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Article

Step Length Asymmetry Predicts Rehabilitation Length in Subacute Post Stroke Patients

1
Faculty of Medicine, Hebrew University of Jerusalem, Jerusalem 9190501, Israel
2
Department of Physical Medicine & Rehabilitation, Hadassah Medical Center, Jerusalem 9765418, Israel
3
Department of Occupational Therapy, Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv 6997801, Israel
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(10), 1995; https://doi.org/10.3390/sym14101995
Submission received: 22 August 2022 / Revised: 18 September 2022 / Accepted: 22 September 2022 / Published: 23 September 2022
(This article belongs to the Special Issue Neuroscience, Neurophysiology and Asymmetry)

Abstract

:
Background: During the rehabilitation of individuals post stroke, evaluations are performed in order to discern the patient’s prognosis and optimize the treatment plan. However, these tests do not focus on gait symmetry, which might be a predictor for rehabilitation outcomes. We aimed to correlate gait symmetry measures of subacute post stroke patients with rehabilitation outcome and find the symmetry measure that best predicts the variability of the rehabilitation duration. A secondary aim was to compare these measures between patients with right and left brain lesions. Methods: We recruited 30 subacute post stroke patients (14 with right side lesion). We collected the following measures: National Institutes of Health Stroke Scale (NIHSS), Functional Independence Measure (FIM), the 10 m walk test (10MWT), Functional Ambulation Categories (FAC), spatial-temporal gait measures, and gait symmetry and variability. Results: We found moderate correlations between the step length symmetry and the length of rehabilitation, NIHSS, FIM, FAC and 10MWT. The symmetry index of the step length predicted the length of the rehabilitation period as it explained 32.1% of its variance (p = 0.001). Discussion: We conclude that a simple test of the step length symmetry might be informative in predicting rehabilitation length in subacute post stroke patients.

1. Introduction

Hemiparetic gait patterns differ from typical gait and might reduce patients’ independence and cause falls. Most hemiparetic gait patterns are affected by muscle spasticity that result in a synergistic pattern of activation during gait [1]. One of the several factors that may influence the abnormal gait pattern post stroke is the legion side. A recent study reported that right hemiplegic patients showed poorer spatiotemporal parameters, yet better kinematic parameters compared to left hemiplegic patients [2]. It was also suggested that gait symmetry worsens in later years post stroke [3]. However, it is uncertain how these measures can help clinicians improve the patient’s prognosis and thereby optimize the treatment plan.
During the rehabilitation of individuals post stroke in the subacute stage, several evaluations are performed in order to discern the patient’s prognosis and formulate the optimal treatment plan. These measures are aimed to produce shorter rehabilitation period, which reduces costs and allow the patients and their families a quick return to daily routine. Most of the conventional motor evaluations are simple to apply, e.g., the Timed Up and Go (TUG) test that requires a chair and a timer. Unfortunately, computerized accurate motor measurements that may procure more complex gait data, such as gait symmetry, are rarely collected due to the low availability of the required technology [4,5]. The limited use of computerized gait analysis might be caused by high costs or non-availability of room space for motion capture systems or alternately, long surface pressure paths, e.g., the GAITRite system. Furthermore, where these systems are available in an allocated space, the therapists might not be inclined to use them due to the technical knowledge required for activation of the system or due to the low availability of a single purchased system in a large rehabilitation center with many clients. Consequently, gait symmetry is mostly reported in research facilities. This is unfortunate since a growing evidence in the literature suggests that gait asymmetry in post stroke patients is a risk
To date, computerized gait analysis systems are becoming more affordable and user friendly. Therefore, the importance of measuring gait symmetry in post stroke population should be further explored as a reliable measurement of the efficacy of interventions. In a review that summarized the findings of 19 studies that quantified gait symmetry measures in individuals post stroke, 10 studies reported gait symmetry improvement following an intervention, while only two studies reported improvement in the raw spatiotemporal data [6]. The authors surmised that gait symmetry measures better reflect motor recovery compared with raw spatiotemporal measures. Since it has been shown that gait symmetry is an important factor to ascertain the efficacy of intervention in post stroke patients, it might prove a predictor of rehabilitation period length. However, this has yet to be reported in the literature.
Our main aims were therefore (i) to correlate gait symmetry measures of subacute post stroke patients with rehabilitation outcomes and (ii) to find the symmetry measure that best predicts the variability of the rehabilitation duration. A third secondary aim was (iii) to compare these measures between patients with right and left brain lesions.

