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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:
[1]

J. BelmonteG. F. Calvo and V. M. Pérez-García, Effective particle methods for front solutions of the Fischer-Kolmogorov equations, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3267-3283.

[2]

R. H. ByrdJ. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Mathematical Programming, 89 (2000), 149-185.

[3]

R. H. ByrdM. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization, 9 (1999), 877-900.

[4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.
[5]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

[6]

E. Fernández-Cara and G. Camacho-Vázquez, Optimal control of some simplified models of tumour growth, International Journal of Control, 84 (2011), 540-550.

[7]

E. Fernández-Cara and L. Prouvée, Optimal control of mathematical models for the radiotherapy of gliomas: The scalar case Comp. Appl. Math. (2016), https://doi.org/10.1007/s40314-016-0366-0. doi: 10.1007/s40314-016-0366-0.

[8] R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, 1987.
[9]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000.

[10]

T. GalochkinaA. Bratus and V. M. Pérez-García, Optimal radiotherapy protocols for low-grade gliomas: Insights from a mathematical model, Math. Biosci., 267 (2015), 1-9.

[11]

M. D. Gunzburger, Perspectives in Flow Control and Optimization Society for Industrial and Applied Mathematics, Philadelphia, 2003.

[12]

J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag, Grundelhren Der Mathematishen Wissenschaften Series, vol. 170,1971.

[13]

A. Martínez-GonzálezG. F. CalvoL. Pérez-Romansanta and V. M. Pérez-García, Hypoxic Cell Waves around Necrotic Cores in Gliobastoma: A Biomathematical Model and its Therapeutic implications, Bull Math Biol., 74 (2012), 2875-2896.

[14]

J. Nocedal and S. J. Wright, Numerical Optimization, Second Edition, Springer Series in Operations Research, Springer, New York, 2006.

[15]

J. PalludL. TaillanderL. CapelleD. FontaineM. PeyreF. DucrayH. Duffau and E. Mandonnet, Quantitative morphological mri follow-up of low-grade glioma: A plead for systematic measurement of growth rates, Neurosurgery, 71 (2012), 729-740.

[16]

J. PalludJ. F. LlitjosF. DhermainP. VarletE. DezamisB. DevauxR. Souillard-ScemamaN. SanaiM. KoziakP. PageM. SchliengerC. Daumas-DuportJ. F. MederC. Oppenheim and F. X. Roux, Dynamic imaging response following radiation therapy predicts long-term outcomes for diffuse low-grade gliomas, Neuro-Oncology, 14 (2012), 496-505.

[17]

V. M. Pérez-García, Mathematical Models for the Radiotherapy of Gliomas, (preprint), 2012.

[18]

V. M. Pérez-GarcíaG. F. CalvoJ. Belmonte-BeitiaD. Diego and L. Pérez-Romansanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921.

[19]

V. M. Pérez-García and A. Martínez-González, Hypoxic ghost waves accelerate the progression of high-grade gliomas, J. Theor. Biol., (to appear. ), 2012.

[20]

V. M. Pérez-GarcíaM. BogdanskaA. Martínez-GonzálezJ. Belmonte-BeitiaPh. Schucht and L. A. Pérez-Romasanta, Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications, Math. Med. Biol., 32 (2015), 307-329.

[21]

L. A. Pérez-Romansanta, J. Belmonte-Beitia, A. Martínez-González, G. F. Calvo and V. M. Pérez-García, Mathematical model predicts response to radiotherapy of grade Ⅱ gliomas, Reports of Practical Oncology and Radiotherapy, 18 (2013), S63.

[22]

L. Prouvée, Optimal Control of Mathematical Models for the Radiotherapy of Gliomas PhD Thesis, 2015.

[23]

R. A. WaltzJ. L. MoralesJ. Nocedal and D. Orban, An interior algorithm for nonlinear optimization that combines line search and trust region steps, Mathematical Programming, 107 (2006), 391-408.

show all references

References:
[1]

J. BelmonteG. F. Calvo and V. M. Pérez-García, Effective particle methods for front solutions of the Fischer-Kolmogorov equations, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3267-3283.

