Force From Cat Soleus Muscle During Imposed Locomotor-Like Movements: Experimental Data Versus Hill-Type Model Predictions
Thomas G. Sandercock and
C. J. Heckman
Department of Physiology, Northwestern University Medical School, Chicago, Illinois 60611
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ABSTRACT |
Sandercock, Thomas G. and C. J. Heckman. Force from cat soleus muscle during imposed locomotor-like movements: experimental data versus Hill-type model predictions. J. Neurophysiol. 77: 1538-1552, 1997. Muscle is usually studied under nonphysiological conditions, such as tetanic stimulation or isovelocity movements, conditions selected to isolate specific properties or mechanisms in muscle. The purpose of this study was to measure the function of cat soleus muscle during physiological conditions, specifically a simulation of a single speed of slow walking, to determine whether the resulting force could be accurately represented by a Hill-type model. Because Hill-type models do not include history-dependent muscle properties or interactions among properties, the magnitudes of errors in predicted forces were expected to reveal whether these phenomena play important roles in the physiological conditions of this locomotor pattern. The natural locomotor length pattern during slow walking, and the action potential train for a low-threshold motor unit during slow walking, were obtained from the literature. The whole soleus muscle was synchronously stimulated with the locomotor pulse train while a muscle puller imposed the locomotor movement. The experimental results were similar to force measured via buckle transducer in freely walking animals. A Hill-type model was used to simulate the locomotor force. In a separate set of experiments, the parameters needed for a Hill-type model (force-velocity, length-tension, and stiffness of the series elastic element) were measured from the same muscle. Activation was determined by inverse computation of an isometric contraction with the use of the same locomotor stimulus pattern. During the stimulus train, the Hill-type model fit the locomotor data fairly well, with errors <10% of maximal tetanic tension. A substantial error occurred during the relaxation phase. The model overestimated force by ~30% of maximal tetanic tension. A nonlinear series elastic element had little influence on the force predicted by a Hill model, yet dramatically altered the predicted muscle fiber lengths. Further experiments and modeling were performed to determine the source of errors in the Hill-type model. Isovelocity ramps were constructed to pass through a selected point in the locomotor movement with the same velocity and muscle length. The muscle was stimulated with the same locomotor pulse train. The largest errors again occurred during the relaxation phase following completion of the stimulus. Stretch during stimulation caused the Hill model to underestimate the relaxation force. Shortening movements during stimulation caused the Hill model to overestimate the relaxation force. These errors may be attributed to the effects of movement on crossbridge persistence, and/or the changing affinity of troponin for calcium between bound and unbound crossbridges, neither of which is well represented in a Hill model. Other sources of error are discussed. The model presented represents the limit of accuracy of a basic Hill-type model applied to cat soleus. The model had every advantage: the parameters were measured from the same muscle for which the locomotion was simulated and errors that could arise in the estimation of activation dynamics were avoided by inverse calculation. The accuracy might be improved by compensating for the apparent effects of velocity and length on activation. Further studies are required to determine to what degree these conclusions can be generalized to other movements and muscles.
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INTRODUCTION |
The complex neuromechanical behaviors of muscle have been studied in considerable detail. Much has been learned, yet a comprehensive model of a muscles' macroscopic properties has not been established. In the motor control literature, most muscle models use some variant of a Hill-type model (Winters 1990
; Zajac 1989
). Despite their wide use, numerous studies have detailed complex muscle properties that are, in general, not predicted by such models. These properties include the persistent extra tension following an active stretch, or decrease in tension following a release (Abbott and Aubert 1952
), yielding (Joyce et al. 1969
), potentiation (Bagust et al. 1974
), the "catchlike" property (Burke et al. 1970
, 1976
), shortening-induced inactivation (Edman 1975
; Edman et al. 1993
), sarcomere inhomogeneity (Edman et al. 1995
), and fatigue (Edwards 1991
). These complex muscle properties are often studied in steady-state conditions, with velocity, length, or stimulus rate sometimes exceeding the physiological range of a muscle. This is necessary to isolate the desired behavior. In comparison, the activation and mechanical conditions that exist in many natural movements may be less extreme, but the inputs are highly dynamic rather than steady state. This makes it difficult to assess whether the complex behaviors noted above are required in a muscle model used for locomotion.
Our long-term goal is to develop a model of the whole muscle based on the mechanical properties of its constituent motor units. This requires a mathematical model of a single muscle fiber or motor unit, capable of predicting force, with fiber length and an impulse train as the inputs to the model. This study was performed to determine whether a Hill-type model could form the basis of such a model. Because Hill-type models exclude the complex muscle properties summarized above, these models can be used to address the issue of whether the excluded properties are important for force generation in normal movement conditions. In other words, the magnitude of the errors in force predictions from a Hill-type model indicates the net contribution of properties such as excess tension and sarcomere heterogeneity.
The goal of the present investigation was to test the hypothesis that a Hill-type model could effectively simulate the generation of muscle force in cat soleus during one particular locomotor condition, slow-speed locomotion. To accomplish this, normal locomotor conditions of activation, length, and velocity for the cat soleus muscle for a single speed of walking were mimicked in a controlled experimental setting in an anesthetized preparation. The whole muscle was stimulated synchronously, mimicking one large motor unit. A Hill-type model was constructed from each muscle's own series elastic element (SE) and length-tension (LT) and force-velocity (FV) behaviors and evaluated to determine how well this model fit the experimental locomotor records. In addition, the effect of a linear versus nonlinear SE on the Hill model predictions was analyzed. The problem of modeling activation was avoided by holding the stimulus pattern constant and estimating activation by inverse computation. A major error due to an interaction between rate of decay of force and velocity of movement was identified. A portion of this data has previously been published in abstract form (Heckman et al. 1993
).
