American Institute of Mathematical Sciences

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May  2019, 24(5): 2017-2038. doi: 10.3934/dcdsb.2019082

Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy

Arturo Alvarez-Arenas 1, , Konstantin E. Starkov 2, , Gabriel F. Calvo 1, and  Juan Belmonte-Beitia 1,,

1. 

Department of Mathematics & MȏLAB-Mathematical Oncology Laboratory, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
2. 

Instituto Politecnico Nacional-CITEDI, Av. de IPN 1310, Nueva Tijuana, Tijuana 22435, B.C., Mexico

* Corresponding author: Juan Belmonte-Beitia

Received  January 2018 Revised  January 2019 Published  March 2019

In this paper, a non-trivial generalization of a mathematical model put forward in [35] to account for the development of resistance by tumors to chemotherapy is presented. A study of the existence and local stability of the solutions, as well as the ultimate dynamics of the model, is addressed. An analysis of different chemotherapeutical protocols using discretization and optimization methods is carried out. A number of objective functionals are considered and the necessary optimality conditions are provided. Since the control variable appears linearly in the associated problem, optimal controls are concatenations of bang-bang and singular arcs. A formula of the singular control in terms of state and adjoint variables is derived analytically. Bang-bang and singular controls from the numerical simulations are obtained where, in particular, singular controls illustrate the metronomic chemotherapy.

Keywords: Optimal control, ultimate dynamics, chemotherapy, singular control, bang-bang control.
Mathematics Subject Classification: Primary: 49K15, 49M37; Secondary: 37C10, 92B05.
Citation: 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
References:
[1]

A. Alvarez-Arenas, A. Podolski-Renic, J. Belmonte-Beitia, M. Pesic and G. F. Calvo, Interplay of Darwinian selection, Lamarckian induction and microvesicle transfer on drug resistance in cancer, Scientific Reports, (submitted).Google Scholar

[2]

P. Bajger, M. Bodzioch and U. Foryś, Overcoming acquired chemotherapy resistance: Insights from mathematical modelling, Communications in Nonlinear Science and Numerical Simulation, (submitted).Google Scholar

[3]

M. Becker and D. Levy, Modeling the transfer of drug resistance in solid tumors, Bulletin of Mathematical Biology, 79 (2017), 2394-2412. doi: 10.1007/s11538-017-0334-x. Google Scholar

[4]

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

[5]

J. Belmonte-Beitia, T. E. Wooley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, Journal of Theoretical Biology, 334 (2013), 1-12. doi: 10.1016/j.jtbi.2013.05.033. Google Scholar

[6]

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. Google Scholar

[7]

M. Bodnar and U. Forys, Two models of drug resistance for low grade gliomas: comparison of the models dynamics, Proceedings of the XXIII National Conference Applications of Mathematics in Biology and Medicine, Politechnika lska, Uniwersytet Warszawski, Gliwice 2017, 37–42.Google Scholar

[8]

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

[9]

M. Calzada, E. Fernández-Cara and M. Marín, Optimal control oriented to therapy for a free-boundary tumor growth model, Journal of Theoretical Biology, 325 (2013), 1-11. doi: 10.1016/j.jtbi.2013.02.004. Google Scholar

[10]

G. Camacho and E. Fernández-Cara, Optimal control of some simplified models of tumour growth, International Journal of Control, 84 (2011), 540-550. doi: 10.1080/00207179.2011.562547. Google Scholar

[11]

C. Carrère, Optimization of an in vitro chemotherapy to avoid resistant tumours, Journal of Theoretical Biology, 413 (2017), 24-33. doi: 10.1016/j.jtbi.2016.11.009. Google Scholar

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R. H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L. Neves de Almeida, A. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation, Cancer Research, 15 (2015), 930-939. doi: 10.1158/0008-5472.CAN-14-2103. Google Scholar

[13]

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. Google Scholar

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M. Doƚbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and Mathematical Methods in Medicine, (2013), ID 567213, 11pp. doi: 10.1155/2013/567213. Google Scholar

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A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

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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. Google Scholar

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R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptative therapy, Cancer Research, 69 (2009), 4894-4903. Google Scholar

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M. Gerlinger and C. Swanton, How Darwinian models inform therapeutic failure initiated by clonal heterogeneity in cancer medicine, British Journal of Cancer, 103 (2010), 1139-1143. doi: 10.1038/sj.bjc.6605912. Google Scholar

