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August  2014, 19(6): 1563-1588. doi: 10.3934/dcdsb.2014.19.1563

Catalog of the optimal controls in cancer chemotherapy for the Gompertz model depending on PK/PD and the integral constraint

Luis A. Fernández 1, and  Cecilia Pola 1,

1. 

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

Received  August 2013 Revised  February 2014 Published  June 2014

We study three optimal control problems associated with Gompertz-type differential equations, including bound control and integral constraints. These problems can be interpreted in terms of planning anticancer therapies. Existence of optimal controls is proved and all their possible structures are determined in detail, by using the Pontryagin's Maximum Principle. The influence of the pharmacokinetics and pharmacodynamics variants, together with the integral constraint, is analyzed. Moreover, the numerical results of some illustrative examples and our conclusions are presented.
Keywords: pharmacokinetics, singular controls., cancer chemotherapy, pharmacodynamics, Optimal control.
Mathematics Subject Classification: 93C15, 49K15, 49M05, 49M37, 92C5.
Citation: 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
References:
[1]

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

[2]

J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments,, Math. Model. Nat. Phenom., 4 (2009), 12.  doi: 10.1051/mmnp/20094302.  Google Scholar

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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.  doi: 10.1002/oca.957.  Google Scholar

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A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Math. Biosci., 222 (2009), 13.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

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L. C. Evans, Partial Differential Equations,, AMS, (1998).   Google Scholar

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K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies,, SIAM J. Appl. Math., 63 (2003), 1954.  doi: 10.1137/S0036139902413489.  Google Scholar

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P. Hartman, Ordinary Differential Equations,, Birkhäuser, (1982).   Google Scholar

[8]

W. Krabs and S. Pickl, An optimal control problem in cancer chemotherapy,, Appl. Math. Comput., 217 (2010), 1117.  doi: 10.1016/j.amc.2010.05.008.  Google Scholar

[9]

A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490.   Google Scholar

[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.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[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.  doi: 10.3934/mbe.2005.2.561.  Google Scholar

[12]

U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy,, Math. Biosci., 206 (2007), 320.  doi: 10.1016/j.mbs.2005.03.013.  Google Scholar

[13]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1994).  doi: 10.1142/9789812832542.  Google Scholar

[14]

J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit,, Math. Biosci., 100 (1990), 49.  doi: 10.1016/0025-5564(90)90047-3.  Google Scholar

[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.  doi: 10.1145/1731022.1731032.  Google Scholar

[16]

G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma,, Bull. Math. Biol., 39 (1977), 317.   Google Scholar

[17]

G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[18]

A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy,, Eur. J. Pharmacol., 625 (2009), 108.  doi: 10.1016/j.ejphar.2009.08.041.  Google Scholar

show all references

References:
[1]

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

[2]

J. Clairambault, Modelling physiological and pharmacological control on cell proliferation to optimise cancer treatments,, Math. Model. Nat. Phenom., 4 (2009), 12.  doi: 10.1051/mmnp/20094302.  Google Scholar

[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.  doi: 10.1002/oca.957.  Google Scholar

[4]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Math. Biosci., 222 (2009), 13.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[5]

L. C. Evans, Partial Differential Equations,, AMS, (1998).   Google Scholar

[6]

K. R. Fister and J. C. Panetta, Optimal control applied to competing chemotherapeutic cell-kill strategies,, SIAM J. Appl. Math., 63 (2003), 1954.  doi: 10.1137/S0036139902413489.  Google Scholar

[7]

P. Hartman, Ordinary Differential Equations,, Birkhäuser, (1982).   Google Scholar

[8]

W. Krabs and S. Pickl, An optimal control problem in cancer chemotherapy,, Appl. Math. Comput., 217 (2010), 1117.  doi: 10.1016/j.amc.2010.05.008.  Google Scholar

[9]

A. K. Laird, Dynamics of tumour growth,, Br. J. Cancer, 18 (1964), 490.   Google Scholar

[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.  doi: 10.3934/mbe.2011.8.307.  Google Scholar

[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.  doi: 10.3934/mbe.2005.2.561.  Google Scholar

[12]

U. Ledzewicz and H. Schättler, Optimal controls for a model with pharmacokinetics maximizing bone marrow in cancer chemotherapy,, Math. Biosci., 206 (2007), 320.  doi: 10.1016/j.mbs.2005.03.013.  Google Scholar

[13]

R. Martin and K. L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy,, World Scientific, (1994).  doi: 10.1142/9789812832542.  Google Scholar

[14]

J. M. Murray, Some optimal control problems in cancer chemotherapy with a toxicity limit,, Math. Biosci., 100 (1990), 49.  doi: 10.1016/0025-5564(90)90047-3.  Google Scholar

[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.  doi: 10.1145/1731022.1731032.  Google Scholar

[16]

G. W. Swan and T. L. Vincent, Optimal control analysis in the chemotherapy of IgG multiple myeloma,, Bull. Math. Biol., 39 (1977), 317.   Google Scholar

[17]

G. W. Swan, Role of optimal control theory in cancer chemotherapy,, Math. Biosci., 101 (1990), 237.  doi: 10.1016/0025-5564(90)90021-P.  Google Scholar

[18]

A. Swierniak, M. Kimmel and J. Smieja, Mathematical modeling as a tool for planning anticancer therapy,, Eur. J. Pharmacol., 625 (2009), 108.  doi: 10.1016/j.ejphar.2009.08.041.  Google Scholar

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