Interaction between crowding and growth in tumours with stem cells: conceptual mathematical modelling

The paper proposes a conceptual modelling of growth of tumours in presence of immortal multipotent cancer stem cells (CSCs) and several lineages of differentiated tumour cells (CCs). The replication of CSCs is assumed symmetric or asymmetric with a prescribed mean ratio and mitosis and apoptosis are taken into account for the CCs aging. Replication can be hindered by the local crowding of the cells. The model is implemented in the framework of 3D cellular automata (CA) whose dynamics is governed by stochastic rules. Simulations are displayed showing the growth of a tumour and the fractions of different lineages and age classes of CCs. Then, an approach that considers the same dynamics of aging, replication, and apoptosis, but studying the time evolution of the fractions of the different lineages and age classes of cells averaged over the total volume is presented. The dynamics is governed by a system of ordinary differential equations (ODEs), hence by deterministic rules. Numerical simulations of the solution of this system show qualitative similarity with the CA results, although the crowding effect is no longer a local effect, but averaged over the total volume. The proof of the mathematical well-posedness of this model is provided.