Mathematical modeling of cancer treatments and the role of the immune system response to tumor invasion.
Despite the success of traditional cancer treatments, a definite cure to several cancers does not exist. Further, the traditional cancer treatments are highly toxic and have a relatively low efficacy. Current research and clinical trials have indicated that virotherapy, a procedure which uses replication-competent viruses to kill cancer cells, is less toxic and highly effective. Some recent studies suggest that the success of combating cancer lies in the understanding of tumour-immune interactions. However, the interaction dynamics of recent cancer treatments with the tumour and immune system response are still poorly understood. In this thesis we construct and analyse mathematical models in the form of ordinary and partial differential equations in order to explain tumour invasion dynamics and new forms of cancer treatment. We use these models to suggest possible measures needed in order to combat cancer. The thesis seeks to determine the most critical biological factors during tumour invasion, describe how the virus and immune system response influences the outcome of oncolytic virotherapy treatment, investigate how drug infusion methods determine the success of chemotherapy and virotherapy, and determine the efficacy of chemotherapy and virotherpy in depleting tumour cells from body tissue. We present a travelling wave analysis of a tumour-immune interaction model with immunotherapy. Here we aim to investigate the existence of travelling wave solutions of the model equations with and without immunotherapy and calculate the minimum wave speed with which tumour cells invade healthy tissue. This investigation highlights the properties which are most vital during tumour invasion. We use the geometric treatment of an apt-phase space to establish the intersection between stable and unstable manifolds. Numerical simulations are performed to support the analytical results. The analysis reveals that the main factors involved during tumour invasion include the tumour growth rate, resting immune cell growth rate, carrying capacity of the resting immune cells, resting cell supply, diffusion rate of the tumour cells, and the local kinetic interaction parameters. We also present a mathematical analysis of models that study tumour-immune-virus interactions using differential equations with spatial effects. The major aim is to investigate how virus and immune responses influence the outcome of oncolytic treatment. Stability analysis is carried out to determine the long term behaviour of the model solutions. Analytical traveling wave solutions are obtained using factorization of differential operators and numerical simulations are carried out using Runge-Kutta fourth order method and Crank-Nicholson methods. Our results show that the use of viruses as a means of cancer treatment can reduce the tumour cell concentration to a very low cancer dormant state or possibly eradicate all tumour cells in body tissue. The traveling wave solutions indicate an exponential increase and decrease in the immune cells density and tumour load in the long term respectively. A mathematical model of chemovirotherapy, a recent experimental treatment which combines virotherapy and chemotherapy, is constructed and analyzed. The aim is to compare the efficacy of three drug infusion methods and predict the outcome of oncolytic virotherapy-drug combination. A comparison of the efficacy of using each treatment individually, that is, chemotherapy and virotherapy, is presented. Analytical solutions of the model are obtained where possible and stability analysis is presented. Numerical solutions are obtained using the Runge- Kutta fourth order method. This study shows that chemovirotherapy may have a higher chance of reducing the tumour cell density in body tissue in a relatively short time. To the best of our knowledge, there has not been a mathematical study on the combination of both chemotherapy and virotherapy. Lastly, the chemovirotherapy model is extended to include spatial distribution characteristics, thus developing a model which describes avascular tumour growth under chemovirotherapy in a two dimensional spatial domain. Numerical investigation of the model solutions is carried out using a multi domain monomial based collocation method and pdepe, a finite element based method in Matlab. This study affirmed that chemovirotherapy may possibly eradicate all tumour cells in body tissue.