Tissue stress measurements with Bayesian Inversion Stress Microscopy

TL;DR

Bayesian Inversion Stress Microscopy (BISM) provides an absolute, tensorial view of tissue stress by inferring the 2D stress field from traction forces while remaining agnostic to tissue rheology. The method enforces force balance with a Gaussian Bayesian framework and, when boundary conditions are imposed, yields isotropic and deviatoric stress components that match independent traction and ablation observations. The authors validate BISM across confined, moving, and heterogeneous geometries, compare it to MSM and Bayesian Force Inference, and demonstrate applicability to ex vivo tumor tissue, highlighting boundary-condition requirements and the potential for 3D extensions via height variation. Overall, BISM offers high-resolution, noninvasive insight into internal tissue mechanics with broad implications for morphogenesis, collective migration, and disease progression.

Abstract

Cells within biological tissue are constantly subjected to dynamic mechanical forces. Measuring the internal stress of tissues has proven crucial for our understanding of the role of mechanical forces in fundamental biological processes like morphogenesis, collective migration, cell division or cell elimination and death. Previously, we have introduced Bayesian Inversion Stress Microscopy (BISM), which is relying on measuring cell-generated traction forces in vitro and has proven particularly useful to measure absolute stresses in confined cell monolayers. We further demonstrate the applicability and robustness of BISM across various experimental settings with different boundary conditions, ranging from confined tissues of arbitrary shape to monolayers composed of different cell types. Importantly, BISM does not require assumptions on cell rheology. Therefore, it can be applied to complex heterogeneous tissues consisting of different cell types, as long as they can be grown on a flat substrate. Finally, we compare BISM to other common stress measurement techniques using a coherent experimental setup, followed by a discussion on its limitations and further perspectives.

Figures (13)

