-
Previous Article
Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion
- DCDS-B Home
- This Issue
-
Next Article
On strong causal binomial approximation for stochastic processes
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. |
| [1] |
Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100 |
| [2] |
Urszula Ledzewicz, Heinz Schättler. The Influence of PK/PD on the Structure of Optimal Controls in Cancer Chemotherapy Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 561-578. doi: 10.3934/mbe.2005.2.561 |
| [3] |
Wei Feng, Shuhua Hu, Xin Lu. Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective. Conference Publications, 2003, 2003 (Special) : 544-553. doi: 10.3934/proc.2003.2003.544 |
| [4] |
Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129 |
| [5] |
Urszula Ledzewicz, Heinz Schättler, Mostafa Reisi Gahrooi, Siamak Mahmoudian Dehkordi. On the MTD paradigm and optimal control for multi-drug cancer chemotherapy. Mathematical Biosciences & Engineering, 2013, 10 (3) : 803-819. doi: 10.3934/mbe.2013.10.803 |
| [6] |
Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040 |
| [7] |
Urszula Ledzewicz, Heinz Schättler, Shuo Wang. On the role of tumor heterogeneity for optimal cancer chemotherapy. Networks and Heterogeneous Media, 2019, 14 (1) : 131-147. doi: 10.3934/nhm.2019007 |
| [8] |
Craig Collins, K. Renee Fister, Bethany Key, Mary Williams. Blasting neuroblastoma using optimal control of chemotherapy. Mathematical Biosciences & Engineering, 2009, 6 (3) : 451-467. doi: 10.3934/mbe.2009.6.451 |
| [9] |
Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 |
| [10] |
Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete and Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 |
| [11] |
Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres, K. A. Jane White. Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1435-1463. doi: 10.3934/mbe.2018066 |
| [12] |
Urszula Ledzewicz, Shuo Wang, Heinz Schättler, Nicolas André, Marie Amélie Heng, Eddy Pasquier. On drug resistance and metronomic chemotherapy: A mathematical modeling and optimal control approach. Mathematical Biosciences & Engineering, 2017, 14 (1) : 217-235. doi: 10.3934/mbe.2017014 |
| [13] |
Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581 |
| [14] |
Urszula Ledzewicz, Heinz Schättler. On optimal singular controls for a general SIR-model with vaccination and treatment. Conference Publications, 2011, 2011 (Special) : 981-990. doi: 10.3934/proc.2011.2011.981 |
| [15] |
Urszula Ledzewicz, Heinz Schättler. Controlling a model for bone marrow dynamics in cancer chemotherapy. Mathematical Biosciences & Engineering, 2004, 1 (1) : 95-110. doi: 10.3934/mbe.2004.1.95 |
| [16] |
Piotr Bajger, Mariusz Bodzioch, Urszula Foryś. Singularity of controls in a simple model of acquired chemotherapy resistance. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083 |
| [17] |
Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266 |
| [18] |
Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028 |
| [19] |
Arturo Alvarez-Arenas, Konstantin E. Starkov, Gabriel F. Calvo, Juan Belmonte-Beitia. Ultimate dynamics and optimal control of a multi-compartment model of tumor resistance to chemotherapy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082 |
| [20] |
M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]