Gerald B. Kasting

Mathematical models of skin permeability: An overview

Mathematical models of skin permeability play an important role in various fields including prediction of transdermal drug delivery and assessment of dermal exposure to industrial chemicals. Extensive research has been performed over the last several decades to yield predictions of skin permeability to various molecules. These efforts include the development of empirical approaches such as quantitative structure-permeability relationships and porous pathway theories as well as the establishment of rigorous structure-based models. In addition to establishing the necessary mathematical framework to describe these models, efforts have also been dedicated to determining the key parameters that are required to use these models. This article provides an overview of various modeling approaches with respect to their advantages, limitations and future prospects.

Visualization of the lipid barrier and measurement of lipid pathlength in human stratum corneum

Detailed models of solute transport through the stratum corneum (SC) require an interpretation of apparent bulk diffusion coefficients in terms of microscopic transport properties. Modern microscopy techniques provide a tool for evaluating one key property—lipid pathway tortuosity—in more detail than previously possible. Microscopic lipid pathway measurements on alkali expanded human SC stained with the lipid-soluble dyes methylene blue, Nile red, and oil red O are described. Brightfield, differential interference contrast, fluorescence, and laser scanning confocal optics were employed to obtain 2-dimensional (2-D) and 3-dimensional (3-D) images. The 2-D techniques clearly outlined the corneocytes. Confocal microscopy using Nile red yielded a well-delineated 3-D structure of expanded SC. Quantitative assessment of the 2-D images from a small number of expanded SC samples led to an average value of 3.7 for the ratio of the shortest lipid-continuous pathway to the width of the membrane. This was corrected for the effect of alkaline expansion to arrive at an average value of 12.7 for the same ratio prior to swelling.

DC electrical properties of frozen, excised human skin.

DC current-voltage relationships and sodium ion transport measurements for human allograft skin immersed in saline buffers have been determined using a four terminal potentiometric method and diffusion cells of our own design. About three-fourths of the skin samples were deemed suitable for study on the basis of their high resistivities and similar j–V characteristics. Most of these samples yielded sodium ion permeability coefficients less than or equal to those reported for human skin in vivo. The current–voltage relationship in these tissues was time dependent, highly nonlinear, and slightly asymmetric with respect to the sign of the applied potential. Skin resistance decreased as current or voltage increased. For current densities less than 15 µA/cm2 and exposure times of 10–20 min, this decrease was almost completely reversible; at higher current densities, both reversible and irreversible effects were observed. The overall dependence of current on voltage was nearly exponential and was satisfactorily described by an equation of the form j ∼ sinh V. Diffusion potentials, sodium ion membrane transference numbers, and sodium ion flux enhancement factors during iontophoresis were measured for skin immersed both in normal saline solutions and in saline solutions of differing concentrations. The sign of the diffusion potentials and the value of the sodium ion transference number (0.51 in normal saline at pH 7.4) indicated a weak permselectivity of the skin for transport of sodium ion versus chloride. At a current density of 71 µA/cm2 and transmembrane potentials in the range of 1.1–1.6 V, the flux enhancement for sodium ion was three to five times greater than that predicted for an uncharged homogeneous membrane according to electrodiffusion theory. For transmembrane potentials less than 0.17 V, agreement of this theory with the data was better but still incomplete.

Application of electrodiffusion theory for a homogeneous membrane to iontophoretic transport through skin

Abstract Two simple models for ionic mass transport across membranes are discussed in the context of iontophoretic delivery of drugs through skin. The constant field model is mathematically the most tractable and offers some insights into the time dependence of iontophoretic transport. However, for thick membranes or for systems in which the total ion concentrations on opposite sides of the membrane differ appreciably, the electroneutrality approximation is more appropriate. Since both of these conditions are likely to be found in skin iontophoresis studies, the electroneutrality model should provide a better starting point for analyzing the details of iontophoresis experiments than does the constant field model. Equations for the diffusion potential, ion transference numbers and partition coefficients and the current-voltage characteristic of the membrane are given, enabling one to calculate ionic fluxes and active/passive flux ratios for a given applied current or voltage. As an example, the flux and transference number of a monovalent drug ion driven across a membrane in the presence of sodium chloride are calculated. Finally, known discrepancies between the predictions of the homogeneous membrane models and available experimental data are examined, and suggestions are made for modifying the theory to resolve these differences.

The microfluidics of the eccrine sweat gland, including biomarker partitioning, transport, and biosensing implications.

