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On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth
On the MTD paradigm and optimal control for multi-drug cancer chemotherapy
1. | Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States, United States |
2. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130 |
References:
[1] |
M. R. Alison and C. E. Sarraf, "Understanding Cancer-From Basic Science to Clinical Practice," Cambridge University Press, 1997. |
[2] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40, 2003. |
[3] |
H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice," (in German), 2nd ed., Deutscher Apotheker -Verlag, Stuttgart, Germany. |
[4] |
M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy," Lecture Notes in Biomathematics, 30, Springer Verlag, 1979. |
[5] |
P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. |
[6] |
M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130. |
[7] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577.
doi: 10.1007/s00285-011-0424-6. |
[8] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637.
doi: 10.1023/A:1016027113579. |
[9] |
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. |
[10] |
U. Ledzewicz, H. Schättler and A. Swierniak, Finite dimensional models of drug resistant and phase specific cancer chemotherapy, J. of Medical Information Technology, 8 (2004), 5-13. |
[11] |
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. |
[12] |
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[13] |
U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[14] |
A. P. Lyss, Enzymes and random synthetics, in "Chemotherapy Source Book" (ed. M. C. Perry), Williams &Wilkins, Baltimore, (1992), 403-408. |
[15] |
J. C. Panetta, Y. Yanishevski, C. H. Pui, J. T. Sandlund, J. Rubnitz, G. K. Rivera, W. E. Evans and M. V. Relling, A mathematical model of in vivo methotrexate accumulation in acute lymphoblastic leukemia, Cancer Chemotherapy and Pharmacology, 50 (2002), 419-428.
doi: 10.1007/s00280-002-0511-x. |
[16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. |
[17] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[18] |
H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, Proc. of the 51st IEEE Conference on Decision and Control, Maui, (2012), 7691-7696. |
[19] |
H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design, Cancer Research, 24 (1964), 1295-1302. |
[20] |
J. Smieja and A. Swierniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 297-305. |
[21] |
G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284.
doi: 10.1016/0025-5564(90)90021-P. |
[22] |
A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172. |
[23] |
A. Swierniak, Some control problems for simplest models of proliferation cycle, Applied mathematics and Computer Science, 4 (1994), 223-233. |
[24] |
A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54.
doi: 10.1142/S0218339095000058. |
[25] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[26] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. |
[27] |
A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139. |
[28] |
A. Swierniak, A. Polanski, M. Kimmel, A. Bobrowski and J. Smieja, Qualitative analysis of controlled drug resistance model - inverse Laplace and semigroup approach, Control and Cybernetics, 28 (1999), 61-75. |
[29] |
A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386.
doi: 10.1016/S0362-546X(01)00184-5. |
[30] |
S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology, J. of Clinical Oncology, 11 (1993), 820-821. |
[31] |
T. E. Wheldon, "Mathematical Models in Cancer Research," Boston-Philadelphia: Hilger Publishing, 1988. |
show all references
References:
[1] |
M. R. Alison and C. E. Sarraf, "Understanding Cancer-From Basic Science to Clinical Practice," Cambridge University Press, 1997. |
[2] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory," Springer Verlag, Series: Mathematics and Applications, 40, 2003. |
[3] |
H. Derendorf, T. Gramatte and H. G. Schaefer, "Pharmacokinetics - Introduction into Theory and Practice," (in German), 2nd ed., Deutscher Apotheker -Verlag, Stuttgart, Germany. |
[4] |
M. Eisen, "Mathematical Models in Cell Biology and Cancer Chemotherapy," Lecture Notes in Biomathematics, 30, Springer Verlag, 1979. |
[5] |
P. Hahnfeldt, J. Folkman and L. Hlatky, Minimizing long-term burden: The logic for metronomic chemotherapeutic dosing and its angiogenic basis, J. of Theoretical Biology, 220 (2003), 545-554. |
[6] |
M. Kimmel and A. Swierniak, An optimal control problem related to leukemia chemotherapy, Scientific Bulletin of the Silesian Technical University, 65 (1983), 120-130. |
[7] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics, J. of Mathematical Biology, 64 (2012), 557-577.
