PATTERN-FORMATION AND SPATIOTEMPORAL IRREGULARITY IN A MODEL FOR MACROPHAGE-TUMOR INTERACTIONS

Solid tumours do not develop as a homogeneous mass of mutant cells, ra ther, they grow in tandem with normal tissue cells initially present, and may also recruit other cell types including lymphatic and endothel ial 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 prope rties 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 homogeneo us 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 dem onstrate the existence of travelling wave solutions connecting the nor mal tissue and tumour steady states, corresponding to a growing tumour , and of the development of a spatial instability behind the wave fron t. Numerical solutions are illustrated in one and two dimensions. We g o on to estimate macrophage motility parameters using data from Boyden chamber experiments. We then extend our model to include macrophage c hemotaxis, that is, their directed movement in response to gradients o f chemicals secreted by tumours. Solutions in one dimension indicate t he possibility of spatiotemporal irregularities within the growing tum our, which are deduced to be the result of a series of bifurcations as the effective domain length increases, leading to a permanently trans ient solution. These results suggest that tumour heterogeneity may ari se, 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. (C) 1997 Academic Press Limited.