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METHODS |
Initial preparation
Data were obtained from 10 adult cats (weight 2-3 kg). Anesthesia was induced with 5% isoflurane in a 1:3 mixture of O2 and N2O. Once a deep level of anesthesia was achieved, as judged by an absence of withdrawal reflexes, the isoflurane concentration was adjusted to between 0.5 and 1.5% to maintain this deep state throughout the surgical procedures for the preparation of the hindlimb and the laminectomy to expose spinal segments L4-S1. The tendon of the soleus muscle was attached to the muscle puller via a small calcaneal bone chip. Before the tendon was freed, the foot was manually dorsiflected, and a thread was tied to the tendons of the soleus and medial gastrocnemius. After the soleus was attached to the puller, alignment of the two threads was defined as maximum physiological length (0 mm). Nerves to all hindlimb muscles were cut, except for the soleus nerve (the denervation included all branches to the lateral gastrocnemius muscle). The animal was then switched from gaseous anesthesia to intravenous pentobarbital sodium anesthesia. This was done gradually to maintain a deep anesthetic state throughout the changeover. The initial cumulative intravenous dose was ~40 mg/kg. Thereafter it was supplemented with 5- to 10-mg doses as needed to maintain stable blood pressure and continued suppression of withdrawal reflexes. Core temperature was maintained at ~37°C by a heated water blanket and 34-35°C in the hindlimb pool by a heat lamp. The animals were killed at the end of the experiment without regaining consciousness by a high intravenous dose of pentobarbital (~100 mg/kg).
Stimulation and recording
The force of the soleus muscle was recorded by a strain gauge in series with the muscle puller, while length was measured by a length transducer within the puller shaft. In three experiments, stimulation of either the L7 or S1 ventral roots were used to generate force in the soleus muscle. The stimulus intensity was set at ~4 times threshold for a twitch force. The resulting tetanic forces varied from ~5 to 10 N, or ~25-50% of the tetanic force generated by stimulation of the entire soleus. In the other seven experiments, both S1 and L7 were stimulated simultaneously to activate the entire soleus muscle. Except for the amplitude of force, no differences were found for the results from whole and partial muscle stimulation.
Locomotor activation and length pattern
It should be emphasized that the goal of this study was not to precisely recreate locomotor conditions, but rather to achieve an approximation that would provide a reasonably natural and highly dynamic set of conditions in which to assess muscle behavior. The limitations in the derivation of the locomotor pattern are given below.
The natural locomotor length patterns for the soleus muscle during slow walking in the cat were based on the data of Goslow et al. (1973)
. Their Fig. 11 was digitized and divided into two segments, each of which was fit with a fifth-order polynomial with the use of least-squares regression (r2 > 0.99 in each case). The two polynomials were then combined to give one smooth waveform and used to create the scaled values for the analog output file, which is shown in Fig. 5. Because of the limitations in the peak amplitude of the puller's maximal excursion, the overall range of motion had to be scaled down by 25%, from 15 to 11 mm. Time was held constant, so velocities were also lower than in the experimental data. The stimulus pattern of course could not accurately reproduce the normal recruitment and rate modulation of motor units. Instead, the entire stimulated filament was treated like a single low-threshold motor unit (pulse duration 0.02 ms; voltage threshold for the filament was typically ~2 V; stimuli were applied at 5-6 times threshold). On the basis of the results of Hoffer et al. (1987)
, it was assumed that the discharge pattern of this "unit" underwent a modulation pattern that was similar to that of the rectified, smoothed electromyogram. Thus the first stimulus slightly preceded stance onset, and the rate then increased for the second interspike interval, and then slowly through five interspike intervals during the stance phase (exact spike times are indicated in Figs. 4 and 5; instantaneous frequencies for these times are given in the legend for Fig. 4). It should be emphasized that each locomotor "cycle" was given as a single trial. In three experiments the effects of repeated locomotor cycles were studied.

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| FIG. 5.
Average experimental results and Hill model fit to the simulated slow walk. Top: soleus length and velocity produced by the muscle puller. Circles: time of each stimulus pulse. Middle: average force, normalized by Po, from all 7 experiments. Solid line: force measured from the soleus. Dashed line: Hill-type model predictions. Bottom: average error (solid line) and SD (- - -) between the Hill model and experimental results.
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| FIG. 4.
Estimate of activation [A(t)] based on an isometric contraction. The soleus muscle was stimulated with the sequence of pulses represented by the circles shown atop the graph. The frequencies defined by the 5 interspike intervals are: 20, 30, 25, 20, and 12 spikes/s, from left to right. Solid line: resulting force. A(t) was calculated with the use of an inverse solution of the Hill model equations. The parameters for the Hill model were determined from an independent set of trials.
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Experimental protocols and data analysis
All simulation patterns and muscle puller length changes were controlled via computer, with the use of digital pulses and digital-to-analog outputs, respectively. The basic protocol was to apply a given length and stimulation pattern to the muscle, wait 15 s, repeat the same length without a stimulus to record the passive force response of the muscle, and then proceed to a different length and stimulation pattern. Forces from locomotor patterns, isovelocity ramps, and various isometric lengths were studied in this manner (see RESULTS). Because of the alternation of active and passive trials, stimulation occurred no more often than once every 30 s. Furthermore, a systematic test for fatigue was applied by periodically monitoring the isometric force generated by the locomotor stimulation. Data were only accepted if the fatigue test remained within 5% of the initial amplitude. The basic data processing was straightforward. Passive forces were eliminated by subtracting the trials without stimulation from the trials with stimulation. Force resolution and repeatability were very good and averaging was not necessary.
Construction of the Hill-type models
Hill's basic model consisted of an elastic element in series with a contractile element. The contractile element was characterized by an FV curve measured during steady-state shortening. This simple model is inadequate for physiological movements that include eccentric contractions and changing stimulus rates. Different investigators have extended the model to suit their particular needs (Winters 1990
; Zajac 1989
). The simplest possible Hill-type model was used. The model is based on the Hill model used by Krylow and Sandercock (1997)
in rat soleus. The model consists of a contractile component in series with an elastic element (Fig. 1). The contractile component is assumed to be modified by activation and the LT curve.
