Characterization of the Stroke-Induced Changes in the Variability and Complexity of Handgrip Force

Introduction: The variability and complexity of handgrip forces in various modulations were investigated to identify post-stroke changes in force modulation, and extend our understanding of stroke-induced deficits. Methods: Eleven post-stroke subjects and ten age-matched controls performed voluntary grip force control tasks (power-grip tasks) at three contraction levels, and stationary dynamometer holding tasks (stationary holding tasks). Variability and complexity were described with root mean square jerk (RMS-jerk) and fuzzy approximate entropy (fApEn), respectively. Force magnitude, Fugl-Meyer upper extremity assessment and Wolf motor function test were also evaluated. Results: Comparing the affected side with the controls, fApEn was significantly decreased and RMS-jerk increased across the three levels in power-grip tasks, and fApEn was significantly decreased in stationary holding tasks. There were significant strong correlations between RMS-jerk and clinical scales in power-grip tasks. Discussion: Abnormal neuromuscular control, altered mechanical properties, and atrophic motoneurons could be the main causes of the differences in complexity and variability in post-stroke subjects.


Introduction
The generation of human movement involves the activation and modulation of muscle force, and these functions are fundamental to performing daily activities. The force production capacity of muscles is an important indicator of motor function. In addition to force production capacity, the ability to modulate and sustain force at certain levels is also critical in daily use of the motor system. During the most-used upper extremity activity, reaching-to-grasp, for example, the ability to grip an object is the mark of the maturation of human motor behavior, requiring the force to be generated precisely at the safety margin predetermined by feedforward modulation [1,2].
Voluntary sensorimotor control function is deteriorated in most stroke survivors. After stroke, impairments such as spasticity [3,4], muscle weakness [5][6][7], increased reaction time [8], co-contraction [9], and contracture [10] lead to motor control dysfunction in patients [11,12]. The dysfunctions have been commonly found to impair force modulation during gripping. Strength is widely used as a measure of handgrip dysfunction. Boissy et al. [6] characterized upper extremity dysfunctions with maximal handgrip strength. Ada et al. [7] claimed that handgrip strength assessment was capable of directing interventions for improving muscle activity and avoiding spasticity. Entropy 2018, 20, x 3 of 10 1000 Hz. Subjects were asked to sit and grasp the dynamometer on the table with their thumb and four fingers on opposite sides. During the experiment, their shoulder should be adducted at approximately 15°-20° of flexion, and their elbow at 90° of flexion.  As seen in Figure 1b, in order to prevent forearm motion, all subjects' forearms were constrained with a belt during power-grip tasks. First, subjects were asked to generate a maximal grip force (MGF) for 5 s three times when the indicator was lit. From the three trials, the largest value of the As seen in Figure 1b, in order to prevent forearm motion, all subjects' forearms were constrained with a belt during power-grip tasks. First, subjects were asked to generate a maximal grip force (MGF) for 5 s three times when the indicator was lit. From the three trials, the largest value of the MGF was obtained for the normalization of grip force during the next submaximal force level tasks. Then, each subject was asked to begin three different kinds of submaximal grip force control Entropy 2018, 20, 377 4 of 11 tasks (power-grip tasks; 25%, 50%, and 75% of MGF). A computer screen was provided to display real-time visual feedback (Figure 1a). On the screen, 25%, 50%, and 75% of the MGF were represented by three stationary horizontal red lines, and the actual force level was represented by a movable horizontal blue bar. The subjects should generate a suitable grip force to make the blue bar reach the red target line and persist for 5 s, during which the interface fed back the errors for grip force control. Each force level was performed three times, and to minimize fatigue, a 30 s rest was allowed for each subject after each trial. In stationary holding tasks, subjects were asked to grip the dynamometer at a height of 20 cm above the desk, and persist for 8 s. The stationary holding task was performed by each subject three times, and a 30 s break between each of the two trials was provided to avoid fatigue. All of the tests were conducted on the affected side of post-stroke subjects, and on the dominant side of the controls. All software used in the two tasks was programmed using LabVIEW (LabVIEW 2012, National Instruments, Austin, TX, USA). The upper-extremity motor impairments of the post-stroke subjects was assessed by clinical scales, namely, FMA-UE [30] and WMFT [31]. The modified Ashworth scale (MAS) was used to assess the information concerning the muscle tone in the upper extremities [32].

