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 and Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028
References:
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N. Andre, E. Pasquier and M. Kavallaris, Metronomic chemotherapy: New rationale for new directions, Nature Reviews Clinical Oncology, 7 (2010), 455-465.

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

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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), 3267-3283. doi: 10.1016/j.cnsns.2014.02.004.

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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), 1878-1886.

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B. Chachuat, Nonlinear and Dynamic Optimization: From Theory to Practice, IC-32: Spring Term 2009, EPFL.

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B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematiques and applications. Springer, Paris, New York, 2003.

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G. Cairncross, et al, Phase III trial of chemoradiotherapy for anaplastic oligodendroglioma: Long-term results of RTOG 9402, Journal Clinical Oncology, 31 (2013), 337-343.

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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), 292-315. doi: 10.1016/j.mbs.2006.05.003.

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M. Dołbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and mathematical methods in medicine, (2013), ID 567213, 11pp.

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A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Bioscience, 222 (2009), 13-26.

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

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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), 1563-1588. doi: 10.3934/dcdsb.2014.19.1563.

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R. Fourer et. al, AMPL: A Modeling Language for Mathematical Programming, Thomson, Second Edition, 2003.

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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), 1-9. doi: 10.1016/j.mbs.2015.05.006.

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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), 1881-1888. doi: 10.1001/jama.2012.12807.

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B. A. Kamen and R. S. Kerbel, The anti-angiogenic basis of metronomic chemotherapy, Nature Reviews Cancer, 4 (2004), 423-436.

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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), 549-556.

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A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[26]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From mtd towards metronomic therapy, Mathematical Modelling of Natural Phenomena, 9 (2014), 131-152. doi: 10.1051/mmnp/20149409.

[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), 307-323. doi: 10.3934/mbe.2011.8.307.

[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), 415-438. doi: 10.3934/dcdsb.2009.12.415.

[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), 129-150.

[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), 803-819. doi: 10.3934/mbe.2013.10.803.

[31]

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

[32]

E. Mandonnet, et.al, Continuous growth of mean tumor diameter in a subset of grade ii gliomas, Annals of Neurology, 53 (2003), 524-528.

[33]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, Journal of Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.

[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), 129-156. doi: 10.1002/oca.756.

[35]

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

[36]

J. D. Olson, E. Riedel and L. M. DeAngelis, Long-term outcome of low-grade oligodendroglioma and mixed glioma, Neurology, 54 (2000), 1442-1448. doi: 10.1212/WNL.54.7.1442.

[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, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.

[38]

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

[39]

E. Pasquier, N. Andre and M. Carre., Metronomics: towards personalized chemotherapy?, Nature Reviews Clinical Oncology, 11 (2014), 413-431.

[40]

E. Pasquier, M. Kavallaris and N. Andre, Metronomic chemotherapy: new rationale for new directions, Nature Reviews Clinical Oncology, 7 (2010), 455-465. doi: 10.1038/nrclinonc.2010.82.

[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), 307-329. doi: 10.1093/imammb/dqu009.

[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.

[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, Fourth edition. "Nauka'', Moscow, 1983.

[44]

N. Pouratian and D. Schiff, Management of low-grade glioma, Current Neurology and Neuroscence Reports, 10 (2010), 224-231. doi: 10.1007/s11910-010-0105-7.

[45]

B. Ribba, et al, A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy, Clinical Cancer Research, 18 (2012), 5071-5080.

[46]

J. Ruiz and G. J. Lesser, Low-grade gliomas, Current Treatments Options in Oncology, 10 (2009), 231-242. doi: 10.1007/s11864-009-0096-2.

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Interdisciplinary Applied Mathematics, Volume 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[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), 1124-1133.

[49]

R. Stupp, et al, Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma, New England Journal Medicine 352 (2005), p987.

[50]

G. W. Swan, Applications of Optimal Control Theory in Biomedicine, Monographs and textbooks in pure and applied mathematics. M. Dekker, New York, 1984.

[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), 195-200.

[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), 344-350.

[53]

A. Viaccoz, A. Lejoubou and F. Ducray, Chemotherapy in low-grade gliomas, Current Opinion in Oncology, 24 (2012), 694-701. doi: 10.1097/CCO.0b013e328357f503.

[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), 25-57. doi: 10.1007/s10107-004-0559-y.

[55]

P. Y. Wen and S. Kesari, Malignant gliomas in adults, New England Journal of Medicine, 359 (2008), 492-507. doi: 10.1056/NEJMra0708126.

[56]

W. Wick, M. Platten and M. Weller, New (alternative) temozolomide regimens for the treatment of glioma}, Neuro-oncology, 11 (2009), 69-79. doi: 10.1215/15228517-2008-078.

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), 385-394. doi: 10.2217/fon.11.11.

[2]

N. Andre, E. Pasquier and M. Kavallaris, Metronomic chemotherapy: New rationale for new directions, Nature Reviews Clinical Oncology, 7 (2010), 455-465.

[3]

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

[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), 3267-3283. doi: 10.1016/j.cnsns.2014.02.004.

[5]

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

[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), 1878-1886.