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D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Mathematical Biosciences and Engineering, 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. Google Scholar

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

[22]

A. P. Krishchenko and K. E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Communication in Nonlinear Sciences and Numerical Simulations, 33 (2016), 174-183. doi: 10.1016/j.cnsns.2015.10.001. Google Scholar

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J. P. LaSalle, An invariance principle in the theory of stability, Center for Dynamical Systems, 1966.Google Scholar

[24]

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. Google Scholar

[25]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307. Google Scholar

[26]

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. Google Scholar

[27]

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. doi: 10.3934/dcdsb.2006.6.129. Google Scholar

[28]

U. Ledzewicz, H. Schättler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering, 10 (2013), 803-819. doi: 10.3934/mbe.2013.10.803. Google Scholar

[29]

U. Ledzewicz, S. Wang, H. Schättler, N. André, M. A. Heng and E. Pasquier, On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach, Mathematical Biosciences and Engineering, 14 (2017), 217-235. doi: 10.3934/mbe.2017014. Google Scholar

[30]

Q. Li, A. Wennborg, E. Aurell, E. Dekel, J. Zou, Y. Xu, S. Huang and I. Ernberg, Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape, Proceedings of the National Academy of Sciences, 113 (2016), 2672-2677. doi: 10.1073/pnas.1519210113. Google Scholar

[31]

D. L. Lukes, Differential Equations: Classical to Controlled, Academic Press New York, 1982. Google Scholar

[32]

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. Google Scholar

[33]

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. Google Scholar

[34]

G. Morgan, R. Ward and M. Barton, The contribution of cytotoxic chemotherapy to 5-year survival in adult malignancies, Clinical Oncology, 16 (2004), 549-560. doi: 10.1016/j.clon.2004.06.007. Google Scholar

[35]

W. Ollier et. al., Analysis of temozolomide resistance in low-grade gliomas using a mechanistic mathematical model, Fundamental & Clinical Pharmacology, 31 (2017), 347-358.Google Scholar

[36]

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. Google Scholar

[37]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1991. doi: 10.1007/978-1-4684-0392-3. Google Scholar

[38]

A. O. Pisco, A. Brock, J. Zhou, A. Moor, M. Mojtahedi, D. Jackson and S. Huang, Non-Darwinian dynamics in therapy-induced cancer drug resistance, Nature Communications, 4 (2013), 2467. doi: 10.1038/ncomms3467. Google Scholar

[39]

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), Interscience Publishers, 1962.Google Scholar

[40]

C. Pouchol, J. Clairambault, A. Lorz and E. Trelat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy, Journal de Mathematiques Pures et Appliques, 116 (2017), 268-308. doi: 10.1016/j.matpur.2017.10.007. Google Scholar

[41]

M. Rabe, et al., A transient population precedes and supports the acquisition of temozolomide resistance in glioblastoma, Nature Communications, (submitted).Google Scholar

[42]

C. Rojas, J. Belmonte-Beitia, V. M. Perez-Garcia and H. Maurer, Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model, Discrete and Continuous Dynamical Systems Series B, 21 (2016), 1895-1915. doi: 10.3934/dcdsb.2016028. Google Scholar

[43]

M. Rosa, A. Podolski-Renic, A. Alvarez-Arenas, J. Dinic, J. Belmonte-Beitia, M. Pesic and V. M. Perez-Garcia, Transfer of drug resistance characteristics between cancer cell subpopulations: A study using simple mathematical models, Bulletin of Mathematical Biology, 78 (2016), 1218-1237. doi: 10.1007/s11538-016-0182-0. Google Scholar

[44]

E. M. Rutter and Y. Kuang, Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1001-1021. doi: 10.3934/dcdsb.2017050. Google Scholar

[45]

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. Google Scholar

[46]

H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, Interdisciplinary Applied Mathematics, Volume 42, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. Google Scholar

[47]

R. L. Siegel, K. D. Miller and A. Jemal, Cancer statistics, CA: A Cancer Journal for Clinicians, 66 (2016), 7-30. Google Scholar

[48]

K. E. Starkov and L. Jimenez-Beristain, Dynamic analysis of the melanoma model: From cancer persistence to Its eradication, International Journal of Bifurcation and Chaos, 27 (2017), 1750151, 11pp. doi: 10.1142/S0218127417501516. Google Scholar

