Determining the Validity of 2D Motion Capture for Estimating Lower Extremity Joint Quasi-Stiffness During Gait in Chronic Stroke Survivors

Abstract

Altered joint stiffness is common after stroke, yet clinically feasible tools to objectively quantify joint stiffness during walking are lacking. Quasi-stiffness, defined as the slope of the joint torque-angle curve, can serve as a surrogate measure of stiffness; however, it typically requires expensive 3D motion capture systems. 2D motion capture is a potential low-cost alternative for measuring quasi-stiffness in the sagittal plane; however, it is unclear if it can accurately estimate quasi-stiffness in patient populations that often exhibit out-of-plane motions. Therefore, in this study, we aimed to identify the minimal data required to accurately estimate joint quasi-stiffness. To do so, we evaluated the agreement between quasi-stiffness measurements obtained from 3D data in fifteen individuals with chronic stroke and from a simulated set of 2D data reconstructed from the 3D coordinates. Lower-extremity kinematic and kinetic data during overground walking were collected using a 3D motion capture system and an embedded force plate. To simulate 2D data, 3D maker data were projected to a simulated camera lens positioned to view sagittal motions, and medio-lateral components of the ground reaction force data were removed. Joint angles and moments at the hip, knee, and ankle were computed for both datasets using inverse dynamics, and quasi-stiffnesses of these joints were estimated during the stance phase. A linear mixed model was used to evaluate the effects of quantification method (2D, 3D) and stroke limb (paretic, non-paretic) on quasi-stiffness. Bland-Altman analyses and Intraclass correlation coefficients (ICCs) were used to evaluate the agreement between 2D and 3D measurements. The results indicated that 2D quasi-stiffness measurements were generally in agreement with the 3D quasi-stiffness measurements (Δ: −0.008–0.007 Nm/deg/kg; ICC: 0.576–0.927 [range]), although the 2D measurements slightly overestimated quasi-stiffness for some joints. Additionally, we found that quasi-stiffness was significantly higher in the paretic limb when the ankle was plantarflexing (Δ: 0.024 Nm/deg/kg) compared to the non-paretic limb. The results of this study suggest that quasi-stiffness can be validly estimated using 2D data, supporting the development of low-cost 2D systems for clinical settings to measure and monitor joint stiffness after stroke.

Keywords

biomechanics rehabilitation therapy function movement

Introduction

Following a stroke, upwards of 80% of survivors will exhibit gait deficits (e.g., reduced walking speed or foot drop) (Chen et al., 2005; Li et al., 2018; Sheffler & Chae, 2015; Woolley, 2001). Gait deficits are commonly attributed to neuromuscular impairments seen following a stroke, such as paresis, spasticity, and muscle co-contraction (Kitatani et al., 2016; Lance, 1980; Sheffler & Chae, 2015). These impairments can limit mobility by affecting the rigidity of the joint. For instance, in stiff knee gait, individuals walk with reduced knee flexion (due to knee muscle spasticity and co-contraction), resulting in a knee joint that appears stiffer (Lee et al., 2025; Li, 2023). On the other hand, individuals with foot drop have an ankle that is too flaccid (due to paresis of the dorsiflexor muscles) (Sheffler & Chae, 2015; Woolley, 2001). In both cases, an underlying impairment alters the stiffness of the joint. These altered joint stiffnesses can be detrimental, as joint stiffness is often leveraged to optimize walking efficiency (Ker et al., 1987; Sawicki et al., 2009). For instance, the ankle joint stores and later releases said energy to help propel the body forward, like a spring (Ker et al., 1987; Sawicki et al., 2009). Hence, alterations in joint stiffness can lead to walking that is energetically less efficient, more strenuous, and increase the risk of secondary complications (e.g., falls, fatigue, and disability) (Batchelor et al., 2012; Kramer et al., 2016; Nadarajah & Goh, 2015; Ramnemark et al., 1998). Thus, joint stiffness should be closely monitored and remediated following a stroke.

Multiple treatment modalities can be applied to address impaired stiffness, including manual physical therapy, bracing, or pharmacological procedures (i.e., Botox injections) (Bressel & McNair, 2002; Lecharte et al., 2020; Picelli et al., 2021). Clinicians use clinical scales to determine which of these treatment modalities to apply and to monitor disease progression. However, these metrics are often qualitative and rely on the clinician's ability to match their observations to an ordinal scale. For instance, the Modified Ashworth Scale (MAS) is an assessment for muscle tone (an impairment related to joint stiffness), in which clinicians manually move a patient's joints at various speeds to assess the resistance to movement (Bohannon & Smith, 1987; Harb et al., 2025). The clinician must then match the resistance they feel to a scale of zero (no increase in muscle tone) to four (affected part is rigid in flexion or extension) based on a written description of this resistance. The qualitative nature of these descriptions does not allow for measuring minute changes in tone, and the instrument has shown only moderate intra- and inter-rater reliability at best, with worse reliability for lower-extremity muscles (Ansari et al., 2008; Blackburn et al., 2002; Meseguer-Henarejos et al., 2018). Further, the MAS is assessed as the patient lies supine on their back in a resting state, whereas recent evidence indicates that joint stiffness differs when a patient is at rest versus when they are walking (Shorter et al., 2021). Hence, it is critical to perform these assessments in a functional manner. However, there is currently no clinically feasible way to measure joint stiffness during functional movements.

While dynamic stiffness is not currently measured in the clinic, researchers have developed various methods to measure joint stiffness during dynamic events. One such method measures the intrinsic stiffness of the joint by applying an external perturbation using a robotic perturbator and measuring the resulting change in torque about the joint (Cubillos et al., 2024; Joshi et al., 2022; Rouse et al., 2013b; Rouse et al., 2014). Using the dynamics of the perturbation (i.e., the angle, velocity, acceleration, and torque response), a system identification can be performed to measure the mechanical impedance of the joint (i.e., the joint's intrinsic stiffness, damping, and inertia that resists the motion when it is perturbed) at the time of the perturbation (Rouse et al., 2013a, 2014; Shorter et al., 2021). While this method provides a direct quantitative measure of joint stiffness and accounts for confounding dynamics that could affect stiffness measurements, it also requires custom robots to apply the perturbation and trained personnel to operate the equipment. Moreover, this method requires hundreds of walking trials to improve measurement fidelity, which may not be feasible for individuals with gait impairments; thus, limiting its application to the laboratory setting.

