MicroBundlePillarTrack.

We developed “MicroBundlePillarTrack” [25] as a robust, open-source software tool for automated segmentation, tracking, and analysis of pillar deflection in beating microbundles imaged using brightfield microscopy. The software processes consecutive frames of a movie, automatically generating two distinct binary masks to delineate each pillar. After segmentation, fiducial markers identified using the Shi-Tomasi corner detection method [52] on the first relaxed frame, are tracked throughout all subsequent frames via a pyramidal implementation of the Lucas-Kanade sparse optical flow algorithm [53,54]. This approach enables precise determination of pillar positions across time, from which both mean directional and absolute displacements are calculated. The software further derives quantitative outputs including microbundle twitch force and stress, as well as temporal metrics such as contraction and relaxation velocities, Full Width at Half Maximum (FWHM), and Full Width at 80% maximum (FW80M), where FWHM and FW80M are defined as the temporal width of the mean absolute displacement curve at 50% and 80% of its peak amplitude, respectively (Fig 4). As measures of contraction duration, FWHM and FW80M are analogous to width-based metrics commonly computed from calcium transients to characterize the duration of calcium activation in engineered cardiac tissues [39,55,56].

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Fig 3. Illustrative 2-dimensional vector fields constructed to demonstrate the types of critical points identified through critical point analysis.

a) isolated first-order critical points; (b) examples of vector fields in which no critical points are detected, demonstrating flows without singularities.

https://doi.org/10.1371/journal.pcbi.1014522.g003

Notably, “MicroBundlePillarTrack” [25] requires no parameter tuning, streamlining the high-throughput analysis of large datasets. Required user input is minimal and limited to frame rate (fps), length scale (m/pixel), pillar stiffness (N/m), and tissue thickness (m). A comprehensive description of the software’s methodology and capabilities can be found in its primary publication [25].

For the present study, we primarily utilize outputs from “MicroBundlePillarTrack” [25] related to pillar force and the temporal metric Full Width at Half Maximum (derived from the pillar mean absolute displacement time series), which become interpretable metrics detailed in the “Interpretable metrics” Section.

MicroBundleCompute.

Our tool “MicroBundleCompute” [24] is also an optical flow-based tracking and analysis software, and is among the few specialized tools available for whole-tissue deformation analysis in microscopy movies of cardiac microbundles [26,57,58].“MicroBundleCompute” is specifically designed for multi-purpose assessment of heterogeneous cardiac microbundle deformation and strain from brightfield and phase contrast videos, and the software has been extensively validated to ensure robust and reliable performance.

Analogous to “MicroBundlePillarTrack” [25], “MicroBundleCompute” [24] automatically generates a binary mask of the tissue and identifies “good features to track” marker points using Shi-Tomasi corner detection [52] within the masked region on the first relaxed frame. These points are then tracked across all frames, employing a sparse optical flow [53,54] approach and segmenting the analysis by individual contraction cycles to mitigate noise accumulation associated with extended temporal tracking. This process yields high-resolution vector maps of tissue displacements throughout the contraction-relaxation cycle, as the tracking code automatically ensures sufficient density and spatial coverage of fiducial markers across the tissue domain, with a minimum of one fiducial marker enforced for every subdivision of 50 pixels of tissue area (see Figs 2b and S1 Fig for representative examples of fiducial marker density and distribution). These vector maps further facilitate the calculation of spatially-averaged Green-Lagrange strain within specified tissue subdomains. Post-processing modules further enhance analytical rigor by allowing for automated rotational alignment of image stacks and tracked data, as well as interpolation of displacement and strain fields onto query grids for spatiotemporal analyses. Importantly, the software is optimized for batch processing and requires minimal user intervention, with only the frame rate (fps) and pixel-to-micrometer conversion factor specified by the user. All other parameters are internally standardized to maintain analytic consistency across large datasets. For an in-depth description of the software’s methods and functionalities, as well as further details on its implementation and usage, please refer to the original publication [24].

In the present study, we utilize the two-dimensional displacement fields generated by “MicroBundleCompute” [24] for direct computation of full-field Green-Lagrange strain, as outlined in the “Strain computation details” Section. These computational outputs serve as the starting point for a substantial subset of the interpretable metrics described in the “Interpretable metrics” Section, supporting rigorous, spatially resolved characterization of contractile tissue mechanics.

Strain computation details.

