American Institute of Mathematical Sciences

Advanced
June  2019, 24(6): 2577-2612. doi: 10.3934/dcdsb.2018266

Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy

Luis A. Fernández and  Cecilia Pola

Dep. Matemáticas, Estadística y Computación, Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain

Received  November 2017 Revised  April 2018 Published  October 2018

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.

Keywords: Optimal control, cancer chemotherapy, pharmacodynamics, pharmacokinetics, singular controls.
Mathematics Subject Classification: 93C15, 49K15, 49M05, 49M37, 92C50.
Citation: Luis A. Fernández, Cecilia Pola. Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2577-2612. doi: 10.3934/dcdsb.2018266
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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[12]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994. doi: 10.1142/2048.  Google Scholar

[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.  Google Scholar

[14]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.   Google Scholar

[15]

M. C. Perry, D. C. Doll and C. E. Freter, Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[4]

L. Cesari, Optimization—Theory and Applications, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[12]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy, World Scientific, Singapore, 1994. doi: 10.1142/2048.  Google Scholar

[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.  Google Scholar

[14]

L. Norton and R. Simon, Tumor size, sensitivity to therapy, and design of treatment schedules, Cancer Treat Rep., 61 (1977), 1307-1317.   Google Scholar

[15]

M. C. Perry, D. C. Doll and C. E. Freter, Perry's The Chemotherapy Source Book, Wolters Kluwer Health, Philadelphia, 2012. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Optimization results for Skipper model on the left (see Table 1) and for the $E_{max}$ model on the right (see Table 2)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Some optimization results for $E_{max}$ model (see Table 2)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Tumor volume with trivial controls for Skipper model (see Table 1)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Tumor volume with trivial controls for $E_{max}$ model (see Table 2)
Figure Options

Download full-size image

Download as PowerPoint slide

Table 3.  Switching times with $G = G_2$, $ k_1 = 4$ and $k_2 = 0.25$
Example Structure S_times $u_{sin}$
$ES_{52o2}$ $u_{max}/0$ 1.4e-01
$EM_{52o2}$ $u_{max}/0$ 2.3e+00
$EM_{52o3}$ $u_{max} / u_{sin}/$
$/0$		 5.6e-01
2.1e+00		 3.4e-01
$EL_{52o2}$ $u_{max}/0$ 2.9e+00
$EL_{52o3}$ $u_{max}/ u_{sin}/$
$/0$		 6.8e-01
2.1e+00		 4.0e-01
$ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02
$EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01
$EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
Table Options

Download as excel

Example Structure S_times $u_{sin}$
$ES_{52o2}$ $u_{max}/0$ 1.4e-01
$EM_{52o2}$ $u_{max}/0$ 2.3e+00
$EM_{52o3}$ $u_{max} / u_{sin}/$
$/0$		 5.6e-01
2.1e+00		 3.4e-01
$EL_{52o2}$ $u_{max}/0$ 2.9e+00
$EL_{52o3}$ $u_{max}/ u_{sin}/$
$/0$		 6.8e-01
2.1e+00		 4.0e-01
$ES_{62o2}$ $u_{max} / u_{sin}$ 1.6e-01 9.9e-02
$EM_{62o2}$ $u_{max} / u_{sin}$ 1.0e+00 5.6e-01
$EL_{62o2}$ $u_{max} / u_{sin}$ 1.4e+00 7.5e-01
Table 1.  Parameters and numerical results with $G = G_1$, $ k_1 = 4$
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$
$ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02
$ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01
$ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00
$ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01
$ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01
$ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00
$ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01
$ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01
$ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00
$EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01
$EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01
$EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00
$EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01
$EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01
$EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01
$EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01
$EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00
$EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01
$EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01
$EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
Table Options