As an alternative, researchers have commonly used quasi-stiffness as a method to estimate joint stiffness. Quasi-stiffness is a simplified measure that is determined by fitting a line to the torque-angle relationship of the joint during movement. Specifically, many motions have phases where the torque-angle relationship is approximately linear, and the slope of this line can serve as an estimation of the stiffness behavior of the joint during this phase (Davis & DeLuca, 1996; Galli et al., 2008, 2018; Hinton et al., 2022; Houdijk et al., 2008; Molitor & Neptune, 2024; Rouse et al., 2013a; Sekiguchi et al., 2012, 2015, 2018; Shamaei et al., 2013a, 2013b, 2013c). Although quasi-stiffness is distinctly different from the intrinsic stiffness of the joint, it approximates intrinsic stiffness when the system dynamics are mostly passive (e.g., from midstance to late stance during gait) (Rouse et al., 2013a). Moreover, there are significantly fewer barriers to using quasi-stiffness in clinical settings because this method does not require custom robotic perturbators but only the kinematics and kinetics of the movement. Further, quasi-stiffness has been shown to be altered in clinical populations and is associated with peak ankle power and gait speed after stroke, indicating its clinical relevance for assessing gait function (Sekiguchi et al., 2012, 2015).

The effect of injury on joint quasi-stiffness during walking has been investigated in several patient populations (e.g., stroke, cerebral palsy, Down syndrome, anterior cruciate ligament reconstruction, and total joint replacement) (Davis & DeLuca, 1996; Galli et al., 2008, 2018; Garcia et al., 2023; Hinton et al., 2022; Houdijk et al., 2008; Sekiguchi et al., 2012, 2015, 2018). In stroke survivors, no single study has comprehensively evaluated quasi-stiffness for the lower extremity during walking. Instead, research has focused solely on the ankle joint (Hinton et al., 2022; Sekiguchi et al., 2012, 2015, 2018), and suggests that quasi-stiffness is increased in the paretic ankle when compared to the non-paretic ankle when dorsiflexing during the early stance phase (Sekiguchi et al., 2015). However, quasi-stiffness has also been seen to be reduced in the paretic ankle when compared to the non-paretic ankle when plantarflexing during the mid-stance phase (Hinton et al., 2022; Sekiguchi et al., 2015, 2018). Such findings highlight the interlimb differences in joint mechanics in stroke patients, and the potential clinical significance of quasi-stiffness in monitoring disease progression and treatment. Nevertheless, quasi-stiffness remains a metric that cannot be readily measured in a clinical setting. Although measuring quasi-stiffness does not require customized robotic equipment, it still requires motion capture and force plates to measure gait biomechanics data (i.e., kinematics and kinetics). This equipment is currently costly; therefore, the cost of performing this analysis must be reduced before quasi-stiffness measurements can be feasibly taken into the clinic.

One potential alternative to reduce the cost and complexity associated with optical motion capture is the use of simplified 2D systems. Two-dimensional motion capture has been widely used in human gait biomechanics and is accurate in measuring sagittal plane kinematics with low-cost webcams (Krishnan et al., 2015; Saner et al., 2017; Washabaugh et al., 2022). However, it is unclear whether two-dimensional motion capture can accurately measure quasi-stiffness. Therefore, in this study, we aimed to identify the minimal data required to accurately estimate joint quasi-stiffness. To accomplish this, we performed a human subjects experiment in stroke survivors, where we measured the quasi-stiffness of the paretic and non-paretic hip, knee, and ankle joints using a 3D motion capture system. We then simulated how a two-dimensional camera and force-plate system would measure quasi-stiffness by projecting the 3D data into the reference frame of a simulated camera viewing the sagittal motion. This approach allowed us to isolate the effects of data reduction on quasi-stiffness accuracy. Moreover, testing on a stroke population allowed us to determine whether 2D data can yield similar conclusions to 3D data regarding the presence or absence of interlimb differences in quasi-stiffness. We hypothesized that, although 2D quasi-stiffness may differ from that of 3D, there would be strong agreement between the two, and the 2D data would yield similar conclusions regarding interlimb differences. If successful, these findings would support the development of dedicated low-cost 2D motion capture systems for measuring quasi-stiffness in clinical settings.

Methods

Fifteen chronic stroke survivors (nine males and six females, Age : 66.1 ± 7.3 years, time since stroke : 6.1 ± 4.4 years) volunteered for this study. Subjects were included if they were aged between 40–75 years, had a unilateral cortical or subcortical stroke at least 6 months prior to the experiment (i.e., chronic stroke), were able to walk independently without assistive devices, and had a Mini Mental State Examination (MMSE) score ≥22. Subjects were excluded if they had a brainstem or cerebellar stroke, traumatic brain injury, an unstable heart condition, uncontrolled diabetes or hypertension, lower-extremity orthopedic or neurologic conditions (other than stroke) that could limit walking ability, significant spatial neglect, recent Botox injection (≤ 3 months), an inability to communicate or sign written consent, or any other medical conditions that would significantly impact the study results. Before participation, each subject reviewed and signed an informed consent document approved by the Wayne State University Institutional Review Board.

Each individual had 36 retroreflective markers placed across their pelvis and legs (16 markers per leg and four on the pelvis). A ten-camera optical motion capture system (Vero, Vicon, Oxford, UK) was used to record the markers’ trajectories. A static trial was first obtained to capture marker positions during neutral standing. Dynamic trials were then recorded to track marker positions while the participants walked overground across a platform with an integrated six-degree-of-freedom force plate (Bertec, Columbus, OH, USA). A total of three walking trials were obtained for each leg (both the paretic and non-paretic), where special consideration was taken to ensure that the desired foot cleanly struck the force plate. The kinematic data were recorded at 100 Hz, and the force plate data were recorded at 2000 Hz. In Vicon Nexus (Version 2.16), the marker trajectories were checked to ensure proper labeling and were gap-filled. Marker coordinate and ground reaction force (GRF) data were both filtered with a 15 Hz low-pass filter before further processing (Tomescu et al., 2018; Welch et al., 2021).

