American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the switching behavior of sparse optimal controls for the one-dimensional heat equation
  • MCRF Home
  • This Issue
  • Next Article
    Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes
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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Evolution of the tumor size -30 Doses
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 4.  The optimal 30 doses -SQP algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The optimal 40 doses -IP algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 7.  The optimal 40 doses -SQP algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  The optimal 60 doses -IP algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 10.  The optimal 60 doses -SQP algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Krzysztof Fujarewicz, Krzysztof Łakomiec. Adjoint sensitivity analysis of a tumor growth model and its application to spatiotemporal radiotherapy optimization. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1131-1142. doi: 10.3934/mbe.2016034
[2]

Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Mesoscopic model for tumor growth. Mathematical Biosciences & Engineering, 2007, 4 (4) : 687-698. doi: 10.3934/mbe.2007.4.687
[3]

Ellina Grigorieva, Evgenii Khailov. Optimal control of a nonlinear model of economic growth. Conference Publications, 2007, 2007 (Special) : 456-466. doi: 10.3934/proc.2007.2007.456
[4]

Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040
[5]

Didier Bresch, Thierry Colin, Emmanuel Grenier, Benjamin Ribba, Olivier Saut. A viscoelastic model for avascular tumor growth. Conference Publications, 2009, 2009 (Special) : 101-108. doi: 10.3934/proc.2009.2009.101
[6]

Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787
[7]

Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082
[8]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[9]

T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 187-201. doi: 10.3934/dcdsb.2004.4.187
[10]

J. Ignacio Tello. On a mathematical model of tumor growth based on cancer stem cells. Mathematical Biosciences & Engineering, 2013, 10 (1) : 263-278. doi: 10.3934/mbe.2013.10.263
[11]

Hyun Geun Lee, Yangjin Kim, Junseok Kim. Mathematical model and its fast numerical method for the tumor growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1173-1187. doi: 10.3934/mbe.2015.12.1173
[12]

Mohammad A. Tabatabai, Wayne M. Eby, Karan P. Singh, Sejong Bae. T model of growth and its application in systems of tumor-immune dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 925-938. doi: 10.3934/mbe.2013.10.925
[13]

Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005
[14]

Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971
[15]

Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
[16]

Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, 2020, 10 (2) : 305-331. doi: 10.3934/mcrf.2019040
[17]

Shaoyong Lai, Yulan Zhou. A stochastic optimal growth model with a depreciation factor. Journal of Industrial & Management Optimization, 2010, 6 (2) : 283-297. doi: 10.3934/jimo.2010.6.283
[18]

Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
[19]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[20]

Peng Zhong, Suzanne Lenhart. Optimal control of integrodifference equations with growth-harvesting-dispersal order. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2281-2298. doi: 10.3934/dcdsb.2012.17.2281

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (118)
  • HTML views (339)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences