Opuscula Math. 37, no. 3 (2017), 403-419
http://dx.doi.org/10.7494/OpMath.2017.37.3.403

 
Opuscula Mathematica

Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics

Maciej Leszczyński
Elżbieta Ratajczyk
Urszula Ledzewicz
Heinz Schättler

Abstract. We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.

Keywords: optimal control, sufficient conditions for optimality, method of characteristics, pharmacodynamic model.

Mathematics Subject Classification: 49K15, 93C15, 92C45.

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  1. B. Bonnard, M. Chyba, Singular Trajectories and their Role in Control Theory, Mathématiques & Applications, vol. 40, Springer Verlag, Paris, 2003.
  2. A. Bressan, B. Piccoli, Introduction to the Mathematical Theory of Control, American Institute of Mathematical Sciences, 2007.
  3. A.E. Bryson (Jr.), Y.C. Ho, Applied Optimal Control, Revised Printing, Hemisphere Publishing Company, New York, 1975.
  4. Z. Chen, J.B. Caillau, Y. Chitour, \(L^1\)-minimization for mechanical systems, SIAM J. on Control and Optimization 54 (2016) 3, 1245-1265.
  5. M.M. Ferreira, U. Ledzewicz, M. do Rosario de Pinho, H. Schättler, A model for cancer chemotherapy with state space constraints, Nonlinear Anal. 63 (2005), 2591-2602.
  6. 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 Research 59 (1999), 4770-4775.
  7. A. Källén, Computational Pharmacokinetics, Chapman and Hall, CRC, London, 2007.
  8. U. Ledzewicz, H. Schättler, Controlling a model for bone marrow dynamics in cancer chemotherapy, Mathematical Biosciences and Engineering 1 (2004), 95-110.
  9. U. Ledzewicz, H. Schättler, The influence of PK/PD on the structure of optimal control in cancer chemotherapy models, Mathematical Biosciences and Engineering (MBE) 2 (2005) 3, 561-578.
  10. P. Macheras, A. Iliadin, Modeling in Biopharmaceutics, Pharmacokinetics and Pharmacodynamics, Interdisciplinary Applied Mathematics, vol. 30, 2nd ed., Springer, New York, 2016.
  11. J. Noble, H. Schättler, Sufficient conditions for relative minima of broken extremals, J. Math. Anal. Appl. 269 (2002), 98-128.
  12. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Macmillan, New York, 1964.
  13. M. Rowland, T.N. Tozer, Clinical Pharmacokinetics and Pharmacodynamics, Wolters Kluwer Lippicott, Philadelphia, 1995.
  14. H. Schättler, U. Ledzewicz, Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38, Springer, New York, 2012.
  15. H. Schättler, U. Ledzewicz, Optimal Control for Mathematical Models of Cancer Therapies, Interdisciplinary Applied Mathematics, vol. 42, Springer, New York, 2015.
  16. H. Schättler, U. Ledzewicz, H. Maurer, Sufficient conditions for strong locak optimality in optimal control problems with \(L_2\)-type objectives and control constraints, Dicrete and Continuous Dynamical Systems, Series B, 19 (2014) 8, 2657-2679.
  17. H.E. Skipper, On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future), Bulletin of Mathematical Biology 48 (1986), 253-278.
  18. G.W. Swan, Role of optimal control in cancer chemotherapy, Mathematical Biosciences 101 (1990), 237-284.
  19. A. Swierniak, Optimal treatment protocols in leukemia - modelling the proliferation cycle, Proc. of the 12th IMACS World Congress, Paris, 4 (1988), 170-172.
  • Maciej Leszczyński
  • Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
  • Elżbieta Ratajczyk
  • Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
  • Urszula Ledzewicz
  • Lodz University of Technology, Institute of Mathematics, 90-924 Lodz, Poland
  • Southern Illinois University Edwardsville, Department of Mathematics and Statistics, Edwardsville, Il, 62026-1653, USA
  • Heinz Schättler
  • Washington University, Department of Electrical and Systems Engineering, St. Louis, Mo, 63130, USA
  • Communicated by Marek Galewski.
  • Received: 2016-09-06.
  • Revised: 2016-10-10.
  • Accepted: 2016-10-10.
  • Published online: 2017-01-30.
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Cite this article as:
Maciej Leszczyński, Elżbieta Ratajczyk, Urszula Ledzewicz, Heinz Schättler, Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics, Opuscula Math. 37, no. 3 (2017), 403-419, http://dx.doi.org/10.7494/OpMath.2017.37.3.403

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