Cardiac Modeling 43 2.1 State of the Art

Modeling and imaging of electrocardiographic activity Karim El Houari

To cite this version:

Karim El Houari. Modeling and imaging of electrocardiographic activity. Signal and Image processing. Université Rennes 1, 2018. English. NNT : 2018REN1S063. tel-02066820

HAL Id: tel-02066820 https://tel.archives-ouvertes.fr/tel-02066820 Submitted on 13 Mar 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE

L’UNIVERSITE DE RENNES 1 COMUE UNIVERSITE BRETAGNE LOIRE

Ecole Doctorale N°601 Mathèmatique et Sciences et Technologies de l’Information et de la Communication Spécialité : Signal, Image, Vision Par Karim EL HOUARI

Modélisation et imagerie électrocardiographiques

Thèse soutenue à RENNES , le 14 Décembre 2018 Unité de recherche : Laboratoire du Traitement du Signal et de l’Image Thèse N° :

Rapporteurs avant soutenance : Composition du jury :

Christian Jutten (PU), GIPSA-LAB Examinateurs Freddy Odille (CR-INSERM), IADI Soﬁane Boudaoud (MCF-HDR), BMBI Christian Jutten (PU), GIPSA-LAB Freddy Odille (CR-INSERM), IADI

Directeur de thèse Laurent Albera (MCF-HDR), LTSI

Co-directeurs de thèse Alfredo Hernández (DR-INSERM), LTSI Christelle Boichon-Grivot (IR), ANSYS

ACKNOWLEDGMENT

First and foremost, I would like to thank Professor Christian Jutten and Associate Professor Freddy Odille for giving their time and expertise to review this manuscript. I am also thankful to them and Associate Professor Sofiane Boudaoud for the interest they demonstrated for this work, and I very much appreciated the constructive discussion that followed my defense. I would like to express my deep and sincere gratitude to my research supervisors in the LTSI laboratory: Associate Professor Laurent Albera, Research Director Alfredo Hernàndez and Research Engineer Amar Kachenoura. Not only their valuable critiques, advice and close guidance were prominent in the progress of my research, but their support, availability and kindness made this whole life experience unforgettable. This project wouldn’t have been possible without the help and financial support of the company Ansys. In particular, I would like to thank Michel Rochette, head of Ansys research team in Lyon and Christelle Boichon-Grivot, Research Engineer within Ansys, first for their management and investment in this project, but also for offering such convenient working conditions. I would also like to extend my thanks to my collegues in Ansys: Valéry Morgenthaler, François Chapuis and Frédéric Thevenon for their valuable help in using Ansys Fluent and Ansys Mechanical software. Many thanks go to Associate Professor Eric Darrigrand from the IRMAR laboratory for receiving me several times in his office and for removing some mathematical modeling ambiguities. I also wish to acknowledge the help provided by my colleagues in the LTSI: Ahmad Karfoul, Siouar Bensaid and Paul Berraute, which saved me a lot of time in understanding and implementing various elements of this PhD such as Finite Element Method operators, data whitening and Morozov discrepancy principle. Special thanks go to Doctor Raphaël Martins and his colleagues of the university hospital of Rennes for their warm welcome in the electrophysiology room, the clinical data they provided, and the valuable explanations of the complex electrical mechanisms behind cardiac arrhythmia. More broadly, I truly consider myself lucky to experience such a pleasant professional environment, this is why I would like to thank all my colleagues in the LTSI for their warm welcome, their help, and the joyful time I have had in and outside the lab. I must also thank my friends and my dear housemates for their moral support and for making these years I spent in Rennes such a thrill.

3 Finally, I owe my deepest gratitude to my family for their unconditional love, support and encouragement throughout my study.

4 RÉSUMÉ SUBSTANCIEL

Cette thèse est le résultat de la collaboration entre le Laboratoire Traitement du Signal et de l’Image (LTSI) et l’entreprise Ansys. Plus précisément, elle regroupe les compétences de deux équipes du LTSI: celles de l’équipe SEPIA en modéllisation de l’électrophysiologie cardiaque, et celles de l’équipe SESAME en termes de méthodes pour la localisation de sources épileptiques de manière non-invasive. ANSYS est une entreprise qui est leader dans le domaine de la simulation numérique de systèmes multiphysiques. Ce travail est destiné à initier une activité de recherche dans le domaine de la modélisation électrique cœur-torse au sein de ces entités.

Motivation

Les maladies cardiovasculaires sont la cause majeure de mortalité dans le monde et représente un fardeau économique et sanitaire conséquent [106]. La moitié des décès causés par ces maladies est dûe à une mort cardiaque subite, qui est un arrêt soudain de la circulation du sang résultant de l’incapacité du cœur à pomper dans le corps [105]. Cette insuffisance cardiaque est la plus part du temps causée par une propagation anormale de l’activité électrique dans le cœur, phénomène connu sous le nom d’arythmie. Parmi les arythmies cardiaques, la Tachycardie Ventriculaire (TV) est la plus mortelle et représente 80% des morts subites. Malgré le développement de plus de 100 anti-arythmogènes, les efforts mis dans la prévention des arythmies ventriculaires sérieuses et fatales restent décevants. L’utilisation des Défibrillateurs Cardiaques Implantables (DCIs) est préconisé à la fois pour la prévention des morts subites chez les patients atteints de maladies structurelles et ceux en cas d’insuffisance de la fraction d’éjection du ventricule. Cependant, l’implantation des DCIs est très coûteuse et n’est pas curative. La perpétuation de la TV dépend de régions pathologiques dont la conduction est relativement lente, donnant la possibilité aux segments adjacents dans le myocarde de regagner de l’excitabilité lorsque l’impulsion se propage lentement dans la région pathologique. Le mélange des conductions normales (tissu sain) et anormales (tissu ischémique) peut causer la formation de circuits de ré-entrée et constitue le substrat arythmogène de la TV. L’ablation par radiofréquence des substrats ventriculaires arythmogènes par cathéterization a montré un grand succès dans le traitement de la TV. Afin de délimiter le circuit critique (isthme), une cartographie

5 Résumé substanciel

électro anatomique est effectuée dans une salle d’électrophysiologie. Les potentiels éléctriques de la surface intramyocardique sont acquis à l’aide d’un cathéter placé à l’intérieur de la cavité cardiaque, créant une cartographie partielle de l’activité électrique du cœur. Cette procédure est invasive, longue et difficile à la fois pour le patient et pour le chirurgien. Les taux de récurrence sont significatifs et le manque de consensus clinique sur la stratégie optimale d’ablation rendent cette option encore plus complexe [5]. D’un autre côté, une étude récente montre que le succès d’une ablation multiplie les chances de survie à l’arythmie par dix [62]. Cependant, la procédure n’aboutit que dans 41 à 89% des cas [189]. C’est pourquoi des avancées sont nécessaires pour une identification plus précise et une meilleure compréhension du substrat arythmogène ventriculaire. Dans ce contexte, un intérêt grandissant pour l’Imagerie ElectroCardioGraphique (IECG) a emergé dans la communauté scientifique. Cette technique consiste à fournir des images 3D de la distribution spatiale de l’information électrique cardiaque, de manière non-invasive. Trois éléments sont requis pour effectuer une procédure d’IECG :

• un modèle géométrique du corps obtenu à partir de données acquises par Imagerie par Résonance Magnétique (IRM) ou par tomographie;

• de multiples mesures de potentiels à la surface du torse qui procurent une version attenuée de l’activité électrique cardiaque;

• un modèle mathématique qui traduit le lien physique entre les sources électriques cardiaques et les potentiels de surface observés.

Bien que l’IECG est prometteuse en théorie, elle demeure une tâche ardure pour trois raisons principales. 1) Elle implique un modèle direct qui traduit de manière adéquate le lien physique entre les ECGs observés et les sources électriques cardiaques à reconstruire. Ce modèle est soumis à plusieurs incertitudes dont: les imprécisions géométriques liées à l’acquisition des images anatomiques et à la déformation du cœur pendant la contraction et la relaxation, les conducitivités électriques des tissus et leur anisotropies, la difficulté de représenter la directionnalité des fibres cardiaques et enfin le bruit d’instrumentation des ECGs. 2) Elle requiert la résolution d’un problème mathématique rendu difficile à cause de son caractère mal-posé. Cette nature provient du faible nombre d’observations comparé nombre de sources à reconstruire mais aussi de l’atténuation de l’activité électrique lorsqu’elle se propage entre le cœur et la surface torse. Ainsi, une solution du problème n’est pas unique, et de petites variations sur les ECGs de surface peuvent engendrer des solutions divergentes. Afin de contourner ce caractère mal-posé, une régularisation est souvent utilisée pour restreindre l’espace des solutions acceptables de manière appropriée et garantir une solution unique au problème régularisé. Plusieurs méthodes et algorithmes ont été proposés dans ce sens, mais des efforts restent à fournir

6 Résumé substanciel dans l’incorporation d’a priori physiologiques sur la solution. 3) La validation clinique chez l’homme reste très limitée puisque l’enregistrement simulanné de l’activité électrique en tout point du cœur est inaccessible à la fois pour des raisons pratiques et éthiques. Une manière d’évaluer des algorithmes inverses est par l’étude de leur performance sur des données simulées par des modèles de propagation électriques cœur-torse. La première étape consiste à modéliser l’excitation et la propagation électrique cardiaque. Un modèle direct est alors utilisé aﬁn de calculer des ECGs de surface correspondant à des cartographies de potentiels simulées connues. Certains ECGs et le modèle direct sont ensuite utilisés comme entrée des méthodes inverses. Les cartographies estimées par ces méthodes sont ﬁnalement comparées aux données de référence. De nombreuses études du problème inverse ont été basées sur des donées simulées. Cependant, les modèles impliqués sont soit trop complexes soit ne produisent pas des ECGs cohérents.

Figure 1: Etapes de résolution du problème inverse en utilisant des données simulées

Objectifs

L’objectif principal de cette thèse est l’évaluation quantitative de méthodes de résolution du problème inverse en ECG en utilisant des données obtenues par des simulations numériques contrôlées. Pour cela, nous procédons d’abord à i) la conception d’un modèle cœur-torse qui trouve un compromis entre réalisme des données générées et complexité, puis à ii) l’exploration de nouvelles méthodes de résolution du problème inverse en ECG et leur comparaison avec les méthodes classiques, en utilisant des données générées par notre modèle. Ce travail s’inscrit dans la perspective de pouvoir évaluer qualitativement les performances de plusieurs algorithmes de résolution du problème inverse sur une multitude de scénarios en termes de géométrie, de conductivité électrique et de pathologies, en utilisant divers modèles de sources.

7 Résumé substanciel

Contributions

La première partie de cette thèse est dédiée à la conception d’un modèle de propagation cœur- torse rapide et simplifié générant des cartographies électriques cardiaques réalistes et des ECGs morphologiques. Ce modèle est bâti sur une géométrie idélisée construite en utilisant le logiciel Ansys. Il implémente le formalisme monodomaine augmenté couplé avec le modèle ionique phénoménologique FitzHugh-Nagumo et utilise la méthode des élements finis pour la discrétisation des équations à dérivées partielles. Les paramètres du modèle sont identifiés par le moyen d’une stratégie évolutionnaire et leur influence est étudiée à travers une méthode de sensibilité de criblage. Une fois les paramètres identifiés dans le cas sain, nous montrons que ce modèle est un outil efficace pour générer des scénarios variés de cœurs sains et pathologiques produisant des ECGs cohérents. De plus, les résultats de l’analyse de sensibilité suggèrent que seuls quelques paramètres de ce modèle sont suffisants à inférer un comportement électrique cardiaque macroscopique à partir d’un ECG donné. Ce travail constitue un cadre contrôlé afin de réaliser des simulations de problème inverse et a été en partie publié dans la conférence internationale IEEE CAMSAP :

•EL HOUARI,K.,KACHENOURA,A.,ALBERA,L.,BENSAID,S.,KARFOUL,A.,BOICHON-GRIVOT,

C.,ROCHETTE,M., AND HERNÁNDEZ, A. A fast model for solving the ecg forward problem based on an evolutionary algorithm. In Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2017 IEEE 7th International Workshop on (2017), IEEE, pp. 1–5

Deuxièmement, nous implémentons et comparons la performance des approches classiques de résolution du problème inverse en ECG en utilisant la formulation en potentiels épicardiques. Les méthodes étudiées sont la famille des méthodes Tikhonov, et des méthodes basées sur une régularisation de norme `1. Nous étudions d’abord l’inﬂuence du pré-traitement des données et celle de l’algorithme d’optimisation utilisé. Enﬁn, une nouvelle régularisation spatiotemporelle est proposée. Elle est basée sur la variation totale entre deux échantillons temporels consécutifs. Cette méthode utilise l’algorithme des directions alternées combinée avec le principe de contradiction de Morozov. Nous montrons que cette approche surpasse les méthodes classiques de résolution du problème inverse en ECG dans plusieurs scénarios.

Une étude préliminaire faisant la comparaison entre méthodes régularisantes basées sur la norme `1 et

`2 a été publiée dans la conférence internationale IEEE SAM:

•EL HOUARI,K.,KACHENOURA,A.,BCRRAUTE, P., BENSAID,S.,KARFOUL,A.,BOICHON-GRIVOT,

C.,HERNÁNDEZ,A.,ALBERA,L., AND ROCHETTE, M. Investigating transmembrane current source formulation for solving the ecg inverse problem. In 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM) (2018), IEEE, pp. 371–375

8 Résumé substanciel

Organisation du manuscrit

Dans le premier chapitre, nous introduisons les notions de base en anatomie et en électrophysiologie cardiaque. Nous décrivons comment l’activité électrique est générée ainsi que la manière dont elle se propage dans le cœur et le torse. Nous expliquons ensuite les principales caractéristiques des ECGs mesurés dans le système standard d’ECG 12-Dérivations. Finalement, les méchanismes et les traitements relatifs à la TV sont présentés ainsi que les les défis reliés à la chirurgie par cathéterization par radiofréquence. Le deuxième chapitre présente le modèle que nous proposons. Nous présentons d’abord un état de l’art des modèles électriques cœur-torse ainsi que des méthodes numériques de discreétisation impliquées. Dans un second temps nous présentons la géométrie conçue et nous décrivons de manière détaillée la dérivation et la discretisation spatiotemporelle des équations du modèle monodomaine. Ensuite, nous présentons l’algorithme évolutionnaire utilisé pour identifier les paramètres et des résultats en termes de cartographies électriques cardiaques et ECGs. Nous abordons par la suite le sujet de la sensibilité du modèle vis à vis de ses paramètres et nous montrons que les paramètres non physiologiques ont une influence faible sur les simulations de propagation cardiaque. Enfin, nous montrons que le modèle est capable de produire des allures d’ECGs réalistes dans les cas d’une TV monomorphe et polymorphe ainsi que dans le cas d’un infarctus du myocarde. Le troisième chapitre porte sur le problème inverse en ECG. Après un état de l’art des méthodes principales utilisées, nous analysons l’influence du pré-traitement des données et nous montrons qu’un blanchiment des données améliore les solutions. Ensuite, nous étudions la performance de différents algorithmes et de plusieurs termes de régularisation. Enfin, nous décrivons notre nouvelle approche basée sur une variation totale spatiotemporelle combinée avec le principe de contradiction de Morozov et nous étudions ses performances dans différents cas sains et pathologiques. Le dernier chapitre contient une discussion générale des travaux présentés et les perspectives reliées en termes de modélisation, des méthodes inverses envisagées et de validation clinique.

TABLEOF CONTENTS

Résumé substanciel 5

List of Tables 14

List of Figures 17

List of Abbreviations 18

Introduction 21

1 Cardiac electrophysiology 25 1.1 Anatomy and function of the heart ...... 25 1.2 Cardiac electrical activity ...... 28 1.2.1 Cardiomyocytes ...... 29 1.2.2 Pacemaker cells ...... 30 1.3 Cardiac conduction system and Electrocardiograms ...... 30 1.4 Electrocardiography ...... 32 1.4.1 History ...... 32 1.4.2 Modern Electrocardiography and 12-Lead ECG ...... 33 1.5 Cardiac arrhythmia ...... 37 1.5.1 Mechanisms underlying ventricular tachycardia ...... 38 1.5.2 Treatment ...... 39 1.5.3 Catheter ablation procedure ...... 40 1.6 Catheter ablation and ECGI challenges ...... 41

2 Cardiac modeling 43 2.1 State of the art ...... 43 2.1.1 Cardiac cell models ...... 44 2.1.2 Macroscopic cardiac source models and extra-cardiac potentials ...... 45 2.1.3 Simulating propagation in the heart ...... 47

11 TABLE OF CONTENTS

2.1.4 Numerical methods ...... 48 2.2 Proposal of a simpliﬁed heart-torso model ...... 49 2.2.1 Heart-torso geometry and meshing ...... 50 2.2.2 Model equations ...... 52 2.2.3 Spatio-temporal discretization ...... 56 2.3 Parameter identiﬁcation using an evolutionary strategy ...... 61 2.3.1 Optimization problem formulation ...... 64 2.3.2 Evolutuionary Algorithms ...... 66 2.3.3 Numerical simulations ...... 67 2.4 Sensitivity analysis and model reduction ...... 78 2.4.1 Morris method ...... 78 2.4.2 Numerical simulations ...... 80 2.5 Pathology simulation ...... 81 2.6 Conclusions ...... 93

3 ECG inverse problem 95 3.1 State of the art ...... 96 3.1.1 Source formulations ...... 96 3.1.2 Classical methods ...... 97 3.2 Methods ...... 100 3.2.1 Data pre-processing ...... 100 3.2.2 Inverse problem formulation ...... 102 3.2.3 Optimization schemes ...... 104 3.2.4 Determination of the regularization parameter ...... 106

3.2.5 A new spatiotemporal approach with two `1 constraints ...... 109 3.3 Numerical experiments ...... 110 3.3.1 Experiment 1: inﬂuence of whitening and regularizers ...... 111 3.3.2 Experiment 2: evaluation of the ADMM combined with the discrepancy principle . 120

3.3.3 Experiment 3: Performance study of the double `1 constraint ...... 122 3.4 Conclusions ...... 123

General conclusion & perspectives 125

A The full heart-torso bidomain model 129

B Quasi-static approximation 133

12 TABLE OF CONTENTS

C Epicardial formulation of the forward problem 135

Bibliography 139

13 LISTOF TABLES

1.1 Normal ECG characteristics ...... 36

2.1 Reference ECG features ...... 68 2.2 Parameters variation domain ...... 69 ref 2.3 Reference features ci and best solution features ci(p) ...... 72

3.1 Variation grid for parameter λ ...... 112 3.2 Variation grid for parameter ρ ...... 112

14 LISTOF FIGURES

1 Etapes de résolution du problème inverse en utilisant des données simulées ...... 7 2 Steps for solving the inverse problem using simulated data ...... 23

1.1 Human heart and body [13] ...... 25 1.2 Human heart anatomy [185] ...... 26 1.3 Heart layers [112] ...... 27 1.4 Systole and Diastole in the atria and the ventricles [112] ...... 28 1.5 Typical myocite action potential ...... 29 1.6 Typical pacemaker cell action potential ...... 30 1.7 Action potential spread and resulting ElectroCardioGram (ECG) [102] ...... 32 1.8 An early commercial ECG device manufactured by Cambridge Scientiﬁc Instrument Company of London, 1911 [193] ...... 33 1.9 Electrodes placement and normal 12-Lead ECG graph [1] ...... 35 1.10 12-Lead ECG angular representation [118] ...... 35 1.11 Typical ECG signal and its components [6] ...... 36 1.12 Cardioinsight ECG recording vest [2] ...... 37 1.13 Ventricular Tachycardia illustration and corresponding ECG signal ...... 39

2.1 Heart-torso geometry ...... 51 2.2 Heart geometry and its seven regions ...... 52 2.3 Heart-torso mesh ...... 53 2.4 SinoAtrial Node location ...... 59 2.5 Heart-torso geometry and 12-Lead ECG electrode conﬁguration ...... 61

∂vm ∂u 2.6 Left: phase plane representation of the nullclines ∂t = 0 (blue), ∂t = 0 (red) and the obtained trajectory (green). Right: transmembrane potential (top) and recovery variable (bottom) ...... 63 2.7 Action potentials generated by the FHN model by region ...... 64 2.8 Target ECG signal and its features ...... 65 2.9 Diagram explaining the basic idea of crossovers and mutations ...... 67

15 LIST OF FIGURES

2.10 Repolarization and repolarization times in an action potential ...... 68 2.11 Boxplot of cost values evolution through generations ...... 70 2.12 Estimated regional transmembrane potentials ...... 71 2.13 Normalized ECG estimated by the EA (red) and reference ECG (blue) ...... 72 2.15 Transmembrane potential (up) and corresponding bipolar potential (down) at node 100 . . 73 2.14 Simulated 12-Lead ECGs ...... 74 2.16 Transmembrane potentials spatial distributions at 9 key instants of the ECG ...... 76 2.17 Local activation times (ms) ...... 77 2.18 Example of Morris method experiment plan in the simple case where P = 2, L = 9 and R = 4 ...... 79 2.19 Sensitivity graphs of parameters by region ...... 83 2.20 Sensitivity graphs when the most influential parameters are omitted ...... 84 2.21 Sensitivity graphs obtained for five different model outputs, each color corresponds to a different ECG reference ...... 85 2.22 Sensitivity graphs obtained for five different model outputs when the most influential parameters are omitted ...... 86 2.23 Separation between the least and the most influential parameters by region ...... 87

0 2.24 Reference (black) and simulated ECGs corresponding to the best p (blue) and the best p (red) ...... 88 2.25 Monomorphic VT simulation ...... 89 2.26 Polymorphic VT simulation ...... 90 2.27 Comaparison of healthy 12-Lead ECGs (blue) and those in the case of myocardial infarction (red) ...... 91 2.28 LAT mapping corresponding to the simulated ischemic condition ...... 92 2.29 LAT mapping for the ﬁrst ectopic beat of the simulated monomorphic VT ...... 92

3.1 Selected times in Lead II for experiments 1 and 2 ...... 111

3.2 Graphical representation of `2 methods performance for raw and whitened data with respect to G and to X ...... 114

3.3 Graphical representation of `1 methods performance for raw and whitened data with respect to G and to X ...... 115

3.4 RE values when using `2 regularization at different times and optimal values of λ in the search grid. Best case scenarios are highlighted in shaded gray ...... 116

3.5 RE values when using `1 regularization at different times and optimal values of λ and ρ in the search grids. Best case scenarios are highlighted in shaded gray ...... 117

16 LIST OF FIGURES

3.6 Estimated cardiac mappings at instant 5 by the different methods when data are whitened with respect to G ...... 118

3.7 Comparison of `2 − V and `1 − V along time when data are whitened with respect to G . 118

3.8 Inﬂuence of the TSVD index on the performance of `2 − V and `1 − V ...... 119

3.9 Comparison of `1 and `2 regularization when the gradient operator is used, for raw and whitened data with respect to G and to X ...... 121

3.10 Comparison of our proposed approach with the `1 − V method when a pre-whitening of G is applied ...... 123

17 LISTOF ABBREVIATIONS

ADMM Alternating Direction Method of Multipliers.

ALM Augmented Lagrangian of Multipliers.

AVN Atrio Ventricular Node.

BEM Boundary Element Method.

CRESO Composite REsidual and Smoothing Operator.

CT Computed Tomography.

EA Evolutionary Algorithm.

ECG ElectroCardioGram.

ECGI ElectroCardioGraphy Imaging.

EDL Equivalent Double Layer.

FDM Finite Difference Method.

FEM Finite Element Method.

FHN FitzHugh-Nagumo.

FVM Finite Volume Method.

GCV Generalized Cross Validation.

IRN Iterative Reweighted Norm.

LAT Local Activation Time.

MRI Magnetic Resonance Imaging.

18 List of Abbreviations

PALM Proximal Alternating Linearized Minimization.

PDE Partial Differential Equation.

SAN Sino-Atrial Node.

SVD Singular Value Decomposition.

TSVD Truncated Singular Value Decomposition.

TV Total Variation.

UDL Uniform Double Layer.

VT Ventricular Tachycardia.

