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

Walid Djema, Catherine Bonnet, Frédéric Mazenc, Jean Clairambault, Emilia Fridman, Pierre Hirsch, François Delhommeau

Research output: Contribution to journalArticlepeer-review


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 nonlinear 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.

Original languageEnglish
Pages (from-to)103-123
Number of pages21
JournalJournal of Theoretical Biology
StatePublished - 14 Jul 2018


  • Cancer dormancy
  • Delay
  • Lyapunov
  • Modeling
  • Nonlinear


Dive into the research topics of 'Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues'. Together they form a unique fingerprint.

Cite this