Download as excel

Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $L(T)/L_0$
$ES_{41a1}$ 0.1 1 2.3e-03 2.4e+00 5.0e-02 1.7e-02
$ES_{41a2}$ 0.1 1 3.0e-02 9.5e-01 5.0e-02 3.8e-01
$ES_{41a3}$ 0.1 1 7.3e-02 0 5.0e-02 1.3e+00
$ES_{51a1}$ 0.1 0.27 1 9.0e-05 2.4e+00 5.0e-02 3.0e-01
$ES_{51a2}$ 0.1 0.27 1 3.0e-02 7.0e-01 5.0e-02 7.0e-01
$ES_{51a3}$ 0.1 0.27 1 6.4e-02 0 5.0e-02 1.3e+00
$ES_{61a1}$ 0.1 0.27 1 2.5e-02 2.4e+00 1.1e+00 5.0e-02 3.0e-01
$ES_{61a2}$ 0.1 0.27 1 3.0e-02 1.8e+00 9.5e-01 5.0e-02 3.8e-01
$ES_{61a3}$ 0.1 0.27 1 7.3e-02 0 0 5.0e-02 1.3e+00
$EM_{41c1}$ 0.1 1 1.2e-01 2.4e+00 5.0e-01 3.9e-01
$EM_{41c2}$ 0.1 1 1.4e-01 1.9e+00 5.0e-01 5.2e-01
$EM_{41c3}$ 0.1 1 1.5e-01 0 5.0e-01 1.0e+00
$EM_{51a1}$ 0.1 0.27 1 0 2.4e+00 5.0e-01 7.6e-01
$EM_{51a2}$ 0.1 0.27 1 5.9e-02 1.4e+00 5.0e-01 8.2e-01
$EM_{61b1}$ 0.1 0.27 1 1.4e-01 2.4e+00 1.1e+00 5.0e-01 7.6e-01
$EL_{41b1}$ 0.1 1 1.1e-01 2.4e+00 7.5e-01 6.8e-01
$EL_{41b2}$ 0.1 1 1.2e-01 0 7.5e-01 1.0e+00
$EL_{51a1}$ 0.1 0.27 1 0 2.4e+00 7.5e-01 8.9e-01
$EL_{51a2}$ 0.1 0.27 1 3.5e-02 1.5e+00 7.5e-01 9.1e-01
$EL_{61b2}$ 0.1 0.27 1 9.3e-02 2.4e+00 1.1e+00 7.5e-01 8.9e-01
Table 2.  Parameters and numerical results with $G = G_2$, $ k_1 = 4$ and $k_2 = 0.25$
Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $ L(T)/L_0$
$ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01
$ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01
$ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00
$ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01
$ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01
$ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00
$ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01
$ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01
$ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00
$EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01
$EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01
$EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00
$EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01
$EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01
$EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01
$EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01
$EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01
$EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01
$EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00
$EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01
$EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01
$EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01
$EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01
$EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
Table Options

Download as excel

Example $\xi$ $\lambda$ T $\hat{\alpha}$ $\int_0^T \bar{u}(t) dt$ $\int_0^T \bar{c}(t) dt$ $L_0/\theta$ $ L(T)/L_0$
$ES_{42a1}$ 0.1 1 1.1e-03 2.4e+00 5.0e-02 4.1e-01
$ES_{42a2}$ 0.1 1 3.0e-02 3.6e-01 5.0e-02 6.5e-01
$ES_{42a3}$ 0.1 1 2.9e-01 0 5.0e-02 1.3e+00
$ES_{52o1}$ 0.1 0.27 1 5.2e-07 2.4e+00 5.0e-02 5.2e-01
$ES_{52o2}$ 0.1 0.27 1 3.0e-02 3.4e-01 5.0e-02 7.1e-01
$ES_{52o3}$ 0.1 0.27 1 2.4e-1 0 5.0e-02 1.3e+00
$ES_{62o1}$ 0.1 0.27 1 1.7e-03 2.4e+00 1.1e+00 5.0e-02 5.2e-01
$ES_{62o2}$ 0.1 0.27 1 3.0e-02 4.6e-01 3.4e-01 5.0e-02 6.7e-01
$ES_{62o3}$ 0.1 0.27 1 2.9e-01 0 0 5.0e-02 1.3e+00
$EM_{42a1}$ 0.1 5 2.7e-03 1.2e+01 5.0e-01 1.5e-01
$EM_{42a2}$ 0.1 5 4.0e-03 1.0e+01 5.0e-01 1.7e-01
$EM_{42a3}$ 0.1 5 4.5e-01 0 5.0e-01 1.3e+00
$EM_{52o1}$ 0.1 0.27 5 4.4e-10 1.2e+01 5.0e-01 1.5e-01
$EM_{52o2}$ 0.1 0.27 5 4.0e-03 5.6e+00 5.0e-01 1.7e-01
$EM_{52o3}$ 0.1 0.27 5 4.5e-02 1.9e+00 5.0e-01 2.8e-01
$EM_{62o1}$ 0.1 0.27 5 4.2e-04 1.2e+01 2.0e+01 5.0e-01 1.5e-01
$EM_{62o2}$ 0.1 0.27 5 4.0e-03 4.7e+00 9.5e+00 5.0e-01 1.9e-01
$EL_{42a1}$ 0.1 5 5.2e-03 1.2e+01 7.5e-01 4.6e-01
$EL_{42a3}$ 0.1 5 2.4e-01 0 7.5e-01 1.1e+00
$EL_{52o1}$ 0.1 0.27 5 8.8e-10 1.2e+01 7.5e-01 4.6e-01
$EL_{52o2}$ 0.1 0.27 5 4.0e-03 6.9e+00 7.5e-01 4.7e-01
$EL_{52o3}$ 0.1 0.27 5 4.5e-02 2.2e+00 7.5e-01 5.6e-01
$EL_{62o1}$ 0.1 0.27 5 7.8e-4 1.2e+01 2.0e+01 7.5e-01 4.6e-01
$EL_{62o2}$ 0.1 0.27 5 4.0e-03 6.0e+00 1.2e+01 7.5e-01 4.8e-01
Table 4.  Comparison Min, Max therapies with $G = G_1$, $ k_1 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 2$ and $L_0 = 0.5\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00
$(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00
$(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00
$(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
Table Options

