A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

Abstract

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, {hat{u}}(hat{{mathbf {x}}},{hat{t}}) that is coupled to the concentration of an immobile extracellular substrate, {hat{s}}(hat{{mathbf {x}}},{hat{t}}). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to {hat{s}}, while a logistic growth term models cell proliferation. The extracellular substrate {hat{s}} is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, c = c_{mathrm{min}}, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, c > c_{mathrm{min}}. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.