An optimal control problem of immuno-chemotherapy in presence of gene therapy

Kaushik Dehingia; Hemanta Kumar Sarmah; Kamyar Hosseini; Khadijeh Sadri; Soheil Salahshour; Choonkil Park; Kaushik Dehingia; Hemanta Kumar Sarmah; Kamyar Hosseini; Khadijeh Sadri; Soheil Salahshour; Choonkil Park

doi:10.3934/math.2021669

AIMS Mathematics

2021, Volume 6, Issue 10: 11530-11549. doi: 10.3934/math.2021669

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Research article

An optimal control problem of immuno-chemotherapy in presence of gene therapy

1.
Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India
2.
Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran
3.
Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
4.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, South Korea

Received: 10 June 2021 Accepted: 26 July 2021 Published: 09 August 2021
MSC : 37M05, 37M10, 37N25, 92B05

This study addresses a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy. The main objective of this study is to understand the optimal effect of immuno-chemotherpay in the presence of gene therapy. The boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system exist and are stable have been derived. An optimal control problem for the system has been constructed and solved to minimize the immuno-chemotherapy drug-induced toxicity to the patient. Amounts of immunotherapy to be injected into a patient for eradication of cancerous tumor cells have been found. Numerical and graphical results have been presented. From the results, it is seen that tumor cells can be eliminated in a specific time interval with the control of immuno-chemotherapeutic drug concentration.
- cancer,
- gene therapy,
- chemotherapy,
- immunotherapy,
- stability,
- optimal control
Citation: Kaushik Dehingia, Hemanta Kumar Sarmah, Kamyar Hosseini, Khadijeh Sadri, Soheil Salahshour, Choonkil Park. An optimal control problem of immuno-chemotherapy in presence of gene therapy[J]. AIMS Mathematics, 2021, 6(10): 11530-11549. doi: 10.3934/math.2021669

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Abstract

This study addresses a cancer eradication model involving effector cells in the presence of gene therapy, immunotherapy, and chemotherapy. The main objective of this study is to understand the optimal effect of immuno-chemotherpay in the presence of gene therapy. The boundedness and positiveness of the solutions in the respective feasible domains of the proposed model are verified. Conditions for which the equilibrium points of the system exist and are stable have been derived. An optimal control problem for the system has been constructed and solved to minimize the immuno-chemotherapy drug-induced toxicity to the patient. Amounts of immunotherapy to be injected into a patient for eradication of cancerous tumor cells have been found. Numerical and graphical results have been presented. From the results, it is seen that tumor cells can be eliminated in a specific time interval with the control of immuno-chemotherapeutic drug concentration.

