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

Optimal control of a two-equation model of radiotherapy

Enrique Fernández-Cara 1,, , Juan Límaco 2, and  Laurent Prouvée 3,

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).

Keywords: Fischer-Kolmogorov model, optimal control, tumor growth, radiotherapy.
Mathematics Subject Classification: Primary: 49J20, 35K20; Secondary: 92C50.
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. Belmonte, G. 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.   Google Scholar

[2]

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

[3]

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

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

[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.   Google Scholar

[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.  Google Scholar

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

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

[10]

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

[11]

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

[12]

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

[13]

A. Martínez-González, G. F. Calvo, L. 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.   Google Scholar

[14]

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

[15]

J. Pallud, L. Taillander, L. Capelle, D. Fontaine, M. Peyre, F. Ducray, H. 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.   Google Scholar

[16]

J. Pallud, J. F. Llitjos, F. Dhermain, P. Varlet, E. Dezamis, B. Devaux, R. Souillard-Scemama, N. Sanai, M. Koziak, P. Page, M. Schlienger, C. Daumas-Duport, J. F. Meder, C. 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.   Google Scholar

[17]

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

[18]

V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego and L. Pérez-Romansanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921.   Google Scholar

[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. Google Scholar

[20]

V. M. Pérez-García, M. Bogdanska, A. Martínez-González, J. Belmonte-Beitia, Ph. 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.   Google Scholar

[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. Google Scholar

[22]

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

[23]

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

show all references

References:
[1]

J. Belmonte, G. 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.   Google Scholar

[2]

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

[3]

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

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

[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.   Google Scholar

[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.  Google Scholar

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

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

[10]

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

[11]

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

[12]

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

[13]

A. Martínez-González, G. F. Calvo, L. 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.   Google Scholar

[14]

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

[15]

J. Pallud, L. Taillander, L. Capelle, D. Fontaine, M. Peyre, F. Ducray, H. 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.   Google Scholar

[16]

J. Pallud, J. F. Llitjos, F. Dhermain, P. Varlet, E. Dezamis, B. Devaux, R. Souillard-Scemama, N. Sanai, M. Koziak, P. Page, M. Schlienger, C. Daumas-Duport, J. F. Meder, C. 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.   Google Scholar

[17]

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

[18]

V. M. Pérez-García, G. F. Calvo, J. Belmonte-Beitia, D. Diego and L. Pérez-Romansanta, Bright solitary waves in malignant gliomas, Phys. Rev. E, 84 (2011), 021921.   Google Scholar

[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. Google Scholar

[20]

V. M. Pérez-García, M. Bogdanska, A. Martínez-González, J. Belmonte-Beitia, Ph. 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.   Google Scholar

[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. Google Scholar

[22]

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

[23]

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

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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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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