 Methodology
 Open Access
 Published:
Predicting nonisometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration
Journal of NeuroEngineering and Rehabilitation volume 10, Article number: 13 (2013)
Abstract
Background
Our previous model of the nonisometric muscle fatigue that occurs during repetitive functional electrical stimulation included models of force, motion, and fatigue and accounted for applied load but not stimulation pulse duration. Our objectives were to: 1) further develop, 2) validate, and 3) present outcome measures for a nonisometric fatigue model that can predict the effect of a range of pulse durations on muscle fatigue.
Methods
A computercontrolled stimulator sent electrical pulses to electrodes on the thighs of 25 ablebodied human subjects. Isometric and nonisometric nonfatiguing and fatiguing knee torques and/or angles were measured. Pulse duration (170–600 μs) was the independent variable. Measurements were divided into parameter identification and model validation subsets.
Results
The fatigue model was simplified by removing two of three nonisometric parameters. The third remained a function of other model parameters. Between 66% and 77% of the variability in the angle measurements was explained by the new model.
Conclusion
Muscle fatigue in response to different stimulation pulse durations can be predicted during nonisometric repetitive contractions.
Background
Functional Electrical Stimulation (FES) protocols use combinations of stimulation parameters (train duration, interpulse interval, pulse duration, and pulse amplitude) to produce functional movements in individuals with paralysis due to stroke or spinal cord injury (SCI). Unlike physiologically induced neuromuscular activation, FES synchronously activates motor units according to their current thresholds relative to the local extracellular current which is dependent on the distance from the electrodes[1, 2]. Consequently, the recruitment of motor units is random[3, 4] as compared to the order followed by the central nervous system, which recruits the smaller, fatigueresistant motor units first and the larger, more fatigable motor units last. When the motor units are activated synchronously the body cannot derecruit motor units as they fatigue and recruit new fresh motor units to replace them[5]. This random recruitment order together with synchronous activation are thought to be two of the major causes for excessive muscle fatigue during FES.
A mathematical model capable of predicting angular excursion, angular velocity, and joint torque during fatiguing contractions as a function of the stimulation parameters could be used to mathematically test combinations of independent and dependent variables to identify stimulation strategies that minimize fatigue. In addition, a validated model could predict force during fatiguing contractions in situations where force cannot be measured easily, such as during general nonisometric leg extensions. The term nonisometric indicates that the joint angle and thus the length of the musculotendon unit continually changes as the muscle contracts and relaxes. The phrase general nonisometric indicates that the leg is free to move solely in response to muscle forces. Although many models of nonisometric nonfatiguing contractions[6–8] and isometric fatiguing contractions[9–14] have been developed, only two models of nonisometric fatiguing contractions in humans appear in the literature[15, 16]. The model by Marion and colleagues[16] is the only one that has been experimentally validated to predict nonisometric fatigue in response to electrical stimulation.
In our previous study[16] we were interested in predicting nonisometric fatigue when the tension per activated motor unit was increased through the application of external loads. A similar situation may occur, for instance, in the spinal cord injured population when the relative resistive torque at the knee as compared to the number of activated motor units in the quadriceps increases as atrophy progresses. We are now interested in determining whether our nonisometric fatigue model can predict angular excursion, angular velocity, and joint torque due to stimulation of the quadriceps muscles at different pulse durations. This interest stems from the following reasons: 1) previous studies suggest that torque output can be predictably controlled and fatigue minimized by simultaneously controlling stimulation pulse duration and frequency during repetitive electrical stimulation[17–19], 2) others have shown the effect of pulse frequency on isometric fatigue and suggest that frequency should be minimized[20, 21], 3) our isometric forcefatigue model accounts for pulse frequency and pattern[10, 20], but neither the isometric nor the nonisometric forcefatigue model account for pulse duration, 4) studies suggest that relative isometric fatigue (compared to the initial torque) does not change with pulse duration[4, 21], therefore pulse duration can be increased to maintain torque, and 5) the relationship between pulse duration and nonisometric fatigue has not been reported, therefore it is unknown whether pulse duration can be increased to maintain torque and/or excursion. Because the overall objective of an ideal FES pulse train is to obtain the desired force and motion while minimizing fatigue, a fatigue model that takes pulse duration into account is required.
The objectives of this study were to: 1) further develop our model of FES nonisometric fatigue to take into account pulse duration, while simultaneously minimizing the number of parameter identification sessions with subjects by minimizing the number of model parameters, 2) experimentally validate the model at different pulse durations, and 3) present outcome measures, such as predicted angular excursion, angular velocity, joint torque, and power (torque (Nm) x angular velocity (rad/s)) due to stimulation, that can be compared over time for different independent variables. For consistency with our previous study, we chose general nonisometric leg extensions to further develop our model of nonisometric muscle fatigue. For reference, the term leg is defined as that section of the lower limb between the knee and ankle.
Methods
Mathematical model
The forcemotionfatigue model developed by Ding and colleagues[7, 10, 16] was used for this study (see Table 1 for definitions of symbols). The forcemotion model[7, 16] describes muscle activation, contraction dynamics, the forceangle relationship, and the forceangular velocity relationship. The input is the time the pulses are delivered, and the output is the force (F) at the ankle predicted for each time point (see Appendix).
