Prediction of Stroke Motor Recovery Using Reflex ... - IEEE Xplore

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IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 20, NO. 6, NOVEMBER 2012

Prediction of Stroke Motor Recovery Using Reflex Stiffness Measures at One Month Mehdi M. Mirbagheri, Member, IEEE, Xun Niu, and Deborah Varoqui

(Invited Paper)

Abstract—This study characterizes the recovery patterns of motor impairment after stroke, and uses neuromuscular measures of the elbow joint at one month after the event to predict the ensuing recovery patterns over 12 months. Motor impairment was assessed using the Fugl–Meyer Assessment (FMA) of the upper extremity at various intervals after stroke. A parallel-cascade system identification technique characterized the intrinsic and reflex stiffness at various elbow angles. We then used “growth-mixture” modeling to identify three distinct recovery classes for FMA. While class 1 and class 3 subjects both started with low FMA, those in class 1 increased FMA significantly over 12-month recovery period, whereas those in class 3 presented no improvement. Class 2 subjects started with high FMA and also exhibited significant FMA improvement, but over a smaller range and at a slower recovery rate than class 1. Our results showed that the one-month reflex stiffness was able to distinguish between classes 1 and 3 even though both showed similarly low month-1 FMA. These findings demonstrate that, using reflex stiffness, we were able to accurately predict arm function recovery in stroke subjects over one year and beyond. This information is clinically significant and can be helpful in developing targeted therapeutic interventions. Index Terms—Fugl–Meyer Assessment (FMA) score, growth mixture model, motor impairment, neuromuscular properties, stroke.

I. INTRODUCTION

M

OTOR impairments following stroke vary widely and are caused by disturbances in descending commands and/or by changes in spinal neuron excitability [1]–[4]. These mechanisms include deficits caused by the direct effects of a decreased excitatory synaptic drive on neuromuscular properties, and also by abnormal changes in neuromuscular properties secondary to stroke [4], [5]. Therefore, the development of motor impairments and/or the progress of stroke recovery might be due to simultaneous changes in these two conflicting mechanisms i.e., brain recovery and secondary effects on neuromuscular properties. Consequently, understanding the contributions Manuscript received December 16, 2011; revised March 15, 2012; accepted May 09, 2012. Date of publication August 01, 2012; date of current version November 02, 2012. The authors are with the Sensory Motor Performance Program, Rehabilitation Institute of Chicago, Chicago, IL 60611 USA, and also with the Department of Physical Medicine and Rehabilitation, Northwestern University, Chicago, IL 60611 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TNSRE.2012.2205943

of these secondary effects to motor impairment is clinically significant, as it provides a basis for the development of effective interventions [6]. There are conflicting views in the literature regarding the relationship between neuromuscular abnormalities and motor impairment. Some studies, for example, have shown a link between spastic hypertonia and disturbed function [7]–[9]. In particular, it has been shown that reducing hypertonia by therapeutic intervention improved function [6], [10]–[14]. Other studies based on clinical observations [15], [16] dispute this relationship, noting that attenuating the hyperexcitability of reflexes does not always promote functional improvement. We believe that these controversies arise mainly from a lack of accurate methods to quantify spasticity and to separate the reflex and intrinsic torques associated with it. Spasticity, which occurs in approximately 40% of the stroke population [17], is manifested as a mechanical abnormality. Our earlier studies have demonstrated no link between clinical measurements of spasticity and the intrinsic and reflex mechanical abnormalities associated with spasticity [18]–[20]. This is supported by more recent studies [21], [22], some of which have even called for an end to the use of some of these clinical measures [21]. In this study, we aimed to address this important question by 1) assessing motor impairment recovery, 2) quantifying neuromuscular abnormalities, and 3) exploring the relationship between these two. First, motor impairment was assessed using the Fugl–Meyer Assessment (FMA) score, shown to be a reliable and valid indicator of stroke recovery [3] particularly for the assessment of upper extremities [23], [24]. The “growth mixture model” was used to track the recovery of FMA over a period of 12 months after stroke. We postulated that there are distinct patterns (termed “classes”) of FMA improvement during stroke recovery. Neuromuscular abnormality associated with upper extremity spasticity was then characterized using a novel system identification technique, based on measurements taken at one month post-stroke. In our earlier study, we used this technique to successfully separate the neural (reflexive) and muscular (intrinsic) components in different spastic populations, including stroke survivors [19], [20], [25], [26]. Finally, the effects of neuromuscular measures at one month post-stroke on FMA recovery class membership was explored using logistic regression [27], [28]. We hypothesized that the trajectory of motor impairment recovery for each patient could