As described in the “MicroBundleCompute” Section, “MicroBundleCompute” [24] is specifically designed to calculate average subdomain strain rather than full-field strain. In this approach, the displacements of all fiducial markers within each subdomain are used to approximate an averaged deformation, from which a single representative strain value is derived rather than a spatially resolved estimate at each marker location. This design choice was motivated by two main considerations: (1) minimizing the impact of imaging artifacts and noise, and (2) facilitating more consistent comparisons across different samples. While the subdomain averaging approach is practical and robust, it inherently restricts the spatial resolution of strain quantification, since the strain in each subdomain is estimated from the available fiducial marker points and requires a minimum number of points to ensure mathematical stability, particularly to avoid singular matrices during computations.

For our current study, we sought higher spatial resolution in strain mapping than what the subdomain-based method can provide. To achieve this, we adopted the approach described by Zimmerman et al. [59] and Benkley et al. [60], which enables computation of a two-dimensional Green-Lagrange strain field. This method estimates the local deformation gradient tensor directly from the tracked positions of randomly distributed particles, utilizing a least-squares fitting procedure on nearest-neighbor vectors combined with a first-order finite difference approximation (S1 Fig). In mathematical terms, the estimated deformation gradient F at a given point can be expressed as follows [59]:

(1)

where represents the vector connecting marker points and in the current (deformed) configuration, while denotes the corresponding vector in the reference (undeformed) configuration. The parameter n specifies the total number of non-collinear neighboring marker points used in the estimation, which, for our analysis, was set to 8 to ensure robust and accurate computation of the local deformation gradient.

Computing the Green-Lagrange strain (E) tensor from the deformation gradient tensor is straightforward:

(2)

where I is a identity matrix.

We provide our Python implementation of this strain computation approach, enabling users to derive strain fields from displacement data obtained via “MicroBundleCompute” [24]. This code, along with all of the scripts used to calculate the defined metrics in the “Interpretable metrics” Section, are made available on GitHub (https://github.com/HibaKob/MicroBundleAnalysis) to allow others to reproduce and build on our methods.

Software failure and data exclusion.

The failure of 100 microscopy movies to be processed successfully can be attributed to specific design requirements and stringent algorithmic constraints within “MicroBundleCompute” [24] and “MicroBundlePillarTrack” [25]. Specifically, two principal categories of movies were excluded: (1) those with blurred frames, which compromise image quality and hinder the accurate identification of fiducial markers essential for robust tracking; and (2) those comprising fewer than three contraction beats, which provide insufficient temporal information for meaningful characterization of contractile dynamics. By enforcing these exclusion criteria, the software ensures that only movies capable of yielding precise and interpretable results are included in subsequent analysis.

Spatial registration of tissue domains.

The images present in this dataset are taken at multiple different angles, and individual tissues vary in size (Fig 2a). To make direct comparisons between these tissues, we began by performing a systematic registration of tissue domains using masks automatically generated by MicroBundleCompute (Fig 2b) [24]. Each mask was rotated so that the tissue’s major axis aligns with the horizontal (column) axis, and a mean tissue mask was then computed from the set of aligned and rotated masks. To ensure standardized subsequent analyses, we performed a grid sensitivity analysis (S2 Fig) on the metrics defined in the “Interpretable metrics” Section, which informed the selection of a regular grid with a resolution of , centered on the mean mask and spanning 80% of its width and height. For each individual tissue, we calculated an affine transformation that maps the example rotated tissue mask onto the mean tissue mask. This transformation was then used to register the reference grid onto the individual tissue, thereby defining a tissue-specific grid. To enable direct comparisons across tissues, the field of interest (either displacement or strain) was interpolated onto each tissue-specific grid using SciPy’s Radial Basis Function (RBF) interpolation (v1.13.1) [61,62]. This workflow is critical for ensuring robust and spatially consistent comparisons across heterogeneous microbundle time-lapse images, enabling meaningful downstream analyses that would otherwise be confounded by variations in tissue orientation, size, and geometry.

It is worth noting that variability in tissue orientation and geometry is an inherent challenge across engineered heart tissue datasets, irrespective of experimental platform. The registration and grid optimization workflow detailed above is therefore not dataset-specific; rather, it is designed to be broadly applicable to alternative tissue geometries and experimental configurations. For datasets derived from distinct microbundle architectures or fabrication platforms, the grid sensitivity analysis outlined in this section would need to be repeated to identify a resolution that adequately captures spatially heterogeneous contractile behavior. The affine registration framework and RBF interpolation pipeline are similarly generalizable, provided that reliable tissue masks can be automatically or semi-automatically generated. Collectively, the methodology presented here establishes a systematic and extensible protocol for spatially consistent cross-tissue comparisons, accommodating the geometric diversity inherent to engineered cardiac tissue datasets produced across different experimental systems.

Principal component analysis.

