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Electromechanical Delay and Reflex Response in Spastic Cerebral Palsy Kevin P. Granata, PhD, Andrea J. Ikeda, MS, Mark F. Abel, MD ABSTRACT. Granata KP, Ikeda AJ, Abel MF. Electromechanical delay and reflex response in spastic cerebral palsy. Arch Phys Med Rehabil 2000;81:888-94. Objective: Electromechanical delay (EMD) and reflex response in patients with spastic cerebral palsy (CP) were quantified and compared with those in normally developing individuals. It was hypothesized that the increased muscle stiffness associated with spasticity must make EMD shorter than the EMD of normally functioning muscles. Design: Electromechanical reflex behavior was assessed in a case-control study. Setting: Motion Analysis and Motor Performance Laboratory, University of Virginia, a tertiary clinical referral center and research facility. Participants: A volunteer sample of 12 children diagnosed with spastic CP and 12 age-matched, normally developing children recruited from the local community and clinical services. Results: EMD in the patients with spasticity was significantly shorter than in the normally developing subjects, 40.5msec and 54.7msec, respectively. The spastic group also had greater reflex activity, rate of force development, and antagonistic muscle activation. Knee flexion angle did not influence EMD in either group. Conclusions: Increased biomechanical stiffness in spastic muscle results in abnormally reduced EMD. Reciprocal excitation of antagonistic cocontraction was uniquely observed in the spastic group, but did not explain the reduced EMD. Key Words: Electromechanical delay; Muscle spasticity; Stiffness; Rehabilitation. 娀 2000 by the American Congress of Rehabilitation Medicine and the American Academy of Physical Medicine and Rehabilitation
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LECTROMECHANICAL Delay (EMD) describes the time between the onset of electromyographic (EMG) activity and the onset of biomechanical force or movement. EMD has been associated with the time required to remove the elastic ‘‘slack’’ from the musculotendinous system,1 ie, muscle contraction lengthens the in vivo structures sufficiently to generate elastic force. As such, EMD is primarily a measure of series elastic stiffness. Research demonstrates that muscle stiffness is greater in individuals with spastic neuropathies.2,3 Consequently, one should expect the EMD in patients with
From the Motion Analysis and Motor Performance Laboratory, Department of Orthopaedic Surgery, University of Virginia, Charlottesville, VA. Submitted July 19, 1999. Accepted in revised form October 19, 1999. Supported in part by a grant from the University of Virginia, Children’s Medical Center. No commercial party having a direct financial interest in the results of the research supporting this article has or will confer a benefit upon the authors or upon any organization with which the authors are associated. Reprint requests to K.P. Granata, University of Virginia, 2270 Ivy Rd, Charlottesville, VA 22903. 0003-9993/00/8107-5739$3.00/0 doi:10.1053/apmr.2000.5578
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spastic motor dysfunctions to be shorter than in normal subjects. Force is developed when muscle contractile elements shorten, thereby stretching the series elastic component transferring force to the tendons and joints.4,5 Stiffness describes the relation between force and stretch length. A mechanically stiff muscle will transmit large forces with very little stretch of the series elastic components. Conversely, a mechanically compliant or lax tissue requires much greater muscle contraction to sufficiently stretch the elastic components and generate measurable force. Compliant tissues require more time from activation until force generation, ie, their EMD is longer. EMD includes time associated with electrochemical muscle excitation, cross-bridge activation, and elastic stretch.6 The predominant component of EMD is the time required to lengthen the elastic components of the musculotendinous structures.1,7 Changes in EMD can be attributed primarily to changes in the stiffness of the series elastic component of muscle. Experiments verify that EMD is shortened when muscle stiffness is increased by pretensioning the muscles.6,8,9 Thus, EMD can describe differences in muscle stiffness in a variety of conditions and populations. Studies of biomechanical muscle stiffness in patients with spastic pathologies revealed that they have significantly more stiffness than normal subjects.2,3,10-12 Two mechanisms have been proposed: (1) increased reflex stiffness, and (2) increased intrinsic stiffness. Reflex significantly contributes to elastic muscle behavior as a result of the force response13-15 and pathologically abnormal reflex amplitude or excitability may lead to increased active muscle stiffness in spastic patients.10,16-23 Increased stiffness has also been measured in the absence of muscle activity in spastic muscles,24-26 indicating abnormal intrinsic2,27 and passive28 muscle stiffness attributed to morphologic adaptation of the involved muscles.24 If stiffness is greater in patients with spastic neuropathies, then EMD must be shorter than normal. EMD may