Modeling the length dependence of isometric force in human quadriceps muscles

Journal of Biomechanics 35 (2002) 919–930

Modeling the length dependence of isometric force in humanquadriceps muscles

Ramu Perumala, Anthony S. Wexlerb, Jun Dingc, Stuart A. Binder-Macleodc,*aDepartment of Mechanical Engineering, University of Delaware, Newark, DE, USA

bDepartment of Mechanical and Aeronautical Engineering, University of California, Davis, CA, USAcDepartment of Physical Therapy, University of Delaware, 301 McKinly Laboratory, Newark, DE 19716, USA

Accepted 5 March 2002

Abstract

Functional electrical stimulation is used to restore movement and function of paralyzed muscles by activating skeletal muscle

artificially. An accurate and predictive mathematical model can facilitate the design of stimulation patterns that produce the desired

force. The present study is a first step in developing a mathematical model for non-isometric muscle contractions. The goals of this

study were to: (1) identify how our isometric force model’s parameters vary with changes in knee joint angle, (2) identify the best

knee flexion angle to parameterize this model, and (3) validate the model by comparing experimental data to predictions in response

to a wide range of stimulation frequencies and muscle lengths. Results showed that by parabolically varying one of the free

parameters with knee joint angle and fixing the other parameters at the values identified at 401 of knee flexion, the model

could predict the force responses to a wide range of stimulation frequencies and patterns at different muscle lengths. This work

showed that the current isometric force model is capable of predicting the changes in skeletal muscle force at different muscle

lengths. r 2002 Published by Elsevier Science Ltd.

Keywords: Catch-like property; Frequency; Functional electrical stimulation

1. Introduction

Functional electrical stimulation (FES) has been usedto restore movement and function of paralyzed musclesby activating the skeletal muscles artificially. However,the practical use of FES has been hindered by the rapidfatigue during this exogenous activation of the muscle(Marsolais and Edwards, 1988). Because the meanstimulation frequency and the pattern of pulses mark-edly affect the force production and fatigue (Bigland-Ritchie et al., 1979; Binder-Macleod et al., 1995,1998a, b), it is important to identify the stimulationfrequency and pattern that maximizes force and mini-mizes fatigue. However, the pattern that maximizesthe force varies with the physiological conditions of themuscle, such as level of fatigue or muscle length (Leeet al., 1999), and varies from person to person (Karuet al., 1995). Numerous experimental tests would be

needed to identify the optimal pattern for each patient.An alternative to experimental testing is the use ofmathematical models that can predict forces and fatigue.Recently, Ding and colleagues reported the developmentof such models for brief isometric contractions (Dinget al., 1998, 2000a, b). However, daily activities likegrasping, walking, and rising from a chair involveconcentric and eccentric contractions of the variousmuscle groups. Hence, there is a need to developmathematical models that can be used to identify theoptimal pattern for muscle activation during non-isometric contractions.A number of mathematical models of the Hill or

Huxley type have been presented to capture the non-isometric behavior of skeletal muscles. Dorgan andO’Malley (1998) developed a non-linear dynamic modelthat could capture non-linear summation, force, andstiffness variation when activating muscles at differentfrequencies of stimulation. It also provided insight intothe internal dynamics of skeletal muscles. However,their model was very complex and needed 24 parametersto be identified to predict the force output of the muscle.

*Corresponding author. Tel.: +1-302-831-8046; fax: +1-302-831-

4234.

E-mail address: [email protected] (S.A. Binder-Macleod).

0021-9290/02/$ - see front matter r 2002 Published by Elsevier Science Ltd.

