Modeling tissue growth with the Stokes equation

Abstract

We present a cell-free continuum model for simulating generalized bulk tissue growth in 3D. We assume that the tissue behaves mechanically as viscous fluid so that its behavior can be described with the Stokes equation with mass sources. The growth is directed by a diffusing morphogen produced by specialized signaling centers, whose positions are established through a reaction-diffusion system coupled with differentiation. We further assume that the tissue interface may be stiff (modeled as surface tension), and that tissue adhesion can vary (modeled as variable viscosity). The numerical validity of the implementation is investigated using test cases with known solutions, and the model dynamics are demonstrated in simulations of idealized tissue growth. The combination of Stokes equation and diffusing morphogens allow the integration of patterning and growth as in real organs systems such as limbs and teeth. We propose that the presented techniques could be useful for simulating and exploring mechanistic principles of tissue growth in various developing organs.