Growth of tumours with stem cells: The effect of crowding and ageing of cells

Mathematical models for the growth of tumours in the presence of stem cells (CSCs) and differentiated tumour cells (CCs) are presented and discussed. The CSCs are assumed to be immortal and multipotent, i.e. capable of generating several possible lineages of CCs that may undergo ageing and apoptosis. Each CC is characterised by two indexes, related to the differentiation lineage and the class of age, respectively. Furthermore, the effect of crowding is taken into account, assuming that mitosis can be hindered by the presence of cells in the vicinity of the would-be mother cell. Two families of models are proposed. First, models based on cellular automata are considered, whose evolution is governed by stochastic rules. Then, by averaging over the cells with the same pair of indexes, we obtain a deterministic model that consists of a system of Ordinary Differential Equations (ODEs) whose unknown functions are the fractions of the cells in each lineage and the class of age. The system presents a basic novelty with respect to the other compartmental models proposed in the literature as it cannot be solved hierarchically because of the presence of the crowding effect. Numerical simulations based on the two families of models give the same qualitative results and, in particular, they evidentiate the occurrence of the tumour paradox: an increased mortality of the CCs may induce a faster growth of the tumour. A final section of the paper is devoted to the case in which the age distribution of the CCs is continuous and not discrete. In this case, an interesting mathematical problem is obtained that consists of one ODE for the fraction of CSCs and m first-order Partial Differential Equations (PDEs); one for each lineage of CCs.