Analysis of Nonlinear mathematical Models of Tumor Growth

Abstract

Tumor growth is basically a hybrid dynamical system. Even same type newlineof tumors varies from patient to patient not only in size but also in temporal newlinebehavior. The complexity in the growth of a tumor is further aggravated by its newlinemulti-directional proliferation. The dynamical nature of a tumor growth can be newlinemathematically modeled and represented in the form of system of nonlinear newlineOrdinary Differential Equation or nonlinear Partial Differential Equation or as newlinean exponential function. newlineThe prime objective of this research work is to study three different newlinetypical nonlinear chaos theory Ordinary Differential Equation Models namely newlineChua, Chen and Lorenz as represents Tumor Model-1, Tumor Model-2 and newlineTumor Model-3. Analyzed the solutions of these Tumor models to obtain newlineconcrete inferences in tumor microenvironment. newlineThe above systems of nonlinear Ordinary Differential Equation Models newlineare solved by employing various techniques namely, Differential Transform newlineMethod, Elzaki Transform Method and Fractional Order Multistage Homotopy newlinePerturbation Method. The system variables x, y, z in the above said newlinenonlinear Ordinary Differential Equation Tumor Model-1, Tumor Model-2 and newlineTumor Model-3 represents Host Cell, Effectors Immune Cell and Tumor Cell newlinerespectively. The Tumor models are analyzed and graphically compared with newlinethe numerical solutions. newlineBasically there are two stages of tumor growth namely Benign and newlineMalignant. The first one corresponds to the initial stage of tumor and the later newlinecorresponds to spiteful stage of the tumor. In Benign stage the rate of growth newlineof the cells in the tumor is slow and in the case of Malignant stage the growth newlineis fast and unpredictable. newlineIn the above three systems of nonlinear Ordinary Differential Equation newlineTumor models, the Tumor Model-1 and Tumor Model-2 corresponds to newlineBenign stage and the Tumor Model-3 corresponds to Malignant stage. newline2 newlineFurther the study is extended to two more nonlinear Partial Differential newlineEquation Models of Tumor growth namely, Anderson Enderling Model and newlineSpatio Temporal System