Document Type


Date of Degree

Summer 2015

Degree Name

PhD (Doctor of Philosophy)

Degree In

Applied Mathematical and Computational Sciences

First Advisor

Bruce P. Ayati


Traumatic injuries lead to articular cartilage lesion formation and result in the development of osteoarthritis. Recent research suggests that the early stage of mechanical injuries involve cell death (apoptosis and necrosis) and inflammation. In this thesis, we focus on building mathematical models to investigate the biological mechanism involving chondrocyte death and inflammatory responses in the process of cartilage degeneration.

Chapter 1 describes the structure of articular cartilage, the process of carti- lage degeneration, and reviews of existing mathematical models. Chapter 2 presents a delay-diffusion-reaction model of cartilage lesion formation under cyclic loading. Computational methods were used to simulate the impact of varying loading stresses and erythropoietin levels. The model is parameterized with experimental results, and is therefore clinically relevant. Due to numerical limitations using delay differential equations, a new model is presented using tools for population dynamics. Chapter 3 presents an age and space-structured model of articular cartilage lesion formation un- der a single blunt impact. Age structure is introduced to represent the time delay in cytokine synthesis and cell transition. Numerical simulations produce similar tempo- ral and spatial patterns to our experimental data. In chapter 4, we extend our model under the cyclic loading setting. Chapter 5 builds a spatio-temporal model adapted from the former models, and investigates the distribution of model parameters using experimental data and statistical methods. Chapter 6 concludes.

Public Abstract

Articular cartilage is a connective tissue that overlays the ends of joints. Healthy articular cartilage can provide a lifetime of support for joint function. However, many causes such as sports injuries, obesity, and aging can induce cartilage degeneration, which may eventually lead to osteoarthritis (OA). Post-traumatic osteoarthritis (PTOA) is the OA that results from traumatic injury.

A challenge for clinical research is the development of biological treatments. Most research has focused on treating osteoarthritis in the middle or late stages of disease. A better understanding of the biological response to acute injuries may provide opportunities for early intervention, which could prevent or delay the development of PTOA.

The early stages of cartilage degeneration includes chondrocytes death (apoptosis and necrosis) and inflammatory responses. In this thesis, a series of mathematical models are developed to investigate the relationship between cell death and inflammation under two scenarios: injuries resulting from a single blunt impact, and excessive loading. The mathematical models bring insight to cartilage degeneration, provide helpful information that cannot be easily obtained from biological experiments, and lead to suggestions for clinical therapies.


publicabstract, articular cartilage, delay-reaction-diffusion, lesion formation, spatio-temporal model, structured model


x, 110 pages


Includes bibliographical references (pages 106-110).


Copyright 2015 Xiayi Wang