Data Analysis
The grip force was filtered with an 8th-order Butterworth low-pass filter (20 Hz). For power-grip tasks, the grip force was cropped into a 3 s window (1 s after timing started, and 1 s before termination) for further analysis, and for the stationary holding tasks, the force signal from the middle 6 s of the 8 s holding period was used.
The smoothness of the grip force was described with normalized RMS-jerk: where J(i) is the jerk of the grip force at i-th sampling instant, which is the third derivative of the signal; and N is the total number of samples. Mag, the denominator, is a normalizing factor proposed by Hogan and Sternad [21], which is the mean maintenance force minus baseline. The complexity of grip force was evaluated with fApEn. The computational process is summarized as follows.
Given a time series with N samples {u(i): can thus be derived for the estimation of fApEn (m, r, N), which is the deviation of φ m from φ m+1 : where the function φ m indicates the averaged logarithm values across j of the averaged m-th similarity function across i for each j (between pairs of vectors X m i and X m j (i = j)). Coefficients m and r determine the gradient and boundary of the membership function, and were empirically set to 2 and 0.15 [33,34], respectively. A discussion of coefficient optimization and a detailed description of fApEn computation have been provided in previous works [33,35]. All data analyses were accomplished using the Matlab signal process toolbox (Matlab R2014a, MathWorks Inc., Natick, MA, USA).

Statistics
The force magnitude, RMS-jerk, and fApEn outcomes of power-grip tasks were examined through two-way analysis of variance (ANOVA), with the assumption that stroke (affected side and age-matched controls) and the force level (25%, 50%, or 75% of MGF) were the two main factors. The results were further tested in a Bonferroni post hoc test as the effect of force level was significant. In addition, as the interaction effect was significant, the effect of stroke was tested with one-way Entropy 2018, 20, 377 5 of 11 ANOVA on the outcomes grouped at each force level. In stationary holding tasks, the force magnitude, RMS-jerk, and fApEn outcomes were tested with a two-tailed t-test to find the effect of stroke in observations. Additionally, Pearson correlation coefficients were applied to investigate the relationship between the clinical scales (WMFT and FMA-UE) and the two measures (RMS-jerk and fApEn) of force modulation, the course of which was followed by a significance test. The significance level of all statistical tests was set at 0.05, and SPSS 19 (SPSS Inc., Chicago, IL, USA) was used to accomplish all statistical computations.

Results
The MGF of post-stroke subjects and age-matched controls ranged from 17.68 to 142 N, and 121 to 246 N, respectively. An example of the force profiles from the two tasks for two subjects (a healthy subject and the affected side of a stroke subject) is shown in Figure 2. As shown in Figure 2a, the force outputs of age-matched controls were higher than those from the affected side of post-stroke subjects at all three force levels. Figure 2a,b display the force profiles from each group during power-grip tasks and stationary holding tasks, respectively. As seen in Figure 2a, the force outputs of age-matched controls were higher than those from the affected side of post-stroke subjects across three submaximal force levels. However, a similar trend was not found in stationary holding tasks, as shown in Figure 2b.
Entropy 2018, 20, x 5 of 10 stroke in observations. Additionally, Pearson correlation coefficients were applied to investigate the relationship between the clinical scales (WMFT and FMA-UE) and the two measures (RMS-jerk and fApEn) of force modulation, the course of which was followed by a significance test. The significance level of all statistical tests was set at 0.05, and SPSS 19 (SPSS Inc., Chicago, IL, USA) was used to accomplish all statistical computations.