Three-Dimensional Joint Angles and Moments

The 3D joint angles and moments were obtained using musculoskeletal modeling software, OpenSim (Version 4.0) (Delp et al., 2007). The OpenSim Gait2392 (i.e., 23 degrees of freedom and 92 muscles) model was scaled using the built-in Scale Tool with the static marker trajectory file and the participant's weight. The scaled model, dynamic marker trajectory files, and ground reaction force files were used with the built-in Inverse Kinematics Tool and Inverse Dynamics Tool to obtain joint angles and moments, respectively. At these respective stages, the kinematic and kinetic data were filtered (6 Hz low-pass) using OpenSim's built-in functionality.

Two-Dimensional Joint Angles

A reduced set of 2D coordinate data was constructed based on the 3D coordinate data from motion capture. For each walking trial, the subject's scaled OpenSim model was used with motion from the 3D dynamic trial to extract point kinematics of the hip, knee, and ankle joint centers, pelvis, and toe using the Analyze Tool. The resulting 3D coordinates (x, y, z) were reduced to 2D by simulating how a camera would see those markers when viewing the sagittal plane of the subject (Figure 1A). For this analysis, we assumed a low-cost 1080p (1920 × 1080) webcam (Logitech c922). The camera's intrinsic matrix (K) (Eq. 1) consisted of f_x, f_y (focal lengths) and c_x, c_y (coordinates of the camera's center) in pixels; these parameters were determined based on our assumed low-cost webcam's properties.

K = [\begin{matrix} f_{x} & 0 & c_{x} \\ 0 & f_{y} & c_{y} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1435.3 & 0 & 939.6 \\ 0 & 1435.1 & 501.4 \\ 0 & 0 & 1 \end{matrix}]

(1)

Figure 1.

A schematic depicting the experimental methods. (A) The 3D data was projected to recreate the view of a simulated 2D camera viewing the sagittal plane. The 3D data were projected from the world frame to the simulated camera’s frame via [R|t], then into a 2D image via K. Red circles represent markers used when creating 2D data, grey markers were used for 3D only. (B-Top) Joint angles were calculated from the 2D camera images. Lines with arrowheads depict the vectors between the markers: posterior superior iliac spine (PSIS) to anterior superior iliac spine (ASIS), hip joint (H) to knee joint (K), knee joint to ankle joint (A), and ankle joint to the toe marker (Toe). A vector perpendicular to the PSIS-ASIS vector was also created and deemed the pelvis reference vector (PRV). The hip angle (θ_Hip) was calculated as the angle between the PRV and the H-K vector. The knee angle (θ_Knee) was calculated as the angle between the H-K vector and the K-A vector. The ankle angle (θ_Ankle) was calculated as the angle between the K-A vector and the A-Toe vector. (B-Bottom) An expanded view of the ankle angle, where an additional offset angle (θ_Offset) was calculated from the static trial.

The camera's extrinsic matrix ([R|t]) represented how the camera was positioned in the global coordinate system in both orientation (R) and position (t) (Eq. 2). Components of the rotation matrix R (i.e., r_ij) indicate the relative relationship from the global coordinate system to the camera. The position vector t allowed us to position the camera at the center of the force plate in x (i.e., 300 mm from the global origin), above the force plate surface in y (i.e., 500 mm from the global origin), and backwards from the force plate in z (i.e., −2750 mm from the global origin). This simulated camera position achieved a sagittal view that allowed for the observation of markers from the foot up to the pelvis throughout the full stride.

[R | t] = [\begin{matrix} r_{11} & r_{12} & r_{13} & t_{x} \\ r_{21} & r_{22} & r_{23} & t_{y} \\ r_{31} & r_{32} & r_{33} & t_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = [\begin{matrix} - 1 & 0 & 0 & 300 \\ 0 & - 1 & 0 & 500 \\ 0 & 0 & 1 & - 2750 \\ 0 & 0 & 0 & 1 \end{matrix}]

(2)

Each 3D point (X_world, Y_world, and Z_world) from the walking trials was then projected to pixels in the camera frame (u,v) by multiplying each point by the intrinsic and extrinsic matrices and an augmented identity matrix $[I | 0]$ (Eq. 3).

[\begin{matrix} u \\ v \\ 1 \end{matrix}] = K [I | 0] [R | t] [\begin{matrix} X_{W o r l d} \\ Y_{W o r l d} \\ Z_{W o r l d} \\ 1 \end{matrix}]

(3)

We then converted the camera pixels (u,v) to meters for the 2D moment calculations. To do so, we multiplied each pixel coordinate by a scale factor. This scale factor was determined by projecting two points that were a known distance apart (i.e., 100 mm) in X_World and Y_World, using Eq 3, and then calculating the ratio of the known distance to the pixel distance (Figure 1B). This analysis reduced the 3D data down to 2D, while also simulating how a potential low-cost system would capture joint kinematics.

A custom MATLAB (Version R2022a) script was used to calculate the hip, knee, and ankle angles based on the 2D coordinates. To calculate the hip angle, three vectors were created. The first vector (i.e., hip-knee vector) was between the hip and knee positions. The second vector (i.e., ASIS-PSIS vector) was between the anterior superior iliac spine (ASIS) and the posterior superior iliac spine (PSIS). The third vector (i.e., pelvis reference vector) was the vector perpendicular to the ASIS-PSIS vector. The hip angle was defined as the angle between the hip-knee vector and the pelvis reference vector. The knee angle was defined as the angle between the hip-knee vector and a vector connecting the knee and ankle positions (i.e., knee-ankle vector). The ankle angle was defined as the angle between the knee-ankle vector and a vector connecting the ankle and toe positions (i.e., ankle-toe vector). This ankle angle calculation assumes that 0° represents when someone is standing in neutral position (i.e., flat foot with vertical shank). To ensure this was the case in the stroke survivors, we calculated an offset angle during each subject's static trial. This ensured that 0° represented true neutral standing in case a subject stood with some degree of dorsi/plantarflexion when at rest. To calculate the offset, we determined the average position of the knee, ankle, and toe markers from our static trial. The angle between the knee-ankle vector and the global y-axis (i.e., vertical) was calculated (θ_Shank) to capture whether or not the shank was vertical. The angle between the ankle-toe vector and a vector perpendicular to the knee-ankle vector was calculated (θ_Foot) to capture the position of the foot relative to the shank (i.e., degree of dorsi/plantarflexion). The offset angle was then calculated as θ_Offset = θ_Shank − θ_Foot. The static offset ankle angle was then added to the ankle angle from the dynamic trials (Figure 1B). This allowed for better agreement between 2D and 3D ankle angle definitions, as traditionally, 3D calculations often assume that a neutral position occurs when the foot is flat (Kim & Won, 2019).

