A two-dimensional mathematical model of non-linear dual-sorption of percutaneous drug absorption
Authors: George K 
Published: July 2005

Abstract
Background
Certain drugs, for example scopolamine and timolol, show non-linear kinetic behavior during permeation process. This non-linear kinetic behavior is due to two mechanisms; the first mechanism being a simple dissolution producing mobile and freely diffusible molecules and the second being an adsorption process producing non-mobile molecules that do not participate in the diffusion process. When such a drug is applied on the skin surface, the concentration of the drug accumulated in the skin and the amount of the drug eliminated into the blood vessel depend on the value of a parameter, C, the donor concentration. The present paper studies the effect of the parameter value, C, when the region of the contact of the skin with drug, is a line segment on the skin surface. To confirm that dual-sorption process gives an explanation to non-linear kinetic behavior, the characteristic features that are used in one-dimensional models are (1) prolongation of half-life if the plot of flux versus time are straight lines soon after the vehicle removal, (2) the decrease in half-life with increase in donor concentration. This paper introduces another feature as a characteristic to confirm that dual-sorption model gives an explanation to the non-linear kinetic behavior of the drug. This new feature is "the prolongation of half-life is not a necessary feature if the plots of drug flux versus time is a non-linear curve, soon after the vehicle removal".

Methods
From biological point of view, a drug absorption model is said to be nonlinear if the sorption isotherm is non-linear. When a model is non-linear the relationship between lag-time and donor concentration is non-linear and the lag time decreases with increase in donor concentration. A two-dimensional dual-sorption model is developed for percutaneous absorption of a drug, which shows non-linear kinetic behavior in the permeation process. This model may be used when the diffusion of the drug in the direction parallel to the skin surface must be examined, as well as in the direction into the skin, examined in one-dimensional models. The dual-sorption model is an initial/boundary value problem which consists of (1) one non-linear, two-dimensional, second-order parabolic equation, (2) boundary conditions, (3) one initial condition. Note that, the number of boundary conditions are, six and four, respectively, if the permeation process under consideration is, during the application of the vehicle and during the removal of the vehicle. Adopting the approach of method of lines, the initial/boundary value problem is transformed into an initial-value problem, which consists of (1) a system of non-linear ordinary differential equations, (2) one initial condition. The system of non-linear ordinary differential equations contains time-dependent non-homogeneous terms, if the permeation process under consideration is, during the application of the vehicle. To solve this initial-value problem, an eight-stage sequential algorithm which is second-order accurate, and requires only tri-diagonal solvers, is developed.

Results
Simulation of the numerical methods described is carried out with various values of the parameter C. The illustrations are given in the form of figures. The concentration profiles are viewed as parabolas along the mesh lines parallel to x-axis or y-axis. The flow rates in different subregions of the skin-region are studied. The shapes of the concentration profiles are examined before and after the steady-state concentration is reached. The concentration reaches steady-state when the flux reaches the steady state. The plots of flux versus time and cumulative amount of drug eliminated into the receptor cell versus time are given.

Conclusion
Based on the various values of the parameter, C, conclusions are drawn about (1) flow rate of the drug in different regions of the skin, (2) shape of the concentration profiles, (3) the time required to reach the steady-state value of the concentration, (4) concentration of the drug in different regions of the skin, when steady-state value of the concentration is reached, (5) the time required to reach the steady-state value of the flux, (6) time required to reach the steady-state value of the concentration of the drug, (7) half-life of the concentration of the drug and (8) lag-time.
A comparison, between this two-dimensional model and the one-dimensional non-linear dual-sorption model that exists in the literature, is done based on (1) the shape of the concentration profiles at various time levels, (2) the time required to reach the steady-state value of the concentration, (3) lag-time and (4) half-life.