Abstract
are lower limb joint angle variability between functional ankle instability (FAI) and healthy controls (CONs) at different running speeds using linear and nonlinear methods. Fifteen males with right-side FAI and fifteen matched CONs ran on a treadmill at self- selected, 20% faster, and 20% slower speeds. From 25 gait cycles, the mean coefficient of variation (CV), Sample Entropy (SampEn), and largest Lyapunov Exponent (LyE) of hip, knee, and ankle angles were computed. A two-way (two groups×three speeds) mixed- design ANOVA was applied (α= 0.05). No significant interaction effects were observed. No significant differences were observed in the CV. SampEn showed group effects: FAI had lower values in hip horizontal, knee sagittal/coronal, and ankle coronal planes, but higher in the hip sagittal plane. Speed effects showed greater SampEn in the ankle sagittal and lower in the hip coronal plane at slow speed. LyE was reduced in FAI for hip, knee, and ankle sagittal planes. Speed effects indicated higher LyE in the knee sagittal and lower in the hip coronal plane at slow speed. FAI showed reduced variability, particularly in the
hip sagittal plane. Speed effects showed greater SampEn in the ankle sagittal and lower in the hip coronal plane at slow speed. LyE was reduced in FAI for hip, knee, and ankle sagittal planes. Speed effects indicated higher LyE in the knee sagittal and lower in the hip coronal plane at slow speed. FAI showed reduced variability, particularly in the sagittal plane, reflecting rigid control. Slower speeds increased ankle and knee sagittal variability but decreased hip coronal variability. Keywords:functional ankle instability; postural control; nonlinear analysis 1. Introduction Lateral ankle ligament sprain is one of the most common musculoskeletal injuries, affecting 45.1–76.8% of athletes [1]. Over half of athletes with prior sprains experience recurrences, contributing to $4–6 billion in annual U.S. medical costs [2]. Up to 70% of ankle sprain subjects develop chronic ankle instability (CAI), which manifests as either mechani- cal ankle instability (MAI) or functional ankle instability (FAI) [3]. MAI is characterized by abnormal joint laxity resulting from ligamentous structural laxity or damage, with surgical repair being the primary intervention [4]. FAI is mainly associated with neuromuscular dysfunction [5], which may lead to muscle weakness [6] and impaired balance control [7], and other complications [8], thereby affecting the individual’s motor control capacity [9]. Recent studies have reported that individuals with CAI not only exhibit increased ankle inversion angles [10,11] but also demonstrate altered proximal joint kinematics, such as increased hip adduction [11], elevated hip flexion angles [12], and reduced knee flexion angles [12]. Furthermore, alterations in joint angles were observed not only during gait but also in side-cutting and jump-landing maneuvers [13]. However, joint angle Entropy2025,27, 1131 https://doi.org/10.3390/e27111131
Entropy2025,27, 1131 2 of 12 magnitudes alone are insufficient to fully reflect neuromuscular control. Therefore, further analysis of variability is required. Variability refers to the fluctuation of gait parameters across consecutive gait cycles, reflecting the nervous system’s motor control capacity and the musculoskeletal system’s adaptability [14]. In individuals with CAI, altered gait variability serves both as a sensitive marker of injury risk and functional deficits, and as a key reference for rehabilitation and recurrence prevention [15–20]. However, conflicting results exist among studies with similar outcome measures [21]. For example, two studies [16,19] collected ankle joint angle variability during running and observed greater variability (calculated via standard deviation, SD) in the frontal plane among CAI individuals, primarily during the early stance phase [16]. This difference is considered a contributing factor to episodes of “giving way” and sensations of instability. However, Kwon et al. [15] observed that during running, CAI individuals exhibited lower joint angle variability in the knee coronal and sagittal planes compared to both healthy controls and copers (defined as those with a history of ankle sprain without developing CAI). Since normal gait should retain a degree of irregularity to accommodate environmental changes [22,23], this suggests that CAI individuals may have a reduced capacity to adapt to task and environmental demands in motor behavior control. Overall, the discrepancies in research results may be attributed to differences in variability calculation methods, gait environments, and other factors, with speed being a key influencing factor [24]. Walking speed has a consistent influence on both kinematic and kinetic parameters of gait in healthy young adults [25]. Increased speed leads to reduced stride time variabil- ity [24] and decreased joint angle variability [26]. For example, Blair et al. [26] demonstrated that higher walking speeds resulted in lower lower-limb joint angle coefficient of varia- tion (CV) in healthy young adults, regardless of whether walking occurred on regular or irregular surfaces. However, numerous studies have confirmed [27–29] that either a decrease or an increase in walking speed can increase movement variability, exhibiting a U-shaped relationship. Gait stability and the most repetitive gait patterns are optimal when speed
