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On the long-time behaviour of age and trait structured population dynamics
Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy
Dep. Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain |
We study a collection of problems associated with the optimization of cancer chemotherapy treatments, under the assumptions of Gomperztian-type tumor growth and that the drug killing effect is proportional to the rate of growth for the untreated tumor (Norton-Simon hypothesis). Classical pharmacokinetics and different pharmacodynamics (Skipper and Emax) are considered, together with a toxicity limit or the penalization of the accumulated drug effect. Existence and uniqueness of the optimal control is proved in some cases, while in others the total amount of drug is the unique relevant aspect to take into account and the existence of an infinite number of optimal controls is shown. In all cases, explicit expressions for the solutions are derived in terms of the problem data. Finally, numerical results of illustrative examples and some conclusions are presented.
References:
[1] |
S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky and P. Hahnfeldt, Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth, PLOS Comput. Biol., 10 (2014), e1003800.
doi: 10.1371/journal.pcbi.1003800. |
[2] |
S. Benzekry, E. Pasquier, D. Barbolosi, B. Lacarelle, F. Barlési, N. André and J. Ciccolini,
Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine, Semin. Cancer Biol., 35 (2015), 53-61.
doi: 10.1016/j.semcancer.2015.09.002. |
[3] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
doi: 10.1007/978-0-387-70914-7. |
[4] |
L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[5] |
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. |
[6] |
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. |
[7] |
C. Faivre, D. Barbolosi, E. Pasquier and N. André,
A mathematical model for the administration of temozolomide: comparative analysis of conventional and metronomic chemotherapy regimens, Cancer Chemother. Pharmacol., 71 (2013), 1013-1019.
doi: 10.1007/s00280-013-2095-z. |
[8] |
L. A. Fernández and C. Pola,
Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1563-1588.
doi: 10.3934/dcdsb.2014.19.1563. |
[9] |
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. |
[10] |
R. Langreth, The mathematics of cancer, Forbes Magazine, 2010, https://www.forbes.com/forbes/2010/0315/opinions-health-cancer-larry-norton-ideas-opinions.html#5469c17c5519 |
[11] |
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. |
[12] |
R. Martin and K. L. Teo,
Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994.
doi: 10.1142/2048. |
[13] |
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. |
[14] |
L. Norton and R. Simon,
Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.
|
[15] |
M. C. Perry, D. C. Doll and C. E. Freter,
Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. |
[16] |
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, 41 (2014), Art. 1, 37 pp.
doi: 10.1145/1731022.1731032. |
[17] |
R. Simon and L. Norton,
The Norton-Simon hypothesis: Designing more effective and less toxic chemotherapeutic regimens, Nat. Clin. Pract. Oncol., 3 (2006), 406-407.
doi: 10.1038/ncponc0560. |
[18] |
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. |
[19] |
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. |
[20] |
T. A. Traina, U. Dugan, B. Higgins, K. Kolinsky, M. Theodoulou, C. A. Hudis and L. Norton,
Optimizing chemotherapy dose and schedule by Norton-Simon mathematical modeling, Breas Dis., 31 (2010), 7-18.
doi: 10.3233/BD-2009-0290. |
show all references
References:
[1] |
S. Benzekry, C. Lamont, A. Beheshti, A. Tracz, J. M. L. Ebos, L. Hlatky and P. Hahnfeldt, Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth, PLOS Comput. Biol., 10 (2014), e1003800.
doi: 10.1371/journal.pcbi.1003800. |
[2] |
S. Benzekry, E. Pasquier, D. Barbolosi, B. Lacarelle, F. Barlési, N. André and J. Ciccolini,
Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine, Semin. Cancer Biol., 35 (2015), 53-61.
doi: 10.1016/j.semcancer.2015.09.002. |
[3] |
H. Brezis,
Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.
doi: 10.1007/978-0-387-70914-7. |
[4] |
L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4613-8165-5. |
[5] |
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. |
[6] |
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. |
[7] |
C. Faivre, D. Barbolosi, E. Pasquier and N. André,
A mathematical model for the administration of temozolomide: comparative analysis of conventional and metronomic chemotherapy regimens, Cancer Chemother. Pharmacol., 71 (2013), 1013-1019.
doi: 10.1007/s00280-013-2095-z. |
[8] |
L. A. Fernández and C. Pola,
Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1563-1588.
