Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors
Электронный научный архив УРФУ
Информация об архиве | Просмотр оригинала| Поле | Значение | |
| Заглавие |
Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors
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| Автор |
Subbotina, N.
Novoselova, N. Krupennikov, E. |
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| Тематика |
CHEMOTHERAPY
INVERSE THERAPY PROBLEMS MALIGNANT TUMOR OPTIMAL THERAPY STRATEGIES THERAPY FUNCTION VIABILITY SET |
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| Описание |
This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending on the time-varying concentration of the drug in the patient’s body. The case of a non-monotonic therapy function with two maxima is studied. It reflects the use of two different therapeutic agents. The state variables of the dynamics are the tumor volume and the amount of the therapeutic agent able to suppress malignant cells (concentration of the drug in the body). The treatment protocol (the rate of administration of the therapeutic agent) is the control in the dynamics. The optimal control problem for this models is considered. It is the problem of the construction of treatment protocols that provide the minimal tumor volume at the end of the treatment. The solution of this problem was obtained by the authors in previous works via the optimal control theory. The form of the considered therapy functions provides a specific structure for the optimal controls. The managerial insights of this structure are discussed. In this paper, the structure of the viability set is described for the model according to the generalized logistic law. It is the set of the initial states of the model for which one can find a treatment protocol that guarantees that the tumor volume remains within the prescribed limits throughout the treatment. The description of the viability set’s structure is based on the optimal control theory and the theory of Hamilton–Jacobi equations. An inverse problem of therapy is also considered, namely the problem of reconstruction of the treatment protocol and identification of the unknown parameter of the intensity of the tumor growth. Reconstruction is carried out by processing information about the observations of the tumor volume dynamics and the measurements of the drug concentration in the body. A solution to this problem is obtained through the use of a method based on the calculus of variations. The results of the numerical simulations are presented herein. © 2023 by the authors.
Ministry of Education and Science of the Russian Federation, Minobrnauka: 075-02-2023-913 The contribution of Nina Subbotina and Evgenii Krupennikov was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2023-913). |
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| Дата |
2024-04-05T16:35:01Z
2024-04-05T16:35:01Z 2023 |
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| Тип |
Article
Journal article (info:eu-repo/semantics/article) |info:eu-repo/semantics/publishedVersion |
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| Идентификатор |
Subbotina, N, Novoselova, N & Krupennikov, E 2023, 'Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors', Mathematics, Том. 11, № 20, 4301. https://doi.org/10.3390/math11204301
Subbotina, N., Novoselova, N., & Krupennikov, E. (2023). Optimal Control Theory and Calculus of Variations in Mathematical Models of Chemotherapy of Malignant Tumors. Mathematics, 11(20), [4301]. https://doi.org/10.3390/math11204301 2227-7390 Final All Open Access, Gold https://www.scopus.com/inward/record.uri?eid=2-s2.0-85175044604&doi=10.3390%2fmath11204301&partnerID=40&md5=f9d5842f6de67e42c83a2ab29dfe1cdb https://www.mdpi.com/2227-7390/11/20/4301/pdf?version=1697440625 http://elar.urfu.ru/handle/10995/130889 10.3390/math11204301 85175044604 001095078400001 |
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| Язык |
en
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| Права |
Open access (info:eu-repo/semantics/openAccess)
cc-by https://creativecommons.org/licenses/by/4.0/ |
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| Формат |
application/pdf
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| Издатель |
Multidisciplinary Digital Publishing Institute (MDPI)
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| Источник |
Mathematics
Mathematics |
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