Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks

Anci Lin; Zhiwen Zhang; Wenju Zhao

cs.LGMar 31, 2026

Cell-induced densification and tether formation in fibrous extracellular matrices with biomimetic physics-informed neural networks

Authors: Anci Lin, Zhiwen Zhang, Wenju Zhao

Organizations: School of Mathematics, Shandong University, Jinan, Shandong, 250100, P.R. China. · Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong SAR, P.R. China. · Materials Innovation Institute for Life Sciences and Energy (MILES), HKU-SIRI, Shenzhen, P.R. China.

Abstract

Nonconvex multi-well energies in cell-induced phase transitions give rise to fine-scale microstructures, low-regularity transition layers and sharp interfaces, all of which pose numerical challenges for physics-informed learning. Here we introduce biomimetic physics-informed neural networks (Bio-PINNs), which implement a near-to-far curriculum by progressively revealing the computational domain away from the cell boundary and combining this schedule with a deformation-uncertainty proxy that concentrates collocation points near evolving transition layers and tether-forming regions. Across single-cell and multicellular benchmarks, Bio-PINNs recover the densified phase more reliably near cell boundaries and in intercellular gaps, while capturing tether morphology more faithfully than representative ungated and residual-driven adaptive baselines.

Figures & tables

Figure 1: Experiment–simulation comparison for mechanically interacting acini. (a) Fluorescence image of two mammary acini embedded in collagen, adapted from Fig. 2B of Shi et al. Shi et al. (2014) (included here with permission from the authors). (b) Bio-PINN solution in the long-distance weak-coupling regime, $d=5r_{c}$ . The numerical panel mirrors the visual structure of the experimental panel: Zoom 1 is placed inside the main field at the bottom and highlights the intercellular corridor, whereas Zoom 2 is placed externally on the right and highlights the pericellular band on the outer cell flank. Both panels display the coexistence of an intercellular mechanically organized corridor and an outer pericellular band. In the numerical panel, colour denotes $J=\det F$ , with smaller $J$ indicating stronger densification.

Figure 2: Cell-induced remodelling motivates the Bio-PINN workflow. A logistic distance gate $g_{\gamma}(x)$ progressively exposes the matrix from the near-cell region to the far field, while a deformation-uncertainty proxy drives an R3 retain, resample, and release update under a fixed collocation budget. Together, these steps focus training on pericellular transition layers and tether-forming regions as the active domain expands.

Figure 3: Near-to-far activation derived from the physical progression of remodelling. The left panel shows a logistic time gate $g_{\beta}(t)=\sigma\!\left(\alpha(\beta-t)\right)$ . The right panel shows its distance-based analogue $g_{\gamma}(\tilde{d})=\sigma\!\left(\alpha(\gamma-\tilde{d})\right)$ , where $\tilde{d}$ is the normalized distance to the cell. The bottom panels show spatial snapshots of $g_{\gamma}(x)=\sigma\!\left(\alpha(\gamma-\tilde{d}(x))\right)$ for increasing $\gamma$ , showing how the active region expands from the pericellular neighbourhood into the surrounding matrix.

Figure 4: Local probing used to detect regions likely to develop transition layers or tethers. Around each representative collocation point $x^{\star}$ , Gaussian probes sample nearby deformations and estimate local variability, for example $\operatorname{Var}_{k}\|\nabla y_{\theta_{i}^{(k)}}(x^{\star})\|_{F}$ . The resulting normalized score $\widetilde{U}_{i}(x^{\star})\in[0,1]$ is largest near evolving interfaces and intercellular interaction regions.

Figure 5: Adaptive collocation loop used in Bio-PINNs. In Step 1, candidate collocation points are proposed in the currently active region and, after gate expansion, in the newly opened shell using low-discrepancy designs. In Step 2, the UQ proxy $\widetilde{U}_{i}$ is evaluated and high-uncertainty points are retained while low-uncertainty points are released. In Step 3, the collocation set is refilled to preserve a fixed budget and the procedure is repeated, so that sampling progressively concentrates on transition layers and emerging tethers while maintaining coverage of the active domain.

Panel of Figure 6 — Figure 6: Single-cell remodelling without second-gradient regularization ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for (a) PINN, (b) RAR-D, (c) R3, and (d) Bio-PINN. The benchmark asks whether the method can recover a densified pericellular band while leaving the surrounding matrix close to the undensified state. Among the methods shown here, Bio-PINN yields the lowest near-cell $J$ values together with a more azimuthally uniform pericellular band.

Panel of Figure 7 — Figure 7: Single-cell remodelling with weak second-gradient regularization ( $\varepsilon_{0}=0.01\,r_{c}$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for (a) PINN, (b) RAR-D, (c) residual-driven R3, and (d) Bio-PINN. The weak $H^{2}$ penalty smooths the finest oscillations across methods, but Bio-PINN still yields a more coherent pericellular band and lower near-cell $J$ values than the baselines shown here.