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MIRBAGHERI et al.: PREDICTION OF STROKE MOTOR RECOVERY USING REFLEX STIFFNESS MEASURES AT ONE MONTH

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be predicted based on neuromuscular measures (particularly reflex stiffness) measured one month after the stroke event. II. METHOD A. Subject Recruitment Twenty-eight hemiparetic stroke survivors were recruited within four weeks of the stroke event; however only 21 (10 female and 11 male) patients ( years) completed the study. Subjects had left- (11) or right-sided (10) lesions. Fourteen sustained primary intracerebral hemorrhage, and seven sustained ischemia. Patients met the following criteria: 1) first stroke, 2) aged 30–80 years, 3) absence of aphasia or cognitive impairment, 4) no deficits in the nonparetic arm, and 5) no major sensory deficits in the paretic arm (as evidenced by an absence of detectable sensory loss on a standard neurological exam). Subjects were assessed prior to each experiment using the modified 6-point Ashworth scale (MAS) for spasticity; only subjects whose elbow flexors had were included [29], [30]. All subjects gave informed consent to the experimental procedures, which were approved by our institutional review board. All subjects were drawn from the inpatient service of the Rehabilitation Institute of Chicago (RIC), and represent a convenience sample of stroke patients at the RIC. B. Motor Recovery Assessment Motor recovery of stroke survivors was evaluated using the FMA for upper-extremity motor function [31], [32], a validated and popular clinical assessment. The FMA evaluation was performed by a physical therapist at the beginning of each experiment. Experiments were repeated at five intervals following stroke onset (i.e., at 1-, 2-, 3-, 6-, and 12-months post-stroke). The effects of recovery time on FMA were assessed using the “growth mixture model” as described below. C. Experimental Apparatus Subjects were seated in an adjustable chair with the forearm secured to a rigid beam via a custom-fitted fiberglass cast. This cast was mounted to the shaft of a position-controlled servo motor, which drove the elbow position to follow a commanded input (Fig. 1). Elbow position and angular velocity were recorded with a potentiometer and tachometer, and torque was recorded using a torque transducer mounted between the beam and the motor shaft. Shoulder abduction was 80 and the elbow axis of rotation was aligned with the axis of the torque sensor and the motor shaft (see [20]). The elbow position where the upper arm and forearm were perpendicular was defined as the neutral position (NP) or zero degrees. Electromyograms (EMGs) were recorded at the short head of biceps, brachoradialis, and triceps long and short heads, using bipolar surface electrodes. All signals were sampled at 1 kHz and low-pass filtered at 200 Hz to prevent aliasing. D. Experimental Procedure We perturbed the paretic elbow joint using pseudorandom binary sequence (PRBS) position inputs with an amplitude of 0.03

Fig. 1. The apparatus, including the joint-stretching motor device, the height adjustable chair, and force and position sensors.