The contractile component during shortening is described by Hill's FV relationship
|
(1)
|
Where VCE is the velocity of the contractile component, F is the force from the muscle, Po is the peak isometric force, and ao and bo are Hill's constants for a particular muscle. Mashima et al. (1972)
extended Hill's model to include lengthening contractions. So, when F is greater than Po, the following equation describes the relationship between force and velocity
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(2)
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Where co and do are constants measured for a particular muscle. The FV parameters were measured during tetanic stimulation at 60 Hz. Figure 2 shows the raw data. For all velocities, force was measured at the same time and length with respect to onset of the isovelocity ramp (Fig. 2, vertical dashed line). Note that the slight difference in isometric force before the isovelocity ramps was due to different starting lengths. Figure 3A shows the resulting FV functions (Eq. 1 and 2) that were fit to these data.

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| FIG. 2.
Raw force-velocity data. The muscle was stimulated with 60-Hz trains lasting from 250 to 850 ms. Bottom: isovelocity ramps (mm) imposed by the muscle puller. Top: resulting muscle force (N). Dashed vertical line: time when force was measured.
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| FIG. 3.
Fit to force-velocity and length-tension data. A: least-squares fit of Hill's equations (Eq. 1) and Mashima's equation (Eq. 2) to the data points (circles) measured from the waveforms shown in Fig. 2. B: least-squares fit of Eq. 3 to length-tension data. Circles: peak force measured during isometric tetani at 100 Hz. Measurements were made every 2 mm. Dashed vertical lines: range of velocities and lengths achieved during the locomotor movement (see Fig. 5) when the muscle is actively contracting.
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The LT function was quantified by fitting the data with a polynomial. First an idealized LT function was fit with a fourth-order polynomial. This polynomial was then scaled by width and length to fit the experimental data
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(3)
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Where g(xm) is the fraction of Po produced at muscle length xm, Lo is the optimal length, and W is the width. The experimental data were obtained by measuring the steady force produced by 100-Hz tetanic stimulation (1 s in duration) at various isometric lengths (spacing 2 mm). Figure 3B shows the raw data and the fit. In all seven experiments, the optimal length for force production (Lo) was 3-8 mm short of physiological maximum (Rack and Westbury 1969
). Note that muscle length was used rather than fiber length. Correcting the data to compensate for tendon and aponeurosis stretch made little difference in cat soleus. Furthermore, because cat soleus has long fibers with a small angle of pinnation, changes in pinnation angle were ignored. Muscle length was assumed to be the sum of the length of the fiber and the SE.
At less than full activation, or at muscle lengths different than Lo, the muscle cannot produce full tetanic tension. The isometric tension produced by the muscle at length xm with activation A(t) is assumed to be
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(4)
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The FV relationship was modified by proportionately scaling the force axis. The maximum velocity of shortening was assumed to remain constant. Thus the constants ao and co were scaled. The scaled FV equations are
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(5)
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(6)
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In most Hill-type models, tendon compliance (including compliance of the aponeurosis) is lumped with crossbridge compliance and represented by an SE (Zajac 1989
). Tendon is known to have nonlinear stress-strain properties, yet, because of the uncertainty of the total SE, and for simplicity, the SE is often assumed to be linear. Unless otherwise noted, a linear SE was used in this paper. The value for this stiffness in each muscle was obtained by applying quick steps 0.5 mm in amplitude (rate of rise 15 ms), measuring the resulting change in force, and dividing this force by the amplitude of the length step. Typical stiffness values obtained with this technique range from 0.4 to 0.7 when normalized by the maximal tetanic force produced by the ventral root stimulation. To test the effect of a nonlinear SE, a piecewise exponential-linear SE was sometimes used
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(7)
|
where fel is the force where the transition from exponential to linear SE occurs, and ae, be, and k are constants.
Activation, A(t), is difficult to define in Hill-type models. To avoid an arbitrary model of the activation process while at the same time generating a physiologically realistic activation pattern, activation was defined with the use of the experimental data. To achieve this, the locomotor stimulation pattern was applied in the isometric state and force was measured (Fig. 4, thick line). Activation was then defined as the input required to cause the Hill model to exactly recreate this force with the use of the previously established parameters. The vertical dashed lines in Fig. 3 show the resulting activation pattern for the isometric force. The rapid time course of the activation pattern in comparison with the force pattern is due to the interaction between the FV function and the changes in muscle fiber length allowed by the SE (see DISCUSSION for consideration of how this might relate to previous definitions of activation).
Because the equations describing the Hill model are one-to-one functions, they can be inverted to solve for A(t). The constants in Eq. 3, 5, 6, and 7 were determined from experimental data as described above. The waveforms F(t) and xm(t) were sampled at 1 kHz during a known stimulus pattern. Equation 6 was used to compute the length of the SE, which, when subtracted from muscle length, yielded an estimate of average fiber length. Fiber length was differentiated to obtain VCE. Equations 5 and 6 were then solved for A(t).
The equations describing the Hill-type model were numerically integrated with the use of a fourth-order Runge-Kutta method (Press et al. 1992
). The inputs to the model were the waveforms xm(t) and A(t). The waveform xm(t) was obtained by digitally sampling muscle length at 1 kHz. A(t) was computed as described above.
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RESULTS |
Locomotor forces compared with the Hill model
The artificial replica of locomotor activation and length conditions for the soleus muscle is shown in Fig. 5, top, and it exhibits the typical features for slow locomotion in ankle extensors. It begins with the flexion phase, consisting of rapid stretch and then shortening. Soleus is quiescent for the majority of this phase but becomes active just before the transition to the stance phase. The initial stance phase consists of a second period of stretch caused by the transfer of weight onto the limb and then concludes with a second period of shortening as the ankle extends before toeoff.
