Mathematical Modelling of Cancer Stem Cells Population Behavior

¹ CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
² CNRS FRE 3377, Laboratoire Epigenetique et Cancer, CEA Saclay 91191 Gif-sur-Yvette, France

^⋆ Corresponding author. E-mail: vincenzo.capasso@unimi.it

Abstract

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questions about their nature, origin, function and their behavior in cell culture. Most of current experiments reporting a dynamics of cancer stem cell populations in culture show the eventual stability of the percentages of these cell populations in the whole population of cancer cells, independently of the starting conditions. In this paper we propose a mathematical model of cancer stem cell population behavior, based on specific features of cancer stem cell divisions and including, as a mathematical formalization of cell-cell communications, an underlying field concept. We compare the qualitative behavior of mathematical models of stem cells evolution, without and with an underlying signal. In absence of an underlying field, we propose a mathematical model described by a system of ordinary differential equations, while in presence of an underlying field it is described by a system of delay differential equations, by admitting a delayed signal originated by existing cells. Under realistic assumptions on the parameters, in both cases (ODE without underlying field, and DDE with underlying field) we show in particular the stability of percentages, provided that the delay is sufficiently small. Further, for the DDE case (in presence of an underlying field) we show the possible existence of, either damped or standing, oscillations in the cell populations, in agreement with some existing mathematical literature. The outcomes of the analysis may offer to experimentalists a tool for addressing the issue regarding the possible non-stem to stem cells transition, by determining conditions under which the stability of cancer stem cells population can be obtained only in the case in which such transition can occur. Further, the provided description of the variable corresponding to an underlying field may stimulate further experiments for elucidating the nature of “instructive" signals for cell divisions, underlying a proper pattern of the biological system.