August  2016, 21(6): 1895-1915. doi: 10.3934/dcdsb.2016028

Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model

1. 

Departamento de Matemáticas, E. T. S. de Ingenieros Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería (IMACI), Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain, Spain, Spain

2. 

Institute of Computational and Applied Mathematics, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Received  May 2015 Revised  March 2016 Published  June 2016

We discuss the optimization of chemotherapy treatment for low-grade gliomas using a mathematical model. We analyze the dynamics of the model and study the stability of solutions. The dynamical model is incorporated into an optimal control problem for which different objective functionals are considered. We establish the existence of optimal controls and give a detailed discussion of the necessary optimality conditions. Since the control variable appears linearly in the control problem, optimal controls are concatenations of bang-bang and singular arcs. We derive a formula of the singular control in terms of state and adjoint variables. Using discretization and optimization methods we compute optimal drug protocols in a number of scenarios. For small treatment periods, the optimal control is bang-bang, whereas for larger treatment periods we obtain both bang-bang and singular arcs. In particular, singular controls illustrate the metronomic chemotherapy.
Citation: Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028
References:
[1]

N. Andre, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?,, Future Oncology, 7 (2011), 385.  doi: 10.2217/fon.11.11.  Google Scholar

[2]

N. Andre, E. Pasquier and M. Kavallaris, Metronomic chemotherapy: New rationale for new directions,, Nature Reviews Clinical Oncology, 7 (2010), 455.   Google Scholar

[3]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems,, Academic Press London; New York [etc.], (1975).   Google Scholar

[4]

J. Belmonte-Beitia, G. F. Calvo and V. M. Pérez-García, Effective particle methods for Fisher-Kolmogorov equations: Theory and applications to brain tumor dynamics,, Communications Nonlinear Science and Numerical Simulation, 19 (2014), 3267.  doi: 10.1016/j.cnsns.2014.02.004.  Google Scholar

[5]

J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed.,, Advances in Design and Control, 19 (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[6]

T. Browder, C. E. Butterfield, B. M. Käling, B. Shi, B. Marshall, M. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer,, Cancer Research, 60 (2000), 1878.   Google Scholar

[7]

B. Chachuat, Nonlinear and Dynamic Optimization: From Theory to Practice,, IC-32: Spring Term 2009, (2009).   Google Scholar

[8]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Mathematiques and applications. Springer, (2003).   Google Scholar

[9]

G. Cairncross, et al, Phase III trial of chemoradiotherapy for anaplastic oligodendroglioma: Long-term results of RTOG 9402,, Journal Clinical Oncology, 31 (2013), 337.   Google Scholar

[10]

L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, T. Neal and K. Yoshida, Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls,, Mathematical Biosciences, 209 (2007), 292.  doi: 10.1016/j.mbs.2006.05.003.  Google Scholar

[11]

M. Dołbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy,, Computational and mathematical methods in medicine, (2013).   Google Scholar

[12]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Mathematical Bioscience, 222 (2009), 13.   Google Scholar

[13]

H. Duffau, Diffuse Low-Grade Gliomas in Adults,, Springer, (2013).  doi: 10.1007/978-1-4471-2213-5.  Google Scholar

[14]

L. A. Fernández and C. Pola, Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint,, Discrete and Continuous Dynamical Systems Series B, 19 (2014), 1563.  doi: 10.3934/dcdsb.2014.19.1563.  Google Scholar

[15]

A. F. Filippov, On certain questions in the theory of optimal control,, SIAM Journal of Control, 1 (1962), 76.   Google Scholar

[16]

W. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Applications of mathematics. Springer, (1975).   Google Scholar

[17]

R. Fourer et. al, AMPL: A Modeling Language for Mathematical Programming,, Thomson, (2003).   Google Scholar

[18]

T. Galochkina, A. Bratus and V. M. Pérez-García, Optimal radiotherapy protocols for low-grade gliomas: Insights from a mathematical model,, Mathematical Bioscience, 267 (2015), 1.  doi: 10.1016/j.mbs.2015.05.006.  Google Scholar

[19]

R. A. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: 10.1038/459508a.  Google Scholar

[20]

J. T. Grier and T. Batchelor, Low-Grade Gliomas in Adults,, The Oncologist, 11 (2006), 681.  doi: 10.1634/theoncologist.11-6-681.  Google Scholar