rad (2 ) and a switching-rate of 150 ms. This input contained power over enough bandwidth to identify the reflex and intrinsic dynamic stiffness [33]. PRBS trials were performed at elbow positions ranging from near-full flexion to near-full extension at 15 increments. All measurements were taken while the subject was relaxed; this was verified using the EMG signals. E. Growth Mixture Model The latent class growth (LCG) model [34]–[36] (also called the Semiparametric Bayesian latent trajectory model [34]–[37]) was applied to all subjects to classify the FMA growth patterns during the 12-month recovery period. In our earlier study, we demonstrated the validity of this model [38]. The LCG model assumes that the population can be divided into a finite number of latent classes (homogeneous inter-subject subpopulations) by inspecting the intra-subject changes. The membership of subjects into classes can be associated with continuous or discrete baseline factors (latent variables). The Bayesian information criterion (BIC) was used to decide the number of latent classes in the LCG model [35], [36], and to compare linear and nonlinear random coefficient regression (RCR) models for the growth pattern of each latent class [34]. A linear RCR and a Michaelis–Menten Model [39] with a nonzero intercept were fit to the growth pattern for each latent class. The model fit which had the smallest BIC among all the others was considered as the best model. Linear RCR Model: The linear RCR model [39] is specified by (1) where is an index for the subject number, is an index for the month post-stroke, corresponds to the latent class number, is the month post-stroke, and is the random residual of the model. Parameters, and are considered for all subjects in subclass , while and vary with subject and time point . Model (1) is similar to a traditional linear regression model except that individual difference is permitted;

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i.e., intercept and slope in each latent class can vary for each individual . Michaelis-Menten Model: The Michaelis–Menten Model [34], [39] was originally proposed to study the kinetics of enzymatic reactions, and has been widely used in cell biology and physiology to examine saturation curves that exhibit a limiting plateau. Since the FMA recovery patterns showed a nonlinear behavior with major changes over the first few months and more modest improvements over later months, consistent with the recovery of stroke found by others [40], we expected the Michaelis–Menten Model (2) to describe the recovery patterns of FMA over one year. The Michaelis–Menten Model with a nonzero intercept is defined as (2) repwhere is the maximum recovery possible, and resents the asymptotic limit that the FMA score can achieve during post-stroke recovery. The constant is the time at which the FMA score has improved to half of the maximum possible improvement (i.e., half of ).

Fig. 2. The FMA score as a function of post-stoke recovery time. The membership of subjects was classified into three latent classes based on the LCG model. The FMA scores were averaged across subjects in the same latent class for each month; the recovery pattern for each latent class was estimated by the standard errors are shown in the figure. Michaelis–Menten Model. Averages

F. System Identification Modeling The neural and muscular contributions to the elbow stiffness were identified using a parallel cascade system identification model comprised of two pathways, i.e., reflex and intrinsic (see details in [33] and [41]). This parallel cascade system identification model has been used repeatedly in many studies, and has been shown to be both reliable and valid [18], [20], [33]. In the intrinsic pathway, the intrinsic stiffness dynamics were estimated in terms of a linear impulse response function (IRF), relating position and torque over a broad range of frequencies. Intrinsic torque was predicted using the intrinsic IRF, and subtracted from the recorded torque to provide the residual or error torque. Reflex dynamics were estimated as a pathway comprised of a delay, a differentiator, a half-wave-rectifier, and linear dynamics, using a Hammerstein identification method [42]. Reflex stiffness dynamics were estimated by determining the IRF that relates the half-waved rectified velocity as the input to the residual torque as the output. The reflex torque was then predicted using the reflex IRF. Linear models were fitted to the estimated intrinsic and reflex IRF curves using the Levenberg–Marquardt nonlinear leastsquares fitting algorithm [43]. The intrinsic and reflex stiffness gains were derived [20], [33], and used to explore the position dependence of mechanical abnormalities associated with spasticity. G. Prediction of FMA Recovery Patterns From Neuromuscular Properties To explore the possible relationship between neuromuscular measures and the FMA, the relationship between the modulation of intrinsic stiffness and reflex stiffness and changes in elbow angle was evaluated at one-month after stroke. Linear regression lines were fit to the intrinsic and reflex stiffness versus elbow angle plots and the slope and intercepts calculated. Our

