Phone: +43 664 1651980 Email: [email protected] Homepage: www.uar-mi.at/staﬀ/mbuchber

Personal Details

Gender: Male Date of birth: 12th of April, 1974 Place of birth: Linz, Austria Citizenship: Austrian

Education

2000–2003 University of Linz, Austria Doctorate study Research Project „SEE-KID“ (Software Engineering Environment for Knowl- edge Based, Interactive Eye Motility Diagnostics)

1994–1998 Upper Austria University of Applied Sciences, Hagenberg Study of Software-Engineering Graduation with honors Diploma Thesis: Design and Implementation of a Client/Server System for a Digital Video Archive

1992–1994 University of Linz, Austria Study of Informatics for 4 semesters

1988–1992 Bundes- Oberstufenrealgymnasium Linz/Honauerstr., Austria High-School with special emphasis on Informatics.

Working Experience since 01/2003 Upper Austrian Research GmbH, Hagenberg, Austria Head of the Research Department for Medical-Informatics

2000–2003 Upper Austrian University of Applied Sciences, Hagenberg, Austria Research Assistant at the Department for Software-Engineering for Medicine

• Project advisor research project „SEE-KID“ (www.see-kid.at), funded by the Austrian Ministry of Science and Technology (FFF) • Collaboration within the establishment of the research department of the Upper Austrian University of Applied Sciences (FORTE), Hagenberg • Collaboration within the establishment of an „Information Marketplace Hagenberg“, funded by the Austrian Ministry of Science and Technology (FFF)

03/2002 Consulting for SKI-DATA AG, Salzburg, Austria Consulting activities in the ﬁelds of project engineering and object-oriented software design using modelling languages (UML).

09/2001 Consulting for Posimis Internet GmbH, Linz, Austria Consulting in the ﬁeld of computer graphics and animation for Internet applications.

2000–2001 Upper Austrian University of Applied Sciences, Hagenberg, Austria Design and Implementation of the Internet presentation of the Polytechnic Uni- versity in Hagenberg. Implementation of a dynamic scripting language for the generation of web pages.

1998–2000 Research Institute for Symbolic Computation (RISC), Hagenberg, Austria

• Collaboration within a project for controlling automated transport system for TMS Voest Alpine, Linz, Austria. • Project manager for the re-engineering of the accounting system for the „Rinderbörse“, Linz, Austria.

1994–1998 AMS-Engineering GmbH, Hagenberg, Austria

• Project manager for diﬀerent project in the ﬁeld of industry automation and data analysis. • Project manager for the project Video X-pert, a Client/Server system for digital video archiving.

1992–1994 Landesbildstelle Oberösterreich, Linz, Austria Programmer in the ﬁeld of computer aided teaching for Geography and English.

Teaching

2002–2003 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Lecturer at the Department for Hardware/Software Systems Engineering, Systems- Engineering IV.

01/2002 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Seminar for object-oriented software design using modelling languages (UML).

2001–2002 Trainer at WIFI Linz, Austria WIFI trainer in Linz und Steyr at the WIFI Fachakademie for Informatics.

• Courses in C/C++ programming • WEB-Designer course for programming in Javascript

2001–2002 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Lecturer at the Department for Software-Engineering for Medicine, Project- Engineering Systems-Engineering: Virtual Surgery.

2000–2001 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Lecturer at the Department for Software-Engineering, Project-Engineering: Data visualization of automation systems using XML. 1999–2001 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Lecturer at the Department for Software-Engineering for Medicine, Project- Engineering: Simulation Expert for Eyes + diagnosis + transposition surgery (SEE++).

1999–2000 Lecturer at the Upper Austrian University of Applied Sciences, Hagenberg, Austria Lecturer at the Department for Software-Engineering. Lecture: Introduction to object-oriented design using modelling languages.

1999 Guest lecturer at the Upper Austrian University of Applied Sciences Guest lectures in Software-Engineering and C++ programming.

1998–2001 University of Linz, Austria Student Tutor for Algorithms and data-structures for technical Mathematics.

Language Knowledge

German native English near native French fair

Programming Skills

Visual Studio, MFC, Borland/Inprise C++ Builder, SNIFF++, Oracle De- signer Tools, Matlab, Mathematica, Labview, GUPTA SQL, ERWin, Rational Software Suite, Select OMT/UML, Objectdomain, Visual J++, J-Builder, Del- phi, Borland Turbo Pascal, Borland C++.

.NET, C/C++, Pascal, Java/Java-Script, Modula-2, Oberon, Python, Perl, SQL, Basic.

Consultations

01/2004– Consultant in the development of the strategic program of the Upper Austrian government for research and technology transfer „Innovatives Oberösterreich 2010+“.

