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Predictive modeling of the evolutionary dynamics of cancer is a challengeissue in computational cancer biology. In this paper, we propose a generalmathematical model framework for the evolutionary dynamics of cancer withplasticity and heterogeneity in cancer cells. Cancer is a group of diseasesinvolving abnormal cell growth, during which abnormal regulations in stem cellregeneration are essential for the dynamics of cancer development. In general,the dynamics of stem cell regeneration can be simplified as a $\mathrm{G_0}$phase cell cycle model, which lead to a delay differentiation equation. Whencell heterogeneity and plasticity are considered, we establish adifferential-integral equation based on the random transition of epigeneticstates of stem cells during cell division. The proposed model highlights cellheterogeneity and plasticity, and connects the heterogeneity with cell-to-cellvariance in cellular behaviors, e.g. proliferation, apoptosis, anddifferentiation/senescence, and can be extended to include genemutation-induced tumor development. Hybrid computations models are developedbased on the mathematical model framework, and are applied to the process ofinflammation-induced tumorigenesis and tumor relapse after CAR-T therapy.Finally, we give rise to several mathematical problems related to the proposeddifferential-integral equation. Answers to these problems are crucial for theunderstanding of the evolutionary dynamics of cancer.

science china. mathematics

Evolutionary dynamics of cancer: from epigenetic regulation to cell  population dynamics -- mathematical model framework, applications, and open  problems