Researches And Numerical Simulations Of Mathematical Models On Tumor Therapeutic Regimens

Cancer constructs a serious threat to human life and health. According to a report released by the world health organization in 2012, there is an estimated 1410 million new cancer cases diagnosed and 8.2 million cancer deaths in the world. Sta-tistical data and related studies have shown that the risk of cancer will significantly increase with age. Hence, cancer treatment has become one of the hot spots in the study of pharmacology, immunology and related life sciences. The paper contains six chapters, in which we investigate the dynamic behaviors of interaction between tumor cells and immune cells, the pulsed immunotherapy and radiotherapy, the pulsed immunotherapy and chemotherapy, and the design of optimal therapeutical regimens on the combined immunotherapy and chemotherapy, respectively.In first chapter, we briefly describe the current epidemic of cancer, the basic methods of cancer treatment, the research status of cancer immune response and cancer treatment effect, the basic concepts and results of ordinary differential equa-tions, impulsive differential equations and optimal control.In second chapter, we divide the immune cell population into two subgroups: the young immune cells and mature immune cell populations, and establish a two-stage ODE model to investigate dynamic features of interaction between immune cells and tumor cells. In addition, we discuss the stabilities of the tumor-free equilib-rium and the tumor-present equilibrium, and give the the corresponding conditions. Further, we apply Hopf bifurcation theorem in three-dimensional differential sys-tems to determine the occurrence of Hopf bifurcation, and use the center manifold theory and the normal form theory to obtain the explicit formulae which determines the stability and the direction of the bifurcating period solutions. Finally, numerical simulations are carried out to illustrate and verify the analytical conclusions.In third chapter, in order to explore the therapeutic effects for different treat-ment regimens, we propose an antitumor model with periodical radiotherapy and immunotherapy. By using Floquent theorem and small-amplitude perturbation skill-s, the sufficient condition for the global stability of the tumor-free periodic solution is derived at first. Furthermore, the fixed point approach is applied to investigate the existence and stability of the nontrivial periodic solution. Moreover, by compar- ing different therapeutic regimens, we achieve the conditions that determine which therapeutic regimen is more effective in diminishing the basic reproduce number. Finally, we perform numerical simulations to illustrate and justify our theoretical findings, and design a combination treatment regimen with reference value and guid-ance significance.To begin with, in fourth chapter, single immunotherapy, single chemotherapy and mixed treatment are discussed, and we then obtain sufficient conditions under which tumor cells will be eliminated ultimately. Furtherly, we analyze the impacts of the least concentration and the half-life of the drug on therapeutic results, and find that increasing the least concentration or extending the half-life of the drug can achieve better therapeutic effects. In addition, since most tumors are resistant to chemotherapy drugs, we consider influence of drug resistance on therapeutic results and propose a new mathematical model to explain the cause of the chemotherapeutic failure with single drug. Based on this, in the end, we explore the therapeutic effects of combination chemotherapy with two drugs, as well as mixed immunotherapy with combination chemotherapy. Numerical simulations indicate that combination chemotherapy is very effective in controlling tumor growth. In comparison, mixed immunotherapy with combination chemotherapy can achieve a better treatment effect.In fifth chapter, a mathematical model is used to investigate the effects of im-munotherapy and chemotherapy. For single immunotherapy and single chemother-apy, the conditions under which the tumor cells will be eliminated eventually are obtained. Optimal control theory is applied to minimize the number of tumor cells and the cost of the combined therapy. We establish the existence of the optimality system and use pontryagin’s maximum principle to characterize the optimal levels of the two control variables. Numerical simulations show that optimal combined treat-ment regimen will depend on the relative weights of each of the control variables in objective functional. In addition, by comparison among single immunotherapy, single chemotherapy and optimal combined treatment, we find that neither im-munotherapy nor chemotherapy alone are adequate to cure a tumor, but optimal combined immunotherapy with chemotherapy are able to eliminate the entire tumor quickly.In sixth chapter, we summarize the current works and make a prospect for the future works.