A mathematical model of curvature controlled tissue growth incorporating mechanical cell interactions

Abstract

Many biological tissues grow at rates that depend strongly on the local geometry of the tissue interface, yet the cellular mechanisms underlying this dependence remain poorly understood. In this work, we develop a two-dimensional mathematical model of tissue growth to investigate how curvature-dependent growth emerges from the complex interactions between cell-scale crowding and mechanical interactions. The tissue interface is represented as a confluent layer of mechanically interacting cells that simultaneously produce new tissue. We formally derive a continuum limit of this discrete model, obtaining a system of partial differential equations governing the coupled evolution of cell density and tissue geometry. Although curvature is not explicitly represented in the discrete model, curvature-dependent tissue growth emerges naturally in the continuum limit through geometric crowding associated with tissue production. The continuum equations also establish a direct relationship between cellular mechanical properties and the effective diffusivity governing mechanical relaxation at the tissue scale. Numerical simulations illustrate that the competition between tissue production and mechanical relaxation can result in a variety of tissue evolutions, including tissue interface smoothing consistent with curvature-controlled in vitro growth in porous scaffolds and bone formation. Finally, the model predicts that the time required for scaffold pore closure scales with pore size but also depends on a pore shape factor, extending previous empirical observations of linear scaling with size. More broadly, this work provides an analytical framework linking cell-scale mechanics to tissue-scale growth and geometry.