Two-Dimensional Joint Moments

For our analysis, we assumed that ground reaction force data were measured using a low-cost 2D force plate that had vertical load cells in each corner and a shear load cell in the center of the plate to measure anterior-posterior forces. With this proposed design, medial-lateral forces cannot be measured; however, this does not affect the force in the anterior-posterior direction, force in the vertical direction, or center of pressure in the anterior-posterior direction (i.e., the components used to calculate moments in the sagittal plane). Hence, we used these components as measured from the human subjects experiment with the 6 DOF force plate to calculate 2D moments.

A set of 2D moments for the hip, knee, and ankle was calculated using inverse dynamics with a custom MATLAB (Version R2022a) script. The analysis followed the procedure outlined by prior analyses (Winter, 2009). In short, the leg was broken down into various segments (i.e., foot, shank, and thigh) and a system of equations (i.e., $\sum F_{x} = m a_{x}$ , $\sum F_{y} = m a_{y}$ , and $\sum M = I α$ ) was solved sequentially for each segment, moving up the kinematic chain, to determine the net moment at each joint.

When solving these equations, the hip, knee, ankle, and toe (i.e., second metatarsal) positions were used to calculate the center of mass (COM) of each segment and the moment arm to each force. These calculations used the following assumptions: (1) The mass of each segment was estimated as a percentage of the subject's total body weight, (2) the location of the COM for each segment was expressed as a ratio relative to the segment's total length, and (3) the inertia (about the COM) of each segment was calculated using the radius of gyration. The reference values for these assumptions were based on previously reported values (Winter, 2009).

Quasi-Stiffness Calculation

For each leg, angle and moment data from the three walking trials were trimmed to the stance phase (i.e., heel strike to toe-off). Using the vertical ground reaction force from the force plate, heel strike was determined as the first point that GRF >20 N, and toe-off was the first point that GRF < 20 N. Each trial was then resampled to 101 points (i.e., percentage of the gait cycle) and averaged together to create an ensemble average.

A custom MATLAB (Version R2022a) program was used to plot the average joint moment against the average joint angle. On the graph, two points were manually selected that corresponded to the region of interest. Linear regression was performed on this sub-selection of data, and the slope of the fitted line was the estimated quasi-stiffness of the joint. This process was repeated for 3D and 2D data. Each joint had two regions of interest, for a total of six quasi-stiffness estimates. For the hip, quasi-stiffness was calculated during the extension stage and the flexion stage of the stance phase. The extension stage was defined as the region from the peak flexion angle during stance to the peak extension angle. The flexion stage was then from the peak flexion angle to toe-off (Figure 2A) (Fang et al., 2023). For the knee, quasi-stiffness was calculated during the flexion stage and extension stage of the stance phase. The flexion stage was defined as the region from the minimum moment after heel strike to the maximum moment. The extension stage was defined as the region from the maximum moment to the minimum moment before toe-off (Figure 2B) (Shamaei et al., 2013c). For the ankle, quasi-stiffness was calculated during the dorsiflexion stage and plantarflexion stage of the stance phase. The dorsiflexion stage was defined as the region from the local minimum angle after heel strike to the maximum moment. The plantarflexion stage was defined as the region from maximum moment to minimum angle (Figure 2C) (Davis & DeLuca, 1996; Molitor & Neptune, 2024; Shamaei et al., 2013a). While these definitions for the regions were followed as closely as possible, some subjects did not follow stereotypical patterns in kinematics and kinetics. In these cases, only the portion of the torque-angle curve that was approximately linear was selected for analysis. When calculating quasi-stiffness during hip flexion and knee flexion, there was one subject for each measurement where quasi-stiffness could not be calculated because the joint never entered a flexion phase within the desired region (Table 1).

Figure 2.

Plots depicting the selected regions of interest used to calculate (3D) and two-dimensional (2D) quasi-stiffness for the (A) hip, (B) knee, and (C) ankle from a single subject. The top plot represents the joint angle, θ (in deg), the middle plot is the joint torque, τ (in Nm), and the bottom plot is the joint torque-angle curve. For the joint torque-angle plot (bottom), the solid black line represents the joint torque against the joint angle curve for one full stride, and the grey portion of the line represents the swing phase where no analysis occurred. Each joint had two regions of interest that were selected. The blue points mark the first selected region, and the orange points mark the second selected region. The solid orange and blue lines represent the fitted linear regression curve over the respective region. For the joint angle and joint torque plots (top and middle), the solid black line represents the angle or torque throughout the gait cycle. The blue and orange line corresponds to the selected regions of interest. The corresponding regions of interest were also highlighted by the blue and orange lines on the joint angle (top) and torque (middle) plots. The grey boxes represent the swing phase, where no analysis was performed. For the hip, the regions of interest were while the hip was extending (blue) and hip flexing (orange). For the knee, the regions of interest were while the knee was flexing (blue) and extending (orange). For the ankle, the regions of interest were when the ankle was dorsiflexing (blue) and plantarflexing (orange). Note that the readers are referred to the online version of this article for color interpretations.

Statistical Analysis

A Linear mixed model (LMM) analysis was performed in IBM SPSS Statistics (Version 30) to evaluate the effect of quantification method (3D, 2D) and stroke-affected limb (paretic, non-paretic) on hip, knee, and ankle quasi-stiffness. Method, limb, and method-by-limb interaction were included as fixed effects, while subject was included as a random effect to account for inter-subject variability. LMM was run for each of the gait phase regions that were investigated. To account for the temporal correlation of repeated observations within subjects, an autoregressive (AR1) covariance structure was used in the model. Model selection was guided by the Akaike Information Criterion (AIC), with lower AIC values indicating better model fit. All models were assessed for convergence to ensure that the model had found a stable and reliable solution and that the estimated parameters were accurate. A significant interaction effect was followed by post hoc comparisons using paired t-tests with Sidak correction to control for multiple comparisons, where appropriate. Note that the interaction effect was our primary interest, as that would suggest that the between-limb differences in quasi-stiffness were affected by the method used to quantify it.

Intraclass correlation coefficient (ICC) analyses were performed in JASP (Version 0.19.3) to evaluate the degree of similarity between the 2D and 3D quasi-stiffness measurements. Specifically, a two-way mixed-effects model with a single rater measurement [i.e., ICC (3,1)] was used. ICC values were interpreted as follows: excellent (> 0.90), good (0.75–0.90), moderate (0.50–0.75), and poor (< 0.50) (Koo & Li, 2016). The agreement was further examined using Bland-Altman plots. Analyses were done for each region of interest. All statistical analyses were performed with a significance level of α = 0.05.

Results

Quasi-stiffness values measured from each subject, limb, joint, and motion can be found in Table 1.

Hip Joint Quasi-Stiffness

During hip extension, there was a significant effect of method, indicating that 2D measurements slightly overestimated quasi-stiffness by 0.004 Nm/deg/kg [F(1,22.767) = 4.733, p = 0.040]. There was no significant effect of limb [F(1,15.939) = 0.832, p = 0.375] or interaction [F(1,35.759) =0.869, p = 0.357]. The paretic and non-paretic limbs showed moderate and excellent agreement, respectively, between 3D and 2D quasi-stiffness measurements (Table 2). The Bland-Altman plot of the paretic (Figure 3A) suggested proportional bias, where the differences between the 2D and 3D measurements increased at higher values of quasi-stiffness. For the non-paretic limb, the Bland-Altman plot (Figure 4A) did not suggest proportional bias between the 2D and 3D quasi-stiffness measurements

Figure 3.

Bland-Altman plots for the paretic limb depicting the agreement between the three-dimensional (3D) and two-dimensional (2D) quasi-stiffness measurements of various regions of interest within the gait cycle: (A) hip extension, (B) hip flexion, (C) knee extension, (D) knee flexion, (E) ankle plantarflexion, and (F) ankle dorsiflexion. Each point represents one subject, where the x coordinate is the mean of the 3D and 2D quasi-stiffness measurements, and the y coordinate is the difference between the 3D and 2D quasi-stiffness measurements. The center dashed line represents the mean difference between 3D and 2D quasi-stiffness for all participants. The upper and lower dashed lines represent the 95% limits of agreement (i.e., mean difference ± 1.96 * standard deviation of the difference).

Figure 4.

Bland-Altman plots for the non-paretic limb depicting the agreement between the three-dimensional (3D) and two-dimensional (2D) quasi-stiffness measurements of various regions of interest within the gait cycle: (A) hip extension, (B) hip flexion, (C) knee extension, (D) knee flexion, (E) ankle plantarflexion, and (F) ankle dorsiflexion. Each point represents one subject, where the x coordinate is the mean of the 3D and 2D quasi-stiffness measurements, and the y coordinate is the difference between the 3D and 2D quasi-stiffness measurements. The center dashed line represents the mean difference between 3D and 2D quasi-stiffness for all participants. The upper and lower dashed lines represent the 95% limits of agreement (i.e., mean difference ± 1.96 * standard deviation of the difference).

Table 1.

2D and 3D Quasi-Stiffness Values for Each Region of Interest for the Paretic and non-Paretic Limb.

Quasi-stiffness (Nm/deg)
			Hip				Knee				Ankle
			Extension		Flexion		Extension		Flexion		PF		DF
Subject	Weight (kg)	LE FM	2D QS	3D QS	2D QS	3D QS	2D QS	3D QS	2D QS	3D QS	2D QS	3D QS	2D QS	3D QS
01 – P	73	28	4.72	4.36	3.16	2.69	2.01	2.30	1.28	1.77	5.95	5.23	6.22	6.18
02 – P	84	28	3.45	3.02	2.53	1.23	4.21	4.94	2.67	2.61	5.81	5.94	4.95	5.75
03 – P	111	33	5.82	4.96	2.75	1.93	4.79	6.01	5.72	6.21	9.29	10.69	6.78	7.95
04 – P	68	21	3.87	2.94	1.71	1.51	2.19	1.84	5.69	6.91	4.22	4.88	4.49	4.85
05 – P	59	28	2.23	1.65	2.60	1.90	2.09	3.32	1.76	2.06	3.65	3.31	3.26	3.71
06 – P	63	27	2.46	1.67	3.03	2.35	2.41	2.92	2.50	1.97	5.08	4.40	12.52	9.06
07 – P	60	20	2.78	2.26	6.66	10.49	1.54	1.68	12.21	5.16	1.29	1.44	2.32	2.22
08 – P	63	30	2.76	2.95	2.03	2.10	2.40	2.61	1.81	2.20	11.73	10.55	9.10	8.66
09 – P	64	26	3.75	5.35	11.32	8.73	3.07	4.26			3.68	3.94	5.30	6.27
10 – P	74	19	4.86	3.65	3.07	3.95	3.11	3.32	4.44	5.18	6.90	4.26	5.26	6.72
11 – P	73	23	2.49	2.09	3.09	2.03	2.90	3.99	2.66	2.57	8.67	8.52	7.66	5.52
12 – P	67	19	2.76	4.02	2.94	0.77	2.54	2.81	3.18	3.49	5.96	7.68	6.37	6.83
13 – P	91	26	3.95	5.01	4.90	3.99	2.69	2.70	1.77	2.67	8.46	8.80	5.58	7.09
14 – P	70	14	3.17	4.11			1.93	2.03	1.16	1.18	1.69	1.71	4.41	2.66
15 – P	73	31	2.52	1.87	4.88	4.05	2.84	3.95	2.14	3.26	10.06	8.24	12.33	6.29
01 – NP	73	28	2.54	2.27	2.76	2.07	1.74	2.09	3.26	2.98	4.24	3.58	4.04	4.35
02 – NP	84	28	4.49	2.79	2.53	2.08	4.80	7.41	3.68	3.70	4.21	3.89	4.90	5.60
03 – NP	111	33	4.38	3.89	1.34	0.43	6.60	5.78	5.87	6.29	5.48	5.63	5.77	6.18
04 – NP	68	21	3.97	3.54	3.75	2.41	3.52	4.06	1.74	3.03	4.59	4.41	3.70	4.10
05 – NP	59	28	2.30	1.66	3.17	2.30	1.76	2.89	1.78	1.81	4.94	4.02	5.27	6.20
06 – NP	63	27	2.09	1.85	3.23	2.28	3.07	4.15	2.46	2.49	3.53	3.32	5.27	5.08
07 – NP	60	20	2.55	1.93	1.50	0.62	2.78	2.26	1.63	2.42	3.24	3.01	2.59	3.12
08 – NP	63	30	2.14	2.33	2.14	1.35	1.82	2.27	1.34	1.36	3.99	3.45	6.03	6.67
09 – NP	64	26	2.63	2.69	3.78	3.77	2.67	3.68	1.25	1.82	5.06	3.38	5.96	5.83
10 – NP	74	19	2.50	2.14	1.64	0.85	3.79	4.28	1.25	0.67	5.78	5.04	3.90	3.73
11 – NP	73	23	1.97	1.65	2.80	1.97	2.18	2.84	2.35	2.55	5.55	4.93	8.61	6.37
12 – NP	67	19	2.36	2.76	1.55	1.38	2.31	2.54	2.41	3.23	3.35	3.68	3.90	4.16
13 – NP	91	26	4.75	2.79	3.12	2.51	2.39	3.10	1.37	2.10	5.95	5.61	9.22	7.65
14 – NP	70	14	8.08	6.81	0.03	0.95	2.19	2.39	1.87	2.84	2.49	2.22	2.49	3.16
15 – NP	73	31	3.14	2.78	2.50	2.21	2.78	3.66	2.00	2.29	4.05	3.68	4.65	5.46