Equation 1 models the ratelimiting step that leads to the formation of strongly bound crossbridges, and it represents the activation dynamics. Equation 2 describes the generation of the instantaneous force (F) near the ankle due to stimulation. It was derived from a Maxwell model of linear viscoelasticity in series with a motor[22]. The terms A and G represent the torqueangle[23] and torqueangular velocity[8] relationships, respectively. A torquepulse duration relationship has not been derived yet for this forcemotion model of nonisometric leg extensions. To meet the overall objective of the current study, to predict the effect of a range of pulse durations on muscle fatigue, the initial nonfatigue torque was measured at the pulse duration of interest just prior to the fatigue test. The MichaelisMenten term, C_{ N }/(K_{ m } + C_{ N }), scaled by A and G, drives the development of force. The last term in Equation (2) accounts for the force decay over two time constants, τ_{ 1 } and τ_{ 2 }. Equation 3 models the dynamics of the leg distal to the knee. The term F_{ M } represents the resistance to knee extension due to the weight of the leg and all other passive resistance about the knee joint, whereas F_{ load } is the load applied at the ankle (e.g. 4.54 kg; see Appendix). The term λ is added to the angle at the knee to ensure that angular acceleration is zero at the beginning of stimulation. Often the resting knee angle is not exactly 90°, λ is the difference.
The fatigue model[10, 16] monitors changes in the three forcemotion model parameters that change with fatigue, A_{ 90 }, K_{ m } and τ_{ 1 }. For each time step the input is instantaneous force (Equation 2) and angular velocity (from angular acceleration in Equation 3) from the forcemotion model (once all force and fatigue model parameters have been identified) for that time step. The output is the A_{ 90 }, K_{ m } and τ_{ 1 } to be used in the forcemotion model at the next time step.
The time constant τ_{ fat } characterizes the rate of change of parameters A_{ 90 }, K_{ m1 }, and τ_{ 1 } from the prefatigue values (A_{ 90,0 }, K_{ m1,0, } and τ_{ 1,0 }) to that in a steady state of fatigue. All of these terms have been reported previously[10, 16, 24] and the same procedures were used here to identify the values.
Parameter identification
The forcemotionfatigue model contains a total of nineteen parameters. Parameters R_{0} and τ_{ c } were held constant at 2 (unitless)[10] and 20 ms[20], respectively (see citations for results showing the derivation of these values). Fourteen of the remaining parameters, A_{ 90 }, a, b, K_{ m }, τ_{ 1 }, τ_{ 2 }, V_{ 1 }, V_{ 2 }, L/I, and F_{ M, } from the forcemotion model and α_{ A }, α_{ Km }, α_{ τ1 }, and τ_{ fat } from the fatigue model, required identification to both develop and validate the model, as well as to generate predictions. These parameters were identified from leg extension measurements, first from the development then from the validation groups of subjects (see Experimental Procedures and Figures 1 and2). The remaining fatigue model parameters, β_{ A }, β_{ Km }, and β_{ τ1 }, were initially identified from measurements and only from the development subjects. Model parameters were identified through minimization of the sum of squares error between the measured and modeled values via a Particle Swarm Optimization algorithm[25] followed by a nonlinear leastsquares algorithm (MatLab®)[26]. Optimizations were repeated several times to confirm that solutions had converged to the “global” minimum.
Preliminary results showed that for many subjects none of the three β parameters were needed for accurate predictions of the measured angles and angular velocities. After careful examination of the subjects that required β, we discovered that only β_{ τ1 } was necessary to predict fatigue in those subjects. Thus, although β_{ A } and β_{ Km } were employed in previous work[16], we postulated that β_{ A } and β_{ Km } were not necessary for modeling nonisometric fatigue; we explored this hypothesis as described in the Results.
Experimental procedures
Equipment and participant setup
Twentyfive healthy subjects, 14 men and 11 women (ages 21–48), with no history of lower extremity orthopedic problems voluntarily participated in this study and signed informed consent agreements. This study was approved by the University of California Human Subjects Review Board. Data from 5 men and 5 women (ages 19–25) from the previous study on predicting fatigue at different loads[16] were also analyzed to further validate the model.
The experimental setup was similar to that described previously[16, 27] (Figure 1). Subjects were seated in a backwardinclined (15° from vertical) chair of an exercise dynamometer (System 2, Biodex Medical Systems, Inc., Shirley, New York). The trunk, hips, and thigh were strapped to the chair, thus fixing the hip angle and limiting leg movement. The ankle was strapped to the lever arm of the dynamometer for the isometric and isovelocity tests. The axis of rotation of the knee joint was aligned with the axis of rotation of the dynamometer. A custom built electrogoniometer with two potentiometers, one positioned at the hip and the other at the knee axis of rotation, was strapped to the lower limb and trunk to measure joint angles. Customized software (LabView 8.0, National Instruments Corporation, Austin, TX) collected the digitized voltage signals at 300 Hz from the dynamometer torque transducer and the electrogoniometer. Two 7.5 cm × 12.5 cm selfadhesive stimulating electrodes (WF35 from http://www.tensproducts.com) were placed on the skin of the right thigh, one at the proximal and the other at the distal end of the quadriceps muscles. The electrode positions were adjusted until both a maximum amplitude and a constant shape of the torquetime curve were achieved at 4 different knee angles, 90°, 65°, 40° and 20° (where full extension was 0˚) and at 3 of the 5 pulse durations to be tested (min, mid, and max) and until the amount of nonplanar movement of the leg during general nonisometric leg extensions was minimized. In some subjects this resulted in the anode being positioned proximal to the cathode.