Results
The MGF of post-stroke subjects and age-matched controls ranged from 17.68 to 142 N, and 121 to 246 N, respectively. An example of the force profiles from the two tasks for two subjects (a healthy subject and the affected side of a stroke subject) is shown in Figure 2. As shown in Figure 2a, the force outputs of age-matched controls were higher than those from the affected side of post-stroke subjects at all three force levels. Figure 2a,b display the force profiles from each group during power-grip tasks and stationary holding tasks, respectively. As seen in Figure 2a, the force outputs of agematched controls were higher than those from the affected side of post-stroke subjects across three submaximal force levels. However, a similar trend was not found in stationary holding tasks, as shown in Figure 2b. Mean force magnitude values in each group across the three submaximal force levels are illustrated in Figure 3a. In post-stroke subjects and age-matched controls, the force outputs increased monotonically as the force level increased. As was revealed from two-way ANOVA, the effect of force level was significant (p < 0.01) with the effect size (ƞ 2 ) of 0.54. The following Bonferroni post hoc test demonstrated a significant difference between each of the two force levels (p < 0.05). The affected side of stroke subjects generated smaller forces at three submaximal force levels than age-matched controls. The effect of stroke was significant (p < 0.01, ƞ 2 = 0.52), however. The interaction effect was also significant (p < 0.01, ƞ 2 = 0.17). Further one-way ANOVA demonstrated that the effect of stroke was significant at all force levels (25%: p < 0.01, 50%: p < 0.01, 75%: p < 0.01). Additionally, the averaged force magnitude values from stationary holding tasks are demonstrated in Figure 3b. The force outputs of age-matched controls (7.19 N) were slightly higher than the affected side (5.57 N), but the effect of stroke was non-significant. Mean force magnitude values in each group across the three submaximal force levels are illustrated in Figure 3a. In post-stroke subjects and age-matched controls, the force outputs increased monotonically as the force level increased. As was revealed from two-way ANOVA, the effect of force level was significant (p < 0.01) with the effect size (η 2 ) of 0.54. The following Bonferroni post hoc test demonstrated a significant difference between each of the two force levels (p < 0.05). The affected side of stroke subjects generated smaller forces at three submaximal force levels than age-matched controls. The effect of stroke was significant (p < 0.01, η 2 = 0.52), however. The interaction effect was also significant (p < 0.01, η 2 = 0.17). Further one-way ANOVA demonstrated that the effect of stroke was significant at all force levels (25%: p < 0.01, 50%: p < 0.01, 75%: p < 0.01). Additionally, the averaged force magnitude values from stationary holding tasks are demonstrated in Figure 3b. The force outputs of age-matched controls (7.19 N) were slightly higher than the affected side (5.57 N), but the effect of stroke was non-significant. RMS-jerk values of the force outputs from each group across the three submaximal force levels are displayed in Figure 4a. RMS-jerk values were greater in the affected side of post-stroke subjects than in age-matched controls. Two-way ANOVA identified a significant effect of stroke (p < 0.01, ƞ 2 = 0.16) and two other non-significant effects: force level and interaction effect. Similarly, in stationary holding tasks, RMS-jerk of the force output was slightly greater in the affected side of post-stroke subjects than in age-matched controls. However, the difference was non-significant, as demonstrated through a t-test (p > 0.05).  