Ultrasound Displacement Tracking Techniques for Post-Stroke Myofascial Shear Strain Quantification

Abstract

Ultrasound shear strain is a potential biomarker of myofascial dysfunction. However, the quality of estimated shear strains can be impacted by differences in ultrasound displacement tracking techniques, potentially altering clinical conclusions surrounding myofascial pain. This work assesses the reliability of four displacement estimation algorithms under a novel clinical hypothesis that the shear strain between muscles on a stroke-affected (paretic) shoulder with myofascial pain is lower than that on the non-paretic side of the same patient. One of the algorithms is Search, which is a common window-based method that determines displacements by searching for maximum normalized cross-correlations within windowed data. OVERWIND-Search, SOUL-Search, and L1-SOUL-Search fine-tune the Search initial estimates by optimizing cost functions comprising data and regularization terms, utilizing L1-norm-based first-order regularization, L2-norm-based first- and second-order regularization, and L1-norm-based first- and second-order regularization, respectively. After initial validation with simulations, these four approaches were evaluated with in vivo data acquired from ten research participants with myofascial post-stroke shoulder pain. SOUL-Search and L1-SOUL-Search most accurately and reliably estimate shear strain relative to our clinical hypothesis, when validated with visual inspection of ultrasound cine loops and quantitative T1ρ magnetic resonance imaging. In addition, L1-SOUL-Search produced the most reliable displacement tracking performance by generating lateral displacement images with smooth displacement gradients (measured as the mean and variance of displacement derivatives) and sharp edges (which enables distinction of shoulder muscle layers). Based on these results and the associated analyses, L1-SOUL-Search emerges as the most suitable option among the four investigated methods to investigate myofascial pain and dysfunction, despite the drawback of slow runtimes, which can potentially be resolved with a deep learning solution.

I. Introduction

Ultrasound shear strain is a potential biomarker of tissue pathologies, such as tumor malignancy [1], [2] and myofascial dysfunction [3]-[7]. Displacement tracking [8], [9] is a critical step in any shear strain imaging framework, which is typically challenged by the ill-posed nature of ultrasound data [10]. Window-based tracking [11]-[15] and energy optimization-based regularized tracking [16]-[22] are two mainstream pathways to accomplish the non-trivial task of displacement estimation from ultrasound data.

A window-based displacement tracking technique [23]-[28] divides a pre-displaced ultrasound frame into several overlapping data kernels. Each window (or kernel) in the pre-displaced frame is sought in the post-displaced frame to find the best match. Maximum normalized cross-correlation [13], [29], [30] or zero-phase-crossing [31]-[33] between pre-displaced and post-displaced ultrasound data segments are typically used as the metric to determine the best match. This window-search process is based on the assumption that all samples in a data window undergo the same amount of spatial displacement [34]. Therefore, segmenting the ultrasound frame into large-sized windows leads to a smooth and accurate displacement estimate [34]. Conversely, small data windows achieve good resolution in displacement estimation by sacrificing spatial smoothness and accuracy [34].

Although window-based algorithms are straightforward, they are generally known to produce noisy displacement results [7]. Energy optimization-based regularized displacement tracking algorithms [35]-[40] aim to resolve this limitation in two sequential steps [8]. First, the displacement tracking problem is modeled as a non-linear regularized cost function. Second, the cost function formulated in the first step is optimized. The cost function attempts to account for the tissue displacement physics and typically comprises data fidelity, displacement continuity, and tissue mechanics-driven physical constraints [21]. Global ultrasound elastography (GLUE) [41] is an energy-based regularized tracking technique that optimizes a non-linear cost function consisting of a data amplitude fidelity term and first-order displacement regularization terms. However, first-order regularization often leads to suboptimal noise suppression and low contrast between different components of a displacement image [42], because it does not represent the true tissue deformation physics [43]. To account for the deformation physics, second-order ultrasound elastography (SOUL) [42] has been proposed that formulates a physicsbased regularizer consisting of both first- and second-order displacement derivatives and resolves the noise-suppression and contrast issues of GLUE. A drawback of both GLUE and SOUL is that they consider the L2-norm of displacement derivatives to formulate respective regularizers, which often blurs boundaries between different tissue types by overpenalizing displacement discontinuities. To strike a balance between displacement continuity and discontinuity, two alternative tracking methods, namely total variation regularization and window-based time delay estimation (OVERWIND) [44] and L1-norm-based SOUL (L1-SOUL) [45], have replaced the L2-norms in GLUE and SOUL, respectively, with L1-norms.

With respect to a clinical application of interest, dysregulation of hyaluronan (HA) homeostasis can lead to HA accumulation in muscles after a stroke, causing muscle stiffness, myofascial dysfunction, and pain [46]. We hypothesized that in patients with post-stroke shoulder pain, the HA accumulation would lead to sticking together of muscles, preventing shear motion between them [46]. From this perspective, Langevin et al. [3], Raghavan et al. [4], and Zhao et al. [5] quantified shear strain in research participants suffering from chronic muscle pain. In each study, a cross-correlation-based window-tracking algorithm supported the clinical hypothesis that muscles experiencing pain are stiffer than otherwise healthy muscles and exhibit lower shear strains. However, these complementary clinical conclusions depend on the findings of a single displacement tracking algorithm per study. Displacement estimation accuracy is instrumental in trustworthy in vivo shear strain quantification, and hence, different tracking techniques can potentially lead to different clinical interpretations.

In this paper, we compare the displacement tracking reliability of a standard cross-correlation-based window-search technique [13] (which we name Search) and three energy-based regularized techniques named OVERWIND-Search, SOUL-Search, and L1-SOUL-Search when calculating shear strains in vivo between the pectoralis major (PMA) and pectoralis minor (PMI) muscles in patients with myofascial post-stroke shoulder pain. OVERWIND-Search, SOUL-Search, and L1-SOUL-Search use OVERWIND [44], SOUL [10], and L1-SOUL [45] to refine the initial displacement field obtained by Search. Our objective is to combine the recent technical and clinical advances noted above to investigate the reliability of displacement tracking results that form the basis of a novel clinical hypothesis that the range (i.e., difference between the maximum and minimum) of measured shear strains between the PMA and PMI muscles in patients with post-stroke shoulder pain are lower on the painful paretic (i.e., stroke-affected) shoulder, relative to the non-paretic (i.e., stroke-unaffected) shoulder of each patient [4]-[7].

A preliminary version of this work [6], comparing Search and SOUL-Search on three patients with post-stroke shoulder pain, was presented and published in SPIE Medical Imaging, San Diego, CA, 2024. The current manuscript investigates four displacement tracking algorithms (Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) instead of only two (Search and SOUL-Search), implemented on simulated data (also referred to as in silico data), which provide ground truth information, followed by investigations with ten patients (i.e., in vivo data). In addition, this manuscript details the comparative implementation methods and presents an indepth results analysis employing comprehensive quantitative metrics. Furthermore, to explore fast and accurate alternatives for displacement estimation, we evaluated the feasibility of using recurrent all-pairs field transforms (RAFT) [47], a deep learning-based optical flow model, for real-time post-stroke myofascial displacement tracking, relative to the best-ranked algorithm determined herein. Relative to previous work, the new contributions of this paper include:

• Highlighting the potential of a conventional window-based technique (Search) and three energy-based regularized techniques (OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) for accurate lateral displacement tracking in the context of a new application (i.e., providing a biomarker for post-stroke myofascial dysfunction).