PII: S 0 0 2 1 - 9 2 9 0 ( 0 2 ) 0 0 0 4 9 - 0

These parameters were muscle specific and theiridentification was difficult and time consuming. Durfeeand Palmer (1994) designed a non-linear non-isometricmuscle model that predicted forces with reasonableaccuracy for a wide range of simultaneously varyingmuscle lengths, velocities, and levels of motor recruit-ment. This model, however, was not capable ofpredicting muscle forces to variations in the activationfrequency. Recently, Ferrarin and Pedotti (2000) devel-oped a mathematical model to predict the active kneejoint torque in response to applied electrical stimuli.Their model, in contrast to the Hill and Huxley typemodels, was based on an autoregressive exogenous inputmodel to identify the relationship between the electricalstimulus and the generated active muscle torque. Theshortcomings of this model were that it did not take theinternal dynamics of the muscle into consideration, wasonly able to predict the torque generated by the kneeextensor muscles in response to a small range ofconstant frequency trains (20–50Hz), and, as stated bythe authors, was not able to account for the non-linearities in the activation dynamics. Bobet et al. (1998)developed a muscle model consisting of two linear, first-order systems separated by a static non-linearity andwas validated on cat soleus and plantaris muscle underisometric conditions. Their model was able to predictthe muscle force in response to a wide range ofstimulation patterns. However, their model was notvalidated under non-isometric conditions and thepredictive ability of the model was not tested for humanmuscles. Thus, the existing models are very complex, donot have the ability to predict force responses undernon-isometric conditions, or they cannot predict forceresponses to a functionally relevant range of frequencies.The present study is the first step in developing a non-

isometric model to predict skeletal muscle forces inresponse to a wide range of stimulation frequencies andpatterns. The purposes of this study are to: (1) identifyhow the Ding and colleagues’ (2000a, b, 2001) isometricforce model’s parameter values vary with muscle length,(2) explore the best knee flexion angle for parameteriz-ing this model, and (3) validate the model by comparingexperimental data to the model’s predictions in responseto a wide range of stimulation frequencies and musclelengths. Future studies will attempt to use this informa-tion to develop a non-isometric force model thatpredicts skeletal muscle force and fatigue duringrepetitive stimulation.

2. Methods

Twelve subjects were recruited for the study. Eachsubject signed an informed consent form approved bythe University of Delaware Human Subject ReviewBoard and participated in one testing session. Six of the

twelve subjects were used to develop the model and eachwas tested at 151, 401, 651, and 901 knee flexion angles(development angles). The remaining subjects were usedto validate the model and were tested at the develop-ment angles and at 301, 551, and 751 of knee flexion(validation angles).

2.1. Equipment and experimental setup

Subjects were seated on a computer controlled(KinCom III 500-11, Chattecx Corporation, Chatta-nooga, TN) dynamometer with their hips flexed toB751(Binder-Macleod et al., 1997). The dynamometer axiswas aligned with the knee joint axis and the forcetransducer pad was positioned anteriorly against thetibia, 4 cm proximal to the lateral malleolus. Two7.62 cm� 12.7 cm self-adhesive electrodes were used tostimulate the muscle. With the knee positioned at 901,the anode was placed proximally over the motor pointof the rectus femoris portion of the quadriceps femorismuscle. The cathode was placed distally over the vastusmedialis motor point with the knee in 151 of flexion tocompensate for skin movement during knee extension(Lee and Binder-Macleod, 2000). The trunk, pelvis, andthigh of the leg being tested were each stabilized withinelastic straps. A Grass S8800 stimulator with anSIU8T stimulus isolation unit (Grass Instruments, WestWarwick, RI) was used for stimulation. The stimulatorwas driven by a personal computer using customizedLabView (National Instruments, Austin, TX) software.Force data from the transducer were sampled at 200Hzusing an analog-to-digital board. The data were thenanalyzed using a custom program written in LabView.

2.2. Mathematical model

As this study was intended to characterize the forceproduced at different isometric muscle lengths, theisometric force model developed by Ding and colleagueswas used (Ding et al., 2000a, b; Ding, 2001). The forcemodel consists of two differential equations:

dCN

dt¼1

tc

Xn

i¼1

Ri exp �t � ti

tc

� ��

CN

tc

ð1Þ

with Ri ¼ 1þ ðR0 � 1Þ exp½�ðti � ti�1Þ=tc� and

dF

dt¼ A

CN

Km þ CN

�F

t1 þ t2CN=ðKm þ CN Þ: ð2Þ

Eq. (1) represents the dynamics of the rate-limiting stepleading to the formation of the Ca2+-troponin complex(CN ; unitless), in which Ri (unitless) accounts for thenon-linear summation of the Ca2+ transient whenstimulated with two closely spaced pulses; tc (ms) isthe time constant controlling the transient shape of CN ; t

(ms) is the time since the beginning of the stimulationtrain; ti (ms) is the time for the ith stimulation train; n is

R. Perumal et al. / Journal of Biomechanics 35 (2002) 919–930920

the number of stimulation pulses before time t in thetrain (its maximum value is six in this study). Eq. (2)represents the development of the mechanical force(F (N)) that is driven by force-producing cross-bridges(strongly bound cross-bridges) and mediated by aMichaelis–Menten term CN=ðKm þ CNÞ: In Eq. (2), A

(N/ms) is the scaling factor for force development, Km isthe sensitivity of strongly bound cross-bridges to CN ; t1(ms) is the time constant of force decline in the absenceof strongly bound cross-bridges, and t2 is the timeconstant of force decline due to the extra frictionbetween actin and myosin resulting from the presence ofcross-bridges.

