Abstract
This paper deals with the optimal control of a mathematical model of glioma progression incorporating the basic facts of the evolution of primary brain tumors. We will consider a model of the simplest possible kind, based on the Fischer-Kolmogorov equations, using ideas from Pérez-García (Mathematical models for the radiotherapy of gliomas (preprint), 2016). The control is the 2n-tuple \((t_1, \ldots , t_n; d_1, \ldots , d_n)\), where \(d_i\) is the i-th applied radiotherapy dose and \(t_i\) is the i-th administration for \(1 \le i \le n\). We search for controls that maximize the survival time, that is, the time at which the tumor mass reaches a critical value \(M_{*}\), over the class of admissible radiation times and doses. We present theoretical and numerical results that justify the relevance of the model and the existence of potential medical applications.
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Acknowledgments
The authors are indebted to the anonymous referees for their very important comments and suggestions. They have led to substantial improvements of the paper. In particular, we thank Referee 2 for several crucial comments concerning in particular the relevance and optimality in practice of the constant-dose strategy.
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Communicated by Dr. Luz de Teresa.
Partially supported by Grant MTM2013-41286-P, MINECO, Spain. Partially supported by CAPES Foundation, BEX 7446/13-6, Ministry of Education of Brazil, Brasília DF 70040-020, Brazil.
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Fernández-Cara, E., Prouvée, L. Optimal control of mathematical models for the radiotherapy of gliomas: the scalar case. Comp. Appl. Math. 37, 745–762 (2018). https://doi.org/10.1007/s40314-016-0366-0
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DOI: https://doi.org/10.1007/s40314-016-0366-0