• Identifying an optimal displacement tracking algorithm to enable robust shear strain estimation as a quantitative biomarker for post-stroke myofascial dysfunction.

• Evaluating ultrasound displacement tracking performance with respect to a novel clinical hypothesis linking myofascial shear strain to post-stroke shoulder pain.

• Demonstrating the feasibility of deep learning-based displacement tracking for real-time shear strain biomarker calculation in myofascial health assessment, relative to the best-performing tracking algorithm identified herein.

The remainder of this manuscript is organized as follows. Section II describes our in silico and in vivo data acquisition processes, followed by descriptions of the four displacement tracking techniques being compared, parameter tuning and benchmarking methods utilized, shear strain quantification based on the displacement tracking results, evaluation metrics, statistical analyses, and methods to comparatively rank the four investigated tracking algorithms. Section III presents related qualitative and quantitative results. Section IV discusses the key findings, limitations, and biomedical impact of this study. Finally, Section V summarizes our major conclusions.

II. Methods

A. Simulated Data

To simulate lateral muscle movement and compare the resulting tracking accuracies with a known ground truth, two controlled simulations were conducted. In the first simulation, opposing lateral strains of 0.5%, 1%, 2%, and 4% were applied to a two-layer phantom. For simplicity, axial tissue motion resulting from these lateral deformations was not simulated. The second simulation aimed to model a more realistic scenario by including a fascial layer between two muscle layers. In this case, both lateral and axial motions were generated from opposing lateral strains of 0.5% applied to the muscle layers. In both simulations, the closed-form equations reported in the Supplementary Material of previous work [48] were used to generate the deformation fields. Pre- and post-deformed ultrasound frames were simulated with Field II [49]. The transmit and sampling frequencies were set to 7.27 MHz and 15 MHz, respectively.

B. In Vivo Imaging of Human Participants

To determine the translatability of simulation results to clinical data, ultrasound data were acquired during movements of painful paretic (i.e., stroke-affected) and non-paretic shoulders. In particular, ten research participants with a history of stroke (referred to as P1-P10, age range: 36-68 years, three female, seven male) underwent 30 degrees of internalexternal shoulder rotation at a periodic rate of 0.5 Hz using a bimanual arm trainer (Mirrored Motion Works Inc., Raleigh, NC), which was operated as designed [50] to provide gentle and controlled passive shoulder rotation. During the periodic rotations, the forearm and wrist rested on the arm trough of the bimanual trainer, ensuring a relatively comfortable examination. A Sawyer robot arm (Rethink Robotics, now Bochum, NRW, Germany) attached to an L15 ultrasound scanner (Clarius Moblie Health, Vancouver, BC) was placed to visualize the PMA and PMI muscles, as illustrated in Fig. 1. The robot arm enabled ultrasound data acquisition with a stabilized ultrasound transducer, thereby preventing unwanted motion of the transducer, which would otherwise influence ultrasound tracking results.

Fig. 1:

Illustration of in vivo ultrasound data acquisition from a research participant with post-stroke shoulder pain.

To perform ultrasound imaging, a 20-second cine loop of envelope-detected ultrasound data was acquired with a frame rate of 24 Hz, per shoulder per participant. The ultrasound scanner sampling frequency was set to 15 MHz, and the transmit frequency was 14 MHz for P1-P6 and 12 MHz for P7-P10. The number of ultrasound frames in each cine loop ranged from 230 to 294. The transducer consisted of 192 elements, and the selected transmit focus ranged from 15 mm to 18.65 mm per acquisition. In addition, T1ρ MRI of the shoulder muscles of a subset of these participants (described in Section II-F) was performed to determine the T1ρ relaxation times, which reflect the relative amount of HA accumulation in the muscles on the paretic and non-paretic sides [51]. The Johns Hopkins Medicine Institutional Review Board (Protocol No. IRB00354876) approved this study with written consent provided by each participant.

C. Displacement Tracking Algorithms

1). Search:

Let $I_1(i,j)\in {\mathbb{R}}^{m\times n}$ and $I_2(i,j)\in {\mathbb{R}}^{m\times n}$ represent two ultrasound envelope frames acquired before and after displacement, where $i$ and $j$ denote axial and lateral positions, respectively. The axial and lateral displacement fields between $I_1$ and $I_2$ can be calculated using displacement tracking techniques. Search [13] is a conventional window-based displacement tracking algorithm that determines axial and lateral displacements, $a_{i,j}$ and $l_{i,j}$, respectively, by locating the spatial lag corresponding to the maximum normalized cross-correlation between kernels within $I_1$ and $I_2$. The displacement fields obtained by Search (i.e., $a_{i,j}$ and $l_{i,j}$) can be refined by OVERWIND [44], SOUL [10], and L1-SOUL [52] through optimization of the regularized cost functions introduced below.

2). OVERWIND-Search Cost Function:

OVERWIND-Search formulates a non-linear cost function $C_{ovs}(\cdot )$ consisting of a data fidelity term and L1-norm-based first-order regularization terms to find the incremental displacement fields $\Delta a_{i,j}$ and $\Delta l_{i,j}$.

$$\begin{gathered} C_{ovs}(\Delta a_{1,1},\dots ,\Delta a_{m,n},\Delta l_{1,1},\dots ,\Delta l_{m,n})= \\ \sum _{j=1}^n\sum _{i=1}^m\{D_{ovs}(i,j,a_{i,j},l_{i,j},\Delta a_{i,j},\Delta l_{i,j})\}+w_{f1}{\alpha }_{1s}{\Vert {\partial }_{y}a\Vert }_1 \\ +w_{f2}{\alpha }_{2s}{\Vert {\partial }_{x}a\Vert }_1+w_{f1}{\beta }_{1s}{\Vert {\partial }_{y}l\Vert }_1+w_{f2}{\beta }_{2s}{\Vert {\partial }_{x}l\Vert }_1 \end{gathered}$$ (1)

where ${\partial }_{y}a$ and ${\partial }_{x}a$ denote the first-order derivatives of the axial displacement field in the axial and lateral directions, respectively, whereas ${\partial }_{y}l$ and ${\partial }_{x}l$ refer to the first-order derivatives of the lateral displacement field in the axial and lateral directions, respectively, and $w_{f1}$, $w_{f2}$, ${\alpha }_{1s}$, ${\alpha }_{2s}$, ${\beta }_{1s}$, and ${\beta }_{2s}$ are tunable regularization weights. $D_{ovs}(\cdot )$ is the data fidelity term defined as:

$$\begin{gathered} D_{ovs}(i,j,a_{i,j},l_{i,j},\Delta a_{i,j},\Delta l_{i,j}) \\ =\frac{1}{(2w_a+1)(2w_l+1)}\sum _{k=-w_a}^{k=w_a}\sum _{r=-w_l}^{r=w_l}[I_1(i+k,j+r)- \\ {I_2(i+k+a_{i,j}+\Delta a_{i,j},j+r+l_{i,j}+\Delta l_{i,j})]}^2 \end{gathered}$$ (2)

where $w_a$ and $w_l$ are tunable parameters, determining the size of the window OVERWIND-Search considers around each sample while constructing the data fidelity term.