Customized software controlled the rate that monophasic pulses were delivered by the Grass S48 stimulator (Grass Technologies, AstroMed, Inc. Product Group, West Warwick, RI) to the electrodes. A constantvoltage transcutaneous system was used to minimize the risk of high current densities that can occur with constantcurrent systems if electrode contact with the skin is reduced. Others, also studying pulse duration, have used a similar system[4, 28]. Stimulus efficacy may have changed with increasing muscle contraction during delivery of the train because the tissues under the skin can move relative to the electrodes as the leg extends and because the current was not held constant, i.e. maximum stimulation of excitable tissue frequently occurs at the beginning of the pulse when current is maximum[29, 30]. Stimulus efficacy also may have changed over time during delivery of repetitive fatiguing trains of pulses because of sweating, which reduces skin impedance, and because of increased blood flow due to increased tissue temperature[31]. However, the parameters for the forcemotionfatigue model are identified from experimental measurements from each subject, therefore the model can and does account for each subject’s muscle response to the stimulation system used for the measurements. An attached SIU8T stimulus isolation unit (Grass Technologies) isolated the electrodes from ground, providing greater safety to the subject.
Testing sessions – general information common to All tests
Each subject participated in 4 to 6 testing sessions. Thirteen subjects were used for model development; the remaining 12 for model validation. Subjects were asked to refrain from strenuous exercise 24 hours prior to each testing session. Successive sessions were separated by at least 48 hours to allow the muscles to recover and again yield the maximum torque measured by the dynamometer prior to fatigue. Prior to, within the consent to participate form, and during the testing sessions, participants were asked to relax their legs so that the stimulation trains could be applied to relaxed quadriceps femoris muscles. The consent form states that the sessions may have to be repeated if they are unable to fully relax their leg. Torque and knee angle were monitored in realtime during the tests. The traces for each had consistent shapes appearing at timed intervals (during electrical stimulation). Volitional activation could be detected easily as alterations to the regularity/uniformity of the traces. Additionally, volitional activation during general nonisometric contractions prevents or alters the pendulum motion of the leg that occurs immediately after the leg drops to the resting position after cessation of a stimulation train. If the realtime traces on the plots or the pendulum motion of the leg looked unusual the test was stopped and the subject was gently reminded to relax.
The stimulation amplitude was set to produce maximum excursion of the freely swinging leg for both the minimum and maximum pulse durations while a 4.54 kg load was strapped to the ankle. This load was applied during all general nonisometric tests. Ten pounds or 4.54 kg was chosen for two reasons: 1) because it was used in previous studies to develop the forcemotion model used in the current study and to identify its parameters[7, 16] and 2) because our previous study[16] and pilot measurements suggested this load would provide measureable declines in force for the desired range of pulse durations during the fatiguing contractions. The stimulation amplitude was set at the voltage that extended the leg to ~15° with two 50 Hz trains, one with 600 μs pulses, trains no shorter than 0.2 seconds, and the other with 170 μs pulses, trains no longer than 0.8 seconds. This assured a maximum range of motion for every subject at all pulse durations, and thus a maximum range of fatigue for model development. When using a Grass stimulator quadriceps force approaches steady state near a pulse duration of 600 μs[4, 32], therefore 600 μs was selected as the maximum pulse duration. The minimum train duration was set at 0.2 seconds so that at least two pulses would be delivered at the lowest frequency tested. Pulse durations shorter than 170 μs were not used because the target excursion could not be reached at shorter durations by all subjects at the maximum pulse amplitude that was limited by the 0.2 second train of 600 μs pulses. Increasing the train duration longer than 0.8 seconds with 170 μs pulses did not increase the excursion of the leg. The pulse amplitude depended on the subject, ranging from 30 to 83 volts.
Both constant (CFT) and variable (VFT) frequency trains, containing equally spaced singlet pulses or an initial doublet (5 ms between pulses within the doublet) followed by equally spaced singlet pulses, respectively, were applied. Previous studies[10] have shown these two types of trains to be effective for identifying the model parameters, in particular 50CFT12.5VFT pairs, where 50 and 12.5 refer to the frequency (Hz) of the singlet pulses. At the beginning of every test, the quadriceps were held isometric and stimulated with twelve 14CFTs (14 Hz pulse frequency), with 0.8 second train durations and 5 seconds between trains, to potentiate the muscle[23]. Twitch responses initially increase during repeated lowfrequency stimulation (staircase phenomenon or twitch potentiation) and after a tetanic contraction (posttetanic potentiation)[33]. The mechanism of force enhancement may be related to phosphorylation of myosin light chains and increased Ca^{2+} sensitivity[34].
Isometric tests
Nonfatigue isometric
Parameters A_{ 90 }, Km, τ_{ 1 }, τ_{ 2 }, a, and b were identified from the nonfatiguing isometric contractions from one testing session. Torque in response to two pairs of testing trains (2 × 50CFT12.5VFT pair) was measured at each of 4 knee angles (15°, 40°, 65°, 90°). The order of the angles varied from session to session and subject to subject. The pulse duration was 600 μs, train duration was 1 second, and the rest between trains was 10 seconds. The muscles rested 4 minutes between angles, which was sufficient because the duty cycle and the number of trains delivered were too low to fatigue the muscles[10]. Measured forces were compared to modeled forces for initial identification of A_{ 90 }, K_{ m }, τ_{ 1 }, and τ_{ 2 }. Parameter A identified from the forcemotion model (Equation 2) and parameter A predicted by the parabolic equation were compared to identify a and b (Equation 2a).
