Deterministic and Stochastic Dynamics of Chronic Myelogenous Leukaemia Stem Cells Subject to Hill-Function-Like Signaling

Based on a discrete Markovian birth-death model including regulated symmetric and asymmetric cell division, we formulate a continuous four-dimensional stochastic (ordinary) differential equation model for the dynamics of Chronic Myelogenous Leukaemia (CML) stem cells in a bone marrow niche involving signaling and competition between active stem cells. Invoking stochastic-deterministic correspondence we then investigate two deterministic subsystems: (a) the competition between active normal and wild-type CML stem cells or also between two developing leukaemic stem cell strains is represented by a two-dimensional equation system, and (b) a three-dimensional model involving both cycling and noncycling normal stem cells as well as cycling wild-type CML stem cells is defined. The four-dimensional equation system finally includes in addition one cycling CML stem cell clone of an anti-CML-drug-resistant mutant. By totally analytic means we discuss the existence and stability of the equilibria of the three systems in the deterministic small noise limit, and establish, by numerical means, connections between these classical results and the original stochastic setting. The robust, stable finite population equilibria can be interpreted as homeostatic equilibria of normal and leukaemic stem cell populations, in the case of the four-dimensional model for the scenario of treatment of the wild-type CML clone with a CML suppressing agent, e.g., imatinib, which leads to the emergence of a resistant CML strain. The four-dimensional model thus represents a common clinical picture. © Springer Basel 2013.