3). SOUL-Search Cost Function:

SOUL-Search devises a non-linear cost function $C_{ss}(\cdot )$ that comprises a data fidelity term and L2-norm-based first- and second-order regularization terms to calculate the incremental displacement fields $\Delta a_{i,j}$ and $\Delta l_{i,j}$.

$$\begin{gathered} C_{ss}(\Delta a_{1,1},\dots ,\Delta a_{m,n},\Delta l_{1,1},\dots ,\Delta l_{m,n})= \\ \sum _{j=1}^n\sum _{i=1}^m\{D_{ss}(i,j,a_{i,j},l_{i,j},\Delta a_{i,j},\Delta l_{i,j})\}+{\alpha }_{1s}{\Vert {\partial }_{y}a\Vert }_2^2+ \\ {\alpha }_{2s}{\Vert {\partial }_{x}a\Vert }_2^2+{\beta }_{1s}{\Vert {\partial }_{y}l\Vert }_2^2+{\beta }_{2s}{\Vert {\partial }_{x}l\Vert }_2^2+w_{s1}{\alpha }_{1s}{\Vert {\partial }_y^{2}a\Vert }_2^2+ \\ {w}_{s1}{\alpha }_{2s}{\Vert {\partial }_x^{2}a\Vert }_2^2+w_{s1}{\beta }_{1s}{\Vert {\partial }_y^{2}l\Vert }_2^2+w_{s2}{\beta }_{2s}{\Vert {\partial }_x^{2}l\Vert }_2^2 \end{gathered}$$ (3)

where ${\partial }_y^{2}a$ and ${\partial }_x^{2}a$ are the second-order derivatives of the axial displacement field in the axial and lateral directions, respectively, whereas ${\partial }_y^{2}l$ and ${\partial }_x^{2}l$ are the second-order derivatives of the lateral displacement field in the axial and lateral directions, respectively. $w_{s1}$ and $w_{s2}$ refer to two tunable weights. $D_{ss}(\cdot )$ denotes the data fidelity term defined as:

$$\begin{gathered} D_{ss}(i,j,a_{i,j},l_{i,j},\Delta a_{i,j},\Delta l_{i,j}) \\ ={[I_1(i,j)-I_2(i+a_{i,j}+\Delta a_{i,j},j+l_{i,j}+\Delta l_{i,j})]}^2 \end{gathered}$$ (4)

4). L1-SOUL-Search Cost Function:

Similar to $C_{ss}$ in Eq. (3), the L1-SOUL-Search cost function $C_{l1ss}$ consists of a data term and first- and second-order regularization terms. However, instead of L2-norm employed in Eq. (3), $C_{l1ss}$ incorporates the L1-norm of displacement derivatives to formulate the regularization terms. $C_{l1ss}$ is defined below.

$$\begin{gathered} C_{l1ss}(\Delta a_{1,1},\dots ,\Delta a_{m,n},\Delta l_{1,1},\dots ,\Delta l_{m,n})= \\ \sum _{j=1}^n\sum _{i=1}^m\{D_{ss}(i,j,a_{i,j},l_{i,j},\Delta a_{i,j},\Delta l_{i,j})\}+w_{f1}{\alpha }_{1s}{\Vert {\partial }_{y}a\Vert }_1 \\ +w_{f1}{\alpha }_{2s}{\Vert {\partial }_{x}a\Vert }_1+w_{f1}{\beta }_{1s}{\Vert {\partial }_{y}l\Vert }_1+w_{f1}{\beta }_{2s}{\Vert {\partial }_{x}l\Vert }_1 \\ +w_{s1}{\alpha }_{1s}{\Vert {\partial }_y^{2}a\Vert }_1+w_{s1}{\alpha }_{2s}{\Vert {\partial }_x^{2}a\Vert }_1+w_{s1}{\beta }_{1s}{\Vert {\partial }_y^{2}l\Vert }_1 \\ +w_{s2}{\beta }_{2s}{\Vert {\partial }_x^{2}l\Vert }_1 \end{gathered}$$ (5)

5). Cost Function Optimization:

The optimizability of (Eqs. 1), (3), and (5) is hindered by the non-linearity of data fidelity terms [i.e., $D_{ovs}(\cdot )$ in Eq. (2) and $D_{ss}(\cdot )$ in Eq. (4)] due to the presence of our optimization variables $\Delta a_{i,j}$ and $\Delta l_{i,j}$ within $I_2(\cdot )$. As detailed in the original publications of OVERWIND [44], SOUL [10], and L1-SOUL [52], the non-linearity challenge in (Eqs. 1), (3), and (5) is addressed by performing the first-order Taylor series expansion of $I_2(\cdot )$ and extracting $\Delta a_{i,j}$ and $\Delta l_{i,j}$ out of $I_2$ to make a linear approximation. Another barrier to the optimization of (Eqs. 1) and (5) is the non-differentiability of the L1-norm due to the presence of the modulus function in its definition (i.e., $\mid x\mid$ is not differentiable at the sharp corner at $x=0$). As discussed in previous work [44], [52], the sharp corner of the modulus function can be smoothed to approximate the L1-norm with its differentiable version, resulting in a differentiability approximation.

After making the linear and differentiability approximations summarized above, the modified versions of $C_{ovs}$, $C_{ss}$, and $C_{l1ss}$ (i.e., $C_{ovs,mod}$, $C_{ss,mod}$, and $C_{l1ss,mod}$, respectively) were iteratively optimized, setting their differentials with respect to $\Delta a_{i,j}$ and $\Delta l_{i,j}$ to zero. Algebraic operations on the resulting equations corresponding to differentiation of each cost function led to the following closed-form representation:

$$(H+D)\Delta d=H_1\mu -Dd$$ (6)

where $H\in {\mathbb{R}}^{2mn\times 2mn}$ and $H_1\in {\mathbb{R}}^{2mn\times 2mn}$ denote symmetric tridiagonal and diagonal matrices, respectively, consisting of the derivatives of $I_2$ originating from its Taylor series expansion and the associated functions, such as squares of the axial and lateral derivatives, and multiplications of axial and lateral derivatives. $D$ is a $2mn\times 2mn$ sparse matrix containing functions of regularization weights, $d\in {\mathbb{R}}^{2mn\times 1}$ is a vector containing the initial displacement estimates provided by Search (i.e., $a$ and $l$) [13], $\Delta d\in {\mathbb{R}}^{2mn\times 1}$ is a vector containing the incremental displacements (i.e., $\Delta a$ and $\Delta l$), and $\mu \in {\mathbb{R}}^{2mn\times 1}$ is a vector that consists of the data residuals (i.e., the sample-wise differences between $I_1$ and $I_2$ warped with $a$ and $l$).