Abbreviations: P (paretic), NP (non-paretic), LE FM (lower extremity Fugl-Meyer), DF (dorsiflexion), PF (plantarflexion), 2D (two-dimensional), 3D (three-dimensional), QS (quasi-stiffness).

Table 2.

Intraclass Correlation Coefficients [ICC(3,1)] Values for Each Region of Interest in the Paretic and non-Paretic Limbs with 95% Confidence Intervals.

	Paretic Side		Non Paretic Side
Region	Point Estimate	95% Confidence Interval	Point Estimate	95% Confidence Interval
Hip Extension	0.576	(0.110,0.834)	0.914	(0.765,0.970)
Hip Flexion	0.855	(0.623,0.949)	0.862	(0.638,0.951)
Knee Extension	0.751	(0.406,0.909)	0.741	(0.386,0.905)
Knee Flexion	0.631	(0.194,0.859)	0.795	(0.492,0.926)
Ankle Plantarflexion	0.927	(0.796,0.975)	0.810	(0.523,0.932)
Ankle Dorsiflexion	0.718	(0.344,0.896)	0.859	(0.632,0.950)

Abbreviations: NA.

During hip flexion, there was a significant main effect of method, indicating that 2D measurements slightly overestimated quasi-stiffness by 0.007 Nm/deg/kg [F(1,23.271) = 5.810, p = 0.024]. There was no significant main effect of limb [F(1,16.004) = 4.045, p = 0.061] or significant interaction [F(1,32.951) = 0.003, p = 0.956]. The quasi-stiffness measurements obtained using the 2D method showed good agreement with those from the 3D method for the paretic and non-paretic limbs (Table 2). The Bland-Altman plot of the paretic limb (Figure 3B) showed no indication of proportional bias. The Bland-Altman plot of the non-paretic limb (Figure 4B) suggested proportional bias.

Knee Joint Quasi-Stiffness

During knee extension, there was a significant main effect of method, indicating that 2D measurements slightly underestimated quasi-stiffness by 0.008 Nm/deg/kg [F(1,22.867) = 25.478, p < 0.001]. There was no significant main effect of limb [F(1,17.356) = 1.712, p = 0.208], or significant interaction [F(1,40.119) = 0.084, p = 0.774]. The paretic and non-paretic limbs exhibited good and moderate agreement between the 3D and 2D quasi-stiffness measurements (Table 2). The Bland-Altman plot of the paretic limb (Figure 3C) suggested proportional bias in that higher quasi-stiffness values had more difference between the 3D and 2D quasi-stiffness measurements. The Bland-Altman plot of the non-paretic limb (Figure 4C) showed no indication of proportional bias.

During knee flexion, there was no significant main effect of method [F(1,17.387) = 0.069, p = 0.796] or limb [F(1,11.307) = 1.567, p = 0.236] or interaction [F(1,34.803) = 0.328, p = 0.570]. The paretic and non-paretic limbs indicated moderate and good agreement between 3D and 2D quasi-stiffness measurements (Table 2). The Bland-Altman plots of the paretic limb (Figure 3D) and non-paretic limb (Figure 4D) suggested no signs of proportional bias.

Ankle Joint Quasi-Stiffness

During ankle plantarflexion, there was a significant main effect of method, indicating that 2D measurements slightly overestimated quasi-stiffness by 0.005 Nm/deg/kg [F(1,24.202) = 7.404, p = 0.012]. There was also a significant main effect of limb, indicating that the paretic limb was stiffer than the non-paretic limb by 0.024 Nm/deg/kg [F(1,14.449) = 5.968, p = 0.028]. There was no significant interaction [F(1,32.266) = 0.506, p = 0.482]. Both the paretic and non-paretic limbs exhibited excellent and good agreement, respectively, between the 3D and 2D quasi-stiffness measurements (Table 2). Bland-Altman plots of the paretic (Figure 3E) and non-paretic limb (Figure 4E) showed no indication of proportional bias.

During ankle dorsiflexion, there was no significant main effect of method [F(1,18.365) = 0.478, p = 0.498] or limb [F(1,12.187) = 3.364, p = 0.091] or significant interaction [F(1,36.924) = 0.823, p = 0.370]. The paretic and non-paretic limbs exhibited moderate and good agreement between the 3D and 2D quasi-stiffness measurements (Table 2). Bland-Altman plots of the paretic (Figure 3F) and non-paretic limb (Figure 4F) showed no indication of proportional bias.