Figure 5 also shows the ensemble average for locomotor forces produced in seven experiments (middle, solid line) and the average Hill model simulation of this force (- - -). Forces are normalized by Po, the maximum isometric force at Lo. The stretch during the flexion phase produced only a small passive force (not shown), but the second phase of stretch when soleus is active produces a substantial force, which then rapidly decreases during subsequent shortening to reach a brief plateau before decaying to zero. In general, the force record in these artificial locomotor conditions closely resembles that actually recorded from soleus in a normal cat during slow locomotion by Walmsley et al. (1978)
. The Hill model closely recreated the initial peak in force but, as the stance phase continues, a progressive trend for overestimation develops and, in fact, the Hill model incorrectly predicts that force should outlast the locomotor length changes and begin to redevelop in the subsequent isometric portion of the trial. The error for the fit of the Hill model to the actual locomotor force (defined simply as the difference between the 2 at each point in time) in all seven experiments is shown in Fig. 5, bottom (solid line; - - - indicate means ± 1 SD). This clearly illustrates the progressive increase in error during the decay and the large postmovement isometric error. It also indicates a tendency for a modest error during the initial rapid rate of rise of force as activation and muscle stretch coincide. Overall, the error is <10% of Po up until about two thirds of the stance phase has been completed, but reaches nearly 30% of Po during the decay phase. In the following sections, potential sources for these errors are considered.
Effects of a nonlinear SE
A linear SE was used in the modeling results presented so far, yet excised tendon is know to have a nonlinear length-force profile. Isolated tendon has a toe region with low stiffness at low force (Zajac 1989
). The aponeurosis has been shown to have properties similar to those of the tendon, so the two may be lumped together as a single element (Scott and Loeb 1995
). In addition, the muscle fibers themselvesshow increased stiffness with greater force because of an increased number of attached crossbridges (Morgan 1977
). In three experiments the effects of nonlinear SEs on Hill model outputs were studied. The results are essentially identical, so only a typical case is reported. Figure 6 shows the Hill model results with the use of three different SEs: 1) a linear SE with stiffness normalized by Po of k = 0.6; 2) a piecewise exponential-linear SE with fel = 0.5Po (Eq. 6) and the linear segment with a normalized stiffness of k = 0.6; and 3) a piecewise exponential-linear SE with fel = 1.0Po and the linear segment with a normalized stiffness of k = 0.6. Figure 6A shows the LT properties of the three SEs. Note that the stiffness of all three elastic elements isthe same at 1.0Po, yet there is a large difference in the toe region. Figure 6B shows the effects of the three SE models on the Hill model prediction of force during the locomotor movement with the locomotor stimulus. The heavy line indicates the experimental force, the thin line the Hill model prediction with the standard SE, and the dashed lines the prediction with the alternative SE models. The different SE models had relatively modest actions on predicted force and therefore none of the SE models had a major impact on the error between the predicted and experimental force.

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| FIG. 6.
Contrast between linear and nonlinear series elastic elements (SEs). A: length-force characteristics of 3 SEs. Each SE has a linear segment with a normalized stiffness of k = 0.6. Two SEs have nonlinear segments with increased compliance in the toe region. B: Hill model predictions with the use of each of the 3 SEs (light solid line and dotted line) compared with the experimental force (heavy solid line). C: predicted fiber lengths for each of the 3 SEs (light solid line, linear SE; dotted lines, nonlinear SEs). Heavy solid line: length of the muscle-tendon unit imposed by the muscle puller. D: stiffness. Again, the light lines represent Hill model predictions and the dots are experimental measurements of stiffness.
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However, the different SE models did impact estimates of fiber length. Figure 6C shows the predicted fiber lengths (XCE in Fig. 1) for the Hill model with the three different SEs. Unlike predicted forces, changes in the tendon model drastically alter predicted muscle length. Figure 6D shows the stiffness versus force relation for one experiment (dots) compared with predictions obtained from the three tendon models in Fig. 6A. The experimental stiffness was measured across a range of forces, with low forces being generated by low stimulation rates. As expected from previous work (Proske and Morgan 1984
; Rack and Westbury 1974
), the data showed that stiffness increases with force. The most nonlinear SE model (dashed line) fits the data the best, whereas the constant stiffness model (solid line) clearly produced large errors at low forces. Thus the most extreme exponential-linear model gave the most accurate prediction of stiffness, but, as noted above, this change in stiffness and the consequent large change in fiber length did not have a major impact on predicted muscle force.
Hill model fit to isovelocity data
A second possible source of error in the Hill model predictions is the continuous variation in muscle velocity, which might induce mechanical history effects not seen in whole muscle isovelocity conditions. To evaluate this possibility, we applied isovelocity ramps at various time points during the locomotor stimulus pattern and compared the results with those for the full locomotor conditions. The isovelocity ramps were selected so that they had a position and velocity equal to a chosen point in the locomotor pattern. The ramps began 100 ms before and ended 100 ms after the chosen point. Note that because of the SE and other history-dependent variables, the common match point does not mean the muscle was in the same state during the two conditions.
Figure 7 shows an example of two such match points. The dotted lines show the locomotor movement and the resulting force. The heavy lines represent the experimental results from the isovelocity ramps and the light lines are the Hill fits to the isovelocity ramps. The Hill parameters and A(t) were estimated in the same way as described in METHODS. First consider Fig. 7A. The selected match point is 576 ms, which is near the beginning of the shortening that follows the initial stretch. The predicted and actual forces correspond well for the first 50 ms or so, but then, just as during the locomotor length changes, the Hill-type model progressively overestimates force and the error is greatest during the decay phase.

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| FIG. 7.
Hill fit to isovelocity ramps during stimulation with the locomotor train. The isovelocity ramps were selected to match the velocity of the locomotor movement at 2 times. A: match at 576 ms when length was increasing (vertical dotted line). B: match at 480 ms. Light dotted lines: locomotor movement and resulting force. Heavy lines: isovelocity ramps and resulting force. Light solid lines: Hill fit to the isovelocity ramps. A(t) was determined by inverse calculation.
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Figure 7B shows results for a match point at 480 ms, which occurs shortly after simulated footfall when the muscle is active and stretched. Again, there is good correspondence for the initial 50-100 ms and then a progressively increasing error develops, becoming quite large during decay. However, the error is in the opposite direction compared with that for the shortening ramp: the predicted force underestimates actual force.
