Rigorous discrete-to-continuum link for multicellular tissues

Develop a rigorous discrete-to-continuum derivation that connects the planar cell vertex model of confluent epithelial monolayers (a discrete, cell-scale description based on vertex positions, cell areas, and perimeters) to a continuous tissue-scale description, specifying the macroscopic governing equations and constitutive relations that faithfully capture viscoelastic relaxation and the effects of geometric stiffness arising from the evolving cell network.

Background

The paper studies viscoelastic relaxation in epithelial monolayers using the cell vertex model and a spectral perspective based on singular-value decomposition. While many continuum approximations exist (e.g., long-wave and homogenization approaches), the authors emphasize that a fully rigorous connection between the discrete cell-scale representation and a continuum tissue-scale description is lacking.

They motivate this open challenge by noting the importance of capturing multiscale mechanical effects—including geometric stiffness due to prestress—and by demonstrating how discrete Laplacian operators provide multiscale bases for evolving fields. Establishing a rigorous discrete-to-continuum link would clarify how these discrete operators and relaxation spectra translate into macroscopic PDEs and constitutive laws.