Principal component analysis (PCA) [63,64] is a widely used unsupervised multivariate analysis technique that transforms complex datasets by projecting them onto a lower-dimensional orthogonal subspace. Typically, the number of retained principal components is far fewer than the original dimensions of the data, resulting in a more concise and informative representation. When applied to vector fields, PCA identifies the primary directions of variation, enabling significant reduction in dimensionality while efficiently filtering out noise and redundancy [65–68]. Moreover, the dominant principal components not only capture the most meaningful patterns within the data, but also provide valuable physical insight into the underlying deformation modes and structures of the system under study [65,67,68].

In this study, PCA was applied to the displacement vector fields within the beating cardiac microbundles, as extracted by “MicroBundleCompute” [24]. Our primary motivation for implementing principal component analysis in this study is twofold. First, we seek to uncover latent patterns within the displacement vector field that may reveal meaningful insights into the contractile behavior of cardiac microbundles. Second, we use PCA in this context to denoise the displacement data to enable additional analysis. By reconstructing each tissue’s displacement vector field using only the first 10 principal components, which capture approximately 93% of the total variance, we effectively remove noise and redundancy, producing cleaner datasets that are better suited for future metric extraction and quantitative analysis.

Given the inherent complexity and spatiotemporal variability of the raw displacement data across samples, two preprocessing steps are required to achieve both objectives. First, we performed temporal homogenization by selecting the displacement field at a single time point corresponding to peak tissue contraction. Next, as detailed in the “Spatial registration of tissue domains” Section, we spatially homogenized the dataset by interpolating the displacement fields of all 670 cardiac microbundle samples onto a unified grid. This careful standardization of both temporal and spatial dimensions ensures consistency across examples.

To organize the data for PCA, we first retained the spatial structure of the grid points and constructed two three-dimensional matrices of size : one for the horizontal (column) and one for the vertical (row) components of the displacement field. In each matrix, the first two dimensions represent the row and column positions on the spatial grid, while the third dimension indexes the 670 tissue samples. We then transformed both 3D matrices into a 2D format suitable for PCA by performing mode-3 unfolding, as described in [69,70]. Specifically, each matrix was independently reshaped into a matrix, where each row corresponds to a tissue sample and each column to a specific spatial displacement feature. We then concatenated the two unfolded matrices, corresponding to the horizontal and vertical displacement components respectively, along the feature axis, resulting in a final data matrix (Q) of size . This organization allows each tissue sample to be represented as a single vector of 1,848 displacement features, enabling a comprehensive and meaningful principal component analysis.

We performed PCA using the Python library scikit-learn (v1.7.1) [71] on the data matrix Q. In accordance with the standard PCA theory, the process begins by centering the data, which involves subtracting the mean from each column to ensure that the analysis captures variance relative to the mean. The subsequent steps differ in implementation from the classical covariance-based derivation, but they yield mathematically equivalent results. Instead of explicitly forming the covariance matrix and performing an eigenvalue decomposition, scikit-learn performs a singular value decomposition (SVD) of the centered data matrix. Given a centered data matrix , it computes ; the principal axes (directions) are the right singular vectors V, and the explained variances are , which are mathematically equivalent to the eigenvectors and eigenvalues of the covariance matrix [72]. This approach avoids explicitly constructing the covariance matrix, which improves numerical stability and is more memory- and compute-efficient for large or high-dimensional datasets. Finally, the principal component scores, or in other words, the transformed variables summarizing the dominant patterns in the data, are obtained as , that is by projecting the centered data onto the principal axes.

Finally, we make our complete implementation of PCA publicly available on GitHub (https://github.com/HibaKob/MicroBundleAnalysis), including detailed steps for matrix unfolding and the construction of the data matrix Q. We present the results of this analysis in the “Principal Component Analysis uncovers primary isotropic contraction mode in tissue displacement” Section.

Critical point analysis.

Analysis of two-dimensional vector field topology has been foundational in fluid dynamics, where it is particularly useful for studying velocity fields in turbulent flows that exhibit intricate and dynamic patterns [73,74]. This approach not only facilitates intuitive visual representation of complex datasets, making them more accessible for human interpretation, but also enables the identification and classification of a broad spectrum of wave phenomena [73–76]. Central to this methodology is critical point analysis, which detects and categorizes local flow patterns and spatial features within vector fields at fixed time points, thereby enabling the identification of spatiotemporal wave phenomena through their evolution in time [75–80]. While rooted in fluid dynamics, the mathematical foundations of critical point analysis are applicable to any continuous vector field, regardless of the underlying physical phenomenon, and its versatility has captured the interest of researchers across a broad range of disciplines. This has led to the adoption and extension of these methods in diverse fields such as neural circuit analysis and brain activity pattern detection [81–83], transcriptomics and cell fate mapping [84–86], and the detection of irregularities in cardiac electrophysiology [87,88].

