Time Delay Effect And Control Optimization Of Tumor-Lymphatic Immune Model

Based on the growth data of tumor cells in different sites and shapes,the establishment of accurate tumor growth models to predict the growth law of tumors has become an important prerequisite for the optimization of tumor diagnosis and treatment.The study of the dynamic properties of tumor-lymphocyte immune model with time delays and the application of control optimization theory to explore the effect of immuno-chemotherapy on tumor growth can provide important theoretical support and reference for the design of effective strategies to inhibit tumor growth in clinic.Firstly,the least square method and SPSS22.0 are used to obtain the parameter estimation formulas for three types of tumor growth models with known and unknown maximum environmental capacities,respectively.Through the parameter estimation formulas and publishing data,the growth rates for 10 types of tumors with high morbidity and mortality are obtained,and the publishing data are fitted by MATLAB.The first three quarters for the initial growth data of tumor cells are used as fitting data and the last quarter is used as prediction data to further verify the fitting effect.The best growth model for various tumor cells is determined by the fitting effect of tumor growth model.Secondly,considering the time delays of lymphocyte maturation and tumor cell proliferation,as well as the Logistic growth of colorectal cancer cells,a tumorlymphatic immune Logistic model with two time delays is established.Taking time delays as parameters,the local stability of equilibria in the model and its more extensive stability state in the delays plane are obtained by using the distribution of the characteristic roots and the stability switching curve method.The conditions and bifurcation properties of the model undergoing Hopf bifurcation at the equilibrium are obtained by applying the bifurcation theory of functional differential equations,the central manifold theorem and the normal form theory.Numerical simulations show the effect of time delays on the stability and periodic oscillation of the tumor-lymphocyte immune model,as well as abundant dynamic behaviors of the model at the double Hopf bifurcation point,such as chaos and other phenomena.Finally,adoptive cell immunotherapy and chemotherapy are introduced into the tumor-lymphocyte immune model,and the optimal control model of immunechemotherapy for anti-tumor is established to minimize the number of remaining tumor cells,the number of killed lymphocytes and the cost of combination therapy.The anti-tumor efficacy of high-dose single immunotherapy and high-dose single chemotherapy are discussed according to the stability condition for the tumor-free equilibrium of the model.The sufficient and necessary conditions for the existence of optimal control are proved by the optimal control theory and the Pantryagin’s maximum principle.Numerical simulations show the therapeutic effect of monotherapy and combination therapy,as well as the drug dose relationship of the optimal combination therapy strategy.