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Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis
1. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899 |
2. | Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653 |
3. | Dept. of Mathematics and Statistic, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States |
References:
[1] |
T. Boehm, J. Folkman, T. Browder and M. S. O'Reilly, Antiangiogenic therapy of experimental cancer does not induce acquired drug resistance, Nature, 390 (1997), 404-407.
doi: 10.1038/37126. |
[2] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer Verlag, Paris, 2003 |
[3] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007. |
[4] |
S. Davis and G. D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis, Current Topics in Microbiology and Immunology, 237 (1999), 173-185. |
[5] |
A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184.
doi: 10.1016/j.mbs.2004.06.003. |
[6] |
A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[7] |
J. Folkman, Antiangiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.
doi: 10.1097/00000658-197203000-00014. |
[8] |
J. Folkman, Angiogenesis inhibitors generated by tumors, Mol. Med., 1 (1995), 120-122. |
[9] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. |
[10] |
R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy, Nature Medicine, 7 (2001), 987-989.
doi: 10.1038/nm0901-987. |
[11] |
R. K. Jain and L. L. Munn, Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents, Principles of Practical Oncology, 21 (2007), 1-7. |
[12] |
M. Klagsburn and S. Soker, VEGF/VPF: The angiogenesis factor found?, Current Biology, 3 (1993), 699-702.
doi: 10.1016/0960-9822(93)90073-W. |
[13] |
R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.
doi: 10.1038/36978. |
[14] |
U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.
doi: 10.1093/imammb/dqp012. |
[15] |
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. |
[16] |
U. Ledzewicz and H. Schättler, A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, Proc. 44th IEEE Conference on Decision and Control, Sevilla, Spain, (2005), 934-939.
doi: 10.1109/CDC.2005.1582277. |
[17] |
U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[18] |
U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57.
doi: 10.1002/oca.814. |
[19] |
U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312.
doi: 10.1016/j.jtbi.2008.02.014. |
[20] |
L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J.of the National Cancer Institute, 58 (1977), 1735-1741. |
[21] |
L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. |
[22] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. |
[23] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[24] |
A. Swierniak, Comparison of six models of antiangiogenic therapy, Applied Mathematics, 36 (2009), 333-348. |
show all references
References:
[1] |
T. Boehm, J. Folkman, T. Browder and M. S. O'Reilly, Antiangiogenic therapy of experimental cancer does not induce acquired drug resistance, Nature, 390 (1997), 404-407.
doi: 10.1038/37126. |
[2] |
B. Bonnard and M. Chyba, "Singular Trajectories and Their Role in Control Theory," Springer Verlag, Paris, 2003 |
[3] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," American Institute of Mathematical Sciences, 2007. |
[4] |
S. Davis and G. D. Yancopoulos, The angiopoietins: Yin and Yang in angiogenesis, Current Topics in Microbiology and Immunology, 237 (1999), 173-185. |
[5] |
A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: Analysis and extensions of the model by Hahnfeldt et al. (1999), Mathematical Biosciences, 191 (2004), 159-184.
doi: 10.1016/j.mbs.2004.06.003. |
[6] |
A. Ergun, K. Camphausen and L. M. Wein, Optimal scheduling of radiotherapy and angiogenic inhibitors, Bulletin of Mathematical Biology, 65 (2003), 407-424.
doi: 10.1016/S0092-8240(03)00006-5. |
[7] |
J. Folkman, Antiangiogenesis: New concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.
doi: 10.1097/00000658-197203000-00014. |
[8] |
J. Folkman, Angiogenesis inhibitors generated by tumors, Mol. Med., 1 (1995), 120-122. |
[9] |
P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: A dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Research, 59 (1999), 4770-4775. |
[10] |
R. K. Jain, Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy, Nature Medicine, 7 (2001), 987-989.
doi: 10.1038/nm0901-987. |
[11] |
R. K. Jain and L. L. Munn, Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents, Principles of Practical Oncology, 21 (2007), 1-7. |
[12] |
M. Klagsburn and S. Soker, VEGF/VPF: The angiogenesis factor found?, Current Biology, 3 (1993), 699-702.
doi: 10.1016/0960-9822(93)90073-W. |
[13] |
R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.
doi: 10.1038/36978. |
[14] |
U. Ledzewicz, J. Marriott, H. Maurer and H. Schättler, Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment, Mathematical Medicine and Biology, 27 (2010), 157-179.
doi: 10.1093/imammb/dqp012. |
[15] |
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. |
[16] |
U. Ledzewicz and H. Schättler, A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors, Proc. 44th IEEE Conference on Decision and Control, Sevilla, Spain, (2005), 934-939.
doi: 10.1109/CDC.2005.1582277. |
[17] |
U. Ledzewicz and H. Schättler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[18] |
U. Ledzewicz and H. Schättler, Analysis of a mathematical model for tumor anti-angiogenesis, Optimal Control Applications and Methods, 29 (2008), 41-57.
doi: 10.1002/oca.814. |
[19] |
U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. of Theoretical Biology, 252 (2008), 295-312.
doi: 10.1016/j.jtbi.2008.02.014. |
[20] |
L. Norton and R. Simon, Growth curve of an experimental solid tumor following radiotherapy, J.of the National Cancer Institute, 58 (1977), 1735-1741. |
[21] |
L. Norton, A Gompertzian model of human breast cancer growth, Cancer Research, 48 (1988), 7067-7071. |
[22] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. |
[23] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[24] |
A. Swierniak, Comparison of six models of antiangiogenic therapy, Applied Mathematics, 36 (2009), 333-348. |
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