INTRODUCTION

This thesis is the result of the collaboration between the “Laboratoire Traitement du Signal et de l’Image” (LTSI) and ANSYS company, within the framework of a CIFRE industrial research agreement. More precisely, this work brings together two teams of the LTSI: the SEPIA team, with expertise in cardiac electrophysiology modeling, and the SESAME team, with expertise on algorithms for non-invasive cerebral source localization. ANSYS is a major company in the ﬁeld of numerical simulation of 3D multiphysics systems. This work is intended to initiate research in the ﬁeld of whole heart and torso modeling and electrocardiography imaging within these entities.

Motivation

Cardiovascular diseases are the leading cause of mortality worldwide and represent a huge health and economic burden [106]. About half of these deaths are due to sudden cardiac death, which is a sudden loss of blood ﬂow resulting from the failure of the heart to effectively pump blood into the body [105]. This failure is in most cases caused by an abnormal propagation of electrical activity within the heart, a phenomenon referred to as cardiac arrhythmia. Among these conditions, Ventricular Tachycardia (VT) is the most life-threatening and is responsible for over 80% of sudden cardiac deaths [105]. VT is associated with a fast, abnormal heart rate that originates in the heart’s lower chambers. Despite the development of over 100 anti-arrhythmic medications, efforts at prevention of serious and fatal ventricular arrhythmia have proved disappointing. Device-based therapies, involving the use of Implantable Cardioverter-Deﬁbrillators (ICDs), are indicated for prevention of sudden cardiac death in patients with a structural disease. However, the implantation of such devices is expensive and non-curative. Perpetuation of VT strongly depends upon pathological regions of relatively slow conduction, enabling adjacent segments in the myocardium to regain excitability whilst the impulse travels slowly through the pathological area. This mixture of normal conduction (healthy tissue) and abnormal conduction (scar tissue) can lead to the formation of a re-entrant circuit and provide a substrate for VT. Catheter radiofrequency ablation of the ventricular arrhythmogenic substrate has been proven to be a successful treatment for VT. In order to delineate the critical circuit (isthmus), conventional

21 Chapter 0 – Introduction electro anatomical mapping is performed in an interventional electrophysiology room. Voltages from the inner myocardial surface are acquired by a catheter placed inside the cardiac cavity, leading to a partial mapping of the electrical activity of the heart. This procedure is invasive, time consuming and challenging. There are signiﬁcant recurrence rates and a lack of clinical consensus on the optimal ablation strategy [5]. A recent systematic review shows that a successful ablation increases the chance of future arrhythmia-free survival ten-fold [62]. However, only 41-89% of patients achieve the procedural endpoint [189]. Therefore, improvements are necessary to better identify and understand the ventricular arrhythmogenic substrate.

In this connection, ElectroCardioGraphy Imaging (ECGI) emerged as a novel tool for providing insight on cardiac electrical activity in a non-invasive fashion. The ultimate goal is to reproduce personalized cardiac spatiotemporal electrical patterns at a high resolution from ECG recordings and the acquisition of the patient’s anatomy. However, although this technique represents a considerable potential for enhancing VT catheter-based therapy, reducing exposure to angiographic X-ray radiation, and reducing the cost and time of the procedures, it remains an arduous task for three main reasons. 1) It requires an adequate model of the transfer between cardiac electrical activity and the ECG signals measured on the torso surface. 2) It requires solving a mathematical inverse problem for which no unique solution exists. 3) Clinical validation in humans is very limited since simultaneous whole heart electrical distribution recordings are inaccessible for both practical and ethical reasons.

One way to evaluate ECGI algorithms is by studying their performance on data simulated by heart-torso propagation models (Figure 2). The ﬁrst step is modeling cardiac electrical excitation propagation. A forward model is then used for the synthesis of torso ECG potentials corresponding to known, simulated cardiac potential mappings. The surface ECGs and the forward model are used as an input to inverse methods. The estimated cardiac mappings by these methods are ﬁnally compared to the reference. Numerous ECG inverse problem studies have been based on simulated data. However, the involved models either suffer from their high complexity due to their high level of detail or do not produce realistic ECGs.

Objectives

The main objective of this thesis is to quantitatively compare existing and new methods for solving the inverse problem in the context of ECGI using a set of data obtained from controlled numerical whole-heart and torso simulations. In order to achieve this goal we proceed to i) the design of a heart-torso electrical model that ﬁnds a trade-off between the realism of the generated data and complexity and ii) a quantitative evaluation of reference-based performance criteria of classical and novel regularization-based ECGI algorithms.

22 Figure 2: Steps for solving the inverse problem using simulated data

Contributions

The first part of this thesis is dedicated to the design of a simplified and fast heart-torso propagation model that generates meaningful cardiac electrical patterns and is able to synthesize ECGs with realistic morphologies. This model is built on an idealized heart-torso geometry using ANSYS software. It implements the augmented monodomain formalism, coupled with the phenomenological FitzHugh-Nagumo (FHN) model and uses the Finite Element Method (FEM) for partial differential equations discretization. In order to produce realistic ECG signals, model parameters are identified by means of an evolutionary strategy and their influence is studied through a screening sensitivity analysis. Once parameters are identified for the healthy case, we show that this model is a reliable tool for generating various scenarios of the healthy and the diseased heart, producing realistic ECG signals. Moreover, sensitivity analysis results suggest that only few parameters of this model are sufficient to infer a desired macroscopic cardiac behavior from a given ECG observed from a patient. This work constitutes a framework for building a controlled environment for ECGI simulations and was partly published in the IEEE CAMSAP international workshop:

•EL HOUARI,K.,KACHENOURA,A.,ALBERA,L.,BENSAID,S.,KARFOUL,A.,BOICHON-GRIVOT,

In the second part, we implement and compare the performance of classical approaches for solving the ECG inverse problem using the epicardial potential formulation. The studied methods are the family of Tikhonov methods, and `1 regularization based methods. We ﬁrst study the inﬂuence of data pre-processing and that of the optimization algorithm used. We also propose a novel spatiotemporal

23 Chapter 0 – Introduction regularization based on the Total Variation (TV) norm of consecutive time samples. This method uses the Alternating Direction Method of Multipliers (ADMM) combined with Morozov’s discrepancy principle. We show that this approach outperforms the classical methods for solving the inverse problem of ECG in several case-scenarios. A preliminary study that compares between `1 et `2 based methods was published in the IEEE SAM international workshop:

•EL HOUARI,K.,KACHENOURA,A.,BCRRAUTE, P., BENSAID,S.,KARFOUL,A.,BOICHON-GRIVOT,

Manuscript organisation

In the first chapter, we introduce basic concepts of anatomy and cardiac electrophysiology. We describe how cardiac electrical activity is generated and propagated through the heart and the torso. We then explain the main characteristics of ECGs recorded in the standard 12-Lead ECG system. Finally, VT mechanisms and its treatment options are presented, as well as the underlying challenges of catheter ablation procedures. The second chapter presents our proposed model. We first review heart-torso electrical models and the involved discretization methods. Secondly, we present the designed geometry and provide a detailed description of monodomain equations derivation and spatio-temporal discretization. Next, we describe the evolutionary algorithm used to identify model parameters and present results in terms of cardiac mappings and ECGs. Thereafter, we address the topic of model sensitivity to its parameters and show that non-physiological parameters have a low influence in cardiac simulations. Finally, we demonstrate that the model is capable of producing clinical patterns of ECGs related to monomorphic and polymorphic VTs as well as myocardial infarction. The third chapter deals with the inverse problem in ECG. After a review of the main methods, we analyze the influence of data pre-processing and demonstrate that data spatial whitening enhances the solutions. Next, we study the performance of different algorithms with different regularization terms. Finally, we introduce our new approach based on spatio-temporal TV norm combined with the discrepancy principle and we study its performance through different case-scenarios. The final chapter provides an overall discussion of this work and its perspectives in terms of modeling, the intended inverse methods to be investigated and clinical validation.

24 CHAPTER 1

CARDIACELECTROPHYSIOLOGY

This chapter describes the clinical context in which this thesis is framed. First, a brief review on the anatomy and function of the heart is provided. In particular, the generation of the cardiac electrical activity and the means for recording it are presented. Mechanisms underlying VT are described as well as the challenges raised by ablation-based procedures.

1.1 Anatomy and function of the heart

The heart is a hollow organ about the size of a closed fist located in the center of the chest, behind the sternum (figure 1.1). It is the most important muscle of the body for being the major actor for a proper blood circulation. Indeed, the heart is a muscle which regularly contracts and relaxes due to an electrical impulse that spreads through the heart tissue, and thus pumps blood through the vessels to the organs, muscles and nerves. The cardiac rhythm has to be sufficiently frequent in order to accomplish this task. In fact, most of human organs cannot survive more than few minutes if they are not supplied with oxygenated blood. An adult heart has a shape of a tilted cone which is about 12cm long, 9cm wide at its

Figure 1.1: Human heart and body [13]

25 Chapter 1 – Cardiac electrophysiology furthest points, 6cm thick, and weighs 300g in average. It is composed of four major chambers, the right atrium, the left atrium, the right ventricle and the left ventricle. The right side of the heart receives blood that has a low concentration in oxygen from veins all over the body, it then pumps the blood through the pulmonary trunk into the lungs where it will become re-oxygenated. The left side of the heart receives oxygenated blood from the lungs, then it is pumped through the aorta back out to the rest of the body through a complex network of arteries, arterioles and capillaries. The veins return the deoxygenated blood to the right atrium and the cycle begins again (ﬁgure 1.2). The heart has four valves, each valve is

Figure 1.2: Human heart anatomy [185] like a one-way door that keeps the blood flowing in the same direction. The valves are made up of two or three small but strong flaps of tissue called leaflets. Leaflets open to allow blood to flow through the valve and close to prevent blood from flowing backward. The openings and closings of the valves are controlled by blood pressure changes within each heart chamber. The tricuspid valve is positioned on the right side between the right atrium and the right ventricle, the pulmonary valve separates the right ventricle from the pulmonary artery, the mitral valve is positioned between the left atrium and the left ventricle, and the aortic valve separates the left ventricle from the aorta. The tricuspid and the mitral valves are called atrioventricular valves, the pulmonic and the aortic valves are called semilunar valves. The heart is composed of three different layers, as shown in figure 1.3:

• the pericardium is a double walled bag that wraps the heart and the blood vessels ﬁlled with ﬂuid that acts as a lubricant for the heart movements. The inner wall of the pericardium is a very dense

26 1.1. Anatomy and function of the heart

connective tissue that spreads the electrical stimulus of the heart;

• the myocardium is the muscle tissue of the heart, it forms a thick layer between the pericardium and the endocardium and is responsible for the heart contraction and relaxation;

• the endocardium is the inner wall of the heart, it protects the heart’s four chambers and is also made with connective tissue.

Figure 1.3: Heart layers [112]

Blood pressure and its circulation rhythm is defined by the precise moments at which ventricles and atria contract and relax. This mechanical activity can be decomposed into two key periods for each chamber: the systole and the diastole, described in figure 1.4. Ventricular contraction causes the atrioventricular valves to close, which signals the beginning of the ventricular systole. The semilunar valves were closed during this period. Continued ventricular contraction increases the pressure above the pressure in the aorta and the pulmonary trunk, causing the semilunar valves to open, which is the late ventricular systole. When the ventricles relax and their pressure drop, blood flowing back towards the relaxed ventricles causes the semilunar valves to close, which is the beginning of ventricular diastole. The atrioventricular valves remain closed during this period. When the pressure in the ventricles becomes lower than the pressure in the atria, the atrioventricular valves open and blood flows into the relaxed ventricles, the atria then contract and complete the ventricular filling, which is the late diastole. As seen in figure 1.4, these mechanical

27 Chapter 1 – Cardiac electrophysiology

Figure 1.4: Systole and Diastole in the atria and the ventricles [112]

phases can be observed in the electrical signal recorded by an electrode on the chest. Indeed, the cardiac mechanical activity is triggered by a complex electrical activity that follows a speciﬁc pattern.

1.2 Cardiac electrical activity

Cardiac cells are filled and surrounded by a ionic solution, mostly sodium Na+, potassium K+, and calcium Ca2+. These charged atoms move between the inside and the outside of the cell through proteins called ion channels. Cells are connected through gap junctions which form channels that allow ions to flow from one cell to another. Two types of cardiac muscle cells can be discriminated: myocytes and pacemaker cells. The heart is mostly constituted of long, tubular spirally oriented myocytes referred to as muscle fibers.

28 1.2. Cardiac electrical activity

1.2.1 Cardiomyocytes

When exposed to electrical or chemical stimuli, small voltage variations occur across myocytes membrane due to ionic exchanges. Theses variations define an action potential that can be described in five steps [122] (figure 1.5):

• phase 4: during its resting period, the cell is polarized. The inside of the cellular membrane is charged negatively if we take the exterior as a reference. The potential difference between the intracellular and the extracellular spaces, called the transmembrane potential, is at this stage about -90mV. This resting state is due to regulated outward leak of K+ and inward current from rectiﬁer channels;

• phase 0: when a cell is stimulated by a current coming from an adjacent cell and its voltage exceeds a certain threshold, sodium fast channels open, which causes a quick depolarization. The potential difference reaches about 30mV. The rise of Ca2+ concentration in the cytoplasm induces changes in the molecules constituting the cell and causes its contraction;

• phase 1: this phase occurs with the inactivation of the fast Na+ channels. A small downward deﬂection of the action potential is due to the movement of K+ ions;

• phase 2: a plateau phase is then observed, in which the cell remains depolarized and contracted, caused by calcium ions ﬂowing into the cell and potassium ions outside of it;

• phase 3: characterized by a slow repolarization, caused by the closing of calcium channels, while potassium channels remain open.

Figure 1.5: Typical myocite action potential

When a cell is depolarized, gap junctions act as low resistive pathways interconnecting cardiomyocytes, causing neighboring cells, in turn, to depolarize and generate an action potential.

29 Chapter 1 – Cardiac electrophysiology

1.2.2 Pacemaker cells

Pacemaker cells on the other hand are self excitable and fire action potentials on their own. This group of cells constitute the natural pacemaker of the heart and determine heart rate in a healthy subject. Pacemaker cells also pass through the same five phases as myocardial cells (figure 1.6). However, they have a different shape and do not require external stimulation to fire action potentials.

Figure 1.6: Typical pacemaker cell action potential

In both types of cells, the period from phase 0 until the end of phase 3 is called the refractory period. It is characterized by the fact that cells cannot produce another action potential during this period. This is due to the closing of sodium channels as transmembrane voltage increases. They remain locked no matter how strong the excitatory stimulus is.

1.3 Cardiac conduction system and Electrocardiograms

Action potentials shape and duration depend on local properties of the cardiac tissue in terms of ionic channels in the cell membrane, electrical conductivity and wavefront curvature. Given cardiac cell properties and cardiac action potentials, a depolarization wave travels through the heart followed by repolarization, at each heartbeat. At a given time instant, the depolarization wave, also called the activation wavefront, separates depolarized cells from those still at rest and is located at the narrow transition region. If we take the extracellular space as a reference, the resting region is uniformly positively charged, whereas the depolarized region is uniformly negatively charged. The depolarization wavefront thus acts as a double layer of current sources or dipoles pointing towards resting cells and originating from depolarized ones. Unlike the depolarization process, repolarization is not a propagating phenomenon. It occurs because action potentials always have a ﬁnite duration and is not caused by neighbouring stimulation. However, if action pulses had the same durations, the repolarization can be also be approximated with a waveform-like phenomenon. This is the case in atria but not in ventricles.

30 1.3. Cardiac conduction system and Electrocardiograms

Indeed, the action potentials of the epicardial cells are shorter than those of endocardial cells, thus acting as a waveform from epicardium to endocardium, contrarily to depolarization. Following the same reasoning as that of the depolarization wave but with opposite charges, the cardiac repolarization process can also be considered as a propagation of a double layer of dipoles, but with opposite polarity than that of the activation wave.

Figure 1.7 describes the initiation and propagation of cardiac electrical activity and the corresponding action potentials in each cardiac region. In a healthy heart, each heartbeat begins in the SinoAtrial Node (SAN) or sinus node, located in the right atrium, which is mostly constituted of pacemaker cells. The generated action potential causes neighbouring cells to depolarize. It then travels following the orientation of the fiber structure of the heart at a velocity that depends on local electrical conductivity. Fibers are organized in a counterclockwise rotation from epi- to endocardium, they have a laminar organization and can be seen as a set of muscle sheets running radially from epi- to endocardium. The electrical signal first spreads through the left atrium wall from the SA node to the AtrioVentricular node (AVN) where it is delayed until the end of atrial contraction. The AVN is also mostly made up of pacemaker cells and can take over electrical propagation when atrial rhythm is disrupted. Action potentials then pass rapidly along atrioventricular bundle which extends from the AVN to the interventricular septum. The atrioventricular bundle, called His bundle, divides into right and left bundle branches with very high conductivity. The depolarization wave is thus rapidly shifted to the apex of each ventricle. Action potentials are then carried by Purkinje fibers from the bundles to the ventricular walls. This propagation of the action potential (figure 1.7) in a wavefront-like manner is responsible for the synchronized mechanical contraction described in 1.1.

When electrodes are placed on the torso surface, the recorded signal with respect to a certain reference on the torso is referred to as the ElectroCardioGram (ECG). Roughly speaking, it measures the contributions of all potential variations in the whole heart scale. Figure 1.7 relates a normal ECG and its key events to the action potential spread at the heart level. One can notice that 3 waves arise from the trace of all the action potentials contributions:

• the P wave which represents the atria depolarization and contraction;

• the QRS complex corresponds to the ventricular depolarization and contraction;

• the T wave represents ventricular repolarization and release.

The durations and the amplitudes of these waves and the linking segments are very important for cardiac pathologies diagnosis, as detailed in section 1.4.

31 Chapter 1 – Cardiac electrophysiology

Figure 1.7: Action potential spread and resulting ElectroCardioGram (ECG) [102]

1.4 Electrocardiography

1.4.1 History

The first time that electrical current was recorded from skeletal muscles was in 1786 by the italian physician Luigi Galvani. Yet, it is only in 1842 thanks to the work of Dr. Carlo Matteucci that cardiac electrical potentials existence is known. He demonstrated that an electrical current accompanies every heartbeat in a frog. The first experiments were conducted in 1878 by John Burden Sanderson and Frederick Page who recorded a frog’s heart electrical current with a capillary electrometer. They were the first to detect what will be later called the QRS complex and the T wave. In 1887, Augustus D. Waller published the first human ECG recorded by electrodes placed on the chest and the back. A breakthrough came in 1895 when Willem Einthoven highlights the five deflections P, Q, R, S and T using a correction formula and an improved electrometer. He then invented the string galvanometer in 1901, which was the first electrocardiograph, and published the first organised presentation of normal and abnormal ECGs in 1906. He was awarded the Nobel prize in 1924 for his work in electrocardiography. The three lead electrocardiogram recorded from the two arms and one foot quickly became a widespread tool for detecting cardiac arrhythmias from the 1900s till the 1930s. At that time, it was commonly admitted that some myocardial infections could not be detected because some electrical information

32 1.4. Electrocardiography were missing in the recordings. The reason was that, in theory, the recorded leads were defined with respect to a remote electrode placed in the infinite medium, which was clearly not the case in practice. In 1934, Dr. Frank N. Wilson introduced the ’Central Terminal’ concept. It consists in considering a reference electrode defined by connecting a 5 kΩ resistor from each terminal of the limb leads to a common point called the ’Central Terminal’. In 1938, the American Heart Association and the Cardiac Society of Great Britain published their recommendation for recording exploring leads from six sites named V1 through V6 across the chest, these were used in addition to the existing three limb leads. In 1942, Emanuel Goldberger adds three signals to the existing 9 ECGs used in clinical practice. This

Figure 1.8: An early commercial ECG device manufactured by Cambridge Scientiﬁc Instrument Company of London, 1911 [193] marked the birth of the standard 12-Lead ECG. Since then, many advances have been made in the instrumentation in terms of device compactness, numerical and electronic improvements. It is now an inevitable tool in any clinical setting and broadened considerably the scope of possibilities in cardiac pathologies diagnosis and monitoring.

1.4.2 Modern Electrocardiography and 12-Lead ECG

Electrocardiography consists in tracing the heart’s electrical activity over a period of time via electrical signals recorded on probes placed on the torso surface. It has revealed itself to be a fundamental tool for cardiac pathologies diagnosis and cardiac monitoring since it is non-invasive, inexpensive and easy

33 Chapter 1 – Cardiac electrophysiology to use. An ECG can help detect:

• arrhythmia: alteration of cardiac electrical system;

• coronary heart disease: obstruction of blood supply in arteries due to fatty substances;

• heart attacks: sudden block of blood supply to the heart;

• cardiomyopathy: enlargement and thickening of cardiac muscle.

The ECG is a tracing of projections of cardiac electrical potentials, called leads, on speciﬁc axes, depending on probes placement. Those leads represent a view of the electrical activity of the heart from a particular angle across the body. The 12-Lead ECG became a standard in clinical practice since the American Heart Association published their recommendation in 1954. It consists in recording signals from 10 electrodes respecting the following placement (see ﬁgure 1.9):

• V1: 4th intercostal space1 to the right of the sternum;

• V2: 4th intercostal space to the left of the sternum;

• V3: midway between V2 and V4;

• V4: 5th intercostal space at the midclavicular line;

• V5: anterior axillary line at the same level as V4;

• V6: midaxillary line at the same level as V4 and V5;

• RL: anywhere above the ankle and below the torso;

• RA: anywhere between the shoulder and the elbow;

• LL: anywhere above the ankle and below the torso;

• LA: anywhere between the shoulder and the elbow.

This conﬁguration enables one to visualize cardiac activity from 12 different angles (ﬁgure 1.10) at each moment of each recorded heartbeat. The observed leads can be decomposed into three main categories:

• 6 precordial leads deﬁned by the 6 electrodes V1, V2, V3, V4 , V5 , V6

• 3 limb leads I, II and III. Lead I is the voltage between the left arm electrode (LA) and the Right Arm electrode (RA): I = LA - RA, lead II is the voltage between the Left Leg electrode (LL) and RA: II = LL - RA and lead III is the voltage between LL and LA: III = LL - LA.

1space between ribs 4 and 5

34 1.4. Electrocardiography

Figure 1.9: Electrodes placement and normal 12-Lead ECG graph [1]

1 1 1 • 3 augmented limb leads: aVR = RA − 2 (LA+LL), aVL = LA − 2 (RA+LL) and aVF = LL - 2 (RA+LA).

Figure 1.10: 12-Lead ECG angular representation [118]

35 Chapter 1 – Cardiac electrophysiology

P wave rate: 60 - 100 bpm 2, variation <10% width <110ms in lead II PR interval 112 to 120ms duration QRS complex duration between 70 and 100ms QT interval less than half the preceding RR 3 interval ST segment slight upward concavity T wave upright in all leads except aVR and V1 symmetrically peaked

Table 1.1: Normal ECG characteristics

As introduced in section 1.2, an ECG recording contains 3 main waves and 4 important segments (ﬁgure 1.11). The duration, amplitude and polarity of those segments and waves observed on recorded signals can directly be exploited to determine cardiac abnormalities.

Figure 1.11: Typical ECG signal and its components [6]

Recent advances in ECG analysis and diagnosis led to the emergence of High Resolution ECG (HRECG) that provides information that cannot be visualized in the 12-Lead ECG. Unlike in High Resolution ElectroEncephaloGraphy (HREEG) where the term high resolution refers to electrodes spatial distribution on the scalp, HRECG stands for computerized signal processing techniques intended to enhance 12-Lead ECGs. Indeed, HRECG is capable of detecting very low amplitude signals in

2beats per minute 3time between 2 consecutive R peaks

36 1.5. Cardiac arrhythmia the ventricles, called late potentials, that are otherwise unobservable when visualizing 12-Lead ECG signals [11]. ECG recording techniques are also taken to another level and can now be obtained from electrodes disposed all over the torso. This practice is not yet standardized and uses a number of electrodes in the range of 60-300 (see ﬁgure 1.12).