2.3. Development of the model

2.3.1. Stimulation protocol

Isometric muscle performance was tested when themuscles were highly potentiated. All trains had sixpulses (Fig. 1), because studies have shown that mostfunctional movements employed a brief activationpattern (Henning and L�mo, 1987; Lee et al., 1999).To study a wide range of stimulation frequencies, eightconstant frequency trains (CFTs) with interpulse inter-vals (IPIs) equal to 10, 30, 50, 70, 90, 100, 110, or 130mswere used; these trains are referred to as CFT10, CFT30,CFT50, CFT70, CFT90, CFT100, CFT110, andCFT130, respectively. Variable frequency trains (VFTs)that elicit the catch-like property of the skeletal muscle(Burke et al., 1970) have been shown to produce fasterrates of rise of force and higher average forces, peakforces (PFs), and force–time integrals (FTIs) duringisometric contractions (Bevan et al., 1992; Binder-Macleod and Barker, 1991; Binder-Macleod et al.,1998a, b; Stein and Parmiggiani, 1993) and augmentedaverage power, peak power, PFs, and excursions duringnon-isometric contractions (Binder-Macleod and Lee,1996; Lee and Binder-Macleod, 2000; Sandercock andHeckman, 1997). For these reasons VFTs were alsostudied. The VFTs had an initial IPI of 5ms, and theremaining IPIs were 10, 30, 50, 70, 90, 110, or 130msIPIs; these trains are referred to as VFT10, VFT30,VFT50, VFT70, VFT90, VFT110, and VFT130, respec-

tively. The sequence of trains within the stimulationprotocol began with the CFT100 and VFT50 followedby a random sequence of the remaining trains. All trainswere then presented in reverse order. Only one train wasdelivered every 10 s to minimize muscle fatigue.

2.3.2. Experimental session

At the beginning of the experimental session, eachsubject performed a maximum voluntary isometriccontraction (MVIC) of their quadriceps femoris muscle,with the knee positioned at 901 of flexion. A burstsuperposition technique was used to ensure that amaximal contraction was being performed (Snyder-Mackler et al., 1993). Subjects were given a rest time of2min. Then, the stimulation intensity was set to producean isometric force equal toB20% of the subject’s MVICusing a 7-pulse CFT10 to activate the muscle. A 12-pulseCFT70 train was then delivered 15 times, with a 5-s restbetween each train to potentiate the muscle. Previousstudies have shown that the 12-pulse CFT70 maximallypotentiated the muscle without causing a measurabledecline in force (Ding et al., 2000a, b) and setting theintensity with the 7-pulse CFT10 at least 75% ofthe muscle was activated (Binder-Macleod and Russ,1999). Testing was then conducted at each of the fourdevelopment angles and the order for testing at each ofthe angles was randomly determined for each subject. Arest period of 5min was provided between testing at eachangle to avoid muscle fatigue. The potentiating trainswere repeated before testing at each angle.

2.3.3. Parameter identification

The force model has six free parameters (R0; tc; A;Km; t1; and t2). Ding and colleagues showed that a fixedvalue of 20ms for tc and 2 for R0 were sufficient for thehuman quadriceps muscles under non-fatigue condition(Ding et al., 2000a, b). In addition, they found that theoptimal values of the remaining four free parameterswere best identified by fitting the model to the measuredforces produced by stimulating the muscle with acombination of VFT50 and CFT100 trains (Ding et al.,2000a, b). For the present study, the values of the fourfree parameters were identified first at a knee flexionangle of 901. t1 was identified as described previously(Wexler et al., 1997). Briefly, after stimulation ceases,CN approaches zero, and Eq. (2) becomes dF=dt ¼�F=t1: By taking the force decay at the end of theforce–time response and performing a linear regressionof ln F vs. t; t1 was obtained from the slope of the fit.The values of remaining three parameters were identifiedby employing the following objective function:

GðA;Km; t2Þ ¼X

p

ðFpredðti;A;Km; t2Þ � F expðtiÞÞ2; ð3Þ

where Fpred; the forces predicted by Eqs. (1) and (2) attime ti; are a function of parameters A; Km; and t2; F exp

VFT50

CFT50

Time (ms)

0 50 100 150 200 250

Fig. 1. Schematic representation of the two stimulation train types

used in the present study. Bottom train (CFT50) is a constant-

frequency train with all IPIs equal to 50ms; top train (VFT50) is a

variable-frequency train with an initial doublet of 5ms and remaining

pulses equally spaced by 50ms. Each train’s name is based on the

duration of the longest IPI within that train.