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K. E. Starkov and A. P. Krishchenko, Ultimate dynamics of the Kirshchner-Panetta model: Tumor eradication and related problems, Physics Letters A, 381 (2017), 3409-3416. doi: 10.1016/j.physleta.2017.08.048. Google Scholar

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X. Sun, J. Bao and Y. Shao, Mathematical modeling of therapy-induced cancer drug resistance: Connecting cancer mechanisms to population survival rates, Scientific Reports, 6 (2016), 22498. doi: 10.1038/srep22498. Google Scholar

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

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C. Tomasetti and D. Levy, An elementary approach to modeling drug resistance in cancer, Mathematical Biosciences and Engineering, 7 (2010), 905-918. doi: 10.3934/mbe.2010.7.905. Google Scholar

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show all references

References:
[1]

A. Alvarez-Arenas, A. Podolski-Renic, J. Belmonte-Beitia, M. Pesic and G. F. Calvo, Interplay of Darwinian selection, Lamarckian induction and microvesicle transfer on drug resistance in cancer, Scientific Reports, (submitted).Google Scholar

[2]

P. Bajger, M. Bodzioch and U. Foryś, Overcoming acquired chemotherapy resistance: Insights from mathematical modelling, Communications in Nonlinear Science and Numerical Simulation, (submitted).Google Scholar

[3]

M. Becker and D. Levy, Modeling the transfer of drug resistance in solid tumors, Bulletin of Mathematical Biology, 79 (2017), 2394-2412. doi: 10.1007/s11538-017-0334-x. Google Scholar

[4]

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

[5]

J. Belmonte-Beitia, T. E. Wooley, J. G. Scott, P. K. Maini and E. A. Gaffney, Modelling biological invasions: Individual to population scales at interfaces, Journal of Theoretical Biology, 334 (2013), 1-12. doi: 10.1016/j.jtbi.2013.05.033. Google Scholar

[6]

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. Google Scholar

[7]

M. Bodnar and U. Forys, Two models of drug resistance for low grade gliomas: comparison of the models dynamics, Proceedings of the XXIII National Conference Applications of Mathematics in Biology and Medicine, Politechnika lska, Uniwersytet Warszawski, Gliwice 2017, 37–42.Google Scholar

[8]

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

[9]

M. Calzada, E. Fernández-Cara and M. Marín, Optimal control oriented to therapy for a free-boundary tumor growth model, Journal of Theoretical Biology, 325 (2013), 1-11. doi: 10.1016/j.jtbi.2013.02.004. Google Scholar

[10]

G. Camacho and E. Fernández-Cara, Optimal control of some simplified models of tumour growth, International Journal of Control, 84 (2011), 540-550. doi: 10.1080/00207179.2011.562547. Google Scholar

[11]

C. Carrère, Optimization of an in vitro chemotherapy to avoid resistant tumours, Journal of Theoretical Biology, 413 (2017), 24-33. doi: 10.1016/j.jtbi.2016.11.009. Google Scholar

[12]

R. H. Chisholm, T. Lorenzi, A. Lorz, A.K. Larsen, L. Neves de Almeida, A. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation, Cancer Research, 15 (2015), 930-939. doi: 10.1158/0008-5472.CAN-14-2103. Google Scholar

[13]

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. Google Scholar

[14]

M. Doƚbniak and A. Świerniak, Comparison of simple models of periodic protocols for combined anticancer therapy, Computational and Mathematical Methods in Medicine, (2013), ID 567213, 11pp. doi: 10.1155/2013/567213. Google Scholar

[15]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Mathematical Biosciences, 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

[16]

S. Eckhouse, G. Lewison and R. Sullivan, Trends in the global funding and activity of cancer research, Molecular Oncology, 2 (2008), 20-32. doi: 10.1016/j.molonc.2008.03.007. Google Scholar

[17]

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. Google Scholar

[18]

R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptative therapy, Cancer Research, 69 (2009), 4894-4903. Google Scholar

[19]

M. Gerlinger and C. Swanton, How Darwinian models inform therapeutic failure initiated by clonal heterogeneity in cancer medicine, British Journal of Cancer, 103 (2010), 1139-1143. doi: 10.1038/sj.bjc.6605912. Google Scholar

[20]

D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Mathematical Biosciences and Engineering, 6 (2009), 573-583. doi: 10.3934/mbe.2009.6.573. Google Scholar

