Stability Analysis of a Model of Interaction Between the Immune System and Cancer Cells in Chronic Myelogenous Leukemia.

We describe here a simple model for the interaction between leukemic cells and the autologous immune response in chronic phase chronic myelogenous leukemia (CML). This model is a simplified version of the model we proposed in Clapp et al. (Cancer Res 75:4053-4062, 2015). Our simplification is based on the observation that certain key characteristics of the dynamics of CML can be captured with a three-compartment model: two for the leukemic cells (stem cells and mature cells) and one for the immune response. We characterize the existence of steady states and their stability for generic forms of immunosuppressive effects of leukemic cells. We provide a complete co-dimension one bifurcation analysis. Our results show how clinical response to tyrosine kinase inhibitors treatment is compatible with the existence of a stable low disease, treatment-free steady state.

Long-term treatment effects in chronic myeloid leukemia.

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.