Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions

Solid tumours do not develop as a homogeneous mass of mutant cells, rather, they grow in tandem with normal tissue cells initially present, and may also recruit other cell types including lymphatic and endothelial cells. Many solid tumours contain a high proportion of macrophages, a type of white blood cell which can have a variety of effects upon the tumour, leading to a delicate balance between growth promotion and inhibition. In this paper we present a brief review of the main properties and interactions of such tumour-associated macrophages, leading to a description of a mathematical model for the spatial interactions of macrophages, tumour cells and normal tissue cells, focusing on the ability of macrophages to kill mutant cells. Analysis of the homogeneous steady states shows that, for this model, normal tissue is unstable to the introduction of mutant cells despite such an immune response, but that the composition of the resulting tumour can be significantly altered. Including random cell movement and chemical diffusion, we demonstrate the existence of travelling wave solutions connecting the normal tissue and tumour steady states, corresponding to a growing tumour, and of the development of a spatial instability behind the wave front. Numerical solutions are illustrated in one and two dimensions. We go on to estimate macrophage motility parameters using data from Boyden chamber experiments. We then extend our model to include macrophage chemotaxis, that is, their directed movement in response to gradients of chemicals secreted by tumours. Solutions in one dimension indicate the possibility of spatiotemporal irregularities within the growing tumour, which are deduced to be the result of a series of bifurcations as the effective domain length increases, leading to a permanently transient solution. These results suggest that tumour heterogeneity may arise, in part, as a natural consequence of the macrophage infiltration. Recent experiments suggest that macrophages may indeed be involved in spatiotemporal variations within some human tumours.