earlier results showed that both and were strongly position-dependent and that this dependency was broadly similar in all subjects [20], [25]. Particularly, both and increased progressively with elbow extension from full flexion to full extension. However, the slope of changes in and with increasing elbow angle as well as the intercept varied considerably among subjects [25], [26]. These linear parameters were used to explore a possible relationship between neuromuscular measures at one month after stroke and FMA recovery patterns over 12 months of recovery. The four stiffness parameters (i.e., intercept and slope of the linear fits of and as a function of elbow angle) were inspected by multinomial logistic regression to identify which were significant predictors for class membership. That is, the ability of these month-1 neuromuscular properties to predict changes in FMA was assessed. A was considered significant for all statistical tests. III. RESULTS A. Recovery of Motor Impairment We studied the progression of changes in the FMA of the upper extremity on the paretic side at five different time points over the year post-stroke, and observed that recovery occurred along three distinct patterns. Thus, the LCG model defined three classes of subjects according to the change in FMA with time. Each subject was assigned to the particular class for which that subject had the highest probability of membership. This resulted in class counts of 6 for class 1, 9 for class 2, and 6 for class 3. Fig. 2 shows the observed and estimated mean FMA for these classes. For class 1, the FMA score at the first month was between 4 and 26, with a median of 6. Four subjects presented improvement in their FMA scores of more than 40 points between the

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PREDICTION

OF THE

TABLE I LATENT SUBCLASSES OF FMA WITH TIME FROM STIFFNESS MEASUREMENTS, AT BASELINE BASED ON MULTINOMIAL LOGISTIC REGRESSION

first and twelfth months. At the twelfth month, the median FMA score was 37, while the maximum was 60. The detailed statistical information for the LCG modeling is described in Table I. The progression in FMA score with time found that the Michaelis–Menten nonlinear model provided a better fit than the linear RCR model for this class; the Michaelis–Menten model explained 89% of the variance, compared to the 65% explained by the linear RCR model. The coefficients involved in the Michaelis–Menten Models were significant . The Michaelis–Menten model indicated the maximum FMA was 60.5 (see Fig. 2, and Table I), half of the maximum improvement possible was reached at month [the value of in (2)], and the starting FMA at the first month was 6.1. All the estimated values based on the model were consistent with the observed FMA scores. For class 2, the FMA score at the first month was between 35 and 62, with a median of 48. All subjects showed an improvement of less than 20 points over the 12-month recovery period.

,

AND

At the twelfth month, the FMA score had a median of 64 (compared to 66, the maximum value that FMA can achieve for the upper extremity), indicating near-complete recovery. As with class 1, the Michaelis–Menten model fit the class 2 data better (88% of the variance explained) than the linear RCR model (41% explained). The coefficients of the Michaelis–Menten Models were significant . The Michaelis–Menten model indicated that the maximum FMA was 65.3, half of the maximum improvement was reached at months, and the starting FMA (at one month post-stroke) was 42.4. The estimated values from the model agreed with the observed FMA scores. For class 3, all subjects had an FMA score of 4 at the first month. Three subjects did not change their scores over the 12 months, one subject improved by one point at the second month and stayed constant for the rest of the 12-month period, and two subjects started presenting an FMA increase at the sixth month and reached FMA scores of 14 and 47 after 12 months. The

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linear RCR model showed that the slope was not significantly different from zero but a significant intercept of 6.5 . As the slope was horizontal (no significant recovery), the Michaelis–Menten model was not fit to this class. Two-way Random Measures ANOVA with categorical factors of Time (five levels) and Class (three levels) confirmed that the interaction of time and class had a significant effect on the FMA . FMA changed significantly with time in Class 1 and Class 2 , but FMA did not change with time in Class 3 . This is consistent with our findings using the growth pattern modeling. Multiple comparisons (post hoc tests with Tukey adjustment) showed that the FMA in Class 2 was higher than the FMA in both Class 1 and Class 3 at each month . The FMA score in Class 1 was higher than in Class 3 at months 6 and 12 , but the FMA in Class 1 was not significantly different from FMA in Class 3 at months 1, 2, and 4 . The saturation curve of FMA score with time has never been modeled in statistics. This study found the Michaelis–Menten model fit the growth pattern for Classes 1 and 2 well, both of which showed significant improvement in FMA score with time. The parameters in the Michaelis–Menten model can be potentially used to predict the maximum recovery extent and speed .

