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, $${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\hat{s}}$$ 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, $$c = c_{\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c > c_{\mathrm{min}}$$ c > c 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.

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion

“…This could be achieved by transforming the time-dependent PDE model into the travelling wave coordinate, true0 z = x − c t , where true0 c is the long-time asymptotic speed of the travelling wave solutions. Properties of the solution of the resulting dynamical system could then be studied in phase space to provide information about the relationship between parameters in the continuum PDE model and the travelling wave speed true0 c and the shape of the travelling wave profile [3,72]. We leave both of these potential extensions for future consideration.…”

“…The discrete model explicitly models how individual cells in a two-dimensional in vitro experiment produce a biological substrate (e.g. biological macromolecules and extracellular material) that is deposited onto the surface of the tissue culture plate [ 14 , 71 , 72 ]. Substrate is produced at a particular rate, and deposited locally by individuals within the simulated population.…”

Quantifying tissue growth, shape and collision via continuum models and Bayesian inference

“…When population pressure is taken into account in continuum models one obtains the Porous-Fisher equation, which replaces the constant diffusion coefficient in the Fisher-KPP equation by a density-dependent function that increases as a power-law of the density. One of the most interesting features about this model is the appearance of compactly supported solutions, which give rise to the sharp invasion fronts observed in tissue formation experiments [10][11][12][13]. Of course, there are additional effects which can play an important role in collective cell motility and have been modelled using extensions of the mentioned equations, such as cell-cell adhesion [14][15][16][17], viscoelastic forces [1,18,19], interactions with the extracellular matrix [20][21][22], heterogeneity in cell size [23,24] and cell-cycle dynamics [25].…”