Building on these advances, we further extend critical point analysis to the study of dynamic cardiac tissue behavior. By applying this technique to PCA-denoised displacement vector fields reconstructed for each tissue, we aim to identify and characterize significant localized spatial patterns underlying tissue contraction and coordination.

Critical points are locations where the vector magnitude is zero, and they are classified into six main types based on the behavior of nearby tangent curves (Fig 3a), with nodes and foci each further distinguished into stable and unstable variants. These primary types include: (1) saddles, characterized by one stable and one unstable axis, often arising from interactions or collisions of different wavefronts; (2) nodes, which act as sinks (stable) or sources (unstable) and represent regions where flow contracts toward or expands from a central point; and (3) foci, around which flow spirals, corresponding to contracting (stable) or expanding (unstable) spiral waves [75,79,89,90].

In our implementation, critical point analysis was performed on the reconstructed displacement vector field for each tissue, using only the first 10 principal components (see “Principal component analysis” Section) to capture the most salient displacement patterns at peak contraction. The reconstructed field was then reshaped into horizontal and vertical displacement components on the unified spatial grid, yielding a denoised two-dimensional displacement vector field for each tissue.

Critical points were subsequently identified and characterized in these reconstructed fields according to established methods from prior literature [78–80]. Specifically, a grid-based Poincaré Index approach was employed [91], under the assumption that the vector field varies piecewise linearly within each grid cell.

Given that our reconstructed displacement field per tissue is represented on a grid, we partitioned the grid into an oriented triangular mesh, and each triangle was evaluated for the presence of a critical point. For each triangle, we calculated the winding number as follows [81]:

(3)

where n = 3 for a triangle, and denotes the angle of the displacement vector at the k^th vertex along the triangle boundary. Here, k indexes the ordered spatial vertices of the triangle, taken in counterclockwise order, with circular indexing such that . The angular differences were wrapped to the interval . Based on the computed winding number for each triangle, we identified three possible scenarios:

• winding number = 1: indicates the presence of a node or focus critical point within the triangle (Fig 3a);
• winding number = -1: indicates the presence of a saddle critical point (Fig 3a);
• winding number = 0: indicates no critical point within the triangle (Fig 3b).

When a nonzero winding number is detected, we used the piecewise linear approximation of the displacement field within the element to locate the critical point, solving for the position where the vector field vanishes in both horizontal (column) and vertical (row) directions. To further characterize the nature of each critical point, we computed the Jacobian matrix J for the corresponding triangle, ensuring it is non-degenerate (). We then solved the characteristic equation to obtain the eigenvalues:

(4)

We then classified the critical points based on the real and imaginary components of the eigenvalues:

• stable node: ,
• unstable node: ,
• stable focus: ,
• unstable focus: ,
• center point: ,

Our complete Python implementation of this critical point analysis, including mesh generation and eigenvalue computation, is available on GitHub: https://github.com/HibaKob/MicroBundleAnalysis. In the “Critical point analysis provides complementary insights to PCA” Section, we show the corresponding findings.

Wasserstein distance.

The Wasserstein distance, also known as Earth Mover’s Distance, is a widely used metric for quantifying the difference between two probability distributions. Originally formulated by Rubner et al. [92,95], the Wasserstein distance measures the minimal “work” required to transform one distribution into another. In vector field analysis [96,97], it has been used to quantitatively compare computational and experimental flow fields under equivalent conditions, thereby providing an interpretable measure of differences between complex vector field patterns.

In this work, we use the Wasserstein distance to quantify the difference between the original displacement field at peak tissue contraction, interpolated onto a tissue-specific grid (see “Spatial registration of tissue domains” Section), and its reconstruction from the first 10 principal components (see “Principal component analysis” Section). This approach allows us to assess the overall similarity between the two vector-valued distributions, where a larger Wasserstein distance indicates greater dissimilarity in the displacement patterns and suggests that the original field contained greater irregularities not preserved in the PCA-based reconstruction (Fig 4b-iv). By comparing the two displacement vector field distributions, the Wasserstein distance provides a measure of reconstruction fidelity beyond point-wise error metrics.

The first Wasserstein distance between two distributions, using the Euclidean norm as the ground metric, is defined as [98]:

(5)

where denotes the set of distributions with marginals u and v, is a transport plan, and is the Euclidean distance between x and y in . In the discrete setting, the Wasserstein distance can be interpreted as the cost of an optimal transport plan required to transform one distribution into the other, where the cost is given by the amount of probability mass moved multiplied by the distance over which it is transported. In this setting, the finite point sets and denote the support set of the probability mass functions u and v, respectively.

