Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues

Abstract : Objective: Modeling and analysis of cell population dynamics enhance our understanding of cancer. Here we introduce and explore a new model that may apply to many tissues. Analyses: An age-structured model describing coexistence between mutated and ordinary stem cells is developed and explored. The model is transformed into a nonlin-ear time-delay system governing the dynamics of healthy cells, coupled to a nonlinear differential-difference system describing dynamics of unhealthy cells. Its main features are highlighted and an advanced stability analysis of several steady states is performed, through specific Lyapunov-like functionals for descriptor-type systems. Results: We propose a biologically based model endowed with rich dynamics. It incorporates a new parameter representing immunoediting processes, including the case where proliferation of cancer cells is locally kept under check by the immune cells. It also considers the overproliferation of cancer stem cells, modeled as a subpopulation of mutated cells that is constantly active in cell division. The analysis that we perform here reveals the conditions of existence of several steady states, including the case of cancer dormancy, in the coupled model of interest. Our study suggests that cancer dormancy may result from a plastic sensitivity of mutated cells to their shared environment, different from that-fixed-of healthy cells, and this is related to an action (or lack of action) of the immune system. Next, the stability analysis that we perform is essentially oriented towards the determination of sufficient conditions, depending on all the model parameters, that ensure either a regionally (i.e., locally) stable dormancy steady state or eradication of unhealthy cells. Finally, we discuss some biological interpretations, with regards to our findings, in light of current and emerging therapeutics. These final insights are particularly formulated in the paradigmatic case of hematopoiesis and acute leukemia, which is one of the best known malignancies for which it is always hard, in presence of a clinical and histological remission, to decide between cure and dormancy of a tumoral clone. 4 E. Fridman is with the Department of Electrical Engineering and Systems at the School of Electrical Engineering, Tel-Aviv, Israel. emilia@eng.tau.ac.il 5 P. Hirsch and F. Delhommeau are with Sorbonne Université, GRC n7, Groupe de Recherche Clinique sur les Myéloproliferations Aiguës et Chroniques, AP-HP, Hôpital Saint-Antoine, 2010 MSC: 93C23, 93C10, 93A30, 93D05, 92B05, 92C50, 80A30, 37L15, 39A30, 39A60. Highlights (to appear before the abstract in JTB template) • Modeling the cell cycle and population cell dynamics taking into account: coexistence between normal and mutated stem cells, overproliferation of cancer stem cells and sensitivity to the immune system. • The study of existence of different steady states, including: unhealthy cells eradication 5 and cancer cells dormancy (i.e. control in dormancy of abnormal cells). • Construction of different types of suitable strict Lyapunov-like functionals for nonlinear delay differential-difference systems. • Deriving stability conditions of steady states in different biological situations: a particular focus on the scenario of cancer dormancy. 10 • Numerical simulations, biological discussions and some therapeutic insights in the
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Journal of Theoretical Biology, Elsevier, 2018, 449, pp.103 - 123. 〈10.1016/j.jtbi.2018.03.038〉
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Walid Djema, Catherine Bonnet, Frédéric Mazenc, Jean Clairambault, Emilia Fridman, et al.. Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues. Journal of Theoretical Biology, Elsevier, 2018, 449, pp.103 - 123. 〈10.1016/j.jtbi.2018.03.038〉. 〈hal-01852154〉

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