[21]

M. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley, (1966).   Google Scholar

[22]

A. S. Jakola, K. S. Myrmel, R. Kloster, S. H. Torp, S. Lindal, G. Unsgard and O. Solheim, Comparison of a strategy favoring early surgical resection vs a strategy favoring watchful waiting in low-grade gliomas,, JAMA, 308 (2012), 1881.  doi: 10.1001/jama.2012.12807.  Google Scholar

[23]

B. A. Kamen and R. S. Kerbel, The anti-angiogenic basis of metronomic chemotherapy,, Nature Reviews Cancer, 4 (2004), 423.   Google Scholar

[24]

A. B. Karim, et.al, A randomized trial on dose-response in radiation therapy of low-grade cerebral glioma: European organization for research and treatment of cancer (EORTC) study 22844., International Journal Radiation Oncology-Biology-Physics, 36 (1996), 549.   Google Scholar

[25]

A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM J. Control Optimization, 15 (1977), 256.  doi: 10.1137/0315019.  Google Scholar

[26]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From mtd towards metronomic therapy,, Mathematical Modelling of Natural Phenomena, 9 (2014), 131.  doi: 10.1051/mmnp/20149409.  Google Scholar

[27]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy,, Mathematical Bioscience and Engineering, 8 (2011), 307.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[28]

U. Ledzewicz, J. Munden and H. Schättler, Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models,, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 415.  doi: 10.3934/dcdsb.2009.12.415.  Google Scholar

[29]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 129.   Google Scholar

[30]

U. Ledzewicz, H. Schättler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Mathematical Bioscience and Engineering, 10 (2013), 803.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[31]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Mathematical and Computational Biology. Chapman & Hall/CRC, (2007).   Google Scholar

[32]

E. Mandonnet, et.al, Continuous growth of mean tumor diameter in a subset of grade ii gliomas,, Annals of Neurology, 53 (2003), 524.   Google Scholar

[33]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques,, Journal of Optimization Theory and Applications, 18 (1976), 235.  doi: 10.1007/BF00935706.  Google Scholar

[34]

H. Maurer, C. Büskens, J. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls,, Optimal Control Applications and Methods, 26 (2005), 129.  doi: 10.1002/oca.756.  Google Scholar

[35]

Newton HB (Ed.), Handbook of Brain Tumor Chemotherapy,, Elsevier, (2006).   Google Scholar

[36]

J. D. Olson, E. Riedel and L. M. DeAngelis, Long-term outcome of low-grade oligodendroglioma and mixed glioma,, Neurology, 54 (2000), 1442.  doi: 10.1212/WNL.54.7.1442.  Google Scholar

[37]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, SIAM Advances in Design and Control, (2012).  doi: 10.1137/1.9781611972368.  Google Scholar

[38]

E. Pasquier and N. Andre, For cancer, seek and destroy or live and let live?,, Nature, 460 (2009), 324.   Google Scholar

[39]

E. Pasquier, N. Andre and M. Carre., Metronomics: towards personalized chemotherapy?,, Nature Reviews Clinical Oncology, 11 (2014), 413.   Google Scholar

[40]

E. Pasquier, M. Kavallaris and N. Andre, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews Clinical Oncology, 7 (2010), 455.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[41]

V. M. Pérez-García, M. Bogdanska, A. Martínez-González, J. Belmonte-Beitia, P. Schucht and L. Pérez-Romasanta, Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications,, Mathematical Medicine and Biology, 32 (2015), 307.  doi: 10.1093/imammb/dqu009.  Google Scholar

[42]

V. M. Pérez-García and L. Pérez-Romasanta, Extreme protraction for low grade gliomas: Theoretical proof of concept of a novel therapeutical strategy,, Mathematical Medicine and Biology, (2015).  doi: 10.1093/imammb/dqv017.  Google Scholar

[43]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes,, International Series of Monographs in Pure and Applied Mathematics, (1983).   Google Scholar

[44]

N. Pouratian and D. Schiff, Management of low-grade glioma,, Current Neurology and Neuroscence Reports, 10 (2010), 224.  doi: 10.1007/s11910-010-0105-7.  Google Scholar

[45]

B. Ribba, et al, A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy,, Clinical Cancer Research, 18 (2012), 5071.   Google Scholar

[46]