The same analysis was performed at different match points with similar results. Error was small for the initial 50-100 ms after ramp onset, but then stretch caused the Hill model to overestimate the force and shortening caused an underestimate. The isovelocity conditions show that these errors do not result simply from the complex variations in velocity during the locomotor length changes.
To determine whether the variations in the locomotor stimulation pattern were an important factor, we tested the ability of the Hill-type model to predict the full time course of force during the trials used to determine the muscle's tetanic FV function. A(t) for the Hill-type model was determined by inverse calculation of an isometric contraction during 60-Hz stimulation from 250 to 850 ms. Figure 8 shows a typical example comparing measured force with a Hill fit in response to two isovelocity ramps during stimulation at 60 Hz. Some of the experimental data from Fig. 2 have been replotted to allow direct comparison with the Hill model. The same analysis was performed on all seven muscles with essentially the same results. Because A(t) was determined by inverse calculation and the LT curve corrects for the different starting length, the model is essentially forced to fit the data before the ramps begin. Except for the initial rise, the calculated A(t) waveform was essentially flat, so it does not influence the later behavior. In addition, because the FV parameters for the Hill-type model were measured from the same data, the model is also essentially forced to fit the data at the time when the FV data were measured (t = 650 ms). Even with these restraints, it is clear that a progressively increasing error develops after 650 ms, much as in the full locomotor conditions (Fig. 5) and in the isovelocity/locomotor stimulation conditions (Fig. 7). As in those conditions, the stretch errors tended to be underestimates and the shortening errors tended to be overestimates. The similarity of these errors suggests mechanisms that do not depend on the existence of the locomotor variations in either stimulation rate or muscle velocity (see DISCUSSION).

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| FIG. 8.
Hill model fit to the force produced by isovelocity ramps during 60-Hz stimulation (250-850 ms). Bottom: superposition of 4 isovelocity ramps. Top 4 graphs: experimental force (heavy lines) vs. the Hill model results (light lines). The Hill model parameters were estimated by the methods described in the text. A(t) was determined by inverse calculation during an isometric contraction at 5 mm with the use of the same 60-Hz train.
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Length-dependent isometric relaxation time accounts for some error observed in the Hill model
The Hill model included the LT relationship (Eq. 3), which was based on parameters measured during 100-Hz stimulation. However, there are two length-related effects that are not accounted for in the model and that thus might account for a substantial portion of the error in the predictions of locomotor force. First, the LT relationship is known to vary with stimulation rate, such that at low stimulus rates, the peak force shifts to longer lengths, and the ascending slope of the LT curve becomes steeper (Rack and Westbury 1969
). At very low stimulation rates this could cause significant errors. Because the simulated locomotor stimulus had an average pulse rate of ~20 pulse/s, which is a relatively high stimulus rate, modifying the LT curve to fit 20 Hz provided a minimal improvement in the model. The second and more important effect is the increase in relaxation rate observed with increasing muscle length (Close 1964
). The model used in this study cannot be easily modified to make relaxation rate a function of length.
Figure 9 shows an experiment used to determine the upper bound for errors introduced by these two length effects. The same locomotor stimulus pattern was repeated during both a locomotor movement and five isometric lengths spanning the distance transversed by the locomotor movement. Figure 9D shows the imposed muscle moments and Fig. 9A shows the resulting forces. Note that the peak isometric force is quite similar (minimal LT effect) but the relaxation rate varies considerably. Activation was computed from each of the six trials and the results are shown in Fig. 9B. The heavy line represents the activation computed from the locomotor movement, and the light lines represent activation computed from the isometric trials (the largest activation resulted from the longest muscle lengths). If the Hill model were a perfect representation of the data, all six waveforms in Fig. 9B would be identical. The waveforms are quite close until relaxation begins (at 600 ms), where they diverge. Activation clearly shows a length dependence. Note that even at the shortest muscle lengths, the isometric trials are distinct from the locomotor trial. Figure 9C shows the Hill model predictions of force during the locomotor movement with the activation waveforms from Fig. 9B serving as the input to the model. An isometric length of
5 mm was used to compute activation for most of the data presented in this paper. Figure 9C shows that when activation was computed with the use of the isometric waveform at
11 mm, the peak relaxation error was reduced by approximately half. This is an unfairly short length, because only the last 50 ms of the locomotor waveform occurred at this length; however, it can be used to provide an upper bound for error produced by length.

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| FIG. 9.
Contribution of length-dependent relaxation times to Hill-type model errors. A: force waveforms resulting from the locomotor stimulus train applied during the locomotor (heavy line) and isometric movements (light lines) at 3, 5, 7, 9, and 11 mm. B: activation computed from the force and length waveforms shown in A and D. Solid circles: time of each stimulus pulse. C: Hill-type model predictions of force during the locomotor movement and the locomotor stimulus train. Light lines: 5 Hill model predictions based on the 5 different isometric activation waveforms shown in B. Heavy line: experimental force replotted from A. D: locomotor and isometric movements plotted together.
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In summary, the length-dependent relaxation rate observed during isometric contractions accounts for some, but less that half of the relaxation error observed in the Hill model.
Effect of consecutive step cycles
It was noticed that low-frequency stimulus trains resulted in variable force waveforms, unless precisely timed rest intervals were interposed between the stimuli. The mechanisms underlying this variability are probably related to the mechanisms responsible for potentiation, fatigue, and the labile force-frequency relationship. Because muscle, particularly locomotor muscle, is not used in isolated bursts with precisely timed rest intervals, it was desirable to determine whether consecutive step cycles would alter force and the Hill model fit.
Consecutive step cycles were studied in three cats. A typical example is shown in Fig. 10. Twenty step cycles were linked together to mimic the one-step per second walking pattern measured by Walmsley and colleagues (see Fig. 2 in Walmsley et al. 1978
). An identical stimulus train was applied in each cycle (the same locomotor-like pattern used previously). In addition, two isometric trials were measured, one 60 s before the first cycle, and one immediately following the 20th cycle, thus allowing activation to be computed for the 1st and the 20th step cycles. Force and length for the complete waveforms are shown in Fig. 10, A and B, respectively. Note the slow decrease in force. The 1st and 20th cycles are replotted in Fig. 10C (heavy lines). There was a 21% decrease in peak force. Similar results were seen in two additional cats, where a fall in peak force of 20% and 18% was measured.