To compute the Wasserstein distance in our analysis, the wasserstein_distance_nd function [99,100], available in SciPy (v1.13.1) [61], was used. This function enables the efficient computation of the Wasserstein distance between N-dimensional discrete distributions. In this implementation, the inputs u_values and v_values correspond to the 2D displacement vectors from the original and reconstructed fields, respectively. The optional inputs u_weights and v_weights specify the associated nonnegative weights of the support points; however, these arguments were not provided in the present analysis, and equal weight (1/M) was therefore assigned to all support points, where M = 924 denotes the number of grid points. For transparency and reproducibility, the complete script for implementing this function, as well as the procedure for reconstructing the displacement fields, is provided on GitHub: https://github.com/HibaKob/MicroBundleAnalysis. We present the results of this analysis in the “Feature correlations reflect both redundancy and novel information” Section.

Global synchrony index (GSI).

In order to quantitatively assess synchronization across multiple neuronal population time series, Li et al. [93] introduced the normalized global synchrony index (GSI). This metric provides an interpretable scale where a value of 0 denotes complete asynchrony among time series, and a value of 1 represents perfect synchrony (Fig 5, left panel). In our dataset, the GSI is well-suited for the quantification of regional synchrony within tissue samples. In particular, the Green-Lagrange strain fields, evaluated on unified grids (see “Spatial registration of tissue domains” Section), display peak strain values that do not necessarily occur at the same time frame across all grid locations.

To apply the global synchrony index to our tissue data, we computed GSI values for each tissue sample and for each of the three Green-Lagrange strain components: (column-column based on row-column descriptions of image data, representing the main axis of tissue contraction), (row-row, normal to the main contraction axis), and (column-row). In order to make the synchrony metric comparable across all 670 tissue examples, we homogenized the temporal profiles of the strain fields by rescaling each time series to span a normalized interval from 0 to 1 time units. For computational consistency, we sampled 25 equally spaced time points within this interval, using steps of 0.04 units. This sampling density was selected as a practical balance between preserving sufficient temporal resolution to capture waveform dynamics and avoiding unnecessary oversampling. At these standardized time points, strain values for each spatial grid location were interpolated using SciPy’s (v 1.13.1) [62] CubicSpline implementation. This approach ensures that the GSI reflects true differences in strain synchrony across spatial regions and tissue samples, independent of variations in tissue beating frequency.

The global synchrony index (GSI) was implemented following the framework established by Li et al. [93], which leverages random matrix theory and equal-time correlation matrix analysis. For each tissue sample, the temporally homogenized strain data were organized as M = 924 strain time series, one per spatial grid location, each sampled at T = 25 normalized time points. The Pearson correlation matrix was then constructed, where , and and denote the strain time series at grid locations i and j, respectively. Thus, each entry of R measures the equal-time temporal association between a pair of spatial locations within the tissue.

We then performed eigenvalue decomposition of R with the largest eigenvalue serving as a robust measure of global synchronization across the tissue [93,101]. To ensure that the GSI values accurately reflect true synchronization rather than statistical artifacts, we normalized using randomized surrogate datasets generated via amplitude-adjusted Fourier transform (AAFT). AAFT effectively preserves the amplitude distribution and approximate autocorrelation structure of each time series while removing genuine equal-time correlations. For each tissue sample, this randomization was repeated 100 times, and the maximum eigenvalue was recorded for each realization. We then calculated the mean () and the standard deviation (SD) of these surrogate eigenvalues to provide a robust estimate of expected synchronization under random conditions.

The normalized GSI value was then calculated as follows:

(6)

where K is the Bonferroni-adjusted critical value from the standard normal distribution to control the overall probability of a Type I error (false positive) for multiple hypothesis tests [102], set to K = 4.42 for an overall significance level of using Z-tests given 924 comparisons.

Considering the inherent randomness in AAFT surrogate generation and to ensure reproducibility, we repeated the entire normalized GSI calculation 100 times for each tissue sample and reported the average normalized GSI value. As demonstrated in Fig 5, the mean measured GSI rapidly stabilizes, converging after approximately 80 iterations. The complete implementation, including scripts for GSI computation and temporal homogenization, is openly available on GitHub: https://github.com/HibaKob/MicroBundleAnalysis. We leverage the results of this analysis in the “Feature correlations reflect both redundancy and novel information” Section.

Average pairwise Dynamic Time Warping (DTW) distance.

Originally developed for speech recognition [94,103,104] and later broadly adopted for pattern analysis in diverse time series applications [105–108], Dynamic Time Warping (DTW) distance provides a robust measure of similarity between two temporal sequences that may differ in speed, phase, or timing. Unlike simple distance measures, DTW flexibly aligns sequences by dynamically “warping” the time axis, identifying the optimal path that minimizes the cumulative disparity between corresponding points in the sequences [105]. The resulting DTW distance reflects the total alignment cost, accounting for shifts and stretching along the time dimension.

