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Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint
1. | Dep. Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain, Spain |
References:
[1] |
L. Cesari, Optimization-Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[2] |
J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments, Math. Model. Nat. Phenom., 4 (2009), 12-67.
doi: 10.1051/mmnp/20094302. |
[3] |
C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Appl. Methods, 32 (2011), 476-502.
doi: 10.1002/oca.957. |
[4] |
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[5] |
L. C. Evans, Partial Differential Equations, AMS, Providence, 1998. |
[6] |
K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971.
doi: 10.1137/S0036139902413489. |
[7] |
P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. |
[8] |
W. Krabs and S. Pickl, An optimal control problem in cancer chemotherapy, Appl. Math. Comput., 217 (2010), 1117-1124.
doi: 10.1016/j.amc.2010.05.008. |
[9] |
A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502. |
[10] |
U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323.
doi: 10.3934/mbe.2011.8.307. |
[11] |
U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal controls in cancer chemotherapy models, Math. Biosci. Eng., 2 (2005), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[12] |
U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Math. Biosci., 206 (2007), 320-342.
doi: 10.1016/j.mbs.2005.03.013. |
[13] |
R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994.
doi: 10.1142/9789812832542. |
[14] |
J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.
doi: 10.1016/0025-5564(90)90047-3. |
[15] |
A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 37 (2010), 1-39.
doi: 10.1145/1731022.1731032. |
[16] |
G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma, Bull. Math. Biol., 39 (1977), 317-337. |
[17] |
G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.
doi: 10.1016/0025-5564(90)90021-P. |
[18] |
A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, Eur. J. Pharmacol., 625 (2009), 108-121.
doi: 10.1016/j.ejphar.2009.08.041. |
show all references
References:
[1] |
L. Cesari, Optimization-Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[2] |
J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments, Math. Model. Nat. Phenom., 4 (2009), 12-67.
doi: 10.1051/mmnp/20094302. |
[3] |
C. L. Darby, W. W. Hager and A. V. Rao, An hp-adaptive pseudospectral method for solving optimal control problems, Optimal Control Appl. Methods, 32 (2011), 476-502.
doi: 10.1002/oca.957. |
[4] |
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci., 222 (2009), 13-26.
doi: 10.1016/j.mbs.2009.08.004. |
[5] |
L. C. Evans, Partial Differential Equations, AMS, Providence, 1998. |
[6] |
K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies, SIAM J. Appl. Math., 63 (2003), 1954-1971.
doi: 10.1137/S0036139902413489. |
[7] |
P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, 1982. |
[8] |
W. Krabs and S. Pickl, An optimal control problem in cancer chemotherapy, Appl. Math. Comput., 217 (2010), 1117-1124.
doi: 10.1016/j.amc.2010.05.008. |
[9] |
A. K. Laird, Dynamics of tumour growth, Br. J. Cancer, 18 (1964), 490-502. |
[10] |
U. Ledzewicz, H. Maurer and H. Schättler, Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy, Math. Biosci. Eng., 8 (2011), 307-323.
doi: 10.3934/mbe.2011.8.307. |
[11] |
U. Ledzewicz and H. Schättler, The influence of PK/PD on the structure of optimal controls in cancer chemotherapy models, Math. Biosci. Eng., 2 (2005), 561-578.
doi: 10.3934/mbe.2005.2.561. |
[12] |
U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy, Math. Biosci., 206 (2007), 320-342.
doi: 10.1016/j.mbs.2005.03.013. |
[13] |
R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994.
doi: 10.1142/9789812832542. |
[14] |
J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit, Math. Biosci., 100 (1990), 49-67.
doi: 10.1016/0025-5564(90)90047-3. |
[15] |
A. V. Rao, D. A. Benson, C. Darby, M. A. Patterson, C. Francolin, I. Sanders and G. T. Huntington, Algorithm 902: GPOPS, a MATLAB software for solving multiple-phase optimal control problems using the Gauss pseudospectral method, ACM Trans. Math. Software, 37 (2010), 1-39.
doi: 10.1145/1731022.1731032. |
[16] |
G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma, Bull. Math. Biol., 39 (1977), 317-337. |
[17] |
G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.
doi: 10.1016/0025-5564(90)90021-P. |
[18] |
A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy, Eur. J. Pharmacol., 625 (2009), 108-121.
doi: 10.1016/j.ejphar.2009.08.041. |
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