in lower lower-limb joint angle coefficient of varia- tion (CV) in healthy young adults, regardless of whether walking occurred on regular or irregular surfaces. However, numerous studies have confirmed [27–29] that either a decrease or an increase in walking speed can increase movement variability, exhibiting a U-shaped relationship. Gait stability and the most repetitive gait patterns are optimal when speed approximates the self-selected walking speed. In pathological conditions, speed alters kinematic variability patterns. For instance, Springer et al. [18] observed that increased speed did not alter stride length variability in individuals with CAI, whereas Wanner et al. [16] reported significantly increased ankle frontal-plane variability in CAI athletes at higher speeds. The use of different metrics may be a source of heterogeneity in research outcomes. In summary, existing studies have investigated lower limb gait kinematic variability in individuals with CAI, yet considerable inconsistencies remain. Furthermore, dynamical sys- tems theory proposes that the human movement system is a complex nonlinear system [23]. The limitation of linear metrics lies in their neglect of the temporal structure embedded within signals. It prevents a conclusive assessment of the dynamic characteristics of the postural control system [30]—a system whose impairment is a hallmark of CAI [7]. There- fore, integrating linear and nonlinear analytical methods can more systematically reveal the mechanisms underlying motor control abnormalities in individuals with CAI [30,31]. Finally, experimental designs focusing solely on the ankle joint have limitations. Given that proximal joint biomechanics are also altered in CAI individuals [32]. Incorporating hip and knee joint analyses may be beneficial. To our knowledge, no study to date has examined the integrated linear and nonlinear variability of the hip, knee, and ankle joints in individuals with FAI across different running speeds. This study aimed to compare the differences in hip, knee, and ankle joint angle variability (using integrated linear and nonlinear analyses) between individuals with FAI and healthy controls under different running speeds, with the research hypotheses that FAI
the differences in hip, knee, and ankle joint angle variability (using integrated linear and nonlinear analyses) between individuals with FAI and healthy controls under different running speeds, with the research hypotheses that FAI
Entropy2025,27, 1131 3 of 12 individuals would exhibit lower joint angle variability and that reduced running speed would increase joint angle variability in FAI individuals. 2. Materials and Methods 2.1. Participant Recruitment Fifteen male participants with unilateral FAI on the right side who met all of the following criteria were recruited from the university [33]: (1) a documented history of at least one severe ankle sprain presenting with pain, swelling, and other inflammatory symptoms that resulted in inability to perform normal daily activities for more than one day; (2) Cumberland Ankle Instability Tool (CAIT) score of less than 24 points [34];(3) two or more episodes of ankle giving way, recurrent sprains, or perceived instability in the affected ankle within the past 12 months; and (4) negative in the anterior drawer test or talar tilt test. They were excluded if they (1) had a history of lower limb fractures or surgical procedures; (2) had acute lower limb injuries, including sprains, within 3 months before the study; (3) were left-leg dominant; (4) had an abnormal arrangement of lower limbs. Fifteen healthy controls (CONs) were recruited if they met the following criteria: (1) Theirright leg was identified as the dominant leg, based on their response to the question: “If you were to kick a ball at a target, which leg would you use?” [35], (2) matched with the FAI group for age, height, body weight, and weekly physical activity time (3) CAIT score >28 points. They were excluded if they had previous lower limb fractures, surgical procedures, or abnormalities in the arrangement of lower limbs. The study was conducted in accordance with the Declaration of Helsinki and was approved by the relevant biomedical research ethics committee. All participants provided written International Physical Activity Questionnaire (IPAQ) and informed consent before the test. 2.2. Experiment Procedure Prior to formal testing, each participant performed a 3 min warm-up run on the treadmill to adapt to the laboratory environment, become familiar with the procedures, and establish a comfortable running speed. The treadmill speed was adjusted in increments of 0.1 km/h according to participant feedback until a self-selected
Questionnaire (IPAQ) and informed consent before the test. 2.2. Experiment Procedure Prior to formal testing, each participant performed a 3 min warm-up run on the treadmill to adapt to the laboratory environment, become familiar with the procedures, and establish a comfortable running speed. The treadmill speed was adjusted in increments of 0.1 km/h according to participant feedback until a self-selected comfortable speed (preferred speed) was identified. This procedure was repeated three times, and the average was recorded as the participant’s preferred speed. Two additional test speeds were defined relative to this baseline: a faster speed (20% above the preferred speed) and a slower speed (20% below the preferred speed) [36]. Each participant then completed three 5 min treadmill runs at the predetermined speeds. The order of speed conditions was randomized using Excel-generated pseudo- random numbers to minimize sequence effects. Participants were instructed to maintain a natural running posture, keep their gaze fixed on the yellow marker, and avoid head movements during the trials, after which the experimenter issued a start signal. Data collection began 30 s after the start signal to allow for adaptation and continued until the completion of the 5 min run. A 5 min rest interval was provided between trials to minimize fatigue. Trials were repeated if interrupted by participant discomfort, external disturbances, marker detachment, or data acquisition failure. For each running condition, 25 consecutive gait cycles were extracted for analysis, starting 30 s after the onset of running. The infrared high-speed optical motion capture system (Arqus500, Qualisys, Gothenburg, Sweden), comprising six cameras, recorded kinematic data at a sampling frequency of 200 Hz (Figure).
