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Article Contents

# Optimal control of a two-equation model of radiotherapy

• * Corresponding author
The first author was partially supported by grant MTM2016-76990-P, DGI-MINECO, Spain.
The third author was partially supported by CAPES Foundation, BEX 7446/13-6, Ministry of Education of Brazil.
• This paper deals with the optimal control of a mathematical model for the evolution of a low-grade glioma (LGG). We will consider a model of the Fischer-Kolmogorov kind for two compartments of tumor cells, using ideas from Galochkina, Bratus and Pérez-García [10] and Pérez-García [17]. The controls are of the form $(t_1, \dots, t_n; d_1, \dots, d_n)$, where $t_i$ is the $i$-th administration time and $d_i$ is the $i$-th applied radiotherapy dose. In the optimal control problem, we try to find controls that maximize, in an admissible class, the first time at which the tumor mass reaches a critical value $M_{*}$. We present an existence result and, also, some numerical experiments (in the previous paper [7], we have considered and solved a very similar control problem where tumoral cells of only one kind appear).

Mathematics Subject Classification: Primary: 49J20, 35K20; Secondary: 92C50.

 Citation:

• Figure 1.  The optimal 30 doses -IP algorithm

Figure 2.  Evolution of the tumor size -30 Doses

Figure 3.  The density of tumor cells (3D global view) -30 Doses

Figure 4.  The optimal 30 doses -SQP algorithm

Figure 5.  The optimal 40 doses -IP algorithm

Figure 6.  The density of tumor cells (3D global view) -40 Doses

Figure 7.  The optimal 40 doses -SQP algorithm

Figure 8.  The optimal 60 doses -IP algorithm

Figure 9.  The density of tumor cells (3D global view) -60 Doses

Figure 10.  The optimal 60 doses -SQP algorithm

Table 1.  The survival times corresponding to IP, SQP and $d_j = d_{\rm st}$

 Experiment IP SQP $d_{\rm st}$ $d_{\rm max}$ 30 doses 214 days 213 days 196 days 212 days 40 doses 254 days 251 days 238 days 250 days 60 doses 358 days 353 days 321 days 350 days

Figures(10)

Tables(1)