Morphological instability of an invasive active-passive interface

TL;DR

This study develops a theoretical morphological phase diagram, and complement these with two-dimensional finite element (FEM) phase-field simulations to track the nonlinear evolution of the interface with results consistent with theoretical predictions and experimental observations.

Abstract

Morphological instabilities of growing tissues that impinge on passive materials are typical of invasive cancers. To explain these instabilities in experiments on breast epithelial spheroids in an extracellular matrix, we develop a continuum phase field model of a growing active liquid expanding into a passive viscoelastic matrix. Linear stability analysis of the sharp-interface limit of the governing equations predicts that the tissue interface can develops long-wavelength instabilities, but these instabilities are suppressed when the active carcinoid is embedded in an elastic matrix. We develop a theoretical morphological phase diagram, and complement these with two-dimensional finite element (FEM) phase-field simulations to track the nonlinear evolution of the interface with results consistent with theoretical predictions and experimental observations. Our study provides a basis for the emergence of interfacial instabilities in active-passive systems with the potential to control them.

Figures (3)

• Figure 1: Viscoelasticity of the extra-cellular matrix regulates the morphological instabilities at an invasive active-passive interface. (a) The top row shows the development of instabilities at the interface when a an active growing spheroid is placed in a liquid like matrix ($\tau\rightarrow 0$). Figure taken from the study of Elosegui-Artola et.al.elosegui2023matrix. (b) Zooming in allows us to focus on the interface. In the sharp-interface limit, we assume that an active growing tissue moves and pushes against a viscoelastic (linear Maxwell model) matrix. The interface, $\eta(x,t)$, is initially flat at $t=0$ (dashed blue line) but can develop instabilites at a later time. In a minimal setting, we assume that the tissue-matrix lies in the $x-y$ plane, along the x-axis extending from $-\infty \leq x \leq \infty$.
• Figure 2: Phase diagram obtained via predictions of the linear stability analysis and non-linear FEM simulations. Morphological phase diagram of the interface as a function of $\tau^*$ and $\mathsf{C}_a$. The blue portion of the phase diagram is the stable regime (i.e $\Re(\Omega(k))<0$) whereas the red portion is the unstable regime (i.e $\Re(\Omega(k))>0$), following (10); the curve that separates the two regime is given by a hyperbola (13). We then use FEM simulations to corroborate our results: the red crosses denote when the interface is unstable whereas the blue triangles correspond to the stable interface. The FEM simulations were based on solutions to (1-4). The ($\tau^{*}, \mathsf{C}_a$) used in the simulations for the stable and unstable snapshots are given by $(100, 0.12)$ and $(0.1, 0.12)$ respectively. All the other parameter values are listed in the SI.
• Figure 3: Phase-field FEM simulations for the evolution of an invasive active-passive interface (a) Two dimensional simulations of a growing tumor liquid spheroid pushing against an elastic matrix. The tissue grows without developing any instabilities consistent with the predictions of our theory (see the panel from left to right). The ($\tau^{*}, \mathsf{C}_a$) used in the simulation is given by $(100, 0.12)$ (b) Same as (a) but with liquid-like matrix. The simulations are consistent with the theoretical predictions which predict the emergence of instabilites in a liquid-like matrix. The inset within each figure shows the experimental micro-graphs from elosegui2023matrix. The ($\tau^{*}, \mathsf{C}_a$) used in the simulations are given by $(0.1, 0.12)$. The equations solved for both set of cases are (\ref{['eqn:pfch']}-\ref{['eqn:maxwell']}) and rest of the parameter values are listed in Table S1.