R. Perumal et al. / Journal of Biomechanics 35 (2002) 919–930 921

Fig. 2. Fittings of VFT50–CFT100 force responses by varying the free parameters at different initial angles for a typical subject used for the pilot

study. (a): Initial fitting to VFT50–CFT100 force responses at 901 knee flexion. These trains were used to identify the free parameter values. (b):

Predicted versus experimental VFT50–CFT100 forces at angles of 151, 401, and 651, when all the parameters are fixed at values obtained at 901.

(c)–(g): Model fitted to the experimental VFT50–CFT100 combination data while freeing A; A � t1; A � t1 � Km; A � t1 � t2; or A � t1 � t2 � Km;respectively (c)–(g) for each knee flexion angle tested. The parameters that were not allowed to vary were set equal to the values obtained when the

fitting was carried out at 901 of knee flexion.


are the experimental forces at time ti; p is the number offorce data points. G was minimized using MATLAB’slarge-scale algorithm, which numerically identifies theoptimal values of A; Km; and t2 by employing thesubspace trust region method based on the interior-reflective Newton method (Coleman and Li, 1994).A pilot study was carried out on three of the six

subjects at knee flexion angles of 901, 651, 401, and 151to determine which of the four free parameters (A; t1; t2;and Km) depend on knee joint angle and whichparameters could be maintained at the 901 values.Because the force produced by the muscle changes withmuscle length (Haffaje et al., 1972), we hypothesizedthat parameter A; the scaling factor for force in Eq. (2),would change with knee angle. The model was fitted tothe experimental VFT50–CFT100 data at each angle byletting parameter A vary and fixing the other parametersat the values obtained at 901 or by varying A and acombination of other parameters (e.g. t1; or t1 and Km).The results of this pilot study showed that the parameterA alone was able to capture most of the changes in theforce produced by the muscle due to changes in the kneeflexion angle (Fig. 2). In addition, A was seen to changein a roughly parabolic manner with respect to the kneeflexion angle (Fig. 3). Hence, the parabolic variation ofA was modeled by the following equation:

A ¼ að90� yÞ2 þ bð90� yÞ þ A90; ð4Þ

where y is the knee joint angle, A90 is the value of A at901, and a and b are constants that needed to beidentified for each subject. Values of a and b for eachsubject were identified in the following manner. First,the value of A and the other free parameters wereidentified at 901 using the VFT50–CFT100 data andprocedures explained above. The value of A at the otherthree angles was found by fitting the experimentalVFT50–CFT100 data and setting t1; t2; and Km equal tothe values obtained when the fitting was carried out atthe 901 knee flexion angle. With the values of A at thefour development angles in hand, best-fit values of a andb were found by minimizing the following function:

G1ða; bÞ ¼X

i

ðApredðyi; a; bÞ � AfitðyiÞÞ2; ð5Þ

where Apred is the value of A predicted from Eq. (4), Afit

is the value of A obtained by fitting the experimentalVFT50–CFT100 combination data at four differentangles, yi is the knee flexion angle, a and b are theconstants in Eq. (4), and i ¼ 1; 2y4 for the fourdevelopmental angles. The value of parameter A canbe predicted at any knee flexion angle by knowing thevalues of a and b and using Eq. (4). Next, we determinedif the 901 knee flexion angle was best for parameterizingthe model for all the six subjects. Parameterization ofthe model was carried out by determining the values ofall the free parameters at the 151, 401, and 651 angles.

This parameterization was carried out as describedabove by fitting the model to the VFT50–CFT100 trainpair experimental data. The predictive ability of themodel parameterized at each of these initial three angleswas then compared to that of the model parameterizedat 901. For the above case, then, Eq. (4) was modified to

A ¼ aðyDEV � yÞ2 þ bðyDEV � yÞ þ ADEV; ð6Þ

where yDEV is any one of the four development anglesand ADEV is the value of A at a particular initial angle atwhich the free parameter values were determined.

