Article:
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T. Fla, F. Rupp & C. Woywod (2014): Bifurcation Patterns in Generalized Models for the Dynamics
of Normal and Leukaemic Stem Cells with Signaling, Mathematical Methods in the Applied Sciences,
accepted.


Abstract:
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The dynamics of normal and leukaemic stem cell clones competing for resources in the same bone marrow niche can be
modeled in a systems biology approach. We derive three related models describing the time evolution of normal and Chronic
Myelogenous Leukaemia (CML) stem cells. The approach is based on the idea that stem cell proliferation is regulated by
feedback mechanisms. We assume here that the growth of stem cell clones can be approximated by Tsallis functions which
depend on population numbers to ensure the communication between different stem cell strains. Of particular interest
is the influence of medication, e.g. by the CML drug imatinib, on the stem cell dynamics, which can be simulated by a
variation of the parameterization of the differential equation systems. The basic 2D model represents the contest between
cycling normal and wild-type CML stem cells. In addition, extension of the basic model (i) by coupled reservoirs of quiescent
stem cells and (ii) by a third stem cell species, which could be interpreted as an imatinib-resistant mutant, is investigated.
We analyze the global dynamics of the corresponding 2D, 3D and 4D equation systems by analytic means and provide
a complete map of the bifurcation landscape that occurs for changing values of the respective signaling strengths and
growth-/ death-parameters. This includes a complete classication of equilibria and their linear and non-linear Lyapunov
stability.