Introduction

This article emphasizes the dynamics of solid tumors which are large enough to be directly observable, at least prior to therapeutic or experimental treatment, e.g. are roughly 1 mm$^3$ or more in diameter ``initially''. Modeling such comparatively late stages is important to carcinogenesis models when comparing to observed outcomes.

Models of tumor growth and treatment based on a small number of ODEs¹ have a very long history, dating back to the equation of exponential growth,

$$dN/dt = \lambda N, \qquad \lambda = const.$$ (1)

Here $N(t)$ is the number of clonogenic cells in a tumor, regarded as so large that small-number ("demographic") fluctuations are negligible and $N$ can be treated as a continuous, deterministic function of the time $t$.

Mathematically speaking, models of this kind, using one or several ODEs, are naive and oversimplified compared to various other kinds of models - reaction-diffusion PDEs are one example among many - used in current mathematical and computer biology and medicine. However, a mathematician who looks in the literature for the models used routinely by experimental biologists and clinicians will see that it is the simplest ODE models which form the foundation of applied biological modelling in practice. The most important case in point is the equation of exponential growth itself; discussions of the value of the Malthusian growth parameter $\lambda$ under various conditions still form a major portion of practical tissue culture and tumor modeling. More generally, it is usually the specification of parameter values for simple ODE models that is the crux of the discussion. Such models aim to capture key features using a small number of adjustable parameters and, equally important, aim to neglect peripheral features judiciously. Often a mathematically quite sophisticated model is basically a marginal elaboration of some simple ODE model.

Consequently, it is useful to consider the present status of simple ODE models. Examples of currently used models will here be given: classic tumor growth models based on a single non-linear ODE; an ODE pair modeling tumor angiogenesis and treatment by angiogenic inhibitors; and a heavily-used ODE pair modeling killing of cells by ionizing radiation. Alternatives will be mentioned very briefly, as will generalizations used for analyzing cell-cycle kinetic effects, multi-compartment systems, stochastic fluctuations in cell number, spatial inhomgeneities and other details or complications.