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Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity
1. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo, 63130, United States |
2. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130 |
References:
[1] |
N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385-394. |
[2] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer Verlag, Series: Mathematics and Applications, Vol. 40, 2003. |
[3] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. |
[4] |
A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. of Math. Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[5] |
R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903.
doi: 10.1158/0008-5472.CAN-08-3658. |
[6] |
R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.
doi: 10.1038/459508a. |
[7] |
J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.
doi: 10.1017/CBO9780511666544. |
[8] |
J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. |
[9] |
R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039.
doi: 10.1158/0008-5472.CAN-05-3077. |
[10] |
J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 76 (2014), 627-653.
doi: 10.1007/s11538-014-9936-8. |
[11] |
O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.
doi: 10.1158/0008-5472.CAN-13-1768. |
[12] |
U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267-276.
doi: 10.1007/978-3-642-12598-0_23. |
[13] |
U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE), 2 (2005), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[14] |
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim., 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[15] |
U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[16] |
J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768-3773. |
[17] |
J. S. Li and N. Khaneja, Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
[18] |
A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399.
doi: 10.1051/m2an/2012031. |
[19] |
A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22.
doi: 10.1007/s11538-014-0046-4. |
[20] |
L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. |
[21] |
L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61. |
[22] |
E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: New rationale for new directions, Nature Reviews|Clinical Oncology, 7 (2010), 455-465.
doi: 10.1038/nrclinonc.2010.82. |
[23] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. |
[24] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York, 2012
doi: 10.1007/978-1-4614-3834-2. |
[25] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Springer, 2015.
doi: 10.1007/978-1-4939-2972-6. |
show all references
References:
[1] |
N. André, L. Padovani and E. Pasquier, Metronomic scheduling of anticancer treatment: The next generation of multitarget therapy?, Future Oncology, 7 (2011), 385-394. |
[2] |
B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, Springer Verlag, Series: Mathematics and Applications, Vol. 40, 2003. |
[3] |
A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007. |
[4] |
A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bull. of Math. Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[5] |
R. A. Gatenby, A. S. Silva, R. J. Gillies and B. R. Frieden, Adaptive therapy, Cancer Research, 69 (2009), 4894-4903.
doi: 10.1158/0008-5472.CAN-08-3658. |
[6] |
R. A. Gatenby, A change of strategy in the war on cancer, Nature, 459 (2009), 508-509.
doi: 10.1038/459508a. |
[7] |
J. H. Goldie, Drug resistance in cancer: A perspective, Cancer and Metastasis Review, 20 (2001), 63-68.
doi: 10.1017/CBO9780511666544. |
[8] |
J. H. Goldie and A. Coldman, A model for resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, 65 (1983), 291-307. |
[9] |
R. Grantab, S. Sivananthan and I. F. Tannock, The penetration of anticancer drugs through tumor tissue as a function of cellular adhesion and packing density of tumor cells, Cancer Research, 66 (2006), 1033-1039.
doi: 10.1158/0008-5472.CAN-05-3077. |
[10] |
J. Greene, O. Lavi, M. M. Gottesman and D. Levy, The impact of cell density and mutations in a model of multidrug resistance in solid tumors, Bull. Math. Biol., 76 (2014), 627-653.
doi: 10.1007/s11538-014-9936-8. |
[11] |
O. Lavi, J. Greene, D. Levy and M. Gottesman, The role of cell density and intratumoral heterogeneity in multidrug resistance, Cancer Research, 73 (2013), 7168-7175.
doi: 10.1158/0008-5472.CAN-13-1768. |
[12] |
U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, M. Diehl, F. Glineur, E. Jarlebring and W. Michiels, Eds., (2010), 267-276.
doi: 10.1007/978-3-642-12598-0_23. |
[13] |
U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE), 2 (2005), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[14] |
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Contr. Optim., 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[15] |
U. Ledzewicz and H. Schättler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[16] |
J. S. Li and N. Khaneja, Ensemble control of linear systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, 3768-3773. |
[17] |
J. S. Li and N. Khaneja, Ensemble control of Bloch equations, IEEE Transactions on Automatic Control, 54 (2009), 528-536.
doi: 10.1109/TAC.2009.2012983. |
[18] |
A. Lorz, T. Lorenzi, M. E. Hochberg, J. Clairambault and B. Berthame, Population adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, ESAIM: Mathematical Modelling and Numerical Analysis, 47 (2013), 377-399.
doi: 10.1051/m2an/2012031. |
[19] |
A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bull. Math. Biol., 77 (2015), 1-22.
doi: 10.1007/s11538-014-0046-4. |
[20] |
L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treatment Reports, 61 (1977), 1307-1317. |
[21] |
L. Norton and R. Simon, The Norton-Simon hypothesis revisited, Cancer Treatment Reports, 70 (1986), 41-61. |
[22] |
E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: New rationale for new directions, Nature Reviews|Clinical Oncology, 7 (2010), 455-465.
doi: 10.1038/nrclinonc.2010.82. |
[23] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964. |
[24] |
H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York, 2012
doi: 10.1007/978-1-4614-3834-2. |
[25] |
H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Springer, 2015.
doi: 10.1007/978-1-4939-2972-6. |
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