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| FIG. 10.
Change in force with repeated locomotor cycles. A: force resulting from 20 concatenated locomotor cycles (shown in B). The locomotor stimulus train (depicted in Fig. 9B) was repeated for each of the cycles. Note that after the 20th cycle the muscle is moved to a length of 5 mm and held isometrically while the locomotor stimulus train is repeated. C: force from the 1st and 20th cycle is replotted ( ). Dotted lines: Hill-type model predictions based on 1) an isometric contraction preceding the 1st cycle by 60 s and 2) the isometric contraction immediately following the 20th cycle.
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The isometric trials preceding and following the 20 step cycles were used to compute activation, and in turn the Hill model fits, for the 1st and 20th cycles. These have been plotted against experimental force in Fig. 10C. Despite falling force, the Hill model predictions were almost as accurate for the 20th cycle as for the 1st cycle. This was because the isometric waveform used to compute activation decreased proportionally to the locomotor waveform. There was a slight increase in the relaxation time of the isometric waveform, so the relaxation error in the Hill model fits was greater for the 20th cycle.
A 20% decrease in force within 20 cycles was a surprising fall in force for the slow, fatigue-resistant soleus. Computer memory limitations, and the structure of the existing computer programs, prevented the use of >20 cycles at time, so it was not possible to determine whether force would continue to fall with repeated step cycles. A slight rest, as short as 15 s, allowed peak force to return to the initial value. To test the viability of the preparation, in one cat a standard fatigue test was run (40-Hz train lasting 0.3 s and repeated every 1.0 s) (Burke et al. 1973
). Force fell by <12% in 10 min. These results underline the complex nature of low-frequency stimulation and fatigue in muscle. Force fell less during the fatigue test, where each train consisted of 12 pulses at 40 Hz, compared with the locomotor stimulus, where each train consisted of six pulses at ~20 Hz. Such effects will make modeling the activation waveform difficult.
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DISCUSSION |
During locomotion, muscles generate force in highly dynamic conditions: a muscle's activation, length, and velocity all change continuously. In this study, the locomotor conditions for the cat soleus muscle during slow walking were approximately replicated with a muscle puller and an electrical stimulation pattern. The measured force is similar to forces measured with the use of a buckle transducer in a freely walking cat (Walmsley et al. 1978
). A Hill-type model, driven by an activation function derived from the isometric response to the locomotor stimulation pattern, was able to provide a reasonably close match to the large peak in the measured locomotor force during the initial rapid stretch at the onset of the stance phase. However, the plateau region of force was overestimated, perhaps because of a mechanical history effect of the previous rapid stretch and release. Furthermore, the decay of force for the Hill-type model was much too slow, because the isometric state used to define the activation function exhibited a slower force decay than occurs in locomotor conditions. Therefore, even in controlled conditions, where the model parameters were measured from the same muscle, the Hill-type model showed substantial errors.
Further experiments were performed to aid in understanding the cause of the model errors, namely, modeling of isovelocity ramps, estimation of activation at different muscle lengths, and changes in activation during repeated locomotor cycles. In the following paragraphs we discuss potential errors in Hill-type models in light of these experiments. The errors have been grouped as 1) errors in the FV relationship; 2) errors in activation, and 3) errors in the SE. At the end of the discussion an attempt is made to rank the sources of error by relative importance.
Errors in the FV relationship
One surprising error is the failure of the model to accurately predict the force resulting from isovelocity releases during tetanic stimulation (Fig. 8). Hill (1938)
showed that when muscle was constrained to shorten at a constant force, the resulting velocity reached a steady state that could be fit by a hyperbola. Similar results were obtained during steady-state isovelocity contractions. Although Edman (1988)
found that near Po the fit could be improved with the use of a second hyperbola, Hill's equation has remained one of the cornerstones of muscle mechanics. However, the wide acceptance of Hill's FV curve for steady-state conditions does not necessarily mean it remains accurate when muscle velocity is changing rapidly. Iwamoto et al. (1990)
demonstrated some departure from Hill's curve during non-steady-state contractions. Because of the series elasticity in the muscle (tendon, aponeurosis, and other structures), steady-state conditions exist only from the time the ramp is partially completed until the end of the ramp. Figure 8 also shows that, during the isometric contraction following the ramp, the experimental force is less than that predicted by the Hill model. This is an example of the persistent deficit in tension observed following an isometric release (Abbott and Aubert 1952
; Marechal and Plaghki 1979
). Edman et al. (1993)
present strong evidence that in frog fibers this is caused by sarcomere inhomogeneity. There is no mechanism in the Hill model to reproduce it unless the muscle is represented by multiple segments in series. Therefore, even during tetanic stimulation when the estimation of activation is not an issue, and during movements that involve only active shortening, there is some error in the Hill model.
Hill's equation does not adapt well to eccentric contractions. Mashima's extensions (Mashima et al. 1972
) do an adequate but far from perfect job characterizing eccentric FV behavior (Krylow and Sandercock 1996). During eccentric contraction the muscle does not really achieve a steady force if velocity is constrained, or a steady velocity if force is constrained. Figure 8 shows the results of two constant velocity stretches. At the start of a stretch, tension rises quickly. This short-term elasticity is believed to result from stretching of the tendon, aponeurosis, and previously attached crossbridges (Proske and Morgan 1984
; Rack and Westbury 1974
). It is followed by a rather abrupt break, or yield, when the crossbridge attachments may break. The force during the +80-mm/s ramp in Fig. 8 is the clearest example of yield in this study. Unlike other reports (Joyce et al. 1969
), a fall in tension was not observed. Following the yield point, tension generally continues to rise but at a much slower rate. Extra tension, that is, greater tension than would be expected from an isometric contraction at the same length, and greater tension than predicted from the Hill model, remains after the ramp stretch. This phenomenon has been observed by many investigators (Abbott and Aubert 1952
; Edman et al. 1982
). Morgan (1990)
recently attributed it to sarcomere popping.
