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

A two-dimensional mathematical model of non-linear dual-sorption of percutaneous drug absorption

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.

A two-dimensional mathematical model of percutaneous drug absorption

BACKGROUND

When 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, r. The values of r depend on the amount of diffusion and the normalized skin-capillary clearance. It is defined as the ratio of the steady-state drug concentration at the skin-capillary boundary to that at the skin-surface in one-dimensional models. The present paper studies the effect of the parameter values, when the region of contact of the skin with the drug, is a line segment on the skin surface.

METHODS

Though a simple one-dimensional model is often useful to describe percutaneous drug absorption, it may be better represented by multi-dimensional models. A two-dimensional mathematical model is developed for percutaneous absorption of a drug, which 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. This model consists of a linear second-order parabolic equation with appropriate initial conditions and boundary conditions. These boundary conditions are of Dirichlet type, Neumann type or Robin type. A finite-difference method which maintains second-order accuracy in space along the boundary, is developed to solve the parabolic equation. Extrapolation in time is applied to improve the accuracy in time. Solution of the parabolic equation gives the concentration of the drug in the skin at a given time.

RESULTS

Simulation of the numerical methods described is carried out with various values of the parameter r. The illustrations are given in the form of figures.

CONCLUSION

Based on the values of r, conclusions are drawn about (1) the flow rate of the drug, (2) the flux and the cumulative amount of drug eliminated into the receptor cell, (3) the steady-state value of the flux, (4) the time to reach the steady-state value of the flux and (5) the optimal value of r, which gives the maximum absorption of the drug. The paper gives valuable information which can be obtained by this two-dimensional model, that cannot be obtained with one-dimensional models. Thus this model improves upon the much simpler one-dimensional models. Some future directions of the work based on this model and the one-dimensional non-linear models that exist in the literature, are also discussed.

Theoretical investigation of drug release from planar matrix systems: effects of a finite dissolution rate

Drug release from planar matrix systems has been investigated with special emphasis on the influence of a finite dissolution rate on the drug release profile. A mathematical model of the drug dissolution and release processes was formulated in terms of two coupled nonlinear partial differential equations (PDEs). These were solved numerically by using well-established FORTRAN routines. An approximate analytical solution, valid during the early stages of the release process, was derived. The analytical solution was compared to the numerical one and to drug release models existing in the literature. From this comparison, it was established that the analytical approximation provided a good description of the major part of the release profile, irrespective of the dissolution rate. Existing literature models, based on instantaneous dissolution, were found to agree with the numerical solution only when drug dissolution proceeded very rapidly in comparison with the diffusion process. Consequently, the new analytical short-time approximation of the drug release complements the formulas existing in the literature, since it provides a superior description of the release of slowly dissolving drugs.

Polymeric controlled drug-delivery systems: perspective issues and opportunities

Although, the drug-delivery system (DDS) concept is not new, great progress has been made recently in the treatment of a variety of diseases. Targeting delivery of drugs to the diseased lesions is one of the most important aspects of DDS. To convey a sufficient dose of drug to the lesion, suitable carriers of drugs are needed. Polymers, which swell and contract in response to external pH levels, are being explored. The research in this area is being carried out all over the world at a great pace. Not only that new developments are emerging in the existing technologies, but also various new technologies are being developed and tested. Consequently, a huge amount of new information is available, which should be compiled and presented in a comprehensive way to benefit large numbers of users in this area as well as to help active research workers in the field. The purpose of this review is to discuss some recent advances and future prospects in controlled drug-delivery technology. The article serves as a useful tool for the beginners as well as for the researchers actively involved in this fascinating area of applied polymer science.

Crystallization of beta-estradiol in an acrylic transdermal drug delivery system

Transdermal drug delivery systems are pharmaceutical products that can deliver controlled doses of drugs from polymeric patches applied on the human skin. The long-term stability of these patches is a critical issue relative to their performance in delivering drugs at a constant rate. Where a drug has been dissolved in the polymeric adhesive patch, crystallization has been reported in several systems. This study uses a variety of characterization tools to determine the physical and chemical nature of the precipitates formed in situ in estradiol patches. Optical microscopy revealed that crystals were formed in a single layer inside the adhesive matrix and that there were two distinctly different morphologies: needle-like crystals and aggregates around the needles. From IR measurements it was evident that estradiol probably was present in more than one crystal form in these patches. Raman microscopy showed that the needle-like crystals contain the adhesive component and the aggregates some modified crystal form of estradiol, indicating that in addition to the drug, the polymeric adhesive also crystallizes during storage.

Simultaneous diffusion and metabolism of betamethasone 17-valerate in the living skin equivalent

Simultaneous diffusion and metabolism of betamethasone 17-valerate was studied using betamethasone 17-valerate, betamethasone 21-valerate, and betamethasone as permeants. These corticosteroids were suspended in silicone adhesive and applied to an artificial living skin equivalent (LSE) for 72 h. When betamethasone was applied, no metabolites were detected in the receptor medium. Conversely, with betamethasone 21-valerate application, only betamethasone but no betamethasone 21-valerate was detected in the receptor medium indicating the metabolism of the latter by skin esterases. When tested with the theory for simultaneous diffusion and metabolism, the result is consistent with the enzyme rate constant in the LSE homogenate measured in a previous study. When betamethasone 17-valerate was applied to the LSE, more than half of the total amount of corticosteroids detected in the receptor medium was unchanged, consistent with the previously reported chemical (as opposed to enzymatic) degradation half-life of about 8 h. This result also indicated that very little metabolism of betamethasone 17-valerate occurred in the skin.