A random walk method for percutaneous drug absorption pharmacokinetics: application to repeated administration of a therapeutic timolol patch

Application of numerical methods for diffusion-based modeling of skin permeation

The application of numerical methods for mechanistic, diffusion-based modeling of skin permeation is reviewed. Methods considered here are finite difference, method of lines, finite element, finite volume, random walk, cellular automata, and smoothed particle hydrodynamics. First the methods are briefly explained with rudimentary mathematical underpinnings. Current state of the art numerical models are described, and then a chronological overview of published models is provided. Key findings and insights of reviewed models are highlighted. Model results support a primarily transcellular pathway with anisotropic lipid transport. Future endeavors would benefit from a fundamental analysis of drug/vehicle/skin interactions.

Repeated applications of a transdermal patch: Analytical solution and optimal control of the delivery rate

The integral transform technique was implemented to solve a mathematical model developed for percutaneous drug absorption. The model included repeated application and removal of a patch from the skin. Fick's second law of diffusion was used to study the transport of a medicinal agent through the vehicle and subsequent penetration into the stratum corneum. Eigenmodes and eigenvalues were computed and introduced into an inversion formula to estimate the delivery rate and the amount of drug in the vehicle and the skin. A dynamic programming algorithm calculated the optimal doses necessary to achieve a desired transdermal flux. The analytical method predicted profiles that were in close agreement with published numerical solutions and provided an automated strategy to perform therapeutic drug monitoring and control.

Drug release from a suspension with a finite dissolution rate: theory and its application to a betamethasone 17-valerate patch

A random walk method for a diffusion equation is applied to the model for a suspension with a finite dissolution rate developed by Ayres and Lindstrom in 1977. In the method, the diffusion of dissolved drug and dissolution of crystal are calculated separately using a simple BASIC program. The random walk method strictly meets the principle of the conservation of mass as the drug amount in each sublayer rather than the concentration at each subinterval is concerned in the ointment. The model is used to analyze the release of betamethasone 17-valerate from a pressure-sensitive silicone adhesive into a sink. The drug release from the 1.50 mg/mL patch shows no substantial discrepancy from that predicted by the classic suspension model assuming an infinite dissolution rate. However, the classic model overestimates the release from the 3.08 and 5.88 mg/mL patches. The disagreement is lessened when the dissolution rate is assumed to be finite. However, the model does not give a perfect explanation because the drug release from the 3.08 and 5.88 mg/mL patches in the early phase is faster than the model predicts.

Significance of viable skin layers in percutaneous permeation and its implication in mathematical models: theoretical consideration based on parameters for betamethasone 17-valerate

The role of viable skin layers (viable epidermis and dermis) is examined by the three-layer model using parameter values for betamethasone 17-valerate. The mathematical three-layer diffusion model indicates that the lag time and half-life after vehicle removal in epidermis and split-thickness skin are longer than those in stratum corneum without viable layers, even if drug flux at steady-state is minimally altered. The theoretical values of the lag time and half-life predicted for epidermis and split-thickness-skin samples are compatible with the values observed in another in vitro study. The results predicted by the three-layer model indicate that the simplified model (single-layer or compartment model), which regards the whole skin as one diffusion layer or one compartment, may be warranted because all three skin layers have the same half-life after vehicle removal. The parameters used in the simplified model are estimated from some of the following values directly obtainable in the experiment: flux from skin and amount in the whole skin at steady-state, lag time, and drug concentration or amount maintained to be unchanged in the donor site. However, the simplified model often cannot resolve some discrepancy between the data and model (e.g., the ratio of the half-life to lag time) even if the data may be explained by the three-layer model.

Percutaneous absorption: a single-layer model

In vitro percutaneous permeation of betamethasone 17-valerate through excised human skin was studied. Pressure-sensitive silicone adhesive containing betamethasone 17-valerate in suspension was used as a vehicle. Steady-state flux through the split-thickness skin was similar to that through the isolated epidermis. However, the lag-time and half-life after removal of the vehicle were longer for the split-thickness skin than from epidermis. At steady state, 37% of the drug in the split-thickness skin was partitioned in dermis. When the kinetic parameters of a simple single-layer model are defined to specify the permeability coefficient and the drug amount in skin at steady state, this model can predict the longer half-life observed for the split-thickness skin sample compared with that for epidermis. The difference between the observed and theoretical values of the half-life after removal of the vehicle was within 23%. On the other hand, the lag-time had a large variation and the simple diffusion model failed to be predictive. A single-layer model described by two or three kinetic parameters may be able to describe percutaneous permeation kinetics even when the processes after the compound permeation through stratum corneum are not negligible. However, it is stressed that none of the kinetic parameters inherent in this simple model directly represents one of the single physicochemical parameters, such as diffusion and partition coefficients and path length of each skin layer.