Fatiguing isometric
Parameters α_{ A }, α_{ Km }, α_{ τ1 }, and τ_{ fat } were identified from the fatiguing isometric contractions from one testing session. One fatiguing stimulation protocol was applied per subject, at the end of a randomly selected testing session. The knee angle was 90° and the pulse duration was 600 μs. Fifteen pairs of testing (50CFT12.5VFT) and 195 fatiguing [33CFT (33 Hz)] trains, a total of 225 trains were applied as follows: 1 pair of testing trains followed by 13 fatiguing trains and then repeating the 15 trains 15 times. All train durations were 1 second. The 50CFT and 12.5VFT in the first pair were each followed by a 10 second rest. All remaining intertrain rests were 1 second. The 33CFT and this duty cycle were chosen because both have been proven effective to fatigue the quadriceps within 10 minutes with minimal discomfort to the participants[10]. The 50CFT12.5VFT pairs, applied after every 13 fatiguing trains, generated the forces used for identification of the isometric fatigue model parameters[10]. The 15 sets of A_{ 90 }, K_{ m }, and τ_{ 1 } parameters, derived by minimizing the error between the forces measured for each pair of testing trains and the forces predicted by the forcemotion model (Equation 2), were compared to the 15 sets of A_{ 90 }, K_{ m }, and τ_{ 1 } parameters predicted by the fatigue model (Equations 4–8) to identify α_{ A }, α_{ Km }, α_{ τ1 }, and τ_{ fat } (Figure 2).
Nonisometric tests
Nonfatigue Nonisometric
Identification of the parameters V_{ 1 } and V_{ 2 } in Equation 2b required isovelocity measurements from one testing session during which the exercise dynamometer extended the leg in passive mode. A previous study showed that forcemotion model predictions were more accurate when parameters V_{ 1 } and V_{ 2 } were identified at 200°/second rather than 125°/second or slower velocities[8]. Therefore, the dynamometer in the current study was set to 150°/second, its maximum velocity in passive mode, and the leg was moved from ~110° to 4°. To obtain only the force due to stimulation, F, it was necessary to collect measurements from leg extensions without and with stimulation[8] as the dynamometer extended the leg from 85° to 20°. This range of motion excluded the acceleration and deceleration tails and is within the general nonisometric range of motion of the leg. Four trains were applied, one per leg extension, two 50CFT12.5VFT pairs with pulse durations of 600 μs and a 10 second rest between each train. Measured forces were compared to the modeled forces (Equation 2) for identification of V_{ 1 }, and V_{ 2 }.
Identification of parameters L/I and F_{ M } (Equation 3) required general nonisometric nonfatiguing measurements immediately before every nonisometric fatiguing session. The leg was released from the dynamometer, a 4.54 kg load was strapped to the ankle, and the leg swung freely. Potentiation trains were applied to the free swinging leg immediately before the general nonisometric measurements. Two pairs of testing trains (2 × 50CFT12.5VFT), each followed by a 10 second rest, were applied to the free swinging leg, immediately prior to the fatiguing trains in the fatigue protocol. The train duration was set to the time needed for the leg with attached 4.54 kg load to extend to 1015° while the thigh was stimulated with a 50CFT at the pulse duration of interest. Measured and modeled angles and angular velocities were compared for every nonisometric session to identify not only the values for L/I and F_{ M } (Equation 3), but also to identify the initial, nonfatigue forcemotion model parameters, A_{ 90,0 }, K_{ m1,0, } and τ_{ 1,0 }, for the fatigue model (Equations 4–8), thereby adjusting for daytoday variability.
Five pulse durations were tested: 170, 200, 250, 400, and 600 μs, one per testing day. Previous studies[4, 32] measured the greatest changes in force at pulse durations between 100 μs and 250 μs, at frequencies used in the current study. The minimum pulse duration in the current study was set to 170 μs because shorter pulse durations frequently did not produce sufficient excursion of the leg at the amplitude set for the subject (as described above). The next higher pulse duration was set to 200 μs because the greatest changes in peak force occurred at the lowest pulse durations. This small increase in pulse duration produced at least a 5% increase in peak force, as was observed in the previous studies. The average train durations were 0.64, 0.51, 0.36, 0.29, and 0.24 seconds for the pulse durations: 170, 200, 250, 400, and 600 μs, respectively. The train duration for a given pulse duration was held constant for all pulse frequencies.
Fatiguing Nonisometric
Five general nonisometric fatiguing stimulation protocols were applied per subject, one per testing day, immediately following the nonfatigue protocol (Figure 1). As with the nonisometric nonfatiguing tests, the leg swung freely with a 4.54 kg load strapped to the ankle and the same five pulse durations were tested: 170, 200, 250, 400, and 600 μs. As with the isometric fatiguing protocol fifteen pairs of testing (50CFT12.5VFT) and 195 fatiguing [33CFT (33 Hz)] trains, a total of 225 trains were applied. The 50CFT and 12.5VFT in the first pair were each followed by a 10 second rest and were used for identification of the initial parameters, A_{ 90,0 }, K_{ m1,0, } and τ_{ 1,0 }, for the fatigue model (Equations 4–8) as was stated in the nonfatiguing nonisometric section. All remaining intertrain rests were 1.2 seconds, the minimum time required for the leg to return to the resting position (80° to 90°) and to manually stop the oscillations with one’s hands. The train duration remained constant during each fatigue protocol, that is, all 225 trains for a specific pulse duration test had the same train duration.