Nm.s/deg ), while those with below this range were members of class 1 and those with above this range belonged to class 3. The statistical tests did not find that either or was a significant predictor for class membership . We then combined each pair of classes (1-2, 2-3, or 1-3) into one new class while using the third class as a reference (i.e., 3, 1, or 2, respectively), and used binary logistic regression to inspect the potential effect of and on this two-group classification. Neither of the stiffness parameters could predict membership of subjects into two classes . C. Prediction of Motor Impairment Recovery As demonstrated in Table I, our logistic analysis showed that the and at one month was a significant predictor for the subclass membership of FMA scores over the 12-month recovery period. The information has clinical significance in that it can be used to predict recovery in terms of FMA, particularly for those subjects with a low FMA score at one month (Classes 1 and 3, Fig. 3). That is, reflex stiffness measures at 1 month can be used to distinguish these recovery patterns, and consequently, can be good predictors of motor impairment recovery in stroke. D. Relation Between FMA and Patient Clinical Characteristics

B. Relation Between Neuromuscular Measures and FMA Score Fig. 3 shows the trend in modulation of reflex and intrinsic stiffness as a function of elbow angular position over the ROM for a typical stroke subject. Both and were described well by a first-order linear model (as indicated by the superimposed solid lines); these results were consistent among the subjects ( for and for ). The parameters of the first-order linear model—i.e., Reflex Intercept , Reflex Slope , Intrinsic Intercept , and Intrinsic Slope —were assessed for their ability to predict recovery class membership, with Class 3 regarded as the reference group. The estimated coefficients for for both Classes 1 and 2 were negative (Table I), indicating that the logit and thus likelihood of Class 1 and Class 2 membership increased when decreased. The logistic analysis showed that at the first month was a significant predictor for FMA recovery class membership. Subjects with between 1 and 1.6 Nm.s/rad (57.3 to 91.7 Nm.s/deg) were more likely to belong to class 2, while subjects with below this range were more likely to belong to class 1 and subjects with above this range were more likely to belong to class 3. The detailed statistical information for the logistic regression analysis is described in Table I. Similarly, the multinomial logistic analysis of showed that the estimated coefficients for classes 1 and 2 were also negative (Table I), indicating that the probability of class 1 and 2 membership increased as decreased. Based on our analysis, measured at the first month was a significant predictor for FMA class membership. Thus, subjects in class 2 had a between 0.81 and 1.34 Nm.s/rad (2659–4399

We further expanded the logistic regression analysis to explore the effects of gender, stroke side, stroke type (ischemic or hemorrhagic), and degree of spasticity (evaluated using the MAS at one month post-stroke) on class membership, using Class 3 as the reference. Our results indicated that none of these patient clinical characteristics was a significant predictor NS for FMA recovery. IV. DISCUSSION A. Motor Impairment Recovery In our earlier study [44], we found two distinct classes of motor recovery. Class 1 started with low FMA values at one month and increased over time, whereas class 2 started with higher values at one month but did not change significantly with time [44]. These findings were interesting and provide several major advances over previous longitudinal studies including 1) tracking FMA for a longer period of time after stroke [23], [45], [46], 2) distinguishing different recovery patterns of FMA rather than a single pattern for all patients, and 3) identifying major kinematic and kinetic measures at one month after stroke as significant predictors for these recovery patterns. However, with only two classes of recovery, we were not able to distinguish between two groups of stroke patients who had a low FMA score at one month post-stroke but presented different recovery patterns: one group presented an improvement over a one-year recovery period (class 1), while the other group exhibited no significant improvement. Our current study addressed this deficit by subdividing these two groups further. Thus, using a new growth mixture model, we identified three classes of motor recovery. Classes 1 and 3 both started with a low FMA at one-month post-stroke but class 1 improved significantly over recovery


Fig. 3. Reflex and intrinsic intercept and slope as a function of elbow position for a typical stroke patient in this study.

time, whereas class 3 did not present any significant improvement. Class 2 started with a high FMA score and showed moderate improvement over one year. B. Motor Impairment and Reflex Stiffness The overall aim of the current study was to determine whether neuromuscular measures at the early stages after stroke are good predictors of motor recovery over a longer period. Particular attention was given to reflex stiffness, which can abnormally increase shortly after stroke [25], [26]. As demonstrated in Table I, our logistic regression analysis shows that we can predict the class membership of the FMA over one year after stroke, based on reflex stiffness measures, i.e., , and , measured within the first month. Recall that subjects in Class 1 and Class 3 could not be differentiated into recovery pattern classes by the baseline FMA scores alone. This indicates these reflex stiffness measures are quantitative predictors of FMA recovery.