Download as excel

Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 7.e-2 2.3e-01 2.9e+00 $(\widehat{OP}_{2MAX})$ 6.9e-01 2.9e+00
$(\widehat{OP}_6)$ 7.e-2 1.3e-01 4.7e+00 $(\widehat{OP}_{3MAX})$ 1.3e-01 4.7e+00
$(\widehat{OP}_5)$ 9.e-2 2.8e-01 2.6e+00 $(\widehat{OP}_{2MAX})$ 7.7e-01 2.6e+00
$(\widehat{OP}_6)$ 9.e-2 1.9e-01 3.8e+00 $(\widehat{OP}_{3MAX})$ 1.9e-01 4.5e+00
Table 5.  Comparison Min, Max therapies with $G = G_2$, $ k_1 = 4$, $ k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 4$ and$L_0 = 0.05\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00
$(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00
$(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00
$(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
Table Options

Download as excel

Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.8e-01 7.9e+00 $(\widehat{OP}_{2MAX})$ 6.7e-01 7.9e+00
$(\widehat{OP}_6)$ 1.e-3 3.7e-01 9.6e+00 $(\widehat{OP}_{3MAX})$ 3.7e-01 9.6e+00
$(\widehat{OP}_5)$ 5.e-3 5.4e-01 4.9e+00 $(\widehat{OP}_{2MAX})$ 1.5e+00 4.9e+00
$(\widehat{OP}_6)$ 5.e-3 7.2e-01 4.3e+00 $(\widehat{OP}_{3MAX})$ 9.3e-01 6.8e+00
Table 6.  Comparison Min, Max therapies with $G = G_2$, $ k_1 = 4$, $ k_2 = 4$, $\lambda = 0.27$, $\xi = 0.1$, $T = 8$ and $L_0 = 0.5\theta$
Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01
$(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01
$(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01
$(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
Table Options

Download as excel

Problem $\hat{\alpha}$ $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$ Problem $ L(T)/L_0$ $\int_0^T \bar{u}(t) dt$
$(\widehat{OP}_5)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{2MAX})$ 4.0e-01 1.9e+01
$(\widehat{OP}_6)$ 1.e-3 3.6e-01 1.9e+01 $(\widehat{OP}_{3MAX})$ 3.6e-01 1.9e+01
$(\widehat{OP}_5)$ 1.e-2 4.1e-01 1.4e+01 $(\widehat{OP}_{2MAX})$ 7.6e-01 1.4e+01
$(\widehat{OP}_6)$ 1.e-2 5.8e-01 1.1e+01 $(\widehat{OP}_{3MAX})$ 7.6e-01 1.4e+01
Table 7.  Catalog for Skipper model ($G = G_1$) under $({\bf{H}}_{\bf{1}}))$
Problem Optimal control
$(OP_1)$ $u_{max}/0$ (*)
$(OP_2)$ $u_{max}/0$
$(OP_3)$ $u_{max}/0$ (*)
Table Options

Download as excel

Problem Optimal control
$(OP_1)$ $u_{max}/0$ (*)
$(OP_2)$ $u_{max}/0$
$(OP_3)$ $u_{max}/0$ (*)
Table 8.  Catalog for $E_{max}$ model ($G = G_2$) under $({\bf{H}}_{\bf{1}}))$
Problem Optimal control
$(OP_1)$ $u_{sin}$
$(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$
$(OP_3)$ $u_{max}/u_{sin}$
Table Options

Download as excel

Problem Optimal control
$(OP_1)$ $u_{sin}$
$(OP_2)$ $u_{max}/0$   $u_{max}/u_{sin}/0$
$(OP_3)$ $u_{max}/u_{sin}$
[1]

Shuo Wang, Heinz Schättler. Optimal control for cancer chemotherapy under tumor heterogeneity with Michealis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2383-2405. doi: 10.3934/dcdsb.2019100
[2]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129
[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. 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
[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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031
[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]

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
[14]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & 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 & Continuous Dynamical Systems - B, 2019, 24 (5) : 2039-2052. doi: 10.3934/dcdsb.2019083
[18]

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
[19]

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 & Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028
[20]

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 & Continuous Dynamical Systems - B, 2019, 24 (5) : 2017-2038. doi: 10.3934/dcdsb.2019082

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (94)
  • HTML views (520)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences