Quiescence as an explanation of Gompertzian tumor growth

In this paper we propose a mathematical model for the growth of solid tumors which employs quiescence as a mechanism to explain characteristic Gompertz-type growth curves. The model distinguishes between two types of cells within the tumor, proliferating and quiescent. Empirical data strongly suggest that the larger the tumor, the more likely it is that a proliferating cell becomes quiescent and the more unlikely it is that a quiescent cell reenters the proliferating cycle. These facts are taken as the basic assumptions of the model. It is shown that these assumptions imply diminishing of the growth fraction (i.e. proportion of proliferating cells), a phenomenon found in most tumors. Three qualitatively different cases are analyzed in detail and illustrated by examples. In the case of a tumor forming a necrotic center the model predicts that the tumor grows monotonically to its ultimate size according to a typical S-shaped Gompertz curve, and that the growth fraction tends to zero. In the case of true quiescence, where the dormant cells retain their capability of becoming proliferating, we distinguish between two types of tumors: one in which only proliferating cells can die and one in which there is mortality among quiescent cells, too. In the first case the predicted tumor growth occurs in the early stages in a way that is very similar to that of tumors forming a necrotic center; the growth fraction still tends to zero, but ultimately the tumor grows without bound. In the second case the tumor grows to a finite limit depending only on the vital rates, while the growth fraction decreases to a strictly positive value.