Travelling wave analysis of a mathematical model of glioblastoma growth

TL;DR

This paper analyzes a mathematical model of glioblastoma growth, deriving analytical expressions for wave speed and shape, revealing effects of phenotypic switching on tumor invasion dynamics.

Contribution

It provides the first analytical expressions for wave speed and shape in a phenotypic switching tumor model, extending previous Fisher equation results.

Findings

Derived closed-form wave speed expression.

Obtained approximate wave front shape.

Showed inverse relationship between wave steepness and speed does not hold.

Abstract

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, PLoS Comp. Bio., 8(6) 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.