Parameters β_{ A } and β_{ Km } were removed from the fatigue model and an equation for β_{ τ1 } (Equation 8) was derived during model development from correlations between the fitted β_{ τ1 } and other forcemotionfatigue model parameters (Objective 1). Predictions for some subjects improved when all three β parameters were set to 0. Therefore, values for β_{ A }, β_{ Km }, and β_{ τ1 } were estimated separately through optimizations where predictions from the fatigue model, containing just one β per optimization, either β_{ A }, β_{ Km }, or β_{ τ1 }, were fit to the fatigue measurements to determine if one or more β parameters could be eliminated. Preliminary results suggested that β_{ A } and β_{ Km } could be removed from the fatigue model. The remaining β_{ τ1 } was initially identified by optimizing the fit between the fatigue model values and the angle and angular velocity fatigue measurements for the 170, 200, and 600 μs pulse duration tests. This fitted β_{ τ1 } was used in the correlations to derive an equation for β_{ τ1 }.
Prediction of outcome measures –experimental data from both the current and previous study
Predicted angular excursion, joint torque due to stimulation, angular velocity, and power (torque (Nm) × angular velocity (rad/s)) were compared over time and under different pulse duration and load conditions. Two pulse durations from the current study and two loads from our previous study[16] were used for the comparisons. From the previous study 4.54 kg and 9.08 kg were chosen. The 4.54 kg was selected because this was used in the current study for all pulse durations and the 9.08 kg was selected because this was the upper limit. From the current study, the pulse durations 600 μs and 170 μs were chosen because these were the lower and upper limits tested. A higher pulse amplitude was required in the current study than in the previous study to extend the leg to ~15° at the lowest pulse duration, 170 μs.
Statistical analysis
To validate the model, the predictive accuracy of the model was determined by analysis of the linear regression coefficient of determination (r^{2}, Objective 2). For each subject and each pulse duration in the current study (170, 200, 250, 400, and 600 μs) or applied load in the previous study[16] (0, 1.82, 4.54, 6.36, and 9.08 kg), the dependent variable was the predicted, and the independent variable was the measured, angular excursion or angular velocity. Both a fixed slope of unity and a yintercept of zero were used. Ideally, if the predictive accuracy of the model were 100%, then the linear regression r^{2} would be unity. Differences in the subjectaveraged r^{2} values between the different pulse durations or applied loads, both for angular velocity and excursion were determined using repeated measures ANOVAs followed by Tukey post hoc tests. A twofactor test was used for the subjects tested in the current study where the independent variables were pulse duration (170, 200, 250, 400, and 600 μs, nonisometric and isometric) and type of subject (development and validation). A onefactor test was used for the subjects tested in the previous study where the independent variable was load (0, 1.82, 4.54, 6.36, and 9.08 kg). In all cases the dependent variable was the r^{2}value.
To present outcome predictions (Objective 3), differences in predicted angular excursion, torque time integral (TTI), joint torque at maximum power, angular velocity at maximum power, and maximum power due to stimulation of the quadriceps were determined using twofactor repeated measures ANOVAs followed by Tukey post hoc tests. The independent variables for the twofactor ANOVAs were pulse duration (170, 200, 250, 400, and 600 μs; measured in the current study) or load (0, 1.82, 4.54, 6.36, and 9.08 kg; measured in the previous study[16]) and contraction number (the first 33CFT and the average of the last seven 33CFTs). The 33 Hz train was chosen because it was used to fatigue the muscle and was the middle frequency train, between the 50 Hz and 12.5 Hz trains. The last seven trains were averaged because the torquetime and angletime curves typically varied more at the end of the fatigue protocol than at the beginning. Additionally, at the beginning of the fatigue protocol there was a 10 second rest just prior to the first 33CFT, whereas only 1.2 seconds separated the remaining trains in the fatigue protocol. The shorter rest time resulted in somewhat increased variability in the starting position and velocity before each contraction. Because the fatigue curve was at steady state when the last set of 33CFTs was applied, the average of the last half of that set adequately represented the last train. The dependent variables were predicted angular excursion, TTI, joint torque at maximum power, angular velocity at maximum power, and maximum power, all due to stimulation. The predicted joint torque was computed by multiplying the force predicted by the forcemotionfatigue model by the moment arm (L) from the knee joint center of rotation to the center of the load applied just proximal to the ankle. In all cases p < 0.05 was considered significant.
Results
Modifications to the fatigue model (objective 1)
Complete data sets were collected on 25 subjects. Preliminary regression analyses of predictions of fatigue using the forcemotionfatigue model from the previous study[16], which used equations for β_{ A }, β_{ Km }, and β_{ τ1 } from the fatigue model (Equations 4–8), showed that although this model accounted for most of the variance in most subjects, predictions for some subjects improved when all three β parameters were set to 0 (not shown). Preliminary results suggested that inclusion of parameter β_{ τ1 } alone, without β_{ A } or β_{ Km }, in the fatigue model could account for fatigue in all the subjects. Angular velocity multiplied by β_{ τ1 } (Equation 8) reduced the impact of fatigue model parameter α_{ τ1 } on the force relaxation time constant τ_{1}. Parameter α_{ τ1 } accounts for the increase in τ_{1} that occurs during isometric fatigue, but in some subjects, parameter τ_{1} changed less during nonisometric fatigue than during isometric fatigue (Figure 3 shows an extreme case). Applying this new fatigue model to measurements from our previous study[16] confirmed that β_{ A } and β_{ Km } were not needed in the fatigue model to predict nonisometric fatigue.
