MATHEMATICAL ANALYSIS AND CHALLENGES ARISING FROM MODELS OF TUMOR GROWTH

Abstract

In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body. Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena. The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.