Discussion

The purpose of this study was to identify the minimal data required to accurately estimate joint quasi-stiffness in a clinical population. To determine this, we performed a human subjects experiment with stroke survivors during overground walking to collect 3D joint angles and moments. The resulting 3D data were projected through a simulated camera lens to obtain a 2D representation of the dataset. Quasi-stiffness was then calculated with both the 3D and 2D data across different portions of the gait cycle for the hip, knee, and ankle joints. Our results showed that the 2D measurements slightly overestimated quasi-stiffness when compared with 3D measurements; however, the 2D quasi-stiffness measurements showed moderate–excellent agreement with 3D quasi-stiffness and yielded similar conclusions to 3D data regarding the presence or absence of interlimb differences in quasi-stiffness. These findings indicate that 2D quasi-stiffness can provide valid insights, and dedicated low-cost 2D motion capture systems could be developed for measuring quasi-stiffness in clinical settings.

Our findings indicate that 2D quasi-stiffness measurements can serve to predict 3D quasi-stiffness measurements, as the ICCs were all moderate, good, or excellent. While there are no studies to our knowledge that have directly compared 3D to 2D quasi-stiffness, many studies have found that 2D systems can accurately measure sagittal plane joint angles when compared to joint angles from 3D motion capture systems data (Ugbolue et al., 2013; Washabaugh et al., 2022; Widhalm et al., 2024). However, these studies were conducted using healthy individuals, which limits the generalization of their findings to clinical populations. In contrast, our analysis was unique in that it used a population of stroke survivors, many of whom had varying levels of impairment (i.e., more variability). With such a diverse group of individuals, we successfully demonstrated agreement between 2D and 3D quasi-stiffness, further suggesting that 2D quasi-stiffness could be used in clinical populations. Given this finding, and prior research indicating a strong relationship between quasi-stiffness and gait function after stroke, there is high potential for quasi-stiffness to be used as a rehabilitation metric to monitor patient progression or to guide treatment.

Additionally, we found that quasi-stiffness differed during ankle plantarflexion between the paretic and non-paretic limbs. While the overall shape of the 2D (Figure 5A) and 3D (Figure 5B) torque-angle curves remained similar during this motion, the paretic and non-paretic limbs exhibited differences in the ranges for joint angle and torque. Specifically, the paretic limb exhibited a smaller change in joint angle than the non-paretic limb. Since joint angle is the denominator in the calculation of quasi-stiffness, dividing by a smaller change in angle resulted in a higher quasi-stiffness for the paretic limb. While the precise mechanisms underlying this observed increase in quasi-stiffness are not fully understood from this study, the findings suggest that increased quasi-stiffness at the ankle may reflect underlying neuromuscular impairments following stroke. These results highlight the potential utility of quasi-stiffness as a clinically relevant metric for monitoring motor impairment and recovery in stroke rehabilitation.

Figure 5.

Plots Depicting the Average Quasi-Stiffness for the Paretic (Blue Curve) and non-Paretic (Orange Curve) Limbs with (A) 2D Data and (B) 3D Data. for all Plots, the x-Axis is the Joint Angle, θ (in deg) and the y-Axis is the Joint Torque, τ (in Nm/kg). Each Plot has the Torque-Angle Curve for the various Joints. the top Plot is the hip Joint, Middle is the Knee Joint, and Bottom is the Ankle Joint. Note That the Readers are Referred to the Online Version of This Article for Color Interpretations.

Unlike prior studies, this study, for the first time, comprehensively evaluated the quasi-stiffness of the hip, knee, and ankle during both flexion and extension phases of the gait cycle. Studies that have examined the ankle during dorsiflexion (Hinton et al., 2022; Sekiguchi et al., 2012, 2015, 2018) have found that the non-paretic ankle was stiffer than the paretic ankle within this region (Hinton et al., 2022; Sekiguchi et al., 2015, 2018). Our findings were trending towards the opposite conclusion from prior studies, as we found that quasi-stiffness tended to be higher in the paretic ankle when compared with the non-paretic ankle, although this was not significant. Potential reasons for this discrepancy could lie in the variability in stroke subjects and impairment levels between studies. Notably, quasi-stiffness has been found to vary due to differences in gait speed (Crenna & Frigo, 2011; Fang et al., 2023; Safaeepour et al., 2014). The participants in our study had higher average gait speed (0.90 ± 0.19 m/s) compared to the average gait speed seen in the other studies [0.41–0.69 (range)] (Hinton et al., 2022; Sekiguchi et al., 2012, 2015, p. 2018), indicating the differences in stroke characteristics between the studies. Further, given that spasticity is velocity-dependent, higher walking speeds in our participants may have led to more spasticity and thus an increase in quasi-stiffness—although this is just a speculation, as we did not characterize spasticity in this study. Overall, the findings of this paper and of the existing literature indicate there are differences in quasi-stiffness between the paretic and non-paretic limb during ankle dorsiflexion and ankle plantarflexion, which reinforces the idea that quasi-stiffness is altered following a stroke and could be useful to monitor patient progress during post-stroke rehabilitation.

While all joints had moderate to excellent agreement between 2D and 3D quasi-stiffness, there are many reasons why the 2D motion would fail to capture the same quasi-stiffness as 3D. First, the 2D image is unable to capture out-of-plane motions. Following a stroke, many individuals have altered gait biomechanics (e.g., decreased knee flexion and ankle dorsiflexion) (Chen et al., 2005; Sheffler & Chae, 2015; Woolley, 2001), and they often compensate with hip hiking and circumduction (Chen et al., 2005; Kerrigan et al., 2000; Stanhope et al., 2014). These compensatory mechanisms occur in the frontal and transverse planes. Hence, a purely 2D (i.e., sagittal plane) system would not capture these movements, potentially adding inaccuracy to the system for individuals who exhibit more out-of-plane motions. Second, a 2D camera is susceptible to perspective distortions, where objects that are closer to the camera appear larger. Because the markers being tracked do not all align on the plane where the camera was calibrated, there will be errors in the reconstructed position due to the assumption that the motion is occurring at a constant depth. This effect can similarly affect how the center of pressure is viewed by the camera. Lastly, we saw that the ankle joint demonstrated the strongest agreement with the 3D system. This likely occurred because any errors in the center of pressure are magnified at more proximal joints as kinetics are propagated up the kinematic chain; hence, knee and hip torques will have larger errors. In summary, given the potential for out-of-plane motions to affect viewing angle and perspective distortion, further investigation is needed to determine which specific gait deviations can affect 2D quasi-stiffness measurements.