J. Ruiz and G. J. Lesser, Low-grade gliomas,, Current Treatments Options in Oncology, 10 (2009), 231.  doi: 10.1007/s11864-009-0096-2.  Google Scholar

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Interdisciplinary Applied Mathematics, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[48]

R. Soffietti, et al, Guidelines on management of low-grade gliomas: Report of an EFNS-EANO task force., European Journal of Neurology, 17 (2010), 1124.   Google Scholar

[49]

R. Stupp, et al, Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma,, New England Journal Medicine 352 (2005), 352 (2005).   Google Scholar

[50]

G. W. Swan, Applications of Optimal Control Theory in Biomedicine,, Monographs and textbooks in pure and applied mathematics. M. Dekker, (1984).   Google Scholar

[51]

W. Taal, et al, Dose dense 1 week on/1 week off temozolomide in recurrent glioma: A retrospective study,, Journal of Neurooncology, 108 (2012), 195.   Google Scholar

[52]

M. Van den Bent, et al, Adjuvant procarbazine, lomustine, and vincristine chemotherapy in newly diagnosed anaplastic oligodendroglioma: Long-term follow-up of eortc brain tumor group study 26951., Journal of Clinical Oncology, 31 (2013), 344.   Google Scholar

[53]

A. Viaccoz, A. Lejoubou and F. Ducray, Chemotherapy in low-grade gliomas,, Current Opinion in Oncology, 24 (2012), 694.  doi: 10.1097/CCO.0b013e328357f503.  Google Scholar

[54]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[55]

P. Y. Wen and S. Kesari, Malignant gliomas in adults,, New England Journal of Medicine, 359 (2008), 492.  doi: 10.1056/NEJMra0708126.  Google Scholar

[56]

W. Wick, M. Platten and M. Weller, New (alternative) temozolomide regimens for the treatment of glioma},, Neuro-oncology, 11 (2009), 69.  doi: 10.1215/15228517-2008-078.  Google Scholar

show all references

References:
[1]

N. Andre, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?,, Future Oncology, 7 (2011), 385.  doi: 10.2217/fon.11.11.  Google Scholar

[2]

N. Andre, E. Pasquier and M. Kavallaris, Metronomic chemotherapy: New rationale for new directions,, Nature Reviews Clinical Oncology, 7 (2010), 455.   Google Scholar

[3]

D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems,, Academic Press London; New York [etc.], (1975).   Google Scholar

[4]

J. Belmonte-Beitia, G. F. Calvo and V. M. Pérez-García, Effective particle methods for Fisher-Kolmogorov equations: Theory and applications to brain tumor dynamics,, Communications Nonlinear Science and Numerical Simulation, 19 (2014), 3267.  doi: 10.1016/j.cnsns.2014.02.004.  Google Scholar

[5]

J. T. Betts, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed.,, Advances in Design and Control, 19 (2010).  doi: 10.1137/1.9780898718577.  Google Scholar

[6]

T. Browder, C. E. Butterfield, B. M. Käling, B. Shi, B. Marshall, M. S. O'Reilly and J. Folkman, Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer,, Cancer Research, 60 (2000), 1878.   Google Scholar

[7]

B. Chachuat, Nonlinear and Dynamic Optimization: From Theory to Practice,, IC-32: Spring Term 2009, (2009).   Google Scholar

[8]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory,, Mathematiques and applications. Springer, (2003).   Google Scholar

[9]

G. Cairncross, et al, Phase III trial of chemoradiotherapy for anaplastic oligodendroglioma: Long-term results of RTOG 9402,, Journal Clinical Oncology, 31 (2013), 337.   Google Scholar

[10]

L. G. de Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, T. Neal and K. Yoshida, Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls,, Mathematical Biosciences, 209 (2007), 292.  doi: 10.1016/j.mbs.2006.05.003.  Google Scholar

[11]

M. Dołbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy,, Computational and mathematical methods in medicine, (2013).   Google Scholar

[12]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Mathematical Bioscience, 222 (2009), 13.   Google Scholar

[13]

H. Duffau, Diffuse Low-Grade Gliomas in Adults,, Springer, (2013).  doi: 10.1007/978-1-4471-2213-5.  Google Scholar

[14]

L. A. Fernández and C. Pola, Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint,, Discrete and Continuous Dynamical Systems Series B, 19 (2014), 1563.  doi: 10.3934/dcdsb.2014.19.1563.  Google Scholar