2. Materials and Methods

We recruited 30 subacute post stroke patients admitted to inpatient rehabilitation, 14 with right side lesion and 16 with left side lesion (see personal characteristics in Table 1). The two groups were matched by age, sex and National Institutes of Health Stroke Scale (NIHSS) score. We did not match for dominant side since previous literature report that individuals post stroke, who had their dominant lower limb affected by the stroke, demonstrated similar impairments, compared with individuals who had their non-dominant lower limb affected [7]. Inclusion criteria were: age > 18, first symptomatic intracerebral haemorrhage or ischemic large vessel occlusion or lacunar stroke, able to stand for at least 2 min and to walk independently or under supervision with or without a walking aid, and able to provide informed consent. Exclusion criteria were orthopaedic injuries or deformations, pre-existing gait abnormalities before the stroke. Ethical approval was granted by the Hadassah medical center Helsinki committee pretrial (approval number 0162-19-HMO). All of the subjects read and signed an informed consent form pretrial.
Tools and protocol: The subjects underwent approximately 1 h of physical therapy and 1 h of occupational therapy every day, 5 times a week. In addition, they receive, when needed, an average of 3 h of speech therapy per week and participated in a variety of group therapy activities according to their individual needs and tolerance. Every patient went through a psychological evaluation for anxiety or depression. An individualized neuropsychological treatment was provided accordingly. The length of rehabilitation was defined from admission to the rehabilitation department to discharge from in-patient or out-patient daycare facility (treatment in an outpatient clinic was not included in the length of rehabilitation reported in this study). Discharge was determined by a multi-disciplinary clinical team that assessed whether the patient reached functional plateau. Since the research was performed in a single rehabilitation department, the criteria did not differ between subjects. The following measures were acquired in a random order on admission and discharge: NIHSS (scores range from 0 to 42, higher scores indicates greater severity [8]. Specifically, the severity of stroke can be defined as mild for values between 1–5, moderate 5–14, severe 15–24, and very severe >25 [9]; the NIHSS has good reliability [10] and validity [11] in post stroke patients), Functional Independence Measure (FIM; Score range from 18 to 126, higher scores indicates greater functional independence [12]; Interrater reliability was found to be acceptable and it was found to have strong construct validity with items on Barthel Index [13]), and Functional Ambulation Categories (FAC; Scored from 0 to 5, where a higher score indicates the ability to ambulate without aids on various terrains [14]; the FAC has good interrater reliability [15], excellent test–retest reliability [14], and excellent concurrent validity [14]). All of the subjects were tested at admission in the Gait and Motion laboratory, where four passive reflective markers (9.5 mm diameter) were attached bilaterally to the calcaneal tuberosity and the distal end of the second metatarsal. The subjects were asked to walk barefoot several times at a comfortable velocity on a 10 m long straight paved path, while 10 motion capture cameras (Qualisys, Gothenburg, Sweden) tracked the 3D coordinates of each marker at a frequency of 120 Hz. A ceiling-mounted safety harness was applied for subjects with balance deficiencies.
Data analysis: For all measures collected on admission and discharge, we calculated the percentage of change of that measure. Initial contact and toe-off times were manually marked using the Qualisys software. The coordinates of the markers and the timings were exported to a self-designed code (LabView 18, National Instruments, Austin, TX, USA), used to calculate all spatio-temporal parameters. Additionally, our code calculated the Symmetry Index (SI) of the spatio-temporal parameters according to the following equation [16]:
S I   [ % ] = X R X L 1 2 ( X R + X L ) · 100
where XR and XL are the gait parameters of the right and left leg, respectively. The SI ranges from ‘0’, denoting symmetry to ‘200’, denoting asymmetry. Additionally, the Coefficient of Variation (CV) was calculated for spatio-temporal parameters by dividing the Standard Deviation (SD) of the parameter by its mean (presented as percentage) [17]. The CV was included in our analyses since previous literature recommended this as a sensitive measure for quantifying gait abnormalities in post-stroke patients [18,19].
Statistical analysis: Statistical analyses were performed using SPSS 27.0 (SPSS Chicago, IL, USA). The Mann–Whitney U test was used due to the small sizes of the two groups (right and left side of injury). The effect size, r, was calculated using the following equation [20]:
r = Z N
Effect sizes were interpreted according to [21]. Correlations were performed using the Spearman correlation test and interpreted according to [22]. Stepwise linear regression was used to explain the variance of the duration of rehabilitation. Statistical significance was considered at p < 0.05.