We found that the 2D data generally overestimated quasi-stiffness slightly, except during knee extension, when the system underestimated quasi-stiffness; however, although these differences were often significant, their magnitude was minimal (≈0.004–0.007 Nm/deg/kg). Moreover, we found that ICC generally had good agreement between the 2D and 3D data, and 2D data yielded similar conclusions to the 3D data. These findings indicate that there is potential to develop dedicated low-cost 2D systems for estimating quasi-stiffness and monitoring rehabilitation progress in the clinic. Such a system would greatly enhance the clinician's ability to estimate joint stiffness compared to currently available tools. Current clinical solutions rely on qualitative metrics (e.g., MAS), which limit the ability to detect subtle changes in joint stiffness and have been found to have poor interrater reliability (Ansari et al., 2008; Blackburn et al., 2002). Additionally, these tools are performed while the patient is at rest (Figure 6A) and do not provide functional joint stiffness measurements. While quasi-stiffness has been proposed to estimate functional joint stiffnesses, current proposed systems are not clinically feasible, as optical motion capture and force plates are used to measure the joint kinematics and kinetics required to calculate quasi-stiffness. This equipment is highly expensive and requires dedicated lab space, greatly reducing the likelihood that a clinic could implement a 3D setup (Figure 6B). On the other hand, a dedicated 2D system could overcome this barrier by reducing equipment costs in a package that is portable. For example, a low-cost 2D system could be built using a simple webcam and 2-axis force plate (Figure 6C). A 2D force plate could be created with an inexpensive load cell at each corner to measure vertical ground reaction forces, along with a single shear load cell in the center to measure anterior-posterior forces. This configuration would allow for force measurements in two directions and allow calculation of the center of pressure in the anterior-posterior direction (Vaughn, 1999). Together, these components would provide the necessary measurements to calculate sagittal plane joint moments. Overall, creation of low-cost 2D systems would greatly increase clinical feasibility by reducing the cost of quasi-stiffness analysis.

Figure 6.

A schematic depicting different ways to measure joint stiffness and the pros and cons of each system in relation to clinical feasibility. The current clinical standard (A) is for clinicians to perform the Modified Ashworth Scale (MAS) where the clinician moves the joint through its range of motion while the patient lies supine at rest. One potential method for measuring joint stiffness is to estimate it with quasi-stiffness. The current system, commonly used to estimate quasi-stiffness (B) utilizes optical motion capture and force plates. An example of a more clinically feasible low-cost 2D system (C) could be comprised of a simple webcam and a 2D force plate. The green circles represent markers that could be placed on the hip, knee, ankle, and toe for the camera to track. Note that the readers are referred to the online version of this article for color interpretations.

There are some limitations to this study. First, the 2D data used in this study was simulated based on data measured from a 3D motion capture system, rather than being collected directly from a 2D system. This was performed to ensure that both 2D and 3D measurements came from the exact same motion, thereby limiting any variability due to differences in the performance between trials. While simulating a 2D camera system did allow us to replicate errors that would arise due to perspective distortion, our simulation did not capture many of the practical limitations when performing 2D motion capture (e.g., marker occlusion, distortion, noise, etc.). Hence, the results of this study should be considered as a proof-of-concept, recognizing that the accuracy estimates represent a best-case scenario for a real-world 2D system, and that any future implementation of a 2D system would require additional validation. Second, Bland-Altman plots were created to determine if there was bias in our 2D quasi-stiffness estimates across the range of quasi-stiffness values. Overall, the 2D and 3D results showed agreement. However, to determine whether two measurement systems are truly interchangeable, the limits of agreement from Bland-Altman plots must fall within a minimal clinically important difference (MCID). Our study cannot confidently make that determination, as no established MCID currently exists for quasi-stiffness. Determining such a threshold would require further longitudinal research to examine how quasi-stiffness changes over the course of treatment and recovery, along with studies on the repeatability and reliability of these measurements. Finally, it is not fully clear if 2D quasi-stiffness will continue to be valid for patients who have severe gait deviations and exhibit more out-of-plane movements. We note that our current analysis had stroke survivors with a wide range of impairment and still showed moderate–excellent agreement between 2D and 3D quasi-stiffness. Further, the sagittal plane is aligned with the patient's direction of movement; hence, even if 2D measurements do not capture out-of-plane motions, they will still contain information that is relevant to the patient's overall function (e.g., gait speed).

In conclusion, our findings indicate moderate–excellent agreement between quasi-stiffness estimates obtained from 2D and 3D systems, and the 2D system was able to detect interlimb differences in quasi-stiffness similar to those identified by the 3D system in stroke survivors. These results support the potential of 2D systems as a valid alternative for measuring quasi-stiffness during the stance phase of the gait in a clinical setting, thereby improving the feasibility of quasi-stiffness evaluation in clinical practice. While this study focused on stroke survivors, the use of 2D systems to measure quasi-stiffness could be extended to other patient populations with gait impairments. Clinicians could potentially use quasi-stiffness as a tool to better prescribe and monitor the treatment for millions of individuals with mobility and gait impairments.

Footnotes

Acknowledgements

None

ORCID iDs

Sierra A Foley

Chandramouli Krishnan

Edward P Washabaugh

CrediT Roles

Sierra A. Foley: Data collection, Data curation; Formal analysis; Investigation; Methodology; Software; Visualization; Writing – original draft; Writing – review and editing. Andrew F. Moul: Recruitment, Writing – review and editing., Chandramouli Krishnan: Conceptualization, Writing – review and editing. Edward P. Washabaugh: Conceptualization; Data collection, Formal Analysis; Methodology; Project administration; Supervision; Writing – original draft; Writing – review and editing.

Funding

This research was supported, in part, with funding from the National Institutes of Health (NIH) (Grant Numbers R41-HD111289 and R01-HD111567).

Eunice Kennedy Shriver National Institute of Child Health and Human Development, (grant number R01-HD111567, R41-HD111289).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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