potentially provide a clinically measurable analog of the stiffness in spastic muscle. The purpose of the present experiment was to quantify the EMD in patients with spastic cerebral palsy (CP) compared with normally developing individuals. Considering the greater musculoskeletal stiffness in the spastic population, we hypothesized that EMD must be shorter in spastic muscles than in normally functioning muscles. METHODS EMD was determined by measuring the latency of knee extensor reflex force relative to the onset of muscle reflex myoactivity. Others have employed voluntary response to audio29 and visual9,30 signals, voluntary cyclic exertions,6,8,31 involuntary response to electrical stimulation,9 and patellar tendon tap as stimuli from which EMD recordings were made. Research suggests that the EMD associated with voluntary exertions is greater than the EMD after involuntary stimuli, but no significant differences between EMD from tendon tap and electrical stimulation have been found.9 Considering that volun-
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tary control may be compromised in patients with spastic dysfunction, we chose to measure EMD from an involuntary reflex stimulus. Vos and colleagues8 conclude that EMD recruited from external electrical stimulation may represent nonphysiologic performance because of the supramaximal stimulation. For these reasons, we chose to use the patellar tendon tap as a stimulus to elicit knee extensor reflex activity and force. The subject population included 12 children with diagnoses of spastic CP and 12 age-matched, normally developing children. The children with CP included 8 with diplegia and 4 with hemiplegia (only the more involved side was tested in this protocol), all were community ambulators (3 required walkers as an assistive device), and all but 1 were born prematurely. Exclusion criteria included severe mental retardation or the presence of concurrent medical conditions that required surgery or hospitalization within 6 months. All subjects and/or their legal guardians signed informed consent approved by the University of Virginia Human Investigations Committee. The participants included 13 girls and 11 boys. The mean ages of the CP and normally developing groups were 11.2 ⫾ 3.2 years and 10.5 ⫾ 2.8 years, respectively. Subjects were seated in an isokinetic dynamometera with the chair’s backrest reclined to 110°. Knee angle was controlled by the dynamometer at isometric angles of 45° and 90° of flexion. The subjects’ legs were attached to the dynamometer arm by a portable load cell and supported by a velcro belt positioned so that the load cell contacted the tibia from an anterior direction at approximately 3cm proximal to the malleoli. A contact switch was taped to the subject’s patellar tendon over a location that would maximally elicit a reflex response. The contact switch recorded the patellar tap contact time and acted as a target for the reflex hammer. Myoelectric activity was recorded from bipolar surface electrodesb placed centrally on the quadriceps and medial hamstrings groups over the muscle bellies midway between the knee and hip. Raw EMG data were amplified and bandpassfiltered from 16Hz to 1500Hz and sampled at 1000Hz.c Knee extensor force was recorded from the portable load celld rigidly attached to the extensor arm of an isokinetic dynamometer and sampled at 1000Hz. Both EMG and force data were lowpassfiltered in software at 500Hz. The portable load cell was used to exploit its sensitivity to small force fluctuations when recording the reflex responses. Subjects were acclimated to the experimental setting until they relaxed, as indicated by the absence of myoelectric activity on a real-time display of knee flexor/extensor EMG activity. Maximal EMG levels were recorded from a series of maximum voluntary contractions (MVC) performed at the start of the experimental session. After a minimum of 5 minutes’ rest and recovery, reflex measurements were recorded. One experimenter distracted the subjects, while a second applied a tendon tap hammer to elicit the reflex response after ensuring that the EMG activities were at resting levels. Ten to 12 stimuli and associated reflex responses were recorded at each knee flexion angle. Independent variables included the subject group consisting of both normal subjects and those with spastic CP, and knee flexion angle (45° and 60°). Dependent variables included the reflex EMD, normalized EMG activity, reflex force (peak), and rate of force onset. Reflex EMG and force responses were determined by identifying signals that exceeded two standard deviations of baseline computed from the preceding 100msec of data. Onset time was determined from an automated algorithm that identified the time at which the response exceeded 5% of the response peak. EMD was defined as the delay from the