3. Results

There were no between-group statistically significant differences in duration from acute stroke to admission, admission scores (NIHSS, FIM, and FAC), and duration of the rehabilitation period (Table 1).

3.1. Percentage of Change from Admission to Discharge

The percentage of change in all measures collected discharge in relation to admission is presented in Table 2. Both groups showed similar reduction in the NIHSS scores, as well as similar increase in their FAC scores at discharge. However, the left side lesion group showed a trend towards greater increase in their FIM scores compared to the right side lesion group.

3.2. Gait Symmetry

There were no between-group statistically significant differences in the following gait parameters of the paretic and non-paretic limbs: step time, stance and swing durations, double support duration, step length, and step width. Additionally, there were no between-group statistically significant differences in the gait velocity, cadence, and SI of the two groups as presented in Table 3.
Since there were no statistically significant between-group differences in the gait SI, we performed correlations for the entire sample size. The statistically significant correlations found are presented in Table 4.
Since the step length SI correlated with the 10MWT, we attributed this result to a possible correlation between the step length SI and performed additional analysis to test our hypothesis. We confirmed that there was a statistically significant correlation between these two measures (r = −0.485, p = 0.007) so that higher step length asymmetry is associated with lower gait velocity.
The best predictor for the length of the rehabilitation period was the SI of the step length, which explained 32.1% of its variance (p = 0.001; See Supplementary Materials. for output). The fitted regression model was: length of rehabilitation = 182.7 + 0.845 • (SI of the step length). So that higher step length asymmetry contributes to longer rehabilitation period. Due to the moderate association between gait velocity and the step length SI, we attempted to predict the length of rehabilitation using the gait velocity, but there was no statistically significant outcome to the regression analysis (p = 0.063).

3.3. Gait Variability

The CV of the two groups for the paretic and non-paretic limbs are presented in Table 5. The main findings are a statistically significant higher CV of the swing duration of the paretic limb and higher CV of the step time of the non-paretic limb in the left side lesion group compared to the right side lesion group.