D. Parameter Tuning and Benchmarking

Following previous myofascial shear strain quantification research [4]-[7], we calculated axial and lateral displacement fields between five simulated envelope-detected ultrasound frame pairs and two consecutive envelope-detected ultrasound frames acquired from non-paretic and paretic shoulders of P1-P10 using Search. The correlation kernel size was 16 × 8 samples (i.e., 0.8 mm × 1.04 mm), and the search window size was 48 × 24 samples (i.e., 2.4 mm × 3.12 mm). Optimal parameter sets for OVERWIND-Search, SOUL-Search, and L1-SOUL-Search were selected based on two consecutive ultrasound envelope frames from the non-paretic shoulder of P3, due to the corresponding Search lateral displacement image, which clearly distinguished PMA from PMI muscles. Axial and lateral displacement estimates between the selected frames were calculated using different parameter combinations of OVERWIND-Search, SOUL-Search, and L1-SOUL-Search. For each of these three tracking techniques, the parameter combination providing a visually smooth lateral displacement image with a clear distinction between PMA and PMI was selected as the optimal parameter set. Table I provides the optimal parameter sets, which were utilized to calculate axial and lateral displacement fields in simulated and in vivo data using OVERWIND-Search, SOUL-Search, and L1-SOUL-Search.

TABLE I:

Applicable OVERWIND-Search, SOUL-Search, and L1-SOUL-Search parameters.

	OVERWIND-Search	SOUL-Search	L1-SOUL-Search
${\alpha }_{1s}$	5	1	3
${\alpha }_{2s}$	1	1	1
${\beta }_{1s}$	2	0.4	1.2
${\beta }_{2s}$	1.5	1.5	1.5
$w_{f1}$	0.01	-	0.001
$w_{f2}$	0.02	-	-
$w_{s1}$	-	5	0.035
$w_{s2}$	-	20	0.175
$w_a$	3	-	-
$w_l$	3	-	-

E. Shear Strain Quantification of In Vivo Data

Two regions of interest (ROIs), sized 5 mm × 5 mm, located at the same lateral positions, were manually selected on the PMA and PMI within the first B-mode image of cine loops acquired per shoulder per participant. The lower boundary of the PMA ROI and the upper boundary of the PMI ROI were placed at 2 mm axial distances from the hyperechoic fascial shear plane. To calculate myofascial shear strain, inter-frame average lateral displacements within the PMA and PMI ROIs were temporally accumulated for each cine-loop acquisition. To estimate the instantaneous shear strains, the cumulative average lateral displacements within the PMA and PMI ROIs (i.e., $l_{\mathrm{c},\text{pma}}$ and $l_{\mathrm{c},\text{pmi}}$, respectively), determined with each displacement tracking technique, were calculated as follows:

$$\text{Shear}\text{strain}=\frac{l_{\mathrm{c},\text{pmi}}-l_{\mathrm{c},\text{pma}}}{D}\times 100\%$$ (7)

where $D$ denotes the axial distance between the ROI centers.

F. Evaluation Metrics

To quantify the in silico lateral displacement image quality and validate expectations for in vivo displacement results, the root-mean-square error (RMSE), mean absolute error (MAE), and mean structural similarity (MSSIM) [53] were employed. In addition, for both in silico and in vivo results, data similarity was quantified using normalized cross-correlation (NCC) between the pre-displaced frame and the post-displaced frame warped with the estimated axial and lateral displacement fields. Displacement estimation smoothness was quantified using four metrics: (1) mean lateral derivative of lateral displacement (${\mu }_{xx}$), (2) mean axial derivative of lateral displacement (${\mu }_{xy}$), (3) variance of the lateral derivative of lateral displacement (${\sigma }_{xx}^2$), and (4) variance of the axial derivative of lateral displacement (${\sigma }_{xy}^2$) within a 5 mm × 5 mm ROI, with the distal boundary placed 2 mm proximal to either the intersection between the simulated muscle layers (when no fascia layer was included), the middle of the fascia layer (when present in simulated data), or the in vivo shear plane (i.e., the hyperechoic region between the PMA and PMI muscles observed in the acquired ultrasound B-mode images). These metrics were chosen to balance the data similarity and smoothness constraints that are critical to any inverse problem solver [54].

To assess in vivo performance, results from each tracking technique were additionally qualitatively evaluated using the visual appearance of 229-293 estimated lateral displacement images per participant. Four visual features were assessed: (1) smoothness, (2) distinction between PMA and PMI, (3) displacement gradient, and (4) sharpness of the fascial shear plane between PMA and PMI. The greater the presence of these features, the better the technique was considered, as these features enable accurate visual assessment of lateral displacement images for clinical decision making. Based on the qualitative appearance of each feature in each assessed image produced, the four tracking techniques were ranked by the first-author (M.A.).

To assess our clinical hypothesis that the range of measured shear strains between PMA and PMI muscles within the paretic shoulder is lower than that within the non-paretic shoulder of a stroke patient, the local maxima and minima of shear strain values corresponding to each acquired cine loop were calculated. The time instances corresponding to local maxima and minima were calculated using MATLAB functions islocalmax and islocalmin, respectively. Then, the differences between consecutive shear strain local maxima and minima (DMM) within a cine loop were calculated, represented as $dif{f}_1$, $dif{f}_2$,…, $dif{f}_n$ in Fig. 2. The median DMM (MDMM) per shoulder per participant per technique was additionally calculated to determine the performance of each displacement tracking technique relative to our clinical hypothesis.

Fig. 2:

Calculation procedure for MDMM. The green curve represents the instantaneous shear strains corresponding to the ultrasound frames acquired from one shoulder of one research participant.

When a particular technique contradicted the clinical hypothesis or when multiple techniques for the same patient produced a different agreement with the clinical hypothesis, the acquired ultrasound cine loops were visually assessed for relative muscle mobility. In addition, T1ρ magnetic resonance imaging (MRI) of the shoulder muscles of these participants was performed, as described in Section II-B.

