Boundedness of solutions of a non-local reaction' diffusion model for adhesion in cell aggregation and cancer invasion

Adhesion of cells to one another and their environment is an important regulator of many biological processes but has proved difficult to incorporate into continuum mathematical models. This paper develops further the new modelling approach proposed by Armstrong et al. (A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243: 98-113, 2006). The models studied in the present paper use an integro-partial differential equation for cell behaviour, in which the integral represents the sensing by cells of their local environment. This enables an effective representation of cell-cell adhesion, as well as random cell movement, and cell proliferation. The authors use this modelling approach to investigate the ability of cell-cell adhesion to generate spatial patterns during cell aggregation. The model is also extended to give a new representation of cancer growth, whose solutions reflect the balance between cell-cell and cell-matrix adhesion in regulating cancer invasion. The non-local term in these models means that there is no standard theory from which one can deduce the boundedness required for biological realism: specifically, solutions for cell density must lie between zero and a positive density corresponding to close cell packing. Here the authors derive a number of conditions, each of which is sufficient for the required boundedness, and they demonstrate numerically that cell density increases above the upper bound for some parameter sets not satisfying these conditions. Finally the authors outline what they regard as the main mathematical challenges for future work on boundedness in models of this type. © Cambridge University Press 2008.