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onset of EMG activity to the onset of reflex force. Hamstring latency was defined as the delay from the onset of quadriceps activation to the onset of hamstring activation. Rate of force development was determined relative to the reflex response force rise from 5% to 50% peak force. Mean values were computed from the multiple reflex responses for each subject at each knee angle. Data processing was blinded to individual subject characteristics and subject category, ie, spastic or normal group. Statistical analyses of variance were performed to determine the significant effect of subject category and knee flexion angle on the results with ␣ ⱕ .05. RESULTS Reflex response profiles (fig 1) differed significantly between subject groups in both response magnitudes and onset timing. Peak reflex EMG recordings from CP patients’ quadriceps muscles were significantly greater than those from normally developing subjects (table 1, fig 2). Patients with spasticity demonstrated increased normalized antagonistic activity in the hamstring muscles compared with the normals (table 1, fig 2), suggesting abnormally high reciprocal excitation in the CP group.32 The timing between quadriceps response and hamstring activation was also influenced by subject group. While hamstring activity onset did not significantly differ from zero in the spastic group, latency was statistically significant in the normally developing subjects. The patients with spastic CP had significantly shorter EMD than the normally developing subjects (fig 3). The mean EMD for the spastic group, 40.5 ⫾ 10.5msec, was 25% shorter than the mean value for the normally developing group, 54.3 ⫾ 14.9msec. Knee angle did not statistically influence EMD. The rate of reflex force increase was significantly slower in the normally developing subjects, 6.2 ⫾ 4.9N/sec, than in patients with spastic CP, 10.0 ⫾ 5.1N/sec. Post hoc analyses revealed that the difference existed only at the 90° knee flexion angle. Neither subject group nor knee angle significantly influenced the rate of EMG onset for hamstrings or quadriceps. DISCUSSION Mean EMD was 47.4 ⫾ 12.7msec measured from the onset of myoelectric activity in the quadriceps muscle group to the onset of knee extensor force. The average value reported in the existing literature is 49.5 ⫾ 30.6msec, with measurements
Fig 1. EMG and knee extensor force responses following a patellar tendon tap in a population of patients with spastic CP and in normally developing subjects. Response curves represent the mean of each population from both knee angles. The patellar tendon tap is at time t ⴝ 0. EMG data were normalized to isometric MVC levels.
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SPASTIC ELECTROMECHANICAL DELAY, Granata Table 1: Statistical Results of EMD and Reflex Behavior as a Function of Subject Category and Knee Flexion Angle Effect Size*
Subject Group
Knee Flexion Angle
(␣ ⬍ .05,  ⫽ .2)
Spastic
Normal
90°
45°
10 32 14 7.3 .36 4.1
40.5 (10.5) 104.4 (63.3) 58.3 (75.7) 4.0 (6.7) .56 (.29) 10.0 (5.1)
54.3† (14.9) 45.3† (17.3) 11.5† (15.5) 10.5† (11.4) .47 (.59) 6.2† (4.9)
49.1 (15.6) 47.6 (49.9) 21.5 (24.6) 9.7 (11.8) .57 (.47) 9.1 (6.8)
45.8 (13.5) 42.9 (35.4) 19.9 (22.2) 5.9 (7.6) .47 (.48) 7.0 (3.9)
EMD (msec) EMG Quadriceps (% MVC) EMG Hamstring (% MVC) Hamstring Latency (msec) Reflex Force (N) Force Rate‡ (N/sec)
Data are reported as mean (standard deviation). * Effect size describes the difference necessary to achieve significance at ␣ ⬍ .05 with statistical power of .80 ( ⫽ .20). † Statistically significant differences between categories. ‡ Force rate was significantly different between subject groups at p ⬍ .05 with statistical power of .75.
ranging from 10msec to 121msec. The reported variability can typically be accounted for by experimental design33 and stimulation method9 (voluntary vs involuntary reaction). Asai and Aoki34 reported the only other measure of EMD in children to date, with an EMD of 77msec resulting from voluntary elbow flexion. In the present study, knee angle did not influence EMD. This finding minimizes the possibility of confounding influences from muscle-length differences between subject groups. Muscle contracture in spastic CP often causes muscles to become less extensible, with associated limitations to joint range of motion. Physical examination of our spastic participants showed that their knee extension with 90° hip flexion was limited to a mean value of 129.7° ⫾ 11.1° (95% confidence interval of ⫾6.2°) compared with normal values typically ranging from 155° to 180°.35,36 Therefore, muscle length as a percentage of the total range may have been systematically different in the spastic population than in normal populations. However, knee angle and associated muscle length did not influence EMD in either subject group. This finding agrees with Norman and Komi,6 who found that joint angle did not affect EMD. Grabiner37 reported that postures associated with shortened muscles resulted in prolonged EMD, but the trend failed to reach statistical significance Thus, results suggest that potential muscle-length differences between groups did not influence the group mean EMD values. The primary finding of the present investigation was that EMD associated with the patellar-tap reflex response was significantly shorter in patients with spastic CP than in normally
Fig 2. Reflex EMG in the quadriceps and antagonistic EMG response in the hamstrings were significantly greater in the patients with spastic CP than in the normal subjects. Peak reflex force responses were not statistically influenced by subject group.