Figure 1.12: Cardioinsight ECG recording vest [2]

1.5 Cardiac arrhythmia

Cardiac arrhythmia is any alteration in the heart’s electrical system that prevents it to beat properly. It causes the heart to beat too fast (tachycardia), too slow (bradycardia), or in any other irregular pattern. These abnormalities can inﬂuence the blood quality that is supplied to the rest of the body, or even prevents it from ﬂowing through the rest of the body. Many types of arrhythmia can be harmless, but they can also be the origin of many annoying symptoms like dizziness, chest pain or even fainting. Other types of arrhythmia are more dangerous and predispose to complications such as:

• stroke: incomplete cardiac cycles allow blood to pool in the atria thus forming clots. If these clots go to the brain it can block a vital artery, thus preventing blood from ﬂowing into the brain;

• heart attack: occurs when the heart is not receiving enough oxygen-rich blood due to the blockage of coronary arteries;

37 Chapter 1 – Cardiac electrophysiology

• heart failure: the heart looses its ability to pump blood around the body properly because it became too weak or too stiff;

• cardiac sudden death (or cardiac arrest): the heart beats dangerously fast, and is unable to pump blood strongly enough. This condition needs emergency treatment and deﬁbrillation.

Arrhythmia affects millions of people. It is a major cause of sudden cardiac death and is responsible for about 15% of all-cause worldwide mortality. Arrhythmia arise from lesions assignable to many heart diseases among which: coronary artery disease, a previous heart attack, high blood pressure and cardiomyopathy. They can also occur in patients with normal hearts. In this case, they can originate from genetic diseases, diabetes or endocrine disorders. Arrhythmia can be categorized by rate (slow, fast), mechanism, duration or by site of origin (atria, ventricles). Over 20 types of arrhythmia can be reported. ECG remains the most efﬁcient way to diagnose arrhythmia. Experts are able to analyze the heart rhythm using the 12-Lead ECG graph and deduce the underlying physiopathology. Ventricular Tachycardia (VT) is the most life-threatening arrhythmia. It is responsible for over 80% of sudden cardiac death.

1.5.1 Mechanisms underlying ventricular tachycardia

The vast majority of patients with VT either have coronary artery disease (ischemic heart disease), heart failure, cardiomyopathy or valvular heart disease. VTs frequently emerge from the activation of pathological pacing sites in the ventricles, additionally to those in the SAN. Two mechanisms can be the origin of VT:

• abnormal automaticity: cells in the ventricle become self-excitable and act as pacemaker cells;

• scar related: most of scar area in the heart is dead and is thus electrically nonconducting. However, some narrow electrical pathways form one or multiple mini re-entrant circuits spontaneously within the ventricular myocardium.

These abnormal impulses increase ventricular contraction rate, which accompanied by already disrupted ventricular function results in an incomplete ﬁlling of the ventricles. Thus, stroke volume and cardiac output are reduced. When VT is sustained and not treated, ventricles start quivering (ventricular ﬁbrillation), which leads to cardiac arrest. VT is characterized by a fast heart rate (>120 bpm) and at least three wide QRS complexes in an ECG row.

38 1.5. Cardiac arrhythmia

Figure 1.13: Ventricular Tachycardia illustration and corresponding ECG signal

1.5.2 Treatment

All forms of treatments for VTs aim at restoring a sinus node. The therapeutic choice depends on the myocardial condition and the risk of recurrence. Three treatment options exist for VT:

• pharmacological: anti-arrhythmic medications are used to decrease conduction velocity or changing the duration of the effective refractory period. However, they come with side effects and can even be proarrhythmic. Despite the development of over 100 antiarrhythmic medications, efforts at prevention of serious and fatal ventricular arrhythmias have proved disappointing;

• Implantable Cardioverter-Deﬁbrillator (ICD): an electronic device is surgically implanted in the chest. It delivers precisely calibrated electrical shocks when needed to restore normal heart rhythm. The ﬁrst ICD was implanted in 1980 and technology evolved considerably since then. Since this therapy does not alter the arrhythmogenic substrate, it does not cure the arrhythmia. However it prevents the cardiac arrest that may follow from a sustained ventricular activation;

• catheter ablation: it consists in inserting a long thin wire through the veins of the leg and targeting the site of the origin of the VT. The site of origin is then eliminated by radiofrequency, laser, thermal or ultrasound stimulation. This concept was introduced in the 1990s with high rate of success. Contrarily to the previous options, catheter ablation is curative, hence the emerging interest and preference for this type of treatment.

39 Chapter 1 – Cardiac electrophysiology

1.5.3 Catheter ablation procedure

Patients are placed under local or general anesthesia to reduce movement and discomfort. Catheters are introduced intravenously through the femoral artery, surgeons guide the catheter and reach the region of interest. Then they proceed to the estimation of a cardiac voltage map, using a multielectrode catheter that enables to record electrical potentials at several positions of the heart’s internal surface. Another catheter is introduced and is used for the ablation and electrical stimulation. The recorded potentials, called electrograms, can either be unipolar or bipolar. Unipolar electrograms are those recorded from one electrode placed on the heart with respect to a reference electrode, typically located remotely from the heart. They provide information for comparing electrical activity over space, but they contain both local information and far fields which are attenuated fields from remote events. A bipolar electrogram is a measure of the potential difference between two closely spaced electrodes (within the mm range) on the heart. They have the advantage of giving a good insight into local activity, with reduced far field. As the mapping catheter is moved on the heart surface, its position and the corresponding electrograms are used to build a virtual 3D image of the heart surface potential values. Additionally, the depolarization time of the inspected region is computed, according to some time reference (R peak of lead V1 for instance). On average the whole procedure lasts 4 hours. The voltage map helps defining ischemic regions by detecting recorded potentials that are of very low amplitude (< 1, 5mV ). The foci responsible for arrhythmia are often located within or in the border of the scar tissue. The ablation target(s) can then be identified by several methods:

• earliest Local Activation Time (LAT): during VT, the depolarization wave originates from a ventricular focus and not from the SAN. Consequently, the tissue zone that depolarizes the earliest is likely to be a successful ablation site;

• pacing: by exciting cardiac tissue using the stimulation catheter, cardiac electrical wavefront is forced to originate from this artiﬁcial pacing site. One way to determine the anatomical site of origin of the VT is by comparing the ECG morphology during artiﬁcial pacing with that of the clinical VT. A successful ablation site would be the one that produces a very close clinical 12-Lead ECG morphology;

• ECG Imaging (ECGI): ECGI is a recent technique of rising interest in clinical practice. It consists in non-invasively reconstructing 3D maps of cardiac electrical information using ECGs recorded by a high resolution vest of electrodes. ECGI aims at predicting the optimal ablation site by ﬁnding the earliest activated regions using the obtained 3D cardiac maps. This solution is still at an early stage because of many underpinning challenges that will be discussed in the next section. Except

40 1.6. Catheter ablation and ECGI challenges

for Cardio-Insight [3], which is the ﬁrst and only ECGI device in the market since its release in 2017 [3], this technique remains in the ﬁeld of active research.

1.6 Catheter ablation and ECGI challenges

Even if catheter ablation is a powerful and the only curative clinical tool for VT, it remains unsatisfactory with 50-70% rates of success, representing a burden in the clinical process both for the surgeon and the patient. This technique requires long and invasive electrophysiological studies conducted through catheter mapping in order to detect the anatomical site of origin of the VT. Also, intracardiac electrical information is only available locally during catheter mapping. This means that each LAT is defined during admittedly similar but yet asynchronous heartbeats. Thus, a LAT map is valid under the assumption that it is the same for two similar heartbeats. When the scar is diffuse, finding the ablation site becomes very challenging since several pathways can be the origin of a VT. Clinical VT morphology is not always available, which prohibits the use of the pacing method. Because of high recurrence rates and a lack of clinical consensus on the optimal ablation strategy, a deeper understanding of the arrhythmogenic substrate and a further insight in cardiac electrical activity in the case of arrhythmia is a milestone in catheter ablation based therapy. In this connection, with the development of ECG recording devices and imaging modalities, a recent and very promising technique referred to as ECGI emerged in the scientific community. This has a great potential for fast, accurate, painless and affordable methods for the localization of cardiac electrical events which would genuinely result not only in more suitable diagnosis but also better procedure planning and easier assessment of ablation success. Three materials are mandatory in order to perform an ECGI procedure on a patient:

• a geometrical/anatomical model of the patient’s torso built from data acquired through Magnetic Resonance Imaging (MRI) or Computed Tomography (CT) scans;

• multiple body surface potentials that provide an attenuated version of cardiac electrical activity;

• a mathematical model that translates the physical relationship between cardiac electrical sources and the observed potentials.

However, even if this approach is in theory promising for its clinical applications, its progress is stunted by three major difﬁculties. The ﬁrst one is that ECGI requires modeling the physical link between the observed torso surface potentials and the cardiac electrical sources to be recovered. This model, referred to as the forward model of electrocardiography, is subject to many uncertainties. The main ones are geometrical inaccuracies due to image acquisition and heart deformation during contraction and release, body tissue

41 Chapter 1 – Cardiac electrophysiology conductivities and anisotropies, personalized cardiac fiber structure and ECG measurement noise. Some studies have addressed the topic of how these uncertainties influence the forward model, but more effort is to be made to quantify how faithful data obtained by forward models can be when compared to real data. The second point is that ECGI requires solving a mathematical inverse problem that is hindered by two common aspects. The first one is the non-uniqueness of the mathematical solution, meaning that several cardiac source configurations could result in the same observed ECG recordings. Besides, due to the attenuation property of the medium separating the heart and the torso surface, small changes in surface potentials arising from modeling and measurement errors can result in arbitrary large changes in the equivalent source solution. Many algorithms were investigated to address this topic, but further efforts can improve precision and reliability. Moreover, each study is conducted on a different data set, which makes algorithms performance comparison difficult. Finally, evaluation methods for the validation of the solution are still very limited. In fact, the only way to truly validate an ECGI procedure in a clinical setup is to compare noninvasive reconstructions with simultaneous whole heart electrical distribution recordings. This type of data is still inaccessible in-vivo for both practical and ethical reasons. Alternatively, validation is either based on comparing measured results from experiments or partial clinical measurements, or based on the comparison of ground truth data obtained by mathematical simulation. In the latter case, computational models are used to generate both cardiac electric sources, the propagation of the depolarization wavefront and the resulting body surface potentials. In this case, validation consists in comparing ground truth sources with the ones reconstructed based on simulated torso surface potentials and the corresponding forward model. This entails producing meaningful cardiac electrical distributions that result in sufficiently realistic ECGs. The difficulties in ECGI raised by the inherent mathematical problem and the need for modeling made ECGI one of the most difficult challenges in the research field of modern electrocardiology. On the one hand, improvements in cardiac modeling are necessary to better understand arrhythmogenesis and is a way to generate synthetic data for the evaluation of inverse algorithms. On the other hand, inverse algorithms are to be compared in different scenarios to assess their performance, and new techniques integrating more physiological knowledge are still to be proposed. In this thesis, we address the first topic by proposing a low-complexity model that simulates realistic cardiac activity and ECGs. Secondly, we compare existing algorithms for solving the ECG inverse problem based on synthetic data generated by our model, and we investigate new approaches that incorporate spatiotemporal a priori on cardiac electrical distributions.

42 CHAPTER 2

CARDIACMODELING

Since the giant step made by Alan Hodgin and Andrew Huxley in the 1950s with their work on the ionic basis of the squid axon action potential and its mathematical model [79], computational modeling of biophysical systems has seen an ever-growing development. In this age of exploding computational power and fast growth of new technologies, modeling and simulation is taken to another level, becoming a useful tool part of clinical diagnosis and treatment [20, 58, 88]. Because cardiovascular disease is the major cause of deaths worldwide, a particular attention is given to cardiac modeling in order to understand physiopathologies at different scales and to provide valuable aid in pre- per- and post-operative procedures. Cardiac modeling is a vast scientific topic that covers disciplines such as mathematics, genetics, chemistry, electricity, mechanics and fluid dynamics. In the context of ECGI, computer models play the essential role of a tool that performs low-cost and repeatable experiments in a controlled environment. Simulation data may provide information that is difficult to obtain experimentally and are extensively used as a means for ECGI algorithms verification [22, 71, 138, 154]. Within the framework of evaluating inverse algorithms using model based data, a convenient model should be able to generate sufficiently realistic cardiac patterns and their corresponding ECGs with sufficiently low complexity in order to conduct various ECGI case scenarios simulations. Numerous ECGI studies and cardiac models can be found in the literature. However, these models are either too complex with simulations times in the range of several hours, or do not produce sufficiently realistic cardiac dynamics. In this chapter, we first review the main cardiac models used in the perspective of ECGI studies. Then, a framework for building a simplified heart-torso propagation model destined to generate various scenario data-sets for inverse problem evaluation is presented.

2.1 State of the art

A famous application of cardiac modeling is the so-called ECG forward problem. The goal is to compute the electric ﬁeld on remote virtual sensors resulting from a given cardiac electrical activity. This is a valuable tool for ECGI. In this context the forward problem usually entails modeling the cardiac

43 Chapter 2 – Cardiac modeling electrical activity at the whole heart scale. Since the true primary electrical cardiac sources are the chemical exchanges between cells and the intracellular and extracellular spaces, an exhaustive model representation would consider each individual cell and their ionic exchanges, as well as its integration into a whole organ and then the whole body. Clearly, this exhaustive model representation is not feasible, knowing that the heart is composed of millions of cells. Therefore a major question in this research ﬁeld is related to the deﬁnition of the appropriate level of model representation.

2.1.1 Cardiac cell models

As stated previously, whole heart simulations cannot be based on an exhaustive representation of all cardiac cells. The cardiac model has thus to be discretized in small pieces of intermediate size between the cell and the organ (tissue-level). Some cellular models can be an approximation of these considered tissues. Since the 1950s, several models with increasing complexity and mathematical sophistication were proposed to describe the cellular action potential. Most modern membrane and cell models are inspired by the Hodgin-Huxley formalism [79] and are derived from the corresponding equation:

∂vm 1 = − (iion + istim) (2.1) ∂t cm

where vm is the membrane potential, cm is the membrane capacitance, iion = ICa + INa + IK ··· is the sum of ionic currents flowing between intracellular and extracellular spaces, and istim is an external stimulus current that causes the myocardial cell to depolarize. Depending on the amount of physiological complexity and detail, cell models can be distinguished into physiological models and phenomenological models. Physiological models tend to model the opening and closing of individual ionic channels and describe how physiological variables evolve during an action potential. They usually address cells of a specific component of the cardiac conduction system, and have been adapted to different mammalian species. The first model of the cardiac cell was proposed by Noble in 1962 [116]. It describes the behavior of a generic Purkinje cell using four variables including the three primary currents, namely calcium, sodium and potassium. Recent models include more detailed information such as gating variables, intracellular and extracellular ionic concentrations. In human cell modeling, the most popular models are Luo and Rudy [99], Priebe and Boeckelmann [136], Ten Tusscher [167] for ventricular cells, and Courtemanche [39] and Nygren [119] for atrial cells, to only cite a few. Phenomenological models on the other hand concentrate on reproducing characteristic shapes of myocardial cells’ ionic currents and action potentials without fully describing the underlying physiology. The FitzHugh-Nagumo [54], the Aliev-Panfilov [4] and the Mitchell-Schaeffer [109] models are good examples of simple models with a

44 2.1. State of the art reasonable number of variables and equations that reproduce physiologically shaped action potentials with very low computational expense. Another advantage of these models is that the correct tuning of their parameters can generate action potentials corresponding to cells of different regions of the cardiac conduction system. Thus, they are much lighter when incorporated in whole heart models compared to physiological models, and, under certain conditions, give results of comparable accuracy [124].

2.1.2 Macroscopic cardiac source models and extra-cardiac potentials

The earliest class of cardiac models that represent cardiac activity were dipole models. The dipole as a cardiac source was introduced by Einthoven in 1913 and consists in considering the heart as two electrical charges of opposite polarities and equal intensity [180]. This model evolved into the moving dipole, oblique dipole and multiple dipoles [32, 59, 82]. The ﬁxed dipole model and its further versions give rather good qualitative interpretations of the ECG and provides a good idea of the physical intuition behind observed potential on the torso. However, physiological meaning and precise quantitative information on cardiac electrical distribution are missing. A popular macroscopic source model that is closer to the heart’s electrical behavior is the Uniform Double Layer (UDL) introduced by Wilson [188]. At each time instant, this model describes cardiac electrical activity by the location, curvature, and direction of the depolarization wavefront that separates depolarized regions from those still at rest. This wavefront is modeled by a boundary carrying dipoles of uniform strength, remote extra-cardiac potentials are computed in an integral equation form based on solid angle theory [81]. This model was mainly used to describe ventricular activity during depolarization. The Equivalent Double Layer (EDL) is a successor to the UDL model and extends the concept of wavefront propagation to describe the repolarization phase [60, 61, 177]. The EDL model relies on the assumption of equal anisotropy between intracellular and extracellular spaces all over the heart. This way, source currents can be expressed as a weighted sum of transmembrane potentials gradients of active myocytes within the myocardium. In a homogeneous heart, all source activity within the heart can be fully described by the distribution of transmembrane potentials on the heart surface S. A basic way to express remote extra-cardiac potentials in an inﬁnite homogeneous external medium is thus given by:

Z 0 −1 0 φ(x , t) = σin(x)vm(x, t)dω(x , x) (2.2) 4πσ S

0 where x is an external observation point, x the source location, σ and σin(x) are respectively the remote and the local intracellular conductivity tensors, vm(x, t) is the transmembrane potential at time t, S is the heart surface and dω(x0, x) denotes the so-called solid angle subtended by small segments of S. Ultimately, equation (2.2) can be re-expressed so that remote potentials φ(x0, t) can be directly linked

45 Chapter 2 – Cardiac modeling to the local activation times over S.

One of the most popular formulations and probably the dominant source model used in the ECG inverse problem is the epicardial surface formulation [8]. Here, the idea is to consider epicardial potentials on a closed surface inside the body as a virtual electrical source. This formulation is based on the principle that in an electrically passive volume, there is a unique relationship between potentials on the outer surface and the potential distribution over an inner closed surface. This means that a potential distribution over any closed surface encompassing the heart fully reﬂects the inner primary currents in the heart, and results in the same ECGs as those produced by these inner primary currents. This surface is ideally chosen to be the epicardium. The resulting forward problem is expressed by the generalized Laplace equation (2.3):

∇ · (σt∇φt) = 0 (2.3) where σt and φt are the extra-cardiac conductivity and potential, respectively. Equation (2.3) reﬂects the passive nature of regions outside the heart. This is completed with Neumann boundary condition on the exterior torso surface and Dirichlet boundary condition on the epicardium: φt = φex where φex are the known epicardial potentials.

Perhaps the most complete and the more physiologically meaningful approach used to model cardiac activity is the bidomain method [108, 108, 149, 172]. First introduced by Schmitt [149] and formulated mathematically by Tung [172], it consists in considering the myocardium as a syncytium of two interpenetrating spaces separated by the cell membrane: the intracellular and the extracellular spaces. This is a simpliﬁcation that can be interpreted by considering average properties of clusters of cells instead of describing each cell individually. This leads to a continuous deﬁnition of extracellular and intracellular potentials and conductivities and allows for the possibility of considering unequal anisotropy ratios. The major Partial Differential Equations (PDEs) of the bidomain model describe the primary currents in the heart volume and are given by:

∂v ∇ · (σ ∇v ) + ∇ · (σ ∇φ ) = χ c m + i + i (2.4) in m in ex m ∂t ion stim

∇ · (σin∇vm) + ∇ · ((σin + σex)∇φex) = 0 (2.5)

where σex and φex are the extracellular conductivity and potential, σin and φin the intracellular conductivity and potential, vm = φin − φex is the transmembrane potential, cm is the membrane capacitance, and χ is the surface to volume ratio representing how much membrane surface area is present per volume of cardiac tissue. istim is a stimulation current that is only non zero in the chosen region from which propagation is initiated. Equation (2.4) is a reaction-diffusion equation that can be seen as a generalization of the cable equation in a 3D space [113]. To compute extra-cardiac potentials,

46 2.1. State of the art this model is completed by the generalized Laplace equation in the extra-cardiac domain 2.3. Eventually, coupling conditions are needed to obtain a full heart-torso model. Note that this full heart-torso model does not only provide a relationship between ECGs and cardiac sources at one time instant but also includes a way to simulate cardiac propagation over time. When equal anisotropy is assumed, the bidomain model reduces to a lower-complexity version called the monodomain model. This model will be treated in more detail in the next section. In the field of ECG simulation based on cardiac models, most works neglect the mechanical character of the heart. Yet, mechanical contraction causes a modification of the distance between BSPM electrodes and the heart and also a change of the electrical activation wavefront. This topic has been addressed in few studies [50, 103], but is still not deeply investigated. A reason why this assumption is still widely used is due to the high complexity of a model involving electromechanical coupling in a detailed realistic anatomical heart. Moreover, there is no consensus on the quantitative effects of cardiac motion on the ECG in real conditions. Also, in the ECGI context, further improvements are still to be made in the case of the fixed heart before focusing on electromechanical models.

2.1.3 Simulating propagation in the heart

Macroscopic source models describe the link between a cardiac configuration and the corresponding remote ECG potential at one time instant. In order to be able to generate a full ECG, it is necessary to simulate electrical propagation in the heart over time. One of the first models consisted in cellular automata [94]. The concept uses a space grid in which each element is an automaton describing the action potential generated by piece of cardiac tissue. An automaton’s behavior depends on a set of rules that link its current state with that of its neighbors. Propagation is initiated at a chosen element and evolves through local coupling of each element of the grid with its neighbors. The main advantage of such a discrete method is its high speed in large scale simulations compared to PDE based models like the bidomain. Very realistic simulations were performed using such method on an anatomically based hearts [94, 186]. However, the shape of the propagation wavefront may be very dependent on the structure of the defined grid. Another limitation comes from the discrete character of such an approach. More precisely, action potentials shape are defined by a small number of states and don’t arise from dynamic cellular electrophysiology. This makes this type of model unsuitable for simulations in which electrical dynamics are important such as in ischemic tissues. Another approach originally conceived for modeling electromagnetic waves propagation is the Huygen’s principle [120, 160]. Because of the wave-like nature of the electrical cardiac wavefront, this model is convenient for a fast generation of a simple temporal pattern of the wavefront evolution. Given the wavefront location and direction at a given time instant, the idea consists in drawing wavelets of

47 Chapter 2 – Cardiac modeling appropriate size around the current location. Regions within these wavelets depolarize in turn in the next time instant. A similar approach consists in solving an eikonal equation instead of Huygen’s parabolic equation. The difference is in the inclusion of a diffusive term in the eikonal equation. Contrarily to Huygen’s method, this approach allows the wavefront speed to be influenced by its curvature, and also gives the wavefront position at all times by a one step calculation [29, 30, 169]. Similarly to cellular automata models, Huygens and eikonal equations models are not built from electrophysiology background. Moreover, it only gives information about the wavefront location at each time step, but don’t provide local electrical features like the transmembrane or the extracellular potential. Although computationally more demanding, the bidomain/monodomain models are the preferred models when the objective is to simulate cardiac propagation with a high degree of accuracy, physiological meaning and realism [12, 15, 96, 130, 132]. Not only do they provide a relationship between cardiac activity and the resulting remote body surface potentials but also include a means to simulate cardiac propagation through reaction-diffusion equations (2.4) and (2.5). These models are usually coupled with some phenomenological model in order to define the term iion in (2.4). The advantages are that they are continuous models, and provide extracellular and transmembrane potentials based on parameters and equations that are meaningful with respect to the physics behind the cardiac propagation process. Activation and recovery times can be computed from transmembrane potentials upstrokes and downstrokes. Additionally, cardiac anisotropy can be taken into account in order to best represent cardiac fibers [53, 78, 133] and thus electrical wave direction.