G. Statistical Analyses and Overall Ranking Methods

For each of the in vivo quantitative metrics (i.e., NCC, ${\mu }_{xx}$, ${\mu }_{xy}$, ${\sigma }_{xx}^2$, ${\sigma }_{xy}^2$, DMM), the Shapiro-Wilk test of normality (5% significance level) determined whether the associated data were drawn from a normal distribution. A post-hoc multiple comparison analysis following the Kruskal-Wallis statistical test (5% significance level) was performed per NCC, ${\mu }_{xx}$, ${\mu }_{xy}$, ${\sigma }_{xx}^2$, and ${\sigma }_{xy}^2$ metric (aggregating the paretic and non-paretic shoulders of P1-P10) and for the DMM metric (disaggregating per paretic and non-paretic shoulders). Median values of each aggregated result per tracking technique were then calculated for each metric. For cases where the medians differed, the statistical significance of any differences in ranks (produced by the post-hoc multiple comparison analysis) of aggregated results were determined.

Any two techniques demonstrating no statistically significant difference were considered to perform similarly (or were considered unable to differentiate between the paretic and non-paretic shoulders with respect to DMM). Otherwise, the medians showing statistically significant differences (when aggregating paretic and non-paretic shoulder results per participant per NCC, ${\mu }_{xx}$, ${\mu }_{xy}$, ${\sigma }_{xx}^2$, and ${\sigma }_{xy}^2$ metric and disaggregating paretic and non-paretic shoulder results per participant per displacement tracking technique for the DMM metric) determined the superior technique and the method most supportive of the clinical hypothesis, respectively. In particular, a greater NCC was considered better, as it represents a more accurate mapping between the pre-displaced and post-displaced frames through the estimated displacement fields. Greater displacement derivative means (i.e., ${\mu }_{xx}$ and ${\mu }_{xy}$) and lower displacement derivative variances (i.e., ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$) were considered better, representing large and consistent displacement gradients, respectively, which are expected from the controlled data acquisition setup. A lower MDMM was interpreted as greater muscle stiffness and myofascial dysfunction.

The four tracking algorithms were ranked from 1 to 4, based on the in vivo metrics (i.e., qualitative observation of the lateral displacement images, NCC, mean ${\mu }_{xx}$ and ${\mu }_{xy}$, mean ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$, and MDMM). Ranks were then averaged across these five metrics to determine the overall rank of each tracking technique. The technique corresponding to the lowest-numbered (i.e., best) rank was determined to be the most promising displacement tracking technique for post-stroke myofascial shear strain quantification.

H. Runtime Challenge and Opportunity

Runtimes required by Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search to calculate the displacement fields between two ultrasound frames were measured. These methods are known to produce slow runtimes that are not suitable for real-time implementation. To explore faster alternatives to the four tracking algorithms, we investigated to potential of recurrent all-pairs field transforms (RAFT) [7], [47], a deep learning-based optical flow model, to estimate frame-to-frame displacements in the non-paretic and paretic sides of P1-P10. A pre-trained RAFT model (trained on FlyingChairs [55] and FlyingThings3D [56] datasets, and fine-tuned on the Sintel [57] dataset) was used, with L1 loss [7], [47] applied during training and fine-tuning. The runtime of RAFT to infer displacements between a frame pair was additionally compared to assess real-time displacement tracking feasibility.

III. Results

A. Simulation Results

1). Simulation Data with Lateral Motion:

Fig. 3 shows ground truth and estimated lateral displacement maps for the simulated dataset with lateral motion, demonstrating visual discrimination of layers with positive and negative displacements. The Search displacement estimates appear noisy. The regularized optimization-based techniques, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search, present smoother displacement images compared to Search. The boundaries between the positive and negative displacement regions are better visualized with Search and OVERWIND-Search estimates than SOUL-Search and L1-SOUL-Search estimates. Search has smoother displacement estimates with 2% deformation relative to 0.5% and 1% deformation. The energy-based techniques yield lower visual bias (compared to the ground truth) in with 2% deformation relative to 0.5% and 1% deformation cases. With 4% deformation, the four techniques fail to generate correct displacement estimates in the high-displacement regions (i.e., left region on the upper layer and right region on the lower layer).

Fig. 3:

Lateral displacement simulation results with lateral motion and 0.5%, 1%, 2%, or 4% lateral deformation.

Table II reports the quantitative metrics rendered by the four competing techniques. OVERWIND-Search, SOUL-Search, and L1-SOUL-Search produce comparable RMSE, MAE, and MSSIM values between estimated and ground truth displacements, resulting in up to 68%, 63.64%, and 117.95% improvements, respectively, relative to Search. These four techniques obtain similar NCC except for the 4% deformation case where Search achieves 9.41% higher NCC than the energy-based techniques. Search renders up to 53 and 168 times higher ${\mu }_{xx}$ and ${\mu }_{xy}$, respectively, relative to OVERWIND-Search, SOUL-Search, and L1-SOUL-Search. In general, L1-SOUL-Search obtains the lowest ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$ among the four techniques.

TABLE II:

Quantitative metrics obtained with the simulated data with lateral motion.

	RMSE (mm)	MAE (mm)	MSSIM	NCC	${\mu }_{xx}$	${\mu }_{xy}$	${\sigma }_{xx}^2$	${\sigma }_{xy}^2$
0.5% deformation
Search	0.25	0.14	0.58	0.95	0.53	0.63	2.64	7.24
OVERWIND-Search	0.08	0.07	0.85	0.95	0.01	0.01	2.05 × 10−5	1.68 × 10−4
SOUL-Search	0.08	0.07	0.85	0.95	0.01	0.02	1.02 × 10−5	3.80 × 10−4
L1-SOUL-Search	0.08	0.07	0.85	0.95	0.01	0.01	2.45 × 10−6	1.17 × 10−4
1% deformation
Search	0.37	0.22	0.39	0.94	1.19	1.68	6.22	20.33
OVERWIND-Search	0.14	0.08	0.84	0.94	0.04	0.01	2.36 × 10−4	3.29 × 10−4
SOUL-Search	0.13	0.08	0.85	0.93	0.03	0.02	1.58 × 10−5	5.45 × 10−4
L1-SOUL-Search	0.13	0.08	0.84	0.93	0.03	0.01	4.97 × 10−6	3.22 × 10−4
2% deformation
Search	0.48	0.26	0.46	0.94	0.12	0.14	0.24	0.92
OVERWIND-Search	0.26	0.14	0.83	0.93	0.03	0.01	1.35 × 10−4	3.19 × 10−4
SOUL-Search	0.24	0.12	0.85	0.91	0.03	0.02	1.81 × 10−5	8.77 × 10−4
L1-SOUL-Search	0.24	0.13	0.85	0.92	0.03	0.01	5.54 × 10−6	1.47 × 10−4
4% deformation
Search	1.06	0.70	0.30	0.93	0.26	0.34	1.17	4.35
OVERWIND-Search	0.88	0.62	0.61	0.88	0.06	0.12	0.0038	0.04
SOUL-Search	0.87	0.65	0.61	0.85	0.04	0.13	0.002	0.02
L1-SOUL-Search	0.88	0.63	0.62	0.86	0.06	0.12	0.0036	0.02

2). Simulation Data with Lateral and Axial Motion:

Fig. 4 presents the ground truth and estimated lateral displacement maps for the simulated dataset containing both lateral and axial motion components. The displacement map produced by Search is noisy and displays prominent vertical artifacts, likely caused by the complex motion pattern in the simulated data. In comparison, the regularized optimization-based methods (i.e., OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) successfully suppress these vertical artifacts and generate cleaner displacement maps. However, Search provides clearer visualization of the boundaries of the fascia-mimicking layer relative to the energy-based techniques.

