An evolutionary model of tumor cell kinetics and the emergence of molecular heterogeneity driving Gompertzian growth

Abstract

A cell-molecular based evolutionary model of tumor development driven by a stochastic Moran birth-death process is developed, where each cell carries molecular information represented by a four-digit binary string, used to differentiate cells into 16 molecular types. The binary string value determines cell fitness, with lower fit cells (e.g. 0000) defined as healthy phenotypes, and higher fit cells (e.g. 1111) defined as malignant phenotypes. At each step of the birth-death process, the two phenotypic sub-populations compete in a prisoner's dilemma evolutionary game with healthy cells (cooperators) competing with cancer cells (defectors). Fitness and birth-death rates are defined via the prisoner's dilemma payoff matrix. Cells are able undergo two types of stochastic point mutations passed to the daughter cell's binary string during birth: passenger mutations (conferring no fitness advantage) and driver mutations (increasing cell fitness). Dynamic phylogenetic trees show clonal expansions of cancer cell sub-populations from an initial malignant cell. The tumor growth equation states that the growth rate is proportional to the logarithm of cellular heterogeneity, here measured using the Shannon entropy of the distribution of binary sequences in the tumor cell population. Nonconstant tumor growth rates, (exponential growth during sub-clinical range of the tumor and subsequent slowed growth during tumor saturation) are associated with a Gompertzian growth curve, an emergent feature of the model explained here using simple statistical mechanics principles related to the degree of functional coupling of the cell states. Dosing strategies at early stage development, mid-stage (clinical stage), and late stage development of the tumor are compared, showing therapy is most effective during the sub-clinical stage, before the cancer subpopulation is selected for growth.