Journal of Mathematical Biology

, Volume 75, Issue 3, pp 733–758 | Cite as

Long-term treatment effects in chronic myeloid leukemia

Article

Abstract

We propose and analyze a simplified version of a partial differential equation (PDE) model for chronic myeloid leukemia (CML) derived from an agent-based model proposed by Roeder et al. This model describes the proliferation and differentiation of leukemic stem cells in the bone marrow and the effect of the drug Imatinib on these cells. We first simplify the PDE model by noting that most of the dynamics occurs in a subspace of the original 2D state space. Then we determine the dominant eigenvalue of the corresponding linearized system that controls the long-term behavior of solutions. We mathematically show a non-monotonous dependence of the dominant eigenvalue with respect to treatment dose, with the existence of a unique minimal negative eigenvalue. In terms of CML treatment, this shows that there is a unique dose that maximizes the decay rate of the CML tumor load over long time scales. Moreover this unique dose is lower than the dose that maximizes the initial tumor load decay. Numerical simulations of the full model confirm that this phenomenon is not an artifact of the simplification. Therefore, while optimal asymptotic dosage might not be the best one at short time scales, our results raise interesting perspectives in terms of strategies for achieving and improving long-term deep response.

Keywords

Chronic myeloid leukemia Renewal model Dominant eigenvalue Structured population dynamic 

Mathematics Subject Classification

92B05 92D25 35Q92 35P15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Univ Lyon, Université Claude Bernard Lyon 1CNRS UMR 5208, Institut Camille JordanVilleurbanne cedexFrance
  2. 2.INRIAVilleurbanneFrance

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