Document Type


Date of Degree

Summer 2012

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Ayati, Bruce

First Committee Member

Ayati, Bruce P

Second Committee Member

Li, Tong

Third Committee Member

Martin, James A

Fourth Committee Member

Mitchell, Colleen

Fifth Committee Member

Ramakrishnan, Prem S


Research in the biomedical sciences is incredibly diverse and often involves the interaction of specialists in a variety of fields. In particular, quantitative, mathematical, and computational methods are increasingly playing significant roles in studying problems arising in biomedical science. This is particularly exciting for mathematical modeling as the complexity of biological systems poses new challenges to modelers and leads to interesting mathematical problems. On the other hand mathematical modeling can provide considerable insight to laboratory and clinical researchers.

In this thesis we develop mathematical representations for three biological processes that are of current interest in biomedical science. A deeper understanding of these processes is desirable not only from the standpoint of basic science, but also because of the connections these processes have with certain diseases. The processes we consider are collective cell motility, bone remodeling, and injury response in articular cartilage. Our goals are to develop mathematical representations of these processes that can provide a conceptual framework for understanding the processes at a fundamental level, that make rigorous the intuition biological researchers have developed about these processes, and that help to translate theoretical and experimental work into information that can be used in clinical settings where the primary concern is in treating diseases associated with the process.


Articular Cartilage, Bone Remodeling, Cell Motility, Level set method, Nonlinear Diffusion, Tumor Invasion


vii, 89 pages


Includes bibliographical references (pages 84-89).


Copyright 2012 Jason Michael Graham