Geodesic
Optimal control for a mathematical model for chemotherapy with pharmacometrics
Maciej Leszczyński1 ; Urszula Ledzewicz1, 2 ; Heinz Schättler3
1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland.
2 Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il 62026-1653, USA.
3 Department of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130 USA.
Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 69.

Voir la notice de l'article provenant de la source EDP Sciences

An optimal control problem for an abstract mathematical model for cancer chemotherapy is considered. The dynamics is for a single drug and includes pharmacodynamic (PD) and pharmacokinetic (PK) models. The aim is to point out qualitative changes in the structures of optimal controls that occur as these pharmacometric models are varied. This concerns (i) changes in the PD-model for the effectiveness of the drug (e.g., between a linear log-kill term and a non-linear Michaelis-Menten type Emax-model) and (ii) the question how the incorporation of a mathematical model for the pharmacokinetics of the drug effects optimal controls. The general results will be illustrated and discussed in the framework of a mathematical model for anti-angiogenic therapy.
DOI : 10.1051/mmnp/2020008

Maciej Leszczyński 1 ; Urszula Ledzewicz 1, 2 ; Heinz Schättler 3

1 Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland.
2 Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Il 62026-1653, USA.
3 Department of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130 USA. 
@article{MMNP_2020_15_a43,
     author = {Maciej Leszczy\'nski and Urszula Ledzewicz and Heinz Sch\"attler},
     title = {Optimal control for a mathematical model for chemotherapy with pharmacometrics},
     journal = {Mathematical modelling of natural phenomena},
     eid = {69},
     publisher = {mathdoc},
     volume = {15},
     year = {2020},
     doi = {10.1051/mmnp/2020008},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2020008/}
}
TY  - JOUR
AU  - Maciej Leszczyński
AU  - Urszula Ledzewicz
AU  - Heinz Schättler
TI  - Optimal control for a mathematical model for chemotherapy with pharmacometrics
JO  - Mathematical modelling of natural phenomena
PY  - 2020
VL  - 15
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2020008/
DO  - 10.1051/mmnp/2020008
LA  - en
ID  - MMNP_2020_15_a43
ER  - 
%0 Journal Article
%A Maciej Leszczyński
%A Urszula Ledzewicz
%A Heinz Schättler
%T Optimal control for a mathematical model for chemotherapy with pharmacometrics
%J Mathematical modelling of natural phenomena
%D 2020
%V 15
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2020008/
%R 10.1051/mmnp/2020008
%G en
%F MMNP_2020_15_a43
Maciej Leszczyński; Urszula Ledzewicz; Heinz Schättler. Optimal control for a mathematical model for chemotherapy with pharmacometrics. Mathematical modelling of natural phenomena, Tome 15 (2020), article  no. 69. doi : 10.1051/mmnp/2020008. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/2020008/

[1] B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory, in In Vol. 40 of Mathématiques Applications. Springer, Paris (2003).

[2] A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, California (2007).

[3] A.E. Bryson, Jr. and Y.C. Ho, Applied Optimal Control, Revised Printing. Hemisphere Publishing Company, New York (1975).

[4] C.S. Chou and A. Friedman, Introduction to Mathematical Biology - Modeling, Analysis and Simulation. Springer Verlag (2016).

[5] M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy. Vol. 30 of Lecture Notes in Biomathematics, Springer, Berlin (1979).

[6] L.A. Fernández, C. Pola Optimal control problems for the Gompertz model under the Norton-Simon hypothesis in chemotherapy Discr. Cont. Dyn. Syst. Ser. B 2019 2577 2612

[7] P. Hahnfeldt, D. Panigrahy, J. Folkman, L. Hlatky Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy Cancer Res 1999 4770 4775

[8] A. Källén, Computational Pharmacokinetics. Chapman and Hall, CRC, London (2007).

[9] H.K. Khalil, Nonlinear Systems, 3rded. Prentice Hall, Upper Saddle River, NJ (2002).

[10] M. Kimmel, A. Swierniak An optimal control problem related to leukemia chemotherapy Sci. Bull. Silesian Tech. Univ 1983 120 130

[11] U. Ledzewicz, H. Maurer and H. Schättler, Minimizing tumor volume for a mathematical model of anti-angiogenesis with linear pharmacokinetics, in Recent Advances in Optimization and its Applications in Engineering, edited by M. Diehl, F. Glineur, E. Jarlebring and W. Michiels. Springer, Heidelberg (2010) 267–276.

