2 cm 2 cm Development of a Hill-Type Muscle Model With Fatigue for the Calculation of the Redundant Muscle Forces using Multibody Dynamics ANDRÉ FERRO PEREIRA Dissertação para obtenção do Grau de Mestre em ENGENHARIA BIOMÉDICA Júri Presidente: Prof. Helder Carriço Rodrigues Orientador: Prof. Miguel Pedro Tavares da Silva Prof. Mamede de Carvalho Vogais: Prof. Jorge Manuel Mateus Martins Prof. João Nuno Marques Parracho Guerra da Costa Outubro 2009 ii Resumo O objectivo deste trabalho é o desenvolvimento de um modelo muscular versátil e a sua implementação de forma robusta e eficiente num código de dinâmica de sistemas multicorpo com coordenadas naturais. São considera- dos dois tipos de modelos: o primeiro é um modelo muscular do tipo Hill que simula o comportamento das estruturas contrácteis, tanto para análises em dinâmica directa como inversa. O segundo é um modelo dinâmico de fadiga muscular que toma em consideração o historial de cada músculo, em termos de força produzida, estimando o seu nível de aptidão física usando um mod- elo multi-compartimentar triplo e a hierarquia de recrutamento muscular. A formulação para as equação do movimento é adaptada, de forma a incluir os modelos descritos, através do método de Newton. Isto permitirá, numa perspectiva de dinâmica directa, que se proceda ao cálculo da cinemática do sistema mecânico resultante para determinadas activações musculares previ- amente conhecidas, ou, numa perspectiva de dinâmica inversa, a computação das activações musculares necessárias para provocarem um movimento artic- ular prescrito. Para ambas estas formulações, os sistemas são redundantes, uma propriedade que é ultrapassada no segundo caso usando um algoritmo de optimização. As metodologias e modelos são aplicados para vários casos de estudo, de forma a avaliar a sua robustez e precisão. Um modelo da extremidade su- perior com sete músculos é considerado para evidenciar a aplicabilidade de um modelo de fadiga muscular num sistema multicorpo. Um segundo mod- elo, que inclui um aparato músculo-esquelético da extremidade inferior com doze músculos, é utilizado com único propósito de calcular as activações musculares. Os resultados associados a estes modelos são apresentados bem como as respectivas conclusões. O trabalho conclui perspectivando eventuais desenvolvimentos futuros. Palavras-chave: Dinâmica multicorpo, dinâmica de contracção muscular, dinâmica de fadiga muscular, forças musculares, activações musculares, op- timização. Abstract The aim of this work is to develop a versatile muscle model and robustly implement it in an existent multibody system dynamics code with natural coordinates. Two different models are included: the first is a Hill-type muscle model that simulates the functioning of the contractile structures both in forward and in inverse dynamic analysis; the second is a dynamic muscular fatigue model that considers the force production history of each muscle and estimates its fitness level using a three-compartment theory approach and a physiological muscle recruitment hierarchy. The existent equations of motion formulation is rearranged to include the referred models using the Newton’s method approach. This allows, in a forward dynamics perspective, for the calculation of the system’s motion that results from a pattern of given muscle activations, or, in an inverse dynamics perspective, for the computation of the muscle activations that are required to produce a prescribed articular movement. In both perspectives the system presents a redundant nature that is overcome in the latter case using an SQP optimization algorithm. The methodologies and models are applied to several case studies to eval- uate their robustness and accuracy. An upper extremity model with seven muscles is designed to evidence the effectiveness of the implementation of a muscle fatigue model in a multibody system. A second model, encom- passing the lower extremity musculoskeletal apparatus with twelve muscles, is proposed for the exclusive calculation of muscle activations. The results are presented and conclusions are discussed. The work concludes with a perspective of possible future developments. Keywords: Multibody dynamics, muscle contraction dynamics, muscle fa- tigue dynamics, muscle forces, muscle activations, optimization. Acknowledgements I would like to start by expressing my deepest gratitude to my supervisor Dr. Miguel Silva, to whom I owe for his inspiration, knowledge, encouragement and patience. This thesis would not have been possible without his wisdom and total dedication. The uncountable meetings and brainstorming sessions were the core of this project and are definitely the highlights of my academic experience, so far. To Prof. Dr. Mamede de Carvalho and Dr. João Costa, for providing their highly motivational medical feedback and points-of-view. A big thanks to Dr. Jorge Martins and to my colleagues Rita Malcata, Pedro Moreira and Daniel