Figure 5a provides the fApEn values of the force outputs from each group across three submaximal force levels. The fApEn values increased monotonically as the force level increased, and the values of the affected side were smaller than those of age-matched controls. Two-way ANOVA revealed that the effects of force level (p < 0.01, ƞ 2 = 0.39) and stroke (p < 0.01, ƞ 2 = 0.19) were both significant, whereas the interaction effect was non-significant (p > 0.05, ƞ 2 = 0.08). Moreover, the following Bonferroni post hoc test also found a significant difference between each of the two force levels (p < 0.05). The fApEn values in stationary holding tasks are provided in Figure 5b. The values of the affected side were significantly smaller than those of age-matched controls (p < 0.05). RMS-jerk values of the force outputs from each group across the three submaximal force levels are displayed in Figure 4a. RMS-jerk values were greater in the affected side of post-stroke subjects than in age-matched controls. Two-way ANOVA identified a significant effect of stroke (p < 0.01, η 2 = 0.16) and two other non-significant effects: force level and interaction effect. Similarly, in stationary holding tasks, RMS-jerk of the force output was slightly greater in the affected side of post-stroke subjects than in age-matched controls. However, the difference was non-significant, as demonstrated through a t-test (p > 0.05). RMS-jerk values of the force outputs from each group across the three submaximal force levels are displayed in Figure 4a. RMS-jerk values were greater in the affected side of post-stroke subjects than in age-matched controls. Two-way ANOVA identified a significant effect of stroke (p < 0.01, ƞ 2 = 0.16) and two other non-significant effects: force level and interaction effect. Similarly, in stationary holding tasks, RMS-jerk of the force output was slightly greater in the affected side of post-stroke subjects than in age-matched controls. However, the difference was non-significant, as demonstrated through a t-test (p > 0.05).  Figure 5a provides the fApEn values of the force outputs from each group across three submaximal force levels. The fApEn values increased monotonically as the force level increased, and the values of the affected side were smaller than those of age-matched controls. Two-way ANOVA revealed that the effects of force level (p < 0.01, ƞ 2 = 0.39) and stroke (p < 0.01, ƞ 2 = 0.19) were both significant, whereas the interaction effect was non-significant (p > 0.05, ƞ 2 = 0.08). Moreover, the following Bonferroni post hoc test also found a significant difference between each of the two force levels (p < 0.05). The fApEn values in stationary holding tasks are provided in Figure 5b. The values of the affected side were significantly smaller than those of age-matched controls (p < 0.05).  Figure 5a provides the fApEn values of the force outputs from each group across three submaximal force levels. The fApEn values increased monotonically as the force level increased, and the values of the affected side were smaller than those of age-matched controls. Two-way ANOVA revealed that the effects of force level (p < 0.01, η 2 = 0.39) and stroke (p < 0.01, η 2 = 0.19) were both significant, whereas the interaction effect was non-significant (p > 0.05, η 2 = 0.08). Moreover, the following Bonferroni post hoc test also found a significant difference between each of the two force levels (p < 0.05). The fApEn  Table 2 shows the Pearson's correlation coefficients relating the clinical scales with the RMS-jerk and fApEn values of the affected side. RMS-jerk was negatively correlated with FMA-UE and WMFT in terms of the force outputs at the three force levels in power-grip tasks (p < 0.05), whereas the correlation was weakened in stationary holding tasks. The correlations between fApEn and the two clinical scales were non-significant in the two tasks.

