Abstract
A numerical approach is described to determine parameters in a system of one-dimensional partial differential equations and coupled ordinary differential equations. The model allows arbitrary transition conditions between separate integration areas for functions and derivatives. The minimum least squares distance of the measured data from the solution of a system of differential equations at designated space values is computed.
A special application model is outlined in detail, that describes the diffusion of a substrate through cutaneous tissue. Metabolic reactions are included in form of Michaelis-Menten kinetics. The goal is to model transdermal drug delivery, where it is supposed that experimental data are available for substrate and metabolic fluxes. Some numerical results are included to show the efficiency of the implemented algorithms.
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References
P. Boderke, K. Schittkowski, M. Wolf, H.P. Merkle (1998), A mathematical model for diffusion and concurrent metabolism in metabolically active tissue,submitted for publication.
J.E. Dennis jr., D.M. Gay, R.E. Welsch (1993), Algorithm 573: NL2SOL-An adaptive nonlinear least-squares algorithm, ACM Transactions on Mathematical Software, Vol. 7, No. 3, pp. 369–383.
M. Dobmann, M. Liepelt, K. Schittkowski, Algorithm 746: PCOMP: A FORTRAN code for automatic differentiation, ACM Transactions on Mathematical Software,Vol. 21,No. 3, pp. 233–266.
R.H. Guy, J. Hadgraft (1988), Physicochemical aspects of percutaneous penetration and its enhancement, Pharmaceutical Research, Vol. 5, No. 12, pp. 753–758.
J. Hadgraft (1979), The epidermal reservoir, a theoretical approach, International Journal of Pharmaceutics, Vol. 2, pp. 265–274.
E. Hairer, S.P. Norsett, G. Wanner (1993), Solving Ordinary Differential Equa-tions I. Nonstiff Problems, Series Computational Mathematics, Springer.
E. Hairer, G. Wanner (1991), Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Series Computational Mathematics, Springer.
S.A.M. Hotchkiss (1992), Skin as a xenobiotic metabolizing organ, in: Process in Drug Metabolism, G.G. Gibson ed., Taylor and Francis Ltd., London, pp. 217–262.
P. Lindström (1983), A general purpose algorithm for nonlinear least squares problems with nonlinear constraints, Report UMINF-103. 83, Institute of Information Processing, University of Umea, Umea, Sweden.
W.B. Pratt, P. Taylor (1990), Principles of Drug Action, Churchill Livingstone, New York.
W. E. Schiesser (1991), The Numerical Method of Lines, Academic Press, Inc., San Diego.
K. Schittkowski (1985), NLPQL: A FORTRAN subroutine solving constrained nonlinear programming problems, Annals of Operations Research, Vol. 5, pp. 485–500.
K. Schittkowski (1988), Solving nonlinear least squares problems by a general purpose SQP-method, in: Trends in Mathematical Optimization, K.-H. Hoffmann, J.-B. Hiriart-Urruty, C. Lemarechal, J. Zowe eds., International Series of Numerical Mathematics, Vol. 84, Birkhäuser, pp. 295–309.
K. Schittkowski (1997a), EASY-FIT: Parameter estimation in dynamical systems-User’s guide, Report, Department of Mathematics, University of Bayreuth, Germany.
K. Schittkowski (1997b), Parameter estimation in one-dimensional time-dependent partial differential equations, Optimization Methods and Software, Vol. 7, No. 3–4, pp. 165–210.
K. Schittkowski (1998), PDEFIT: A FORTRAN subroutine for estimating parameters in systems of partial differential equations to appear: Optimization Methods and Software.
I. Steinsträsser (1994), The organized Ha CaT cell culture sheet: A model approach to study epidermal peptide drug metabolism, Dissertation, Pharmaceutical Institute, ETH Zürich, Switzerland.
M. Wolf (1994), Mathematisch-physikalische Berechnungs-and Simulationsmodelle zur Beschreibung and Entwicklung therapeutischer Systeme, Habilitation, Faculty of Mathematics and Natural Sciences, University of Bonn, Germany.
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Schittkowski, K. (2000). Parameter Estimation in a Mathematical Model for Substrate Diffusion in a Metabolically Active Cutaneous Tissue. In: Yang, X., Mees, A.I., Fisher, M., Jennings, L. (eds) Progress in Optimization. Applied Optimization, vol 39. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0301-5_22
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DOI: https://doi.org/10.1007/978-1-4613-0301-5_22
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