Mathematical models for tumours with cancer stem cells

The paper deals with mathematical modelling of the growth of tumours characterized by the presence of cancer stem cells (CSC). These cells are assumed to be immortal and multi-potent, in the sense that they can generate either CSC or ordinary cancer cells (CC). We present a synthetic review of some models that have been proposed in the literature and we report the development of new original models, to open a complete multi-approach window on modellization of the problem. The mathematical models are presented without technical details, but numerical simulations are displayed that exhibit the so-called “tumour growth paradox” that consists in the fact that tumours with a larger death rate for the CC may grow faster than tumours with a lower death rate for the same class of cells. The occurrence of this paradox has important consequences also on the strategy of treatment of tumours.