Discussion
In this study, grip control impairments were investigated by comparing post-stroke subjects and age-matched controls in terms of the variability and complexity of grip forces recorded in power grip and precision grip. The objective of this study was to find comprehensive measures of hand motor deficiencies in force modulation. As Ada et al. [7] suggested, maximal voluntary grip force reduction in the affected side was indicative of the weakness in force production that resulted from lost muscle cross-sectional area, and the reduction of motor units [5]. The significantly reduced force outputs at three force levels of the affected side indicated similar findings to those in previous studies.

The Variability of Force Modulation after Stroke
RMS-jerk reflects the variability of grip forces from the view of smoothness. The stroke-induced decrease in the smoothness of kinematic signals was previously reported in tasks with reflection [4,14]. These changes in smoothness may be visual feedback-dependent and could arise from the central  Table 2 shows the Pearson's correlation coefficients relating the clinical scales with the RMS-jerk and fApEn values of the affected side. RMS-jerk was negatively correlated with FMA-UE and WMFT in terms of the force outputs at the three force levels in power-grip tasks (p < 0.05), whereas the correlation was weakened in stationary holding tasks. The correlations between fApEn and the two clinical scales were non-significant in the two tasks.

Discussion
In this study, grip control impairments were investigated by comparing post-stroke subjects and age-matched controls in terms of the variability and complexity of grip forces recorded in power grip and precision grip. The objective of this study was to find comprehensive measures of hand motor deficiencies in force modulation. As Ada et al. [7] suggested, maximal voluntary grip force reduction in the affected side was indicative of the weakness in force production that resulted from lost muscle cross-sectional area, and the reduction of motor units [5]. The significantly reduced force outputs at three force levels of the affected side indicated similar findings to those in previous studies.

The Variability of Force Modulation after Stroke
RMS-jerk reflects the variability of grip forces from the view of smoothness. The stroke-induced decrease in the smoothness of kinematic signals was previously reported in tasks with reflection [4,14]. These changes in smoothness may be visual feedback-dependent and could arise from the central nervous system. In studies of gripping [18], reaching [36,37] and multi-joint arm movement [38], greater neuromotor noise was reported in subjects after stroke. Neuromotor noise causes more errors in force output, and post-stroke subjects would rely more on feedback control to correct the errors [39]. As reported by Kim et al. [14], the decrease in smoothness was accompanied by an increase in the number of force corrections. The mechanical properties of the affected joints are other factors that lead to decreased smoothness. The deficient passive mechanical properties of joints were reported in post-stroke subjects while they were performing constant velocity stretch [40], sinusoidal excitation [41], and pendulum test [42]. Biomechanical model indices (stiffness, damping, viscoelasticity, and stretch reflex) were found to be significantly correlated to the Ashworth scale [40]. Changes in mechanical properties hinder precise force generation, and ultimately increase the variability in force output [42].

The Complexity of Force Modulation after Stroke
Entropy analysis of physiological signals has been widely applied as an indicator of frailty in aging [23,43], fatigue [44], stroke-induced hemiplegia [12,13], etc. A decrease in entropy values indicates a decrease in complexity or an increase in regularity. The reduced fApEn values in the affected side are in agreement with previous studies involving complexity analysis in chronic stroke [12,13]. According to Sethi et al.'s [12] study, the ApEn values of upper extremity kinematics were significantly reduced in post-stroke subjects during functional reach-to-grasp [12]. Lodha et al. [13] also observed stroke-related decreases in the ApEn values of isometric force in wrist-finger extension. Kang and Cauraugh [45] summarized these studies and suggested that stereotypic movements and underlying abnormal synergies were the causes that led to compromised motor adaptability across different task requirements [45,46]. Another mechanism attributed to the stroke-induced decrease in complexity is related to deficient central drive, the subsequent loss of alpha motor neurons, and, consequently, a decrease in the number and activation of the motor unit [34].

Clinical Implications
Although semi-quantitative clinical scales have been widely applied, ceiling effects have often been reported, especially for subjects with fine motor control changes [11]. The significant and strong correlations between RMS-jerk and clinical scales in power-grip tasks demonstrated the potential of RMS-jerk as a quantitative indicator of motor impairments. The significant stroke effect and non-significant force level effect in power-grip tasks also provided evidence of the stable difference between the two groups in RMS-jerk. Although there were non-significant correlations between the clinical scales and fApEn values, fApEn, as a nonlinear outcome measurement, describes the regular patterns of force modulation in the time domain, which reflect the expression of internal randomness rather than force amplitude [12,22]. The parameter could advance our understanding of motor impairments from a different aspect. Accordingly, the descriptions provided by RMS-jerk and fApEn involve two different perspectives of force modulation, which could reflect post-stroke impairments relatively comprehensively.

Conclusions
Two parameters, RMS-jerk and fApEn, were applied to analyze force modulation recorded in power and precision grip. These indicators described stroke-induced disabilities from the view of variability and complexity. Both indicators have the potential to be applied in a clinical setting for quantitative motor function evaluation.