modeling muscle: basics · muscle fiber length fio - maximum isometric force produced at the...
TRANSCRIPT
John H. Challis - Modeling in Biomechanics
6A-1
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)
John H. Challis - Modeling in Biomechanics
6A-2
KEY PROPERTIES
Passive
Different LengthMuscle Fibers Active
ActivationDynamics
DifferentVelocities
Tendon Passive Force/Length
Muscle Moment ArmsJoint
Passive Moment Profile
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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MODEL REPRESENTATION
% Max.isometrictension
Length of contractile element
L0
100
75%
50%
25%
- Lengthen Shorten +VELOCITY
Maximumtension
FORCE
Extension
Force
John H. Challis - Modeling in Biomechanics
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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).
John H. Challis - Modeling in Biomechanics
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MODEL REPRESENTATIONSchematically the muscle mode components can berepresented as follows
Contractile Component
Elastic Component
Damping/Viscous Component
CE OR
John H. Challis - Modeling in Biomechanics
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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)
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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
John H. Challis - Modeling in Biomechanics
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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
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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 .==
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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 .
∫∫∫∫
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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
John H. Challis - Modeling in Biomechanics
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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.
John H. Challis - Modeling in Biomechanics
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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.