A key advantage of DTW distance over standard Euclidean distance is its resilience to temporal misalignments. Euclidean-based comparisons are highly sensitive to even minor offsets: for example, if one time series is delayed or shifted relative to another, Euclidean distance will exaggerate their difference despite underlying similarity. In contrast, DTW distance accurately quantifies true similarity by aligning corresponding features, making it especially well-suited for biological time series data where phase variability is common.

In this study, we leverage the DTW distance (Fig 4b-iii) to quantitatively assess waveform heterogeneity within each tissue sample using an average pairwise DTW distance metric. For every tissue, we first normalized each of the three Green-Lagrange strain components (, , and ) such that all strain time series amplitudes are scaled between −1 and 1 at each spatial grid location. This normalization ensures meaningful cross-tissue comparisons by removing inter-tissue amplitude differences that could otherwise inflate the DTW metric, while preserving the intrinsic waveform differences among time series within a given tissue. We then computed the DTW distance for every possible pair among the 924 normalized time series per tissue sample, utilizing the Python implementation provided by the DTAIDistance library [109]. The average pairwise DTW distance was defined as follows:

(7)

where N is the total number of time series per tissue. Interpretively, a lower average pairwise DTW distance indicates that most time series within the tissue are temporally alike, with waveform patterns that align closely even if offset or stretched in time; a higher value signifies greater heterogeneity, capturing pronounced differences in strain dynamics, timing, or shape that persist despite optimal alignment. All code and workflow for calculating the average pairwise DTW distance are openly accessible on GitHub: https://github.com/HibaKob/MicroBundleAnalysis. We present the results of this analysis in the “Feature correlations reflect both redundancy and novel information” Section.

Additional metrics.

In addition to the metrics detailed in “Wasserstein distance,” “Global synchrony index (GSI),” and “Average pairwise Dynamic Time Warping (DTW) distance” Sections, several informative measures are readily available through the core functionalities of the software tools “MicroBundleCompute” [24] and “MicroBundlePillarTrack” [25]. These common metrics offer complementary insights into fundamental aspects of tissue structure, function, and dynamics and include:

• Tissue Aspect Ratio – calculated as the ratio of tissue length to width, with both dimensions extracted from segmented tissue masks. Length is defined as the edge-to-edge distance along the tissue’s major axis, while width is measured at the midpoint perpendicular to the major axis.
• Tissue Curvature (m) – quantified as the mean curvature of the two free tissue edges. For each edge, curvature is determined by fitting a circle to the contour points and taking the reciprocal of its radius.
• Beat Frequency (Hz) – determined by analyzing the mean absolute displacement time series and calculating the average number of beats per second. Individual beats are identified as intervals between two consecutive valleys (zero-crossings) in the displacement curve.
• Full Width at Half Maximum (FWHM) (frames) – measured from the pillar mean absolute displacement time series as the number of frames between the points where displacement amplitude equals half of its peak value (Fig 4b-i).
• Pillar Mean Peak Force (N) – obtained by averaging the absolute peak contraction force measured at each pillar during tissue contraction events.
• Tissue Mean Peak Absolute Displacement (pixels) – calculated by taking the mean of the absolute displacement values at maximal contraction across the tissue.
• Strain Peak Average Asynchrony (frames) – computed as the mean difference between the frame at which peak mean absolute displacement occurs and the frame of peak Green-Lagrange strain (, , or ) at each grid location, providing an aggregate measure of temporal offset across the tissue (Fig 4b-ii).

In this work, displacement-based measurements are reported in pixels rather than physical length units, and temporal measurements are reported in frames rather than physical time units. This choice is justified by the fact that all image sequences in the dataset were acquired under identical imaging settings, at a uniform spatial resolution of and an image acquisition rate of 65 Hz. Converting pixel-based displacements to physical units is straightforward: dividing by the pixel size yields displacements in . Similarly, converting frame-based temporal measurements to physical time units requires dividing by the frame rate, which in this case is 65 frames/s, yielding measurements in seconds. Neither conversion alters any comparisons or conclusions within this dataset, as both amount to scaling all values by a constant factor. For cross-platform applicability, however, where imaging settings and pixel sizes may differ across systems, conversion to physical units is recommended to ensure meaningful comparability. The proposed framework readily supports this conversion, provided the pixel size and frame rate of the imaging setup are known.

All code for extracting these direct metrics from our software tools, together with the complete post-processing workflow, is openly available on GitHub: https://github.com/HibaKob/MicroBundleAnalysis. We present the results of these metrics in the “Feature correlations reflect both redundancy and novel information” Section.