2.3.4. Data analysis

For all data collected, the two occurrences of each ofthe stimulation trains used in the testing protocol wereaveraged to minimize the effects of previous activationhistory on the muscle’s response to each train type. Pilotdata on three of the six subjects were analyzed, asdescribed in the above section, to determine thevariation of free parameters with the four developmentangles. The data collected on six subjects were then usedto determine the best development angle to parameterize

0

2

4

6

8

10

12

0 15 30 45 60 75 90

Knee Flexion Angle (Degrees)

A(N

/ms)

a =-0.00071b =-0.02962

0

0.5

1

1.5

2

2.5

3

0 15 30 45 60 75 90

Kneee Flexion Angle (Degrees)A

(N/m

s)

(a)

(b)

Fig. 3. (a) Values of A averaged across six subjects used for the

development portion of the study. Parameter A varies in a parabolic

manner with the knee flexion angle and has maximum value around

701. The bars above the data points are the standard error bars. (b)

The parabolic variation of parameter A for typical subject used for the

validation portion of the study. The solid diamond at an angle of 401

knee flexion indicates the value of A obtained by parameterizing the

model at 401; the other solid diamonds indicate the values of A

obtained by fitting the model to the CFT100–VFT50 experimental

data collected at each joint angle and fixing all the other parameters at

the values obtained at 401. The solid line indicates the values of A

obtained by using Eq. (6). For this subject the values of a and b are

�0.00071 and �0.02926, respectively.


the model. Here, all the free parameters were determinedat one particular development angle using the objectivefunction given by Eq. (3). Parameter A was thendetermined for the remaining development angles byfitting the model data to the VFT50–CFT100 experi-mental data at that particular development angle. The

best parameterizing angle was then determined by theability of the model to predict the force responses to awide range of stimulation frequencies at the four angles.For each subject and for each development angle, thePFs and the FTIs were calculated for each train tested.The experimental PFs and FTIs were then plottedagainst the predicted PFs and FTIs, respectively, foreach of the angles. Linear regression trend lines, withintercepts of zero, were used to compare force responsesacross all stimulation patterns and across all four angles,and to determine how well the model predicted thePFs and the FTIs. A perfectly accurate model wouldhave a slope and R2 of one (the identity line). The R2s ofthe PFs and the FTIs for the four angles, with theparameterization carried out at one particular develop-ment angle, were averaged. These averaged R2s werethen compared across angles to find the best develop-ment angle to parameterize the model.

2.3.5. Results of model development

For the six subjects tested, parameter A showed aparabolic variation with the knee flexion angle and hada maximum value between 551 and 751 of knee flexion

Table 1

Averaged correlation coefficients between experimental and predicted

force for a wide range of stimulation patterns

Knee flexion angle Parameterizing angle

151 401 651 901

FTI PF FTI PF FTI PF FTI PF

151 0.90 0.82 0.41 0.75 0.28 0.72 0.46 0.73

401 0.74 0.66 0.86 0.89 0.69 0.82 0.76 0.81

651 0.70 0.70 0.86 0.90 0.77 0.84 0.79 0.83

901 0.75 0.64 0.85 0.90 0.79 0.82 0.79 0.70

Avg 0.77 0.71 0.75 0.86 0.64 0.80 0.70 0.77

Parameterization was carried out at each development angle. The

value of parameter A at the non-parameterizing development angles is

determined by fitting the model data to the VFT50–CFT100

experimental data.

Yes No

No

Yes

Randomize order of angles to betested

Developmentangle?

Potentiate

Stimulate muscle using 2 pairsof CFT100-VFT50, with 10s

delay between each train.

Stimulate muscle with 32 testing trains.

Rest 2 minutes. Rest 5 minutes.

All sevenangles tested?

End of experiment.

Fig. 4. Schematic representation of the experimental procedure for validation study. Each subject was tested with just two different stimulation

trains (VFT50 and CFT100) at four knee flexion angles of 151, 401, 651, and 901 (development angles). At three knee flexion angles of 301, 551, and

751 (validation angles) subjects were stimulated with 32 different trains to test the predictive ability of the model (see text for details).


