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Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine

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  • In this paper, we reconstruct a mathematical model of therapy by CAR T cells for acute lymphoblastic leukemia (ALL) With injection of modified T cells to body, then some signs such as fever, nausea and etc appear. These signs occur for the sake of cytokine release syndrome (CRS). This syndrome has a direct effect on result and satisfaction of therapy. So, the presence of cytokine will be played an important role in modelling process of therapy (CAR T cells). Therefore, the model will include the CAR T cells, B healthy and cancer cells, other circulating lymphocytes in blood, and cytokine. We analyse stability conditions of therapy Without any control, the dynamic model evidences sub-clinical or clinical decay, chronic destabilization, singularity immediately after a few hours and finally, it depends on the initial conditions. Hence, we try to show by which conditions, therapy will be effective. For this aim, we apply optimal control theory. Since the therapy of CAR T cells affects on both normal and cancer cell; so the optimization dose of CAR T cells will be played an important role and added to system as one controller $ u_{1} $ . On the other hand, in order to control of cytokine release syndrome which is a factor for occurrence of singularity, one other controller $ u_{2} $ as tocilizumab, an immunosuppressant drug for cytokine release syndrome is added to system. At the end, we apply method of Pontryagin's maximum principle for optimal control theory and simulate the clinical results by Matlab (ode15s and ode45).

    Mathematics Subject Classification: Primary: 34A34, 49K15; Secondary: 92Bxx.

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  • Figure 1.  Expression of CD19 and other B cell markers on B lineage cells

    Figure 2.  Graph of solutions $ x(t), y(t), z(t), w(t), v(t) $

    Figure 3.  Graph of solutions $ x(t), y(t), z(t), w(t), v(t) $ with a new initial condition

    Figure 4.  Graph of optimal solutions $ x^{*}(t), y^{*}(t), z^{*}(t), w^{*}(t), v^{*}(t)$ for 1 month

    Figure 5.  Graph of optimal controls

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