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2018, 8(1): 63-80. doi: 10.3934/naco.2018004

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

 1 Ferdowsi University of Mashhad, Mashhad, Iran 2 Interdisciplinary Centre for Scientific Computing (IWR) and BIOQUANT Ruprecht-Karls-Universitat, Heidelberg, Germany

* Corresponding author

Received  December 2016 Revised  January 2018 Published  March 2018

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).

Citation: Reihaneh Mostolizadeh, Zahra Afsharnezhad, Anna Marciniak-Czochra. Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004
##### References:
 [1] R. J. Brentjens, Adoptive Therapy of Cancer with T cells Genetically Targeted to Tumor Associated Antigens through the Introduction of Chimeric Antigen Receptors (CARs): Trafficking, Persistence, and Perseverance American society of Gene and cell therapy, 14th Annual Meeting, 2011. [2] F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, Journal of Theoretical Biology, 247 (2007), 723-732. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1990. [4] L. G. DE Pillis and A. E. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. Theor. Medicine., 3 (2001), 79-100. [5] L. G. DE Pillis, W. Gu and A. E. Radunskayab, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841-862. [6] L. G. DE Pillis and et al., Mathematical model creation for cancer chemo-immunotherapy, J. Computational and Mathematical Methods in Medicine, 10 (2009), 165-184. [7] F. R. Gantmacher, Applications of the Theory of Matrices, New York: Wiley, 2005. [8] S. A. Grupp et al., Chimeric antigen receptor-modified T cells for acute lymphoid leukemia, New Engl. J. Med., 16 (2013), 1509-1518. [9] UL. Heinz Schattler, Geometric Optimal Control Theory, Methods and Examples, Springer. New York, 2012. [10] M. I. Kamien and N. L. Schwartz, Dynamic Optimization: The Calulus of Variations and Optimal Control in Economics and Management, North-Holland, 1991. [11] M. Kalos, B. L. Levine, D. L. Porter and et al., T cells with chimeric antigen receptors have potent antitumor effects and can establish memory in patients with advanced leukemia, Sci. Transl. Med., 3 (2011), 95-73. [12] JN. Kochenderfer, WH. Wilson, JE. Janik and et al., Eradication of B-lineage cells and regression of lymphoma in a patient treated with autologous T cells genetically engineered to recognize CD19, Blood, 2010. [13] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Second Edition, Springer, Berlin, 1998. [14] D. W. Lee and et al. T cells expressing CD19 chimeric antigen receptors for acute lymphoblastic leukaemia in children and young adults: a phase 1 dose-escalation trial, Lancent J., 385 (2015), 517-528. [15] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall/CRC, Boca Raton, FL, 2007. [16] E. Maino and et al., Modern immunotherapy of adult B-Lineage acute lymphoblastic leukemia with monoclonal antibodies and chimeric antigen receptor modified T cells, Mediterr J Hematol Infect Dis., 7 (2015). [17] A. Marciniak-Czochra, T. Stiehl, A. Ho, W. Jager and W. Wagner, Modeling of Asymmetric Cell Division in Hematopoietic Stem Cells-Regulation of Self-Renewal Is Essential for Efficient Repopulation, Mary Ann Liebert Inc, New Rochelle, 2009. [18] SL. Maude and et al. Chimeric antigen receptor T cells for sustained remissions in leukemia, New Engl. J. Med., 16 (2014), 1507-1517. [19] R. Mostolizadeh, Mathematical model of Chimeric Antigene Receptor(CAR)T cell therapy, Preprint. [20] Pontryagin, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt. Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962. [21] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer Academic Publishers, 2000. [22] MP. Velders, S. T. Horst and WM. Kast, sProspects for immunotherapy of acute lymphoblastic leukemia, Nature Leukemia, 15 (2001), 701-706. [23] X. Wang, Solving optimal control problems with MATLAB: Indirect methods, Technical report, ISE Dept., NCSU, 2009. [24] J. P. Zbilut, Unstable Singularities and Randomness: Their Importance in the Complexity of Physical, Biological and Social Sciences, Elsevier Press, 2004.

