Parameter Identification and Model Verification in Systems of Partial Differential Equations Applied to Transdermal Drug Delivery

K. Schittkowski: Mathematics and Computers in Simulation, Vol. 79, 521-538
Abstract: The purpose of this paper is to present some numerical tools which facilitate the interpretation of simulation or parameter estimation results and which allow to compute optimal experimental designs. They help to validate mathematical models describing the dynamical behavior of a biological, chemical, or pharmaceutical system, without requiring a priory knowledge about the physical or chemical background. Although the ideas are quite general, we will concentrate our attention to systems 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 sys\-tem of partial differential equations at designated space values is computed.

A special application model serves as a case study and is outlined in detail. We consider the diffusion of a substrate through cutaneous tissue, where metabolic reactions are included in form of Michaelis-Menten kinetics The goal is to simulate transdermal drug delivery, where it is supposed that experimental data are available for substrate and metabolic fluxes. Numerical results are included based on laboratory data to show typical steps of a model validation procedure, i.e., the interpretation of confidence intervals, the compliance with physical laws, the identification and elimination of redundant model parameters, the computation of optimum experimental designs and the determination of a minimum set of experimental time values.

To download a preprint, click here: pe_valid.pdf

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