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On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth
1. | Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653, United States |
2. | Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130 |
References:
[1] |
M. M. Al-Tameemi, M. A. J. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the immune system: Consequences of brief encounters,, Biology Direct, 7 (2012).
doi: 10.1186/1745-6150-7-31. |
[2] |
N. Bellomo and N. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells,, Physics of Life Reviews, 5 (2008), 183. Google Scholar |
[3] |
N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system,, Mathematical and Computational Modelling, 32 (2000), 413.
doi: 10.1016/S0895-7177(00)00143-6. |
[4] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003).
|
[5] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences, (2007).
|
[6] |
G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting,, Annual Review of Immunology, 22 (2004), 322.
doi: 10.1146/annurev.immunol.22.012703.104803. |
[7] |
A. Friedman, Cancer as multifaceted disease,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1.
doi: 10.1051/mmnp/20127102. |
[8] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer Verlag, (1983).
|
[9] |
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. Google Scholar |
[10] |
T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology,", W. H. Freeman, (2006). Google Scholar |
[11] |
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. of Mathematical Biology, 37 (1998), 235.
doi: 10.1007/s002850050127. |
[12] |
C. M. Koebel, W. Vermi, J. B. Swann, N. Zerafa, S. J. Rodig, L. J. Old, M. J. Smyth and R. D. Schreiber, Adaptive immunity maintains occult cancer in an equilibrium state,, Nature, 450 (2007), 903.
doi: 10.1038/nature06309. |
[13] |
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295. Google Scholar |
[14] |
U. Ledzewicz, M. Faraji and H. Schättler, On optimal protocols for combinations of chemo- and immunotherapy,, Proceedings of the 51st IEEE Proceedings on Decision and Control, (2012), 7492.
doi: 10.1109/CDC.2012.6427039. |
[15] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem,, Proceedings of the 8th AIMS Conference, (2010), 971.
|
[16] |
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.
doi: 10.1007/s00285-011-0424-6. |
[17] |
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.
doi: 10.3934/mbe.2005.2.561. |
[18] |
A. Matzavinos, M. Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour,, Mathematical Medicine and Biology, 21 (2004), 1. Google Scholar |
[19] |
A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220.
doi: 10.1016/j.physd.2005.06.032. |
[20] |
A. d'Onofrio, Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy,, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1375.
doi: 10.1142/S0218202506001571. |
[21] |
A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS,, Chaos, 31 (2007), 261.
doi: 10.1016/j.chaos.2005.10.006. |
[22] |
A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,, Mathematical and Computational Modelling, 47 (2008), 614.
doi: 10.1016/j.mcm.2007.02.032. |
[23] |
A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa,, Chaos, 41 (2009), 875.
doi: 10.1016/j.chaos.2008.04.014. |
[24] |
A. d'Onofrio and A. Ciancio, Simple biophysical model of tumor evasion from immune system control,, Physical Review E, 84 (2011).
doi: 10.1103/PhysRevE.84.031910. |
[25] |
A. d'Onofrio, A. Gandolfi and A. Rocca, The cooperative and nonlinear dynamics of tumor-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings,, Cell Proliferation, 42 (2009), 317.
doi: 10.1111/j.1365-2184.2009.00595.x. |
[26] |
D. Pardoll, Does the immune system see tumors as foreign or self?,, Annual Reviews of Immunology, 21 (2003), 807. Google Scholar |
[27] |
E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455.
doi: 10.1038/nrclinonc.2010.82. |
[28] |
K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch'' regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. of Clinical Oncology, 23 (2005), 939.
doi: 10.1200/JCO.2005.07.093. |
[29] |
L. G. de Pillis, A. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. Google Scholar |
[30] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964).
|
[31] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples,", Springer Verlag, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[32] |
N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917. Google Scholar |
[33] |
J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. of Clinical Investigations, 117 (2007), 1137.
doi: 10.1172/JCI31405. |
[34] |
H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, J. of Theoretical Biology, 227 (2004), 335.
doi: 10.1016/j.jtbi.2003.11.012. |
[35] |
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. Google Scholar |
show all references
References:
[1] |
M. M. Al-Tameemi, M. A. J. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the immune system: Consequences of brief encounters,, Biology Direct, 7 (2012).
doi: 10.1186/1745-6150-7-31. |
[2] |
N. Bellomo and N. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells,, Physics of Life Reviews, 5 (2008), 183. Google Scholar |
[3] |
N. Bellomo and L. Preziosi, Modelling and mathematical problems related to tumor evolution and its interaction with the immune system,, Mathematical and Computational Modelling, 32 (2000), 413.
doi: 10.1016/S0895-7177(00)00143-6. |
[4] |
B. Bonnard and M. Chyba, "Singular Trajectories and their Role in Control Theory,", Springer Verlag, 40 (2003).
|
[5] |
A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control,", American Institute of Mathematical Sciences, (2007).