The parameter β_{ τ1 } could be expressed as a function of parameters in the nonisometric force and isometric fatigue models. This is shown in Equation 9:
where A_{ 90,0 }, F_{ M }, and V_{ 1 } are nonfatigue forcemotion model parameter values from the day of the nonisometric fatigue session of interest, τ_{ 1,0,iso } is the nonfatigue forcemotion model parameter value from the isometric fatigue session, and α_{ τ1 } and τ_{ fat } are fatigue model parameter values from the isometric fatigue session. The equation for β_{ τ1 } (Equation 9) in the current study is different from the equation in the previous study (Equation 8a) (11) because in the previous study three β parameters, β_{ A }, β_{ Km }, and β_{ τ1 }, were used in the fatigue model. All three were identified simultaneously when fitting the fatigue model (Equations 4–8) predictions to the fatigue measurements to obtain the fitted β values used in the correlations to derive the equations for β. In the current study only one β parameter, β_{ τ1 } (Equation 8), was used and identified when fitting the fatigue model predictions to the measurements, therefore the fitted β_{ τ1 } in the current study was different from that in the previous study. Because β_{ τ1 } could be estimated from equation 9, nonisometric fatigue measurements were not needed to predict nonisometric fatigue.
Predictions of fatigue validated the model (objective 2)
Both measured and predicted angular velocity and excursion showed the greatest fatigue at the highest load, shortest pulse duration, and longest train duration (Figure 4, Objective 2). Train duration was a confounding factor, but was consistent across both studies. Predicted velocity and excursiontime curves were within one standard deviation of measured curves, with the exception of the first 1.5 minutes of the 0 kg load tests (Figure 4B, D).
Comparison of predictions to measurements through linear regression analyses (Figure 5) indicated that the new nonisometric forcemotionfatigue model accounted for between 66% and 77% of the variability in nonisometric fatigue for different clinically relevant pulse durations (170, 200, 250, 400, or 600 μs) with 4.54 kg applied to the ankle (Figure 6A). Predictions of measurements from our previous study[16] indicated that the new model also explained between 67% and 81% of the variability in nonisometric fatigue for different applied loads (0, 1.82, 4.54, 6.36, or 9.08 kg) when stimulating with 600 μs pulses (Figure 6B). Recall that the model development measurements were collected only in the current study and only at 170, 200, and 600 μs. All other measurements were used only for model validation. The predictions for the isometric measurements exceeded those for the nonisometric measurements (0.0001<p< 0.02, Figure 6B), accounting for >85% of the variability in isometric fatigue.
Outcome measures that can be predicted and compared (objective 3)
Torque at the knee due to stimulation of the quadriceps cannot be measured directly during general nonisometric leg extensions because the leg is not attached to any device that might resist its natural motion. However, this torque can be predicted by our forcemotionfatigue model. In this way, angular excursion, joint torque, angular velocity, and power due to stimulation can be compared over time and under different conditions (Figure 7, Objective 3). The predicted dependent variables showed significant fatigue (contraction number as the independent variable) at both loads and both pulse durations. With applied load or pulse duration as the independent variable, differences between the two applied loads or two pulse durations were not always significant. The predicted initial maximum power was not significantly different between the two loads or between the two pulse durations. The predicted angular velocity at maximum power was significantly less at the highest load and lowest pulse duration, while the predicted initial joint torque at maximum power was significantly greater at the highest load and lowest pulse duration. The initial angular excursion at 170 μs pulse duration was significantly less than at 600 μs (Figure 7A). Keep in mind that the train duration was set such that the 50CFT, not necessarily the 33CFT, produced the maximum excursion at each pulse duration or load.
Discussion
The key findings in the current study were that

(a)
pulse duration was not explicitly needed in the fatigue model; its effects on fatigue were captured by its effects on force,

(b)
two of three β parameters could be eliminated from our previous fatigue model without loss of predictive value with current and previous data sets,

(c)
the remaining β parameter is expressed completely as a function of values already measured, so, effectively, no additional parameters were added to the fatigue model,

(d)
the new forcemotionfatigue model accounted for 6677% and 6781% of the variability in the nonisometric measurements from the current and previous study, respectively, and

(e)
the model can be used to compare the power, angular velocity, angular excursion, and joint torque due to stimulation produced during fatiguing nonisometric contractions under different testing conditions.
The fatigue model was simplified by eliminating the parameters β_{ A } and β_{ Km } from the fatigue model and generating a new equation for β_{ τ1 } (Equation 9) as a function of existing forcemotionfatigue model parameters. Because β_{ τ1 } was multiplied by negative angular velocity, the β_{ τ1 } term reduced the effect of α_{ τ1 }, bringing τ_{ 1 } closer to its prefatigue value (Figure 3). In some subjects the difference between the prefatigue and fatigue twitch relaxation times was minimal during nonisometric contractions. For some subjects, the twitch relaxation time during nonisometric fatiguing contractions was less than during isometric fatiguing contractions.
Nonisometric fatigue measurements were not needed to predict nonisometric fatigue. In total, all but 5 parameters (A_{ 90 }, K_{ m }, τ_{ 1 }, L/I and F_{ M }), from both the force and fatigue models were identified from measurements collected during one testing session. The remaining 5 parameters were identified from prefatigue general nonisometric leg extension measurements from each nonisometric fatigue testing session.
