Avascular tumour dynamics and necrosis

We consider the dynamic growth of a tumour, concentrating on the possible development of a necrotic region and examine some simple tumour geometries in detail. The growth and death rates of the cells in the viable rim of the tumour are taken to be determined by the local oxygen concentration. Crucially the cell motion is determined by the forces generated by cell affinity, by cell interaction and by the need to get the waste products of cell death, primarily water, out of the tumour and products for cell growth, again primarily water, into the tumour. A consolidation type model with surface tension on the cells, slow viscous flow of the cells and porous media flow of the extracellular water is derived. The dynamic behaviour of this model is examined. Considering the very simple case where resistance to extracellular water flow dominates the problem, the model accounts naturally for the formation of a necrotic region. In regions where the extracellular water pressure gets too large, the cells are assumed to be ripped from the extracellular matrix and die. This model contrasts significantly from previous models which typically assume a necrotic region exists and that its behaviour is primarily governed directly by oxygen concentration. Here, the stress determines the necrotic region behaviour and this is affected by the oxygen only indirectly through the cell growth and death rates. The predicted time-dependent growth of one-dimensional, and spherical tumours are illustrated by numerical calculations.