Interrelationships and redundancy among extracted metrics.

We next investigated the relationships among the extracted metrics in order to assess their informational value and identify potential redundancy. Our goal was to determine whether all metrics are essential, or if some exhibit sufficiently high inter-correlation to warrant exclusion. To this end, we evaluated multicollinearity among the metrics using two complementary measures: the condition number, which provides an overall assessment of collinearity, and the variance inflation factor (VIF), which pinpoints the specific sources. [114].

The condition number is derived as the maximum of the condition indices, which themselves are computed from the eigenvalues () of the scaled correlation (or covariance) matrix of the remaining metrics in a regression model as where are the eigenvalues of the correlation matrix of the predictors, and is the largest eigenvalue. Then for each metric, VIF is calculated as , where is the coefficient of multiple determination from regressing metric i against the remaining metrics.

Commonly accepted thresholds for weak collinearity are a condition number less than 10 and VIF values below 5 [114]. Table 2 summarizes the multicollinearity analysis. The results demonstrate that, while the complete set of 16 metrics displays weak global collinearity (condition number below 10), certain strain-derived metrics exhibit elevated VIF values. When metrics related to and are excluded, both the condition number and variance inflation factors for the remaining metrics drop well below critical thresholds–except for “Tissue Mean Peak Absolute Displacement” and “Pillar Mean Peak Force”–suggesting minimal collinearity. Based on this assessment, we restrict our downstream analyses to the 10 key metrics identified in Table 2.

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Table 2. Multicollinearity was assessed using condition number and variance inflation factor. Collinearity is generally low, with condition numbers for both the full set of 16 metrics (8.04) and the 10 selected metrics (5.67) well below the threshold of 10. The 10 selected metrics are highlighted in bold font. Notable multicollinearity is confined to strain-derived metrics; excluding those linked to and yields consistently low VIF values for the remaining metrics, except for “Tissue Mean Peak Absolute Displacement” and “Pillar Mean Peak Force,” which are highly inter-correlated.

https://doi.org/10.1371/journal.pcbi.1014522.t002

As a next step, we investigated pairwise relationships among all metrics. We first calculated the Spearman’s rank correlation coefficient () [115], which captures monotonic relationships regardless of linearity, for every metric pair. Recognizing, as Anscombe’s quartet [116] exemplifies, that summary statistics alone can mask nuanced patterns in the data, we complemented the correlation analysis with pairplots for each metric combination to visually examine their associations. In addition, we included Kernel Density Estimate (KDE) plots to assess the distribution and variability of each metric. The width of the KDE curve reflects the degree of variability: wider curves indicate greater variance, while narrow curves suggest more closely grouped observations. Moreover, skewness in the KDE curves may indicate data asymmetry or the presence of outliers.

To streamline interpretation and avoid redundancy inherent in symmetric correlation matrices, we condensed the results in Fig 10, focusing on the 10 representative metrics with minimal collinearity identified in Table 2. The figure is organized as a matrix: the diagonal elements feature KDE plots illustrating each metric’s distribution, the upper triangle displays Spearman’s correlation coefficients, and the lower triangle contains pairplots visualizing the relationships. Comprehensive results for all 16 metrics are similarly presented in S3 Fig.

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Fig 10. Comprehensive visualization of pairwise relationships among metrics.

The upper triangle displays Spearman’s correlation coefficients () along with the corresponding p-values, providing both the strength and statistical significance of monotonic associations between metrics. The lower triangle features pairplots that visualize the joint distribution of each metric pair. Kernel Density Estimate (KDE) plots along the diagonal illustrate the distribution of individual metrics, highlighting variation in data spread.

https://doi.org/10.1371/journal.pcbi.1014522.g010

Surveying the matrix, the upper triangle reveals that most metric pairs show negligible () or weak () monotonic associations [117,118], suggesting that the metrics largely capture distinct aspects of tissue behavior. Moderate () and strong () correlations are relatively rare and limited to several metric clusters. Noteworthy examples of strong positive correlations include “Tissue Mean Peak Absolute Displacement,” “Pillar Mean Peak Force,” “Wasserstein Distance PC10,” and “ GSI.” In contrast, “Beat Frequency,” “Full Width at Half Maximum,” and “ Average Pairwise DTW Distance” form a cluster of negative correlations. Nevertheless, perfect correlations are absent; the strongest observed Spearman coefficient is between “Tissue Mean Peak Absolute Displacement” and “Pillar Mean Peak Force” (), underscoring that each metric retains unique informational value.