The predictive ability of our new nonisometric forcemotionfatigue model (0.66 <= r^{2} <= 0.77 for pulse duration and 0.67 <= r^{2} <= 0.81 for applied load) tended to be higher than that of our previous nonisometric forcemotionfatigue model (0.56 < r^{2} <= 0.76 for applied load)[16], though lower than that of our isometric (r^{2} >0.85) forcefatigue model (Figure 6). The predictions in the current study for the measurements collected in the previous nonisometric modeling study[16] tended to be more accurate than those in the previous study because 1) the baseline angle or velocity for every stimulation train (contraction) delivered during the fatigue protocol on a given day in the current study was the initial value before the first train in the fatigue protocol, whereas in the previous study the baseline was an average of the initial angles or velocities before each of the 225 trains, and these sometimes deviated from the baseline of the resting leg and 2) the time between the last potentiation train and the first train in the fatigue protocol was reduced in the current study compared to the previous study, which reduced the magnitude of force enhancement that often occurred within the first few trains in the fatigue protocol. Insufficient potentiation explains why the measured fatigue tended to be less than the predicted fatigue for the 0 kg load (Figure 4) because the forcemotionfatigue model had no provision for potentiation.
A number of factors may explain why the isometric forcefatigue model accounted for more of the variability in the isometric measurements (Figure 6; 8692%) than the nonisometric forcemotionfatigue model could account for in the nonisometric measurements. These include the following:

(1)
In the isometric case, all model parameters were identified from force measurements at one knee angle, 90°. In the nonisometric case, the forcelength relationship model parameters were identified from force measurements at 4 different angles and the isovelocity and free model parameters were identified from angle and angular velocity measurements at the angles between ~85° (resting) and ~12° (nearly full extension).

(2)
In the isometric case, the electrode position relative to the nerves and muscles beneath was nearly constant from the beginning to the end of a fatiguing protocol. In the nonisometric case, both the skin and muscles moved as the leg extended, and maximum extension depended on pulse frequency and extent of fatigue, and therefore the amount of movement may have varied from train to train.

(3)
In the isometric case, the leg remained in the sagittal plane. In the nonisometric case the leg may have moved out of the sagittal plane as it fatigued.

(4)
In the isometric case, the potentiation protocol given just prior to the fatigue protocol minimized the force enhancement that often occurred during the first few trains in the fatigue protocol. In the nonisometric case, the potentiation protocol was not as effective at reducing the force enhancement that occurred with the shortest train durations and highest velocities, perhaps due to the continual and rapid change in myofiber or myofibril conformation.

(5)
In the isometric case, the initial force immediately before every contraction in the fatigue protocol was the same. In the nonisometric case, the initial angle and angular velocity was not always identical because we manually stopped and released the leg after each fatiguing extension, allowing for some human error.
To reach the desired excursion, as pulse duration decreased, train duration increased; as applied load increased, train duration increased. Train duration was therefore a confounding factor in our results, interacting with pulse duration and applied load (Figure 4). Taken together, the results of both studies suggest that the higher the duty cycle of the train, the greater the fatigue (constant rest time between trains: 1.2 and 1.3 seconds). This has been observed by others[35]. It may seem that holding the train duration the same across all pulse durations (or loads) would have led to a clearer interpretation of the measurements, but then maximum excursion would not have been constant across trials. Both the 12.5 Hz VFT and 50 Hz CFT trains were required to identify the prefatigue forcemotion model parameters. Considering that the leg was free to move, the maximum train duration was limited by the highest frequency train at the longest pulse duration. Holding the train durations constant would have resulted in significantly different angular excursions among pulse durations, thus creating a different confounding factor. Additionally, our objective was to validate fatigue predictions from excursion and velocity measurements. Using the same prefatigue excursion (at 50 Hz) for every pulse duration provided the largest range of excursion between the prefatigue and final fatigue measurements.
Comparing initial and final outcome measures in response to different independent variables, such as applied load or pulse duration, could help a therapist determine which stimulation parameters are most desirable for the patient and task. If higher joint torque is required (e.g. to strengthen the muscles), then a pulse duration of 170 μs is preferable to 600 μs (see Figure 7). On the other hand, if maintaining the highest level of power over the greatest length of time is the goal, then a pulse duration of 600 μs is preferable to 170 μs. There was no significant difference in the initial maximum power between the two pulse durations however, the final maximum power for the 170 μs was less than that for 600 μs.
The isometric force model has been shown to perform equally well for both ablebodied and SCI subjects, requiring only minor modifications to the parameter identification procedures for the SCI subjects[28, 36]. The new nonisometric forcemotionfatigue model, also validated to account for different loads per activated muscle (which could occur if atrophy progresses), may be equally robust, where similar minor modifications to the parameter identification procedures would pertain to this model. The maximum force generating ability of the muscles could be estimated from peak twitch force measurements as described by Ding, et al. (2005)[36]. The stimulation amplitude could be set as described in the current study, but would not exceed a level consistent with 50% of the force generating ability of the muscles. The isometric experimental protocol and identification of the isometric force model parameters could be similar to that described by Ding, et al. (2005)[36]. The nonisometric experimental protocol could be similar to that described in the current study, but the pulse frequencies would be as described by Ding, et al. (2005)[36]. The model parameters are subject specific, identified by fitting the model to the experimental measurements obtained from one testing session; therefore the current procedure for identifying these model parameters may require only minor modifications for the nonisometric forcemotionfatigue model to predict fatigue in SCI subjects. Spastic measurements would be excluded.