Listed in Who’s Who in Science and Engineering, 7th Edition, 2003-2004. Publications

Reviewed Publications M. Buchberger, T. Kaltofen, S. Priglinger, R. Hörantner, Construction and Application of an Object-Oriented Computer Model for Simulating Ocular Po- sitioning Defects, Journal of the Austrian Ophthalmologic Society Nr. 17/4, Springer-Verlag, pp. 151-157, Vienna, Austria, 2003. R. Hörantner, M. Buchberger, T. Kaltofen, S. Priglinger, Diﬀerentialdiagnose vertikaler Schielformen bedingt durch schräge Augenmuskeln und Pulleys, Jour- nal of the Austrian Ophthalmologic Society Nr. 17/4, Springer-Verlag, pp. 158-163, Vienna, Austria, 2003. M. Buchberger, Ein biomechanisches Modell der Augenmotilitõt, Journal of the Austrian Ophthalmologic Society Nr. 16/4, Springer-Verlag, pp. 176-182, Vi- enna, Austria, 2002.

Conference Papers M. Buchberger, T. Kaltofen, An Ophthalmologic Diagnostic Tool Using MR Im- ages for Biomechanically-Based Muscle Volume Deformation, Proc. of Medical Imaging 2003, Image Processing; Milan Sonka, Univ. of Iowa (USA); J. Michael Fitzpatrick, Vanderbilt Univ. (USA), San Diego, USA, 2003. S. Priglinger, R. Hörantner, M. Buchberger, T. Kaltofen, Functional Topogra- phy in an Eye Model, Proceedings of Development and Perspectives in Visual Processing and Eye Movements, The Heidelberg Meeting on Eye Movements, International Symposium of the German Ophthalmological Society, Heidelberg, Germany, 2002. M. Buchberger, Ein dynamisches, volumserhaltendes 3D-Deformationsmodell von extraokularen Muskeln, basierend auf der Analyse von MRI-Daten, Proc. 1. Jahrestagung der Österreichischen Wissenschaftlichen Gesellschaft für Telemedi- zin, pp. 37, Innsbruck, Austria, 2001. M. Buchberger, H. Mayr, SEE-KID: Software Engineering Environment for Knowledge-based Interactive Eye Motility Diagnostics, Proc. International Sym- posium on Telemedicine, EU Human Potential Programme, pp. 67-79, Gothen- burg, Sweden, 2000.

Conference Presentations R. Hörantner, M. Buchberger, T. Kaltofen, S. Priglinger, Myopia Alta, a SEE++ case study, Consilium Strabologicum Austriacum, Innsbruck, Austria, 2003. M. Buchberger, T. Kaltofen, Ein computerunterstütztes biomechanisches Mod- ell zur Vorbereitung, Planung und Simulation von Strabismus-Operationen, 44. Jahrestagung der österreichischen Ophthalmologischen Gesellschaft, Salzburg, Austria, 2003. M. Buchberger, T. Klatofen Computer-based Simulation in Medical Informatics, Talk at the Brown-Bag Symposium, Smith Kettlewell Eye Research Institute, San Francisco, USA, 2003. M. Buchberger, T.Kaltofen, S. Priglinger, R. Hörantner, SEE-KID and EOM- Modelling, The Heidelberg Meeting on Eye Movements, International Sympo- sium of the German Ophthalmological Society, Heidelberg, Germany, 2002. M.Buchberger, J. Hildebrandt, T. Kaltofen MR-Cinematographie der Augen- motilität, Consilium Strabologicum Austriacum, St. Pölten, Austria, 2001.

References

These persons are familiar with my professional qualiﬁcations and my character:

Univ.-Prof. Dipl.-Ing. Dr. Roland Wagner Thesis supervisor Phone: +43 732 2468 8791 FAW-University of Linz Fax: +43 732 2468 9308 Altenberger Straße 69 Email: [email protected] A-4040 Linz, Austria

Univ.-Doz. Dipl.-Ing. Dr. Thomas Haslwanter Thesis advisor Phone: +41 1 255 3996 Institute for Theoretical Physics, Fax: +41 1 255 4507 ETHZ and Dept. of Neurology University Hospital Zurich Email: [email protected] Frauenklinikstr. 26 8091 Zurich, Switzerland

Univ.-Prof. Dipl.-Ing. Dr. Witold Jacak Thesis advisor Phone: +43 7236 3888 2000 Upper Austrian University of Applied Sciences Fax: +43 7236 3888 99 Hauptstr. 117 Email: [email protected] A-4232 Hagenberg, Austria

Prim. Prof. Dr. Siegfried Priglinger Thesis advisor Phone: +43 732 7897 1300 Convent Hospital of the „Barmherzigen Brüder“ Fax: +43 732 7897 1099 Seilerstätte 2 Email: [email protected] A-4020 Linz, Austria Prim. Univ.-Doz. DDr. Armin Ettl Collaborator Phone: +43 2742 300 2869 Hospital St. Pölten Fax: +43 2742 300 3285 Propst Führer Str. 4 Email: [email protected] 3100 St. Pölten, Austria

Univ.-Doz. Dr. Franz Fellner Collaborator Phone: +43 732 6921 26701 Radiologic Institute of the Wagner Jauregg Hospital Fax: +43 732 6921 26704 Wagner-Jauregg-Weg 15 Email: [email protected] A-4020 Linz, Austria