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Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy

ABSTRACT: Stability Analysis of a Mathematical Model for Glioma-Immune Interaction under Optimal Therapy Subhas Khajanchi Abstract We investigate a mathematical model using a system of coupled ordinary differential equations, which describes the interplay of malignant glioma cells, macrophages, glioma specific CD8+T cells and the immunotherapeutic drug Adoptive Cellular Immunotherapy (ACI). To better understand under what circumstances the glioma cells can be eliminated, we employ the theory of optimal control. We investigate the dynamics of the system by observing biologically feasible equilibrium points and their stability analysis before administration of the external therapy ACI. We solve an optimal control problem with an objective functional which minimizes the glioma cell burden as well as the side effects of the treatment. We characterize our optimal control in terms of the solutions to the optimality system, in which the state system coupled with the adjoint system. Our model simulation demonstrates that the strength of treatment u1(t) plays an important role to eliminate the glioma cells. Finally, we derive an optimal treatment strategy and then solve it numerically. Keywords: malignant gliomas; stability analysis; optimal control; adoptive cellular immunotherapy

SUBMITTER: Mohammad Umer Sharif Shohan

PROVIDER: BIOMD0000000891 | BioModels | 2020-01-03

REPOSITORIES: BioModels

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Project description:This ordinary differential equation model simulating the interactions between tumor and immune cells is detailed in the publication: Ahmed M. Makhlouf, Lamiaa El-Shennawy, Hesham A. Elkaranshawy, "Mathematical Modelling for the Role of CD4+T Cells in Tumor-Immune Interactions", Comput Math Methods Med. 2020 Feb 19;2020:7187602. doi: 10.1155/2020/7187602 Comment: This no treatment model is described by equations 1-7 of the publication manuscript. Abstract: Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8+T cells, but they neglected the role of CD4+T cells. Other models were constructed to study the role of CD4+T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge–Kutta method failed to solve it. Consequently, the “Adams predictor-corrector” method is used. The results reveal that the patient’s immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4+T cells can be an alternative to both CD8+T cell therapy and cytokines in some cases. Moreover, CD4+T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4+T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8+T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients’ treatment intervention strategies.

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Khajanchi2019 - Stability Analysis of a Mathematical Model forGlioma-Immune Interaction under OptimalTherapy

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