
-
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
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 |
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 [
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.
|
[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.
|
[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.
|
[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. 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.
|
[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-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.
|
[14] |
J. Nocedal and S. J. Wright,
Numerical Optimization, Second Edition, Springer Series in Operations Research, Springer, New York, 2006. |
[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.
|
[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.
|
[17] |
V. M. Pérez-García, Mathematical Models for the Radiotherapy of Gliomas, (preprint), 2012. |
[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.
|
[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-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.
|
[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. 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.
|
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.
|
[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.
|
[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.
|
[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. 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.
|
[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-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.
|
[14] |
J. Nocedal and S. J. Wright,
Numerical Optimization, Second Edition, Springer Series in Operations Research, Springer, New York, 2006. |
[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.
|
[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.
|
[17] |
V. M. Pérez-García, Mathematical Models for the Radiotherapy of Gliomas, (preprint), 2012. |
[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.
|
[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-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.
|
[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. 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.
|










Experiment | IP | SQP | ||
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 | ||
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] |
Hawraa Alsayed, Hussein Fakih, Alain Miranville, Ali Wehbe. Optimal control of an Allen-Cahn model for tumor growth through supply of cytotoxic drugs. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022003 |
[3] |
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 |
[4] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
Rudolf Olach, Vincent Lučanský, Božena Dorociaková. The model of nutrients influence on the tumor growth. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2607-2619. doi: 10.3934/dcdsb.2021150 |
[8] |
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 |
[9] |
Karam Allali, Sanaa Harroudi, Delfim F. M. Torres. Optimal control of an HIV model with a trilinear antibody growth function. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 501-518. doi: 10.3934/dcdss.2021148 |
[10] |
Dongxi Li, Ni Zhang, Ming Yan, Yanya Xing. Survival analysis for tumor growth model with stochastic perturbation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5707-5722. doi: 10.3934/dcdsb.2021041 |
[11] |
Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2729-2749. doi: 10.3934/dcdss.2020457 |
[12] |
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 and Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082 |
[13] |
Ștefana-Lucia Aniţa. Optimal control for stochastic differential equations and related Kolmogorov equations. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022023 |
[14] |
Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373 |
[15] |
Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011 |
[16] |
T.L. Jackson. A mathematical model of prostate tumor growth and androgen-independent relapse. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 187-201. doi: 10.3934/dcdsb.2004.4.187 |
[17] |
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 |
[18] |
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 |
[19] |
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 |
[20] |
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 |
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]