A Mathematical Model for the Onset of Avascular Tumor Growth in Response to the Loss of P53 Function

Abstract

We present a mathematical model for the formation of an avascular tumor based on the loss by gene mutation of the tumor suppressor function of p53. The wild type p53 protein regulates apoptosis, cell expression of growth factor and matrix metalloproteinase, which are regulatory functions that many mutant p53 proteins do not possess. The focus is on a description of cell movement as the transport of cell population density rather than as the movement of individual cells. In contrast to earlier works on solid tumor growth, a model is proposed for the initiation of tumor growth. The central idea, taken from the mathematical theory of dynamical systems, is to view the loss of p53 function in a few cells as a small instability in a rest state for an appropriate system of differential equations describing cell movement. This instability is shown (numerically) to lead to a second, spatially inhomogeneous, solution that can be thought of as a solid tumor whose growth is nutrient diffusion limited. In this formulation, one is led to a system of nine partial differential equations. We show computationally that there can be tumor states that coexist with benign states and that are highly unstable in the sense that a slight increase in tumor size results in the tumor occupying the sample region while a slight decrease in tumor size results in its ultimate disappearance.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

“…Cell proliferation, death and pressure have also been considered (e.g. [3, 22, 39, 59, 73, 84–87, 92, 93, 95, 96, 120, 129, 146, 186, 189, 220, 222, 272, 283–285, 287, 298, 328, 336, 346, 414, 420, 457, 468, 493, 496, 521, 536, 545]). Linear and weakly nonlinear analyses have been performed to assess the stability of spherical tumours to asymmetric perturbations (e.g.…”

Network systems biology for targeted cancer therapies

“…Therefore, in addition to the general topological static analyses that indicate the error and attack tolerance of metabolic networks from a global view[25], powerful kinetic models such as ODE or FBA can be set up to trace the network response against changes in enzyme activity and compound concentration. Examples include the model developed for predicting the onset of avascular tumor growth among cells in response to the loss of p53 function[46], as well as the model for developing hypoxia-inducible factor-1α (HIF-1α)-based therapies[47].…”