Date of Award

2018

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Xinfeng Liu

Abstract

This dissertation focuses on three projects. In Chapter 1, we derive and implement the compact implicit integration factor method for numerically solving partial differential equations. In Chapters 2 and 3, we generalize and analyze a mathematical model for the nonlinear growth kinetics of breast cancer stem cells. And in Chapter 4, we develop a novel mathematical model for the HER2 signaling pathway to understand and predict breast cancer treatment. Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another challenge is finding the matrix of high order space discretization, especially near the boundaries. In Chapter 1, we extend IIF method to high order discretization for spatial derivatives through an example of reaction diffusion equation with fourth order accuracy, while the computational cost and storage are similar to the general second order cIIF method. The method can also be efficiently applied to deal with other types of partial differential equations with both homogeneous and inhomogeneous boundary conditions. Direct numerical simulations demonstrate the efficiency and accuracy of the approach. Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to aid in designing experiments to develop novel therapeutic strategies against cancer. In Chapter 2, by using theory of functional and ordinary differential equations, we study the existence and stability of non-linear growth kinetics of breast cancer stem cells. First we provide a sufficient condition for the existence and uniqueness of the solution for non-linear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a non-trivial steady state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions which allow for this criteria to be satisfied. Next we apply these theorems to a special case of the system of functional differential equations that has been used to model non-linear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for developing novel strategies for cancer treatment. In Chapter 3, the moment stability of nonlinear stochastic systems of breast cancer stem cells with time-delays is investigated. First, based on the technique of the variation- of-constants formula, we obtain the second order moment equations for the nonlinear stochastic systems of breast cancer stem cells with time-delays. By the comparison principle along with the established moment equations, we can get the comparative systems of the nonlinear stochastic systems of breast cancer stem cells with time-delays. Then moment stability theorems are established for the systems with the stability properties for the comparative systems. Based on the linear matrix inequality (LMI) technique, we next obtain a criteria for the exponential stability in mean square of the nonlinear stochastic systems for the dynamics of breast cancer stem cells with time-delays. Finally, some numerical examples are presented to illustrate the efficiency of the results. Over-expression of human epidermal growth factor receptor 2 (HER2) plays a role in regulation of cancer stem cell (CSC) population in breast cancer. Current cancer therapy includes drugs that block HER2, however, patients can develop anti-HER2 drug resistance. Downstream of HER2 is nuclear factor κB (NFκB). The aberrant regulation of NFκB leads to cancer growth, which makes it a promising target for cancer therapy, especially for those who have developed resistance to anti-HER2 treatment. In Chapter 4 we develop a novel mathematical model that represents the dynamics of the HER2 signaling pathway. By integrating experimental data with model simulations, we discover that interleukin-1 (IL1), which is downstream of HER2, is responsible for NFκB activation. We perform global sensitivity analysis on the model to identify key reactions. Our modeling effort shows that IL1 is critical in NFκB regulation, especially in the absence of HER2, making it a potential target in treating breast cancer for patients who have developed resistance to anti-HER2 drugs.

Rights

© 2018, Sameed Ahmed

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