Fick’s Second Law of Diffusion has long been employed to describe in vitro percutaneous permeation experiments: Numerical solution in percutanous drug absorption Thesis, USM, Malaysia

University Universiti Sains Malaysia (USM)
Subject Numerical solution in percutanous drug absorption Thesis
  1. Introduction

Fick’s Second Law of Diffusion has long been employed to describe in vitro percutaneous permeation experiments where the steady state is eventually attained (see [27–29]). When the donor cell concentration, C, is maintained at a constant value, the drug flux per unit area from the donor cell to the receptor cell, through the skin, approaches the steady-state flux, Jss , which is

related to C by the formula
Jss ¼ Kp C;

where Kp is the permeability coefficient. When Fick’s Law prevails, and provided the skin may be regarded as a simple diffusion membrane and the skin–receptor cell-boundary clearance may be assumed to be very large, the permeability coefficient may also be written in the form
Kp ¼ KmDs=Ls; ð2Þ

where Km is the vehicle (donor cell)-skin partition coefficient and Ds is the diffusion coefficient of a drug through the skin which has thickness Ls . Experimental values for Km and Ds for various compounds have been estimated using (1) and (2) in [27–29].

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It is only recently, however, that this simple diffusion membrane model has been applied to processes in which a finite dose is applied topically [1,5,8,10,18,20,26,30] as well as to processes after removal [16,17,20,32]. I

n finite-dose processes, two parameters, the diffusion parameter, kd, and the partition parameter, d , [25] (called the apparent length of diffusion in [18]), are employed.
These are defined by
kd ¼ Ds=L2
d ¼ KmLs;
respectively, and are used in pharmacokinetics when a process approaching the steady state is under consideration. The lag-time, TL , has been estimated to be TL ¼ 1=ð6kdÞ; this estimate was used by Barry [2], Crank [6] and Kubota and Twizell [21]. The parameters kd and d are connected to Kp by the formula
Kp ¼ kd‘d.

Sometimes, in the description of finite-dose percutaneous-absorption kinetics, the diffusivity of a drug in the vehicle, of which the thickness is Lv , is taken to be very large and the skin-receptor cell clearance (or skin-capillary clearance for in vivo conditions) to be very large also [1,5,10,20,26,30]. A few papers describe the diffusion processes of finite doses of a drug, assuming
that the diffusion coefficient of the drug, Dv , through the vehicle has a finite value (see [13,18,19,22]) and that the skin-receptor cell (capillary) clearance per unit area, per unit con- centration excess at the skin surface, CL, also has a finite value [9,13,15–19,32]. When this clearance process is incorporated into the model the normalized skin-capillary clearance, kc , found by the formula
kc ¼ CL=Ls,

is used to describe the pharmacokinetic processes [8,13,15–19,32]. In addition, it is only in very few papers [14,19] that models which regard the skin to be a simple diffusion membrane, consider repeated applications of finite doses of a drug to the same site, though some experimental data associated with the repeated application of a drug [3,4] and the analysis using a certain com-
partment model [9] are available.

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The repeated topical application of a finite dose of a drug is common practice in dermatology and orthopaedics [3,4]. Even for those drugs which produce systemic effect(s) when applied to the skin, the pharmacokinetics after a repeated dose to the same site are worthy of study [9,19,22,23].
Thus, to model the repeated dose using a diffusion model should have important clinical appli-cations. Usually, a drug is applied on a continuous basis and is administered with sufficient frequency that a significant amount persists in the skin when a subsequent dose is applied. When the drug is applied with a fixed dose at a constant dosing interval, the maximum amount in the skin following the second and subsequent doses of the drug is invariably higher than the amount due to the first application. The build-up of drug in the skin is countered by a concurrent increase in the flux of drug to the receptor site.
Therefore, the accumulation of drug in the skin proceeds at a decreasing rate as more and more doses are applied, until the ‘‘steady-state’’ levels of the amount of drug inthe skin and of the flux to the receptor site are reached. In the steady state, the minimum and maximum values of the flux of the drug to the receptor site during a given dosing interval, should be the same as the corresponding values in the preceding or succeeding dosing interval. Similar observations relating to the intravenous injection of infusion and oral administration of a drug have been made in Chapter 3 of the book by Gibaldi and Perrier [7].

The purpose of this paper is to model the repeated topical application of a drug by partial differential equations (PDEs). These are solved using finite-difference methods which are second-order accurate in space and time [33]. The results of numerical experiments will reveal the effects of the change in the application time (the time interval after the application of the vehicle to the skin until its removal) while the dosing interval (the time between consecutive applications) is fixed. Results relating to a change in the dosing interval with fixed application time are also given.

In addition, comparisons are made of results calculated using the ‘‘random walk method’’, em- ployed by Kubota [14] and Kubota et al. [19] to model the repeated application of a finite dose of a drug to the skin, with results obtained using the finite-difference methods[33].

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