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.
Introduction: The invasive nature of tumors and their ability to metastasize to distant organs remain among the primary causes of cancer-related mortality. A central mechanism underlying this process is collective cell migration, through which groups of tumor cells coordinately invade surrounding tissue [1,2]. This mode of invasion is characterized by strong mechanical coupling, sustained tissue-level deformation, and emergent collective dynamics.
A variety of experimental and theoretical approaches have been used to study collective tumor invasion. Experimental studies have focused on identifying histological, biochemical, and phenotypic signatures associated with collective migration [3]. Complementing these, agent-based and particle-based models have been developed to resolve cell-scale mechanics, adhesion, and rearrangements during invasion [4][5][6]. At larger scales, continuum descriptions have treated tissues as active materials, employing hydrodynamic theories of active polar and nematic fluids [7][8][9][10], chemotactic fronts [11], and related formulations to study interfacial instabilities, tissue fluidization, and fingering phenomena [8,[12][13][14]. These continuum approaches have also been extended to include environmental effects through mechanically imposed boundary stresses [15,16].
Recent experiments [17] have added a new dimension to this picture by demonstrating that the viscoelastic properties of the extracellular matrix play a decisive role in regulating tumor invasion. In particular, the interface of an actively growing breast epithelial tumor spheroid remains flat or spherical when embedded in an elastic matrix, but becomes unstable to longwavelength undulations when the matrix is more liquidlike. This behavior can be understood by comparing the * lmahadev@g.harvard.edu* Zooming in allows us to focus on the interface. In the sharpinterface limit, we assume that an active growing tissue moves and pushes against a viscoelastic (linear Maxwell model) matrix. The interface, η(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 -∞ ≤ x ≤ ∞.
matrix stress-relaxation timescale with the tissue growth timescale: if the matrix stress relaxes slowly, this suppresses interfacial instabilities, whereas if the matrix relaxes quickly, there is a propensity for enhancing instabilities. These experimental observations were rationalized using agent-based cellular simulations that explicitly model cell-matrix interactions and matrix remodeling [17]. While these simulations successfully reproduce the observed phenomenology, they do not yet provide arXiv:2602.10375v1 [cond-mat.soft] 10 Feb 2026 a general theoretical framework that links matrix viscoelasticity, active growth, and interfacial stability in a transparent and predictive manner.
Here, we develop a continuum phase-field hydrodynamic theory, together with finite-element simulations, to provide such a framework. We build on phase-field methods for modeling interfacial dynamics in complex fluids and soft matter [18][19][20], by coupling interfacial motion to hydrodynamics and active stresses in growing tissues interacting with viscoelastic environments to explain the experimentally observed behavior of the spheroid-matrix interface in both the linear and nonlinear regimes. Diffuse-interface model of an active tissue invading a viscoelastic matrix: We model a growing epithelial spheroid invading a viscoelastic extracellular matrix using a diffuse-interface (phase-field) formulation that unifies growth, active forcing, matrix viscoelasticity, and interfacial capillarity within a single continuum framework. In Fig. 1a, we show the evolution of the instability of a growing epithelial spheroid invading the extracellular matrix, and note that the basic spatiotemporal fields that need to be tracked include the active tissue phase and passive viscoelastic matrix, characterized by a velocity field u(r, t), and the stress-field σ(r, t). The diffuseinterface (phase-field) is modeled via the scalar order parameter ϕ(x, t) ∈ [-1, 1] which distinguishes the active tissue (ϕ = +1) from the passive matrix (ϕ = -1), with a narrow interfacial region of thickness ϵ, assumed to be smaller than all other length scales. In addition to resolving the microscopic physics in a thermodynamically consistent manner, the phase-field model is much easier to solve numerically without the need to separately track the interfaces.
The phase-field, ϕ, evolves according to a Cahn-Hilliard like equation,
modified to account for the advection of the phase(s) by the fluid with velocity u(x, t), and a growth term given by G
The form of the growth has been chosen so that growth near the interface (i.e. ϕ ≈ 0) is promoted. Here the double-welled free energy density
), with κ ch having dimensions of energy density, and M charact
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