Wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations

Abstract

Linear di¬usion is an established model for spatial spread in biological systems, including movement of cell populations. However, for interacting, closely packed cell populations, simple di¬usion is inappropriate, because di¬erent cell populations will not move through one another: rather, a cell will stop moving when it encounters another cell. In this paper, I introduce a nonlinear di¬usion term that re®ects this phenomenon, known as contact inhibition of migration. I study this term in the context of two competing cell populations, one of which has a proliferative advantage over the other; this is motivated by the very early stages of solid tumour growth. I focus in particular on travelling-wave solutions, corresponding to moving interfaces between the two cell populations. Numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value, and I present linear analysis that explains the selection of wave speeds by initial conditions. I obtain an approximation to the shape of these waves for high speeds, and show that the minimum speed arises via quite new behaviour in the travelling-wave equations, with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum. Exploiting this structure, I use singular perturbation theory to investigate the wave shape for speeds close to the minimum.

A Multiphase Model Describing Vascular Tumour Growth

“…We simplified the resulting system of equations and boundary conditions, obtaining the parabolic equation (30) and the hyperbolic equation (31) describing the evolution of the volume fractions of the tumour cell and blood vessel phases. The former equation is similar to that used by other authors as a starting point for modelling tumour growth (Gatenby and Gawlinski (1996), Sherratt (2000)). We illustrated one possible growth regime by setting all the drag coefficients equal.…”

“…Secondly, the cell-cell interactions manifest themselves as a nonlinear diffusion term in (30), which will transport cells down gradients in cellular density. We note also that other authors (for example Gatenby and Gawlinski (1996) and Sherratt (2000)) use nonlinear diffusion equations as the starting point for their modelling; by contrast we start by modelling the stresses in the tumour constituents and arrive at a situtation in which cells move by a diffusion-like process.…”

Soliton-like phenomena in one-dimensional cross-diffusion systems: a predator–prey pursuit and evasion example

“…Many models of spatial dynamics of populations take taxis into account, and its importance has been recognized in modelling various biological and ecological processes, including propagation of epidemics, bacterial population waves, aggregation in the cellular slime mold Dictyostelium discoideum, dynamics of planktonic communities and of insect populations [2,4,5]. The existence of travelling waves, and also stationary spatially-inhomogeneous structures, in interacting populations with taxis has been demonstrated experimentally and theoretically [6,7,8,9,10,11].…”

Justification for wavefront propagation in a tumour growth model with contact inhibition

“…With regard to interaction between the normal and tumour cell populations, diffusion has also been used very successfully in models for spatial spread (Sherratt 1993;Gatenby & Gawlinski 1996;Sherratt 2000). Diffusion terms in reactiondiffusion equations of tumour growth are broadly divided into two categories.…”

“…For tumour growth, viewed usually as a competition process between tumour cells and surrounding normal tissue cells, numerous mathematical models have recently been investigated. These include some examples based on ordinary differential equations (ODEs) modelling tumour growth and tumour-host interaction as competing populations (Michelson & Leith 1991;Gatenby 1995), and reactiondiffusion systems modelling the dispersal behaviour of tumour cell growth (Sherratt 1993;Gatenby & Gawlinski 1996;Sherratt 2000) and also some particularly novel models that reflect the cancer evolution and its interaction with the immune system (Sherratt 1993;Bellomo & Preziosi 2000).…”