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Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy
1. | Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653 |
2. | Institut für Numerische und Angewandte Mathematik, Universität Münster, D-48149 Münster, Germany |
3. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899 |
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] |
C. Büskens, "Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen," Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. |
[5] |
A. d'Onofrio, Rapidly acting antitumoral anti-angiogenic therapies, Physical Review E, 76 (2007), Art. No. 031920. |
[6] |
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. |
[7] |
A. D'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95.
doi: 10.1093/imammb/dqn024. |
[8] |
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. |
[9] |
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. |
[10] |
J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.
doi: 10.1097/00000658-197203000-00014. |
[11] |
J. Folkman, Angiogenesis inhibitors generated by tumors, Molecular Medicine, 1 (1995), 120-122. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.
doi: 10.1038/36978. |
[17] |
R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515.
doi: 10.1093/carcin/21.3.505. |
[18] |
I. A. K. Kupka, The ubiquity of Fuller's phenomenon, in "Nonlinear Controllability and Optimal Control" (H. Sussmann, Ed.), Marcel Dekker, (1990), 313-350. |
[19] |
U. Ledzewicz, H. Maurer and H. Schättler, Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, Proc. 48th IEEE Conference on Decision and Control, Shanghai, China, (2009), 2280-2285. |
[20] |
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. |
[21] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. |
[22] |
U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206.
doi: 10.1142/S0218339002000597. |
[23] |
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. |
[24] |
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), 945-950.
doi: 10.1109/CDC.2005.1582277. |
[25] |
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. |
[26] |
U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206 (2007), 320-342.
doi: 10.1016/j.mbs.2005.03.013. |
[27] |
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. |
[28] |
U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[29] |
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: Appliations and Methods, 26 (2005), 129-156.
doi: 10.1002/oca.756. |
[30] |
H. Schaettler, U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Mathematical Biosciences and Engineering (MBE), this volume, 355-369. |
[31] |
A. Swierniak, Modelling combined angiogenic and chemo-therapy, Proc. of the Fourteenth National Conference on Applications of Mathematics in Biology and Medicine, Leszno, Poland, (2008), 127-133. |
[32] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[33] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 357-368. |
[34] |
M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering," Birkhäuser, Boston, 1994. |
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] |
C. Büskens, "Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen," Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. |
[5] |
A. d'Onofrio, Rapidly acting antitumoral anti-angiogenic therapies, Physical Review E, 76 (2007), Art. No. 031920. |
[6] |
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. |
[7] |
A. D'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Mathematical Medicine and Biology, 26 (2009), 63-95.
doi: 10.1093/imammb/dqn024. |
[8] |
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. |
[9] |
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. |
[10] |
J. Folkman, Antiangiogenesis: new concept for therapy of solid tumors, Annals of Surgery, 175 (1972), 409-416.
doi: 10.1097/00000658-197203000-00014. |
[11] |
J. Folkman, Angiogenesis inhibitors generated by tumors, Molecular Medicine, 1 (1995), 120-122. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
R. S. Kerbel, A cancer therapy resistant to resistance, Nature, 390 (1997), 335-336.
doi: 10.1038/36978. |
[17] |
R. S. Kerbel, Tumor angiogenesis: Past, present and near future, Carcinogensis, 21 (2000), 505-515.
doi: 10.1093/carcin/21.3.505. |
[18] |
I. A. K. Kupka, The ubiquity of Fuller's phenomenon, in "Nonlinear Controllability and Optimal Control" (H. Sussmann, Ed.), Marcel Dekker, (1990), 313-350. |
[19] |
U. Ledzewicz, H. Maurer and H. Schättler, Bang-bang and singular controls in a mathematical model for combined anti-angiogenic and chemotherapy treatments, Proc. 48th IEEE Conference on Decision and Control, Shanghai, China, (2009), 2280-2285. |
[20] |
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. |
[21] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, Journal of Optimization Theory and Applications - JOTA, 114 (2002), 609-637. |
[22] |
U. Ledzewicz and H. Schättler, Analysis of a cell-cycle specific model for cancer chemotherapy, J. of Biological Systems, 10 (2002), 183-206.
doi: 10.1142/S0218339002000597. |
[23] |
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. |
[24] |
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), 945-950.
doi: 10.1109/CDC.2005.1582277. |
[25] |
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. |
[26] |
U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Mathematical Biosciences, 206 (2007), 320-342.
doi: 10.1016/j.mbs.2005.03.013. |
[27] |
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. |
[28] |
U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[29] |
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: Appliations and Methods, 26 (2005), 129-156.
doi: 10.1002/oca.756. |
[30] |
H. Schaettler, U. Ledzewicz and B. Cardwell, Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis, Mathematical Biosciences and Engineering (MBE), this volume, 355-369. |
[31] |
A. Swierniak, Modelling combined angiogenic and chemo-therapy, Proc. of the Fourteenth National Conference on Applications of Mathematics in Biology and Medicine, Leszno, Poland, (2008), 127-133. |
[32] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[33] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 357-368. |
[34] |
M. I. Zelikin and V. F. Borisov, "Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering," Birkhäuser, Boston, 1994. |
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