  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.
