A PDMP to model the stochastic influence of quiescence dynamics in blood cancers

Abstract: In this article, we will see a new approach to study the impact of a small microscopic population of cancer cells on a macroscopic population of healthy cells, with an example inspired by pathological hematopoiesis. Hematopoiesis is the biological phenomenon of blood cells production by differentiation of cells called hematopoietic stem cells (HSCs). We will study the dynamics of a stochastic $4$-dimensional process describing the evolution over time of the number of healthy and cancer stem cells and the number of healthy and mutant red blood cells. The model takes into account the amplification between stem cells and red blood cells as well as the regulation of this amplification as a function of the number of red blood cells (healthy and mutant). A single cancer HSC is considered while other populations are in large numbers. We assume that the unique cancer HSC randomly switches between an active and a quiescent state. We show the convergence in law of this process towards a piecewise deterministic Markov process (PDMP), when the population size goes to infinity. We then study the long time behaviour of this limit process. We show the existence and uniqueness of an absolutely continuous invariant probability measure with respect to the Lebesgue's measure for the limit PDMP, previously gathered. We describe the support of the invariant probability and show that the process converges in total variation towards it, using theory develop by M. Benaim et al. We finally identify the invariant probability using its infinitesimal generator. Thanks to this probabilistic approach, we obtain a stationary system of partial differential equation describing the impact of cancer HSC quiescent phases and regulation on the cell density of the hematopoietic system studied.