2.1.4 Numerical methods

Because the heart and the body shapes are highly irregular, analytical solutions cannot be obtained for PDEs. Instead, the studied medium is discretized into a mesh of small elementary pieces that serve to obtain an approximated numerical solution. Several methods can be employed, they can be classiﬁed into surface methods such as the Boundary Element Method (BEM), and volume methods like the Finite Difference Method (FDM), the Finite Element Method (FEM) and the Finite Volume Method (FVM). The choice of these methods usually arises from the chosen diffusion model. Double layer and epicardial source descriptions usually use the BEM to solve the forward problem in isotropic cases, whereas bidomain/monodomain propagation in the heart volume is performed using the FEM. Surface methods consist in discretizing only the interfaces of interest in the studied medium, namely the heart surface and the torso surface. Rarely, inner structures interfaces like lungs and bones are also included in order to take into account the non-homogeneity of the volume conductor [163]. However, volume methods are the only way to include electrical anisotropy in the heart and torso. Indeed, volume discretization is

48 2.2. Proposal of a simplified heart-torso model not only performed on the interfaces but also in the volume between them. This allows to set different conductivity tensor values as a function of the position in the 3D space. The simplest and most straightforward method is the FDM, it consists in partitioning the volume into a uniform cubic grid. Partial derivatives are approximated with finite differences. Then, convergence of the numerical scheme is obtained when the distance diminishes. FDM is a simple and very intuitive method, but it is inadequate when dealing with irregular geometries [55]. Although it has been successfully used in some cardiac modeling studies [144, 145], it is not as adapted as the FEM [31, 83, 91] and the BEM [8, 51, 63, 134, 163], mainly because very high computational costs are required when using the FDM, in order to obtain an equivalent accuracy as FEM and BEM, when using realistic anatomies. The FEM and the BEM are quite similar in the sense that they both solve a weak form of the original PDE. This weak formulation consists in minimizing a weighted residual expression of the original problem [93]. The difference occurs mainly in the number of unknowns involved. Since the FEM is a volume method, the number of degrees of freedom is equal to the number nodes in the whole volume mesh, which is obviously larger than that of the BEM. The matrices involved in the BEM forward problem are thus much smaller than those in the FEM. However, FEM matrices contain very few non-zero elements whereas BEM matrices are fully populated, which makes both approaches computationally comparable [150]. As the FEM, the FVM method is also based on a finite element volume mesh. The implementation is simpler and it is ideally used in flux conservation contexts since conservation is solved for each element of the mesh. FVM is more accurate than FEM in applications where flux is the important variable but is more computationally demanding. Hence, FVM is generally more used in fluid dynamics and FEM is more adapted to structural mechanics. For this reason, FVM is more rarely used in cardiac electrical simulations, but seems to give comparable results [37, 38]. To sum up, each method has its advantages and drawbacks and the method to employ depends on the targeted application [72]. Interestingly, combination methods were also used [53, 137, 162] in order to overcome the associated limitations. Ultimately, all these numerical methods provide a linear expression that links cardiac sources and the observed ECGs.

2.2 Proposal of a simpliﬁed heart-torso model

In this thesis, one important objective is to design a heart-torso model that is capable of generating meaningful data that can be used to evaluate ECG inverse problem algorithms. Many works were devoted to simulate ECGs with a high level of detail including ﬁber orientation, high resolution meshes, novel heart-torso coupling techniques and anatomically detailed geometries [15, 27, 33, 84, 90, 96, 126, 129, 131–134, 148, 170, 171]. Nevertheless, when the goal is to evaluate, compare performance and

49 Chapter 2 – Cardiac modeling study parameters influence on ECGI algorithms, these models are far too complex and computationally demanding. Moreover, only few works are able to synthesize a complete and meaningful set of 12-Lead ECG signals, since most of them only represent the ventricular activity. An interesting approach for building a model that produces fairly good ECGs in an idealized geometry was presented in [161]. This work shows that simplified models, even in idealized geometries, are able to produce complete ECGs with coherent cardiac patterns. Even so, the presented model in [161] is still computationally expensive due to the bidomain approach, and uses a phenomenological ionic model whose parameters tuning is not described. Moreover, the bidomain model is suitable for anisotropic cardiac tissues, which was not the case of the considered piecewise homogeneous model. In this chapter, we propose a simplified, fast, and evolutive model that is capable of simulating various scenarios of cardiac mappings. To this end, we choose the adapted monodomain model which is a modification of the bidomain approach with lower complexity [12]. The monodomain model is coupled with the phenomenological FitzHugh-Nagumo (FHN) model for describing transmembrane currents. Parameters are defined for each region of the cardiac conduction system. This model includes both atria and ventricles, and is able to synthesize meaningful ECGs while keeping a complexity level that makes feasible the application of screening sensitivity analysis and parameter identification methods. Contrarily to most studies in the field of ECG simulation based on whole heart models, we also pay a particular attention to model equations derivation and spatiotemporal discretization.

2.2.1 Heart-torso geometry and meshing

Inspired by [161], we designed a new 3D geometry of heart-torso model (figure 2.1). It consists of an elliptic-shaped torso and lungs and an ellipsoid heart that contains the main cardiac regions of the cardiac conduction system: atria, Atrio-Ventricular Node (AVN), ventricles (endocardium and epicardium), bundle of His, bundle branches and Purkinje fibers (see figure 2.2). Since electrical connection between atria and ventricles is only allowed through the AVN in the real heart, an empty space was created between the two chambers so they are physically connected only through the AVN. Blood chambers and the valve plane were considered as vacuum when simulating cardiac propagation. We integrate an isotropic but regionally heterogeneous structure representation, without adding fiber orientation information. Although fibers orientation is of great interest when the purpose is to represent cardiac activity in the most realistic way, its influence on inverse algorithms performance is still to be proved. More precisely, in-silico, the estimation of cardiac activity is to be compared with the one that produced the simulated ECGs. Whether it was based on isotropic or anisotropic model is not likely to be influential on reconstruction performance.

50 2.2. Proposal of a simpliﬁed heart-torso model

Figure 2.1: Heart-torso geometry

A major improvement in our geometry with respect to [161] is that we add another layer at the ventricular level to distinguish between the epicardium and the endorcardium. In fact, the difference in electrical properties of ventricular action potentials from the inner and outer layers is a major point to the generation of appropriate ECG signals, and particularly the T-wave. More precisely, action potentials duration diminishes smoothly as one moves from the outer layer to the inner layer of the heart. This gives the effect of a repolarization wavefront traveling from the outside to the inside of the heart, contrarily to depolarization.

This geometry is built using ANSYS DesignModeler software after a series of elementary operations. It is then exported to ANSYS Mechanical in which a meshing step is performed before conducting FEM based simulations. A ﬁrst order tetrahedral uniform mesh is generated. It resulted in 4374 nodes and 17418 elements in the heart domain and in 43760 nodes and 244901 elements of 1cm side length for

51 Chapter 2 – Cardiac modeling

Figure 2.2: Heart geometry and its seven regions the whole geometry (ﬁgure 2.3).

2.2.2 Model equations

The bidomain and the monodomain models are widely used to simulate cardiac electrical propagation in current research and are largely accepted for their physiological relevance. Before providing the monodomain equations used in our model, we first give some insight on the bidomain model and explain how it reduces to the lower complexity monodomain model. The bidomain model is based on the assumption that cardiac tissue can be partitioned into two separate conducting media: the intracellular space, located inside the cardiac cells, and the extracellular space that connects cells between them. These two spaces are separated by the cells membranes through which current flows from one space to another. At the macroscopic scale, a homogenization process [114, 117] of the bidomain model considers that these two spaces overlap so that each point of the 3D space has intracellular and extracellular variables. The bidomain model arises from Maxwell’s equations under quasi-static conditions. The quasi-static assumption consists in neglecting the temporal dynamics of the electrical and the magnetic field in the body. Indeed, considering the size of the human torso and the range of physiological variables in it, capacitive, inductive, and propagation effects of these fields can

52 2.2. Proposal of a simpliﬁed heart-torso model

Figure 2.3: Heart-torso mesh largely be ignored. A detailed derivation of the bidomain equations and their coupling with the torso domain is provided in appendix A. The quasi-static approximation is formally justiﬁed in appendix B. The full heart-torso bidomain model provides the relationship between potentials in the heart domain, namely the extracellular potentials φex and the transmembrane potentials vm, and the potentials in the torso domain φt. The resulting equations within the heart domain H can be given in the following form:

∂u + f(v , u) = 0 (2.6) ∂t m ∂v ∇ · (σ ∇v ) + ∇ · (σ ∇φ ) = χ c m + i (v , u) + i (2.7) in m in ex m ∂t ion m stim

∇ · (σin∇vm) + ∇ · ((σin + σex)∇φex) = 0 (2.8)

where φex is the extracellular potential, σin and σex are the intracellular and extracellular electrical conductivities, respectively, vm = φin − φex is the transmembrane potential that measures the electrical potential difference between the intracellular and the extracellular space, cm is the membrane capacitance, χ is the surface to volume ratio representing how much membrane surface area is present per volume of cardiac tissue. istim is a stimulation current that is only non zero in the chosen region from which propagation is initiated. iion is a function of the transmembrane potential vm and an introduced variable u and represents the ionic current exchanges between intra- and extracellular spaces. This variable u is a slow variable and characterizes the dynamics of ion channels opening and closing. The deﬁnition of the expression iion(vm, u) and the function f in equation (2.6) are deﬁned according to the chosen ionic model. Equation (2.7) represents the reaction-diffusion dynamics that occur through the

53 Chapter 2 – Cardiac modeling membranes, and equation (2.8) represents the charge conservation in the heart domain. In the torso domain T , which refers to the extracardiac region in the body, the electrical potentials φt are governed by the generalized Laplace equation:

∇ · (σt∇φt) = 0 (2.9)

The heart and the torso domain equations need to be completed by appropriate boundary conditions on the surface separating the heart and the torso, and on the body exterior surface. These boundary conditions are expressed by: On the heart surface ∂H

φt = φex (2.10)

n∂H · (σt∇φt) = n∂H · (σex∇φex) (2.11)

On the exterior torso surface ∂T

n∂T · (σt∇φt) = 0 (2.12)

where n∂H is the unit outward pointing normal vector to the heart surface and n∂T is the unit outward pointing normal vector to the torso surface. Equations (2.10) and (2.11) represent the ﬂux and potential continuity between the heart and the torso domain, and equation (2.12) reﬂects the fact that no current leaves the body, as the medium outside the body is vacuum.

From equations (2.7) and (2.8), variables φex and vm are fully coupled. Moreover, variables φex and φt are coupled through boundary conditions (2.10) and (2.11). This implies that the computation of variables φex, φt and vm cannot be performed independently. The discretization of the bidomain heart-torso problem thus results in a huge linear system to be solved for the three variables at each time step. Moreover, although the coupling between φex and φt has been assessed [101], no clear dependence between φt, φex and vm has been proved in physiological studies.

Alternatively, the monodomain model is a good compromise between computational complexity and physiological meaning [128]. Directly arising from the bidomain concept, the monodomain is based on an equal anisotropy assumption in the myocardium. This assumption lies in the existence of a constant

λ such that: σex = λσin everywhere in H. Equation (2.8) can then be rewritten as follows:

∇ · (σin∇vm) + (1 + λ)∇ · (σin∇φex) = 0 (2.13)

54 2.2. Proposal of a simpliﬁed heart-torso model

Substituting ∇ · (σin∇φex) in (2.7) based on the previous equation gives:

λ ∂v ∇ · (σ ∇v ) = χ c m + i (v , u) + i (2.14) 1 + λ in m m ∂t ion m stim

An additional boundary condition is considered, stating that transmembrane current doesn’t leave H.

n∂H · (σin∇vm) = 0 (2.15)

This expression has no physiological justiﬁcation, but is convenient in the sense that variables vm on the one hand, φex and φt on the other hand are decoupled. Consequently, the two bidomain reaction-diffusion equations (2.7) and (2.8) linking φex and vm reduce to only one reaction-diffusion equation that only involves vm. More importantly, bidomain and monodomain models seem to provide very close results in whole heart simulation studies [128, 133].

Finally, an appropriate choice for defining f and iion(vm, u) is required in (2.6) and (2.7). Since physiological models are two complex for whole heart simulations for our intended application, phenomenological models are used instead. The oscillatory FitzHugh-Nagumo model [54] adds only one ODE equation of the type (2.6) to the system. It is continuous, analytically tractable and presents interesting uniqueness and existence properties [17, 34, 65]. Also, it presents the nice advantage of a graphical analysis of its dynamics using a phase plane representation [141]. Moreover, it is capable of precisely fitting physiological action potentials and has been successfully used in 3D cardiac propagation models [43, 142, 161]. Within the FHN framework, the function f in (2.6) and the term iion(vm, u) in (2.7) are defined as follows:

vm − b2 vm − b2 iion(vm, u) = kc1(vm − b2)(a1 − )(1 − ) + kc2u(vm − b2) (2.16) a2 a2 vm − b2 f(vm, u) = −ke( − du − b1) (2.17) a2 where a1, a2, b1, b2, c1, c2, d, e and k, are non physiological parameters, but their tuning can result in characteristic shapes of cardiac action potentials of different tissues [44]. To summarize, the ﬁnal system of our proposed full heart-torso model is given by:

55 Chapter 2 – Cardiac modeling

In the heart domain H In the heart domain H

∂u + f(v , u) = 0 (2.18) ∂t m λ ∂v ∇ · (σ ∇v ) = χ c m + i (v , u) + i (2.19) 1 + λ in m m ∂t ion m stim

On the heart surface ∂H

φt = φex (2.20)

n∂H · (σt∇φt) = n∂H · (σex∇φex) (2.21)

n∂H · (σin∇vm) = 0 (2.22)

In the volume conductor domain T

∇ · (σt∇φt) = 0 (2.23)

On the exterior torso surface ∂T

n∂T · (σt∇φt) = 0 (2.24)

where f and iion(vm, u) are deﬁned according to (2.16) and (2.17).

2.2.3 Spatio-temporal discretization

In this section, we provide the spatio-temporal discretization of the previous system. As stated previously, reaction-diffusion equations (2.18) and (2.19) can be solved ﬁrst. Thereafter, extracellular potentials φex and φt can be computed using the obtained values of vm. We ﬁrst focus on reaction-diffusion equations in H. Regarding the temporal integration, a semi-implicit Euler scheme is used. Equations (2.18) and (2.19) become:

(n) ! (n+1) (n) vm − b2 (n) u = u + ∆tke − du − b1 (2.25) a2 ! λ v(n+1) − v(n) ∇ · (σ ∇v(n+1)) = χ c m m + i (v(n), u(n+1)) + i (2.26) 1 + λ in m m ∆t ion m stim where ∆t is the time resolution and n the current time step. Since cardiac electrical activity is to be described in the whole heart volume, the FEM is adequate to our problem. The FEM is a numerical

56 2.2. Proposal of a simpliﬁed heart-torso model method to solve equations with partial derivatives. It uses Galerkin variational analysis [164], and is based upon two main procedures: 1) the generation of a ﬁnite element mesh that results in a set M of

M discrete points i ∈ M called nodes whose coordinates are (xi)i∈M and N finite elements, and 2) the definition of shape functions, usually referred to as test functions, defined on reference elements. The discretization process in the FEM is achieved through the projection of the continuous field of interest onto the basis formed by the M defined shape functions. In the rest of the section, we denote by MH and MT the set of nodes indexes in the heart and in the torso, respectively, and by MH and MT the corresponding number of nodes. FEM resolutions start with writing the problem in a weak form, the PDEs are linearized in each finite element so that an approximate solution is obtained on the nodes of the mesh. Since (2.25) does not involve space derivatives, it is simply solved for each node defined in H separately. The FEM is applied on (2.26) only. The weak form is a weighted residual obtained by multiplying by some weighting function ψ and taking the integral over the studied domain. The weak form of (2.26) reads:

Z Z λ (n+1) (n+1) (n) ∆t ∇ · (σin∇vm )ψdx = χcm(vm − vm )ψdx 1 + λ H H Z (2.27) (n) (n+1) +∆t χ iion(vm , u ) + istim ψdx H where dx is the elementary volume displacement. Note that ψ can be any differentiable function, and that this weak form is equivalent to its original strong form (2.26). Applying Green’s divergence theorem to the left hand side of (2.27) yields:

Z Z Z (n+1) (n+1) (n+1) ∇ · (σin∇vm )ψdx = − (σin∇vm ) · ∇ψdx + (σin∇vm ) · n∂Hψds (2.28) H H ∂H where ds is the the elementary surface displacement. This formula can be seen as an integration by parts in multiple dimensions. Because of the boundary condition (2.22), the second term in the right hand side vanishes. The discretization of the equation comes from the interpolation of field variables over a finite element of the mesh by a set of shape functions (γi)i∈MH . These functions are associated to the nodes of the mesh and are defined such that: γi(xj) = 1 when i = j, and decrease to reach zero as x gets far from xi. We choose the simple case where functions γi : H → [0, 1] are piecewise linear such that: γi(xj ) = δij, ∀j ∈ MH where δij is the Kronecker product. This results in a discretized expression of the field variable vm: X vm(x) = vmiγi(x) (2.29)

i∈MH

where (vmi)i∈MH are the coordinates of the continuous ﬁeld vm in the basis formed by the shape functions (γi)i∈MH . If we replace in (2.27) the weighting function ψ, yet unspeciﬁed, by a shape function

57 Chapter 2 – Cardiac modeling

γk corresponding to some node k ∈ MH, and inject the canonical form (2.29), we get:

Z Z λ X (n+1) X (n+1) (n) −∆t vmi (σin∇γi) · ∇γkdx = (vmi − vmi ) χcmγiγkdx 1 + λ H H i∈MH i∈MH Z X (n) (n+1) +∆t iion(vmi , ui ) + istim χγiγkdx (2.30) H i∈MH

Finally, writing the last equation for each value of k ∈ {1..NH} results in the following linear system:

λ − ∆t A v(n+1) = D (v(n+1) − v(n)) + ∆tD i (v(n), u(n+1)) + i (2.31) 1 + λ σin m χcm m m χ ion m stim

MH MH th where vm ∈ R and u ∈ R are vectors whose i coefﬁcient correspond to the values vm(xi) and u(xi), Aσin is the so-called stiffness matrix, and Dχ and Dχcm are the so-called damping matrices. (n+1) Putting the unknowns on the left-hand side gives a system to be solved at each time step for vm :

λ (∆t A + D )v(n+1) = D v(n) − ∆tD i (v(n), u(n+1)) + i (2.32) 1 + λ σin χcm m χcm m χ ion m stim

The square matrices in (2.32) are defined by the following inner products: R Aσin = H(σin∇γi) · ∇γkdx i∈MH R k∈MH Dχcm = H χcmγiγkdx i∈MH R k∈MH Dχ = H χγiγkdx i∈MH k∈MH These matrices present nice computational properties. The most trivial one is that they are symmetric. Secondly, if nodes i and k do not belong to the same finite element, the corresponding coefficient in these matrices equals zero. Consequently, these matrices are sparse (contain few non-zero elements). The third important property is that they are positive semi-definite [127]. The linear system can be solved by iterative methods, but since only the right-hand side is different at each time step and the system matrix remains unchanged, decomposition techniques are more suitable for solving (2.32). Because of the properties described above, Cholesky Decomposition [179] is well adapted.

In order to solve the PDE system (2.32) and (2.26), an initial stimulation needs to be applied to initiate the propagation process. To do so, the stimulation current istim is applied on a chosen node isan in the exterior wall of the right atrium as the SAN region of the designed heart (ﬁgure 2.4). This stimulation is applied for a certain period tstim, which is just the right duration necessary for this node to ﬁre an action potential. The complete procedure for solving the PDE system is given in Algorithm 1.

Now that transmembrane potentials vm are available, we describe how to obtain extracellular λ potentials within the myocardium and those of the rest of the volume. Adding (2.23) to 1+λ × (2.13)

58 2.2. Proposal of a simpliﬁed heart-torso model

Figure 2.4: SinoAtrial Node location

Algorithm 1 Algorithm for computing vm based on the monodomain formalism (0) 0 (0) 0 Initialization: vm = vm, u = u , istim = 0, istim(isan) 6= 0

Assemble matrices Aσin , Dχcm and Dχ λ Compute the system matrix A = ∆t 1+λ Aσin + Dχcm for n = 1 : T do if n∆t > tstim then istim(isan) = 0 end if (n) (n+1) (n) vmi −b2 (n) Compute u = u + ∆tke − du − b1 ∀i ∈ MH i i a2 i

(n+1) (n) (n+1) Compute the right-hand side b = Dχcm vm − ∆tDχ iion(vm , u ) + istim (n+1) Solve Avm = b using Cholesky decomposition end for

59 Chapter 2 – Cardiac modeling reﬂects the charge conservation in the whole heart-torso domain:

λ ∇ · (σ ∇φ ) + ∇ · (σ ∇v ) + λ∇ · (σ ∇φ ) = 0 (2.33) t t 1 + λ in m in ex

Taking the weak form and applying the divergence theorem gives:

λ Z Z Z (σin∇vm) · ∇ψdx + λ (σin∇φex) · ∇ψdx + (σt∇φt) · ∇ψdx = 1 + λ H H T λ Z Z Z (σin∇vm) · n∂Hψds + λ (σin∇φex) · n∂Hψds − (σt∇φt) · n∂Hψds (2.34) 1 + λ ∂H ∂H ∂H Z + (σt∇φt) · n∂T ψds ∂T

In the last equation, since the torso domain has two boundaries, the divergence theorem applied to this domain results in two surface terms. One is related to the outer torso surface, and the other is related to the shared boundary with the heart. The right-hand side of (2.34) equals zero. Indeed, the ﬁrst and the last term of the right hand side vanish because of boundary conditions (2.22) and (2.24). The sum of the second and the third term in the right-hand side equals zero because of: i) the outward pointing normal of T and H being of opposite signs, ii) the equal anisotropy assumption σex = λσin and iii) boundary conditions expressed in (2.21) and (2.24). Following the same discretization scheme as for the reaction-diffusion equation, one gets the following linear system:  (n)   (n)  φex λ −Aσin vm Aσ¯   =   (2.35) (n) 1 + λ φt 0

(n) (n) where φex and φt are the discretized versions of the continuous variables φex and φt, and Aσ¯ is the stiffness matrix of the whole heart-torso volume. σ¯ is an equivalent conductivity that equals λσin in H and σt in T . The solution of the system (2.35) can only determined to within an arbitrary additive constant. Indeed,

Aσ¯ is one-rank deficient and its kernel is given by the space of uniformly valued vectors. A physical explanation is that any uniform potential distribution over the entire domain satisfies the conservation law since the resulting current equals zero. To overcome this problem, one solution is to set a potential reference at a chosen node of the volume mesh, which reduces the system size (originally N × N) to (N − 1) × (N − 1). After implementing the full-heart torso model, the next step of this work is to be able to simulate realistic ECGs signals for a sinus-rhythm pattern. To compute the resulting ECGs from the model, 10 points are chosen on the torso surface whose potentials are used to compute the standard 12-Lead ECGs. The locations of these sensors are defined according to the 12-Lead ECG recording system and

60 2.3. Parameter identiﬁcation using an evolutionary strategy

are given in ﬁgure 2.5. The 12-Lead ECGs are computed from the values in φt at these locations based on section 1.4.2. The potential reference is set as in clinical practice at the RF electrode.

Figure 2.5: Heart-torso geometry and 12-Lead ECG electrode conﬁguration

The heart-torso model equations were implemented in Matlab. After importing the created mesh from ANSYS Mechanical, Finite Element matrices were assembled according to [25]. In order to capture cardiac action potential events, ∆t was ﬁxed to 0.1ms in equation (2.32). Then, the obtained transmembrane potentials vm are sub-sampled at a resolution of 1ms and used in (2.35) for computing

φex and φt. Given some set of parameter values, the simulation time required for the full model is about 70s on a computer with Intel Xeon 3.47 Ghz processor. Our model depends on many parameters that need to be tuned to obtain coherent surface ECGs. The difﬁculty lies mainly in the non-physiological character of the FHN model parameters. A naive way to proceed is to use parameters values from [161]. However, since the geometry, the mesh, the model and the solver are different, this approach failed in giving realistic ECGs. Instead, we choose a different approach to identify model parameters.

2.3 Parameter identiﬁcation using an evolutionary strategy

In order to produce coherent cardiac activation patterns and realistic ECGs using our model, two elements must ﬁt physiological knowledge: i) the action potentials shapes produced by each deﬁned

61 Chapter 2 – Cardiac modeling region and ii) the cardiac propagation wavefront speed in each region. A natural way to proceed is by first defining the FHN ionic model parameters that result in the desired action potential shapes and then focus on the other parameters introduced by the monodomain model. In order to obtain different action potential shapes, we first consider the FHN model applied to an isolated tissue. The corresponding equations are given by:

∂v m = −i (v , u) − i (2.36) ∂t ion m stim ∂u vm − b2 = ke( − du − b1) (2.37) ∂t a2 where:

vm − b2 vm − b2 iion(vm, u) = kc1(vm − b2)(a1 − )(1 − ) + kc2u(vm − b2) a2 a2

Here, the stimulation current istim plays the role of an inward current originating from neighboring nodes. Because the FHN model only depends on two variables, it has the advantage of visualizing the solution of the previous ODE system in a phase plane representation. This is done by plotting the so-called

∂vm ∂u nullclines [44], which are the curves deﬁned by ∂t = 0 and ∂t = 0 in the plane where the x-axis is represented by vm and the y-axis by u. The corresponding nullclines uvm and uu are obtained by

∂vm ∂u considering ∂t = 0, ∂t = 0 and expressing u as a function of vm in each equation (2.37) and (2.37):

−c1 vm − b2 vm − b2 istim uvm = (a1 − )(1 − ) + (2.38) c2 a2 a2 kc2(vm − b2) 1 vm − b2 uu = ( − b1) (2.39) d a2

This concept is illustrated in figure 2.6 for a given parameter set. The graphical representation of the FHN model has the advantage to provide a direct interpretation of model parameters and their influence on the nullclines, and thus on the generated action potential. Following physiological knowledge and based on nullclines theory, a set of FHN parameters is defined for each region of the heart model (figure 2.7).