doi: 10.3934/dcdsb.2014.19.1563. |
[9] |
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. |
[10] |
R. Langreth, The mathematics of cancer, Forbes Magazine, 2010, https://www.forbes.com/forbes/2010/0315/opinions-health-cancer-larry-norton-ideas-opinions.html#5469c17c5519 |
[11] |
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. |
[12] |
R. Martin and K. L. Teo,
Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994.
doi: 10.1142/2048. |
[13] |
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. |
[14] |
L. Norton and R. Simon,
Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.
|
[15] |
M. C. Perry, D. C. Doll and C. E. Freter,
Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. |
[16] |
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, 41 (2014), Art. 1, 37 pp.
doi: 10.1145/1731022.1731032. |
[17] |
R. Simon and L. Norton,
The Norton-Simon hypothesis: Designing more effective and less toxic chemotherapeutic regimens, Nat. Clin. Pract. Oncol., 3 (2006), 406-407.
doi: 10.1038/ncponc0560. |
[18] |
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. |
[19] |
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. |
[20] |
T. A. Traina, U. Dugan, B. Higgins, K. Kolinsky, M. Theodoulou, C. A. Hudis and L. Norton,
Optimizing chemotherapy dose and schedule by Norton-Simon mathematical modeling, Breas Dis., 31 (2010), 7-18.
doi: 10.3233/BD-2009-0290. |
Example | Structure | S_times | |
1.4e-01 | |||
2.3e+00 | |||
5.6e-01 2.1e+00 |
3.4e-01 | ||
2.9e+00 | |||
6.8e-01 2.1e+00 |
4.0e-01 | ||
1.6e-01 | 9.9e-02 | ||
1.0e+00 | 5.6e-01 | ||
|
1.4e+00 | 7.5e-01 |
Example | Structure | S_times | |
1.4e-01 | |||
2.3e+00 | |||
5.6e-01 2.1e+00 |
3.4e-01 | ||
2.9e+00 | |||
6.8e-01 2.1e+00 |
4.0e-01 | ||
1.6e-01 | 9.9e-02 | ||
1.0e+00 | 5.6e-01 | ||
|
1.4e+00 | 7.5e-01 |
Example | T | |
|
|
| |||
0.1 | 1 | 2.3e-03 | 2.4e+00 | 5.0e-02 | 1.7e-02 | |||
0.1 | 1 | 3.0e-02 | 9.5e-01 | 5.0e-02 | 3.8e-01 | |||
0.1 | 1 | 7.3e-02 | 0 | 5.0e-02 | 1.3e+00 | |||
0.1 | 0.27 | 1 | 9.0e-05 | 2.4e+00 | 5.0e-02 | 3.0e-01 | ||
0.1 | 0.27 | 1 | 3.0e-02 | 7.0e-01 | 5.0e-02 | 7.0e-01 | ||
0.1 | 0.27 | 1 | 6.4e-02 | 0 | 5.0e-02 | 1.3e+00 | ||
0.1 | 0.27 | 1 | 2.5e-02 | 2.4e+00 | 1.1e+00 | 5.0e-02 | 3.0e-01 | |
0.1 | 0.27 | 1 | 3.0e-02 | 1.8e+00 | 9.5e-01 | 5.0e-02 | 3.8e-01 | |