Fig. 4:

Lateral displacement simulation results with axial and lateral motion components and 0.5% lateral deformation.

Table III summarizes the quantitative metrics produced by the four techniques under investigation. OVERWIND-Search, SOUL-Search, and L1-SOUL-Search produce identical RMSE and MAE values and comparable MSSIM scores, yielding improvements of 77.14%, 65%, and up to 115%, respectively, over Search. Search and the three energy-based techniques obtain similar NCC values. However, Search renders up to 200 and 241.03 times higher ${\mu }_{xx}$ and ${\mu }_{xy}$, respectively, relative to the energy-based methods. L1-SOUL-Search renders the lowest ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$ among the four competing techniques.

TABLE III:

Quantitative metrics obtained with the simulated dataset with axial and lateral motion components.

	RMSE (mm)	MAE (mm)	MSSIM	NCC	${\mu }_{xx}$	${\mu }_{xy}$	${\sigma }_{xx}^2$	${\sigma }_{xy}^2$
Search	0.35	0.20	0.40	0.94	1.34	1.88	8.03	26.48
OVERWIND-Search	0.08	0.07	0.86	0.93	0.0086	0.0078	1.99 × 10−5	1.04 × 10−4
SOUL-Search	0.08	0.07	0.84	0.93	0.0067	0.015	1 × 10−5	3.04 × 10−4
L1-SOUL-Search	0.08	0.07	0.85	0.93	0.0074	0.0089	1.84 × 10−6	6.32 × 10−5

B. Qualitative In Vivo Assessment

Fig. 5 shows representative pre-displaced B-mode images and the corresponding lateral displacement images applied to the non-paretic shoulders of P1-P10 estimated by the four investigated tracking methods. The B-mode images reveal the fascial shear planes between PMA and PMI muscles. The fascial shear planes are also visible in the displacement images. Search produces the noisiest displacement images. The energy-based techniques OVERWIND-Search, SOUL-Search, and L1-SOUL-Search render visually smoother and less noisy displacement images than Search. In addition, the displacement images estimated by OVERWIND-Search, SOUL-Search, and L1-SOUL-Search clearly distinguish PMA and PMI muscles with alternating colors (blue and red). Although SOUL-Search displacement images are generally smoother than OVERWIND-Search, L1-SOUL-Search exhibits a smoother lateral displacement gradient in the lateral direction than SOUL-Search. In addition, possibly due to L1-norm regularization, OVERWIND-Search and L1-SOUL-Search yield visually sharper shear planes than SOUL-Search. The displacement tracking quality of the four techniques is visibly worse with P2 when compared to that of the remaining participants, likely due to irregular motion patterns.

Fig. 5:

Representative pre-displaced B-mode images and the corresponding lateral displacement results obtained from P1-P10. The green dashes delineate the hyperechoic fascial shear plane between PMA and PMI for P1. The B-mode images are shown with 20 dB dynamic range.

Based on overall lateral displacement image quality, the ranks of Search, SOUL-Search, OVERWIND-Search, and L1-SOUL-Search are 4, 2, 2, and 1. There is a tie between SOUL-Search and OVERWIND-Search because SOUL-Search performs better than OVERWIND-Search concerning image smoothness, whereas OVERWIND-Search outperforms SOUL-Search regarding fascia boundary sharpness.

C. Quantitative In Vivo Assessment

Fig. 6 shows the NCC between the pre-displaced and post-displaced frames, warped by the displacements calculated when the four tracking techniques were applied to the non-paretic shoulder. Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search perform similarly, with median NCC values (across non-paretic and paretic shoulders of P1-P10) of 0.96, 0.96, 0.96, and 0.96, respectively. Therefore, in terms of NCC, the four tracking techniques tie at rank 1.

Fig. 6:

Mean and standard deviation of NCC between predeformed and warped post-deformed frames acquired from the non-paretic shoulders of participants P1 through P10.

Fig. 7 shows the ${\mu }_{xx}$ and ${\mu }_{xy}$ obtained with the four tracking techniques. In the case of ${\mu }_{xx}$, the post-hoc analysis following the Kruskal-Wallis test indicates that the multiple possible combinations of techniques produce statistically significant differences ($p<=1.71\times {10}^{-15}$) from each other except for SOUL-Search and L1-SOUL-Search ($p=0.53$). However, all combinations of techniques exhibit statistically significant ($p<=1.72\times {10}^{-4}$) differences for ${\mu }_{xy}$. As SOUL-Search and L1-SOUL-Search yield a statistically significant difference for one of the mean displacement derivative metrics (${\mu }_{xy}$), we assign different ranks to these two tracking techniques based on the median values. Due to its noisy displacement estimates, Search produces the greatest displacement derivatives, with median ${\mu }_{xx}=1.51$ and median ${\mu }_{xy}=3.02$ (across the non-paretic and paretic shoulders of P1-P10). Among the energy-based techniques, L1-SOUL-Search generally produces greater ${\mu }_{xx}$ and ${\mu }_{xy}$ values than OVERWIND-Search and SOUL-Search, which can be related to the smooth gradient of the L1-SOUL-Search lateral displacement images (Fig. 5). The corresponding median ${\mu }_{xx}$ and ${\mu }_{xy}$ across the aggregated non-paretic and paretic shoulders of P1-P10 were 6.1 × 10⁻³ and 1.83 × 10⁻², respectively, for OVERWIND-Search, 5.2× 10⁻³ and 2.55 × 10⁻², respectively, for SOUL-Search, and 5.5 × 10⁻³ and 2.41 × 10⁻², respectively, for L1-SOUL-Search. Displacement derivatives, when averaged across median ${\mu }_{xx}$ and ${\mu }_{xy}$, corresponding to Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search are 2.27, 1.22 × 10⁻², 1.53 × 10⁻², and 1.48 × 10⁻², respectively. Based on the median displacement derivatives, Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search rank 1, 4, 2, and 3, respectively.

Fig. 7:

Mean lateral derivatives of lateral displacements (${\mu }_{xx}$) and mean axial derivatives of lateral displacements (${\mu }_{xy}$) between consecutive envelope-detected ultrasound frames, acquired from the non-paretic shoulders of P1-P10.

Fig. 8 shows the variance of displacement derivatives (i.e., ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$). Search produces up to 6 orders of magnitude larger ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$ values than the energy-based tracking techniques due to its noisy displacement estimates. In P8, SOUL-Search and L1-SOUL-Search produce up to 2 orders of magnitude larger ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$ values relative to those produced by the other participants. Averaging across the median ${\sigma }_{xx}^2$ and ${\sigma }_{xy}^2$ reported above results in 31.06, 1.76 × 10⁻⁴, 3.36 × 10⁻⁴, and 1.99 × 10⁻⁴, for Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search, respectively. Based on these values, Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search rank 4, 1, 3, and 2, respectively.

