Models of cancer growth

In this paper we present a simple mathematical model of cancer growth. The model is discrete, it is a linear map (T) on three dimensional Eucledian vector space. But for some values of the parameters there exists a linear vector field on three dimensional Eucledian vector space whose time one map is T. We can analyze what happens to the flow of this vector field on one of the coordinate hyperplanes (see Fig. 5 in Sect. 4). In words this figure depicts, that if you have a bad ratio of growth inhibitor to growth factor cancer grows and if you have a good ratio cancer is eradicated. We can modify the dynamical system to model giving chemo therapy or immune therapy and discuss the consequences. We fit the model predictions to a Gompertz function and this gives a good fit for some values of the parameters (see Figs. 1, 2, 3 and 4). A main topic of the present paper is a mass action kinetic model for cancer growth. This gives a three dimensional ODE model, for which the first orthant is positively invariant. We discretize the linearization of this nonlinear ODE and with appropriate parameter definitions we recover the model T of Sect. 1. For some parameter values the model T induces a two dimensional map of the plane, see Sect. 5 .