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developing subjects (fig 3, table 1). The only other studies of EMD associated with central nervous system disorders to date were performed by Pedersen and coworkers38 and Stelmach and coworkers,39 who compared patients with Parkinson’s disease to asymptomatic adults. In both studies, the Parkinson’s group demonstrated reduced EMD compared with the normals. Pederson38 observed mean EMD in the Parkinson’s patients of 9.6msec, while Stelmach reported mean EMD values of 68msec compared with normal control values of 10.8msec and 74msec, respectively. The remarkably brief EMD reported by Pederson38 may be attributable to experimental methods (low inertial mass of the index finger from which they measured EMD involving the m. adductor pollicis, and supramaximal electrical excitation of the ulnar nerve to elicit the response). We found statistically shortened EMD in the CP population relative to age-matched normals, but the shortened EMD in the Parkinson’s patients of Pedersen38 and Stelmach39 failed to reach statistical significance. This difference in findings may point to notable differences in age of the subject populations, neuromuscular nature of the disorder (rigidity vs spasticity), and developmental adaptations in our population of children with perinatal brain injury versus those suffering from adult-onset neuropathy. Further research is necessary to quantify the unique neuromuscular behaviors that characterize developmental neuropathy versus adult-onset or traumatically induced neuropathy. Research measuring muscle force in response to sinusoidal perturbations supports our conclusion that EMD is significantly shorter in patients with spasticity. During passive flexion and extension of a joint at low frequency, the musculoskeletal system typically resists the stretch of the muscles.2 The reflex response modifies the response, increasing the resistive force at
Fig 3. EMD was significantly lower in the CP group than in the normal group.
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low frequencies of motion. As the frequency of oscillation increases, response latency causes the reflex force to appear later in the motion sequence, resulting in constructive and destructive cancellation (phase-interference) of the resistive force.10,11 Consequently, the measured resistance to motion is related to the frequency of oscillation and the reflex response latency, of which EMD is a significant component. Reduced EMD in spastic muscle suggests that the frequency of peak response must be higher in patients with spasticity than in healthy normal control subjects. Gottlieb et al2 showed that the resonance frequency in spastic muscles tended to occur at higher frequencies than normal. Results from our present study concur with previous measurements of spastic muscle behavior. The source of the reduced EMD in the CP patients may be related to a variety of physiologic and biomechanical mechanisms, including increased rate of force development, differences in fiber-type concentration, differences in fiber recruitment, increased biomechanical muscle stiffness, and increased cocontraction. On review of the data, we find that stiffness is the most likely cause of shortened EMD in the spastic muscles. Although the rate at which force develops (force rate) can affect measured EMD,33 it did not influence the statistical differences in EMD between our subject groups. The normal population had a slower reflex force rate, indicating that they required more time to reach a threshold of 5% peak force compared with the spastic population. Assuming linear behavior, the time necessary to reach the 5% threshold can be calculated from force rate and peak force response (table 1). The reduced force rate in the normal subjects contributed .99msec to the 14-msec EMD difference between groups. Relaxing the linear assumptions may increase the contribution of force rate to EMD, but the effect is negligible when compared with the statistical effect size. Clearly, the population differences in EMD were not significantly influenced by force rates. Shortened EMD in the spastic muscles cannot be explained by fiber-type recruitment patterns. The normal group responded to the tendon tap with quadriceps reflex activity at 12% of MVC, while the spastic group responded with reflex activity greater than 100% maximal voluntary isometric activation (fig 1). This maximal excitation may have recruited greater numbers of fast-twitch motor units40 in the spastic muscles, thereby accounting for the increased rate of force onset in the patient group and potentially contributing to reduced EMD.30,41-43 However, force rate did not significantly contribute to group mean differences in EMD. Thus, group differences in fiber-type recruitment most likely contributed little to the group differences in EMD. Differences in muscle fiber-type compositions in normal versus spastic muscle has been reported, but did not contribute to EMD results. Histologic assays illustrate increased concentrations of fatigue-resistant (slow-twitch) fibers in spastic muscle compared with normal.44-47 Because increased concentrations of slow-twitch fibers increase EMD,30,41-43 one would conclude that spastic patients must demonstrate abnormally elongated EMD; however, in the present study, EMD was shorter in the spastic group. Potentially increased concentrations of slowtwitch fibers in spastic muscles cannot explain our results. In fact, fiber-type characteristics may have attenuated the difference between subject groups. The most probable cause of shortened EMD in the spastic population was increased musculotendinous stiffness. The predominant component of EMD is the time required to lengthen the elastic components of the musculotendinous structures,1,7 suggesting that EMD is related to biomechanical
