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

Abstract

This research paper proposes and discusses a conceptual modelling of both growth of tumours in presence of immortal multipotent cancer stem cells (CSCs) and of 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 in the vicinity of the mother cell. The model is implemented in the framework of 3D cellular automata (CA) whose dynamics is governed by stochastic rules. Some 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 with the aim to study 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 also averaged over the total volume. The Appendix provides the proof of the mathematical well-poscdness of this model in a general framework.