Panel of Figure 8 — Figure 8: Two-cell remodelling without second-gradient regularization ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for a long-distance regime (top row; $d=5\,r_{c}$ , weak coupling) and a short-distance regime (bottom row; $d=2.5\,r_{c}$ , strong coupling). The long-distance case probes weak coupling between cells, whereas the short-distance case probes tether formation in the intercellular gap. Panels (a,e) correspond to PINN, (b,f) to RAR-D, (c,g) to residual-driven R3, and (d,h) to Bio-PINN. Among the methods shown here, Bio-PINN produces a more contiguous densified band next to each cell and in the interaction region, with less spurious structure in the surrounding matrix. In particular, panel (d) recovers the same pair of motifs emphasized in Fig. 1 : an intercellular corridor and an outer pericellular band, whereas panel (h) sharpens the intercellular low- $J$ region into a tether.

Panel of Figure 9 — Figure 9: Three-cell remodelling stress test without second-gradient regularization ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is compared across methods for a long-distance regime (top row; weak coupling) and a short-distance regime (bottom row; strong coupling). The colour scale indicates whether the densified phase is reached around each cell and in the multicell interaction core. Bio-PINN reaches lower near-cell and interaction-core $J$ values than the baselines while keeping the surrounding matrix closer to the undensified state.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Default setting
Network	3 hidden layers, width 128, CELU activation
Optimizer	Adam ( $10^{-3}$ , $\beta_{1}{=}0.9$ , $\beta_{2}{=}0.999$ ); LR $\times 0.9$ /10k iters
Collocation budget	$N=10^{4}$ points, updated by R3
Quadrature	QMC (Hammersley) for bulk terms
Gating	$g_{\gamma_{i}}(x)=\sigma(\alpha(\gamma_{i}-\tilde{d}(x)))$ ; $\alpha{=}5.0$ , $\gamma_{0}{=}-0.5$ , $\Delta_{\max}{=}0.05$
R3 policy	Retain fraction $\rho=0.5$

Appendix

Table S1 : Default hyperparameters used in all experiments. Network architecture, optimizer settings, collocation budget, quadrature, gating schedule, R3 policy, and UQ-proxy parameters.

Panel of Figure 11 — Figure S1 : Effect of the resampling period $P$ in the single-cell non-regularized regime ( $\varepsilon_{0}=0$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ).

Panel of Figure 12 — Figure S2 : Effect of network depth $h_{l}$ in the single-cell non-regularized regime ( $\varepsilon_{0}=0$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) for different hidden-layer depths.

Panel of Figure 13 — Figure S3 : Effect of network width in the single-cell non-regularized regime ( $\varepsilon_{0}=0$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) for different hidden-layer widths.

Panel of Figure 14 — Figure S4 : Effect of the UQ probe count $m_{\mathrm{uq}}$ in the single-cell non-regularized regime ( $\varepsilon_{0}=0$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ).

Panel of Figure 15 — Figure S5 : Effect of the UQ probe variance $\rho_{\mathrm{uq}}$ in the single-cell non-regularized regime ( $\varepsilon_{0}=0$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ).

Panel of Figure 16 — Figure S6 : Effect of the resampling period $P$ in the single-cell weakly regularized regime ( $\varepsilon_{0}=0.01\,r_{c}$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ).

Panel of Figure 17 — Figure S7 : Effect of network depth $h_{l}$ in the single-cell weakly regularized regime ( $\varepsilon_{0}=0.01\,r_{c}$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) for different hidden-layer depths.

Panel of Figure 18 — Figure S8 : Effect of network width in the single-cell weakly regularized regime ( $\varepsilon_{0}=0.01\,r_{c}$ ). Each panel shows the Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) for different hidden-layer widths.

Panel of Figure 19 — Figure S9 : Effect of the resampling period $P$ in the two-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top) and the long-distance regime ( $d=5\,r_{c}$ ; bottom).

Panel of Figure 20 — Figure S10 : Effect of network depth $h_{l}$ in the two-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top row) and the long-distance regime ( $d=5\,r_{c}$ ; bottom row).

Panel of Figure 21 — Figure S11 : Effect of network width in the two-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top row) and the long-distance regime ( $d=5\,r_{c}$ ; bottom row).

Panel of Figure 22 — Figure S12 : Effect of the UQ probe variance $\rho_{\mathrm{uq}}$ in the two-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top row) and the long-distance regime ( $d=5\,r_{c}$ ; bottom row).

Panel of Figure 23 — Figure S13 : Effect of the resampling period $P$ in the three-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top row) and the long-distance regime ( $d=5\,r_{c}$ ; bottom row).

Panel of Figure 24 — Figure S14 : Effect of the UQ probe variance $\rho_{\mathrm{uq}}$ in the three-cell non-regularized regime ( $\varepsilon_{0}=0$ ). The Jacobian determinant $J=\det F$ (with $F=\nabla y_{\theta}$ ) is shown for the short-distance regime ( $d=2.5\,r_{c}$ ; top row) and the long-distance regime ( $d=5\,r_{c}$ ; bottom row).

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Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to

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reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to

2\mathrm{D}{+}t

systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering

2\mathrm{D}{+}t

reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.

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