Deterministic and Stochastic Dynamics of Chronic Myelogenous Leukaemia Stem Cells Subject to Hill-Function-Like Signaling

  • Tor Flå
  • Florian Rupp
  • Clemens WoywodEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 35)


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


Stem Cell Chronic Myelogenous Leukaemia Stem Cell Population Stem Cell Niche Normal Stem Cell 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



C.W. would like to thank the Mohn Foundation, the Centre for Theoretical and Computational Chemistry (CTCC) at the University of Tromsø and the Research Council of Norway (Grant Nr. 177558/V30) for continued support.

We also acknowledge computational resources provided by Norwegian High Performance Computing (NOTUR).

Special thanks go to Jürgen Scheurle, who brought F.R. back to academia.


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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of TromsøTromsøNorway
  2. 2.Lehrstuhl für Höhere Mathematik und Analytische MechanikTechnische Universität München, Fakultät für MathematikGarchingGermany
  3. 3.Centre for Theoretical and Computational Chemistry (CTCC), Department of ChemistryUniversity of TromsøTromsøNorway

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