# American Institute of Mathematical Sciences

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March  2018, 8(1): 117-133. doi: 10.3934/mcrf.2018005

## Optimal control of a two-equation model of radiotherapy

 1 Dpto. EDAN e IMUS, Universidad de Sevilla, Spain 2 Instituto de Matemática e Estatística, Universidade Federal Fluminense, Brazil 3 Instituto de Matemática e Estatística, Universidade do Estado do Rio de Janeiro, Brazil

* Corresponding author

Received  April 2017 Revised  August 2017 Published  January 2018

Fund Project: 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).

Citation: Enrique Fernández-Cara, Juan Límaco, Laurent Prouvée. Optimal control of a two-equation model of radiotherapy. Mathematical Control & Related Fields, 2018, 8 (1) : 117-133. doi: 10.3934/mcrf.2018005
##### References:

show all references

##### References:
The optimal 30 doses -IP algorithm
Evolution of the tumor size -30 Doses
The density of tumor cells (3D global view) -30 Doses
The optimal 30 doses -SQP algorithm
The optimal 40 doses -IP algorithm
The density of tumor cells (3D global view) -40 Doses
The optimal 40 doses -SQP algorithm
The optimal 60 doses -IP algorithm
The density of tumor cells (3D global view) -60 Doses
The optimal 60 doses -SQP algorithm
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
 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
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