Cancer Modeling: A Personal Perspective

Abstract

C ancer modeling comes in a wide variety of styles. Indeed, it can involve almost any type of applied mathematics. My personal favorite approach is the use of probability models to understand how genetic mutations lead to cancer progression, metastasis, and resistance to therapy. Ordinary differential equations can be used to study the growth of tumor cell populations, often leading to a conclusion of Gompertzian growth [21]. PDE models using cell densities and nutrient concentrations as state variables can be used to analyze various spatiotemporal phenomena; see [13]. Individual and agent-based models that treat cells as discrete objects with predefined rules of interaction can offer an improvement over PDE methods in some situations, such as the study of angiogenesis, the development of new blood vessels to bring nutrients to a growing tumor [1]. For a comparison of individual-based and continuum approaches in one particular example, see [4]. Agent-based systems are one of many computationally intensive methods [24] and are often components of multiscale models (see [16], [6], and [8]). Rather than spend the entire article in the land of generalities with random pointers to the literature, I will next give a description of a useful, simple, and flexible model: multitype branching processes. The types represent stages in the cancer progression. For example, in colon cancer, type 1 cells have one copy of the gene APC inactivated, type 2 cells have both copies inactivated, type 3