A Hill-type model has no specific structure to reproduce yield. The FV curve has a different shape on the positive and negative side of zero velocity. This, plus a possible nonlinear SE, are the only features to distinguish the shapes of the force waveforms for positive and negative isovelocity ramps. As Fig. 8 shows, other than the sign, the Hill model predictions for positive and negative ramps are quite similar. Because experimental data in this study do not show a pronounced yield point, i.e., a fall in force, this does not appear to be a major problem with the model. The results of this study generally agree with those of Lin and Rymer (1993)
, who examined the effect of isovelocity ramps on cat soleus during activation by the crossed-extension reflex. They found that following the yield point, force either held constant or continued to rise slowly, and concluded that this was a stable state that could be used in control of movements.
The same sources of error that were observed in the Hill fit to isovelocity ramps are likely to exist in a Hill fit to locomotor movements. Certainly similar errors are observed in Fig. 7, showing isovelocity ramps driven by a locomotor stimulus train. The locomotor movement has a rapidly changing velocity, so the phenomena that produced an excess or deficit in tension following a ramp may well contribute to the error in the Hill fit. Because the mechanism is not understood, it is difficult to speculate as to the magnitude of the errors produced under different conditions.
Errors inherent in the use of activation to drive a
Hill model
To use a Hill-type model in circumstances where the muscle is not fully activated, it is necessary to have some sort of input to the model representing activation. This was implemented by having the force axis of the FV curve scaled by A(t). Such a model separates activation dynamics from the FV relationship and SE. This structure is similar to Hill's concept of "active state" (Hill 1949
). It has been demonstrated to be at least partially incorrect, because Jewell and Wilkie (1960)
showed that the duration of the active state changed with movement and muscle length.
Some investigators have used activation to scale the velocity axis, as well as the force axis, of the FV curve (Zajac 1989
). In some experiments in this study the maximum velocity of shortening was scaled by activation, and it was found that, overall, the model error was not improved. However, the tests were not extensive enough to conclude that scaling of the velocity axis is undesirable in all circumstances. Podolin and Ford (1986)
showed that in skinned frog fibers shortening velocity is independent of the level of activation. This is consistent with scaling only the force axis.
A(t) in Fig. 1 does not have a specific physical representation. It is strongly tied to Ca2+ transients but also likely to be influenced by the other rate reactions involved in crossbridge cycling as well. The structure of the model shown in Fig. 1 artificially separates the activation dynamics from the FV relationship and SE. This leads to difficult, possibly insurmountable errors. To accurately predict muscle force, a model may need to independently vary the rate of crossbridge attachment or detachment. Movement of a muscle alters the rate at which crossbridges break and reform. During an isovelocity movement this rate is reflected in the FV relationship. However, when the stimulus is started or stopped, or when fiber velocity changes rapidly, the crossbridges may not be in a steady state and errors may arise. The rate of decay at the end of a stimulus may depend on the state of the crossbridges when the stimulus was stopped. For example, the stretch shown in Fig. 7 leads to a prolongation in the decay of force compared with the locomotor movement.
Shortening-induced deactivation is another phenomenon not readily predicted by a Hill-type model. When a partially activated muscle is shortened, the resulting force is depressed by an amount dependent on the magnitude of the movement and not on the speed or load (Edman 1975
; Ekelund and Edman 1982
). This may be caused by the decreased ability of troponin to bind calcium when crossbridges are unattached compared with the attached state. Thus, when a partially activated muscle is moved, existing crossbridges break, calcium is released from troponin, and, at least in the partially activated muscle, a considerable time is needed for troponin to rebind calcium. This may account for the inability of the model to predict the shortened relaxation time during the locomotor movement (Fig. 5).
Sarcomere inhomogeneity may also alter the rate of force decay. This could appear as an inaccuracy in A(t) when it is computed by inverse computation. Caputo et al. (1994)
recorded free calcium in the myoplasm of frog fibers during stretch and shortening. Stretch or shortening ramps during the plateau of an isometric tetanus had no detectable effect on the calcium during the movement, yet stretch prolonged force, and shortening sped the decay of force. Either stretch or shortening during the relaxation phase of a tetanus increased the decay of force. Caputo et al. attributed this to nonuniform length changes across the fiber, which increased the rate of decay. If a similar mechanism is responsible for the variable decay rates observed in this study (see Figs. 5 and 7), then it would appear as an inaccuracy in A(t), even though velocity may have no effect on calcium dynamics.
Muscle length is known to alter activation. In this study the force axis of the FV curve was scaled by the LT relationship (Eq. 4). This corrects for the number of crossbridges available to produce force according to the sliding filament (Gordon et al. 1966
). The LT relationship is also known to vary with stimulus rate (Rack and Westbury 1969
). This is at least partially caused by length-dependent changes in sensitivity to calcium concentration (Martyn et al. 1993
). This is seen in Fig. 9, where the tetanic LT relationship does not fully correct for length changes at lower stimulus rate.
Modeling of activation
The problems with the concept of the active state suggest that muscle models should be based on crossbridge interactions. Zahalak and colleagues (Zahalak 1981
; Zahalak and Ma 1990
) have presented a distributed moment model that is based on Huxley's crossbridge model and uses a potentially more accurate model of activation. Unfortunately a simple method to estimate the parameters from experimental data has not been developed, so the model has not seen widespread use or testing. Other crossbridge-type models have generally not been applied to whole muscle.
As an alternative to crossbridge models, further modifications of A(t) in Hill-type models could be considered. Figure 9 suggests that activation may be altered by fiber length and fiber velocity. Conceptually, having fiber length modulate activation is reasonable because there is evidence that fiber length influences calcium dynamics either by intracellular release (Close and Lannergren 1984
) or by length-dependent calcium sensitivity (Balnave and Allen 1996
; Martyn et al. 1993
). However, there is no evidence that fiber velocity directly effects calcium dynamics (except for release from troponin during relaxation); rather, it probably alters crossbridge dynamics or sarcomere inhomogeneity (see previous text). Shue et al. (1995)
recently found improvements in a Hill-type model when activation was modified by muscle velocity. Such a change could improve the fit of the model and is an important area for further study.
