Interactions between a uniformly proliferating tumour and its surroundings: Stability analysis for variable material properties

We examine the stability properties of a planar tumour growing into neighbouring material and assess their dependence on the viscosities (providing a possible measure of the degree of malignancy of tumour cells) and rates of cellular proliferation of the tumour and surrounding tissue. An abundant supply of nutrient is assumed and Darcy’s law and Stokes flow are considered in describing the constitutive behaviour of the tissue.Using a combination of linear-stability analysis and numerical methods, the evolution of the advancing tumour boundary is determined. It is shown that when the deformation is governed by Darcy’s law, the surface is stable when the viscosity of the tumour is greater than that of the surroundings, but highly unstable when the surroundings are more viscous than the tumour. When the deformation is governed by Stokes flow, the surface is always unstable regardless of the viscosity ratio. We discuss the implications of these results and suggest possible extensions to this work.