#### Date of Award

Spring 2000

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

John A. Adam

#### Committee Member

John Mark Dorrepaal

#### Committee Member

John Heinbockel

#### Committee Member

Constance Schober

#### Committee Member

Toby Barco

#### Abstract

A theoretical model for the existence of a Critical Size Defect (CSD) in certain animals is the focus of the majority of this dissertation. Adam [1] recently developed a one-dimensional model of this phenomenon, and chapters I–V address the exist the CSD in a two-dimensional model and a three-dimensional model. The two dimensional (or 1-d circular) model is the more appropriate for a study of CSD's. In that model we assume a circular wound of uniform depth and develop a time-independent form of the diffusion equation relevant to the study of the CSD phenomenon. It transpires that the range of CSD sizes for a reasonable estimate of parameter values is 1mm-1cm. More realistic estimates await the appropriate experimental data.

The remainder of this dissertation is devoted to two phenomenological models describing the spread of cancer and the effects of the immune system on that spread. In chapter VI, Tumor Immunity, a PDE similar to Fisher's equation is analyzed in terms of the equilibrium points and their linear stability and similarities are noted with the Spruce-Budworm problem of Ludwig et al (and summarized by Strogatz). This chapter concludes with a standard phase plane analysis of a traveling wave solution. Chapter VII, Tunneling, introduces a novel and hopefully useful way of looking at cancer growth and the immune system. In the governing differential equation, the cancer cell number represents the independent variable, while the dependent variable is related to the probability of achieving that size cell number. (The square of the dependent variable is the probability). By analogy with quantum mechanics, the idea is introduced that the immune system (represented by a rectangular barrier of height V) may not in all cases prevent the cancer from “penetrating” the barrier i.e. tunneling through. The governing differential equation and boundary conditions represent a classical eigenvalue problem which may be thought of here as a “semi-classical” version of the time-independent Schröinger equation. Examples are provided which show considerable variation in the effectiveness of the “immune barrier” towards limiting the numerical growth of cancer cells.

#### DOI

10.25777/w8r1-3m35

#### ISBN

9780599754577

#### Recommended Citation

Arnold, Julia S..
"Diffusion Problems in Wound Healing and a Scattering Approach to Immune System Interactions"
(2000). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/w8r1-3m35

https://digitalcommons.odu.edu/mathstat_etds/9

#### Included in

Applied Mechanics Commons, Immunology and Infectious Disease Commons, Mathematics Commons