MODELING MUSCLE: BASICS · muscle fiber length FIO - maximum isometric force produced at the optimum length of the muscle fibers LF - length of the muscle fibers LFO - length at which

John H. Challis - Modeling in Biomechanics

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MODELING MUSCLE: BASICS

Lecture Overview

• Key Properties

• Model Representation

• Example I – Alexander, (1989)

• Example II – Challis and Kerwin (1994)

• Example III – Hof (1991)

• Options

• Model Selection

“This model will be a simplification and an idealization,and consequently a falsification. It is to be hoped thatthe features retained for discussion are those ofgreatest importance in the present state of knowledge.”Alan Turing (1952)


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KEY PROPERTIES

Passive

Different LengthMuscle Fibers Active

ActivationDynamics

DifferentVelocities

Tendon Passive Force/Length

Muscle Moment ArmsJoint

Passive Moment Profile


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KEY PROPERTIES

The force produced by the muscle model ( mF ) can bedescribed using the following function

( ) ( )fffm VFLFFaF 21max ...=

Wherefa - normalized degree of activation of muscle

fibers.

maxF - maximum isometric force muscle canproduce

( )fLF1 - normalized force length relationship ofmuscle,

( )fVF2 normalized force-velocity relationship ofmuscle.


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MODEL REPRESENTATION

% Max.isometrictension

Length of contractile element

L0

100

75%

50%

25%

- Lengthen Shorten +VELOCITY

Maximumtension

FORCE

Extension

Force


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MODEL REPRESENTATION

Potential Model ComponentsModels of muscle normally include some of thefollowing components.

Contractile Component – normally representing(some) properties of the muscles (force-length, force-velocity, activation dynamics).

Parallel Elastic Component – normally a linearelastic component representing elastic material inparallel to the muscle fibers (connective tissue).

Series Elastic Component – normally a linearelastic component representing elastic properties ofmaterial in series with the contractile component(tendon, muscle cross-bridge elasticity).


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MODEL REPRESENTATIONSchematically the muscle mode components can berepresented as follows

Contractile Component

Elastic Component

Damping/Viscous Component

CE OR


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MODEL REPRESENTATIONMuscle models have the a varying number of modelcomponents. The more complicated representations isthat of Hatze (1981). The model includes the followingcomponents,

PS – parallel sarcomere elasticityCE – contractile elementBE – cross-bridge elasticitySE – series elastic elementPE – Parallel element of muscle (having bothelasticity and viscous damping)


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EXAMPLE I

Source: Alexander, R.M. (1990) Optimum take-offtechniques for high and long jumps. PhilosophicalTransactions of the Royal Society, Series B, 329, 3-10

Model Components• Force-velocity• (Series elastic component)

Equation Inputs• Data derived from cadavers implied relatively fixed

moment arm.• Starts at velocity of zero with maximum activation.

Model Parameters• Determined from cadaver data and experimental

observations.

Model Validation• Comparison of model output with reality.


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EXAMPLE IThe model “runs” in at a given horizontal velocity andplants its leg and then jumps. The computer model wasrun with different horizontal velocities and knee anglesat plant, jump height was computed for each of theseconditions.

Equation

θθθθθθθθθθθθθθθθ��

��

.CTT

MAX

MAXMAX +

−=

A moment-angular velocity relationship, based on Hill’s1938 equation.

x,y a

aφ

θ

F0

y

x


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EXAMPLE II

Source: Challis, J.H., and Kerwin, D.G. (1994)Determining individual muscle forces during maximalactivity: Model development, parameter determination,and validation. Human Movement Science 13:29-61.

Model ComponentsForce-lengthForce-velocitySeries elastic component


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EXAMPLE IIThe model of the force-length relationship used wasthat of Hatze (1981)

−−=

21exp.

SKQFF IOI

FO

FLLQ =

WhereIF - maximum isometric tension at a given

muscle fiber lengthIOF - maximum isometric force produced at the

optimum length of the muscle fibersFL - length of the muscle fibersFOL - length at which the muscle fibers exert

their maximum tension (optimum length)and SK is a constant specific for each muscle

where SK > 0.28.


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EXAMPLE IIThe model of the force-velocity relationship of Hill(1938) was adopted

( ) ( ) aVb

aFbVb

VVaFF

I

F

FMAXV −

++=

+−= ..