The second step consists in tuning the parameters introduced by the monodomain, namely χ, cm and σin. Although the FHN model is a controllable oscillatory tool for generating various action potential shapes in the isolated cell, it is another matter when integrated in a 3D reaction-diffusion model. The manual tuning of monodomain parameters requires running Algorithm 1 and solving (2.35) for each different parameter combination. Moreover, it is difﬁcult to identify interactions between monodomain and FHN parameters in the whole heart system. An adequate choice of a global set of parameters

62 2.3. Parameter identiﬁcation using an evolutionary strategy

∂vm ∂u Figure 2.6: Left: phase plane representation of the nullclines ∂t = 0 (blue), ∂t = 0 (red) and the obtained trajectory (green). Right: transmembrane potential (top) and recovery variable (bottom)

should take into account the mesh resolution, parameters interaction and thus system spatio-temporal numerical stability. Thus, the resulting ECGs based on this ﬁrst attempt were consequently unsatisfying.

Alternatively, another way to proceed is by seeking parameter values of the whole model that produce ECGs that are close to reference sinus-rhythm ECGs. As seen previously, the ECG signal can be decomposed into three waves resulting directly from the cardiac depolarization and repolarization wavefronts: the P wave, the QRS complex and the T wave that have known characteristics in the case of normal rhythm. This can help deﬁning a measure of correctness of the ECGs simulated by our model.

Let us denote by p the set of parameters to identify, namely the FHN model parameters deﬁned in equations ( 2.16) and ( 2.17), completed with cm, σin and χ. The vector p is deﬁned by:

p = {σini, χ, cmi, a1i, a2, b1i, c1i, c2i, di, ei, ki} where i is the index of the considered region. It is noteworthy since our model is isotropic and regionally homogeneous, the conductivity tensor σini reduces to a scalar parameter to be identiﬁed for each region.

Also, parameters a2 and χ take the same value for all regions of the heart. In fact, χ is the surface to volume ratio. It is more related to the mesh resolution rather than cardiac regional characteristics.

Regarding a2, the justiﬁcation comes from the fact that this parameter regulates action potentials

63 Chapter 2 – Cardiac modeling

Figure 2.7: Action potentials generated by the FHN model by region amplitude, which is supposed to be roughly the same (around 30mV) everywhere in the healthy heart. The equal anisotropy parameter λ is not identified since it is a scaling factor that is independent from space and thus the considered region, and whose influence is absorbed in the value of σin. Moreover, we impose that epicardium parameters take the same values as those of the endocardium, except for parameter e that mainly regulates action potential duration. This parameter is denoted eepi for the epicardium. Indeed, we assume the potentials of inner and outer layer of the epicardium to be of the same shape but with different durations. Parameter b2i represents the resting potential and is fixed to −85mV for all regions.

2.3.1 Optimization problem formulation

An appropriate cost function needs to be deﬁned in order to solve the underlying optimization problem. ref Our approach consists in maximizing a similarity measure between a real ECG ecgII and the one simulated by our model ecgII (p), constrained by the signal waves durations and relative amplitudes. We deﬁne the cost function based on the normalized scalar product between the two signals. Apart

64 2.3. Parameter identiﬁcation using an evolutionary strategy from providing a similarity measure, this choice also overcomes scaling issues. Equality constraints are imposed based on features measured on the target ECG. The chosen features are: the durations of ref ref ref ref the P wave (c1 ), of the QRS complex (c2 ), of the T wave (c3 ) and of the PR interval (c4 ), the ref maximum peak amplitude ratios between the QRS complex and the P wave (c5 ), and that between ref the QRS complex and the T wave (c6 ). Figure 2.8 shows the reference ECG and the corresponding features.

Figure 2.8: Target ECG signal and its features

The optimization problem can then be expressed as follows:

ref !2 ∗ | < ecgII (p), ecgII > | p =argmin 1 − ref (2.40) p ||ecgII (p)||2||ecgII ||2 s.t: c(p) = cref

where cref is the vectorized form of the features deﬁned above, and c(p) those measured on the simulated ECG with parameter set p.

The minimization problem (2.40) can be re-expressed as the unconstrained minimization of a

65 Chapter 2 – Cardiac modeling

Lagrangian function deﬁned by:

ref !2 6 (< ecgII (p), ecgII >) X ref 2 L(p, λ) = 1 − ref + λi(ci(p) − ci ) (2.41) ||ecgII (p)||2||ecgII ||2 i=1 where the so-called Lagrange multipliers λ = (λi)i∈{1..6} were empirically ﬁxed at values that ensure the same order of magnitude between these non-homogenous constraints. The optimization problem addressed in our context is non-convex, non-linear, non-differentiable and doesn’t have a unique solution. Indeed, the system we are studying is weakly observable, meaning that the internal states (cardiac signals) cannot be inferred directly from only one or only few target ECGs. Moreover, the size of the parameter set to identify exceeds 60. The purpose here is to ﬁnd a solution that is satisfying regarding the constraints that have been imposed in equation (2.40). An appropriate algorithm for solving this problem is one that does not require the existence of the derivative of the cost function with respect to parameters, adapted to complex problems with potentially multiple local optima, and robust. For these reasons, heuristic-based approaches, particularly Evolutionary Algorithms (EAs) [19, 80] are more suitable than classical deterministic algorithms.

2.3.2 Evolutuionary Algorithms

EAs are guided random search approaches that originate from genetic algorithms [147]. They are inspired by the evolution theory in which the strongest organisms are the ones that survive in the future generations. We suppose that the solution of problem (2.41) can be encoded by a set of parameters, called genes. Each set p is represented by a vector of genes called chromosome. A generation can be defined by a population of several chromosomes. Following the principle of the survival of the j fittest, the algorithm starts from an initial population P op0 of N chromosomes p0, j ∈ {1..N}. Then, it evolves from one generation P opi−1 to the next P opi through genetic operations referred to as mutations and crossovers. These operations are applied to a subset of P opi−1, giving birth to new 0 0 chromosomes P opi−1. Only the N fittest chromosomes in P opi−1 and P opi−1 are kept to constitute the next population P opi. In the original definition of genetic algorithms, genes are discrete values encoded in one bit. EAs are an extension to this concept in which genes are real valued. This entails defining an appropriate variation domain with an upper and a lower boundary for each parameter. The idea behind crossovers and mutations is described in figure 2.9 and the corresponding definitions are given by:

• Crossovers: they are applied on several couples of chromosomes randomly selected from the current generation. Crossovers consist in interchanging genes from two parent chromosomes in

66 2.3. Parameter identiﬁcation using an evolutionary strategy

order to give birth to two chromosome children. Each child thus contains genes from both parents.

• Mutations: they are applied on a random subset of the current population. For each selected chromosome, it consists in random perturbation of some genes values.

Figure 2.9: Diagram explaining the basic idea of crossovers and mutations

Convergence and robustness properties of EAs strongly depend on the cost function, the appropriate encoding of individuals, and the choice of genetic operators (see [89] for further reading). A pseudo-code of the used EA is given in Algorithm 2.

Algorithm 2 EA pseudo-code

Initialization: Provide a population P op0 of N individuals and evaluate their associated cost i = 1 while i < Npopulations do Select the best individuals from P opi−1 0 Create new individuals P opi−1 using crossover and mutation 0 Evaluate each individual of P opi−1 0 Create P opi by keeping the N best individuals of P opi−1 ∪ P opi−1 i = i + 1 end while

2.3.3 Numerical simulations

This paragraph first describes the simulation steps followed in order to obtain realistically shaped 12-Lead ECGs and the corresponding cardiac propagation pattern. Then, we provide and discuss quantitative and qualitative results in terms of convergence of the EA, ECGs and action potential shapes and activation times mappings. Simulation protocol The simulated surface ECGs are obtained following three main steps. In the first step, a variation domain for each parameter in p is defined. Indeed, in order to run an EA, a necessary prerequisite is the definition of the variation domain for each gene. To do so, a local study for each parameter is conducted. Given a vector set p, we predefine a large range of values for each parameter. We vary only one parameter in its predefined range and execute Algorithm 1 until a degenerate solution is found. This is done successively for each parameter, and the upper and lower values for which

67 Chapter 2 – Cardiac modeling degenerate solutions are obtained are taken to be the upper and lower bounds of the corresponding parameter variation domain. In the second step, Algorithm 2 is run in order to identify the best parameter vector p∗ according to the cost function (2.40). The last step consists in running Algorithm 1 using the optimal parameter set p∗ to obtain and visualize the corresponding cardiac mappings, and solving (2.35) to compute the 12-Lead ECGs. We run Algorithm 2 for 35 generations of 70 chromosomes. The mutation probability is set to 0.25 and the selection rate is of 8%. The initial population is constituted of 38 manually chosen individuals, the other 32 are chosen randomly in the deﬁned parameter space. Regarding cost function computations, ref ref reference features c in (2.41) are directly extracted from ecgII and their values are given in table 2.1.

ref ref ref ref ref ref c1 c2 c3 c4 c5 c6 0.098 0.052 0.175 0.115 0.120 0.160

Table 2.1: Reference ECG features

Figure 2.10: Repolarization and repolarization times in an action potential

The P, QRS and T waves of the simulated surface ECG used in (2.41) were segmented directly from the generated action potentials vm. The P wave duration is computed as the difference between the time

68 2.3. Parameter identification using an evolutionary strategy of last depolarization in the atrial region with that of the first depolarization in the same region. Similarly, the QRS complex duration is computed as the difference between the time of the last depolarization in the ventricular region with that of the first depolarization in this region. Finally, the T wave duration is obtained by substracting the time instant of the last repolarization in the ventricles with that of the first repolarization. Depolarization and repolarization times are defined as the instant with maximum slope and minimum slope in the action potential, respectively (see figure 2.10). Amplitude ratios are directly computed on the simulated lead II. An infinite cost is assigned when at least 50% of nodes do not fire action potentials, or when the solution diverges due to numerical instability.

Results After completing the ﬁrst step, the obtained lower and upper bounds for each parameter are provided in table 2.2.

Parameters σini χ cmi a1i a2 b1i c1i c2i di ei ki Lower bound 4 0.01 0.15 0.09 0.01 -0.09 2.5 0.9 -0.5 0.0025 810 Upper bound 80 1 1.5 0.14 0.16 0.05 2.9 1.5 1.1 0.015 1400

Table 2.2: Parameters variation domain

Regarding the EA behavior, figure 2.11 shows the boxplot of cost values variability (y-axis) through generations (x-axis). The red bar indicates the median of cost values at each generation, and the bottom and top edges of the box indicate the 1st and 3d percentiles, respectively. The “whiskers” above and below the box show the locations of the minimum and maximum values, and the outliers are plotted individually using the “+” symbol. A rapid decrease in the median cost occurs in the first generations and reaches a minimum cost of 0.1367 in generation 35. The first generation contains randomly selected chromosomes in the variation domain of p. This population contains parameter sets that are inadequate and produce infinite costs. It is interesting to see that these cases are rejected starting from generation 2. One can also notice that cost values are very close and don’t evolve between generations 15 and 21, but a jump occurs around generation 22. This leads to think that the algorithm went out of a local minimum and that it is capable of exploring better solutions. We display in figure 2.12 the obtained action potential in each region. Clearly, the general shape of action potentials and their durations globally match physiological knowledge. Only bundle branches action potentials are quite long compared to their average normal duration. In fact, since it is a relatively small region in the heart domain, action potentials duration in this region do not contribute much in the ECG shape. This means that information about their repolarization phase is difficult to integrate in the cost function and any variations in the repolarization phase of this region won’t influence the simulated ECG shape. However, the depolarization wave speed is important and is well identified since it regulates

69 Chapter 2 – Cardiac modeling

Figure 2.11: Boxplot of cost values evolution through generations

70 2.3. Parameter identiﬁcation using an evolutionary strategy the PR interval, which is a feature that is integrated in the cost function.

Figure 2.12: Estimated regional transmembrane potentials

Figure 2.13 shows a comparison between the reference (real) ECG and the best solution provided by the EA. One can see that duration constraints have been respected, as well as the general characteristic shape of the P wave, QRS complex and the T wave of the ECG signal. This is conﬁrmed quantitatively ref () by a similarity value ref that reaches 89%, and features values that are close to the ||ecgII (p)||2||ecgII ||2 reference as shown in table 2.3. Note that adding an additional layer to the ventricles in the heart geometry enables to produce a T wave with an appropriate duration because of the difference of repolarization times between the ventricular layers. Without the additional ventricular layer, the repolarization wave in the ventricles would naturally be of the same speed as that of depolarization, which would give a sharp T wave as one can observe in [161]. The T wave amplitude in the simulated ECG is though of higher amplitude compared to the reference. This can be explained by a strong gradient between the two ventricular layers. Indeed, since the action potential duration does not vary smoothly in the ventricular region as the repolarization front moves from the epicardium to the endocardium, an abrupt potential difference

71 Chapter 2 – Cardiac modeling

Figure 2.13: Normalized ECG estimated by the EA (red) and reference ECG (blue) occurs in the boundary seperating the two layers, causing a T wave of higher amplitude. In this study, the influence of the electrical axis of the heart is not investigated, which explains the difference in the S wave. In fact, the observed surface ECGs depend on the heart orientation, which was not among parameters to identify. As long as heart orientation is fixed, any model, no matter how realistic, may not be able to exactly reproduce ECGs resulting from a different geometry configuration, especially regarding RS patterns [115].

ref ref ref ref ref ref c1 c1(p) c2 c2(p) c3 c3(p) c4 c4(p) c5 c5(p) c6 c6(p) 0.098 0.068 0.052 0.046 0.175 0.187 0.115 0.117 0.120 0.10 0.160 0.115

ref Table 2.3: Reference features ci and best solution features ci(p)

Figure 2.14 depicts the simulated 12-Lead ECGs. Although the identiﬁcation process was performed only on lead II, the general shapes of the rest of the 12-Lead ECGs agree well with morphological sinus-rythm ECGs in terms of polarity and durations: The P wave polarity is positive in all ECGs except in aVR, and its duration is normally less or equal to 110ms. The QRS complex is represented by a

72 2.3. Parameter identification using an evolutionary strategy positive deflection with a large, upright R in leads I, II, V4 - V6 and a negative deflection in aVR. Its duration is less or equal to 120 ms. The ST segment is isoelectric, sloping upwards to the T wave. The T wave polarity is the same as that of the QRS except in V2. During electrophysiologial studies, the potential maps are constructed by measuring bipolar potentials on the heart surface. The simulated extracellular potentials φex can be seen as unipolar leads measured with respect to the defined reference RF. Bipolar leads can be obtained by subtracting two extracellular potentials at two neighboring nodes. An example of a simulated bipolar potential is given in figure 2.15. We find the same characteristic sharp signal shape at the local activation instant.

Figure 2.15: Transmembrane potential (up) and corresponding bipolar potential (down) at node 100

For visualization purposes, the data generated by the model in Matlab are exported back to ANSYS Mechanical. Figure 2.17 presents the spatial distribution of transmembrane potential mappings at the heart volume for 9 key events in the ECG. The activation pattern follows the characteristic sequence in the normal conduction system. A depolarization wave is initiated at the SAN region and spreads through the left and the right atrium (1-3). It is delayed in the AVN and spreads through bundle branches (3-4). A rapid depolarization occurs in the Purkinje ﬁbers (4-5), ventricles depolarize from the inner layer to the outer layer (6-7) while atria are repolarizing. Finally, the repolarization wave goes from the epicardium to the endocardium (7-9). Local Activation Time mappings provide valuable functional and structural information about the initiation and the propagation pattern of cardiac electrical activity.

73 Chapter 2 – Cardiac modeling

Figure 2.14: Simulated 12-Lead ECGs

74 2.3. Parameter identiﬁcation using an evolutionary strategy

From data simulated by our model, the Local Activation Time of each node can be obtained from the corresponding transmembrane potential as seen in ﬁgure 2.10. These LATs can then be visualized for all nodes at the heart level as shown in ﬁgure 2.17.

75 Chapter 2 – Cardiac modeling

Figure 2.16: Transmembrane potentials spatial distributions at 9 key instants of the ECG

76 2.3. Parameter identiﬁcation using an evolutionary strategy

Figure 2.17: Local activation times (ms)

77 Chapter 2 – Cardiac modeling

2.4 Sensitivity analysis and model reduction

Since our model depends on many non-physiological parameters whose influence may be negligible in the simulation of cardiac propagation, it is of interest to perform a sensitivity analysis of our model with respect to its input parameters. An advantage lies in the fact that the parameter set on which the model depends may be reduced in order to only keep parameters that are the most influential in the parameter identification process. Moreover, in a concern of providing a quantitative measure of the validity of the defined cost function (2.41) and parameters influence on the model, it is relevant to perform this study on this same function. Several sensitivity analysis methods exist and can be distinguished in two categories. Local sensitivity methods rely on derivatives taken at a single point. They analyze the impact of one parameter on the cost function at a time, while other parameters are fixed. This type of methods is unadapted when dealing with complex models with discontinuities like ours. Moreover, it doesn’t provide information about how model parameters interact with each other. Global sensitivity methods study model variations by exploring the whole parameter space, usually by using monte carlo iterations. They are more costly than local methods, but do not require assumptions on the studied function, and provide information on parameters global influence and interaction. Interestingly, screening methods do not perform an exhaustive study of the parameter space. Instead, they consider several inputs configuration and evaluate the effects on the outputs when elementary perturbations are applied on the inputs, for each configuration.

2.4.1 Morris method

One of the simplest and most popular screening approaches is the Morris method [111]. It is a one at the time approach that is well adapted when the model depends on a high number of parameters. The Morris method is a statistical analysis of model variations with respect to each parameter. The assumption of a one at the time design is that if model parameters undergo elementary changes, those that cause the largest variation in the model output are the most inﬂuential.

Let us consider a model output L that depends on set of P parameters p = (p1, p2, ..., pP ), a regular grid that partitions each parameter variation domain into L levels, and an elementary displacement δ in the grid that is the same for all parameters. The first step of the method is to define random trajectories in the parameter space in which only one parameter at the time is varied. This consists in sampling (0) (0) a random point p in the defined grid. From this point, only one parameter pi , i ∈ {1..P } is varied by δ to create a new point p(1). From p(1) this operation is performed for all resting parameters. This results in P + 1 points such as only one parameter varies between p(i−1) and p(i). This scheme is then

78 2.4. Sensitivity analysis and model reduction repeated for R − 1 different starting points so that R trajectories are created. This experiment plan ends (i,j) with R × (P + 1) different parameter sets (p )i∈{0,..,P }. A simple case where P = 2, L = 9 and R = 4 j∈{1,..,R} is presented in ﬁgure 2.18.

Figure 2.18: Example of Morris method experiment plan in the simple case where P = 2, L = 9 and R = 4

The second step consists in evaluating each of the created parameter sets by the model and computing output variations corresponding to elementary displacements. The variation of the model th output for an elementary displacement in parameter pi (i ∈ {1, .., P }) in the j trajectory (j ∈ {1, .., R}) is given by: d(i,j) = L(p(ki,j)) − L(p(ki−1,j)) (2.42)

ki−1,j ki,j where ki and ki − 1 are the indexes for which only pi varied between p and p .

The third step consists in computing the mean µi and variance σi of elementary effects on all

79 Chapter 2 – Cardiac modeling

trajectories, for each parameter pi. These quantities are given by:

R 1 X µ = | d(i,j) | (2.43) i R j=1 v u R u 1 X (i,j) 2 σi = t (d − µi) (2.44) R j=1

Finally, a graphical analysis where µi is reported on the x-axis and σi on the y-axis enables to study the inﬂuence of each parameter:

• Parameters with low inﬂuence are close to the origin (0,0).

• Parameters with linear inﬂuence on the output are close to the x-axis.

• Parameters with non-linear inﬂuence or with strong interaction with others are close to the y-axis.

2.4.2 Numerical simulations

In this study, the parameter set p and its variation domain are the same as those defined in section 2.3, to which we add lungs, torso and blood conductivities denoted by σlungs, σtorso and σblood respectively. The output model value L is the cost function defined in equation (2.41). We choose: R = 3000, L = 50 and δ represents two unitary displacements in the uniform grid. The obtained sensitivity graphs are presented in figure 2.19 and are categorized by region for better visualization. Each point on the graph corresponds to each parameter values µi and σi. The first remark is that extra-cardiac and blood conductivities are the least influential. Indeed, since the whole domain is piece-wise homogeneous and isotropic, extra-cardiac conductivities act almost as a scaling factor in the computation of the ECG. Our cost function L does not include an amplitude constraint but rather constraints on waves relative amplitudes. This explains why the output is not sensitive to these conductivities. Secondly, the regions with the highest mean value are those that occupy the largest space in the heart volume, namely the atria and ventricles. The contribution of parameters in small regions (AVN, His and bundle branches) is negligible compared to the influence of the global parameter χ. This can be explained by the fact that χ acts on all regions, and more specifically on atria and ventricles, whose contributions are the most important in the ECG. When the most influential parameters are omitted (figure 2.20), a global remark is that parameters influence is not the same depending on cardiac regions. For instance, in the AVN region, parameter b1 has a small ref relative influence than in the His regions. Since our model output L depends on a reference ECG ecgII , these results may be compromised and may depend on the defined ECG reference. In order to assess

80 2.5. Pathology simulation the previous results for different references, the sensitivity analysis was performed on four different other ECGs of healthy cases. This results in five different cost functions L(k), k ∈ 1..5, each corresponding (k) (k) to one of the chosen ECGs. The corresponding statistics µi σi were computed based on the same (i,j) parameter samples (p )i∈{0,..,P }. The obtained graphs are given in figure 2.21 and figure 2.22. j∈{1,..,R} These last results show that parameters sensitivity is globally the same for the different references. Thus, for the considered cost function, the cost variability regarding each parameter is globally the same for different ECG signals. Put differently, the contributions of model parameters regarding the defined cost function are the same independently from the targeted ECG. Two conclusions can be drawn from this analysis: i) this is an encouraging indicator in terms of reproducibility of the result shown in figure 2.13 and ii) some parameters influence is negligible regarding ECGs simulated by our model. One way to assess the second statement is by running the EA in order to identify a parameter set

0 p that is a reduced version of p based on this sensitivity analysis. We first determine the regional parameters that have that highest influence. To do so, we normalize mean and standard deviation values regionally and we separate the least influential and the most influential parameters by a quadrant

0 of radius 0.4 as shown in figure 2.23. We define p by keeping only parameters that are out of ref the quadrant. Then, we run the EA to identify this new set parameter with respect to ecgII . The non-influential parameters are fixed to their values in the best estimation in the first generation. The

0 identiﬁed parameters p and p result in very close ECGs, which justiﬁes the parameter set reduction.

