Mathematical models of prevascular spheroid development and catastrophe-theoretic description of rapid metastatic growth/tumor remission

Abstract

A brief survey is provided of deterministic models of tumor growth and development over the last three decades. The evolution of these models has proceeded from basic phenomenological and empirical descriptions, through both time-dependent and time-independent diffusion models (largely within the diffusive equilibrium approximation). This includes a study of the diffusion of growth inhibitors. The stability of spheroid models to small perturbations is discussed, and also recent applications of nonlinear elasticity theory and differential geometry to possible staging and grading of cancers. Finally, an excursion is made into catastrophe theory, wherein it is suggested that the cusp catastrophe (in particular) may provide a qualitative description of rapid, almost spontaneous (i) growth of metastases, or (ii) tumor remission (both occurring under certain restrictive conditions).