Geodesic
Optimal strategies of CAR T-Cell therapy in the treatment of leukemia within the Lotka-Volterra predator-prey model
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 3, pp. 43-58 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

A controlled mathematical model of leukemia treatment is considered. The model is based on the three-dimensional Lotka–Volterra predator–prey model, which describes a recently developed leukemia treatment technology called Chimeric Antigen Receptor (CAR) T-cell therapy, and is given on a fixed time interval by a system of four differential equations. The equations describe the interaction between populations of healthy and cancer cells, CAR T-cells, and cytokines. The CAR T-cells act as predators, while healthy and cancer cells act as prey. The CAR T-cell therapy leads to serious side-effects associated with the rapid growth of cytokines, and therefore their dynamics is also included in the model. The model also contains two bounded control functions reflecting the intensity of the therapy (the first control) and the intensity of administration of drugs that suppress the activity of the immune system (the second control). We study the problem of minimizing the objective function related to the number of cancer and healthy cells, as well as cytokines, both at the final moment of a given time interval and during this entire interval. The Pontryagin maximum principle is applied for the analysis of the problem; it is used to establish the bang-bang nature of an optimal first control and to estimate the number of its switchings. It is shown that an optimal second control is a constant function on the entire time interval. The BOCOP-2.2.1 environment is used for the numerical analysis of the problem. The results of numerical calculations are presented, demonstrating various types of optimal protocols for CAR-T therapy.
Keywords: leukemia, nonlinear control system, optimal control, Pontryagin maximum principle, bang-bang control, switching function, generalized Rolle theorem. 
@article{TIMM_2021_27_3_a3,
     author = {N. L. Grigorenko and E. N. Khailov and E. V. Grigorieva and A. D. Klimenkova},
     title = {Optimal strategies of {CAR} {T-Cell} therapy in the treatment of leukemia within the {Lotka-Volterra} predator-prey model},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {43--58},
     year = {2021},
     volume = {27},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2021_27_3_a3/}
}
TY  - JOUR
AU  - N. L. Grigorenko
AU  - E. N. Khailov
AU  - E. V. Grigorieva
AU  - A. D. Klimenkova
TI  - Optimal strategies of CAR T-Cell therapy in the treatment of leukemia within the Lotka-Volterra predator-prey model
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2021
SP  - 43
EP  - 58
VL  - 27
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2021_27_3_a3/
LA  - ru
ID  - TIMM_2021_27_3_a3
ER  - 
%0 Journal Article
%A N. L. Grigorenko
%A E. N. Khailov
%A E. V. Grigorieva
%A A. D. Klimenkova
%T Optimal strategies of CAR T-Cell therapy in the treatment of leukemia within the Lotka-Volterra predator-prey model
%J Trudy Instituta matematiki i mehaniki
%D 2021
%P 43-58
%V 27
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2021_27_3_a3/
%G ru
%F TIMM_2021_27_3_a3
N. L. Grigorenko; E. N. Khailov; E. V. Grigorieva; A. D. Klimenkova. Optimal strategies of CAR T-Cell therapy in the treatment of leukemia within the Lotka-Volterra predator-prey model. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 3, pp. 43-58. http://geodesic.mathdoc.fr/item/TIMM_2021_27_3_a3/

[1] Heymach J. [et all.], “Clinical cancer advances 2018: annual report on progress against cancer from the American Society of Clinical Oncology”, J. Clin. Oncol., 36:10 (2018), 1020–1044 | DOI

[2] June C.H., Sadelain M., “Chimeric antigen receptor therapy”, N. Engl. J. Med., 379:1 (2018), 64–73 | DOI

[3] Hopkins B., Tucker M., Pan Y., Fang N., Huang Z., “A model-based investigation of cytokine storm for T-cell therapy”, IFAC-PapersOnLine, 51:19 (2018), 76–79 | DOI

[4] Barros L.R.C., Rodrigues B.J., Almeida R.C., “CAR-T cell goes on a mathematical model”, J. Cell Immunol., 2:1 (2020), 31–37 | DOI

[5] Konstorum A., Vella A.T., Adler A.J., Laubenbacher R.C., “Addressing current challenges in cancer immunotherapy with mathematical and computational modelling”, J. Roy. Soc. Interface, 14:131 (2017), 20170150, 1–10 | DOI

