Article:
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T. Flå, F. Rupp & C. Woywod (2013): Deterministic & Stochastic Dynamics of
Chronic Myelogeneous Leukaemia Stem Cells with Hill-Function Like Signaling,
in A. Johann et al. (Eds., 2013): Recent Trends in Dynamical Systems -
Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings
in Mathematics & Statistics (Vol. 35), Springer-Verlag, pp. 221-263.


Abstract:
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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:
(i) 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 (ii) a three-dimensional model involving both
cycling and noncycling normal stem cells as well as cycling wild-type CML stem
cells is defined. The fourdimensional 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.