How accurate is mechanobiology? A statistical test of cell force

TL;DR

This work tackles the lack of quantified measurement uncertainty in image-based mechanobiology by recasting force reconstruction as a one-step inverse problem that directly links image data to force maps. It develops a Bayesian framework to compute per-point covariances and credible regions, enabling hypothesis testing with p-values and visualization of uncertainty via Main Credible Alternatives (MCAs). The approach is demonstrated on Traction Force Microscopy and Active-Nematics experiments, showing how image noise and ill-posedness propagate into force estimates and how statistical tests can assess changes, background significance, and feature presence. By integrating uncertainty quantification with hypothesis testing, the work provides a principled path to assess statistical and practical significance of observed mechanobiological force patterns, with potential applicability to broader computational-imaging contexts.

Abstract

Mechanobiology is gaining more and more traction as the fundamental role of physical forces in biological function becomes clearer. Forces at the microscale are often measured indirectly using inverse problems such as Traction Force Microscopy because biological experiments are hard to access with physical probes. In contrast with the experimental nature of biology and physics, these measurements do not come with error bars, confidence regions, or p-values. The aim of this manuscript is to publicize this issue and to propose a first step towards a remedy therefor in the form of a general reconstruction framework. We also show that this opens the door to hypothesis testing of seemingly abstract experimental questions.

Figures (5)

• Figure 1: A General Framework for Image-Based Mechanobiology.Top row: application of the framework to Traction Force Microscopy (TFM). Left to right: one image (green) of the two required for TFM capturing the fluorescent beads in the substrate with the cell (magenta) overlaid; second image (green, substrate at rest) of the two with the first one overlaid (magenta, substrate under traction) to help with the visualization of movement; displacement measurements resulting from our framework applied to TFM (colorbar range 0-2.1); continuum model used for TFM; traction-force measurements resulting from our framework applied to TFM (range 0-0.16^-1). Second row: same as Top but for taking measurements in Active-Nematics (AN) systems. The colorbar ranges are 0-2.9 . ^-1 for the velocity, and 0-5.6⋅ 10^-2^-1 . ^-1 for the (relative) nematic forces. Third row: General mathematical formulation (see text) of the framework. The red arrows represent the two steps of classical techniques, e.g. for TFM. The blue arrows represent how our framework links the image data directly to the measurement of interest and creates a feedback loop while taking the noise into account. Bottom row: Examples of other image-based measuring techniques that can be reformulated into our framework using different systems and models (e.g., Zener or Jeffrey models for viscoelastic solids or liquids).
• Figure 2: Visualization of the uncertainty of the measurements via the variance and MCAs.Top row: TFM. Left to right: fluorescence image of the beads in the substrate (grey scale) with the cell overlaid (green); variance of the measurement to be compared with the distribution of beads; measurement; and three Main Credible Alternatives (MCAs) to the measurement. Colorbar ranges: 0-8.2⋅ 10^-4^-2 and 0-0.16^-1. Bottom row: same as Top but for an AN system. The colorbar ranges are 0-1.5⋅ 10^-4^-2 . ^-2 and 0-5.6⋅ 10^-2^-1 . ^-1. Cyan arrows point at relevant differences with respect to the reconstructed measurements.
• Figure 3: Hypothesis tests for the significance of force changes after some event. (Question 1.) The tests show that some events are not significant enough with respect to the ill-posedness and image noise. Top row: TFM. Colorbar ranges are 0-0.16^-1. Left to right: force maps at 0, 40 (non-significant change), and 400 (significant change) as the cell establishes after seeding. Bottom row: same as Top but for an AN system. The colorbar range is 0-2.8⋅ 10^-2^-1 . ^-1.
• Figure 4: Hypothesis tests for the significance of features in the measurements. (Question 3.) The tests show that the existence of some force patches is uncertain when noise and ill-posedness are accounted for. Top row: TFM (range 0-0.16^-1). Left to right: A map of the magnitude $||\mathbf{f}^\star_{\varphi_{1,2}}||_2(\mathbf{x})$ of the measurement overlaid on the cell with circles delimiting two regions of interest (ROIs) for reference. Zoom-ins around one of the ROIs showing a comparison between the original reconstructed force field and a version where we inpainted the ROI (change not significant). Zoom-ins around the other ROI for which the inpainting did result in a significant change. Bottom row: same as Top but for an AN system (range 0-5.6⋅ 10^-2^-1 ^-1).
• Figure 5: Rapid decay of the eigenvalues of the Hessian-based generalized eigenvalue problem in \ref{['eq:eigenvalue_problem']}. The ordered index is simply the index of the eigenvalues after sorting them.