Tumours with cancer stem cells: A PDE model

• A new reaction–diffusion model for a tumour comprising stem- and ordinary cancer cells.
• Existence of classical solutions and numerical simulations.
• Tumour paradox: higher death rate of non-stem cells can induce faster tumour spread.

The role of cancer stem cells (CSC) in tumour growth has received increasing attention in the recent literature. Here we stem from an integro-differential system describing the evolution of a population of CSC and of ordinary (non-stem) tumour cells formulated and studied in a previous paper, and we investigate an approximation in which the system reduces to a pair of nonlinear coupled parabolic equation. We prove that the new system is well posed and we examine some general properties. Numerical simulations show more on the qualitative behaviour of the solutions, concerning in particular the so-called tumour paradox, according to which an increase of the mortality rate of ordinary (non-stem) tumour cells results asymptotically in a faster growth.