Optimal control of a two-equation model of radiotherapy

Enrique Fernández-Cara; Juan Límaco; Laurent Prouvée

doi:10.3934/mcrf.2018005

Article Contents

2018, Volume 8, Issue 1: 117-133. Doi: 10.3934/mcrf.2018005

This issue Previous Article Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes Next Article On the switching behavior of sparse optimal controls for the one-dimensional heat equation

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
* Corresponding author

Received: April 2017

Revised: August 2017

Published: January 2018

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.

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Abstract

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).

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

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Figure 1. The optimal 30 doses -IP algorithm

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Figure 2. Evolution of the tumor size -30 Doses

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

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Figure 4. The optimal 30 doses -SQP algorithm

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Figure 5. The optimal 40 doses -IP algorithm

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

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Figure 7. The optimal 40 doses -SQP algorithm

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Figure 8. The optimal 60 doses -IP algorithm

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

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Figure 10. The optimal 60 doses -SQP algorithm

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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

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References

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