4. Discussion

Our main finding shows that the SI of the step length is a significant predictor of the duration of rehabilitation in subacute post stroke patients. This predictor also correlates with other various outcome measures at discharge (NIHSS, FIM, FAC, 10MWT). A secondary outcome of this study shows higher CV in the swing duration of the paretic limb and the step time of the non-paretic limb for the group with left side lesion compared to the group with right side lesion.
Our results show statistically significant moderate correlations between the step length SI and the length of rehabilitation, NIHSS, FIM, FAC and 10MWT scores at discharge (Table 4). The two temporal SI (step time and stance duration) had a weak correlation with only one of the aforementioned outcome measures, while the base width SI had no correlations at all with the aforementioned outcome measures (Table 4). These findings, coupled with the capacity of the step length SI to predict the length of the rehabilitation duration, point to the importance the step length SI for prognosis of newly admitted individuals post stroke in rehabilitation departments. The highest correlation coefficient found was between the step length SI and the 10MWT (r = 0.561), meaning that greater step length asymmetry is associated with longer time to complete the 10MWT. Since further analysis showed that higher step length asymmetry is also associated with lower gait velocity, this explains its association with the 10MWT scores. A possible explanation to the association between step length SI and gait velocity is provided by a previous study that showed that step length asymmetry correlates with propulsive forces generated by the paretic leg [23]. The authors suggested that the severity of muscular impairments might provide an explanation for the asymmetrical patterns and consequently, the reduced gait velocity. The aforementioned explanation can also relate to our finding of a relation between the step length SI and the FAC score at discharge. Specifically, the greater gait asymmetry associated with lower FAC might also related to the reduced gait velocity, as a previous study demonstrated very high correlation between gait velocity and FAC scores in individuals post stroke at various time points [14]. Although the correlations between the step length SI and gait-related outcome measures, such as the 10MWT and FAC, can be explained via the SI relation to gait velocity, it is important to note that the gait velocity was not a predictor of the length of rehabilitation, yet the SI was a predictor of it.
Another correlation found in our study was between the step length SI and FIM at discharge. Although, to the best of our knowledge, this is the first study to show such an association, a previous study showed high correlation between symmetry of knee kinematics and FIM in chronic post stroke patients [24]. Though there might be differences in the gait pattern between subacute and chronic post stroke patients, our finding can be explained via basic biomechanics, so that the previously reported reduced knee angle of the paretic limb may limit the travel distance of the limb during the swing phase, thus reducing the step length and causing asymmetry. A fourth association of the step length SI was found with the NIHSS at discharge. To the best of our knowledge, there is no previous record of this finding. The aforementioned correlations between the step length SI and the four outcome measures at discharge (10MWT, FAC, FIM, NIHSS) might contribute to our understanding of prediction explain the prediction of the length of rehabilitation by the SI.
In our study, we measured the gait characteristic of the subjects once in admission in order to predict the length of the rehabilitation period, we cannot account for their improvement, or lack of improvement, during the rehabilitation period. Previous literature show no improvement in post stroke gait asymmetry between admission and discharge [25]. Additionally, our data collection period ended when the patients were discharged from daycare facility. Therefore, we have no knowledge regarding change in gait characteristics in the consequent months following discharge. However, based on published reports, one might assume that their gait symmetry did not change after discharge. A recent study showed that the step length symmetry of 61 post stroke patients did not change in a follow up examination that was performed 6 months following discharge [26]. The authors assumed that the lack of improvement in detectable change in walking velocity and gait symmetry in most subjects might relate to the discontinuation of focused gait interventions after discharge [26]. The explanation is supported by evidence of improvement in spatial gait symmetry in chronic post stoke patients who underwent exercises via a split-belt treadmill [27,28].
Since the legion side is one of the several factors that may influence the abnormal gait pattern, we compared gait and rehabilitation parameters between two groups, right and left side of the lesion. Interestingly, there were no between-groups statistically significant differences in the percentage of change in all measures collected on discharge in relation to admission (NIHSS, FAC, FIM, and 10MWT) and no differences in the gait velocity, cadence and SI. The effect sizes calculated for these analyses ranged between 0.011 and 0.355, indicating low to medium effect size [21]. However, we found higher CV of the swing duration of the paretic limb and higher CV of the step time of the non-paretic limb in the left side lesion group compared to the right side lesion group. Effect sizes for these results were 0.372 and 0.425, which are considered medium-large effect sizes [21]. We assume that erratic swing duration of the paretic limb invoked an erratic step time of the non-paretic limb in the left side lesion group. This finding is with accordance to a recent finding showing that left hemiplegic patients show worse segmentary motor performances [1]. Previous literature shows that right side legion affects vertical orientation, body sway, stance control, and weight shift [29], whereas left side legion is affects intersegmental coordination, direction, and linearity of movement [30]. It was also reported that left side legion produces deficits in planning well-coordinated movements [31]. Since we did not match the groups by type and extent of the legion, these differences should be interpreted with caution.
The study limitations include small, heterogeneous sample sizes of the two groups. Additionally, although the rehabilitation treatment plan was administered in a single rehabilitation facility by a constant staff of clinicians, the treatment differed between patients as it was fitted according to the patient’s condition and needs. Additionally, there were no hard-set criteria for patient’s discharge, since it was assessed in the conventional manner dictated in the rehabilitation department. Another limitation might have been produced by the multiple comparisons so that alpha correction such as the Bonferroni correction might have been applied resulting in non-significance in our correlations and between-group differences. However, this correction would have had no effect on our main finding of the regression analysis for the prediction of the rehabilitation length. It should also be noted that for non-significant findings (p < 0.05), most effect sized were very small so that no strong conclusion can be determined. Finally, although the subjects were provided with rest during data acquisition, they might have been fatigue. If so, we do not expect this limitation to affect the main results of this study since previous findings report no effect of fatigue on gait symmetry in individuals post stroke [32].