[15]

A. F. Filippov, On certain questions in the theory of optimal control,, SIAM Journal of Control, 1 (1962), 76.   Google Scholar

[16]

W. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, Applications of mathematics. Springer, (1975).   Google Scholar

[17]

R. Fourer et. al, AMPL: A Modeling Language for Mathematical Programming,, Thomson, (2003).   Google Scholar

[18]

T. Galochkina, A. Bratus and V. M. Pérez-García, Optimal radiotherapy protocols for low-grade gliomas: Insights from a mathematical model,, Mathematical Bioscience, 267 (2015), 1.  doi: 10.1016/j.mbs.2015.05.006.  Google Scholar

[19]

R. A. Gatenby, A change of strategy in the war on cancer,, Nature, 459 (2009), 508.  doi: 10.1038/459508a.  Google Scholar

[20]

J. T. Grier and T. Batchelor, Low-Grade Gliomas in Adults,, The Oncologist, 11 (2006), 681.  doi: 10.1634/theoncologist.11-6-681.  Google Scholar

[21]

M. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley, (1966).   Google Scholar

[22]

A. S. Jakola, K. S. Myrmel, R. Kloster, S. H. Torp, S. Lindal, G. Unsgard and O. Solheim, Comparison of a strategy favoring early surgical resection vs a strategy favoring watchful waiting in low-grade gliomas,, JAMA, 308 (2012), 1881.  doi: 10.1001/jama.2012.12807.  Google Scholar

[23]

B. A. Kamen and R. S. Kerbel, The anti-angiogenic basis of metronomic chemotherapy,, Nature Reviews Cancer, 4 (2004), 423.   Google Scholar

[24]

A. B. Karim, et.al, A randomized trial on dose-response in radiation therapy of low-grade cerebral glioma: European organization for research and treatment of cancer (EORTC) study 22844., International Journal Radiation Oncology-Biology-Physics, 36 (1996), 549.   Google Scholar

[25]

A. J. Krener, The high order maximal principle and its application to singular extremals,, SIAM J. Control Optimization, 15 (1977), 256.  doi: 10.1137/0315019.  Google Scholar

[26]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From mtd towards metronomic therapy,, Mathematical Modelling of Natural Phenomena, 9 (2014), 131.  doi: 10.1051/mmnp/20149409.  Google Scholar

[27]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy,, Mathematical Bioscience and Engineering, 8 (2011), 307.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[28]

U. Ledzewicz, J. Munden and H. Schättler, Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models,, Discrete and Continuous Dynamical Systems Series B, 12 (2009), 415.  doi: 10.3934/dcdsb.2009.12.415.  Google Scholar

[29]

U. Ledzewicz and H. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem,, Discrete and Continuous Dynamical Systems Series B, 6 (2006), 129.   Google Scholar

[30]

U. Ledzewicz, H. Schättler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy,, Mathematical Bioscience and Engineering, 10 (2013), 803.  doi: 10.3934/mbe.2013.10.803.  Google Scholar

[31]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Mathematical and Computational Biology. Chapman & Hall/CRC, (2007).   Google Scholar

[32]

E. Mandonnet, et.al, Continuous growth of mean tumor diameter in a subset of grade ii gliomas,, Annals of Neurology, 53 (2003), 524.   Google Scholar

[33]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques,, Journal of Optimization Theory and Applications, 18 (1976), 235.  doi: 10.1007/BF00935706.  Google Scholar

[34]

H. Maurer, C. Büskens, J. H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls,, Optimal Control Applications and Methods, 26 (2005), 129.  doi: 10.1002/oca.756.  Google Scholar

[35]

Newton HB (Ed.), Handbook of Brain Tumor Chemotherapy,, Elsevier, (2006).   Google Scholar

[36]

J. D. Olson, E. Riedel and L. M. DeAngelis, Long-term outcome of low-grade oligodendroglioma and mixed glioma,, Neurology, 54 (2000), 1442.  doi: 10.1212/WNL.54.7.1442.  Google Scholar

[37]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control,, SIAM Advances in Design and Control, (2012).  doi: 10.1137/1.9781611972368.  Google Scholar

[38]

E. Pasquier and N. Andre, For cancer, seek and destroy or live and let live?,, Nature, 460 (2009), 324.   Google Scholar

[39]

