Mathematical modelling and simulation of the foot with specific application to the Achilles tendon

In this thesis, the development of an anatomically meaningful musculoskeletal model of the human foot with specific application to the Achilles tendon is presented. An in vivo experimental method of obtaining parameter values for the mechanical characteristics of the Achilles tendon and the gastrocnemius muscle is presented incorporating a Hill-type muscle model. The incentive for this work has been to enable the prediction of movement with regard to Achilles tendon motion of healthy volunteers, in order to then compare it with the movement of a pathologic gait and help in preventing Achilles tendon injuries. There are relatively few mathematical models that focus on the characterisation of the human Achilles tendon as part of a muscle-tendon unit in the literature. The mechanical properties of the Achilles tendon and the muscles connected to the tendon are usually calculated or predicted from muscle-tendon models such as the Hill-type muscle models. A significant issue in model based movement studies is that the parameter values in Hill-type muscle models are not determined by data obtained from in vivo experiments, but from data obtained from cadaveric specimens. This results in a complication when those predictive models are used to generate realistic predictions of human movement dynamics. In this study, a model of the Achilles tendon-gastrocnemius muscle is developed, incorporating assumptions regarding the mechanical properties of the muscle fibres and the tendinous tissue in series. Ultrasound images of volunteers, direct measurements and additional mathematical calculations are used to determine the initial lengths of the muscle-tendon complex as well as the final lengths during specific movements of the foot and the leg to parameterise the model. Ground reaction forces, forces on specific joints and moments and angles for the ankle are obtained from a 3D motion capture system. A novel experimental marker placement for the Achilles tendon is developed and generated in the 3D motion capture system. Movement dynamics of the foot are described using Newton’s laws, the principle of superposition and a technique known as the method of sections. Structural identifiability analyses of the muscle model ensured that values for the model parameters could be uniquely determined from perfect noise free data. Simulated model dynamics are fitted to measured movements of the foot. Model values are obtained on an individual subject basis. Model validation is performed from the experimental data captured for each volunteer and from reconstruction of the movements of specific trajectories of the joints, muscles and tendons involved in those movements. The major output of this thesis is a validated model of the Achilles tendon-gastrocnemius muscle that gives specific parameters for any individual studied and provides an integral component in the ultimate creation of a dynamic model of the human body. A new approach that was introduced in this thesis was the coupling of the Achilles tendon force from the musculoskeletal model to the muscle-tendon model and the non-linearity approach studied through a motion capture system. This approach and the new Achilles tendon marker placement is to the best of the author's knowledge, novel in the field of muscle-tendon research