5. Conclusions

We conclude that a simple test of the step length symmetry is informative in predicting rehabilitation length in subacute post stroke patients. Further studies should test this finding in a larger sample sized population and explore the predictive value of this measure in other populations, e.g., individuals following traumatic brain injury or individuals with multiple sclerosis. Additionally, incorporating gait training focused on gait asymmetry, e.g., using treadmill [33,34] or overground [35] virtual biofeedback, should be further explored, combined with muscle strengthening [36].

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/sym14101995/s1, Figure S1: SPSS Output of the Regression Analysis.

Author Contributions

Conceptualization, I.S. and Y.O.; methodology, I.S, Y.O. and N.K.; software, M.S. and S.P.; validation, S.P. and M.S.; formal analysis S.P., M.S.; investigation, I.S., Y.O. and N.K.; resources, I.S.; data curation, Y.O., N.K. and M.S.; writing—original draft preparation, I.S. and S.P.; writing—review and editing, I.S., Y.O., N.K., M.S. and S.P.; visualization, S.P.; supervision, I.S.; project administration, Y.O. and N.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

The study was conducted in accordance with the Declaration of Helsinki, and approved by the Hadassah medical center Helsinki committee (approval number 0162-19-HMO).

Informed Consent Statement

Informed consent was read and signed by each subject pretrial.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Table 1. Patient characteristics. Values are presented as mean ± standard deviation. NIHSS = National Institutes of Health Stroke Scale; FIM = Functional Independence Measure; FAC = Functional Ambulation Categories.
Table 1. Patient characteristics. Values are presented as mean ± standard deviation. NIHSS = National Institutes of Health Stroke Scale; FIM = Functional Independence Measure; FAC = Functional Ambulation Categories.
CharacteristicRight Side Lesion
(n = 14)
Left Side Lesion
(n = 16)
p
Age (years)54.7 ± 10.9
Range 39–71
63.0 ± 13.9
Range 37–81
0.096
Sex11 males; 3 females10 males; 6 females0.918
Days from acute stroke to beginning rehabilitation16.3 ± 6.722.4 ± 16.10.496
Length of rehabilitation (days)208.8 ± 75.6231.1 ± 68.30.380
NIHSS on admission (0–42)7.7 ± 3.611.0 ± 5.80.091
Motor FIM on admission (13–91)83.5 ± 17.471.2 ± 23.30.067
FIM on admission (18–126)81.8 ± 16.969.1 ± 23.60.058
FAC on admission (0–5)1.4 ± 1.31.4 ± 1.60.758
Table 2. The percentage of change in all measures collected on discharge in relation to admission, and the actual measures on discharge. The measures presented are the National Institutes of Health Stroke Scale (NIHSS), Functional Independence Measure (FIM), and Functional Ambulation Categories (FAC). Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
Table 2. The percentage of change in all measures collected on discharge in relation to admission, and the actual measures on discharge. The measures presented are the National Institutes of Health Stroke Scale (NIHSS), Functional Independence Measure (FIM), and Functional Ambulation Categories (FAC). Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
Measure Right Side Lesion
(n = 14)
Left Side Lesion
(n = 16)
pr
NIHSS%56.3 ± 22.864.6 ± 14.30.262−0.208
mild 0–5
moderate 6–14 *
9 mild, 5 moderate7 mild, 9 moderate0.089-
Motor FIM%166.3 ± 62.5192.4 ± 78.40.462−0.139
(13–91)74.4 ± 13.570.6 ± 14.80.533−0.114
FIM%134.8 ± 21.1159.1 ± 39.90.065−0.355
(18–126)108.4 ± 14.0101.3 ± 17.80.298−0.190
FAC%179.2 ± 36.5221.3 ± 110.50.660−0.106