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stiffness. Corcos and associates33 concluded that EMD is inversely proportional to stiffness—ie, increased stiffness raises the force generated per unit of stretch, thereby requiring less time to achieve the threshold of measurable force. Experiments verify that EMD is shortened when muscle is pretensioned6,8,9 and pretensioning increases muscle stiffness.13,14,48 From this finding, it can be concluded that EMD is shortened with increased musculotendinous stiffness,33 and reduced EMD in the spastic patients may reflect this increased stiffness. This conclusion supports that of other researchers who have found increased biomechanical stiffness in spastic muscle.2,3,25,26,49 Despite this preponderance of evidence, no published reports directly link stiffness and EMD with empirical data. Shortened EMD in the spastic muscles may have been associated with either intrinsic and/or reflex stiffness. Intrinsic muscle stiffness identifies the behavior of the musculotendinous system before reflex activation50 and is related to tissue characteristics25,26 and active pretension.51,52 Research2,25,26 suggests that intrinsic muscle stiffness is abnormally increased in spastic muscle. Reflex activation adds to the mechanical stiffness,53 contributing up to 40% of the total stiffness in isometric conditions.54 Our results illustrate significantly increased reflex activation in the spastic population (table 1). Shortened EMD in the spastic population may be related to either intrinsic or reflex stiffness, or both. Distinguishing between intrinsic or reflex stiffness contributions to EMD cannot be ascertained from the data. Patients with spastic CP had significantly greater levels of normalized hamstring activity, ie, antagonism, in response to the patellar tendon tap than the normal group (fig 2). Coactivity measured in the hamstrings may result from reciprocal excitation or from a remote possibility of cross-talk radiating from the quadriceps muscles. Myklebust and colleagues32 observed simultaneous activity in both the flexor and extensor muscles in response to a myotatic stretch stimulus about the ankle, particularly in patients with perinatal spastic syndrome. They concluded that the pathologic spinal circuitry promoted reciprocal excitation rather than reciprocal inhibition typically demonstrated in normally developing subjects. Our results showed similar trends. Hamstring reflex response at 58% of MVC occurred simultaneously with the quadriceps activity in the spastic group. Although the normally developing subjects also demonstrated hamstring activation, the response was attenuated, 12% MVC, and significantly delayed relative to the quadriceps activity by 10.5msec. Low-amplitude cross-talk from the quadriceps was observed in the hamstring signals in some of the normal subjects. However, it is unlikely that measured hamstring activity in the spastic group could be explained from quadriceps cross-talk given the excessive magnitude of the myoelectric response. Thus, our results are similar to others who have observed antagonistic activity and reciprocal excitation in patients with spastic CP. Antagonistic reflex coactivation may have attenuated the group mean difference in EMD. Muscle stiffness increases in the presence of antagonistic cocontraction because of the balance of forces and the relation between muscle force and stiffness.55-57 However, antagonistic contraction in the hamstrings will reduce the extensor moment measured at the load cell, thereby delaying the onset of measurable force and elongating the EMD (see Appendix). For these reasons, antagonistic cocontraction fails to explain the reduced EMD in spastic muscle behavior. Results may have been limited by the relative uncontrolled nature of the tendon-tap stimulus. Considering the reduced Arch Phys Med Rehabil Vol 81, July 2000
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voluntary motor control available to the CP patients, we chose to examine the contraction behavior associated with involuntary reflexes. It was impractical to employ a constant tendon tap force because the stimulus threshold varied from one subject to the next. Because some evidence exists that the tendon-tap stimulus strength may influence the response magnitude,58,59 one might be tempted to suggest that the magnitude of the stimulus may have confounded our EMD results. Was shorter duration EMD associated with increased force of impact? On the contrary, we observed that reflexes were elicited in the spastic patients with very light impact, while most of the normal subjects required significant effort to achieve a response. If the variability in stimulus magnitude has influenced our results, it attenuated the differences between groups. Another potential