Fig. 8:

Variance of lateral derivatives of lateral displacements (${\sigma }_{xx}^2$) and variance of axial derivatives of lateral displacements (${\sigma }_{xy}^2$) between consecutive envelope-detected ultrasound frames, acquired from non-paretic shoulders of P1-P10.

Fig. 9 shows the MDMM obtained with the four tracking techniques. Per tracking technique, MDMM is generally lower on the paretic relative to non-paretic shoulder, which is consistent with our clinical hypothesis, with the exception of three patients (i.e., P4 with Search, P7 with the four tracking techniques, and P9 with three of the four tracking techniques, including Search). Post-hoc analysis of the Kruskal-Wallis test reveals that the DMM values corresponding to the paretic shoulders (aggregated across P1-P10) and non-paretic shoulders (aggregated across P1-P10) yield statistically significant differences with OVERWIND-Search ($p=4.04\times {10}^{-9}$), SOUL-Search ($p=1.10\times {10}^{-10}$), and L1-SOUL-Search ($p=3.45\times {10}^{-7}$) techniques. However, Search does not produce statistically significant differences between the paretic and non-paretic DMM values ($p=0.77$).

Fig. 9:

Median local maxima to local minima shear strain differences (MDMM) for non-paretic and paretic shoulders of P1-P10 obtained by Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search.

In P4, Search produces a greater MDMM on the paretic shoulder (11.74%) rather than non-paretic shoulder (6.92%) shoulder (Fig. 9), which contradicts the clinical hypothesis. However, the associated ultrasound cine loop is consistent with the clinical hypothesis, depicting greater mobility on the non-paretic rather than paretic shoulder of P4. Similarly, T1ρ MRI is consistent with the clinical hypothesis, as the shoulder muscles of P4 revealed greater HA accumulation on the paretic shoulder (i.e., 28.42 ms T1ρ relaxation time) relative to the non-paretic (24.09 ms T1ρ relaxation time) shoulder. These findings are inconsistent with the result produced by Search, yet consistent with the results achieved by the remaining three tracking techniques.

In P7, the four techniques produce greater MDMM on the paretic shoulder rather than the non-paretic shoulder (Fig. 9). The corresponding ultrasound cine loops support this result by depicting greater mobility on the paretic rather than non-paretic shoulder of P7. T1ρ MRI similarly supports this finding by revealing greater HA accumulation on the non-paretic (29.35 ms T1ρ relaxation time) rather than the paretic (27.04 ms T1ρ relaxation time) shoulder.

In P9, with the exception of OVERWIND-Search, the tracking techniques contradict the hypothesis that MDMM is lower on the paretic rather than non-paretic shoulder. However, the corresponding ultrasound cine loop revealed greater muscle mobility on the paretic rather than non-paretic shoulder of P9, which is consistent with the findings of Search, SOUL-Search, and L1-SOUL-Search. T1ρ MRI was not performed on P9 due to technical issues with the MRI machine on the scheduled day of examination and unavailability of the participant afterward.

Based on DMM, MDMM, and the above clinical hypothesis assessments, SOUL-Search and L1-SOUL-Search tie at rank 1, whereas OVERWIND-Search ranks 3 due to an incorrect clinical inference on P9 based on the visual inspection of the ultrasound cine loop. Search ranks 4th, given no statistically significant differences between paretic and non-paretic DMM values ($p=0.77$) and incorrect MDMM estimates in P4 based on visual assessments of ultrasound cine loops and associated T1ρ MRI values.

D. Overall Assessment

The mean ranks corresponding to Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search are reported in Table IV. These mean ranks consider the individual ranks for qualitative appearances of displacement images, NCC, mean and variances of displacement derivatives (i.e., ${\mu }_{xx}$, ${\mu }_{xy}$, ${\sigma }_{xx}^2$, and ${\sigma }_{xy}^2$), and assessments of clinical hypothesis. Achieving the lowest overall rank, L1-SOUL-Search is considered the most promising displacement tracking technique among the four investigated methods for post-stroke myofascial shear strain quantification.

TABLE IV:

Mean ranks based on qualitative and quantitative features investigated and clinical hypothesis assessments

	Mean Rank
L1-SOUL-Search	1.6
SOUL-Search	1.8
OVERWIND-Search	2.2
Search	2.8

E. Deep Learning as a Solution to Accelerate Runtimes

Using a 6th generation Intel Core i5 CPU with 32 GB RAM, the Search algorithm required approximately 29 seconds to estimate displacements between two consecutive 36.2 mm × 49.92 mm ultrasound frames, resulting in a total processing time of approximately 116 minutes for the full 240-frame video acquired from the non-paretic side of P3. The refinement steps introduced by OVERWIND-Search, SOUL-Search, and L1-SOUL-Search added approximately 12.01 s, 20.50 s, and 28.07 s per frame-pair (see Table V), respectively, leading to additional processing times of 52, 84, and 110 minutes, respectively, beyond that required by Search. Given these runtimes, none of these techniques are suitable for real-time implementation of shear-strain-based biomarker calculation in post-stroke shoulder pain assessment.

TABLE V:

Computation time required to estimate displacements between two ultrasound frames of size 36.2 mm × 49.92 mm.

	Execution time
Search	29 s
OVERWIND-Search	41.01 s
SOUL-Search	49.50 s
L1-SOUL-Search	57.07 s
RAFT	0.25 s

The RF data that produced the images in Fig. 5 were processed with RAFT for the 10 patients included in our study. To offer representative examples of the range of outputs obtained, the best (P10) and worst (P3) outcomes are shown in Fig. 10, based on the four visual characteristics described in Section II-F and comparison with the displacement images in Fig. 5 produced by the best-ranked method (i.e., L1-SOUL-Search). When shown on the same color scale, the negative displacements produced by RAFT with P3 are not as prominent as the corresponding L1-SOUL-Search image shown in Fig. 5 (reproduced in Fig. 10 for convenience). RAFT also produced more positive displacements in the lower left of the image from P3, which is considered correct, based on the controlled experimental design and assessment of the corresponding motion in the associated ultrasound cine loop (and when considering the L1-SOUL-Search errors obtained with 4% positive lateral displacement, relative to the ground truth in Fig. 3). Conversely, the displacement patterns in the RAFT and L1-SOUL-Search images produced with P10 are generally consistent.

Fig. 10:

Worst (P3) and best (P10) outcomes of lateral displacement results obtained with RAFT (a potential deep learning solution to slow runtimes), relative to corresponding L1-SOUL-Search displacement images from Fig. 5 (reproduced for the convenience of direct comparison).