|
[6] |
G. P. Dunn, L. J. Old and R. D. Schreiber, The three ES of cancer immunoediting,, Annual Review of Immunology, 22 (2004), 322.
doi: 10.1146/annurev.immunol.22.012703.104803. |
[7] |
A. Friedman, Cancer as multifaceted disease,, Mathematical Modelling of Natural Phenomena, 7 (2012), 1.
doi: 10.1051/mmnp/20127102. |
[8] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer Verlag, (1983).
|
[9] |
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. Google Scholar |
[10] |
T. J. Kindt, B. A. Osborne and R. A. Goldsby, "Kuby Immunology,", W. H. Freeman, (2006). Google Scholar |
[11] |
D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. of Mathematical Biology, 37 (1998), 235.
doi: 10.1007/s002850050127. |
[12] |
C. M. Koebel, W. Vermi, J. B. Swann, N. Zerafa, S. J. Rodig, L. J. Old, M. J. Smyth and R. D. Schreiber, Adaptive immunity maintains occult cancer in an equilibrium state,, Nature, 450 (2007), 903.
doi: 10.1038/nature06309. |
[13] |
V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bulletin of Mathematical Biology, 56 (1994), 295. Google Scholar |
[14] |
U. Ledzewicz, M. Faraji and H. Schättler, On optimal protocols for combinations of chemo- and immunotherapy,, Proceedings of the 51st IEEE Proceedings on Decision and Control, (2012), 7492.
doi: 10.1109/CDC.2012.6427039. |
[15] |
U. Ledzewicz, M. Naghnaeian and H. Schättler, Dynamics of tumor-immune interactions under treatment as an optimal control problem,, Proceedings of the 8th AIMS Conference, (2010), 971.
|
[16] |
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.
doi: 10.1007/s00285-011-0424-6. |
[17] |
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.
doi: 10.3934/mbe.2005.2.561. |
[18] |
A. Matzavinos, M. Chaplain and V. A. Kuznetsov, Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour,, Mathematical Medicine and Biology, 21 (2004), 1. Google Scholar |
[19] |
A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,, Physica D, 208 (2005), 220.
doi: 10.1016/j.physd.2005.06.032. |
[20] |
A. d'Onofrio, Tumor-immune system interaction: Modeling the tumor-stimulated proliferation of effectors and immunotherapy,, Mathematical Models and Methods in Applied Sciences, 16 (2006), 1375.
doi: 10.1142/S0218202506001571. |
[21] |
A. d'Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS,, Chaos, 31 (2007), 261.
doi: 10.1016/j.chaos.2005.10.006. |
[22] |
A. d'Onofrio, Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy,, Mathematical and Computational Modelling, 47 (2008), 614.
doi: 10.1016/j.mcm.2007.02.032. |
[23] |
A. d'Onofrio, Cellular growth: Linking the mechanistic to the phenomenological modeling and vice-versa,, Chaos, 41 (2009), 875.
doi: 10.1016/j.chaos.2008.04.014. |
[24] |
A. d'Onofrio and A. Ciancio, Simple biophysical model of tumor evasion from immune system control,, Physical Review E, 84 (2011).
doi: 10.1103/PhysRevE.84.031910. |
[25] |
A. d'Onofrio, A. Gandolfi and A. Rocca, The cooperative and nonlinear dynamics of tumor-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings,, Cell Proliferation, 42 (2009), 317.
doi: 10.1111/j.1365-2184.2009.00595.x. |
[26] |
D. Pardoll, Does the immune system see tumors as foreign or self?,, Annual Reviews of Immunology, 21 (2003), 807. Google Scholar |
[27] |
E. Pasquier, M. Kavallaris and N. André, Metronomic chemotherapy: new rationale for new directions,, Nature Reviews$|$ Clinical Oncology, 7 (2010), 455.
doi: 10.1038/nrclinonc.2010.82. |
[28] |
K. Pietras and D. Hanahan, A multitargeted, metronomic, and maximum-tolerated dose "chemo-switch'' regimen is antiangiogenic, producing objective responses and survival benefit in a mouse model of cancer,, J. of Clinical Oncology, 23 (2005), 939.
doi: 10.1200/JCO.2005.07.093. |
[29] |
L. G. de Pillis, A. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. Google Scholar |
[30] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", MacMillan, (1964).
|
[31] |
H. Schättler and U. Ledzewicz, "Geometric Optimal Control: Theory, Methods and Examples,", Springer Verlag, (2012).
doi: 10.1007/978-1-4614-3834-2. |
[32] |
N. V. Stepanova, Course of the immune reaction during the development of a malignant tumour,, Biophysics, 24 (1980), 917. Google Scholar |
[33] |
J. B. Swann and M. J. Smyth, Immune surveillance of tumors,, J. of Clinical Investigations, 117 (2007), 1137.
doi: 10.1172/JCI31405. |
[34] |
H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy,, J. of Theoretical Biology, 227 (2004), 335.
doi: 10.1016/j.jtbi.2003.11.012. |
[35] |
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. Google Scholar |
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