[21]

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

[22]

A. P. Krishchenko and K. E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Communication in Nonlinear Sciences and Numerical Simulations, 33 (2016), 174-183. doi: 10.1016/j.cnsns.2015.10.001. Google Scholar

[23]

J. P. LaSalle, An invariance principle in the theory of stability, Center for Dynamical Systems, 1966.Google Scholar

[24]

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. Google Scholar

[25]

U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Mathematical Biosciences and Engineering, 8 (2011), 307-323. doi: 10.3934/mbe.2011.8.307. Google Scholar

[26]

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. Google Scholar

[27]

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. doi: 10.3934/dcdsb.2006.6.129. Google Scholar

[28]

U. Ledzewicz, H. Schättler, M. R. Gahrooi and S. M. Dehkordi, On the MTD paradigm and optimal control for multi-drug cancer chemotherapy, Mathematical Biosciences and Engineering, 10 (2013), 803-819. doi: 10.3934/mbe.2013.10.803. Google Scholar

[29]

U. Ledzewicz, S. Wang, H. Schättler, N. André, M. A. Heng and E. Pasquier, On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach, Mathematical Biosciences and Engineering, 14 (2017), 217-235. doi: 10.3934/mbe.2017014. Google Scholar

[30]

Q. Li, A. Wennborg, E. Aurell, E. Dekel, J. Zou, Y. Xu, S. Huang and I. Ernberg, Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape, Proceedings of the National Academy of Sciences, 113 (2016), 2672-2677. doi: 10.1073/pnas.1519210113. Google Scholar

[31]

D. L. Lukes, Differential Equations: Classical to Controlled, Academic Press New York, 1982. Google Scholar

[32]

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. Google Scholar

[33]

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. Google Scholar

[34]

G. Morgan, R. Ward and M. Barton, The contribution of cytotoxic chemotherapy to 5-year survival in adult malignancies, Clinical Oncology, 16 (2004), 549-560. doi: 10.1016/j.clon.2004.06.007. Google Scholar

[35]

W. Ollier et. al., Analysis of temozolomide resistance in low-grade gliomas using a mechanistic mathematical model, Fundamental & Clinical Pharmacology, 31 (2017), 347-358.Google Scholar

[36]

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. Google Scholar

[37]

L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1991. doi: 10.1007/978-1-4684-0392-3. Google Scholar

[38]

A. O. Pisco, A. Brock, J. Zhou, A. Moor, M. Mojtahedi, D. Jackson and S. Huang, Non-Darwinian dynamics in therapy-induced cancer drug resistance, Nature Communications, 4 (2013), 2467. doi: 10.1038/ncomms3467. Google Scholar

[39]

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), Interscience Publishers, 1962.Google Scholar

[40]

C. Pouchol, J. Clairambault, A. Lorz and E. Trelat, Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy, Journal de Mathematiques Pures et Appliques, 116 (2017), 268-308. doi: 10.1016/j.matpur.2017.10.007. Google Scholar

[41]

M. Rabe, et al., A transient population precedes and supports the acquisition of temozolomide resistance in glioblastoma, Nature Communications, (submitted).Google Scholar

[42]

C. Rojas, J. Belmonte-Beitia, V. M. Perez-Garcia and H. Maurer, Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model, Discrete and Continuous Dynamical Systems Series B, 21 (2016), 1895-1915. doi: 10.3934/dcdsb.2016028. Google Scholar

[43]

M. Rosa, A. Podolski-Renic, A. Alvarez-Arenas, J. Dinic, J. Belmonte-Beitia, M. Pesic and V. M. Perez-Garcia, Transfer of drug resistance characteristics between cancer cell subpopulations: A study using simple mathematical models, Bulletin of Mathematical Biology, 78 (2016), 1218-1237. doi: 10.1007/s11538-016-0182-0. Google Scholar

[44]

E. M. Rutter and Y. Kuang, Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer, Discrete and Continuous Dynamical Systems Series B, 22 (2017), 1001-1021. doi: 10.3934/dcdsb.2017050. Google Scholar

[45]

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. Google Scholar

[46]

H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, Interdisciplinary Applied Mathematics, Volume 42, Springer, New York, 2015. doi: 10.1007/978-1-4939-2972-6. Google Scholar

[47]

R. L. Siegel, K. D. Miller and A. Jemal, Cancer statistics, CA: A Cancer Journal for Clinicians, 66 (2016), 7-30. Google Scholar

