Numerical Analysis of Mathematical Models Originating in Biological and Physical Applications

Abstract

This thesis presents a comprehensive exploration of tumor growth through the development of advanced mathematical models that integrate biological phenomena and therapeutic interventions. Emphasizing numerical techniques, such as finite volume, finite element, and discontinuous Galerkin methods ensures accurate simulations and contributes valuable insights into optimizing cancer treatment strategies. This work spans multiple chapters, each contributing unique insights into tumor dynamics and treatment strategies. Chapter 1 provides an overview of the research and sets the context for the subsequent chapters. Chapter 2 introduces a mathematical model for avascular tumor growth with chemotaxis in a two-phase medium. The model uses conservation laws to derive a system of nonlinear partial differential equations involving cell volume fraction, cell velocity, and nutrient concentration, solved using finite volume and finite element methods. Additionally, numerical simulation suggest model parameters such as chemotaxis and diffusion coefficients plays an important role in controlling tumor progression. Chapter 3 advances the work done in Chapter 2 by incorporating drug transport equation and utilizing optimal control theory to develop a treatment strategy that minimizes tumor size while reducing chemotherapy toxicity. This study employs semigroup theory and truncation methods to establish the existence and uniqueness of solutions for the model. A sequence of numerical simulations demonstrates the effectiveness of optimized chemotherapy strategies, highlighting the significance of numerical optimization in developing personalized cancer treatments. Chapter 4 focuses on the interaction between tumor growth and the immune system through a mathematical modeling. This model, incorporating tumor cell density, immune cell populations (CD4+ and CD8+ T cells), and nutrient content, uses predator-prey dynamics to capture tumor-immune interactions. Numerical solutions are achieved using a combination of finite volume and finite element methods, effectively simulating the complex dynamics of tumor-immune interactions. This chapter underscores the role of numerical methods in accurately depicting biological systems. Chapter 5 explores a nonlinear model of tumor cell populations, emphasizing the dynamics of proliferative and quiescent phases. The model is discretized using discontinuous Galerkin and Runge-Kutta methods, with numerical simulations confirming the model’s accuracy and demonstrating the impact of various parameters. Chapter 6 summarize the thesis and also shed light on some future direction of the present work.