TOWARDS A STOCHASTIC TWO-COMPARTMENT MODEL IN TUMOR GROWTH

Abstract

A stochastic model of tumor growth incorporating several key elements of the growth processes is presented. Generalizing a previous work by the authors, two one-dimensional diffusion processes representing populations of proliferating and quiescent cells are obtained. Their forms turn out by their relation with total tumor population analysed in (1). The proposed model is able to incorporate the effects of the mutual interactions between the two subpopulations. It is also used to simulate the effects of two kinds of time-dependent therapies: non-specific cycle and specific cycle drugs. Moreover, the first-exit-time problem is analyzed to study cancer evolution in the presence of a time-dependent therapy. 1 Introduction and background Tumor is one of the main causes of death in our so- ciety so, in the last twenty years a lot of attempts have been made to describe the tumor kinetic. Cancer cells population consists of a combination of proliferating, quiescent and dead cells that determine tumor growth based on surrounding environmental conditions (cf. (5)). Furthermore, since experimental data show the existence of more or less intense random fluctuations in tumor growth, in a previous work (see (1)) the authors provided a stochastic generalization of the Gompertz law in order to model monoclonal tumor growth. So tumor size is described by means of a one-dimensional diffusion process X(t) and the first exit time problem (FET) for X(t) from a region D has been analysed. In particular, D is restricted by two absorbing boundaries representing healing threshold and the carrying capacity. A first natural generalization consists of including all the essential biological phenomena of a cellular population. To this aim, following Kozusko and Bajzer (cf. (7)), we split tumor population in two subpopulations: proliferating and quiescent cells. In this direction, in (7), the authors proposed a deterministic model to describe tumor dynamics, assuming that the transition rates between proliferating and quiescent populations depend on the total population size. By imposing that the total population is governed by the Gompertz law, Kozusko and Bajzer resolved analytically the model and obtained the dynamics of prolif- erating and quiescent populations as functions of the whole population size. Following this approach, we describe proliferating and quiescent populations via two new diffusion pro- cesses, generally time-non-homogeneous, connected to the process X(t). The FET problem for such processes through suitable boundaries is considered. The followed approach per- mits to analyse the effect of different therapies on tumor cells. In Section 2 we will review the main characteristics of the model proposed in (1) in which the effect of a therapy is seen as a moderation term of the growth rate of the tumor cells. In Section 3 we generalize the deterministic model by Kozusko and Bajzer by obtaining two diffusion processes P(t) and Q(t) representing the populations of proliferating and quies- cent cells, respectively. Finally, in Section 4 the effect of a therapy on X(t) is investigated