The Research Of Dynamic Modeling And Inhibitory Strategies For Tumor Growth

Tumor growth is a highly complex evolutionary process.Establish a mathematical model to simulate tumor growth,study the law of tumor development and predict the development trend of tumors have important theoretical guiding significance for formulating new treatment plans.Tumor growth can be regarded as a nonlinear dynamical system,bifurcation and chaos are very important core problems in nonlinear systems,and whether the tumor can be treated depends on the stability of this system.In this paper,the nonlinear mathematical model describing tumor growth is improved.Through the dynamic analysis and simulation of the model,the treatment strategies of inhibiting tumor growth are explored.The specific research contents are as follows:(1)Chemotherapy effect on the steady-state solution of the mathematical model of cancer cells.Based on Lotka-Volterra competition equation,a two-dimensional nonlinear mathematical model containing tumor cells and healthy host cells is constructed to analyze the effect of chemotherapy on tumor growth.Firstly,according to the Lyapunov stability theory,the stability of the equilibrium points in the system is discussed and the conditions of global asymptotic stability are given.Secondly,the theoretical results are verified by numerical simulations.Finally,the effects of chemotherapeutic drug concentrations and drug-to-tumor cell killing coefficients on the system dynamics are further explored.Bifurcation analysis showns that whether chemotherapeutic drugs can completely eradicate cancer cells depends on the killing coefficient of cancer cells without considering cell mutation and drug resistance.Therefore,for tumor therapy,enhancing the killing rate of tumor cells is more effective than increasing the drug concentration.(2)Dynamics analysis in a tumor-immune system with chemotherapy.Firstly,based on the model of Letellier et al,the effect of chemotherapy on the steady-state solution in tumor-immune dynamic system is studied and the existence and stability of the equilibrium solution in the system are discussed.By calculating Lyapunov exponent and Lyapunov dimension,the system exhibits chaotic dynamics with the selected parameter set.Secondly,the bifurcation diagrams of the system with respect to the concentration of chemotherapeutic drugs show that with the increase of drug concentration,the system has experienced from chaotic phenomenon to period-doubling oscillation,from perioddoubling oscillation to limit cycle oscillation with period 1,and then from limit cycle oscillation to stable state evolution.Finally,the numerical results show that the drug can effectively inhibit tumor growth as the concentration increases.However,high concentrations of chemotherapeutic drugs cause non-negligible damage to immune cells while destroying tumor cells,resulting in the failure of treatment.Therefore,the research results show that immunotherapy and chemotherapy are only the regulation and inhibition of this chaotic phenomenon,and it is difficult to achieve a state of complete cure.(3)Bifurcation and chaos analysis of tumor growth.Three-dimensional nonlinear models including healthy host cells,immune cells and tumor cells were deeply analyzed to determine how the dynamics of tumor growth is controlled by some key parameters.Firstly,by varying the competition coefficient between healthy host cells and tumor cells,a Hopf bifurcation occurs in this system.Then,the existence of Hopf bifurcation in the system is discussed by using bifurcation theory and the conditions for Hopf bifurcation are given.Finally,the direction of bifurcation and the stability of bifurcation periodic solution are analyzed by using normalization and central manifold theorem,and the specific expression of bifurcation stability coefficient is derived.Hopf bifurcation leads to the generation of limit cycle.Through the numerical analysis of the continuity of limit cycle,the system produces period doubling bifurcation,which leads to the generation of chaotic attractor.Finally,we divide the parameter space,give the region where the chaotic attractor exists,and discuss the method of suppressing tumor growth(eliminating chaos).(4)Adaptive therapy of metastatic melanoma.Adaptive therapy seeks to use intratumoral competition to avoid or delay the emergence of drug resistance in cancer treatment.Therefore,we develop a mathematical model containing both cell populations of sensitive and drug-resistant tumor cells,calibrated data from patients with metastatic melanoma,and predicted the outcome of adaptive therapy.It has been shown that adaptive treatment can prolong the survival of patients by comparing the two schemes of continuous treatment and adaptive treatment.The numerical results show that adaptive therapy can strive for a longer survival time for patients when the tumor load can be fully reduced to allow the cessation of drug use,the competition between sensitive cells and drug-resistant cells is strong enough and the rate of drug-induced drug resistance is low enough.In addition,prolonging tumor treatment leave can enhance intratumoral competition and improve the efficacy of adaptive therapy.