A further problem with modeling activation is the complexity of excitation-contraction coupling process in a muscle. In some additional modeling experiments in this study, a heuristic model of activation was developed by the use of an impulse train to drive a pair of nonlinear differential equations. Provided the muscle remained in the same state, the results were similar to, but no better than, the inverse calculations. With repeated stimulation, the parameters had to be altered to match the changing state of the muscle. Muscle exhibits potentiation (Bagust et al. 1974
), the catchlike property (Burke et al. 1970
), and fatigue, all of which are poorly understood and possibly difficult to model. Figure 10 shows that following 20 repeated locomotor cycles there was a substantial change in activation. Computing activation by back calculation avoided this problem but requires an isometric contraction recorded when the muscle was in the same state. The change in activation over time will surely add errors to a Hill-type model that predicts force from a pulse train.
Compliance of the SE
In the simulated locomotor movement, the peak force lags behind the changes in muscle velocity (see Fig. 5). The explanation for this is that the SE allows the change in length of muscle fibers to be substantially different than that for the whole muscle. Griffiths (1991)
and Hoffer et al. (1989)
have shown that, in slow locomotion in the cat, the muscle fibers of the medial gastrocnemius muscle actually shorten throughout the stretch at the beginning of the stance phase, because all of the stretch is taken up by the tendon and aponeurosis. Fiber lengths in soleus have not yet been measured in locomotor conditions, and it is not known whether the same dramatic difference occurs. However, the Hill model predicts (Fig. 6C) that most of the stretch is taken up by the SE. This is true for all three SEs shown in the Fig. 6.
Tendon is known to have a nonlinear LT profile (Zajac 1989
), but because of uncertainty about the actual form of the exponential for the cat soleus muscle, a linear SE was used for most calculations. Figure 6B demonstrates that the use of a nonlinear SE has little effect on the force predicted by the Hill model. This is surprising and may result from the relatively high stiffness of the soleus tendon. In muscles with more compliant tendons the toe region may have a larger influence on predicted force. Also, during the simulated slow walk, shortening velocities only reached about
50 mm/s when the muscle was active, yet cat soleus has a maximal velocity or shortening of approximately
180 mm/s (Walmsley et al. 1978
and Fig. 3). If the muscle was tested with movements having velocities closer to the limit, the tendon stiffness could become more important.
Although not important for the prediction of muscle force, there are circumstances when the use of a nonlinear SE could be important in a model. Muscle stiffness is often an important variable. Although the stiffness of the muscle tendon complex depends on both the tendon and the number of attached crossbridges (Morgan 1977
), and is not well represented in a Hill-type model, simply adding a nonlinear SE to a Hill-type model can improve its accuracy (Schultz et al. 1991
). Figure 6D shows that the nonlinear SE improved the fit to measured stiffness. Also, the total length of the muscle tendon complex is often an important variable and, as Fig. 6A shows, this is dramatically influenced by the selection of a nonlinear tendon. In a muscle with a steep LT curve, accurate prediction of fiber length and stretch of the SE may be necessary.
Conclusions
A slow walk in cat soleus was approximately replicated with a muscle puller and an electrical stimulation pattern. A critical view of the Hill model revealed several problems in the prediction of force during these locomotor conditions. The most troublesome error was the prolonged relaxation time predicted for the locomotor movement (see Fig. 5). This was the largest error (almost 30% of Po) and would seriously degrade models of whole limb movement because the delayed relaxation would oppose contraction of agonist muscles. It may be related to shortening-induced deactivation (Ekelund and Edman 1982
) resulting from the changing affinity of troponin for calcium between bound and unbound crossbridges. Second in significance is the effect of length on activation (see Fig. 9). This error probably results from changes in calcium sensitivity at different sarcomere lengths (Balnave and Allen 1996
; Martyn et al. 1993
). A third error is the persistent excess and deficit in tension (Fig. 8), probably caused by sarcomere inhomogeneities (Edman et al. 1993
; Morgan 1990
). This error appears to be small, but may have contributed to the Hill model error after the initial stretch. Furthermore, sarcomere inhomogeneities may effect relaxation times, and thus contribute to the most significant error observed in this study. It should be emphasized that these results only apply to the specific locomotor conditions studied. Hill model errors from the soleus at different locomotor speeds, and of other muscles in general, could be quite different. It must also be stressed that we avoided some errors in predicting activation by the use of the method of inverse calculation.
Few, if any, studies assess Hill model errors in practical applications. Hill-type models continue to receive widespread use, so knowledge of their associated errors is important. The model presented in this paper had many advantages over most situations in which Hill models are used. Activation was estimated by back calculation, which is usually not possible. The parameters were also estimated from the same muscle in which the model was tested. Still, errors of up to 30% of Po were observed. Although further testing is needed in other muscles and under a wider range of conditions, similar errors can be expected in most applications.
Error in Hill-type models might be diminished by allowing fiber length and fiber velocity to modulate activation. However, such heuristic modifications may result in a model so complex that a crossbridge model would offer a better solution.
 |
ACKNOWLEDGEMENTS |
The authors gratefully acknowledge K. D. Paul for the development of the data collection software, for assistance with the initial surgical preparation, and for assistance with preparation of the manuscript.
This project was supported by National Institute of Arthritis and Musculoskeletal and Skin Diseases Grant AR-41531 and National Science Foundation Grant BES-9419528.
 |
FOOTNOTES |
Address for reprint requests: T. G. Sandercock, Dept. of Physiology, M211, Ward 5-295, Northwestern University School of Medicine, 303 E. Chicago Ave., Chicago, IL 60611.
Received 21 March 1996; accepted in final form 9 December 1996.
 |
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