limitation to the study was the use of voluntary maximal exertions to normalize reflex EMG data. Variability in the level of effort during the MVC exertions may contribute to variability in the EMG, but did not influence EMD data. As an example, it is unlikely that the reflex response in the spastic group was truly supramaximal, ie, 104%, indicating that these patients generated MVC data less than maximum. However, all the subjects—both CP and healthy normal control groups—understood the instructions and were capable of performing this isometric task. The MVC levels were determined from a series of 3 exertions in each direction, 3 knee flexions, and 3 knee extensions, to reduce error. We are confident that the mean reflex response in the spastic group, a value more than twice the level demonstrated by normally developing subjects and statistically greater ( p ⬍ .005), cannot be explained by variability in the MVC trials. Furthermore, the MVC exertions did not influence the primary focus of this effort, because EMD was abnormally lower in the population of children with spastic CP. Experimental results supported our hypothesis that EMD was shorter in patients with spastic CP than in age-matched normals. The relation between musculotendinous stiffness and EMD suggests a potentially causative effect, and helps to explain published results of others who have measured abnormal stiffness behavior in spastic muscles. Future research will validate the biomechanical cause of reduced EMD in spastic patients and investigate the clinical applicability of using EMD when assessing spasticity. CONCLUSIONS EMD in a population of children with spastic CP was significantly shorter than in normally developing children. The magnitude of the myoelectric reflex response and reciprocal excitation were also significantly different between groups, but the force responses were not statistically different. Group mean differences in rate of force development, muscle fiber-type concentration, and fiber recruitment failed to explain the differences in EMD. Reciprocal excitation of antagonistic cocontraction was uniquely observed in the spastic group, but could not account for their reduced EMD. Results suggest that abnormally high biomechanical stiffness in the musculotendinous unit was the primary factor contributing to reduced EMD in spastic muscle. Further research is necessary to investigate the clinical applicability of using EMD to quantify stiffness in spastic muscle and to evaluate the progression of pathologic stiffness. Acknowledgment: The authors thank Ms. Traci Martellotta for her assistance in the acquisition and processing of the data. Arch Phys Med Rehabil Vol 81, July 2000
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APPENDIX To illustrate the influence of antagonistic cocontraction on EMD, a simple model of muscle contraction has been developed. For the purposes of the discussion above, the muscle can be modeled as a contractile element in series with a stiffness element, ie, a spring. Assuming the series elastic element behaves linearly over the range of interest, the quadriceps force, FQ, can be represented as FQ(t) ⫺ FQ(t0) ⫽ kQL˙t where L˙ is the rate of contraction in the contractile element, FQ (t0 ) is the muscle tonic level before the contraction, and t is time. Muscle stiffness is linearly related to the tension,60 and can be described as the product of FQ (t) and a proportionality coefficient q. The system response to a constant rate of contraction L˙ can be described as FQ(t) ⫽
FQ(t0) 1 ⫺ qL˙t
In the absence of antagonistic cocontraction, the quadriceps force, FQ, develops at a rate inversely proportional to time (fig 4). Antagonistic contraction can be represented as a force in the hamstrings, FH, proportional to the tension in the quadriceps muscle FH ⫽ FQ(t) Arch Phys Med Rehabil Vol 81, July 2000
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where is less than 1 and greater than zero. The hamstring force acts in opposition to the quadriceps force, and the equilibrium relation now becomes FQ(t) ⫺ FQ(t0) ⫽ kQL˙t ⫺ FQ(t) Solving for the response in the presence of antagonistic cocontraction yields FQ(t) ⫽
FQ(t0) 1 ⫺ qL˙ (1 ⫺ )t
The response of the antagonistic system behaves similarly to the unopposed response, but the force rate is reduced by the (1 ⫺ ) coefficient (fig 4). If the threshold for force detection is set at a constant value of FT, then the EMD in the antagonistic system is greater than in the unopposed system. The description provided in this analysis is simple and limited by the assumption of equivalent muscle contraction rates, L˙, in the agonist and antagonist muscle groups. This simple analysis can be expanded to include more biomechanical realism, but research61 suggests that the response of the model
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Fig 4. Muscle force response as modeled in the appendix, representing the time-dependent behavior without antagonism, and the behavior including antagonistic contraction at a level of ⴝ .25. EMD with antagonistic activity, EMD, is longer than without, EMD0.
to antagonistic cocontraction will continue to demonstrate slower response times and increased EMD. Thus, the increased antagonistic activity seen in the spastic population cannot explain their abnormally reduced EMD.