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Our findings also address an important controversy regarding the relationship between spasticity reflex and motor impairment. One view in the literature has demonstrated a relationship between spastic hypertonia and disturbed function [7]–[9], whereas the other view, based on clinical observation [15], [16], does not support this relationship. Our results reject the latter view by demonstrating that reflex stiffness serves as a good predictor for motor impairment recovery in stroke. We believe that these controversies mainly arise from a lack of accurate tools for quantifying spasticity and for separating the reflex and intrinsic torques associated with it. Spasticity is manifested as a mechanical abnormality, but is clinically evaluated by the Ashworth scale, which has been shown to have no link to the intrinsic and reflex mechanical abnormalities associated with spasticity [18]–[20]. Developing and using a novel system identification model enabled us to quantify the mechanical abnormality and separate its intrinsic and reflex components [20], [25], [33]. Our quantification of the reflex stiffness proves to be useful to discriminate different recovery patterns at one month post-stroke. It seems that with a comparable FMA score, patients presenting a lower reflex stiffness at one-month have a higher potential recovery gain. We also believe that these controversies could be due to the fact that these studies have investigated different features of reflex behavior. Since stroke can trigger several modifications to the neuromuscular system that profoundly influence voluntary movement, it is important not only to examine the magnitude of reflex stiffness but also to study abnormal modulation of reflexes during function. However, due to technical limitations, it is currently impractical to perturb the joints during spontaneous movement and obtain valid data. Therefore, we are obliged to perform these measurements with subjects attached to the testing device, and then to explore the relationship between the abnormal modulation of reflex stiffness versus joint angle (represented by and ) with FMA recovery patterns. represents an indirect measure of reflex threshold, while quantifies the abnormal modulation of reflex stiffness as a function of joint angle. C. Prediction of Motor Recovery—Implications for Rehabilitation A better understanding of the contributions of different impairment mechanisms at different times after stroke is a prerequisite for the development of effective therapies. This understanding can be achieved, in part, by determining the recovery patterns of motor impairment after stroke. Our findings are the first to identify several classes of recovery for FMA over 12 months post-stroke and to predict the recovery patterns even beyond 12 months. Our results demonstrate three FMA recovery patterns over one year. The FMA score at one-month post-stroke showed two distinct levels of upper limb motor impairment in our group of hemiplegic patients: patients with mild to moderate motor impairment (class 2, ) and patients with severe motor impairment (classes 1 and 3, ). However, the low one-month FMA scores for patients with severe motor impairment (class 1 versus class 3) limit our ability to distinguish recovery patterns based on the FMA score alone. However, our

