Modeling growth of a heterogeneous tumor.

It has long been recognized that the growth of tumor population depends on the initial age distribution of the cells in the tumor and the age-dependent cellular birth rate. Deterministic dual-cell models have been available for sometime; these models take into account the effects of the resultant cell heterogeneity. Nevertheless, these models ignore various variables significantly affecting the growth, such as those characterizing the cells' inherent properties and environmental factors. Uncertainties, or fluctuations, arise when the growth is simulated with the models. Stochastic analysis of these fluctuations is the focus of the current work. Two types of cells are visualized to proliferate separately and to transform mutually during the process. The master equations of the system have been formulated through probabilistic population balance around a particular state by considering all mutually exclusive events. The governing equations for the means, variances, and covariance of the random variables have been derived through the system-size expansion of these nonlinear master equations. The stochastic pathways of the two different types of cells have been numerically simulated by the algorithm derived from the master equation for two different physical situations, one without and, the other, with the chemotherapeutic treatment. The results of the current study illuminate the significance of stochastically modeling the responses of the tumor to a variety of medicinal treatments: The coefficient of variation of the malignant cells' population magnifies with time under chemotherapeutic regimens. Consequently, the impact of the uncertainties in the exact number of malignant cells as expressed by this coefficient of variation is highly unpredictable. For example, it becomes increasingly uncertain if or how fast these cells will reactivate to become a full-blown carcinogenic tumor after treatment.