References

[1]	R. Coletti, L. Leonardelli, S. Parolo, L. Marchetti, A QSP model of prostate cancer immunotherapy to identify effective combination therapies, Sci. Rep-UK., 10 (2020), 9063. doi: 10.1038/s41598-020-65590-0
[2]	R. Coletti, A. Pugliese, L. Marchetti, Modeling the effect of immunotherapies on human castration-resistant prostate cancer, J. Theor. Biol., 509 (2021), 110500. doi: 10.1016/j.jtbi.2020.110500
[3]	S. Wilson, D. Levy, A mathematical model of the enhancement of tumour vaccine eﬃcacy by immune therapy, Bull. Math. Biol., 74 (2012), 1485–1500. doi: 10.1007/s11538-012-9722-4
[4]	F. Frascoli, P. S. Kim, B. D. Hughes, K. A. Landman, A dynamical model of tumour immunotherapy, Math. Biosci., 253 (2014), 50–62. doi: 10.1016/j.mbs.2014.04.003
[5]	V. T. DeVita Jr., E. Chu, A history of cancer chemotherapy, Cancer Res., 68 (2008), 8643–8653. doi: 10.1158/0008-5472.CAN-07-6611
[6]	L. G. De Pillis, K. R. Fister, W. Gu, T. Head, K. Maples, T. Neal, et al., Optimal control of mixed immunotherapy and chemotherapy of tumors, J. Biol. Syst., 16 (2008), 51–80. doi: 10.1142/S0218339008002435
[7]	S. Khajanchi, D. Ghosh, The combined effects of optimal control in cancer remission, Appl. Math. Comput., 271 (2015), 375–388.
[8]	S. Sharma, G. P. Samanta, Dynamical behaviour of a tumor-immune system with chemotherapy and optimal control, J. Nonlinear Dyn., 2013 (2013), 608598.
[9]	D. Maurici, P. Hainaut, TP53 gene and P53 protein as targets in cancer management and therapy, Biotech., 12 (2001).
[10]	D. A. Yardley, Drug resistance and the role of combination chemotherapy in improving patient outcomes, Int. J. of Breast Cancer, 2013 (2013), 137414.
[11]	M. Zhang, O. B. Garbuzenko, K. R. Reuhl, L. Rodriguez-Rodriguez, T. Minko, Two-in-one: Combined targeted chemo and gene therapy for tumor suppression and prevention of metastases, Nanomed., 7 (2012), 185–197. doi: 10.2217/nnm.11.131
[12]	T. Lin, L. Zhang, J. Davis, J. Gu, M. Nishizaki, L. Ji, et al., Combination of TRAIL gene therapy and chemotherapy enhances antitumor and antimetastasis effects in chemosensitive and chemoresistant breast cancers, Mol. Ther., 8 (2003), 441–448. doi: 10.1016/S1525-0016(03)00203-X
[13]	L. M. Cannon, M. J. Hernandez, R. Zurakowski, Modeling and analysis of gene-therapeutic combination chemotherapy for pancreatic cancer, IFAC Proc. Vols., 44 (2011), 14217–14222. doi: 10.3182/20110828-6-IT-1002.03162
[14]	N. Bellomo, N. K. Li, P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci., 18 (2008), 5930–646.
[15]	R. P. Araujo, D. L. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol., 66 (2004), 1039–1091. doi: 10.1016/j.bulm.2003.11.002
[16]	V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295–321. doi: 10.1016/S0092-8240(05)80260-5
[17]	D. Kirschner, J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235–252. doi: 10.1007/s002850050127
[18]	M. Kolev, E. Kozlowska, M. Lachowicz, A mathematical model for single cell cancer-immune system dynamics, Math. Comput. Model., 41 (2005), 1083–1095. doi: 10.1016/j.mcm.2005.05.004
[19]	N. M. Berezhnaya, Interaction between tumor and immune system: The role of tumor cell biology, Exp. Oncol., 32 (2010), 159–166.
[20]	H. Gonzalez, C. Hagerling, Z. Werb, Roles of the immune system in cancer: From tumor initiation to metastatic progression, Genes Dev., 32 (2018), 1267–1284. doi: 10.1101/gad.314617.118
[21]	F. A. Rihan, M. Safan, M. A. Abdeen, D. A. Rahman, Qualitative and computational analysis of a mathematical model for tumor-immune interactions, J. Appl. Math., 2012 (2012), 475720.
[22]	S. S. Musa, S. Qureshi, S. Zhao, A. Yusuf, U. T. Mustapha, D. He, Mathematical modeling of COVID-19 epidemic with effect of awareness programs, Infect. Dis. Model., 6 (2020), 448–460.
[23]	Z. Memon, S. Qureshi, B. R. Memon, Assessing the role of quarantine and isolation as control strategies for COVID-19 outbreak: A case study, Chaos Solitons Fractals, 144 (2021), 110655. doi: 10.1016/j.chaos.2021.110655
[24]	N. H. Sweilam, S. M. Al-Mekhlafi, A. O. Albalawi, D. Baleanu, On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach, Adv. Differ. Equ., 2020 (2020), 528. doi: 10.1186/s13662-020-02982-6
[25]	L. G. de Pillis, K. R. Fister, W. Gu, C. Collins, M. Daub, D. Gross, et al., Mathematical model creation for cancer chemo-immunotherapy, Comput. Math. Methods Med., 10 (2009), 165–184. doi: 10.1080/17486700802216301
[26]	S. T. R. Pinho, F. S. Bacelar, R. F. S. Andradea, H. I. Freedman, A mathematical model for the effect of anti-angiogenic therapy in the treatment of cancer tumours by chemotherapy, Nonlinear Anal. Real World Appl., 14 (2013), 815–828. doi: 10.1016/j.nonrwa.2012.07.034
