ALPHA-TOCOPHEROL ACETATE PERMEATION 93 THEORY Bellantone et al. (16) described in detail the mathematical model used to estimate the permeability coefficients in this study. The model corresponds to the experimental setup used in this study (modified Franz diffusion cells), in which the drug leaves an unstirred donor, crosses through a membrane of thickness hand cross-area A) and accumulates in a stirred receiver for which sink conditions are maintained. Initially, the drug concen­ tration C 0 in the donor is uniform, while the membrane and receiver are void of drug. Fick's laws were used to give equations for the rate of accumulation of the drug in the receiver. (Table I contains a description of the symbols and abbreviations used in this paper.) OBTAINING THE PERMEABILITY OF A RATE-LIMITING MEMBRANE Here, the transport across the membrane is the rate-controlling step. The general equa­ tion for the cumulative amount of drug released into the receiver M at time t is given as an infinite series. However, if the times used for the data points are not too large (see below), the equation can be simplified to give A AT ATA Co Cd cm D Dd Dm g h HPLC K L M N pm r R t T UV V Table I Definitions of Symbols and Abbreviations Used in This Paper Area Alpha-tocopherol Alpha-tocopherol acetate Initial drug concentration Donor drug concentration Membrane drug concentration Diffusion coefficient Diffusion coefficient of donor Diffusion coefficient of membrane Acceleration due to gravity Thickness of membrane High-performance liquid chromatography Membrane/donor partition coefficient Distance Amount release Avogadro's number Membrane permeability Radius of the spherical particle Molar gas constant Time Absolute temperature (K) Ultraviolet Velocity Viscosity of solvent Density of the spherical particle Density of solvent (1)

94 where and JOURNAL OF COSMETIC SCIENCE h2 � 2 =_ 4Drn and erfc u = 21. J:' exp( -w2)dw is the complementary error function. Here, D rn denotes the permeability of the drug through the membrane, D d denotes the diffusion coefficient of the drug in the donor medium, and K represents the membrane/donor partition coefficient. The parameter a has units of mass per square root of time, and �2 has units of time. The specific requirements for the simplification of equation 1 are that the times at which the samples are taken satisfy t 2-3 � 2 , and less than -1/3 of the drug initially in the donor has left the donor (16). The parameters a and � can be determined from the experimental release data by performing a nonlinear regression using equation 1. Detailed information on the method used to find good initial estimates of the parameters a and � is reported by Bellantone et al. (16). The permeability P m of the membrane can be estimated as (16): KD m a� P --h-= m 2�( 4AC0 �- a-v;:) (2) In this study, A = 1.76 cm2, C0 = 50,000 µg/ml, � ranged from 3-5 hr 112 , and Dd 0.01 cm2/hr in all cases. For the ATA/isopropyl myristate formulations, a - 10,000, and equation 2 was used to calculate Pm. For formulations other than the ATA/isopropyl myristate, a ranged from 100 to 500 µg/hr 112 , and less than 2% error is introduced by simplifying equation 2 as a,v;: pm= 8�AC 0 (3) The ATA diffusion coefficients in the various donors were estimated by two methods. In the first method, cellulose permeation data was used to obtain the diffusion coefficient in the donor, and was employed when it was possible to assay the drug in the receiver. The second method used the relative viscosities of the media to estimate the diffusion coefficient and was employed in cases where permeation data was not easily evaluated. The methods are described below. Obtaining D d using cellulose permeation data. Here, cellulose membranes are used in release experiments because they are thin and highly permeable. Thus, a pseudo-steady state develops in the membrane quickly, and the release of drug is primarily controlled by the donor region behavior. If the donor and receiver media are the same (to avoid solvent drag effects in the membrane), this model can be used to obtain information about the diffusion coefficient in the donor D d· The cumulative amount of drug released is given by (17):