0.1 | 0.27 | 1 | 7.3e-02 | 0 | 0 | 5.0e-02 | 1.3e+00 | |
0.1 | 1 | 1.2e-01 | 2.4e+00 | 5.0e-01 | 3.9e-01 | |||
0.1 | 1 | 1.4e-01 | 1.9e+00 | 5.0e-01 | 5.2e-01 | |||
0.1 | 1 | 1.5e-01 | 0 | 5.0e-01 | 1.0e+00 | |||
0.1 | 0.27 | 1 | 0 | 2.4e+00 | 5.0e-01 | 7.6e-01 | ||
0.1 | 0.27 | 1 | 5.9e-02 | 1.4e+00 | 5.0e-01 | 8.2e-01 | ||
0.1 | 0.27 | 1 | 1.4e-01 | 2.4e+00 | 1.1e+00 | 5.0e-01 | 7.6e-01 | |
0.1 | 1 | 1.1e-01 | 2.4e+00 | 7.5e-01 | 6.8e-01 | |||
0.1 | 1 | 1.2e-01 | 0 | 7.5e-01 | 1.0e+00 | |||
0.1 | 0.27 | 1 | 0 | 2.4e+00 | 7.5e-01 | 8.9e-01 | ||
0.1 | 0.27 | 1 | 3.5e-02 | 1.5e+00 | 7.5e-01 | 9.1e-01 | ||
0.1 | 0.27 | 1 | 9.3e-02 | 2.4e+00 | 1.1e+00 | 7.5e-01 | 8.9e-01 |
Example | T | |
|
|
| |||
0.1 | 1 | 2.3e-03 | 2.4e+00 | 5.0e-02 | 1.7e-02 | |||
0.1 | 1 | 3.0e-02 | 9.5e-01 | 5.0e-02 | 3.8e-01 | |||
0.1 | 1 | 7.3e-02 | 0 | 5.0e-02 | 1.3e+00 | |||
0.1 | 0.27 | 1 | 9.0e-05 | 2.4e+00 | 5.0e-02 | 3.0e-01 | ||
0.1 | 0.27 | 1 | 3.0e-02 | 7.0e-01 | 5.0e-02 | 7.0e-01 | ||
0.1 | 0.27 | 1 | 6.4e-02 | 0 | 5.0e-02 | 1.3e+00 | ||
0.1 | 0.27 | 1 | 2.5e-02 | 2.4e+00 | 1.1e+00 | 5.0e-02 | 3.0e-01 | |
0.1 | 0.27 | 1 | 3.0e-02 | 1.8e+00 | 9.5e-01 | 5.0e-02 | 3.8e-01 | |
0.1 | 0.27 | 1 | 7.3e-02 | 0 | 0 | 5.0e-02 | 1.3e+00 | |
0.1 | 1 | 1.2e-01 | 2.4e+00 | 5.0e-01 | 3.9e-01 | |||
0.1 | 1 | 1.4e-01 | 1.9e+00 | 5.0e-01 | 5.2e-01 | |||
0.1 | 1 | 1.5e-01 | 0 | 5.0e-01 | 1.0e+00 | |||
0.1 | 0.27 | 1 | 0 | 2.4e+00 | 5.0e-01 | 7.6e-01 | ||
0.1 | 0.27 | 1 | 5.9e-02 | 1.4e+00 | 5.0e-01 | 8.2e-01 | ||
0.1 | 0.27 | 1 | 1.4e-01 | 2.4e+00 | 1.1e+00 | 5.0e-01 | 7.6e-01 | |
0.1 | 1 | 1.1e-01 | 2.4e+00 | 7.5e-01 | 6.8e-01 | |||
0.1 | 1 | 1.2e-01 | 0 | 7.5e-01 | 1.0e+00 | |||
0.1 | 0.27 | 1 | 0 | 2.4e+00 | 7.5e-01 | 8.9e-01 | ||
0.1 | 0.27 | 1 | 3.5e-02 | 1.5e+00 | 7.5e-01 | 9.1e-01 | ||
0.1 | 0.27 | 1 | 9.3e-02 | 2.4e+00 | 1.1e+00 | 7.5e-01 | 8.9e-01 |
Example | T | |
|
|||||
0.1 | 1 | 1.1e-03 | 2.4e+00 | 5.0e-02 | 4.1e-01 | |||
0.1 | 1 | 3.0e-02 | 3.6e-01 | 5.0e-02 | 6.5e-01 | |||
0.1 | 1 | 2.9e-01 | 0 | 5.0e-02 | 1.3e+00 | |||
0.1 | 0.27 | 1 | 5.2e-07 | 2.4e+00 | 5.0e-02 | 5.2e-01 | ||
0.1 | 0.27 | 1 | 3.0e-02 | 3.4e-01 | 5.0e-02 | 7.1e-01 | ||
0.1 | 0.27 | 1 | 2.4e-1 | 0 | 5.0e-02 | 1.3e+00 | ||
0.1 | 0.27 | 1 | 1.7e-03 | 2.4e+00 | 1.1e+00 | 5.0e-02 | 5.2e-01 | |
0.1 | 0.27 | 1 | 3.0e-02 | 4.6e-01 | 3.4e-01 | 5.0e-02 | 6.7e-01 | |