[27]	A. Ghaffari, M. Nazari, F. Arab, Optimal finite cancer treatment duration by using mixed vaccine therapy and chemotherapy: State dependent Riccati equation control, J. Appl. Math., 2014 (2014), 363109, 1–9.
[28]	Z. Liu, C. Yang, A mathematical model of cancer treatment by radiotherapy followed by chemotherapy, Math. Comput. Simul., 124 (2016), 1–15. doi: 10.1016/j.matcom.2015.12.007
[29]	R. T. Guiraldello, M. L. Martins, P. F. A. Mancera, Evaluating the efficacies of maximum tolerated dose and metronomic chemotherapies: A mathematical approach, Phys. A: Stat. Mech. Appl., 456 (2016), 145–156. doi: 10.1016/j.physa.2016.03.019
[30]	L. Pang, L. Shen, Z. Zhao, Mathematical modelling and analysis of the tumor treatment regimens with pulsed immunotherapy and chemotherapy, Comput. Math. Methods Med., 2016 (2016), 6260474.
[31]	D. S. Rodrigues, P. F. A. Mancera, T. Carvalho, L. F. Gonçalves, A mathematical model for chemoimmunotherapy of chronic lymphocytic leukemia, Appl. Math. Comput., 349 (2019), 118–133.
[32]	W. F. F. M. Gil, T. Carvalho, P. F. A. Mancera, D. S. Rodrigues, A mathematical model on the immune system role in achieving better outcomes of cancer chemotherapy, Trends Comput. Appl. Math., 20 (2019), 343–357.
[33]	M. A. Alqudah, Cancer treatment by stem cells and chemotherapy as a mathematical model with numerical simulations, Alex. Eng. J., 59 (2020), 1953–1957. doi: 10.1016/j.aej.2019.12.025
[34]	A. Tsygvintsev, S. Marino, D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer, Mathematical Methods and Models in Biomedicine, Springer, New York, 2013, pp. 367–385.
[35]	F. A. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim, M. A. Abdeen, Delay differential model for tumour-immune response with chemoimmunotherapy and optimal control, Comput. Math. Methods Med., 2014 (2014), 982978.
[36]	A. d'Onofrio, U. Ledzewicz, H. Maurer, H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13–26. doi: 10.1016/j.mbs.2009.08.004
[37]	S. Khajanchi, Stability analysis of a mathematical model for glioma-immune interaction under optimal therapy, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 269–285. doi: 10.1515/ijnsns-2017-0206
[38]	M. Leszczynski, U. Ledzewicz, H. Schattler, Optimal control for a mathematical model for chemotherapy with pharmacometrics, Math. Model. Nat. Phenom., 15 (2020), 69. doi: 10.1051/mmnp/2020008
[39]	P. Khalili, R. Vatankhah, Derivation of an optimal trajectory and nonlinear adaptive controller design for drug delivery in cancerous tumor chemotherapy, Comput. Biol. Med., 109 (2019), 195–206. doi: 10.1016/j.compbiomed.2019.04.011
[40]	M. Najafi, H. Basirzadeh, Optimal control homotopy perturbation method for cancer model, Int. J. Biomath., 12 (2019), 1950027. doi: 10.1142/S179352451950027X
[41]	F. A. Rihan, S. Lakshmanan, H. Maurer, Optimal control of tumour-immune model with time-delay and immuno-chemotherapy, Appl. Math. Comput., 353 (2019), 147–165.
[42]	A. Bukkuri, Optimal control analysis of combined chemotherapy-immunotherapy treatment regimens in a PKPD cancer evolution model, Biomath., 9 (2020), 2002137. doi: 10.11145/j.biomath.2020.02.137
[43]	P. Das, S. Das, R. K. Upadhyay, P. Das, Optimal treatment strategies for delayed cancer-immune system with multiple therapeutic approach, Chaos Solitons Fractals, 136 (2020), 109806. doi: 10.1016/j.chaos.2020.109806
[44]	D. Lestari, R. Dwi Ambarwati, A local stability of mathematical models for cancer treatment by using gene therapy, Int. J. Model. Optim., 5 (2015), 202–206. doi: 10.7763/IJMO.2015.V5.462
[45]	D. H. Margarit, L. Romanelli, A simple model for control of tumor cells, J. Biol. Syst., 23 (2015), S33–S41. doi: 10.1142/S0218339015400033
[46]	D. L. Lukes, Differential equations: Classical to controlled, Mathematics in Science and Engineering, Academic Press, New York, 1982.
[47]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Process, Gordon and Breach, 1962.
[48]	S. Qureshi, A. Yusuf, A new third order convergent numerical solver for continuous dynamical systems, J. King Saud Univ. Sci., 32 (2020), 1409–1416. doi: 10.1016/j.jksus.2019.11.035
[49]	Z. Odibat, D. Baleanu, Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives, Appl. Numer. Math., 156 (2020), 94–105. doi: 10.1016/j.apnum.2020.04.015
[50]	S. Qureshi, A. Yusuf, S. Aziz, On the use of mohand integral transform for solving fractional-order classical caputo differential equations, J. Appl. Math. Comput. Mech., 19 (2020), 99–109. doi: 10.17512/jamcm.2020.3.08

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