2.5 Pathology simulation

We show in this section that our model is capable of simulating different pathology scenarios. In the case of VT, an abnormal activation occurs in the myocardium at a fast rate due to one or several ectopic sources. An ECG showing a VT is characterized by a large QRS and a heart rate > 120bpm. Two main classes of VT can be distinguished. Monomorphic VTs are generated from an increased automacity of a single point in the ventricle, or due to a reentry circuit within the ventricle. In this case, each beat of the ECG looks the same. Polymorphic VTs occur is caused by abnormal ventricular repolarization. This can be viewed as if several ectopic sources are activated at each heartbeat. Consequently, the ECG beats are different. To simulate monomorphic VT and polymorphic by our model, either a single node in the epicardium is stimulated to initiate abnormal propagation for successive heartbeats, or alternating sites between endo- and epicardium are stimulated to initiate different pathological heartbeats. In both cases, VT starts after a healthy ECG and the stimulation rate is ﬁxed to 200 bpm. Figures 2.25 and 2.26 show the simulation results of a monomorphic and a polymorphic VT, respectively. In both cases, the QRS complex is of less amplitude than the ﬁrst healthy beat and is larger. A LAT mapping of an ectopic

81 Chapter 2 – Cardiac modeling beat is given in figure 2.29 where the blue part in the ventricles is the activation region. VT usually occurs in patients with acute myocardial infarction. In this case, a large scar in the ventricular region is present and is electrically passive. The ECG of patients with ventricular infarction is characterized by an ST segment elevation or depression. To simulate such conditions, a region close to the septum was forced to maintain the same action potential value during the full simulated heartbeat. This region can be seen in figure 2.28 in dark blue in the ventricular region. The obtained 12-Lead ECGs are shown in figure 2.27. An elevation of the ST segment in leads II, III, aVL, aVF V1 and V2 is clearly visible. These findings match their clinical usage as a diagnostic criterion of myocardial infarction.

82 2.5. Pathology simulation

Figure 2.19: Sensitivity graphs of parameters by region

83 Chapter 2 – Cardiac modeling

Figure 2.20: Sensitivity graphs when the most inﬂuential parameters are omitted

84 2.5. Pathology simulation

Figure 2.21: Sensitivity graphs obtained for ﬁve different model outputs, each color corresponds to a different ECG reference 85 Chapter 2 – Cardiac modeling

Figure 2.22: Sensitivity graphs obtained for ﬁve different model outputs when the most inﬂuential parameters are omitted 86 2.5. Pathology simulation

Figure 2.23: Separation between the least and the most inﬂuential parameters by region

87 Chapter 2 – Cardiac modeling

0 Figure 2.24: Reference (black) and simulated ECGs corresponding to the best p (blue) and the best p (red)

88 2.5. Pathology simulation

Figure 2.25: Monomorphic VT simulation

89 Chapter 2 – Cardiac modeling

Figure 2.26: Polymorphic VT simulation

90 2.5. Pathology simulation

Figure 2.27: Comaparison of healthy 12-Lead ECGs (blue) and those in the case of myocardial infarction (red) 91 Chapter 2 – Cardiac modeling

Figure 2.28: LAT mapping corresponding to the simulated ischemic condition

Figure 2.29: LAT mapping for the ﬁrst ectopic beat of the simulated monomorphic VT

92 2.6. Conclusions

2.6 Conclusions

In this chapter, we present a low-resolution 3D model that produces realistic cardiac patterns and morphological ECGs. This model is built upon the monodomain formalism coupled with the phenomenological FitzHugh-Nagumo ionic model and enables to perform heart-torso simulations in less than three minutes on a Intel Xeon 3.47 Ghz processor computer. Parameters were identified using an evolutionary algorithm that optimizes a similarity measure between simulated ECGs and a defined reference, observed from a patient. Such quick simulations allow for the investigation of model sensitivity to its parameters and reveal the model parameters with higher influence. To our knowledge, this kind of sensitivity analysis applied to a whole heart and torso model is original. Finally, we show that this model is capable of generating different pathology scenarios that can be used for evaluating ECGI algorithms. These modeling choices offer a controllable environment for easily studying the influence of multiple aspects such as geometry variations, inclusion of anisotropy in the volume conductor and the relationships between various cardiac patterns and the resulting ECGs.

CHAPTER 3

ECG INVERSEPROBLEM

In broad terms, the ECG inverse problem can be defined as the determination of cardiac electrical function from body surface potential recordings. In this sense, even the visual analysis of a 12-Lead ECG graph can be considered as an empirical inverse problem that uses previously documented information. In the rest of this chapter, we denote by ECG inverse problem the mathematical manipulation of ECG recordings in order to obtain a 3D mapping of cardiac electrical activity. Solving the ECG inverse problem requires a set of ECG measurements acquired on the torso surface and a forward model that links these observables to cardiac electrical sources in the considered geometry. Naively, one would simply want to invert the forward model to obtain cardiac sources from ECGs. However, such solutions will undoubtedly be inappropriate due the ill-posed nature of the ECG inverse problem. This term stems from a definition given by Hadamard [73] and designates any problem for which one of the following conditions is not satisfied: 1) a solution exists, 2) the solution is unique and 3) the solution’s behavior changes continuously with the inputs. In the case of the ECG inverse problem, both conditions 2) and 3) are dissatisfied. In fact, solution non-uniqueness arises from the small number of observations compared to the number of cardiac electrical sources to be recovered. Solutions instability on the other hand is due to the attenuation of electrical activity when propagating from the heart to the body surface. As a result, several cardiac mappings can correspond to the same observed ECGs, and small modeling uncertainties can result in large deviations in the obtained solutions. When the forward operator is linear, this is translated by its low-rank and ill-conditioned character. To circumvent this obstacle, it is necessary to mathematically constrain appropriately the solution space. This can certainly guarantee the uniqueness of the solution but can still lead to inappropriate solutions if the penalty doesn’t adequately fit the physiological nature of a cardiac electrical mapping. Numerous methods have been proposed for solving the ECG inverse problem, but efforts are still to be made in finding penalties that best match cardiac electrical physiology. Moreover, studies comparing these methods performance under unified conditions in terms of source models, forward models and considered pathologies are still missing. In this chapter, we first review the main approaches for solving the ECG inverse problem. Then, we introduce the selected methods in terms of data pre-processing, the different regularizers choice and

95 Chapter 3 – ECG inverse problem implementation, the determination of an adequate regularization parameter value and the optimization algorithms. We ﬁnally provide results of the numerical experiments performed in healthy and pathological cases.

3.1 State of the art

3.1.1 Source formulations

The ECG inverse problem can be formulated differently depending on the chosen cardiac source model and consequently the underlying forward model. The main macroscopic cardiac source models are presented in section 2.1.3. From an ECGI perspective, the choice of an adequate source model is even more important than that of simulating ECGs by forward calculations. Indeed, although several source formulations can result in similar ECG patterns, the source spatio-temporal nature strongly affects the a priori to be imposed when solving the inverse problem. Moreover, some types of models involve electrical sources that cannot be measured in practice, which renders validation difﬁcult. The earliest family of source models consisted of a distribution of equivalent current dipoles in the heart [108, 166, 176]. These models served very well ever since for a qualitative explanation of ECG waveforms. However, it is hard to capture the electrical events in the whole heart when it is represented by only few dipoles [40, 46, 108]. The inverse problem based on UDL and EDL types of models is referred to as activation time imaging [40, 85, 86]. In these cases, one directly seeks the depolarization time corresponding to the upstroke of transmembrane potentials in the regions of the heart. This approach is of great clinical interest because of the coherence of cardiac activation times, but requires solving a non-linear inverse problem and does not provide the temporal information of cardiac potentials. Moreover, it only focuses on cardiac electrical activity neighboring the electrical wavefront and thus does not take into account activity that is far from it. More recently, efforts have been made in directly recovering the transmembrane potentials in the heart volume [76, 107]. Nevertheless, the underlying inverse problem is severely ill-posed since the number of sources to be recovered are that of nodes in the whole heart mesh. Also, these potentials are difﬁcult to measure in clinical routines, which limits its use in clinical procedures. The preferred source formulation in the ECG inverse problem is in terms of epicardial potentials [67, 104]. Indeed, this last formulation provides substantial information of the local electrical activity everywhere on the epicardial surface. This formulation enables the recovery of late potentials which are of major importance in identifying slow conduction pathways responsible for VT. Local activation times can eventually be computed as the time when the local potential reaches an optimum. Most importantly, these potentials are those that are measured and used in invasive clinical routines of electrophysiological studies.

96 3.1. State of the art

3.1.2 Classical methods

Regarding methodological aspects, several approaches for solving the inverse problem can be found. The most classical regroups regularization based methods. Originally proposed by Tikhonov [168], these methods consist in ﬁnding the best ﬁt to data solution in the least squares sense while considering an additional penalty term. This additive penalty term, referred to as the regularization term, aims at reducing the inverse solution’s high frequency content. The resulting formulation is given by:

∗ 1 2 z = argmin ||G(z) − x||2 + λ||R(z)||p (3.1) z 2 where z represents the sources to be recovered, G is the forward model, x is the set of observed ECGs, R is the regularization operator and λ a regularization parameter that balances between the amount of data ﬁdelity and regularization. In Tikhonov regularization, p = 2 and R is often taken as the identity, the spatial gradient or the spatial Laplacian operator. This method presents the huge advantage of providing an analytical solution when G and R are linear operators, which is mostly the case in this context. Although this approach has been extensively used in the ECG inverse problem [26, 56, 97, 155, 157, 184], it is often criticized for not taking into account the physiological nature of the sought solution and providing over-smoothed solutions due to the use of the `2 norm in the penalty term. The rising interest in sparsity-based approaches in the signal processing community led to investigating

`1-based regularization in the ECG inverse problem (p = 1), and more particularly Total Variation (TV) regularization. TV regularization appeared in image denoising and is based on the principle that when signals have spurious details, the integral of their absolute gradient (TV) is high [143]. Thus, minimizing the TV of a signal removes unwanted details and enhances important information like contours. Since the `1 norm is not differentiable, analytical solutions cannot be obtained and this formulation requires the use of iterative algorithms. The most widely used algorithm in TV regularization is the Iterative Reweighted Norm (IRN) [41, 157, 158, 190]. Other studies have investigated other types of optimization methods, such as in [182] where inequality constraints were applied and the problem was solved by an interior point method, or in [28] where the Alternating Direction of Multipliers (ADMM) was used. In these studies, TV performed better than the more classical Tikhonov regularization. Interestingly, the study in [139] shows that at some time courses when sources are neither focal nor diffused, `p norm regularization when 1 < p < 2 may be more appropriate. However, it is still unclear how a link between the appropriate value of p and the observed ECG can be established. In the same spirit of regularization, Wang et. al [181, 183] proposed penalty terms that arise from the FEM variational deﬁnition of the notions of identity, gradient and Laplacian. The choice of the regularization parameter value is critical in regularization-based methods and a major difﬁculty lies in its determination. Several classical methods

97 Chapter 3 – ECG inverse problem attempt to provide a balance between the solution and regularization norm. Among these, the most straightforward is the optimal criterion solution that consists in minimizing an error criterion between the exact epicardial solution and each of the regularized solutions obtained by testing values of λ in a defined dense grid. This method is clearly not feasible clinically since it requires the knowledge of the true solution but is convenient when the purpose is to evaluate inverse methods performance. The highly 2 used L-curve method plots the data fidelity term ||G(zλi ) − x||2 against ||R(zλi )||p obtained for a set of estimated solutions zλi with parameter λi and often results in a characteristic L-shaped curve. The optimal λi in the L-curve is chosen to be the one corresponding to the corner of this curve since it ideally balances between the two terms in the problem (3.1)[75]. Similarly to the L-curve, the Generalized Cross Validation criterion (GCV) tests a set of values for λ and chooses the optimal parameter as the one that minimizes a trade-off function between the data fidelity term and the regularization term [66]. The Composite REsidual and Smoothing Operator (CRESO) is also commonly used and chooses the parameter value that generates the first local maximum of the difference between the partial derivatives of the two terms in problem (3.1) with respect to λ [31]. In contrast with these a posteriori methods, a priori methods rely on prior knowledge on the system such as noise variance in the case of the discrepancy principle [135]. All these methods may provide convenient regularized solutions, but they are very time consuming since they require defining and testing values in an exhaustive search space for λ. Of major interest are methods that choose λ in an adaptive fashion, such techniques are successfully implemented in other fields but haven’t been investigated in the context of ECG inverse problem [77].

Another well-known family of methods consists in approximating the forward operator G to a better conditioned version G˜. In practice, this consists in considering a truncated version of the Singular Value Decomposition (SVD) of G in which small singular values are rejected [74]. This has the effect of keeping only the modes that are informative and thus diminishes numerical instability. In this case, the regularization lies in truncating the singular values of G, and the sources are obtained by computing the pseudo-inverse of G˜ without adding penalty terms. First proposed by Greensite for ECGI [68], this approach gained popularity and several versions of this approach can be found [26, 69, 156]. Although conceptually different, the classical Truncated SVD method (TSVD) can be formally comparable to Tikhonov regularization in the sense that they both ﬁlter the high frequency components of the solution [74]. By analogy, the truncation level plays the role of the regularization parameter λ in Tikhonov methods.

In regularized and SVD-based solutions, much attention has been given to methods that incorporate knowledge both on the spatial and the temporal character of the sources [100]. The most common way to do so is by spatially regularizing the difference between two consecutive potentials spatial distributions. This method is known as Twomey regularization [121], and postulates that the sought solution does

98 3.1. State of the art not signiﬁcantly change between consecutive time samples. This has also been achieved by combining TSVD as a temporal whitening and imposing explicit spatial constraints [21, 154]. Finally, using several constraints as temporal and spatial regularization as in [22, 23, 49] seems to produce much more realistic temporal patterns of the reconstructed signals, while spatial coherence is also respected.

Alternatively to deterministic methods, some studies solve the inverse problem from a Bayesian perspective [9]. In this case, one seeks to estimate the Bayesian maximum a posteriori of a probabilistic problem in which the sources and the noise are random variables. Van Oosterom was the ﬁrst to introduce such an approach in ECGI [173–175]. Few publications appeared later [151–153, 192], but it is not as well established as regularization methods in ECGI community. The main reason lies in the challenge raised by the difﬁculty in determining the mean, the covariance, and an adequate probabilistic model for describing the noise and the sources.

Apart from the methods reported above, another main class of methods lie in solving the ECG inverse problem by using cardiac propagation models. This emerging approach addresses the ECG inverse problem by a parameter identification strategy. Rather than directly estimating the electrical sources, it consists in finding the optimal set of parameters of a cardiac propagation model that results in the simulated ECGs that are the closest to those observed. In [98], the problem is reformulated as an optimal control problem where ECGs are simulated using the bidomain formalism and the control is the natural onset of cardiac activation. In [64], a personnalized monodomain model with the Mitchell–Schaeffer’s ionic model were used to train a machine learning algorithm in order to identify the onset activation location and cardiac tissue conductivity. In [140] an adaptive inverse monodomain operator was used to reconstruct potentials. In a quite similar way, authors in [36] use a Kalman filter and mechanical observations to identify parameters of a bidomain electrical model coupled with a mechanical model. In [16], a genetic algorithm is employed to identify ionic parameters of the Mitchell-Schaeffer model using a bidomain formalism and a model reduction method. This type of approaches is promising since it relies on well established models for simulating cardiac propagation. In this case, regularization may be seen as constraining the solution space by that of the possible solutions of the forward problem. However, they suffer from their high computational cost since they either rely on simulating one or several full heartbeats at each algorithm iteration, or use an extensive database computed in advance. Moreover, these methods require knowing the onset activation of the cardiac cycle beforehand, or estimate it during the inverse procedure, which leads to an even more complex problem. Another limitation comes from the fact that these methods may not be able to match the reference cardiac temporal patterns simply because the involved models may not offer the possibility to do so. For these reasons, classical methods are still preferred because they offer a solution space that is more likely to contain the true solution.

99 Chapter 3 – ECG inverse problem

3.2 Methods

In this thesis, we choose the epicardial source model for solving the inverse problem. From the model proposed in chapter 2, the forward problem in (2.35) used to compute ECGs is equivalent to that in terms of heart surface extracellular potentials. Indeed, cardiac extracellular potentials distribution over any closed surface encompassing the heart fully reﬂects the inner primary currents within the heart. Consequently, these epicardial potentials obtained by solving equation (2.35) result in the same ECGs as those computed originally from this same equation. The discretization of the corresponding forward problem using the FEM is detailed in Appendix C and is expressed by the linear system:

X = GZ (3.2) where Z ∈ RN×T is a matrix whose ith row represents the time course of epicardial potential values at the ith node on the heart surface, X ∈ RM×T is a matrix in which the ith row represents the time course of the potential observed at the ith deﬁned sensor on the torso surface, and G ∈ RM×N is the linear instantaneous transfer matrix that translates the attenuation effect of the volume conductor separating sensors and the source surface. This section ﬁrst presents the data pre-processing step that we investigate, we then provide the ECG inverse problem formulation that we consider, the classical regularizers employed in our practical context, and the optimization algorithms used to solve the inverse problem. We choose the Alternating Direction Method of Multipliers (ADMM) for solving the `1 norm-based ECG inverse problem. Indeed, the ADMM offers a good compromise between convergence rate and numerical complexity. The application of Morozov discrepancy principle combined with ADMM for `1 regularization is investigated in order to adaptively determine an adequate value of λ, and we extend this method to the `2 case. Finally, we propose a new spatiotemporal approach that exploits two

`1-based regularization terms.

3.2.1 Data pre-processing

In addition of the raw data case, two types of data pre-processing are investigated, one where the system GZ = X is whitened with respect to the transfer matrix G, and one where this system is whitened with respect to the data X. The prewhitening step is commonly used in various signal processing applications and is intended to decorrelate data and to remove artifacts [7, 87, 146]. The whitening process is achieved through a Singular Value Decomposition (SVD) which consists in factorizing a matrix in a

MC ×NC unitary basis [42]. Let C ∈ R be a rectangular matrix such that NC > MC , the SVD of C consists

100 3.2. Methods in writing this matrix in the form:

M XC C = U S V T = σ u vT C C C i Ci Ci i=1

MC ×MC NC ×MC where UC ∈ R and VC ∈ R are matrices with orthonormal column vectors uCi and vCi respectively, and SC is a diagonal matrix containing the singular values σi > 0 in decreasing order. This decomposition is unique up to an orthogonal transformation. We now deﬁne the whitening process for −1 T C by applying the linear transform: C˜ = SC UC C where C˜ is the whitened version of C. Since UC T is unitary, the previous expression results in C˜ = VC . The whitened version C˜ is now constituted with orthonormal row vectors, which is an interesting property when dealing with ill-posed problems or highly correlated data.

Singular values ranking can be interpreted as the amount of information contained in C. Therefore, high singular values are those that correspond to the relevant amount of information on the structure of C while low singular values select singular vectors corresponding to high frequencies or artifacts in

C. As a result, when inverting SC , low singular values are highly ampliﬁed and the artifacts contribution may be dominant relatively to the principal components of C. To overcome this problem, a manipulation called Truncated SVD (TSVD) is often performed in order to substitute C by a low-rank approximation of its SVD [191]. The TSVD of C at rank k simply consists in keeping only the k highest values and their corresponding singular vectors in the SVD and is given by:

C = U S V T k Ck Ck Ck (3.3) k X = σ u vT i Ci Ci i=1

where the matrices UCk and VCk are derived from the k ﬁrst singular vectors of UC and SC , respectively.

Note that k is chosen to be lower than MC and the case k = MC corresponds to the original SVD. The corresponding truncated whitening transform C˜ is then given by C˜ = V T where artifacts are no longer k k Ck ampliﬁed.

Transfer matrix whitening The whitening transform of the system (3.2) with respect to the transfer

101 Chapter 3 – ECG inverse problem matrix G is obtained from the TSVD of G and is given by:

X˜ = S−1 U T X (3.4) G Gk Gk = S−1 U T GZ Gk Gk = V T Z Gk

= G˜ kZ

˜ ˜ where Gk is the truncated whitening transform of G and XGk the matrix of whitened ECG observations. Data matrix whitening The whitening transform of the system (3.2) with respect to the data matrix X is obtained from the TSVD of X and is given by:

X˜ = S−1 U T X (3.5) k Xk Xk = V T Xk = S−1 U T GZ Xk Xk ˜ = GXk Z

˜ ˜ where Xk is the truncated whitening transform of X and GXk the corresponding new transfer matrix. These two processing steps may seem similar in the sense that they both express components of the ﬁrst dimension (which corresponds to potentials spatial distribution of observations) of matrices G and X in an orthonormal manner. The whitening transform can thus be seen as a means of treating each observation with equal importance.

3.2.2 Inverse problem formulation

Without loss of generality, we consider in the rest of this section the vector system x = Gz corresponding to the case of one time sample, but the formulations and the algorithms are equivalently applied to matrix equations (3.4) and (3.5). Then we formulate the epicardial inverse problem in the traditional regularization scheme at one time instant as following:

∗ 1 2 λ p z = argmin ||Gz − x||2 + ||R(z − z0)||p (3.6) z 2 p where z ∈ RN and x ∈ RN are respectively the vector of epicardial potentials and the vector of observed potentials on the torso surface at a given time instant, z0 is the estimated solution at the previous time instant, R ∈ RR×M is a linear transform that maps the sources onto some deﬁned transform domain,

|| · ||p is the classical deﬁnition of the p norm and λ is the regularization parameter. This optimization

102 3.2. Methods problem has the advantage of being convex when p ≥ 1 and can be solved using convex optimization methods.

p The adequate choice of a regularizer ||R(z − z0)||p is intrinsically linked to prior knowledge on the spatiotemporal character of the sought solution. Cardiac potential maps at a given time instant reflect the electrical excitation wave within the heart. This means that at a given time, most of the sources should be of low amplitude, except for those that are on or close to the activation wavefront. Depending on the considered time instant and pathology, the electrical wave can be either extended or focal, and can be assumed to present either a smooth variation of potentials amplitudes between activated and unactivated regions on the heart surface, or abrupt changes in amplitude (e.g ischemia). In the case where only spatial regularization is applied (z0 = 0), these natures can be enforced by imposing spatial smoothness or sparsity on the source character, either directly in the source domain or in a transformed domain. This justifies the choice of || · ||p as being the `1 to enforce sparsity on the source structure and the `2 norm to impose smoothness. R is taken to be either the identity matrix I (zero order regularization) which limits the total magnitude of the solution, the gradient operator V (first order regularization) that limits the solution steepness, or the Laplacian L (second order regularization) that restricts steepness change. V maps the node space onto the edges space and plays the role of a spatial gradient. It is defined as following:   1 if n = ne,1  Ve,n = −1 if n = ne,2 (3.7)   0 otherwise

th where ne,1 and ne,2 are the indexes of the nodes of the heart surface sharing the e edge, and th where Ve,n is the (e, n) component of matrix V . V is constructed so that each of its rows computes the amplitude difference between two nodes belonging to the same edge of the heart surface mesh. Following this scheme, we deﬁne the Laplacian as the difference between two neighboring edges. The notion of numerical neighborhood between two edges is considered in the sense that neighboring edges are all the couples of edges sharing one node. Formally, L can be computed as the product L = AV where V is the numerical gradient and A is given by:

  1 if e = ei,1  Ai,e = −1 if e = ei,2 (3.8)   0 otherwise

with i representing an index in the set of couples of edges (ei,1, ei,2) sharing a node. To our knowledge, this is the ﬁrst time that a formal deﬁnition of the numerical Laplacian is provided. Regarding the temporal character of the sources, a legitimate assumption consists in considering

103 Chapter 3 – ECG inverse problem that the temporal change of heart surface potentials distribution is limited. This is referred to as Twomey 2 regularization that considers the regularization term ||z − z0||2. This approach is commonly found in the literature for the `2 norm.

