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

Predictive modeling of the evolutionary dynamics of cancer is a challenging issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer, including plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulation of stem cell regeneration is essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a G0 phase cell cycle model, which leads to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity;connects the heterogeneity with cell-to-cell variance in cellular behaviors(for example, proliferation, apoptosis, and differentiation/senescence);and can be extended to include gene mutation-induced tumor development. Hybrid computational models are developed based on the mathematical model framework and are applied to the processes of inflammationinduced tumorigenesis and tumor relapse after chimeric antigen receptor(CAR)-T cell therapy. Finally, we propose several mathematical problems related to the proposed differential-integral equation. Solutions to these problems are crucial for understanding the evolutionary dynamics of cancer.