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 Zh. Vychisl. Mat. Mat. Fiz., 2009, Volume 49, Number 11, Pages 1907–1919 (Mi zvmmf4778)

Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A. V. Antipov, A. S. Bratus'

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

Abstract: A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.

Key words: mathematical model of optimal chemotherapy, optimal control problem, numerical methods.

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English version:
Computational Mathematics and Mathematical Physics, 2009, 49:11, 1825–1836

Bibliographic databases:

UDC: 519.626

Citation: A. V. Antipov, A. S. Bratus', “Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor”, Zh. Vychisl. Mat. Mat. Fiz., 49:11 (2009), 1907–1919; Comput. Math. Math. Phys., 49:11 (2009), 1825–1836

Citation in format AMSBIB
\Bibitem{AntBra09} \by A.~V.~Antipov, A.~S.~Bratus' \paper Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a~heterogeneous tumor \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2009 \vol 49 \issue 11 \pages 1907--1919 \mathnet{http://mi.mathnet.ru/zvmmf4778} \transl \jour Comput. Math. Math. Phys. \yr 2009 \vol 49 \issue 11 \pages 1825--1836 \crossref{https://doi.org/10.1134/S0965542509110013} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000272464100001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-71549145339} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bratus A.S., Fimmel E., Todorov Y., Semenov Y.S., Nuernberg F., “On strategies on a mathematical model for leukemia therapy”, Nonlinear Anal. Real World Appl., 13:3 (2012), 1044–1059
2. V. A. Srochko, “On solving the optimization problem for chemotherapy process in terms of the maximum principle”, Russian Math. (Iz. VUZ), 56:7 (2012), 55–59
3. V. A. Srochko, “Ekstremalnye rezhimy upravleniya v zadache optimizatsii protsessa terapii”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 2012, no. 3, 113–119
4. Fimmel E., Semenov Yu.S., Bratus A.S., “On optimal and suboptimal treatment strategies for a mathematical model of leukemia”, Math. Biosci. Eng., 10:1 (2013), 151–165
5. Bratus A., Todorov Y., Yegorov I., Yurchenko D., “Solution of the Feedback Control Problem in the Mathematical Model of Leukaemia Therapy”, J. Optim. Theory Appl., 159:3 (2013), 590–605
6. I. E. Egorov, “Optimalnoe pozitsionnoe upravlenie v matematicheskoi modeli terapii zlokachestvennoi opukholi s uchetom reaktsii immunnoi sistemy”, Matem. biologiya i bioinform., 9:1 (2014), 257–272
7. Dimitriu G., Lorenzi T., Stefanescu R., “Evolutionary Dynamics of Cancer Cell Populations Under Immune Selection Pressure and Optimal Control of Chemotherapy”, Math. Model. Nat. Phenom., 9:4 (2014), 88–104
8. Yegorov I., Todorov Y., “Synthesis of Optimal Control in a Mathematical Model of Tumour-Immune Dynamics”, Optim. Control Appl. Methods, 36:1 (2015), 93–108
9. Bratus A.S., Kovalenko S.Yu., Fimmel E., “On Viable Therapy Strategy For a Mathematical Spatial Cancer Model Describing the Dynamics of Malignant and Healthy Cells”, Math. Biosci. Eng., 12:1 (2015), 163–183
10. Bratus A., Samokhin I., Yegorov I., Yurchenko D., “Maximization of Viability Time in a Mathematical Model of Cancer Therapy”, Math. Biosci., 294 (2017), 110–119
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