3.2.3 Optimization schemes

`2 regularization In the Tikhonov case (p = 2), an analytical solution of (3.6) is obtained from 1st order optimality conditions by ﬁnding the solution that cancels the cost function’s partial derivative. The corresponding solution is given by:

∗ T T −1 T T z = (G G + λT T ) (G x + λT T z0) (3.9)

`1 regularization When p = 1, this doesn’t hold because of the non-differentiability of the `1 norm. The Alternating Direction Method of Multipliers (ADMM) has received considerable attention recently and is particularly suited for such problems. It consists in using the principle of variable splitting of the Douglas-Rachford algorithm [45] with the Augmented Lagrangian Multipliers (ALM) method [95] and uses partial updates for solving simpler subproblems. A major advantage of the ADMM in our optimization problem is that it offers the possibility of using proximal approaches for handling the `1 norm. In the rest of this paragraph, we consider only spatial regularization (z0 = 0). The ﬁrst step is rewriting problem (3.6) for p = 1 in the following equivalent constrained form:

∗ 1 2 (z) = argmin ||Gz − x||2 + λ||w||1 (3.10) (z) 2

s.t: Rz = w

Although this reformulation may seem trivial, the problem can now be tackled using methods of constrained convex optimization, in this case the ALM method, and offers the possibility for solving separately for z and for w. The augmented Lagrangian for (3.10) is given by:

1 ρ L (w, z, µ) = ||Gz − x||2 + λ||w|| + uT (Rz − w) + ||Rz − w||2 (3.11) ρ 2 2 1 2 2 where u is the scaled Lagrangian multiplier and ρ the penalty parameter. The advantage of “augmenting” the Lagrangian by adding the last term in equation (3.11) is that the method ALM converges under far more general conditions than the classical Lagrangian, including cases where the objective function is not strictly convex [18]. This equation can be equivalently written in a scaled form when considering

104 3.2. Methods

1 µ = ρ u: 1 ρ ρ L (w, z, µ) = ||Gz − x||2 + λ||w|| + ||Rz − w + µ||2 − ||µ||2 (3.12) ρ 2 2 1 2 2 2 2

A pair ((w, z), µ) is said to be a saddle point for Lρ if:

M×R×R ∀(z, w, µ) ∈ R Lρ((w, z), µ) ≤ Lρ((w, z), µ) ≤ Lρ((w, z), µ) (3.13)

Based on duality theory and the saddle point theorem [187], if there exists µ ∈ RR such that ((w, z), µ) is a saddle point for Lρ, then (w, z) solves problem (3.10). Conversely, if (w, z) is a solution to problem (3.10) and the constraint Rz = w is satisﬁed, then there is a µ ∈ RR such that ((w, z), µ) is a saddle point for Lρ. An optimal point Lρ of the Lagrangian can thus be approached either by solving the primal ∗ ∗ problem (z, w) = argmin Lρ(w, z, µ) or its dual problem µ = argmax Lρ((w, z), µ). Interestingly, (z,w) µ the optimal point for the primal and the dual problem coincide in the saddle point Lρ((w, z), µ) when the cost function and the constraint are strictly convex. Moreover, a solution to the saddle point problem provides a solution to (3.10). Following the ALM method, the solution of the saddle point problem is given by alternating the joint minimization with respect to (z, w) (primal problem) and the maximization with respect to µ (dual problem). In the ADMM, the joint minimization is avoided and is instead performed alternatively with respect to z and w, one after another. Convergence proofs of the ADMM algorithm can be found in [18] and update rules are given by alternating the optimization with respect to each variable. We have at iteration k + 1:

(k+1) 1 2 ρ (k) (k) 2 z = argmin ||Gz − x||2 + ||Rz − w + µ ||2 (3.14) z 2 2 (k+1) ρ (k+1) (k) 2 w = argmin λ||w||1 + ||Rz − w + µ ||2 (3.15) w 2 (k+1) ρ (k+1) (k+1) 2 ρ 2 µ = argmax ||Rz − w + µ||2 − ||µ||2 (3.16) µ 2 2

The update of z is obtained by canceling the partial derivative of Lρ with respect to z, and the update of µ is performed using the gradient ascent method. Since equation (3.15) involves the minimization of a non-smooth function, a proximity operator is used in the update of w. The proximity operator has ﬁrst been introduced in [110], it is frequently used in non-differentiable optimization problems and more particularly in TV denoising. The proximity operator proxf,β(·) of a convex function f and parameter β is deﬁned by: 1 2 proxf,β(w) = argmin ||w − y||2 + βf(y) (3.17) y 2 The idea behind this operator is that it compromises between minimizing f and being near to y. It can be viewed as a generalization of projections onto convex sets [123]. In the case where f is the `1 norm,

105 Chapter 3 – ECG inverse problem

∗ R the elements wi of w = proxf,β(w) ∈ R are given by:

wi wi = (|wi| − β)+ (3.18) |wi| where (·)+ = max(·, 0). Finally the ADMM algorithm for solving (3.10) is given in Algorithm 3. It is noteworthy that it was however proved recently in [24] that the ADMM algorithm may fail to converge and a randomized version of it was proposed in [165], which was shown to converge in expectation under some technical assumptions. Note that the use of such kind of randomized block-coordinate ﬁxed point algorithms was investigated in [35] where almost sure convergence properties were established. Basically, the randomized form of ADMM is similar to (3.14-3.16) but, instead of updating the components according to their index order 1, 2, they are updated following the order σk(1), σk(2), where σk : {1, 2} → {1, 2} is a permutation which is randomly chosen at each iteration k. Finally, it can be noticed that another technique for dealing with criteria split in a sum of more than two functions is to resort to parallel versions of ADMM [52, 125]. Note that the matrix that has to be inverted in the ADMM is GT G + ρRT R. In comparison with the Tikhonov solution, each iteration of the ADMM requires inverting the same matrix except that ρ in the ADMM plays the role of parameter λ in Tikhonov regularization.

Algorithm 3 ADMM algorithm Initialize variables z(0) = 0, w(0) = 0, µ(0) = 0 Compute and store P = GT G + ρRT R (k+1) (k) ||z −z ||2 While (k) < ||z ||2 z(k+1) = P −1 GT x + ρRT (w(k) − µ(k)) (k+1) (k+1) (k) w = prox`1,λ/ρ(Rz + µ ) µ(k+1) = µ(k) + Rz(k+1) − w(k+1) end

3.2.4 Determination of the regularization parameter

As explained in section 3.1.1, the regularization parameter plays a signiﬁcant role in regularization-based solutions. In this paragraph, we describe the optimal criterion method and introduce a new iterative method for determining λ based on the discrepancy principle. ∗ Optimal criterion We deﬁne the optimal parameter value λi by the one that minimizes the Relative Error (RE) in euclidean norm between the exact, known epicardial solution zˆ and the estimated solution z∗ using λ by the algorithms described previously: λi i

∗ ||z − zˆ||2 λi (3.19) zˆ

106 3.2. Methods

Adaptive choice based on the ADMM and the discrepancy principle The discrepancy principle is a method that determines a value for λ when the noise variance is available [70]. It states that satisfying 2 solutions are the ones lying in the set S = {z : ||Gz−x||2 ≤ c} where c is computed from noise variance. However, in regularized least square problems, it is very time-consuming to ﬁnd an appropriate value for λ, and Newton type methods are often used outside the original optimization scheme [159]. As shown in [77], an additional splitting in the ADMM used in TV denoising combined with the discrepancy principle results in a closed form for the update of the regularization parameter at each iteration. We use this strategy for a variety of sparse regularization terms and extend it to Tikhonov based methods. Let us reformulate problem (3.6) for p = 2 by adding the constraint y = Gz:

∗ λ 2 1 2 z = argmin ||x − y||2 + ||Rz||2 (3.20) z 2 2 s.t y = Gz

The scaled Lagrangian is given by:

λ 1 ρ ρ L (z, y, µ) = ||x − y||2 + ||Rz||2 + ||Gz − y + µ||2 − ||µ||2 (3.21) ρ 2 2 2 2 2 2 2 2

The update of variable y is obtained by canceling the corresponding partial derivative of Lρ and is given by: λx + ρ(Gz + µ) y(k+1) = (3.22) λ + ρ

Since the constraint states that y = Gz and that the update of y depends on λ, a convenient value 2 for λ according to the discrepancy principle can be obtained by solving the equation ||y(λ) − x||2 = c. Inserting the update of y (3.22) in the last expression gives a closed form for λ at iteration k + 1:

ρ||y(k) − Gz(k+1) − µ(k)|| λ(k+1) = √ 2 − ρ (3.23) c

(k+1) 2 This update needs to be performed only in the case where ||y − x||2 > c. Following the same ADMM scheme and inserting the update (3.23) results in the following algorithm:

In the same spirit, `1 regularization problems can be reformulated using two constraints:

∗ 2 z = argmin ||y − x||2 + λ||w||1 (3.24) (z)

s.t: Rz = w

Gz = y

107 Chapter 3 – ECG inverse problem

Algorithm 4 ADMM algorithm and discrepancy principle for Tikhonov regularization Initialize variables z(0) = 0, w(0) = 0, µ(0) = 0 and λ(0) Compute and store P = ρGT G + RT R (k+1) (k) ||z −z ||2 While (k) < ||z ||2 z(k+1) = ρP −1 GT (y(k) − µ(k)) (k+1) (k) 2 If ||Gz + µ − x||2 ≤ c λ(k+1) = 0 Else ρ||y(k)−Gz(k+1)−µ(k)|| (k+1) √ 2 λ = c − ρ end (k+1) λx+ρGz(k+1) y = λ(k+1)+ρ µ(k+1) = µ(k) + Gz(k+1) − y(k+1) end

The associated augmented Lagrangian involves two multipliers µ1 and µ2 and is given by:

λ 1 L (z, w, y, µ , µ ) = ||x − y||2 + ||w|| (3.25) ρ 1 2 2 2 2 1 ρ ρ + 1 ||Rz − w + µ ||2 − 1 ||µ ||2 2 1 2 2 1 2 ρ ρ + 2 ||Gz − y + µ ||2 − 2 ||µ ||2 2 2 2 2 2 2 and the resulting algorithm is given in Algorithm 5.

Algorithm 5 ADMM algorithm and discrepancy principle for sparse regularization (0) (0) (0) (0) (0) (0) Initialize variables z = 0, y = 0, w = 0, µ1 = 0, µ2 = 0 and λ T T Compute and store P = ρ2G G + ρ1R R (k+1) (k) ||z −z ||2 While (k) < ||z ||2 (k+1) −1 T (k) (k) T (k) (k) z = P ρ1R (w − µ1 ) + ρ2G (y − µ2 ) (k+1) (k) 2 If ||Gz + µ2 − x||2 ≤ c λ(k+1) = 0 Else ρ ||y(k)−Gz(k+1)−µ(k)|| (k+1) 2 √ 2 2 λ = c − ρ2 end (k+1) (k+1) λx+ρ2Gz y = (k+1) λ +ρ2 (k+1) (k+1) (k) w = prox`1,1/ρ1 (Rz + µ ) (k+1) (k) (k+1) (k+1) µ1 = µ1 + Gz − y (k+1) (k) (k+1) (k+1) µ2 = µ2 + Rz − w end

108 3.2. Methods

3.2.5 A new spatiotemporal approach with two `1 constraints

As stated in paragraph 3.1.1, methods that employ spatiotemporal constraints are of rising interest. Here we extend the use of Twomey regularization to the `1 regularization in a transform domain. We consider it as a complement to spatial regularization, leading to the following penalty term ||V (zt − zt−δt)||1. We consider it as a complement to spatial regularization and we express the inverse problem as:

∗ 1 2 z = argmin ||Gz − x||2 + λ1 (λ2||V z||1 + (1 − λ2)||V (z − z0)||1) (3.26) z 2 where z0 is the estimated solution at the previous time sample and V is the numerical gradient deﬁned in

(3.7). The two regularization parameters λ1 and λ2 are deﬁned such that λ1 represents the total amount of regularization in the optimization problem and λ2 ∈ [0, 1] represents the relative contribution of the two regularization terms. The term V (z −z0) reﬂects the difference between the spatial gradients at two consecutive time samples and its `1 norm is intended to penalize abrupt temporal changes in the space spanned by V . Following the same scheme as previously for adaptively choosing the regularization parameter and handling the `1 norm, we re-express problem (3.26) as:

∗ λ1 2 z = argmin ||y − x||2 + λ2||w1||1 + (1 − λ2)||w2||1 (3.27) (z) 2

s.t: V z = w1

V (z − z0) = w2

Gz = y

The associated scaled augmented Lagrangian is given by:

λ L (z, w , w , µ , µ ) = 1 ||y − x||2 + λ ||w || + (1 − λ )||w || (3.28) ρ 1 2 1 2 2 2 2 1 1 2 2 1 ρ ρ + 1 ||V z − w + µ ||2 − 1 ||µ ||2 2 1 1 2 2 1 2 ρ ρ + 2 ||V z − V z − w + µ ||2 − 2 ||µ ||2 2 0 2 2 2 2 2 2 ρ ρ + 3 ||Gz − y + µ ||2 − 3 ||µ ||2 2 3 2 2 3 2

The corresponding ADMM algorithm is given in Algorithm 6.

109 Chapter 3 – ECG inverse problem

Algorithm 6 ADMM algorithm for two spatiotemporal `1 constraints (0) (0) (0) (0) (0) (0) (0) Initialize variables z = 0, y = 0, w1 = 0, w2 = 0, µ1 = 0, µ2 = 0, µ3 = 0 T T Compute and store P = ρ3G G + (ρ1 + ρ2)V V (k+1) (k) ||z −z ||2 While (k) < ||z ||2 (k+1) −1 T T (k) (k) T (k) (k) z = P ρ3G (y − µ3) + ρ1V (w1 − µ1 ) + ρ2V (V z0 + w2 − µ2 ) (k+1) (k) 2 If ||Gz + µ3 − x||2 ≤ c λ(k+1) = 0 Else ρ ||y(k)−Gz(k+1)−µ(k)|| (k+1) 3 √ 2 2 λ = c − ρ3 end (k+1) (k+1) λ1x+ρ3Gz y = (k+1) λ1 +ρ3 (k+1) (k+1) (k) w1 = prox`1,λ2/ρ1 (V z + µ1 ) (k+1) (k+1) (k) w2 = prox`1,(1−λ2)/ρ2 (V z − V z0 + µ2 ) (k+1) (k) (k+1) (k+1) µ1 = µ1 + V z − w1 (k+1) (k) (k+1) (k+1) µ2 = µ2 + V z − V z0 − w2 (k+1) (k) (k+1) (k+1) µ3 = µ3 + Gz − y end

3.3 Numerical experiments

This section describes the numerical experiments that we conduct using the methods presented in section 3.2 based on the data generated by our model as described in chapter 2. Simulation protocol In a first numerical experiment, we perform a comprehensive study in a sinus-rhythm cardiac cycle, the performance of spatial regularization based methods depending on i) the regularization operator, ii) the norm used (`1 or `2), iii) the data whitening and iv) the parameters introduced by the optimization problem. In a second experiment, we investigate the performance of the used methods when λ is iteratively estimated using the ADMM combined with the discrepancy principle and compare it to that of the best case scenarios with respect to experiment 1. These two experiments are conducted on 9 key different instants selected from Lead II of the standard 12-Lead ECG as shown in figure 3.1. In the third experiment, we evaluate the proposed approach and compare its performance to that of classical regularization methods in the sinus-rhythm scenario used in experiments 1 and 2 but also in the cases of an ectopic beat and an ischemic beat. The source space is defined by the closed heart surface mesh which consists of 2111 nodes. Epicardial potentials on this surface are obtained by first running Algorithm 1 for the considered cardiac (n) scenario, solving (2.35), and then extracting from φex the coefficients that correspond to the heart surface. The set of observations consists of 256 points randomly selected on the torso surface. The corresponding potentials x are obtained through the equation x = Gz where G is constructed as explained in Appendix C. It is noteworthy that data are not affected by noise in this study, and that we

110 3.3. Numerical experiments evaluate the investigated methods under ideal modeling conditions. In the rest of the section we denote by `p − R the method that employs the `p norm (either `1 or `2) and the regularization operator R (either the identity I, the gradient V or the Laplacian L). In our experience the ADMM algorithm for our problem converges in less than 300 iterations and we

fixed the maximum number of iterations Niterations accordingly. The convergence criterion if fixed to a value of 10−3, the value of c used with the discrepancy principle and corresponding to noise variance is fixed to 10−13. Methods performance is studied according to the RE between the estimated solution and the exact solution, which is commonly accepted to be a good performance indactor.

Figure 3.1: Selected times in Lead II for experiments 1 and 2

3.3.1 Experiment 1: inﬂuence of whitening and regularizers

This experiment studies the performance of the different combinations in terms of pre-processing and spatial regularization. The regularization parameter λ is chosen based on the optimal criterion deﬁned by the expression (3.19) and varied in a set of 13 values non uniformly distributed that we give in table 3.1. The penalty parameter ρ introduced when using the ADMM is also deﬁned in a grid provided in table 3.2. At this stage, we only apply a classical SVD when a whitening is performed and do not

111 Chapter 3 – ECG inverse problem

consider its truncated version. For more readability, we ﬁrst provide the results separately for the `2 and the `1 norms and then we compare them through their best case scenarios.

10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 0.01 0.1 1 2 4 6 8 10

Table 3.1: Variation grid for parameter λ

0.001 0.005 0.01 0.015 0.1 0.5 1 2 4 6 8 10 15 20 40 100 1000

Table 3.2: Variation grid for parameter ρ

Figures 3.2 and 3.3 present results of the experiment at the considered different time instants in a graphical form when the `2 and `1 norms are used, respectively. The ﬁrst remark is that the considered pre-processing enhance considerably the results in terms of RE. The whitening with respect to G and the one with respect to X seem to perform equivalently both for the `1 and `2 norms. The second remark is that for both norms, the Laplacian and gradient operators perform better than the identity, whether data are pre-processed or not. Still, results are in general slightly better when the gradient is chosen. If one looks along the time dimension, it is noticeable that the methods do not perform equivalently depending on the time course of cardiac activity.

For a closer look, we provide RE values in tables 3.4 and 3.5 for methods using the `2 and `1 norms, respectively. Each row RE of these tables represents the performance of one method through time. It is interesting to see that REs are globally higher during atrial activity (instants 1-3) than during ventricular activity (instants 4-9), which suggests that the inverse problem is harder during atrial activity, and that these spatial regularizations are best suited for epicardial mappings in which most of nodes are active, as it is the case during ventricular activity. These tables also provide the optimal values of λ and ρ

(when applied). In the case of `2, very low amount of regularization needs to be considered to achieve good solutions in the case of raw data and data whitened with respect to G. This can be explained by the difference in the low energy contained in the matrix GT G compared to that of RT R. Indeed, when inverting GT G + λRT R in the Tikhonov solution, it is advisable that the second term is of lower energy T ||G G||2 −3 in order to achieve good data ﬁdelity. In our case T = 5.6 × 10 , which explains the need for a ||R R||2 low value of λ. In the case of the ADMM, the parameter ρ plays the role of λ while inverting matrix P in Algorithm 3. When whitening with respect to data is applied, the optimal value of λ is much higher. This is due to the matrix inversion involved in the computing of the whitening matrix. The lowest singular value is in the range of 10−13, which means that the whitened version of G has a much higher energy than the operator R. In table 3.5, one can see that the optimal value of λ is globally as low as for the

112 3.3. Numerical experiments

`2 norm, but it is shown that the optimal value varies in time when a whitening with respect to G is applied. Values of ρ vary more and are much higher in the raw data case, and even higher in the case of whitening with respect to X, the same reasoning applies as for λ. In figure 3.6, we represent the cardiac epicaridal mapping of the reference and the estimated solutions at time 5 when a pre-whitening with respect to G is applied. The estimated mappings are globally in line with the quantitative results: `1 and `2 regularization give similar performance, and the estimation is more influenced by the regularization operator R than the used norm. Moreover, results obtained when using the gradient and the Laplacian are very close whereas the solution obtained when using the identity does not correspond to the reference. In figure 3.7 we present a graphical performance comparison between the `1 and the `2-based methods. Since the best results are those obtained by using the gradient for both norms and since data pre-whitening gives very close results for the `1 norm and better results for `2 when the whitening is with respect to G, we only show this particular case in the

figure. Clearly, the spatial constraint involving the gradient and the `1 norm performs better than when using the `2 during atrial activity. Their performance is quite similar during the other time instants. This better performance in instants 1-3 can be explained by the wave-like aspect of the epicardial mapping at these times. Since only atria are activated during this period, a sharp difference of amplitude is present in the space neighboring the electrical wave while the rest of the source domain is not active. This sparse nature is thus better expressed using an `1 norm and the sharp amplitude variation is better represented when using the gradient. During ventricular activity, a higher RE is obtained at instant 6. In fact, since most ventricular tissue is active, it is harder to find abrupt changes and the term V z has many more high amplitude components than in the other instants. In order to study the influence of a TSVD, we chose different truncation indexes and we applied a

TSVD on G. We use methods `1 − V and `2 − V to solve each corresponding problem and we report the result in ﬁgure 3.8. The chosen indexes values are 256, 250, 240, 200 and 150. Clearly, the RE increases when the index k decreases. Although the last singular values are very low (approximately 10−13), all the information in G is required for solving the ill-posed inverse problem. This may be explained by G being full row rank, which means that as the truncation index decreases, substantial information in the matrix is dropped.

113 Chapter 3 – ECG inverse problem

Figure 3.2: Graphical representation of `2 methods performance for raw and whitened data with respect to G and to X 114 3.3. Numerical experiments

Figure 3.3: Graphical representation of `1 methods performance for raw and whitened data with respect to G and to X 115 Chapter 3 – ECG inverse problem

Figure 3.4: RE values when using `2 regularization at different times and optimal values of λ in the search grid. Best case scenarios are highlighted in shaded gray

116 3.3. Numerical experiments

Figure 3.5: RE values when using `1 regularization at different times and optimal values of λ and ρ in the search grids. Best case scenarios are highlighted in shaded gray

117 Chapter 3 – ECG inverse problem

Figure 3.6: Estimated cardiac mappings at instant 5 by the different methods when data are whitened with respect to G

Figure 3.7: Comparison of `2 − V and `1 − V along time when data are whitened with respect to G

118 3.3. Numerical experiments

Figure 3.8: Inﬂuence of the TSVD index on the performance of `2 − V and `1 − V

119 Chapter 3 – ECG inverse problem

3.3.2 Experiment 2: evaluation of the ADMM combined with the discrepancy principle

The first experiment shows that the optimal value of λ is in most cases the lowest value. However, although a lower bound of 10−6 is low, it may be possible that the true optimal value of λ is even lower. Moreover, it has been shown that the optimal value for this parameter varies with time in the case of whitening with respect to G. In this experiment we investigate the use of the discrepancy principle combined with the ADMM and we compare its performance to solutions obtained with the optimal criterion. Since the gradient is the best regularizer we only compare the `1 − V and the `2 − V methods for the different pre-processings. The parameter ρ1 in Algorithm 5 is fixed according to the optimal values obtained in experiment 1 and parameters ρ2 in Algorithms 5 and ρ in Algorithm 4 are fixed to 1 in order to keep a unitary weight of the term GT G while inverting matrix P . In figure 3.9 the blue bars correspond to `2 methods and the red bars correspond to `1 methods. Shaded colors indicate the use of the discrepancy principle with ADMM and dark colors correspond to solutions estimated by the optimal criterion and presented in the first experiment. This experiment uses the methods `1 − V and the `2 − V since they give the best results according to experiment 1. In the case of raw data, the discrepancy principle case largely outperforms the case of the optimal criterion when the `1 norm is used. On the contrary, it gives higher errors when a whitening of data is applied in instants 1-3.

The discrepancy principle applied to the `2 norm gives similar results to those obtained with the optimal criterion. When the `1 norm is used, the ADMM-discrepancy method gives better results at instants 1-3 when a whitening with respect to G compared to the case of whitening with respect to X. An opposite tendency can be noticed at times 4-9 where whitening G performs better. Depending on the chosen pre-processing, the ADMM with the discrepancy principle gives fairly comparable results to those obtained with an exhaustive search of parameters λ and ρ. Interestingly, the ADMM algorithm with the discrepancy principle converges much faster and stops in less than 20 iterations. This method thus offers a good compromise between reasonable estimation accuracy and computational complexity.

120 3.3. Numerical experiments

Figure 3.9: Comparison of `1 and `2 regularization when the gradient operator is used, for raw and whitened data with respect to G and to X 121 Chapter 3 – ECG inverse problem

3.3.3 Experiment 3: Performance study of the double `1 constraint

This experiment evaluates our approach, which is a spatiotemporal `1-based method that uses two constraints and determines λ1 adaptively with the discrepancy principle. This method is evaluated under a sinus-rhythm scenario, a VT scenario (with corresponding LAT in figure 2.29) and an ischemic scenario (with corresponding LAT in figure 2.28). Since these pathologies originate from ventricular activity, we compare our method with the method from experiment 2 that performs best during ventricular activity and uses the discrepancy principle, namely the `1 − V method with a pre-whitening applied on G. For consistency we apply the same pre-processing before running our method. In experiment 2, since the value of penalty parameters was fixed according to the previous experiment, we also study the influence of one of these parameters on the solutions obtained by the two selected methods. In algorithm 5, two penalty parameters ρ1 and ρ2 are used. We choose to vary only parameter ρ1 since it is involved both in the inversion of P and in the proximal operator. In Algorithm 6 used in the case of two `1 constraints, the influence of parameter ρ1 in Algorithm 5 is split into two parameters denoted ρ1 and ρ2 in Algorithm

6. For any chosen ρ1 in Algorithm 5, we assign the half of its value to each of parameters ρ1 and ρ2 in Algorithm 6. In the case of two constraints, parameter λ2 is assigned to 0.5 in order to give equal inﬂuence to each regularization term. For each cardiac scenario and for each chosen value of ρ1 in table 3.2, we run the two selected methods for a full cardiac cycle which consists of approximately 400 time samples. We compute the RE of each method at each time instant and we compute the mean RE value with respect to the temporal dimension. Results are reported in table 3.10. Except when ρ1 equals 1000, for which both errors are the highest, our proposed approach performs better than the `1 − V method in all cardiac conditions. Since the `1 − V with a whitening of data with respect to G is the best scenario, it is very likely that our approach outperforms methods presented in experiments 1 and 2. Moreover, the lowest reported error corresponds to the ischemic case, which is not surprising since potentials in the ischemic region are almost constant during the cardiac cycle, and that this is particularly the type of the source character that is enforced in our method.

