# Mathematical Modelling of Cancer Stem Cells Population Behavior

E. Beretta; V. Capasso; N. Morozova

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 1, page 279-305
- ISSN: 0973-5348

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topBeretta, E., Capasso, V., and Morozova, N.. "Mathematical Modelling of Cancer Stem Cells Population Behavior." Mathematical Modelling of Natural Phenomena 7.1 (2012): 279-305. <http://eudml.org/doc/222434>.

@article{Beretta2012,

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.},

author = {Beretta, E., Capasso, V., Morozova, N.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {cancer stem cells; delay differential equations; qualitative behavior; stability; oscillations},

language = {eng},

month = {1},

number = {1},

pages = {279-305},

publisher = {EDP Sciences},

title = {Mathematical Modelling of Cancer Stem Cells Population Behavior},

url = {http://eudml.org/doc/222434},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Beretta, E.

AU - Capasso, V.

AU - Morozova, N.

TI - Mathematical Modelling of Cancer Stem Cells Population Behavior

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/1//

PB - EDP Sciences

VL - 7

IS - 1

SP - 279

EP - 305

AB - 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.

LA - eng

KW - cancer stem cells; delay differential equations; qualitative behavior; stability; oscillations

UR - http://eudml.org/doc/222434

ER -

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