Travelling wave analysis of a mathematical model of glioblastoma growth

• We show that the rate of growth in an individual-based model of brain tumour growth can be approximated by a closed form expression that depends crucially on the rate of phenotypic switching.
• Using singular perturbation methods we also derive an approximate expression of the wave front shape.
• These new analytical results agree with simulations of the cell-based model.
• We show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.

In this paper we analyse a previously proposed cell-based model of glioblastoma (brain tumour) growth, which is based on the assumption that the cancer cells switch phenotypes between a proliferative and motile state (Gerlee and Nelander, 2012). The dynamics of this model can be described by a system of partial differential equations, which exhibits travelling wave solutions whose wave speed depends crucially on the rates of phenotypic switching. We show that under certain conditions on the model parameters, a closed form expression of the wave speed can be obtained, and using singular perturbation methods we also derive an approximate expression of the wave front shape. These new analytical results agree with simulations of the cell-based model, and importantly show that the inverse relationship between wave front steepness and speed observed for the Fisher equation no longer holds when phenotypic switching is considered.