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##### References:
 [1] R. J. Brentjens, Adoptive Therapy of Cancer with T cells Genetically Targeted to Tumor Associated Antigens through the Introduction of Chimeric Antigen Receptors (CARs): Trafficking, Persistence, and Perseverance American society of Gene and cell therapy, 14th Annual Meeting, 2011. [2] F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, Journal of Theoretical Biology, 247 (2007), 723-732. [3] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, second edition, 1990. [4] L. G. DE Pillis and A. E. Radunskaya, A mathematical tumor model with immune resistance and drug therapy: an optimal control approach, J. Theor. Medicine., 3 (2001), 79-100. [5] L. G. DE Pillis, W. Gu and A. E. Radunskayab, Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841-862. [6] L. G. DE Pillis and et al., Mathematical model creation for cancer chemo-immunotherapy, J. Computational and Mathematical Methods in Medicine, 10 (2009), 165-184. [7] F. R. Gantmacher, Applications of the Theory of Matrices, New York: Wiley, 2005. [8] S. A. Grupp et al., Chimeric antigen receptor-modified T cells for acute lymphoid leukemia, New Engl. J. Med., 16 (2013), 1509-1518. [9] UL. Heinz Schattler, Geometric Optimal Control Theory, Methods and Examples, Springer. New York, 2012. [10] M. I. Kamien and N. L. Schwartz, Dynamic Optimization: The Calulus of Variations and Optimal Control in Economics and Management, North-Holland, 1991. [11] M. Kalos, B. L. Levine, D. L. Porter and et al., T cells with chimeric antigen receptors have potent antitumor effects and can establish memory in patients with advanced leukemia, Sci. Transl. Med., 3 (2011), 95-73. [12] JN. Kochenderfer, WH. Wilson, JE. Janik and et al., Eradication of B-lineage cells and regression of lymphoma in a patient treated with autologous T cells genetically engineered to recognize CD19, Blood, 2010. [13] Y. Kuznetsov, Elements of Applied Bifurcation Theory, Second Edition, Springer, Berlin, 1998. [14] D. W. Lee and et al. T cells expressing CD19 chimeric antigen receptors for acute lymphoblastic leukaemia in children and young adults: a phase 1 dose-escalation trial, Lancent J., 385 (2015), 517-528. [15] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall/CRC, Boca Raton, FL, 2007. [16] E. Maino and et al., Modern immunotherapy of adult B-Lineage acute lymphoblastic leukemia with monoclonal antibodies and chimeric antigen receptor modified T cells, Mediterr J Hematol Infect Dis., 7 (2015). [17] A. Marciniak-Czochra, T. Stiehl, A. Ho, W. Jager and W. Wagner, Modeling of Asymmetric Cell Division in Hematopoietic Stem Cells-Regulation of Self-Renewal Is Essential for Efficient Repopulation, Mary Ann Liebert Inc, New Rochelle, 2009. [18] SL. Maude and et al. Chimeric antigen receptor T cells for sustained remissions in leukemia, New Engl. J. Med., 16 (2014), 1507-1517. [19] R. Mostolizadeh, Mathematical model of Chimeric Antigene Receptor(CAR)T cell therapy, Preprint. [20] Pontryagin, The Mathematical Theory of Optimal Processes, Translated from the Russian by K. N. Trirogoff, edited by L. W. Neustadt. Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962. [21] S. P. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer Academic Publishers, 2000. [22] MP. Velders, S. T. Horst and WM. Kast, sProspects for immunotherapy of acute lymphoblastic leukemia, Nature Leukemia, 15 (2001), 701-706. [23] X. Wang, Solving optimal control problems with MATLAB: Indirect methods, Technical report, ISE Dept., NCSU, 2009. [24] J. P. Zbilut, Unstable Singularities and Randomness: Their Importance in the Complexity of Physical, Biological and Social Sciences, Elsevier Press, 2004.
Expression of CD19 and other B cell markers on B lineage cells
Graph of solutions $x(t), y(t), z(t), w(t), v(t)$
Graph of solutions $x(t), y(t), z(t), w(t), v(t)$ with a new initial condition
Graph of optimal solutions $x^{*}(t), y^{*}(t), z^{*}(t), w^{*}(t), v^{*}(t)$ for 1 month
Graph of optimal controls
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