Non-invasive and accurate access of biomarkers remains a holy grail of the biomedical community. Human eccrine sweat is a surprisingly biomarker-rich fluid which is gaining increasing attention. This is especially true in applications of continuous bio-monitoring where other biofluids prove more challenging, if not impossible. However, much confusion on the topic exists as the microfluidics of the eccrine sweat gland has never been comprehensively presented and models of biomarker partitioning into sweat are either underdeveloped and/or highly scattered across literature. Reported here are microfluidic models for eccrine sweat generation and flow which are coupled with review of blood-to-sweat biomarker partition pathways, therefore providing insights such as how biomarker concentration changes with sweat flow rate. Additionally, it is shown that both flow rate and biomarker diffusion determine the effective sampling rate of biomarkers at the skin surface (chronological resolution). The discussion covers a broad class of biomarkers including ions (Na(+), Cl(-), K(+), NH4 (+)), small molecules (ethanol, cortisol, urea, and lactate), and even peptides or small proteins (neuropeptides and cytokines). The models are not meant to be exhaustive for all biomarkers, yet collectively serve as a foundational guide for further development of sweat-based diagnostics and for those beginning exploration of new biomarker opportunities in sweat.

Ionic mass transport through a homogeneous membrane in the presence of a uniform electric field

Abstract The time-dependent theoretical solution to ionic mass transport through a uniform membrane under the influence of a uniform electric field is derived. Equivalent lag times and steady-state fluxes are then determined. The enhancement ratio of ionic flow (at a given voltage) to passive flow is shown to be proportional to the voltage for large augmenting voltages, and to decrease exponentially for large retarding voltages. Positive enhancement factors can be as large as 120 for triply charged ions, with one volt imposed across a membrane. The lag times decrease with increasing voltage magnitudes, and can (for triply charged ions, with one volt imposed across a membrane) be as small as 1/20 of the passive lag time.

Theoretical models for iontophoretic delivery

Abstract Several mathematical models are presented, each of which offers insights into the factors governing iontophoretic drug delivery. None of the models are sufficiently developed to describe all of the phenomena observed experimentally; in fact, none of them can with certainty be claimed to be the right starting point for describing these phenomena. But each allows testable predictions to be made, and future experiments designed with these predictions in mind will ultimately distinguish the useful approaches from those which are less so. As in other scientific endeavors, the utility of the theories lies in their ability to focus experimental efforts on the right questions, thereby improving the efficiency of the discovery process.

Design and performance of a spreadsheet-based model for estimating bioavailability of chemicals from dermal exposure☆

A comprehensive transient model of chemical penetration through the stratum corneum, viable epidermis and dermis formulated in terms of an Excel™ spreadsheet and associated add-in is presented. The model is a one-dimensional homogenization of underlying microscopic transport models for stratum corneum and dermis; viable epidermis is treated as unperfused dermis. The models salient features are a detailed structural description of the skin layers, a combination of first-principles based transport equations and empirical partition and diffusion coefficients, and the capability of simulating a variety of exposure scenarios. Model predictions are compared with representative in vitro skin permeation data obtained from the literature using as summary parameters total absorption (Q(abs)), maximum flux (J(max)) and skin permeability coefficient (k(p)). The results of this evaluation demonstrate the current state-of-the-art in prediction of transient skin absorption and highlight areas in which further elaborations are needed to obtain satisfactory predictions.

Kinetics of finite dose absorption through skin 1. Vanillylnonanamide

Despite the considerable success in predicting the steady-state dermal absorption rates of chemical compounds from large reservoirs applied to skin, correspondingly little progress has been made in predicting the absorption rate and extent for small doses of topically applied compounds. In the latter case, steady-state absorption rates are generally not obtained, and rapid evaporation or penetration of the dose solvent makes application of permeability coefficient models problematic. This report presents a new analysis of the finite dose problem in terms of a diffusion model with three parameters-a characteristic time for diffusion, h2/D; a skin solubility factor, S(m)h; and a capacity factor for absorption of the dose during the dry down period, M*. These parameters can be related to the molecular weight and oil and water solubilities of the permeant in a manner similar to models describing steady-state absorption from saturated solutions. Some variation of the parameter values based on the chemical nature and volume of the dose solvent is anticipated. The applicability of the model is demonstrated by analyzing the in vitro absorption rates of varying doses of vanillylnonamide (VN, synthetic capsaicin) applied to excised human skin from propylene glycol. The analysis shows that a three-parameter model that assigns all of the resistance to transport to diffusion through the stratum corneum is able to explain most of the significant features of VN absorption through skin.

Improving the Sensitivity of in Vitro Skin Penetration Experiments

The institution of a readily-implemented sample screening and data handling procedure for in vitro skin penetration studies yields substantial improvements in sensitivity for distinguishing between formulations, treatments, penetrants, etc. The procedure involves four steps: 1) prescreen the tissue samples to determine their intrinsic permeability; 2) apply treatments using a randomized complete block (RGB) design, with blocking by tissue permeability; 3) apply a variance-stabilizing transformation to the penetration data, followed by outlier testing; and 4) analyze the transformed data according to an RGB analysis of variance, using tissue permeability as the blocking variable. For penetration studies in which high sample variability is a concern, the above procedure commonly yields a sensitivity advantage of several-fold versus alternative methods of comparison.