Residual stress generation and necrosis formation in multi-cell tumour spheroids

We consider how cell proliferation and death generate residual stresses within a multi-cell tumour spheroid (MCTS). Previous work by Jones and co-workers [8] has shown that isotropic growth in a purely elastic MCTS produces growth induced stresses which eventually become unbounded, and hence are physically unrealistic. Since viscoelastic materials show stress relaxation under a fixed deformation we consider the effect of the addition of a small amount of viscosity to the elastic system by examining formation of equilibrium stress profiles within a Maxwell type viscoelastic MCTS. A model of necrosis formation based upon that proposed by Please and co-workers (see [16] [17] [18]) is then presented in which necrosis forms under conditions of adverse mechanical stress rather than in regions of extreme chemical stress as is usually assumed. The influence of rheology on necrosis formation is then investigated, and it is shown that the excessive stress generated in the purely elastic tumour can be relieved either by the addition of some viscosity to the system or by accounting for an inner necrotic interface with an appropriate stress boundary condition.