E. Pasquier, N. Andre and M. Carre., Metronomics: towards personalized chemotherapy?,, Nature Reviews Clinical Oncology, 11 (2014), 413.   Google Scholar

[40]

E. Pasquier, M. Kavallaris and N. Andre, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews Clinical Oncology, 7 (2010), 455.  doi: 10.1038/nrclinonc.2010.82.  Google Scholar

[41]

V. M. Pérez-García, M. Bogdanska, A. Martínez-González, J. Belmonte-Beitia, P. Schucht and L. Pérez-Romasanta, Delay effects in the response of low grade gliomas to radiotherapy: A mathematical model and its therapeutical implications,, Mathematical Medicine and Biology, 32 (2015), 307.  doi: 10.1093/imammb/dqu009.  Google Scholar

[42]

V. M. Pérez-García and L. Pérez-Romasanta, Extreme protraction for low grade gliomas: Theoretical proof of concept of a novel therapeutical strategy,, Mathematical Medicine and Biology, (2015).  doi: 10.1093/imammb/dqv017.  Google Scholar

[43]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes,, International Series of Monographs in Pure and Applied Mathematics, (1983).   Google Scholar

[44]

N. Pouratian and D. Schiff, Management of low-grade glioma,, Current Neurology and Neuroscence Reports, 10 (2010), 224.  doi: 10.1007/s11910-010-0105-7.  Google Scholar

[45]

B. Ribba, et al, A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy,, Clinical Cancer Research, 18 (2012), 5071.   Google Scholar

[46]

J. Ruiz and G. J. Lesser, Low-grade gliomas,, Current Treatments Options in Oncology, 10 (2009), 231.  doi: 10.1007/s11864-009-0096-2.  Google Scholar

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples,, Interdisciplinary Applied Mathematics, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar

[48]

R. Soffietti, et al, Guidelines on management of low-grade gliomas: Report of an EFNS-EANO task force., European Journal of Neurology, 17 (2010), 1124.   Google Scholar

[49]

R. Stupp, et al, Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma,, New England Journal Medicine 352 (2005), 352 (2005).   Google Scholar

[50]

G. W. Swan, Applications of Optimal Control Theory in Biomedicine,, Monographs and textbooks in pure and applied mathematics. M. Dekker, (1984).   Google Scholar

[51]

W. Taal, et al, Dose dense 1 week on/1 week off temozolomide in recurrent glioma: A retrospective study,, Journal of Neurooncology, 108 (2012), 195.   Google Scholar

[52]

M. Van den Bent, et al, Adjuvant procarbazine, lomustine, and vincristine chemotherapy in newly diagnosed anaplastic oligodendroglioma: Long-term follow-up of eortc brain tumor group study 26951., Journal of Clinical Oncology, 31 (2013), 344.   Google Scholar

[53]

A. Viaccoz, A. Lejoubou and F. Ducray, Chemotherapy in low-grade gliomas,, Current Opinion in Oncology, 24 (2012), 694.  doi: 10.1097/CCO.0b013e328357f503.  Google Scholar

[54]

A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[55]

P. Y. Wen and S. Kesari, Malignant gliomas in adults,, New England Journal of Medicine, 359 (2008), 492.  doi: 10.1056/NEJMra0708126.  Google Scholar

[56]

W. Wick, M. Platten and M. Weller, New (alternative) temozolomide regimens for the treatment of glioma},, Neuro-oncology, 11 (2009), 69.  doi: 10.1215/15228517-2008-078.  Google Scholar

[1]

Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547

[2]

Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443

[3]

Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611

[4]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279

[5]

Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control & Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005

[6]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129

[7]

Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451

[8]

M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511

[9]

Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803

[10]

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

[11]

Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014

[12]

Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066

[13]

M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223

[14]

Marek Bodnar, Monika Joanna Piotrowska, Magdalena Urszula Bogdańska. Mathematical analysis of a generalised model of chemotherapy for low grade gliomas. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2149-2167. doi: 10.3934/dcdsb.2019088

[15]

Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1257-1275. doi: 10.3934/mbe.2015.12.1257

[16]

Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100

[17]

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

[18]

Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266

[19]

Alain Bensoussan, John Liu, Jiguang Yuan. Singular control and impulse control: A common approach. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 27-57. doi: 10.3934/dcdsb.2010.13.27

[20]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (3)

[Back to Top]