• Figure 1: Graphical abstract
• Figure 2: Bayesian Inference Stress Microscopy workflow: Experimental and analytical pipeline of BISM. Top: Requirements and assumptions of the BISM algorithm. Force balance and experimental data from TFM allow computation of the most probable corresponding stress tensor. Middle: Cartoon illustrating the main steps of BISM implementation. The first step is monitoring the substrate displacement to calculate 2D traction force components. Then, BISM is applied to obtain the 2D stress tensor which includes different components. Bottom: Representative example of BISM analysis of a MDCK monolayer on a 15 kPa soft (PDMS) substrate. [Scale bar: $100 \,\mu$m.]
• Figure 3: Stress inference in a confined system: MDCK cell monolayer in a square domain of lateral extension $L = 500 \,\mu$m Peyret2019. a) Phase contrast image of the confining domain. b, c) Color maps of the components of the traction force field $t_x^{\mathrm{true}}$, $t_y^{\mathrm{true}}$ measured by TFM (in kPa). d, e, f) Color maps of the inferred isotropic stress $\sigma_{\mathrm{iso}}^{\mathrm{inf}}$ and deviatoric stress components $\sigma_{\mathrm{xy}}^{\mathrm{inf}}$, $\sigma_{\mathrm{d}}^{\mathrm{inf}}$ (in kPa.$\mu$m). Here the zero-stress condition is imposed on the four boundaries. g, h) Comparison of measured ($t^{\mathrm{true}}_x$, $t^{\mathrm{true}}_y)$ and inferred ($t^{\mathrm{inf}}_x$, $t^{\mathrm{inf}}_y$) values of components of the traction force vectors (blue crosses), computed from the inferred stress field $\vec{t}^{\mathrm{inf}} = \mathrm{div} \, \sigma^{\mathrm{inf}}$. The bisectrix $y = x$ is plotted as a red line for comparison. The coefficient of determination is $R^2_t = 1.0$. We find excellent agreement between stress averages computed in the cell domain and the true values obtained from moments of the traction force field, thus confirming that BISM provides an absolute measurement of stresses in confined domains: compare $\langle \sigma_{\mathrm{iso}}^{\mathrm{true}} \rangle = 7.77$ kPa and $\langle \sigma_{\mathrm{iso}}^{\mathrm{inf}} \rangle = 7.76$ kPa; then $\langle \sigma_{\mathrm{xy}}^{\mathrm{true}} \rangle = 565$ Pa and $\langle \sigma_{\mathrm{xy}}^{\mathrm{inf}} \rangle = 568$ Pa; and $\langle \sigma_{\mathrm{d}}^{\mathrm{true}} \rangle = -420$ Pa and $\langle \sigma_{\mathrm{d}}^{\mathrm{inf}} \rangle = -416$ Pa. i, j, k) Time evolution of the monolayer. We plot i) the mean tension; j) the cell density against time. Panel k) shows the mean tissue tension as a function of mean traction force norm (blue circles), for the same duration ($30$ h). The red line is the linear regression of data points, with a slope of $15.5\,\mu$m, of the order of a typical cell diameter. [Scale bar: $100 \,\mu$m (a).]
• Figure 4: Stress inference in star-shaped tissue: MDCK cell monolayer. a) Phase contrast image of the cell island. b, c) Color maps of the components of the traction force field $t_x^{\mathrm{true}}$, $t_y^{\mathrm{true}}$ measured by TFM (in kPa). d, e, f) Color maps of the inferred isotropic stress $\sigma_{\mathrm{iso}}^{\mathrm{inf}}$ and deviatoric stress components $\sigma_{\mathrm{xy}}^{\mathrm{inf}}$, $\sigma_{\mathrm{d}}^{\mathrm{inf}}$ (in kPa.$\mu$m). Here the zero-stress condition $\sigma_{ij} \, n_j=0$ is imposed on the four boundaries of the smallest rectangular domain encompassing the cell domain. g, h) Comparison of measured ($t^{\mathrm{true}}_x$, $t^{\mathrm{true}}_y)$ and inferred ($t^{\mathrm{inf}}_x$, $t^{\mathrm{inf}}_y$) values of components of the traction force vectors (blue crosses), computed from the inferred stress field $\vec{t}^{\mathrm{inf}} = \mathrm{div} \, \sigma^{\mathrm{inf}}$. The bisectrix $y = x$ is plotted as a red line for comparison. The coefficient of determination is $R^2_t = 1.0$. As in Fig. \ref{['fig:confined:MDCK']}, we find excellent agreement between stress averages computed in the cell domain and the true values obtained from moments of the traction force field: compare $\langle \sigma_{\mathrm{iso}}^{\mathrm{true}} \rangle = 1.56$ kPa and $\langle \sigma_{\mathrm{iso}}^{\mathrm{inf}} \rangle = 1.57$ kPa; $\langle \sigma_{\mathrm{xy}}^{\mathrm{true}} \rangle = 58$ Pa and $\langle \sigma_{\mathrm{xy}}^{\mathrm{inf}} \rangle = 69$ Pa; $\langle \sigma_{\mathrm{d}}^{\mathrm{true}} \rangle = -45$ Pa and $\langle \sigma_{\mathrm{d}}^{\mathrm{inf}} \rangle = -47$ Pa. [Scale bar 100 $\mu$m (a).]
• Figure 5: Stress inference in a system with a moving boundary.a) Brightfield image representative of a MDCK WT cells wound healing assay. The free edge of the monolayer is moving in the direction of the white arrow. b, c) Color maps of the components of the traction force field $t_x^{\mathrm{true}}$, $t_y^{\mathrm{true}}$ measured by TFM (in kPa). d, e, f) Color maps of the inferred isotropic stress $\sigma_{\mathrm{iso}}^{\mathrm{inf}}$ and deviatoric stress components $\sigma_{\mathrm{xy}}^{\mathrm{inf}}$, $\sigma_{\mathrm{d}}^{\mathrm{inf}}$ (in kPa.$\mu$m). Here the zero-stress condition $\sigma_{ij} \, n_j=0$ is imposed only on the top boundary, while the three other edges are not subject to a boundary condition. g, h, i) The same stress components are inferred under the same conditions, but within a subdomain that intersects the moving front. j, k, l) Comparison of measured stress components in the cropped region depending on whether it was computed using the whole field of view ("all") or only the cropped region ("cropped"). The bisectrix $y = x$ is plotted as a red line for comparison. [Scale bar 80 $\mu$m (a).]