WhereVF - maximum possible force at a given muscle

fiber velocitya,b - constantsVMAX - maximum speed of shortening of the

fibersVF - current speed of shortening of the fibersand IF is the maximum isometric tensionpossible at a given muscle fiber length.

a.VMAX = b.FI


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EXAMPLE IISeries elasticity was considered to reside only in thetendon. The model of tendon used in this studyassumed the stress-strain relationship of tendon to belinear, therefore:-

+=

EAFLLT

MTRT .

0.1.

WhereTL - length of the tendonTRL R - resting length of the tendonMF - force exerted by the muscle on the tendonTA - cross-sectional area of the tendon

and E is Young's modulus of elasticity fortendon.


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EXAMPLE II

Primary Assumptions of Muscle Model• The stress-strain relationship of tendon is linear.• Muscle fiber elasticity was insignificant compared with

tendon elasticity.• The moment at the joint caused by the passive

structures crossing the joint, and joint friction wasinsignificant compared with that produced by themuscles.

• During the studied activity there was no co-contraction of antagonist muscles.

• The various elements of the model are adequatelyrepresented by the equations used to describe them.


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EXAMPLE IIIf the muscle fibers are not pennated, and parallelcomponents produce little force then

MT FF =

As the model assumed the stiffness of the tendon wasconstant for all lengths of the tendon then

T

M

T

TdLdF

dLdFK ==

Where K is the stiffness of the tendon.

The rate of change of the muscle force is equal to theproduct of tendon stiffness and the rate of change oftendon length, therefore

dtdL

dLdF

dtdF

dtdF T

T

TTM .==


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EXAMPLE IIThe rate of change in tendon length is equal to thedifference between muscle velocity and muscle fibervelocity

FMT VV

dtdL −=

Re-arrangement of Hill’s equation gives( )

( ) baFaFbV

M

IF −

+−=

Substitution gives the following

( )FMM VVk

dtdF −=

This ordinary differential equation can be solved using avariable step-size fifth order Runge-Kutta technique.


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EXAMPLE II

Equation Inputs• Data derived from cadavers allow determination of

muscle velocity.• If the movement starts from stationary then the

velocity of the muscle fibers is known (zero).

Model Parameters• Determined using an experimental procedure.

Model Validation• An elbow flexion was simulated driven using the

muscle model, and compared with reality.


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EXAMPLE IIMUSCLE MODEL SIMULATION

( )FMTM VLagFF ,,==

CONTRACTILEELEMENT

MODELV - VM F V . KT T

ML MV

aV V FF T T F T

��

.

∑=

=NM

iiiTiJ RFM .

∫∫∫∫


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EXAMPLE III

Source: Hof and Van Den Berg (1981) EMG to Forceprocessing parts I-IV. Journal of Biomechanics14:747-792.

Model ComponentsForce-lengthForce-velocitySeries elastic componentParallel elastic componentActivation dynamics

Model Parameters• Determined using an experimental procedure.

Model Validation• Comparison of model predicted ankle joint moments

with reality.


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OPTIONS

Force–Length – linearize, ascending or descendinglimb only, plateau

Force–Velocity – linearize, ignore

SEC – ignore, linear or exponential. Include cross-bridge or just tendon.

Muscle PEC or Joint Elasticity.

ActivationBang-BangKnownEstimated from EMGDetermined from a control routine


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MODEL SELECTION• Complexity and completeness

• Problem of model parameter determination(accessibility)

• Compensating errors

Option 1Start from simplest model and add complexity untilmodel reflects reality

Option 2Start from complex model and remove complexity untilmodel no longer reflects reality, previous level ofcomplexity was most appropriate.


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REVIEW QUESTIONS1) What are the different methods via which muscle

models allow for specifying muscle activation?

2) Give the structure of a muscle model you havestudied (name the author of the work). Detail whatthe components of the model correspond tobiologically. What is the major element(s) which ismissing from the model?

3) What are the locations and properties of thefollowing• Contractile Element• Parallel Elastic• Series Elastic

4) What are the implications of the SEC for humanmovement?

5) With examples outline the relative merits of simpleversus a complex muscle model.

MODELING MUSCLE: BASICS · muscle fiber length FIO - maximum isometric force produced at the optimum length of the muscle fibers LF - length of the muscle fibers LFO - length at which

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