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quantification of the reflex stiffness proves to be useful to discriminate between these two classes at one month post-stroke. It appears that, between patients with a comparable FMA score, those presenting a lower reflex stiffness at one month have a high chance of motor recovery but those with a high reflex stiffness initially may not obtain a significant motor improvement. This early differentiation between the two recovery patterns for post-stroke patients with severe impairments is crucial to determine the most beneficial therapies for the patient. Indeed, research on neurologic rehabilitation has not yet established which types of physical therapy lead to better intrinsic recovery of the upper extremity. There is a conflict in the literature regarding whether promoting motor recovery of normal patterns of movement is more efficient than orienting the patient to a motor compensation strategy, i.e., development and reinforcement of new patterns of movements based on patient capacity [47]–[49]. It seems that when a patient suffers from severe impairments, physical therapies are often more oriented to develop compensation strategies than to work on the recovery of “normal” patterns of movement [50]. However, compensation strategies can limit functional gain compared to recovery [49], [51], [52]. On the other hand, a therapy based on the restoration of a normal pattern of movement in a patient with severe deficits could limit or delay his ability to develop autonomy [49]. Thus, the therapy provided to the post-stroke patient should be appropriate to the degree of motor impairment. Based on our results, more accurate quantification of the level of impairment could be made using the parallel-cascade identification technique to assess the reflex stiffness as a complement to the FMA. Assessment of the reflex stiffness early after stroke (i.e., one month), combined with the FMA score, could improve the prognostic power of upper-limb motor impairment in stroke patients—particularly for patients presenting severe impairment. Moreover, this quantification of neuromuscular properties would be helpful to determine an appropriate physical therapy regimen for the patient (i.e., restoration towards a normal movement pattern or reinforcement of a compensation strategy). Our results indicated a nonlinear pattern of recovery for the FMA for patients in classes 1 and 2. This is consistent with data reported by others [23], [24], [53], indicating that major recovery of arm motor functionality can be obtained during the first few months after the event. Both of these classes showed significant recovery over a 12 month period. However, the fastest improvement occurs in patients who present lower FMA scores and low values of reflex stiffness at one month (i.e., class 1). A comparison of class 1 and class 2 showed that subjects in class 1 exhibited an improvement of 200%–520% in the FMA score over their one-month score over a 12-month recovery period, whereas for class 2 FMA improvement peaked at 25%, which occurred after 12 months. The rate of FMA improvement (calculated by differentiating (2) and evaluating over the range 1–12 months) showed that while the recovery rate for class 1 was 32% lower than for class 2 at month 1, it was consistently higher after two months of recovery, with the recovery rate for class 1 at 12 months being 4.5 times the rate for class 2. In other words, subjects following the class 1 recovery pattern will, after two or more months, experience a greater FMA improvement per month than subjects in class

2. Furthermore, major improvements in class 2 scores were limited to the first 2–3 months after stroke, whereas class 1 presented progressive recovery over the entire 12 months. These findings indicate that subjects with both severe motor impairment and low reflex stiffness at the early stages of stroke had a higher chance of motor recovery (relative to their initial state) than subjects with minimal or mild motor impairment. Thus, immediate and intensive appropriate physical interventions may have substantial impact on the recovery of stroke survivors who show class 1 features. This recovery potential remains to be validated. Our analysis was based on a relatively small sample (N = 21) which is a limitation of the study. A larger stroke population will be needed to confirm the potential clinical applications of these findings. D. Methodological Considerations Post-stroke recovery has been found to be heterogeneous [54], i.e., there is a considerable degree of individual variability in recovery patterns [54], [55]. Therefore, averaging across the total stroke population may not appropriate to adequately characterize the motor recovery patterns over time. Classifying stroke patients into classes and modeling the growth pattern for each class thereafter can play an important role in the appropriate evaluation of clinical interventions. In the current study, it was shown that the latent class growth and random coefficient regression models were able to distinguish the recovery pattern of patients over a one-year period after stroke; these models can make predictions about the longitudinal improvement in motor function. Floor and ceiling effects are always present in the assessment of neurological recovery. It has been noted that the FMA scale is subject to a ceiling effect for the hand and leg [8], [10]. To provide powerful discrimination of the clinical changes following stroke, it has been recommended that other more sensitive assessments, e.g., the Chedoke–McMaster Disability Inventory, be used together with FMA once FMA reaches the recovery plateau [54], [56]. On the other hand, the magnitude of the recovery plateau and the time that it takes to reach the recovery plateau require exact and unambiguous definitions, because they might vary among patients. To fulfill such a goal, a growth model should have the capability to capture the ceiling effect of the assessment. To the best of our knowledge, the Michaelis–Menten equation has not been introduced into any rehabilitation studies to model this effect. The Michaelis–Menten equation was originally used to model the rate at which enzymatic reactions occur based on the concentrations of the chemical reagents [39]. The analogy in the nonlinear growth pattern between the stroke recovery accessed by FMA and enzymatic reactions encouraged us to introduce this type of modeling to the context of stroke. We attempted to inspect the statistical significance of the individual parameter estimates rather than just performing a simple curve fit. Based our results of this study, we can conclude that the Michaelis–Menten equation can be used to model the FMA over time once patients have been stratified into several latent classes according to their growth pattern. In recovery classes in which improvement was statistically significant (classes 1 and 2), the Michaelis–Menten equation provided


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