0.1 | 0.27 | 1 | 2.9e-01 | 0 | 0 | 5.0e-02 | 1.3e+00 | |
0.1 | 5 | 2.7e-03 | 1.2e+01 | 5.0e-01 | 1.5e-01 | |||
0.1 | 5 | 4.0e-03 | 1.0e+01 | 5.0e-01 | 1.7e-01 | |||
0.1 | 5 | 4.5e-01 | 0 | 5.0e-01 | 1.3e+00 | |||
0.1 | 0.27 | 5 | 4.4e-10 | 1.2e+01 | 5.0e-01 | 1.5e-01 | ||
0.1 | 0.27 | 5 | 4.0e-03 | 5.6e+00 | 5.0e-01 | 1.7e-01 | ||
0.1 | 0.27 | 5 | 4.5e-02 | 1.9e+00 | 5.0e-01 | 2.8e-01 | ||
0.1 | 0.27 | 5 | 4.2e-04 | 1.2e+01 | 2.0e+01 | 5.0e-01 | 1.5e-01 | |
0.1 | 0.27 | 5 | 4.0e-03 | 4.7e+00 | 9.5e+00 | 5.0e-01 | 1.9e-01 | |
0.1 | 5 | 5.2e-03 | 1.2e+01 | 7.5e-01 | 4.6e-01 | |||
0.1 | 5 | 2.4e-01 | 0 | 7.5e-01 | 1.1e+00 | |||
0.1 | 0.27 | 5 | 8.8e-10 | 1.2e+01 | 7.5e-01 | 4.6e-01 | ||
0.1 | 0.27 | 5 | 4.0e-03 | 6.9e+00 | 7.5e-01 | 4.7e-01 | ||
0.1 | 0.27 | 5 | 4.5e-02 | 2.2e+00 | 7.5e-01 | 5.6e-01 | ||
0.1 | 0.27 | 5 | 7.8e-4 | 1.2e+01 | 2.0e+01 | 7.5e-01 | 4.6e-01 | |
0.1 | 0.27 | 5 | 4.0e-03 | 6.0e+00 | 1.2e+01 | 7.5e-01 | 4.8e-01 |
Example | T | |
|
|||||
0.1 | 1 | 1.1e-03 | 2.4e+00 | 5.0e-02 | 4.1e-01 | |||
0.1 | 1 | 3.0e-02 | 3.6e-01 | 5.0e-02 | 6.5e-01 | |||
0.1 | 1 | 2.9e-01 | 0 | 5.0e-02 | 1.3e+00 | |||
0.1 | 0.27 | 1 | 5.2e-07 | 2.4e+00 | 5.0e-02 | 5.2e-01 | ||
0.1 | 0.27 | 1 | 3.0e-02 | 3.4e-01 | 5.0e-02 | 7.1e-01 | ||
0.1 | 0.27 | 1 | 2.4e-1 | 0 | 5.0e-02 | 1.3e+00 | ||
0.1 | 0.27 | 1 | 1.7e-03 | 2.4e+00 | 1.1e+00 | 5.0e-02 | 5.2e-01 | |
0.1 | 0.27 | 1 | 3.0e-02 | 4.6e-01 | 3.4e-01 | 5.0e-02 | 6.7e-01 | |
0.1 | 0.27 | 1 | 2.9e-01 | 0 | 0 | 5.0e-02 | 1.3e+00 | |
0.1 | 5 | 2.7e-03 | 1.2e+01 | 5.0e-01 | 1.5e-01 | |||
0.1 | 5 | 4.0e-03 | 1.0e+01 | 5.0e-01 | 1.7e-01 | |||
0.1 | 5 | 4.5e-01 | 0 | 5.0e-01 | 1.3e+00 | |||
0.1 | 0.27 | 5 | 4.4e-10 | 1.2e+01 | 5.0e-01 | 1.5e-01 | ||
0.1 | 0.27 | 5 | 4.0e-03 | 5.6e+00 | 5.0e-01 | 1.7e-01 | ||
0.1 | 0.27 | 5 | 4.5e-02 | 1.9e+00 | 5.0e-01 | 2.8e-01 | ||
0.1 | 0.27 | 5 | 4.2e-04 | 1.2e+01 | 2.0e+01 | 5.0e-01 | 1.5e-01 | |
0.1 | 0.27 | 5 | 4.0e-03 | 4.7e+00 | 9.5e+00 | 5.0e-01 | 1.9e-01 | |
0.1 | 5 | 5.2e-03 | 1.2e+01 | 7.5e-01 | 4.6e-01 | |||
0.1 | 5 | 2.4e-01 | 0 | 7.5e-01 | 1.1e+00 | |||
0.1 | 0.27 | 5 | 8.8e-10 | 1.2e+01 | 7.5e-01 | 4.6e-01 | ||
0.1 | 0.27 | 5 | 4.0e-03 | 6.9e+00 | 7.5e-01 | 4.7e-01 | ||
0.1 | 0.27 | 5 | 4.5e-02 | 2.2e+00 | 7.5e-01 | 5.6e-01 | ||
0.1 | 0.27 | 5 | 7.8e-4 | 1.2e+01 | 2.0e+01 | 7.5e-01 | 4.6e-01 | |