[2]

R. H. ByrdJ. C. Gilbert and J. Nocedal, A trust region method based on interior point techniques for nonlinear programming, Mathematical Programming, 89 (2000), 149-185.

[3]

R. H. ByrdM. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization, 9 (1999), 877-900.

[4] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1990.
[5]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.

[6]

E. Fernández-Cara and G. Camacho-Vázquez, Optimal control of some simplified models of tumour growth, International Journal of Control, 84 (2011), 540-550.

[7]

E. Fernández-Cara and L. Prouvée, Optimal control of mathematical models for the radiotherapy of gliomas: The scalar case Comp. Appl. Math. (2016), https://doi.org/10.1007/s40314-016-0366-0. doi: 10.1007/s40314-016-0366-0.

[8] R. Fletcher, Practical Methods of Optimization, John Wiley and Sons, 1987.
[9]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications Translations of Mathematical Monographs, 187. American Mathematical Society, Providence, RI, 2000.

[10]

T. GalochkinaA. Bratus and V. M. Pérez-García, Optimal radiotherapy protocols for low-grade gliomas: Insights from a mathematical model, Math. Biosci., 267 (2015), 1-9.

[11]

M. D. Gunzburger, Perspectives in Flow Control and Optimization Society for Industrial and Applied Mathematics, Philadelphia, 2003.

[12]

J. -L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag, Grundelhren Der Mathematishen Wissenschaften Series, vol. 170,1971.

[13]

A. Martínez-GonzálezG. F. CalvoL. Pérez-Romansanta and V. M. Pérez-García, Hypoxic Cell Waves around Necrotic Cores in Gliobastoma: A Biomathematical Model and its Therapeutic implications, Bull Math Biol., 74 (2012), 2875-2896.

[14]

J. Nocedal and S. J. Wright, Numerical Optimization, Second Edition, Springer Series in Operations Research, Springer, New York, 2006.

[15]

J. PalludL. TaillanderL. CapelleD. FontaineM. PeyreF. DucrayH. Duffau and E. Mandonnet, Quantitative morphological mri follow-up of low-grade glioma: A plead for systematic measurement of growth rates, Neurosurgery, 71 (2012), 729-740.

[16]

J. PalludJ. F. LlitjosF. DhermainP. VarletE. DezamisB. DevauxR. Souillard-ScemamaN. SanaiM. KoziakP. PageM. SchliengerC. Daumas-DuportJ. F. MederC. Oppenheim and F. X. Roux, Dynamic imaging response following radiation therapy predicts long-term outcomes for diffuse low-grade gliomas, Neuro-Oncology, 14 (2012), 496-505.

[17]

V. M. Pérez-García, Mathematical Models for the Radiotherapy of Gliomas, (preprint), 2012.

[18]

V. M. Pérez-GarcíaG. F. CalvoJ. Belmonte-BeitiaD. Diego and L. Pérez-Romansanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921.

[19]

V. M. Pérez-García and A. Martínez-González, Hypoxic ghost waves accelerate the progression of high-grade gliomas, J. Theor. Biol., (to appear. ), 2012.

[20]

V. M. Pérez-GarcíaM. BogdanskaA. Martínez-GonzálezJ. Belmonte-BeitiaPh. Schucht and L. A. Pérez-Romasanta, Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications, Math. Med. Biol., 32 (2015), 307-329.

[21]

L. A. Pérez-Romansanta, J. Belmonte-Beitia, A. Martínez-González, G. F. Calvo and V. M. Pérez-García, Mathematical model predicts response to radiotherapy of grade Ⅱ gliomas, Reports of Practical Oncology and Radiotherapy, 18 (2013), S63.

[22]

L. Prouvée, Optimal Control of Mathematical Models for the Radiotherapy of Gliomas PhD Thesis, 2015.

[23]

R. A. WaltzJ. L. MoralesJ. Nocedal and D. Orban, An interior algorithm for nonlinear optimization that combines line search and trust region steps, Mathematical Programming, 107 (2006), 391-408.

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
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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