[12] U. Ledzewicz, H. Maurer, H. Schättler Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy Math. Biosci. Eng 2011 3-7 323

[13] U. Ledzewicz, H. Maurer, H. Schättler Optimal combined radio- and anti-angiogenic cancer therapy J. Optim. Theory Appl 2019 321 340

[14] U. Ledzewicz, H. Moore Dynamical systems properties of a mathematical model for the treatment of CML Appl. Sci 2016 291

[15] U. Ledzewicz, H. Moore Optimal control applied to a generalized Michaelis-Menten model of CML therapy Dicr. Cont. Dyn. Syst. Ser. B 2018 331 346

[16] U. Ledzewicz, H. Schättler Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy J. Optim. Theory Appl 2002 609 637

[17] U. Ledzewicz, H. Schättler The influence of PK/PD on the structure of optimal control in cancer chemotherapy models Math. Biosci. Eng 2005 561 578

[18] U. Ledzewicz, H. Schättler Antiangiogenic therapy in cancer treatment as an optimal control problem SIAM J. Control Optim 2007 1052 1079

[19] U. Ledzewicz, H. Schättler Controlling a model for bone marrow dynamics in cancer chemotherapy Math. Biosci. Eng 2004 95 110

[20] U. Ledzewicz, H. Schättler Singular controls and chattering arcs in optimal control problems arising in biomedicine Control Cybern 2009 1501 1523

[21] U. Ledzewicz, H. Schättler Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments J. Optim. Theory Appl 2012 195 224

[22] M. Leszczyński, The Role of Pharmacometrics in Optimal Controls Problems for Mathematical Models of Cancer Therapies. Ph.D. thesis, Lodz University of Technology, Lodz, Poland (2019).

[23] M. Leszczyński, E. Ratajczyk, U. Ledzewicz, H. Schättler Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics Opuscula Math 2017 403 419

[24] M. Leszczyński, U. Ledzewicz, H. Schättler Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics Discr. Cont. Dyn. Syst. Ser. B 2019 2315 2334

[25] P. Macheras and A. Iliadin, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, in Vol. 30 of Interdisciplinary Applied Mathematics, 2nd ed. Springer, New York (2016).

[26] R. Martin and K.L. Teo, Optimal Control of Drug Administration in Cancer Chemotherapy. World Scientific Press, Singapore (1994).

[27] A. D’Onofrio, U. Ledzewicz, H. Maurer, H. Schättler On optimal delivery of combination therapy for tumors Math. Biosci 2009 13 26

[28] L.G. De Pillis, A. Radunskaya A mathematical tumor model with immune resistance and drug therapy: an optimal control approach J. Theor. Med 2001 79 100

[29] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes. Macmillan, New York (1964).

[30] M. Rowland and T.N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia (1995).

[31] H. Schättler and U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, Vol. 38, Springer, New York (2012).

[32] H. Schättler and U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, Vol. 42, Springer, New York (2015).

[33] H. Schättler, U. Ledzewicz, H. Maurer Sufficient conditions for strong locak optimality in optimal control problems with L2 -type objectives and control constraints Dicr. Cont. Dyn. Syst. Ser. B 2014 2657 2679

[34] S. Shimoda, K. Nishida, M. Sakakida, Y. Konno, K. Ichinose, M. Uehara, T. Nowak, M. Shichiri Closed-loop subcutaneous insulin infusion algorithm with a short-acting insulin analog for long-term clinical application of a wearable artificial endocrine pancreas Front. Med. Biol. Eng 1997 197 211

[35] H.E. Skipper On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future) Bull. Math. Biol 1986 253 278

[36] G.W. Swan, Applications of Optimal Control Theory in Medicine, Marcel Dekker, New York (1984).

[37] G.W. Swan General applications of optimal control theory in cancer chemotherapy IMA J. Math. Appl. Med. Biol 1988 303 316

[38] G.W. Swan Role of optimal control in cancer chemotherapy Math. Biosci 1990 237 284

[39] A. Swierniak Optimal treatment protocols in leukemia - modelling the proliferation cycle Proc. of the 12th IMACS World Congress, Paris 4 1988 170 172

[40] A. Swierniak Cell cycle as an object of control J. Biol. Syst 1995 41 54

Cité par Sources :