Density-Dependent Diffusion Models with Biological Applications

Abstract

Diffusion is defined as the movement of a substance down a concentration gradient. The physics of diffusion is well described by Fick’s law. A density-dependent diffusion process is a one in which the diffusion coefficient is a function of the localized density of the diffusing substance. In my thesis I analyze the density-dependent diffusion behavior of two independent processes of biological interest. The first is tumor growth and invasion. The second is single-file diffusion, which in turn has a considerable biological significance, since it has been recently used to model transitions of proteins through DNA molecules. Tumor invasion of normal tissue is a complex process, involving cell migration and proliferation. It is useful to mathematically model tumor invasion in an attempt to find a common, underlying mechanism by which primary and metastatic cancers invade and destroy normal tissue. In our approach we make no assumptions about the process of carcinogenesis, that is, we are not modeling the genetic changes which result in transformation, nor do we seek to understand the causes of these changes. Similarly, we do not attempt to model the large-scale morphological features of tumors such as central necrosis. Rather, we concentrate on the microscopic-scale population interactions occurring at the tumor-host interface, reasoning that these processes strongly influence the clinically-significant manifestations of invasive cancer. We analyze a reaction-diffusion model due to Gawlinski, Gatenby, and others of the acid-mediated tumor invasion mechanism that incorporates the ion concentration as a reaction factor affecting the tumor growth and invasion process. It also adds density-dependent diffusion parameters to the reaction terms yielding independent reaction-diffusion equations for the normal, tumor, and acid populations. For reasonable biological parameters we study the fixed-points of the partial differential equations central of the model and their stability. The fixed-points determine the balance reached by the normal, tumor, and acid populations. As for the second application we present on density-dependent diffusion processes, we consider a model for single-file diffusion that is relevant in a variety of biological processes, for example ion transport in channels. The model is of mathematical and physical as well as biological interest because it exhibits an anomalously-slow tracer diffusion fundamentally different from diffusion without the single-file restriction. We carry out extensive computer simulations to study the role of particle adhesion and space availability (hard-core exclusion) in the model. Both tracer (tagged-particle) and bulk or collective diffusion are considered. Tracer diffusion focuses on the diffusion of the individual particles relative to their starting points, whereas bulk diffusion focuses on the diffusion of the particle distribution as a whole. The nature of the diffusion depends strongly on the initial particle distribution, and both homogeneous and inhomogeneous (for example Gaussian) distributions are considered. For all these models a density-dependent diffusion behavior is confirmed by studying the time evolutions of the moments and widths of these distributions.