Achilles tendon model development

In Chapter 4 the development of the mathematical, mechanical, musculoskeletal model of

the Achilles tendon was described. The model is anatomically meaningful; it describes the

anatomy and physiology of the body components incorporated as realistically as possible.

Inverse dynamics in conjunction with Newton's second and third laws were used in order to

compute forces and moments of force indirectly from the kinematics and inertial properties of

the lower bodies studied. The method used to numerically compute the internal kinetics of

planar human movements was presented in Section 4.1. In order to calculate the forces and

moments at specific joints, body kinematics and anthropometric data are applied. Three

important principles were involved in this method; i) Newton’s second law, ii) the principle of

superposition and iii) a technique known as the method of sections. A single net force and a

single moment of force, that can be measured from experiments, were produced in order to

solve the unknown Achilles tendon force. Also forward dynamics were used to determine the

resultant motion of bodies [41].

The present model is sensitive to its input data. Errors in marker locations, segment inertial

properties, joint centre estimates, segment accelerations, joint forces and ground reaction forces

affect the joint moment data. Some of these errors are less significant than others. An example

mentioned in Robertson [41], states that ground reaction forces during locomotion tend to

dominate stance-phase kinetics and measuring them accurately prevents the majority of

accuracy problems. However, assumption made for the model and limitations that occurred do

not invalidate the model data, but they do limit the extent of interpretation.

In Section 4.2, a two segment model of the human foot and leg was presented that describes

the development of the musculoskeletal model of the Achilles tendon. Newton’s laws and the

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mathematically model the skeletal system along with the Achilles tendon. In Section 4.3 the

method used to model human skeletal muscle dynamics using Hill-type muscle models was

portrayed. A gastrocnemius muscle-Achilles tendon model was presented and the derivation of

the system equations that characterise the muscle-tendon model were given. The purpose of this

study is to acquire mechanistic knowledge of the gastrocnemius muscle-Achilles tendon

complex behaviour during specific movements in humans through mathematical modelling.

Accurate model outcomes were very important in order to study how particular muscles

contribute to movement coordination and propose or assess interventions to prevent foot

injuries. Therefore, a validated model of the Achilles tendon that could provide specific

parameters for a specific person studied, was a main goal of this thesis. The derivation of the

Achilles tendon force from the system equations for the musculoskeletal model of the Achilles

tendon and the implementation of that force in the muscle-tendon model in order to derive the

parameters of the model as well as the non-linear approach of the behaviour of the Achilles

tendon studied through a motion capture system is, to the best of the authors knowledge, novel

in the field of muscle-tendon research.

The variability in Achilles tendon stiffness between subjects strengthens the need for the

development of experimentally measured subject-specific tendon properties. These properties

would be the input parameters of the models that would help to improve the accuracy of

musculoskeletal models. Hill-type muscle models and musculoskeletal simulations have broad

application to biomechanical research and experimental investigations of musculoskeletal

pathologies [129]. However, the accuracy of these simulations directly depends on the values

used in the models, i.e. the natural resting length of muscles and tendons. Despite significant

variability in these tendon properties between individuals many studies use generic values for

the tendon properties. That is why in this thesis the model developed, used specific data for

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In order to obtain the appropriate specific inputs for each individual to then be incorporated

in the model, anthropometric data were used. Ultrasound and Gait Laboratory experiments were

used to obtain the natural resting lengths of the Achilles tendon and the gastrocnemius muscle

as well as the change in length of the muscle and the tendon when an individual performed

specific movements. A novel marker placement, which is a combination of the validated

existing Plug-in Gait model with new markers added, was created in order to acquire the

necessary data to compute the natural resting lengths and the elongation and shortening lengths

of the muscle-tendon model. Cross validation of the lengths was performed through the lengths

obtained from the Ultrasound experiments conducted at the UHCW. EMG experiments also

provided information about the activation of the muscles involved in the movements studied.

This information was valuable for the development of the model and the assumptions for the

description of the contractile force input.

Structural identifiability analysis of the modified Hill-type muscle model was performed and

concluded that all the parameters of the model are identifiable and can describe the movement

of the muscle-tendon complex. Then, parameter estimation was performed in the time domain

for each volunteer in order to acquire a set of fitted values for the parameters b, k1, and k2 that

were the unknown ones in our model. Parameter estimation of the model led to satisfactory

agreement between measured and simulated data for the specific movements that were studied.

All results were found to lie in the accepted range of values mentioned in the literature. A non-

linear model was also investigated which showed that the solution for the quadratic Equation

(4.11) when simulated gave a better approach to the experimental data than the simulation of

the solution of the cubic equation.

The general approach and the model developed was proven to be suitable for the leg muscles

and the Achilles tendon, so we can speculate that other muscle groups can be parameterised and

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