Entropy2025,27, 1131 4 of 12 Figure 1.Subjects were formally tested for running. 2.3. Data Processing Gait cycle identification was performed in Qualisys Track Manager, using the peak value of the heel marker along the anterior–posterior axis as the criterion. Subsequently, all participants’ files were imported into Visual-3D (C-Motion, Inc., Rockville, MD, USA) for data processing, where virtual modeling of the trunk and lower limb joints was performed. Following model construction, the data were filtered using a 4th-order Butterworth low- pass filter with a cutoff frequency of 8 Hz. Based on the established model, Euler angles of the lower limb joints were calculated, from which the joint angle data required for this study were extracted and interpolated to 101 points [37]. Previous studies employing the CV have demonstrated that calculating variability for partitioned gait phases can offer additional clinically relevant information. Nevertheless, computing variability across the entire gait cycle remains an efficient and reliable approach, even in studies of analogous neurological conditions [38]. Furthermore, to maintain con- sistency in data structure with the nonlinear metrics, the CV in this study was calculated based on the entire gait cycle. The CV is calculated as the standard deviation divided by the mean (absolute mean) [39]. The specific formula is as follows: CV Mean Joint Angles=mean ` SD mean ´ ×100% (1) This study adopts the Sample Entropy (SampEn) algorithm proposed by Richman et al. [40] to quantitatively assess the complexity and irregularity of joint kinematics. The SampEn calculation formula is as follows: SampEn(m, r)=−ln A(m+1, r) B(m, r) (2) Here, m refers to the embedding dimension, which represents the dimensionality of the vectors when reconstructing the time series, and r is the tolerance, typically a multiple of the standard deviation of the time series, used to measure the similarity between two time sequences. The parameter was set to 2, and r was set to 0.2 times the standard deviation [40]. The largest Lyapunov Exponent (LyE) was calculated using the Rosenstein algorithm to quantify gait dynamic stability by computing the LyE (λmax) of lower limb joint angle time series data [41]. The LyE was
series, used to measure the similarity between two time sequences. The parameter was set to 2, and r was set to 0.2 times the standard deviation [40]. The largest Lyapunov Exponent (LyE) was calculated using the Rosenstein algorithm to quantify gait dynamic stability by computing the LyE (λmax) of lower limb joint angle time series data [41]. The LyE was obtained based on a linear fitting procedure: by performing a linear regression of ln di(k) against K, the slope of the fitted line represents the estimated
Entropy2025,27, 1131 5 of 12 λmax (Figure), where d i(k) denotes the distance between neighboring points in phase space at time K, and b is the intercept of the fitted line. The specific formula is as follows: ln di(k)≈λmaxK+b (3) Figure 2.Schematic of the fitting process of the maximum Lyapunov exponent based on the mean logarithmic dispersion. Based on the formula, custom codes were developed and executed in Matlab (R2023b, MathWorks, Inc., Natick, MA, USA) to calculate the LyE of hip, knee, and ankle joint angles in the sagittal, coronal, and transverse planes [41]. The Matlab function was implemented as: lambda_1 = lyarosenstein (data, m, tau, p, maxiter, fs). Here, lambda_1 denotes the calculated LyE; lyarosenstein is the function implementing Rosenstein’s method; data represents the input time-series of lower limb joint angle signals; m is the embedding dimension for reconstructing the phase space, which was determined using the False Nearest Neighbors (FNNs) algorithm and was found to be 4 for the current dataset;τis the time delay that defines phase space reconstruction, which was calculated using the Average Mutual Information (AMI) method and was found to be 29; p refers to the mean period of joint angles, derived via Fourier transform; maxiter indicates the maximum iteration count (set to 500) to control computational accuracy; and fs is the sampling frequency (100 Hz). After low-pass filtering at 8 Hz, downsampling from 200 Hz to 100 Hz did not significantly affect the LyE [42]. 2.4. Statistical Analyses Statistical analyses were performed using SPSS 27.0 (IBM SPSS Statistics for Win- dows, Version 27.0., IBM Corp., Armonk, New York, NY, USA). The Shapiro–Wilk test, Levene’s test, and Mauchly’s test of sphericity were first applied to assess data normality, homogeneity of variance, and sphericity, respectively. If the data met the assumptions of normality and homogeneity, a two-way mixed-design ANOVA (version 27) was conducted. In the presence of an interaction effect, simple effect analyses were performed; in the absence of interaction, Bonferroni post hoc tests were applied to examine main effects. All statistical results are reported as Mean±SD, withαset at 0.05. Effect sizes for ANOVA were evaluated
the data met the assumptions of normality and homogeneity, a two-way mixed-design ANOVA (version 27) was conducted. In the presence of an interaction effect, simple effect analyses were performed; in the absence of interaction, Bonferroni post hoc tests were applied to examine main effects. All statistical results are reported as Mean±SD, withαset at 0.05. Effect sizes for ANOVA were evaluated using partial eta squared (η 2), with thresholds defined as large (η 2 ≥0.14), medium (0.14 >η 2 ≥0.06), and small (0.06 >η 2 ≥0.01) [43]. Cohen’sdwas interpreted as follows:≥0.80, large; 0.50 to 0.79, moderate; 0.20 to 0.49, small; and < 0.20, trivial [44]. For non-normally distributed or heteroscedastic data, Aligned Rank Transform (ART) ANOVA was employed.