122 3.4. Conclusions

Figure 3.10: Comparison of our proposed approach with the `1 − V method when a pre-whitening of G is applied

3.4 Conclusions

In this chapter, we proceed to a comprehensive study of regularization-based methods for solving the ECG inverse problem. We study the inﬂuence of the algorithms, the regularization, data pre-processing and the method for choosing an adequate regularization parameter. We show that both whitenings with respect to G and X give better results when an adequate value of the regularization parameter is chosen. Although performance of the methods depends on the time course at which the inverse problem is solved, the source character seems to be generally be better represented when using the gradient operator and the `1 norm. The choice of the regularization parameter is critical and often requires deﬁning a dense grid of search and a lot of computations. To circumvent this problem we combine the

ADMM with the discrepancy principle in order to adaptively estimate this parameter in the case of `1 regularization and we adapt this scheme to Tikhonov regularization. Results show that this algorithm succeeds in reproducing similar performance to that when the regularization parameter is chosen with an optimal criterion. Finally we propose a new spatiotemporal regularization based on two `1 constraints and show that it outperforms the best method among those that are compared through the study in normal and pathological cases.

123

GENERALCONCLUSION &

PERSPECTIVES

The present work had as main objective to perform a quantitative comparison of existing and new methods for solving the inverse problem of the ECG, using a set of data obtained from controlled numerical whole-heart and torso simulations. Four main contributions can be highlighted from this work. The first contribution consists in the proposal of a simplified, fast, controllable and evolving whole-heart and torso model, destined to perform quantitative evaluations of inverse methods on simulated data. ECGs generation by this model is made fast by the use of the monodomain formalism and the FHN phenomenological model. Region-dependent model parameters, characterizing the electrical properties of the main anatomical structures of the heart are used in order to best represent the propagation of cardiac electrical activity. With the proposed model, the simulation of whole-hart cardiac electrical activity propagation and the corresponding ECG synthesis for a whole cardiac beat is performed in less than 3 minutes on an Intel Xeon 3.47 Ghz processor computer. We have shown that this model is capable of synthesizing realistic normal ECG signals under sinus rhythm, as well as realistic pathological activation patterns representing VT and the effect of myocardial ischemia. A second contribution, which is a direct consequence of the above-mentioned model properties, is the parameter identification process that has been used in order to generate ECG signals that best fit a given observed reference. We show that, based on the proposed cost function, it is possible to reproduce the main temporal features and the general aspect of an observed, real ECG. The third contribution of this work is a sensitivity analysis of the monodomain model with respect to its parameters. We identified the FHN parameters having the highest influence on the generated cardiac electrical activity. The above-mentioned parameter identification process succeeds in giving the same solution when only these highly-sensitive parameters are identified. Building from these three first contributions, we consider the proposed modelling approach as an interesting signal generator in the context of ECG inverse methods evaluation. The proposed model is built on a evolutive perspective. The use of the FEM formalism allows us to consider anisotropic structures and the simplified geometry offers the possibility to easily study the influence of geometrical errors in the ECG inverse problem. Moreover, this model allows for the possibility of considering

125 Chapter 3 – General conclusion & perspectives several source formulations when dealing with the inverse problem. Indeed, it provides volumetric transmembrane potentials as well as epicardial potentials, LATs can be computed from transmembrane potentials in order to consider an UDL or an EDL source model. Furthermore, the applied evolutionary method has the potential to be used as, or included within an inverse algorithm, in which one seeks model parameters that result in the best ﬁt to reference data.

The fourth contribution of this work is the proposal of a novel spatiotemporal approach for solving the ECG inverse problem and its performance comparison with classical inverse regularization-based methods on data simulated by our model. This part of the work first analyzes the influence of several aspects on the solution, namely: i) data-preprocessing using a whitening transform that is either performed on the transfer matrix G or the data matrix X, ii) the regularization operator used, iii) the regularization norm and iv) the penalty parameters. When `1 regularization is considered, we employ the ADMM to solve the optimization problem for its good compromise between numerical complexity and computation accuracy. To our knowledge, no study has ever used the ADMM in the context of ECGI in a classical regularization formulation. We showed that a pre-processing step, either performed on the data or on the transfer matrix enhances considerably the estimated solutions in terms of relative error. Moreover, we find that the TV performs better than the other classical methods, meaning that it is better suited to the epicardial source character than the Laplacian and the Identity. We also introduce the discrepancy principle combined with the ADMM algorithm in order to adaptively determine a value for the regularization parameter at each iteration of the ADMM. This has been used previously in the context of

TV denoising, we adopt the same scheme and we adapt this method to `2 regularization. Results show that the discrepancy principle gives similar results than when using the optimal criterion for choosing the regularization parameter. Finally, we propose a novel approach that uses two `1 constraints that reﬂects both the spatial and the temporal character of the cardiac electrical source. The ﬁrst constraint is the TV norm of the source and the second one is the extension of Twomey regularization to an `1 regularization term. We study the performance of these methods on sinus-rhythm and two pathological cases and we show that the proposed method outperforms the best spatial regularization based method.

The perspectives of this work can be distinguished between the cardiac modeling part and the inverse problem part. Regarding the modeling part, the sensitivity analysis is to be extended in order to explore model parameters that can be linked to pathological cases. This can serve to better identify methodologies for simulating certain cardiac pathologies. We have shown that our model is capable of simulating ischemia and idiopathic VTs, the next step is to simulate re-entrant pathways in order to assess the possibility of recovering such phenomena in an inverse procedure. In order to do so, an improved spatial resolution should be included in the model. Finally, further improvements on the computational costs associated with this model can be obtained by optimizing some implementation

126 aspects. Indeed, numerical simulations are performed using Matlab, which is an interpreted language. Thus, re- implementing the full heart-torso propagation model in a compiled language such as C++ would incredibly reduce simulation time. Another improvement lies in the use of parallel computing for the update of the latent variable in the FHN model. In fact, since this variable is computed independently for each node of the heart mesh, considering a parallel computing instead of the present sequential implementation would also reduce simulation time. A preliminary parallel version of the model has been implemented in the Ansys Mechanical software. However, this version still doesn’t run in its optimal speed due to some software architecture issues. The major one is that the current version reloads the model matrices at each iteration time, which is an unnecessary step that precludes the model from reaching the desired computation speed. Regarding the ECG inverse problem, further work is warranted on investigating new regularization strategies that best take into account the spatiotemporal character of the source. A ﬁrst way to do so is by combining assumptions on the sparse temporal character and the smooth spatial character of epicardial potentials through the use of mixed norms constraints. These mixed norms have been successfully applied in other contexts such as in EEG, where the `1,2 mixed norm is often used to promote spatially focal sources with smooth temporal estimates [10]. Another interesting approach lies in considering the notion of social sparsity which has proven to exploit well the structured sparsity of signals depending on local neighborhoods [92, 178]. This approach might be a substitute to the TV norm in our context in order to better represent the notion of sparsity in a transformed domain. We also intend to investigate regularization that exploits our propagation model in order to enforce the propagating nature of cardiac electrical activity. Future work also includes the comparison of the ADMM with other algorithms that handle `1-based regularization (e.g Proximal Alternating Linearized Minimization (PALM) [14] and the Iterative Reweighted Norm (IRN) [158]) in terms of convergence rate and speed, numerical complexity and estimation accuracy. In this study, inverse ECG simulations were conducted without the presence of noise. The results presented in this work might be compromised in the presence of noise and methods robustness might not be the same. Another perspective is to consider real data originating from electrophysiological procedures. Ongoing work with the Rennes University Hospital offers the possibility of using invasive data obtained during surgical procedures as well as patients anatomy through CT scans. This will be an opportunity to compare the performance of inverse methods that have been obtained using data simulated by our model with those of real case geometries and electrophysiological data. We have shown in chapter 3 that methods performance depend on the studied time instant, which means that an adequate choice of the regularization parameter should be made depending on the considered time instant. An idea to be investigated is the automatic tuning of λ depending on features extracted from the observations.

127

APPENDIX A

THEFULLHEART-TORSOBIDOMAIN

MODEL

Like any electromagnetic system, the electrical activity in the body is governed by Maxwell’s equations:

ρ ∇ · E = (A.1) 0 ∇ · B = 0 (A.2) ∂B ∇ × E = − (A.3) ∂t ∂E ∇ × B = µ J + µ (A.4) 0 0 0 ∂t where ∇· and ∇× are divergence and curl operators, respectively. E and B represent the electric and magnetic fields, respectively. µ0 and 0 are the vacuum permeability and permittivity, respectively. ρ is the electric charge density, and finally J is the current density. Although cardiac sources are time-varying, the temporal variation of the electrical and the magnetic field they generate can be neglected to a good approximation. Indeed, considering the size of the human torso and the range of physiological variables in it, capacitive, inductive, and propagation effects of these fields can largely be ignored. This assumption is referred to as the quasi-static approximation and is treated in more detail in appendix B. Under these conditions, temporal derivatives of E and B are neglected and simplified Maxwell equations are given by:

ρ ∇ · E = (A.5) 0 ∇ · B = 0 (A.6)

∇ × E = 0 (A.7)

∇ × B = µ0J (A.8)

129 Based on vector analysis, (A.7) implies that E derives from a scalar electrical potential φ: E = −∇φ, and ∇ · (∇ × B) = µ0∇ · J = 0. The latter means that there is no charge accumulation everywhere in the studied domain: any current entering a closed volume must be equal to the current going out of this volume. Besides, Ohm’s local law writes: J = σE = −σ∇φ. The charge conservation is then given by the generalized Laplace equation: ∇ · (σ∇φ) = 0 (A.9) where σ is the conductivity tensor.

Following the Bidomain concept, intracellular and extracellular currents ﬂow through each macroscopic point in the heart domain H so that the total current density in the heart Jh is the sum of intracellular Jin and extracellular Jex current densities: ∇ · (Jh) = ∇ · (Jin + Jex) = 0. Introducing the corresponding potentials φin and φex and conductivity tensors σin and σex, this charge conservation equation becomes:

∇ · (σin∇φin) + ∇ · (σex∇φex) = 0 (A.10)

Equation (A.10) states that the inward current in a domain must be equal to the outward current of the other domain. Since these currents ﬂow through cellular membranes, they can be expressed as a sum of a capacitive term and a ionic term according to Hodgin-Huxley formalism such that:

∂v ∇ · (σ ∇φ ) = −∇ · (σ ∇φ ) = χ c m + i + i (A.11) in in ex ex m ∂t ion stim where vm = φin − φex denotes the membrane potential, cm is the membrane capacitance, iion is the sum of ionic currents ﬂowing between intracellular and extracellular spaces. χ is a ratio coefﬁcient that adapts the originally surface membrane current to a volume current. The variables of interest are vm and

φex because they are measurable during clinical procedures. Using: φin = vm + φex and rewriting (A.10) and (A.11) gives the two main equations of the bidomain model for variables vm and φex:

∂v ∇ · (σ ∇v ) + ∇ · (σ ∇φ ) = χ c m + i + i (A.12) in m in ex m ∂t ion stim

∇ · (σin∇vm) + ∇ · ((σin + σex)∇φex) = 0 (A.13) where the left hand side terms are the reaction terms which represent regions passive reaction to neighboring activity, and the right hand side is the diffusion term that symbolizes the active membrane activity. In general, the transmembrane ionic current iion is deﬁned as a function of vm and some other variable u, possibly vector valued. u usually satisﬁes an Ordinary Differential Equations (ODEs) system of the form: ∂u + f(v , u) = 0 (A.14) ∂t m

130 where f is some chosen function depending on the considered ionic model. Regarding the passive extra-cardiac region T , the absence of primary currents is expressed by the generalized Laplace equation:

∇ · (σt∇φt) = 0 (A.15) where φt and σt are respectively the potentials and the conductivity in the torso domain, outside the heart. In order to link the source domain and the volume conductor variables, coupling conditions between the two domains are needed. Current continuity at the boundary ∂H separating the heart and the torso links torso current density Jt and cardiac current density Jh by:

n∂H · Jt = n∂H · (Jin + Jex) (A.16)

where n∂H is the unit outward pointing normal vector to the heart surface. A common and realistic assumption consists in considering that the intracellular domain is insulated from the extra-cardiac regions, while the extracellular domain is contiguous to regions outside the heart.

This is mathematically formulated by: n∂H · Jin = 0 at the boundary ∂H and leaves:

n∂H · Jt = n∂H · Jex (A.17)

The ﬁnal form of boundary conditions on the surface ∂H separating the heart and the body domain is given by the following Dirichlet and Neumann boundary conditions:

φt = φex (A.18)

n∂H · (σt∇φt) = n∂H · (σex∇φex) (A.19)

Equations (A.18) and (A.18) thus translate potential and ﬂux continuity respectively on the boundary

∂H. This means on the one hand that σt and φt can be considered as an extension to the extra-cardiac regions of cardiac extracellular variables σex and φex. On the other hand, transmembrane potentials and primary current sources are only deﬁned in the heart domain. Finally, since the body is surrounded by air, whose conductivity is assumed to be zero, no current ﬂows outside the torso. This leads to the last Neumann boundary condition on the torso surface ∂T :

n∂T · (σt∇φt) = 0 (A.20)

where n∂H is the unit outward pointing normal vector to the torso surface.

131 In the heart domain H

∂u + f(v , u) = 0 (A.21) ∂t m ∂v ∇ · (σ ∇v ) + ∇ · (σ ∇φ ) = χ c m + i (v , u) + i (A.22) in m in ex m ∂t ion m stim

∇ · (σin∇vm) + ∇ · ((σin + σex)∇φex) = 0 (A.23)

On the heart surface ∂H

φt = φex (A.24)

n∂H · (σt∇φt) = n∂H · (σex∇φex) (A.25)

In the volume conductor domain T

∇ · (σt∇φt) = 0 (A.26)

On the exterior torso surface ∂T

n∂T · (σt∇φt) = 0 (A.27)

132 APPENDIX B

QUASI-STATIC APPROXIMATION

The quasi-static approximation consists in neglecting the partial derivatives in the Maxwell equations system, meaning: ∂BB ∂EE ≈ ≈ 0 ∂t ∂t

To validate this approximation for the electric ﬁeld, we consider a frequency band between 0 and 1kHz, which corresponds to the frequency domain of what we observe in ECG. The chosen values for tissue 5 −7 −1 permittivity and conductivity in the human body are respectively: ≈ 3.5 × 10 0 = 3.1 × 10 F.m and σ ≈ 0.11S.m−1 [57]. We start by verifying that: ∂EE || || ||σEE|| ∂t

In the harmonic state, the electric ﬁeld can be expressed as following:

2πjft E(xx, t) = E0(x)e

Where E0(x) is the amplitude of the ﬁeld and where x denotes the space point variable, characterized by its three Cartesian coordinates (x,y,z). ∂EE In this case ∂t = 2πfE0(x). The ratio between the two norms is reduced to:

2πf = 2.8 × 10−4 1 σ

Consequently, the approximation is veriﬁed for the electric ﬁeld, and equation (A.4) becomes:

∇ × B = µ0J

The second hypothesis is: ∂BB ≈ 0 ∂t

−2πjft We consider, without loss of generality, a planar electrical wave E(rr, t) = E0(r)e . Applying ∇× to

133 (A.3) gives:

∂ ∇ × ∇ × E = − (∇ × B) ∂t = ∇(∇ · E) − ∆E

Where ∆ is the Laplace operator. Since there is no current ﬂow in the torso domain: ∇ · E = 0, which implies:

∂ (∇ × B) = ∆E ∂t

Moreover, Ohm’s law is expressed by:

∂PP ∂EE J = σEE + = σEE + ( − ) (B.1) ∂t 0 ∂t

(A.4) and (B.1) give:

∂ (∇ × B) = 2iπfµ (σ + 2iπf)E ∂t 0 = ∆E

1 ∂2E The propagation equation for the electric ﬁeld is: ∆E − c2 ∂t = 0, where c is the celerity of the light, which implies:

1 2iπfµ (σ + 2iπf)E − (2iπf)2E = 0 0 c2

Since c = λf, we ﬁnally have: s 2π λ = |2iπfµ0(σ + 2iπf)|

Which gives a wavelength of λ = 290m for the electric ﬁeld. The human body dimensions are much smaller than λ, which veriﬁes the quasi-static approximation. We can then write:

∇ × E = 0

134 APPENDIX C

EPICARDIALFORMULATIONOFTHE

FORWARD PROBLEM

This appendix describes how torso surface potentials can be obtained from heart surface potential distributions. We recall that the epicardial formulation of the forward problem is given by the generalized Laplace equation completed with Dirichlet boundary condition on the heart surface ∂H and Neumann boundary condition on the exterior torso surface ∂T : On the heart surface ∂H

φt = φex (C.1)

In the volume conductor domain T

∇ · (σt∇φt) = 0 (C.2)

On the exterior torso surface ∂T

n∂T · (σt∇φt) = 0 (C.3)

The weak form of the system is given by multiplying equation (C.2) by a shape function ψ and integrating in the studied domain, namely the volume conductor T :

Z ∇ · (σt∇φt)dx = 0 (C.4) T

135 Applying Green’s divergence theorem yields:

Z Z Z ∇ · (σt∇φt)dx = − (σt∇φt) · ∇ψdx + (σt∇φt) · n∂T ψds (C.5) T T ∂T Z + (σt∇φt) · n∂Hψds ∂H = 0

where n∂T is the unitary outward pointing normal with respect to the outer torso boundary ∂T and the inner torso boundary (heart surface) n∂H, respectively. Because of boundary condition (C.3), the second term on the right hand side vanishes. Moreover, since the potential distribution on the heart surface ∂H, the weighting function ψ is chosen to be equal to zero on this surface, which cancels the last term. The equation in the volume conductor domain becomes:

Z Z Z ∇ · (σt∇φt)dx = − (σt∇φt) · ∇ψdx + (σt∇φt) · n∂T ψds (C.6) T T ∂T Z + (σt∇φt) · n∂Hψds ∂H

The discretization of the field variable φt over a finite element of the volume conductor mesh by the first order basis functions (γi)i∈{1,..,NH} (defined in section 2.2.3) gives:

X φt(x) = φtiγi(x) (C.7)

i∈MT

By injecting this expression in equation (C.6) and replacing ψ by a shape function γk we have:

X Z φti (σt∇γi) · ∇γkdx = 0 (C.8) T i∈MT

Now if we evaluate k for each node index in MT except those on the heart surface (for which potentials are known through (C.1)), we get the linear system:

  φ∂H A t = 0 (C.9)  T\∂H  φt

∂H where φt is the discretized version of volume conductor potentials on the heart surface ∂H (known T\∂H through Dirichlet boundary condition (C.1)), φt is the discretized version of (unknown) potentials in the rest of the volume conductor T\∂H, and A is rectangular matrix whose column indexes correspond to the set MT of nodes in the whole volume conductor domain T , and whose rows correspond to the

136 set MT\∂H nodes in the volume conductor domain but outside the heart surface. Its coefﬁcients are given by: Z A = (σt∇γi) · ∇γkdx T i∈MT\∂H k∈MT Separating the known and the unknown potentials in the system (C.9) by splitting A gives:

    φ∂H h i φ∂H A t = t ⇐⇒ CφT\∂H = −Bφ∂H = 0 (C.10)  T\∂H  BC  T\∂H  t t φt φt where C is a square matrix relating nodes in the domain T\∂H and B is a rectangular matrix whose ∂H rows correspond to nodes in T\∂H and whose columns correspond to nodes in ∂H. Since φt is known, potentials in the rest of the volume conductor domain T can be obtained by solving the linear system. Matrix C is the stiffness matrix in the discretization of the epicardial forward model. The Dirichlet boundary condition ensures that this forward problem is well posed in the sense that matrix C is positive deﬁnite, and thus invertible. Consequently, a unique solution exists and is obtained by inverting C. Torso potentials in the volume conductor are thus linked to heart surface potentials at each time instant through the linear system: T\∂H ∂H φt = Dφt (C.11) where D = C−1B is the transfer matrix between the source potentials on the heart surface and those T\∂H outside this surface. Since only few torso potentials are measured in reality, only a sub-vector of φt is available. This can be modeled by a selection matrix S whose entries are ones and zeros that extracts only the sub-vector x of available torso potentials such that:

∂H x = SDφt (C.12)

We get the ﬁnal system: ∂H x = Gφt (C.13) where G = SD is the ill-conditioned transfer matrix relating the observations and the source potentials that is used in chapter 3.

137

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Titre : Modélisation et Imagerie Électrocardiographiques

Mot clés : problème inverse, ECG, arythmies, modèle cardiaque, MEF

Résumé : L’estimation des solutions du prob- temps, nous avons conçu un modèle cœur-torse lème inverse en ElectroCardioGraphie (ECG) basse-résolution qui génère des cartographies car- représente un intérêt majeur dans le diagnostic diaques et des ECGs réalistes dans des cas sains et la thérapie d’arythmies cardiaques par cathéter. et pathologiques. Ce modèle est bâti sur une Ce dernier consiste à fournir des images 3D de la géométrie cœur-torse simplifiée et implémente le distribution spatiale de l’activité électrique du cœur formalisme monodomaine en utilisant la Méthode de manière non-invasive à partir des données des Éléments Finis (MEF). Les paramètres ont anatomiques et électrocardiographiques. D’une été identifiés par une approche évolutionnaire et part ce problème est rendu difficile à cause de leur influence a été analysée par une méthode de son caractère mal-posé. D’autre part, la validation criblage. Dans un second temps, une nouvelle des méthodes proposées sur données cliniques approche pour résoudre le problème inverse a été reste très limitée. Une alternative consiste à proposée et comparée aux méthodes classiques évaluer ces méthodes sur des données simulées dans les cas sains et pathologiques. Cette méth- par un modèle électrique cardiaque. Pour cette ode utilise un a priori spatiotemporel sur l’activité application, les modèles existants sont soit trop électrique cardiaque ainsi que le principe de con- complexes, soit ne produisent pas un schéma de tradiction afin de trouver un paramètre de régulari- propagation cardiaque réaliste. Dans un premier sation adéquat.

Title: Modeling & Imaging of Electrocardiographic Activity

Keywords: inverse problem, ECG, arrhythmia, cardiac model, FEM

Abstract: The estimation of solutions of the terns. As a first step, we designed a low-resolution inverse problem of ElectroCardioGraphy (ECG) heart-torso model that generates realistic cardiac represents a major interest in the diagnosis and mappings and ECGs in healthy and pathological catheter-based therapy of cardiac arrhythmia. The cases. This model is built upon a simplified heart- latter consists in non-invasively providing 3D im- torso geometry and implements the monodomain ages of the spatial distribution of cardiac electrical formalism by using the Finite Element Method activity based on anatomical and electrocardio- (FEM). Parameters were identified using an evolu- graphic data. On the one hand, this problem is tionary approach and their influence was analyzed challenging due to its ill-posed nature. On the other by a screening method. In a second step, a hand, validation of proposed methods on clinical new approach for solving the inverse problem was data remains very limited. Another way to proceed proposed and compared to classical methods in is by evaluating these methods performance on healthy and pathological cases. This method uses data simulated by a cardiac electrical model. For a spatiotemporal a priori on the cardiac electrical this application, existing models are either too activity and the discrepancy principle for finding an complex or do not produce realistic cardiac pat- adequate regularization parameter.