0.1 | 0.27 | 5 | 4.0e-03 | 6.0e+00 | 1.2e+01 | 7.5e-01 | 4.8e-01 |
Problem | |
|
Problem | | ||
7.e-2 | 2.3e-01 | 2.9e+00 | |
6.9e-01 | 2.9e+00 | |
7.e-2 | 1.3e-01 | 4.7e+00 | |
1.3e-01 | 4.7e+00 | |
9.e-2 | 2.8e-01 | 2.6e+00 | 7.7e-01 | 2.6e+00 | ||
9.e-2 | 1.9e-01 | 3.8e+00 | 1.9e-01 | 4.5e+00 |
Problem | |
|
Problem | | ||
7.e-2 | 2.3e-01 | 2.9e+00 | |
6.9e-01 | 2.9e+00 | |
7.e-2 | 1.3e-01 | 4.7e+00 | |
1.3e-01 | 4.7e+00 | |
9.e-2 | 2.8e-01 | 2.6e+00 | 7.7e-01 | 2.6e+00 | ||
9.e-2 | 1.9e-01 | 3.8e+00 | 1.9e-01 | 4.5e+00 |
Problem | |
|
Problem | | ||
|
1.e-3 | 3.8e-01 | 7.9e+00 | |
6.7e-01 | 7.9e+00 |
|
1.e-3 | 3.7e-01 | 9.6e+00 | |
3.7e-01 | 9.6e+00 |
5.e-3 | 5.4e-01 | 4.9e+00 | 1.5e+00 | 4.9e+00 | ||
|
5.e-3 | 7.2e-01 | 4.3e+00 | 9.3e-01 | 6.8e+00 |
Problem | |
|
Problem | | ||
|
1.e-3 | 3.8e-01 | 7.9e+00 | |
6.7e-01 | 7.9e+00 |
|
1.e-3 | 3.7e-01 | 9.6e+00 | |
3.7e-01 | 9.6e+00 |
5.e-3 | 5.4e-01 | 4.9e+00 | 1.5e+00 | 4.9e+00 | ||
|
5.e-3 | 7.2e-01 | 4.3e+00 | 9.3e-01 | 6.8e+00 |
Problem | |
|
Problem | | ||
1.e-3 | 3.6e-01 | 1.9e+01 | |
4.0e-01 | 1.9e+01 | |
1.e-3 | 3.6e-01 | 1.9e+01 | |
3.6e-01 | 1.9e+01 | |
1.e-2 | 4.1e-01 | 1.4e+01 | 7.6e-01 | 1.4e+01 | ||
1.e-2 | 5.8e-01 | 1.1e+01 | 7.6e-01 | 1.4e+01 |
Problem | |
|
Problem | | ||
1.e-3 | 3.6e-01 | 1.9e+01 | |
4.0e-01 | 1.9e+01 | |
1.e-3 | 3.6e-01 | 1.9e+01 | |
3.6e-01 | 1.9e+01 | |
1.e-2 | 4.1e-01 | 1.4e+01 | 7.6e-01 | 1.4e+01 | ||
1.e-2 | 5.8e-01 | 1.1e+01 | 7.6e-01 | 1.4e+01 |
Problem | Optimal control |
Problem | Optimal control |
Problem | Optimal control |
Problem | Optimal control |
[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] |
Luis A. Fernández, Cecilia Pola. Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1563-1588. doi: 10.3934/dcdsb.2014.19.1563 |
[8] |
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 |
[9] |
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 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
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 |
[15] |
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 |
[16] |
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 |
[17] |
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 |
[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 |
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