Overall, the outputs of the results produced by RAFT were generally consistent with expected displacement patterns based on our controlled experimental design and the associated tissue mechanics. In addition to the promising qualitative performance, with an NVIDIA TITAN Xp GPU (12 GB memory), RAFT required approximately 0.25 seconds to estimate displacements between two consecutive 36.2 mm × 49.92 mm ultrasound images (Table V), completing the entire 240-frame video from the non-paretic side of P3 in approximately 1 minute. Therefore, potential solutions for real-time displacement tracking implementations include using the trained RAFT model as is or potentially fine tuning RAFT results with tracking algorithm results (e.g., L1-SOUL-Search) or ground truth simulation results during network training.

IV. Discussion

This work qualitatively and quantitatively investigates the displacement estimation reliability of a correlation kernelbased method (i.e., Search) and three energy-functionoptimization-based tracking techniques (i.e., OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) to quantify muscle shear strain. To our knowledge, this is the first comparative assessment of displacement tracking reliability in the context of post-stroke muscle stiffness. Search produces a characteristic noisy appearance of lateral displacement images in silico and in vivo, which is noticeably reduced by the three energy-function-optimization-based techniques OVERWIND-Search, SOUL-Search, and L1-SOUL-Search (Figs. 3-5). Among the energy optimization techniques, L1-SOUL-Search yields the best performance in terms of noise suppression. In addition, the gradient of the estimated lateral displacement field along the lateral direction is more pronounced with L1-SOUL-Search in validations with in vivo datasets (Figs. 5). These visual features are supported by the quantitative metrics reported in Tables II and III for simulated data and in Figs. 7 and 8 for the in vivo datasets.

Search fails to reveal displacement patterns consistent with expectations based on the controlled experimental design in the cases of P1 and P2. This limitation may be attributed to (1) the lower displacement range observed in the pectoralis muscles for these participants (Fig. 5) and/or (2) complex tissue motion which can corrupt lateral displacement estimates (demonstrated by the worse 0.5% lateral deformation results achieved with Search with the addition of axial motion components in Fig. 4 and Table III, relative to Fig. 3 and Table II, respectively). In contrast, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search successfully produce displacement estimates for P1 and P2 that align with the experimental setup, highlighting the robustness of these energyfunction-optimization-based approaches in challenging motion scenarios. Comparison of the 0.5% lateral deformation results produced by these energy-based approaches in Tables II-III similarly reveal a remarkable ability to maintain robustness in the presence of more complex tissue motion.

Our promising results demonstrate that the energy-functionoptimization-based algorithms do more than merely smooth displacement estimates, as they investigate ultrasound data and tissue deformation physics simultaneously to reveal true displacement patterns across diverse clinical scenarios. Clinical advantages of these energy-based methods over Search include overall denoising of the displacement images, clearer visualization of the shear plane, better assessment of interfacial slippage (or restriction), and assistance with identifying abnormal tissue motion (e.g., due to post-surgical fibrosis [58], chronic tendinopathy [59], or scar adhesions [60]), where interlayer mobility is clinically relevant. Potential clinical advantages of Search include helping to ensure that the visualized shear plane is real (Fig. 4) or that local abnormalities are evident (e.g., during dry needling treatments on patients in which a local twitch response is desired [61]). A deep learning solution can additionally be considered for clinical deployment, depending on the specific task and the associated time frame in which biomarker results are needed (e.g., a few hours vs. one minute).

Despite noted differences in tracking quality, the investigated tracking algorithms generally support the clinical hypothesis that MDMM on the painful paretic shoulder of a post-stroke shoulder pain patient is lower than the MDMM on the non-paretic shoulder (Fig. 9). Therefore, each investigated technique can potentially be used to make an objective clinical decision based on MDMM. However, this decision is likely to be questioned with the qualitatively noisier displacement images produced by Search, which is supported by the lack of statistically significant differences between the DMM values on the paretic and non-paretic shoulders (i.e., $p=0.77$).

The parameters associated with the three energy-function-optimization-based techniques required tuning (Table I) to ensure optimal performance. Although this parameter-tuning requirement added manual intervention to the myofascial shear strain estimation procedure, tuning was performed only once for two consecutive ultrasound frames collected from the non-paretic shoulder of P3 (Section II-D). It is promising that the same parameter sets obtained from the tuning process worked well for five simulated ultrasound frame pairs and for thousands of ultrasound frame pairs acquired from non-paretic and paretic shoulders of ten research participants. The generalizability of OVERWIND-Search, SOUL-Search, and L1-SOUL-Search parameters to new research participants is promising for clinical deployment of our shear strain quantification approach. In addition, this approach can potentially be extended to other muscle groups or other diagnostic tasks that rely on displacement tracking (e.g., novel approaches to ultrasound elastography [21]).

One potential limitation of our comparison is that OVERWIND-Search and L1-SOUL-Search optimize the L1-norm by approximating its differentiable version (Section II-C5), sacrificing the full potential of L1-norm regularization. In the future, this limitation can be mitigated by employing an alternating direction method of multipliers (ADMM)-based L1-norm optimization algorithm [18], [54], [62], [63] in myofascial shear strain quantification. In addition, the energy-function-optimization-based displacement tracking techniques employed in this work (OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) are unsuitable for real-time or nearreal-time implementation (Table V) due to obtaining the initial displacement estimates using an exhaustive window search technique, followed by iterative refinement. In contrast, deep learning-based methods like RAFT offer a promising solution (see Fig. 10 and Table V), providing faster displacement estimates and showing potential for real-time applications. Future work will extensively investigate a multitude of deep learning-based displacement tracking techniques [7], [64]-[68], which are generally faster than traditional algorithms [69], for the myofascial shear strain quantification task.

V. Conclusion

The displacement tracking reliability of a window-based algorithm (Search) and three energy function optimization-based algorithms (OVERWIND-Search, SOUL-Search, and L1-SOUL-Search) were investigated with respect to post-stroke myofascial shear strain quantification. While the investigated tracking techniques generally support a novel clinical hypothesis that the range of shear strains between PMA and PMI muscles on the paretic shoulder of a stroke patient is lower than the shear strain range on the non-paretic shoulder, SOUL-Search and L1-SOUL-Search produce the most accurate and reliable shear strain estimates among the ten research participants (relative to the clinical hypothesis). In addition, L1-SOUL-Search produced the most reliable displacement tracking performance by producing smooth lateral displacement images with sharp boundaries and by reducing the median variance of displacement derivatives from 31.06 to 1.99×10⁻⁴ with respect to the previously implemented Search algorithm. These findings collectively support the integration of L1-SOUL-Search into large clinical studies to investigate the interplay between myofascial stiffness and chronic pain. While Search, OVERWIND-Search, SOUL-Search, and L1-SOUL-Search are not suitable for real-time myofascial tracking, deep learning-based methods (e.g., RAFT) may help mitigate this limitation. In addition to related implications for other diagnostic applications that rely on ultrasound displacement tracking, this work provides a supportive foundation to enhance our understanding of musculoskeletal health, leading to the possible development of novel biomarkers and rehabilitation strategies in this area.