MATHEMATICAL MODELING OF MALIGNANT BRAIN TUMOR WITH T11 TARGET STRUCTURE

Abstract

Clinicians and oncologists believe that the malignant gliomas are highly di usive and invasive brain tumors and it has the ability to evade the surrounding normal tissues. Recently, a signi cant amount of attention has been given on exploring the chemotherapies and immunotherapies to eliminate the glioma cells but it has limited success due to the sequestered location beyond the blood-brain-barrier. Numerous mathematical notions and techniques have been used to study the highly nonlinear interactions between gliomas and immune system and these, in turn, has enhanced the eld of mathematics itself. In this thesis, we construct a mathematical model which comprises a system of ve coupled nonlinear ordinary di erential equations (ODEs) to study the qualitative and quantitative analysis of malignant gliomas and immune system interactions, by considering the role of immunotherapeutic drug/agent T11 target structure. The present work is an amalgamation of applied mathematics and biology. The mathematical model has been studied both analytically and numerically. The model system undergoes sensitivity analysis, that determines which state variables are most sensitive to the given parameters and the parameters are estimated from the experimental data procured by our collaborator Prof. Swapna Chaudhuri, Department of Laboratory Medicine, School of Tropical Medicine, Kolkata, India. Next, we add four discrete time delays, namely, interaction delays, to asses its e ect on the interaction between macrophages, cytotoxic T-lymphocytes, immunosuppressive factor TGF- and malignant gliomas. Although delays are present in all biological process, in our case they have no signi cant in uence on the model dynamics. We also employed deterministic optimal control treatment strategy by using i immunotherapeutic drug T11 target structure. The biomedical goal of the application of an optimal treatment strategy for brain tumor is to reduce the glioma cells burden and maximize the amount of drug concentration and increase the cell count of macrophages and cytotoxic T-lymphocytes. Once the optimal control problem is constructed via the de nition of the Lagrangian function, we employ the classical approach of calculus of variation with certain adjustments. This leads to a system of nonlinear ordinary di erential equations, associated with adjoint, state variables and a control variable which is the rate at which the drug is given to the brain tumor patients'. We also established the uniqueness of the quadratic control in a su ciently small time window. Computer simulations were used in all the aspects for model veri cation and validation, which highlight the importance of T11 target structure in brain tumor therapy.