(0–5)4.0 ± 1.04.0 ± 0.70.859−0.032
* Chi-square test was applied.
Table 3. The velocity, cadence, and Symmetry Indices (SI) of the gait parameters. Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
Table 3. The velocity, cadence, and Symmetry Indices (SI) of the gait parameters. Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
MeasureRight Side Lesion
(n = 14)
Left Side Lesion
(n = 16)
pr
Velocity (m/s)0.42 ± 0.220.27 ± 0.180.070−0.331
Cadence (steps/s)72.5 ± 21.161.3 ± 19.50.220−0.224
Step time SI49.5 ± 29.139.5 ± 33.60.228−0.220
Stance duration SI18.0 ± 9.817.6 ± 21.70.244−0.213
Double support duration SI65.7 ± 49.379.9 ± 79.10.803−0.045
Step length SI49.5 ± 53.844.7 ± 43.20.835−0.038
Base width SI8.5 ± 5.810.3 ± 11.30.950−0.011
Table 4. The coefficient of correlation (r) and p values are presented (r, p). Correlations shown are between the Symmetry Indices (SI) and rehabilitation outcomes (all outcome measures were collected at discharge). National Institutes of Health Stroke Scale (NIHSS), Functional Independence Measure (FIM), and Functional Ambulation Categories (FAC), 10 m walk test (10MWT).
Table 4. The coefficient of correlation (r) and p values are presented (r, p). Correlations shown are between the Symmetry Indices (SI) and rehabilitation outcomes (all outcome measures were collected at discharge). National Institutes of Health Stroke Scale (NIHSS), Functional Independence Measure (FIM), and Functional Ambulation Categories (FAC), 10 m walk test (10MWT).
Length of RehabilitationNIHSSFIMFAC10MWT
Step time SI0.379, 0.0430.347, 0.065−0.252, 0.180−0.345, 0.0620.351, 0.119
Stance duration SI−0.168, 0.384−0.346, 0.0660.383, 0.036 0.282, 0.132−0.141, 0.542
Step length SI0.508, 0.0050.523, 0.004−0.492, 0.006−0.523, 0.0030.561, 0.008
Base width SI0.059, 0.760−0.047, 0.809−0.007, 0.971−0.064, 0.739−0.293, 0.197
Table 5. The Coefficient of Variation (CV) for the gait parameters. Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
Table 5. The Coefficient of Variation (CV) for the gait parameters. Values are presented as mean ± standard deviation. The p-value and effect size, r, are shown.
LimbMeasureRight Side Lesion
(n = 14)
Left Side Lesion
(n = 16)
pr
Paretic limbStep time7.2 ± 4.113.1 ± 11.20.114−0.288
Stance duration13.5 ± 12.418.0 ± 17.80.454−0.137
Swing duration8.2 ± 6.419.7 ± 24.40.042−0.372
Double support duration40.5 ± 32.224.8 ± 37.10.280−0.197
Step length12.5 ± 9.016.2 ± 21.70.454−0.137
Base width12.7 ± 8.818.2 ± 13.80.212−0.228
Non-paretic limbStep time8.2 ± 5.213.9 ± 7.80.020−0.425
Stance duration15.6 ± 10.912.1 ± 14.80.114−0.288
Swing duration6.1 ± 6.810.7 ± 8.40.053−0.353
Double support duration23.1 ± 15.616.8 ± 22.80.616−0.092
Step length21.2 ± 22.022.1 ± 21.30.708−0.050
Base width12.8 ± 7.518.6 ± 10.00.096−0.304
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Schwartz, I.; Ofran, Y.; Karniel, N.; Seyres, M.; Portnoy, S. Step Length Asymmetry Predicts Rehabilitation Length in Subacute Post Stroke Patients. Symmetry 2022, 14, 1995. https://doi.org/10.3390/sym14101995

AMA Style

Schwartz I, Ofran Y, Karniel N, Seyres M, Portnoy S. Step Length Asymmetry Predicts Rehabilitation Length in Subacute Post Stroke Patients. Symmetry. 2022; 14(10):1995. https://doi.org/10.3390/sym14101995

Chicago/Turabian Style

Schwartz, Isabella, Yonah Ofran, Naama Karniel, Martin Seyres, and Sigal Portnoy. 2022. "Step Length Asymmetry Predicts Rehabilitation Length in Subacute Post Stroke Patients" Symmetry 14, no. 10: 1995. https://doi.org/10.3390/sym14101995

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