(Fig. 3(a)). The correlation coefficients between themeasured and predicted PFs and FTIs in response to awide range of stimulation patterns, averaged acrosssubjects for each development angle, showed that the401 knee flexion angle had the best averaged R2 for thePFs and a good averaged R2 for the FTIs (Table 1).Although the average R2 (0.77) for the FTI with theparameterization carried out at 151 was slightly betterthan the corresponding R2 (0.75) with the parameteriza-tion carried out at 401, we chose the 401 knee flexionangle as the optimal angle for parameterizing the modelas this angle predicted the PFs (R2 ¼ 0:86) much betterthan the other parameterizing angles.

2.4. Validation of the model

The model was validated by determining its ability topredict forces at angles and for subjects other than thosealready tested. Data were collected on six new subjects atthe development angles and at knee flexion angles of 301,551, and 751 (validation angles). The CFT100–VFT50train pair, with a 10-s delay between each train, wasadministered at the four development angles to determineA; t1; t2; Km; a; and b: At the validation angles, thestimulation protocol (described in Section 2.3.1) contain-ing a wide range of CFT and VFT frequencies were usedto stimulate the muscle. The order of the seven angles(four development and three validation angles) to be testedwas randomized for each subject. Before testing at eachangle, the muscle was potentiated with 15, 12-pulse CFT70trains with a 5-s delay between each train. A 2-min restperiod was provided at the end of testing each develop-ment angle and a rest period of 5min after each validationangle to prevent the onset of fatigue (Fig. 4). The values ofparameters A; t1; t2; and Km were found by parameteriz-ing the model at 401 of knee flexion. The values of a and b

were found as described in Section 2.3.3 (see Eq. (6)).

2.4.1. Data analysis

The predictive ability of the model was tested bycomparing the forces predicted by the model at thevalidation angles with the experimental forces. Linearregression trendlines were used to determine how wellthe model predicted the FTIs and the PFs for the trainstested. The intercept of the trendline was set to zero. Aperfectly accurate model would have both a slope andR2 of one. Paired t-tests were used to compare theexperimental and predicted FTIs and PFs. Comparisonswere considered significant if pp0:05:

3. Results

Fig. 3(b) shows the fitting of Eq. (6) for a typicalsubject used for the validation study. The model wasparameterized at 401 of knee flexion and the parameter

values for a and b of Eq. (6) were calculated using thedevelopment angles with the value of yDEV in Eq. (6) setequal to 401. From Eq. (6), the values of parameter A

were determined at the three validation angles. Sub-jective evaluations showed that the model was able topredict the force responses of the muscles to a widerange of stimulation frequencies for each of thevalidation angles (Fig. 5). In general, the model over-estimated the FTIs and the PFs at the higher stimulationfrequencies (IPIsp50ms) for 301 and 751 of knee flexion(Fig. 6). Also, at these knee flexion angles, the modelgenerally overestimated the FTIs at low frequencies(IPIsX100ms). For 551 knee flexion, the pattern wasmore random; the model overestimated the FTIs forCFT10, CFT130, VFT10, and VFT30 and underesti-mated the FTIs for CFT100, VFT70, and VFT90. Incontrast, the model underestimated the PFs for CFT50,CFT70, VFT50, VFT70, VFT90, and VFT110.The correlation coefficients between the experimental

and predicted forces showed that the model accountedfor B63% and B81% (average values for the threevalidation angles) of the variances in the experimentalFTIs and PFs, respectively (Fig. 7). The model predictedthe PFs and the FTIs better at smaller angles than atlarger angles of knee flexion (e.g., 301 predictions werebetter than the 751 predictions). Also, the modelpredicted the PFs better than the FTIs for all threevalidation angles.

4. Discussion

The present study shows that the model developed byDing and colleagues (Ding et al., 2000a, b; Ding, 2001)has the ability to predict the force responses ofmuscles at different muscle lengths. The only modifica-tion of the model required is to allow parameter A;the scaling factor in force development of the model tovary with knee flexion angle. Thus, the force–lengthrelationship can be modeled indirectly through therelationship between parameter A and the knee flexionangle.Ding and colleagues previously tested the ability of

their two-step, isometric force model (Ding et al., 1998)to predict the force responses of human quadricepsmuscles at both 901 and 151 of knee flexion. Theyparameterized the model at each muscle length andfound that forces could be predicted in response to awide range of stimulation frequencies at both 901 and151 of knee flexion. For the present model the values oft1; Km; and t2 were obtained at 401 of knee flexion, whilethe parabolic behavior of A was modeled from fourdifferent muscle lengths. With this form of parameter-ization, the current force model predicted the forceresponse to a wide variety of stimulation frequencies atany muscle length.


