Optimal control of mathematical models for the radiotherapy of gliomas: the scalar case. (English) Zbl 1397.49064
Summary: 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 consider a model of the simplest possible kind, based on the Fischer-Kolmogorov equations, using ideas from V. M. Pérez-García [“Mathematical models for the radiotherapy of gliomas”, Preprint]. The control is the 2\(n\)-tuple \((t_1, \dots , t_n; d_1,\dots, d_n)\), where \(d_i\) is the \(i\)-th applied radiotherapy dose and \(t_i\) is the \(i\)-th administration for \(1 \leq i \leq n\). 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.
MSC:
| 49S05 | Variational principles of physics |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 92C50 | Medical applications (general) |
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