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Mathematical Modelling of Cancer Stem Cells PopulationBehavior

Published online by Cambridge University Press:  25 January 2012

E. Beretta ,
V. Capasso  and
N. Morozova
E. Beretta
Affiliation:
CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
V. Capasso*
Affiliation:
CIMAB (InterUniversity Centre for Mathematics Applied to Biology, Medicine and Environment) Dipartimento di Matematica, Universitá degli Studi di Milano, 20133 Milano, Italy
N. Morozova
Affiliation:
CNRS FRE 3377, Laboratoire Epigenetique et Cancer, CEA Saclay 91191 Gif-sur-Yvette, France
*
⋆Corresponding author. E-mail: vincenzo.capasso@unimi.it
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Abstract

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questionsabout their nature, origin, function and their behavior in cell culture. Most of currentexperiments reporting a dynamics of cancer stem cell populations in culture show theeventual stability of the percentages of these cell populations in the whole population ofcancer cells, independently of the starting conditions. In this paper we propose amathematical model of cancer stem cell population behavior, based on specific features ofcancer stem cell divisions and including, as a mathematical formalization of cell-cellcommunications, an underlying field concept. We compare the qualitative behavior ofmathematical models of stem cells evolution, without and with an underlying signal. Inabsence of an underlying field, we propose a mathematical model described by a system ofordinary differential equations, while in presence of an underlying field it is describedby a system of delay differential equations, by admitting a delayed signal originated byexisting cells. Under realistic assumptions on the parameters, in both cases (ODE withoutunderlying field, and DDE with underlying field) we show in particular the stability ofpercentages, provided that the delay is sufficiently small. Further, for the DDE case (inpresence of an underlying field) we show the possible existence of, either damped orstanding, oscillations in the cell populations, in agreement with some existingmathematical literature. The outcomes of the analysis may offer to experimentalists a toolfor addressing the issue regarding the possible non-stem to stem cells transition, bydetermining conditions under which the stability of cancer stem cells population can beobtained only in the case in which such transition can occur. Further, the provideddescription of the variable corresponding to an underlying field may stimulate furtherexperiments for elucidating the nature of “instructive" signals for cell divisions,underlying a proper pattern of the biological system.

Keywords

cancer stem cellsdelay differential equationsqualitative behaviorstabilityoscillations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 279 - 305
Copyright
© EDP Sciences, 2012

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