(a)

CFT10

0

50

100

150

200

0 200 400 600 800

For

ce(N

)

(i)

VFT10

0

50

100

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200

0 200 400 600 800

For

ce(N

)

(b)

CFT30

0

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0 200 400 600 800

For

ce(N

)

(j)

VFT30

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0 200 400 600 800

For

ce(N

)

(c)

CFT50

0

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0 200 400 600 800

For

ce(N

)

(k)

VFT50

0

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0 200 400 600 800

For

ce(N

)

(d)

CFT70

0

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0 200 400 600 800

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ce(N

)

(l)

VFT70

0

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0 200 400 600 800

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ce(N

)

(e)

CFT90

0

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ce(N

)

(m)

VFT90

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ce(N

)

(f)

CFT100

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ce(N

)

(n)

VFT110

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ce(N

)

(g)

CFT110

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0 200 400 600 800

For

ce(N

)

(o)

VFT130

0

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0 200 400 600 800Time (ms)

For

ce(N

)

(h)

CFT130

0

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0 200 400 600 800TIme (ms)

For

ce(N

)

Predicted

Experimental


Durfee and Palmer’s (1994) muscle model was notcapable of predicting the force response to variations inthe activation frequency of the muscle and their linearforce–length relationship could not account for muscleforces beyond the optimal length. In contrast, thepresent model predicts the force response to variationsin the activation frequency and accounts for muscleforces beyond the optimal length through the parabolic

force–length relationship. Dorgan and O’Malley (1998)used Hatze’s (1977) force–length relationship, whichuses a combination of exponential and sine functions todescribe the variation of muscle force with musclelength. This relationship was muscle fiber type specificand hence a number of constants needed to bedetermined for each fiber type. Therefore, the force–length relationship in Dorgan and O’Malley’s model

Fig. 5. Experimental and predicted forces from a typical subjects’ quadriceps muscle at a knee flexion angle of 301 (same subject presented in

Fig. 3(a). The model parameters were determined by first parameterizing the model at 401 of knee flexion and then the value of A was determined at

301 of knee flexion by Eq. (6) (see text for details). (a)–(h): Comparison of measured and predicted forces in response to CFTs with IPIs of 10, 30, 50,

70, 90, 100, 110, and 130ms, respectively. (i)–(o): Comparison of measured and predicted forces in response to VFTs with IPIs of 10, 30, 50, 70, 90,

110, and 130ms, respectively. Force model parameters were: A ¼ 1:783; t2 ¼ 196:88; Km ¼ 0:336; t1 ¼ 38:85; and R0 ¼ 2:

Force-time Integral (N-ms)

(a)

0

10

20

30

40

50

60

70

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

Experimental

Model

**

* ***

* *

Peak Force (N)

(d)

0

50

100

150

200

250

300

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

* *

*

(b)

0

10

20

30

40

50

60

70

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

**

*

* *

(e)

0

50

100

150

200

250

300

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

* **

**

30 ̊

55 ̊

75 ̊

(c)

0

10

20

30

40

50

60

70

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

*

**

*

(f)

0

50

100

150

200

250

300

C10

C30

C50

C70

C90

C10

0

C11

0

C13

0

V10

V30

V50

V70

V90

V11

0

V13

0

* *

*

Fig. 6. Bar graphs of the mean (þstandarderror) experimental and predicted FTIs (a)–(c) and PFs (d)–(f) of six subjects used for validation of themodel in response to CFTs with interpulse intervals of 10, 30, 50, 70, 90, 100, 110, and 130ms (C10, C30, C50, C70, C90, C100, C110, and C130) and

in response to VFTs with interpulse intervals of 10, 30, 50, 70, 90, 110, and 130ms (V10, V30, V50, V70, V90, V110, and V130) at knee flexion angles

of 301 (a) and (d), 551 (b) and (e), and 751 (c) and (f). ¼ pp0:05:


needed to be both muscle specific and subject specific,which makes it difficult to implement for a real timeFES-based system. The current model uses a parabolicequation to describe the variation of force with musclelength and is not fiber type specific. Although a modelsimilar to the current one was proposed and analyzed byStein and O&guzt .oreli (1982), no details of comparison ofmodeled data with experimental data were presented.The present study has shown that by allowing only A;