Lopes for their help in the whole project process. To Dr. Marko Ackermann, Dr. Maury Hull, Dr. Maxime Raison and Dr. Ting Xia for their kindness in providing their papers and work. To the University of Washington for providing the Musculoskeletal Images from the "Musculoskeletal Atlas: A Musculoskeletal Atlas of the Human Body" by Carol Teitz, M.D. and Dan Graney, Ph.D. To all of my friends that supported me throughout these University years. Nevertheless, some of them should be mentioned, due to their true friendship and comfort: Nadir Abu-Samra, Diogo Almeida, Ana Barradinhas, André Bento, João Cabaça, Luís Cabecinha, Akshay Chaudry, Fábio Coelho, João Fayad, Artur Ferreira, Daniel Fitas, Diogo Geraldes, Maria João Lascas, Gonçalo Marcelo, Daniel Martins, André Medeiros, Diana Nunes, João Maia de Oliveira, Susana Palma, Manuel Rosa, Rafael Rosário, João Lála dos Santos, Ricardo Serrano, César Silveira, João Venes. To all my family, namely to my parents, my sister, my grandmother, cousins and Maria. The deepest acknowledgment goes to my mother. I thank her for being the person who always was there for me, who motivated me all the way through, and granted that I was guided by the same values of ambition and drive that I recognise in herself. Indeed, this work is dedicated to her. Para a minha mãe, Maria Isabel. To my mother, Maria Isabel. iv Contents List of Figures vii List of Tables xi List of Symbols xiii Glossary xv 1 Introduction 1 1.1 Motivation . .1 1.2 Objectives . .2 1.3 Literature review . .3 1.4 Contributions . .7 1.5 Thesis Organization . .8 2 Musculoskeletal System Modelling 11 2.1 Skeletal Muscle Anatomy and Physiology . 12 2.1.1 Skeletal Muscle Anatomy . 12 2.1.2 Skeletal Muscle Physiology . 18 2.2 Dynamics of the Muscle Tissue . 20 2.2.1 Activation Dynamics . 21 2.2.2 Contraction Dynamics . 23 2.2.3 Muscle Fatigue . 31 2.3 Discussion . 37 v CONTENTS 3 Multibody Dynamics 39 3.1 Kinematics . 40 3.2 Equations of motion . 43 3.3 Generic muscle forces . 44 3.4 Inverse Dynamics . 50 3.5 Optimization . 54 3.6 Forward Dynamics . 58 3.7 Discussion . 60 4 Biomechanical Models 63 4.1 Muscle model verification . 63 4.2 Muscle Fatigue . 67 4.3 Human Gait . 74 4.4 Discussion . 88 5 Conclusions and Future Developments 93 5.1 Conclusions . 93 5.2 Future Developments . 95 A Apollo – Hill-type muscles manual 99 A.1 MDL File . 99 A.2 Simulation file . 101 B MHILL Data Visualizer manual 103 References 109 vi List of Figures 2.1 Arrangement of the skeletal muscle structure, from an external level to a molecular level [53]. 13 2.2 Representation of the penation angle α between muscle fibers and ten- dons [42]. 14 2.3 Detail of a myosin molecule (A) and an actin filament (B) [53]. 15 2.4 Illustration of the cross-bridges formed by the connections of actin fila- ments with a myosin filament. Adapted from Reference [53]. 15 2.5 Detail of a muscle fiber [62]. 16 2.6 Neural muscular junction with three different detail levels [62]. 17 2.7 Contraction regulation mechanism by the troponin-tropomyosin complex, dependent on the concentration level of Ca2+ [62]. 18 2.8 Crossbridge theory steps [62]. 19 2.9 Relation between twitch frequency and effective muscle contraction [1]. 21 2.10 Scheme of muscle tissue dynamics, with a series model of Activation Dynamics and Contraction Dynamics. 21 2.11 Activation dynamics model consistency [20]. 23 2.12 Hill-type muscle models. 24 2.13 Discrete length-tension diagram of the contractile contribution of a single fully activated sarcomere [53] (a) and the same relationship scaled to the whole muscle [1] (b). 26 2.14 Force-length relationship of the passive element [1]. 27 2.15 Force-velocity relationship [18]. 28 2.16 Activation scaling evidence: Force-length and force-velocity relationships for different levels of muscle activation a(t) [18]. 28 vii LIST OF FIGURES 2.17 Block diagram of the muscle contraction dynamics model implemented in the forward dynamics multibody system routines. 30 2.18 Block diagram of the muscle contraction dynamics model implemented in the inverse dynamics multibody system routines. 30 2.19 Curve based on Rohmert’s relationship between %MVC (percentage of maximum voluntary contraction) and endurance time (minutes) [56]. 32 2.20 Three-compartment theory flowchart. 33 2.21 Muscle recruitment hierarchy pile chart [59]. 36 3.1 Muscle representation for a biomechanical system. 45 3.2 Application of a force Fm to pointo p, belonging to the rigid body defined by points i and j and vectors u and v [1]. 46 m 3.3 The different representations of a generic muscle force Fp ......... 48 3.4 Muscle force representation for the 3 via-point model in Figure 3.1. 50 3.5 Used optimization methodology. 56 3.6 Direct integration algorithm flowchart for a forward dynamics problem [1]. 60 4.1 Mechanical system with muscles mX and mY for Hill-type muscle model verification purposes. 64 4.2 Activations pattern for muscles mX and mY ...............