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Preface
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
| 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 |
In this paper, a non-trivial generalization of a mathematical model put forward in [
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 and Continuous Dynamical Systems - B,
2019, 24
(5)
: 2017-2038.
doi: 10.3934/dcdsb.2019082
![]()
References:
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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). |
| [2] |
P. Bajger, M. Bodzioch and U. Foryś, Overcoming acquired chemotherapy resistance: Insights from mathematical modelling, Communications in Nonlinear Science and Numerical Simulation, (submitted). |
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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. |
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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, 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. |
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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. |
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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. |
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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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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. |
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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. |
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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. |
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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. |
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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. 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. |
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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. |
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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.
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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.
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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. |
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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. |
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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. |
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J. P. LaSalle, An invariance principle in the theory of stability, Center for Dynamical Systems, 1966. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
D. L. Lukes, Differential Equations: Classical to Controlled, Academic Press New York, 1982. |
| [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. |
| [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. |
| [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. |
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W. Ollier et. al., Analysis of temozolomide resistance in low-grade gliomas using a mechanistic mathematical model, Fundamental & Clinical Pharmacology, 31 (2017), 347-358. |
| [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. |
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L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
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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. |
| [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. |
| [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. |
| [41] |
M. Rabe, et al., A transient population precedes and supports the acquisition of temozolomide resistance in glioblastoma, Nature Communications, (submitted). |
| [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. |
| [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. |
| [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. |
| [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. |
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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. |
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R. L. Siegel, K. D. Miller and A. Jemal,
Cancer statistics, CA: A Cancer Journal for Clinicians, 66 (2016), 7-30.
|
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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. |
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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. |
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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. |
| [51] |
G. W. Swan, Applications of Optimal Control Theory in Biomedicine, Monographs and textbooks in pure and applied mathematics. M. Dekker, New York, 1984. |
| [52] |
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. |
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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. |
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J. X. Zhou, A. O. Pisco, H. Qian and S. Huang, Non-equilibrium population dynamics of phenotype conversion of cancer cells, PLOS ONE, 9 (2014), e110714. |
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). |
| [2] |
P. Bajger, M. Bodzioch and U. Foryś, Overcoming acquired chemotherapy resistance: Insights from mathematical modelling, Communications in Nonlinear Science and Numerical Simulation, (submitted). |
| [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. |
| [4] |
D. J. Bell and D. H. Jacobson, Singular Optimal Control Problems, Academic Press London; New York [etc.], 1975. |
| [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. |
| [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. |
| [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. |
| [8] |
B. Bonnard and M. Chyba, Singular Trajectories and Their Role in Control Theory, Mathematiques and Applications. Springer, Paris, New York, 2003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [18] |
R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden,
Adaptative therapy, Cancer Research, 69 (2009), 4894-4903.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [23] |
J. P. LaSalle, An invariance principle in the theory of stability, Center for Dynamical Systems, 1966. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
D. L. Lukes, Differential Equations: Classical to Controlled, Academic Press New York, 1982. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [37] |
L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 1991.
doi: 10.1007/978-1-4684-0392-3. |
| [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. |
| [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. |
| [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. |
| [41] |
M. Rabe, et al., A transient population precedes and supports the acquisition of temozolomide resistance in glioblastoma, Nature Communications, (submitted). |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [47] |
R. L. Siegel, K. D. Miller and A. Jemal,
Cancer statistics, CA: A Cancer Journal for Clinicians, 66 (2016), 7-30.
|
| [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. |
| [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. |
| [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. |
| [51] |
G. W. Swan, Applications of Optimal Control Theory in Biomedicine, Monographs and textbooks in pure and applied mathematics. M. Dekker, New York, 1984. |
| [52] |
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. |
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A. Wächter and L. T. Biegler,
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doi: 10.1007/s10107-004-0559-y. |
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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
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
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
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
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
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
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
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] |
| 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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