Entropy2025,27, 1131 6 of 12 3. Results 3.1. Participants A total of 30 participants were recruited, with detailed demographic characteristics presented in Table. Table 1.Participant characteristics. FAI Patients (n= 15) Controls (n= 15) p Age (years) 22.00 ±2.56 23.67 ±1.05 0.537 Height (cm) 179.00 ±4.63 176.73 ±3.51 0.142 Body mass (kg) 72.70 ±7.76 70.03 ±4.95 0.271 BMI (kg/m 2 ) 22.66±1.88 22.41 ±1.22 0.671 CAIT (scores) 18.00 ±4.54 29.33 ±0.98 0.006 * IPAQ (MT-min/W) 5796 ±2361 6212 ±1124 0.419 Slower speed (km/h) 5.41 ±0.34 5.51 ±0.41 0.473 Preferred speed (km/h) 6.77 ±0.42 6.90 ±0.51 0.452 Faster speed (km/h) 8.13 ±0.50 8.29 ±0.61 0.438 * Significant differences. 3.2. Coefficient of Variation No significant or main effect of group was observed (Table). The main effect of speed was significant in the knee joint sagittal plane (F(2,56) = 3.498,p= 0.035, partialη 2= 0.077). Post hoc comparisons with Bonferroni correction revealed that none of thep-values reached the adjusted significance level (p> 0.0167). Table 2.Mean CV results for joint angle at different running speeds and in different groups (%). Plane Group Faster Speed Preferred Speed Slower Speed Hip Sagittal FAI 0.686 ±0.363 0.661±0.214 0.854±0.504 CON 0.690 ±0.146 0.664±0.186 0.807±0.463 Hip Coronal FAI 0.680 ±0.462 0.463±0.121 0.595±0.229 CON 0.568 ±0.123 0.536±0.153 0.579±0.140 Hip Horizontal FAI 1.014 ±0.541 0.809±0.227 1.091±0.603 CON 0.805 ±0.272 0.896±0.340 0.809±0.161 Knee Sagittal &,@ FAI 1.830 ±0.604 1.925±0.728 2.592±1.432 CON 1.840 ±0.608 1.782±0.546 2.120±0.942 Knee Coronal FAI 0.440 ±0.180 0.384±0.114 0.531±0.328 CON 0.419 ±0.082 0.422±0.187 0.420±0.133 Knee Horizontal FAI 0.709 ±0.258 0.686±0.210 0.948±0.551 CON 0.741 ±0.209 0.757±0.261 0.695±0.154 Ankle Sagittal FAI 0.940 ±0.446 0.954±0.411 1.179±0.602 CON 1.087 ±0.315 0.992±0.319 1.068±0.462 Ankle Coronal FAI 0.713 ±0.325 0.695±0.284 0.831±0.305 CON 0.835 ±0.433 0.733±0.246 0.774±0.321 Ankle Horizontal FAI 0.723 ±0.238 0.763±0.252 0.918±0.356 CON 0.810 ±0.243 0.785±0.202 0.818±0.285 @ Significant difference between preferred and slower speeds. & Significant difference between faster and slower speeds. 3.3. Sample Entropy No significant interaction effects were observed (Table). The main effect of group was significant in the hip sagittal plane (F(1,28) = 4.515,p= 0.037, partialη 2= 0.051), hip horizontal plane (F(1,28) = 4.055,p= 0.047, partialη 2= 0.046), knee sagittal plane (F(1,28)
Description
This study compares gait variability in individuals with functional ankle instability at varying running speeds.