The KDE plots along the diagonal confirm that metrics such as “Tissue Curvature” and “ Strain Peak Average Asynchrony” are relatively consistent across conditions, while others like “Pillar Mean Peak Force” and “Full Width at Half Maximum” display greater variability and occasional outliers. The pairplots in the lower triangle further validate the absence of pronounced non-monotonic or anomalous relationships, providing visual confirmation that the data structure is well-represented by the computed Spearman correlations.

Prediction-based assessment of metric redundancy.

To provide an alternative perspective on metric redundancy, we designed an exhaustive prediction-based strategy using machine learning methods. From the set of 10 metrics, we selected one as a held-out target and used the remaining 9 metrics as input features for prediction. For these 9 metrics, we considered all possible feature combinations of size k (), resulting in a total of 511 unique subsets (). For each subset, we trained a multilayer perceptron (MLP) using the scikit-learn Python library (v1.7.1) [71], employing a parameter grid search to optimize model depth (up to three layers) and the hyperparameters alpha and learning_rate_init, selecting the configuration with the lowest validation loss. Fixed common hyperparameters include activation = ‘relu’, solver = ‘adam’, num_epochs = 1000, patience = 15, lr_patience = 8, decay_factor = 0.5, and random_state = 42. The dataset was split into 60%, 20%, and 20% for training, validation, and test, respectively. This analysis was performed for two representative metrics with notably high inter-correlation and elevated VIF values: “Pillar Mean Peak Force” and “ GSI.”

The outcomes of this predictive analysis are summarized in Fig 11, where we evaluated the reconstruction accuracy using the R² score on the test dataset. Each outcome is color-coded based on the highest Spearman’s correlation coefficient identified among the input features. In general, we observed that the predictive performance improves as the number of input features increases, particularly when those features include variables with strong Spearman correlation to the held-out metric. This trend demonstrates that access to more relevant features enhances the model’s ability to capture the underlying structure of the data. However, after a certain point, the addition of more features yields only marginal gains, and the R² curve plateaus. This plateauing effect indicates that once the most influential variables have been incorporated into the model, further expansion of the input set provides diminishing returns, highlighting redundancy among certain features. For example, a model trained with 2 metrics, with one being “Tissue Mean Peak Absolute Displacement,” can predict “Pillar Mean Peak Force” with a similar accuracy to one trained on all 9 features.

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Fig 11. Summary of prediction performance for all multilayer perceptron (MLP) models.

These models were trained using different numbers and combinations of input features, with (a) “Pillar Mean Peak Force” and (b) “ GSI” held out as target variables. For each case, the R² value of the neural network on test data is shown as a function of the number of input features. Results are color-coded according to the maximum Spearman’s correlation coefficient observed among the selected input features, highlighting the impact of feature inter-correlation on model performance. Insets provide a closer view of regions with higher R².

https://doi.org/10.1371/journal.pcbi.1014522.g011

This analysis reveals intricate, non-linear relationships between metrics that transcend the scope of standard correlation measures. Importantly, subsets of features with moderate maximum Spearman correlation () can still achieve substantial predictive accuracy. Future studies employing larger datasets, rigorously defined metrics, and advanced machine learning techniques will be pivotal in disentangling the connections among diverse tissue characteristics, such as the links between structural properties and heterogeneous contractile function. Such approaches can also help determine the minimum set of metrics required for comprehensive characterization of tissue dynamics. Ultimately, these strategies will provide critical insights for experimental design and support more informed and targeted investigations.

Note on the dangers of analytical flexibility.

A critical implication of our statistical analysis is the risk of inadvertently engaging in data dredging, or p-hacking [125]. This occurs when researchers explore a wide array of analytical choices such as varying the selection of variables, data subsets, or preprocessing decisions, and selectively report only those results that achieve statistical significance. In our dataset, this risk may take the form of highlighting only significant findings, repeatedly modifying data-cleaning criteria (for example, removing or retaining outliers), or performing multiple subgroup analyses but disclosing only those yielding p-values below 0.05. Notably, these behaviors may arise inadvertently rather than from intentional misconduct. Nevertheless, they undermine statistical validity and inflate the likelihood of false positives, presenting spurious outcomes as genuine discoveries.

In the present work, our aim is not to advance specific biological conclusions but to illustrate, in a conclusion-agnostic manner, the potential range of analytical comparisons that can be performed with these types of data, and highlight the methodological considerations that accompany them. As larger and more complex datasets become increasingly common, we encourage researchers to carefully document and justify key analytical decisions, consider pre-specifying their primary analysis strategy (for example, through preregistration), and transparently report relevant analytical alternatives where practical. Such practices need not require exhaustive reporting of every possible analysis, but fostering transparency in the analytical workflow can help minimize false discoveries and strengthen the reproducibility and robustness of scientific findings [126–128].