[6] Valentinuzzi D., Jeraj R., “Computational modelling of modern cancer immunotherapy”, Phys. Med. Biol., 65:24 (2020), 24TR01, 1–22 | DOI

[7] Leon-Triana O., Sabir S., Calvo G.F., Belmonte-Beitia J., Chulian S., Martinez-Rubio A., Rosa M., Perez-Martinez A., Ramirez-Orellana M., Perez-Garcia V.M., “CAR T-cell in B-cell acute lymphoblastic leukaemia: insights from mathematical models”, Commun. Nonlinear Sci., 94 (2021), 105570 | DOI | Zbl

[8] Perez-Garcia V.M., Leon-Triana O., Rosa M., Perez-Martinez A., CAR T cells for T-cell leukemias: insights from mathematical models, 26 Apr 2020, 20 pp., arXiv: 2004.14291

[9] Sahoo P. [et al.], “Mathematical deconvolution of CAR T-cell proliferation and exhaustion from real-time killing assay data”, J. Roy. Soc. Interface, 17:162 (2019), 20190734, 1–10 | DOI

[10] Nichelatti M., “A mathematical model for the chimeric antigen receptor T cell (CAR-T) therapy as a Lotka-Volterra system”, J. Math. Stat. Res., 3:3 (2020), 1–4 | DOI

[11] Khailov E.N., Klimenkova A.D., Korobeinikov A., “Optimal control for anti-cancer therapy”, Extended abstracts spring 2018, Trends in mathematics, 11, eds. A. Korobeinikov, M. Caubergh, T. Lazaro, J. Sardanyes, Birkhauser, Basel, 2019, 35–43 | DOI | Zbl

[12] Grigorenko N.L., Khailov E.N., Klimenkova A.D., Korobeinikov A., “Program and positional control strategies for the Lotka-Volterra competition model”, Stability, Control and Differential Games (Proceedings of the International Conference “Stability, Control, Differential Games” (SCDG2019)), eds. A. Tarasyev, V. Maksimov, T. Filippova, Springer Nature, Cham, 2020, 39–49 | DOI | Zbl

[13] Grigorenko N.L., Khailov E.N., Grigoreva E.V., Klimenkova A.D., “Optimalnye strategii lecheniya rakovykh zabolevanii v matematicheskoi modeli konkurentsii Lotki - Volterry”, Tr. In-ta matematiki i mekhaniki UrO RAN, 26:1 (2020), 71–88 | DOI

[14] Mostolizadeh R., Afsharnezhad Z., Marciniak-Czochra A., “Mathematical model of chimeric anti-gene receptor (CAR) T cell therapy with presence of cytokine”, Numer. Algebr. Control Optim., 8:1 (2018), 63–80 | DOI | Zbl

[15] Khailov E., Grigorieva E., Klimenkova A., “Optimal CAR T-cell immunotherapy strategies for a leukemia treatment model”, Games, 11:4 (2020), 53, 1–26 | DOI

[16] Pillis L.G., Radunskaya A., “A mathematical tumor model with immune resistance and drug therapy: an optimal control approach”, J. Theoret. Medicine, 3 (2001), 79–100 | DOI | Zbl

[17] Pillis L.G., Radunskaya A., “The dynamics of an optimally controlled tumor model: a case study”, Math. Comput. Model., 37 (2003), 1221–1244 | DOI | Zbl

[18] Bazykin A.D., Nelineinaya dinamika vzaimodeistvuyuschikh populyatsii, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevskii institut kompyuternykh issledovanii, Moskva; Izhevsk, 2003, 368 pp.

[19] Li E.B., Markus L., Osnovy teorii optimalnogo upravleniya, Nauka, M., 1972, 576 pp.

[20] Vasilev F.P., Metody optimizatsii, Faktorial Press, M., 2002, 824 pp.

[21] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970, 720 pp.

[22] Dmitruk A.V., “A generalized estimate on the number of zeros for solutions of a class of linear differential equations”, SIAM J. Control Optim., 30:5 (1992), 1087–1091 | DOI | Zbl

[23] Kuzenkov O.A., Ryabova E.A., Matematicheskoe modelirovanie protsessov otbora, Izd-vo Nizhnegorod. un-ta, N. Novgorod, 2007, 324 pp.

[24] Bonnans F., Martinon P., Giorgi D., Grelard V., Maindrault S., Tissot O., Liu J., BOCOP 2.2.1 - user guide, [e-resource], August 8, 2019 http://bocop.org