[7]

B. Chachuat, Nonlinear and Dynamic Optimization: From Theory to Practice, IC-32: Spring Term 2009, EPFL.

[8]

B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematiques and applications. Springer, Paris, New York, 2003.

[9]

G. Cairncross, et al, Phase III trial of chemoradiotherapy for anaplastic oligodendroglioma: Long-term results of RTOG 9402, Journal Clinical Oncology, 31 (2013), 337-343.

[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), 292-315. doi: 10.1016/j.mbs.2006.05.003.

[11]

M. Dołbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and mathematical methods in medicine, (2013), ID 567213, 11pp.

[12]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Bioscience, 222 (2009), 13-26.

[13]

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

[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), 1563-1588. doi: 10.3934/dcdsb.2014.19.1563.

[15]

A. F. Filippov, On certain questions in the theory of optimal control, SIAM Journal of Control, 1 (1962), 76-84.

[16]

W. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of mathematics. Springer, New York, Berlin, Heidelberg, 1975.

[17]

R. Fourer et. al, AMPL: A Modeling Language for Mathematical Programming, Thomson, Second Edition, 2003.

[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), 1-9. doi: 10.1016/j.mbs.2015.05.006.

[19]

R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509. doi: 10.1038/459508a.

[20]

J. T. Grier and T. Batchelor, Low-Grade Gliomas in Adults, The Oncologist, 11 (2006), 681-693. doi: 10.1634/theoncologist.11-6-681.

[21]

M. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.

[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), 1881-1888. doi: 10.1001/jama.2012.12807.

[23]

B. A. Kamen and R. S. Kerbel, The anti-angiogenic basis of metronomic chemotherapy, Nature Reviews Cancer, 4 (2004), 423-436.

[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), 549-556.

[25]

A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optimization, 15 (1977), 256-293. doi: 10.1137/0315019.

[26]

U. Ledzewicz and H. Schättler, A review of optimal chemotherapy protocols: From mtd towards metronomic therapy, Mathematical Modelling of Natural Phenomena, 9 (2014), 131-152. doi: 10.1051/mmnp/20149409.

[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), 307-323. doi: 10.3934/mbe.2011.8.307.

[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), 415-438. doi: 10.3934/dcdsb.2009.12.415.

[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), 129-150.

[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), 803-819. doi: 10.3934/mbe.2013.10.803.

[31]

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

[32]

E. Mandonnet, et.al, Continuous growth of mean tumor diameter in a subset of grade ii gliomas, Annals of Neurology, 53 (2003), 524-528.

[33]

H. Maurer, Numerical solution of singular control problems using multiple shooting techniques, Journal of Optimization Theory and Applications, 18 (1976), 235-257. doi: 10.1007/BF00935706.

[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), 129-156. doi: 10.1002/oca.756.

[35]

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

[36]

J. D. Olson, E. Riedel and L. M. DeAngelis, Long-term outcome of low-grade oligodendroglioma and mixed glioma, Neurology, 54 (2000), 1442-1448. doi: 10.1212/WNL.54.7.1442.

[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, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.

[38]

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

[39]

E. Pasquier, N. Andre and M. Carre., Metronomics: towards personalized chemotherapy?, Nature Reviews Clinical Oncology, 11 (2014), 413-431.

[40]

E. Pasquier, M. Kavallaris and N. Andre, Metronomic chemotherapy: new rationale for new directions, Nature Reviews Clinical Oncology, 7 (2010), 455-465. doi: 10.1038/nrclinonc.2010.82.

[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), 307-329. doi: 10.1093/imammb/dqu009.

[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.

[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, Fourth edition. "Nauka'', Moscow, 1983.

[44]

N. Pouratian and D. Schiff, Management of low-grade glioma, Current Neurology and Neuroscence Reports, 10 (2010), 224-231. doi: 10.1007/s11910-010-0105-7.

[45]

B. Ribba, et al, A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy, Clinical Cancer Research, 18 (2012), 5071-5080.

[46]

J. Ruiz and G. J. Lesser, Low-grade gliomas, Current Treatments Options in Oncology, 10 (2009), 231-242. doi: 10.1007/s11864-009-0096-2.

[47]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Interdisciplinary Applied Mathematics, Volume 38, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2.

[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), 1124-1133.

[49]

R. Stupp, et al, Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma, New England Journal Medicine 352 (2005), p987.

[50]

G. W. Swan, Applications of Optimal Control Theory in Biomedicine, Monographs and textbooks in pure and applied mathematics. M. Dekker, New York, 1984.

[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), 195-200.

[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), 344-350.

[53]

A. Viaccoz, A. Lejoubou and F. Ducray, Chemotherapy in low-grade gliomas, Current Opinion in Oncology, 24 (2012), 694-701. doi: 10.1097/CCO.0b013e328357f503.

[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), 25-57. doi: 10.1007/s10107-004-0559-y.

[55]

P. Y. Wen and S. Kesari, Malignant gliomas in adults, New England Journal of Medicine, 359 (2008), 492-507. doi: 10.1056/NEJMra0708126.

[56]

W. Wick, M. Platten and M. Weller, New (alternative) temozolomide regimens for the treatment of glioma}, Neuro-oncology, 11 (2009), 69-79. doi: 10.1215/15228517-2008-078.

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