[48]

K. E. Starkov and L. Jimenez-Beristain, Dynamic analysis of the melanoma model: From cancer persistence to Its eradication, International Journal of Bifurcation and Chaos, 27 (2017), 1750151, 11pp. doi: 10.1142/S0218127417501516. Google Scholar

[49]

K. E. Starkov and A. P. Krishchenko, Ultimate dynamics of the Kirshchner-Panetta model: Tumor eradication and related problems, Physics Letters A, 381 (2017), 3409-3416. doi: 10.1016/j.physleta.2017.08.048. Google Scholar

[50]

X. Sun, J. Bao and Y. Shao, Mathematical modeling of therapy-induced cancer drug resistance: Connecting cancer mechanisms to population survival rates, Scientific Reports, 6 (2016), 22498. doi: 10.1038/srep22498. Google Scholar

[51]

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

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Figure 1.  Phase portrait of the orbits of system (4). Examples of convergent trajectories to $ P_2 $ for $ x_{0}+y_{0}+z_{0}\leq K $. Parameters used to calculate the phase portrait are given in Table 1
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Figure 2.  Optimal solutions for the objective $ J_{0, 1}(u) $ and $ M = 1/6 $. a) Optimal control $ u^*(t) $ (blue curve) and switching function $ \phi $ (dashed red line) satisfying the bang-bang control law (12). b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 1 $ month
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Figure 3.  Optimal solutions for the objective $ J_{0, 1}(u) $ and $ M = 5 $. a) Optimal control $ u^*(t) $ (blue curve) and switching function $ \phi $ (dashed red line) satisfying the bang-singular-bang control law (24). b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 30 $ months
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Figure 4.  Suboptimal protocol for the objective $ J_{0, 1}(u) $ and $ M = 5 $. a) Suboptimal bang-bang control. b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 30 $ months
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Figure 5.  Optimal solutions for the objective $ J_{1, 0}(u) $ and $ M = 1/6 $. a) Optimal control $ u^*(t) $ (blue curve) and switching function $ \phi $ (dashed red line) satisfying the bang-bang control law (12). b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 1 $ month
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Figure 6.  Optimal solutions for the objective $ J_{1, 0}(u) $ and $ M = 5 $. a) Optimal control $ u^*(t) $ (blue curve) and switching function $ \phi $ (dashed red line) satisfying the bang-singular-bang control law (26). b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 30 $ months
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Figure 7.  Suboptimal protocol for the objective $ J_{1, 0}(u) $ and $ M = 5 $. a) Suboptimal bang-bang control. b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 30 $ months
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Figure 8.  Suboptimal protocol for the objective $ J_{1, 0}(u) $ and $ M = 5 $. a) Suboptimal bang-bang control. b) Sensitive cells $ x(t) $ c) Damaged cells $ y(t) $ d) Resistant cells $ z(t) $. The time horizon was fixed to $ T = 30 $ months
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Table 1.  Values of the biological parameters for the system (4)
Variable Value Units Reference
$\rho_s$ 0.000385 day$^{-1}$ [35]
$\alpha_s$ 0.0382 L day/g [35]
$\mu_d$ 0.00219 day$^{-1}$ [35]
$\mu_r$ 0.000544 day$^{-1}$ [35]
$\rho_r$ 0.000385 day$^{-1}$ [35]
$\gamma$ 0.000136 day$^{-1}$ Estimated
$k_s$ 0.474 [35]
$P_0$ 40 mm [35]
$x(0)$ $k_s P_0$ mm [35]
$y(0)$ 0 mm [35]
$z(0)$ 0 mm [35]
$K$ 120 mm [35]
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Variable Value Units Reference
$\rho_s$ 0.000385 day$^{-1}$ [35]
$\alpha_s$ 0.0382 L day/g [35]
$\mu_d$ 0.00219 day$^{-1}$ [35]
$\mu_r$ 0.000544 day$^{-1}$ [35]
$\rho_r$ 0.000385 day$^{-1}$ [35]
$\gamma$ 0.000136 day$^{-1}$ Estimated
$k_s$ 0.474 [35]
$P_0$ 40 mm [35]
$x(0)$ $k_s P_0$ mm [35]
$y(0)$ 0 mm [35]
$z(0)$ 0 mm [35]
$K$ 120 mm [35]
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