the scaling factor in force development, to change withmuscle length, the model could capture the changes inforce with muscle length. It should be noted that in thisstudy forces were measured by a force transducer placedabove the ankle joint. Parameter A; which scales theforces measured at the transducer, is therefore afunction of the knee joint torque and the distance fromthe knee joint axis at which the force transducer isplaced. The knee joint torque is a function of thepatellar tendon force, which is generated by thequadriceps muscle, and the lever arm. Both the patellar

tendon force and the lever arm are affected by the kneejoint angle. Our results showed that A changed in aparabolic manner with knee flexion angle and had apeak value when the knee was flexed to an angle between651 and 701. This is consistent with previous studies(Lindahl and Movin, 1968; Roland and Friedemann,2001) that have shown that the knee joint torque reachesa maximum value at around 701 of knee flexion. Theusual explanation for the increase in the knee jointtorque between 01 and 701 is the lengthening of thequadriceps muscle with increasing knee flexion, whichyields an optimization of actin-myosin overlap (Cutts,1988; Gordon et al., 1966; Herzog and ter Keurs, 1988).Also, both A and force decreased when the knee wasflexed more than 701. This decay in the knee joint torquehas been shown by other investigators (Buff et al., 1988;Huberti et al., 1984). At least two possible reasons existfor this decrease in A and force. The first is that aquadriceps muscle length may be reached for whichthe sarcomere length is beyond the region of optimal

(a)

FTI

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Experimental (N-ms)

Pre

dic

ted

(N-m

s)

(d)

PF

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Experimental (N)

Pre

dic

ted

(N)

(b)

FTI

Slope = 0.96

R2 = 0.67

Slope = 0.92

R2 = 0.50

Slope = 0.98

R2 = 0.71

Slope = 1.16

R2 = 0.72

Slope = 0.92

R2 = 0.82

Slope = 1.06

R2 = 0.89

0

20

40

60

80

100

0 20 40 60 80 100

Experimental (N-ms)

Pre

dic

ted

(N-m

s)

(e)

PF

0

100

200

300

400

0 100 200 300 400

Experimental (N)

Pre

dic

ted

(N)

30 ̊

55 ̊

75 ̊

(c)

FTI

0

20

40

60

80

100

120

0 20 40 60 80 100 120

Experimental (N-ms)

Pre

dic

ted

(N-m

s)

(f)

PF

0

100

200

300

400

500

0 100 200 300 400 500

Experimental (N)

Pre

dic

ted

(N)

Fig. 7. Comparison of FTIs and PFs between experimental and predicted forces for six subjects at knee flexion angles of 301, 551, and 751. (a)–(c):

plots of predicted vs. experimental FTI at the validation angles. (d)–(f): plots of predicted vs. experimental PF at the validation angles.


actin–myosin overlap (Herzog and ter Keurs, 1988).However, there is little evidence from direct determina-tions of human quadriceps sarcomere length that themuscle is being stretched beyond the point of optimalactin–myosin overlap when the knee is flexed to 901(Cutts, 1988; Roland and Friedemann, 2001). Althoughsome alterations in sacromere length have been reportedfor the human femoris muscle (Herzog and ter Keurs,1988), these were confined to rather extreme hip andknee joint angles, yielding further evidence that sarco-mere length cannot explain the observed changes in A

and force between 701 and 901 of knee flexion. Thesecond explanation relates to the complex transmissionof quadriceps force to produce knee joint torque(Roland and Friedemann, 2001). Buff and colleagues(1988) have shown that there is a significant decline inthe force measured at the patellar ligament as comparedto the force introduced at the quadriceps tendon abovethe patella beyond 701 of knee flexion. They attributedthis decrease to the fact that patello-femoral joint actsmore like a balance beam than a pulley. Also, the lengthof the lever arm changes during flexion. These factorsmay explain the decline in the value of A and forcebeyond 701 of knee flexion.

5. Conclusion

This work elucidates the effects of knee joint angle onthe isometric forces predicted by the model developed byDing and colleagues. The isometric forces producedover a wide range of joint angles can be predicted byscaling one model parameter, A; as a function of jointangle. Subsequent work will use this information todevelop a non-isometric model that can predict theforces and length changes resulting from electricalactivation of skeletal muscle.

Acknowledgements

The authors would like to thank Drs. ThomasBuchanan and Sunil Agarwal for their helpful com-ments regarding the previous version of the manuscript.This study was supported by the National InstitutesHealth Grant HD 36797.

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Modeling the length dependence of isometric force in human quadriceps muscles

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