Our model has the potential to help physical therapists design stimulation protocols for patients in rehabilitation programs and to help researchers improve the task performance of FES systems[19, 32, 37–39]. The isometric forcefatigue model was extensively validated to account for the effect of different pulse frequencies and patterns on fatigue[10, 20]. The nonisometric forcemotionfatigue model has been validated to account for different applied loads and pulse durations, and these have resulted in a validation of different train duty cycles. From these model validations we learned that frequency, pulse pattern, pulse duration, and applied load are not explicitly needed in the fatigue model. Their effects on fatigue can be captured by their effects on force. Therefore, the nonisometric forcemotionfatigue model should be able to predict unique combinations of stimulation parameters for different subjects, such that each subject can achieve a desired outcome, such as maintaining a functional level of power for a useful period of time (e.g. Figure 7). The nonisometric forcemotion model[39] and the isometric fatigue model[40] have been used in a similar manner in other studies. The forcemotionfatigue model, with all model parameters identified for the task, could either mathematically test combinations of stimulation parameters until the desired outcome is obtained, or could be fit to an experimental force or trajectory (using an optimization algorithm) to generate optimal stimulation patterns that yield the force or trajectory for the desired length of time (see Maladen, et al.[39]).
Because this nonisometric forcemotionfatigue model would be capable of generating subjectspecific and taskspecific stimulation patterns that can maintain a desired force and motion for a desired length of time into the future, it has the potential for use as a feed forward model in FES systems[41]. If a system either does not use a feed forward model or requires more immediate real time output, then this model could be used to test the performance of the system prior to patient use. The model could generate a series of taskspecific optimal stimulation patterns, and these patterns could be compared to the real time FES system selections to optimize the system.
Because ablebodied subjects were tested in our study, there is a small chance that volitional contractions occurred during stimulation. However, it is unlikely that volitional contractions, if present, substantively affected our results. The forcemotionfatigue model has been shown to successfully predict fatigue in response to different frequencies and pulse patterns for numerous subjects over many years[10, 27, 36, 40]. This indicates that the signaltonoise ratio has been high, where here the stimulated contractions correspond to the signal and the volitional contractions correspond to the noise. In all cases, several testing sessions were performed on each subject, and each session was separated by 48 hours.
Conclusion
Pulse duration was not explicitly needed in the fatigue model; its effects on fatigue were captured by its effects on force. The nonisometric forcemotionfatigue model from our previous study[16] was simplified to predict nonisometric fatigue both at different applied loads and at different pulse durations. Parameters β_{ A } and β_{ Km } in the previous version of the fatigue model were eliminated and a new equation for the parameter β_{ τ1 } was derived. The β_{ τ1 } was solely a function of existing model parameters; therefore measurements of nonisometric fatigue are not needed to predict nonisometric fatigue. From 66% to 77% of the variability in the nonisometric measurements for different pulse durations was explained by the new forcemotionfatigue model. This new nonisometric forcemotionfatigue model can be used to predict angular excursion, angular velocity, joint torque, or power due to stimulation at different time intervals during repetitive contractions. This could assist with rehabilitation exercises and with the design and testing of new FES control systems.
Appendix
Derivation of the equation of motion
As described by Perumal, et al[8] the instantaneous moment about the knee center of rotation was derived from the free body diagram of the leg shown in Figure 8.
The equation of motion derived from the free body diagram for isovelocity extensions is:
where F_{ ext } = F_{ Bio } is the force component of the torque measured by the Biodex dynamometer and
Thus,
where F is the force just proximal to the malleoli exerted by the quadriceps through the knee joint in response to stimulation. It is defined as the instantaneous force near the ankle due to stimulation in Table 1. Previous passive force measurements on healthy subjects showed that (see Perumal, et al[8]):
where R is an intermediate variable.
Letting
and substituting equation 10d into 10b yields
where F_{ M } is obtained by fitting the function F_{ M } cos(θ) to force data collected during passive leg extensions where the quadriceps are relaxed and the dynamometer alone extends the leg.
The equation of motion for the general nonisometric leg extensions is:
Angular acceleration is no longer zero. F_{ ext }= the component of F_{ load } (applied ankle weight) that resists the contractile force of the quadriceps. The parameter L/I is a lumped parameter encompassing more than length and moment of inertia. Previous estimates of L/I using anthropometric data revealed differences from the values estimated through optimization[7, 8]. The differences may be the result of: 1) identifying L/I and F_{ M } simultaneously during optimization and 2) assuming that acceleration and/or applied weight have no effect on the forcemotion model (Equation 2), keeping in mind that F in the equation of motion is predicted from the forcemotion model (Equation 2). Therefore, L/I represents a more generalized parameter.
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Acknowledgments
The authors thank the undergraduate students CherylLynn Chow, George Marcotte, and Eddie Pham for their technical assistance. This work was supported in part by grants from the National Institute for Disability Related Research (NIDRR, Award Number H133G0200137) and NIH (R01 HD03858208, Robotic Exoskeletons, FES, and Biomechanics: Treating Movement Disorders).
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MSM, ASW, and MLH conceived of the study, participated in its design and coordination, and drafted and edited the manuscript. MSM wrote the software, tested the participants, performed the optimizations, simulations, and statistical analyses. All authors read and approved the final manuscript.
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Marion, M.S., Wexler, A.S. & Hull, M.L. Predicting nonisometric fatigue induced by electrical stimulation pulse trains as a function of pulse duration. J NeuroEngineering Rehabil 10, 13 (2013). https://doi.org/10.1186/174300031013
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Keywords
 Functional electrical stimulation (FES)
 Nonisometric
 Muscle fatigue
 Mathematical model
 Pulse duration