doi: 10.1007/s00285-011-0424-6. |
[8] |
U. Ledzewicz and H. Schättler, Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy, J. of Optimization Theory and Applications - JOTA, 114 (2002), 609-637.
doi: 10.1023/A:1016027113579. |
[9] |
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. |
[10] |
U. Ledzewicz, H. Schättler and A. Swierniak, Finite dimensional models of drug resistant and phase specific cancer chemotherapy, J. of Medical Information Technology, 8 (2004), 5-13. |
[11] |
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. |
[12] |
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46 (2007), 1052-1079.
doi: 10.1137/060665294. |
[13] |
U. Ledzewicz and H. Schaettler, Singular controls and chattering arcs in optimal control problems arising in biomedicine, Control and Cybernetics, 38 (2009), 1501-1523. |
[14] |
A. P. Lyss, Enzymes and random synthetics, in "Chemotherapy Source Book" (ed. M. C. Perry), Williams &Wilkins, Baltimore, (1992), 403-408. |
[15] |
J. C. Panetta, Y. Yanishevski, C. H. Pui, J. T. Sandlund, J. Rubnitz, G. K. Rivera, W. E. Evans and M. V. Relling, A mathematical model of in vivo methotrexate accumulation in acute lymphoblastic leukemia, Cancer Chemotherapy and Pharmacology, 50 (2002), 419-428.
doi: 10.1007/s00280-002-0511-x. |
[16] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," MacMillan, New York, 1964. |
[17] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples," Springer Verlag, 2012.
doi: 10.1007/978-1-4614-3834-2. |
[18] |
H. Schättler, U. Ledzewicz, S. Mahmoudian Dehkordi and M. Reisi Gahrooi, A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, Proc. of the 51st IEEE Conference on Decision and Control, Maui, (2012), 7691-7696. |
[19] |
H. E. Skipper, Perspectives in cancer chemotherapy: Therapeutic design, Cancer Research, 24 (1964), 1295-1302. |
[20] |
J. Smieja and A. Swierniak, Different models of chemotherapy taking into account drug resistance stemming from gene amplification, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 297-305. |
[21] |
G. W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences, 101 (1990), 237-284.
doi: 10.1016/0025-5564(90)90021-P. |
[22] |
A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proceedings of the 12th IMACS World Congress, Paris, 4 (1988), 170-172. |
[23] |
A. Swierniak, Some control problems for simplest models of proliferation cycle, Applied mathematics and Computer Science, 4 (1994), 223-233. |
[24] |
A. Swierniak, Cell cycle as an object of control, J. of Biological Systems, 3 (1995), 41-54.
doi: 10.1142/S0218339095000058. |
[25] |
A. Swierniak, Direct and indirect control of cancer populations, Bulletin of the Polish Academy of Sciences, Technical Sciences, 56 (2008), 367-378. |
[26] |
A. Swierniak, U. Ledzewicz and H. Schättler, Optimal control for a class of compartmental models in cancer chemotherapy, Int. J. Applied Mathematics and Computer Science, 13 (2003), 357-368. |
[27] |
A. Swierniak, A. Polanski and M. Kimmel, Optimal control problems arising in cell-cycle-specific cancer chemotherapy, Cell Proliferation, 29 (1996), 117-139. |
[28] |
A. Swierniak, A. Polanski, M. Kimmel, A. Bobrowski and J. Smieja, Qualitative analysis of controlled drug resistance model - inverse Laplace and semigroup approach, Control and Cybernetics, 28 (1999), 61-75. |
[29] |
A. Swierniak and J. Smieja, Cancer chemotherapy optimization under evolving drug resistance, Nonlinear Analysis, 47 (2000), 375-386.
doi: 10.1016/S0362-546X(01)00184-5. |
[30] |
S. D. Weitman, E. Glatstein and B. A. Kamen, Back to the basics: the importance of concentration $\times$ time in oncology, J. of Clinical Oncology, 11 (1993), 820-821. |
[31] |
T. E. Wheldon, "Mathematical Models in Cancer Research," Boston-Philadelphia: Hilger Publishing, 1988. |
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