Application Of Mathematical Modeling In Cancer Immunology

Cancer is a major disease that seriously threatens human life and social development.The use of scientific methods to prevent and control cancer has became one of the most important public health problems in the world.In recent decades,with the transformation of disease patterns and the aging trend of the population,the incidence and mortality of cancer are increasing and the cancer prevention and treatment are facing a severe situation in our country.In this thesis,based on the biological data of cancer and the principles of cancer immunology,we set up a series of mathematical models to investigate the interac-tion between taumor cells and immune cells,as well as chemotherapeutic drug.In addition,we analyze the dynamic behaviors of these models by using the theoretical methods of ordinary differential equations and dynamic systems,and propose targeted treatment s-trategies through sensitivity analysis.The first chapter is the introduction of the thesis,which mainly introduces the re-search background,treatment methods and research status of cancer,and then briefly explains the main work and innovations of this thesis.In the second chapter,we establish a mathematical model of tumor-immune cell interaction.In order to study the growth and development of tumor cells under immune surveillance more clearly,the model mainly considers the most representative immune cells in the host immune system.Therefore,we only consider two types of immune cells,namely,natural killer(NK)cells representing innate immunity and CD8+cytotoxic T lymphocytes(CTLs)representing adaptive immunity.According to the experimental and clinical results,several parameters are fixed to simplify the model.Then the local geometric properties such as the existence and stability of the equilibrium points of the reduced three-dimensional model are analyzed.Also,we give a detailed description of the process of solving the existence of the equilibrium points of the model,judging the stability of the equilibrium points,and how to take the values of the parameters in the numerical simulations.Furthermore,numerical simulations are performed to verify the conditions of stability of equilibrium points by using MAT LAB.Finally,a sensitivity analysis is performed on the parameters of the model.Numerical simulation results show that the host imnune system alone cannot completely effectively control the development of tumor cells,and the CD8+cytotoxic T lymphocytes(CTLs)play an important role in tumor immune monitoring.In the third chapter,we consider the most common treatment method for cancer-chemotherapy.Chemotherapy is short for chemotherapy drugs,which is achieved by using chemotherapy drugs to kill cancer cells.But chemotherapy has inevitable side ef-fects on the host.Therefore,this chapter expands the model of the previous chapter and establishes an ordinary differential equation(ODE)mathematical model of four ordinary differential equations including tumor growth,immune function and chemotherapy.This four-dimensional model is composed of natural killer(NK)cells,CD8+cytotoxicity T lymphocytes(CTLs),tumor cells and chemotherapy drug.Unlike previous literatures on chemotherapy,the input of chemotherapeutic drug in our model is a constant input rather than a variate over time.In addition,the dose-response kinetics is represented by the mass action term instead of the exponential decay term.We locate the equilibrium points with-in the model parameters and judge their stability to study the dynamic behaviors of the differential equation model.Then,we give the results of sensitivity analyses and numeri-cal simulations of each parameter of the model.In addition,we simulate the time-varying quantitative curves of the tumor cells under different immune intensities.The parameter sensitivity results show that the tumor mortality induced by chemotherapeutic drugs and the drug decay rate have a greater impact on the final number of tumor cells.Numerical simulations highlight the importance of CD8+cytotoxic T lymphocytes(CTLs)activity in tumor chemotherapy,which suggests that immunotherapy based on CD8+cytotoxic T lymphocytes(CTLs)should be developed and tried.Furthermore,numerical simulations of different immune intensities show that the same concentration of the same chemother-apy drug has different effects on tumor cells in different hosts.Considering the degree of drug that the host body can be tolerated,the strategies for chemotherapeutic drugs should vary from person to person.These results point out the direction for us to formulate the best treatment plan and develop immunotherapy,and have certain practical significance.In the fourth chapter,we consider the drug resistance.Drug resistance is one of the most difficult problems in the current clinical practice to successfully treat tumors.In or-der to overcome drug resistance,we need to understand the mechanism of drug resistance and its effect to the development of tumors.In this chapter,we modify the previously combined immune-tumor model and introduce a mathematical model of drug resistance in the immune-tumor environment.The difference from the models in the previous t-wo chapters is that we do not distinguish between specificity and non-specificity of the immune system,but consider all types of immune effector cells in the immune system(including macrophages,natural killers(NK)cells,cytotoxicity T lymphocytes,helper T cells,regulatory T cells,etc.).Moreover,we divide the tumor cells into two categories:one is the drug-sensitive tumor cells that are sensitive to chemotherapeutic drugs;the oth-er is the drug-resistant tumor cells that are resistant to chemotherapeutic drugs.Therefore,we develop a mathematical model of tumor growth in the form of ordinary differential e-quations(ODEs),which includes the effector immune cells,the drug-sensitive tumor cells,the drug-resistant tumor cells and the chemotherapy drug.In this model,we assume that the drug-resistant tumor cells are caused by tumor cell mutations,and the mutation rate is a constant ?.In order to clarify the effect of drug on tumor cells,we respectively study the existence and stability of the equilibrium points between the drug-free model and the drug-present model,and perform numerical simulations and parameter sensitivity analyses of the model.These results of sensitivity analyses and numerical simulation-s demonstrate that the stability of the tumor-free equilibrium point can be achieved by increasing the killing rate of effector immune cells against drug-sensitive tumor cells,re-ducing the regression rate of chemotherapeutic drug and increasing the constant input of drug.Finally,we conclude the current works and make a prospect for the future works.