Mathematical models of diffusion through and near skin appendages: Hair follicle and eccrine sweat gland pathways
Dancik, Yuri H
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Quantitative modeling of the transport of topically applied chemical substances into skin appendages and the surrounding skin tissue is important to topical and transdermal drug delivery, as well as risk assessment of chemical exposure, because skin appendages---in particular hair follicles and eccrine sweat glands---can contribute significantly to skin permeability by providing shunt pathways through the epidermal barrier. In addition, an understanding of follicular delivery per se may be of considerable importance in the treatment of diseases originating in the pilosebaceous unit and of interest to the cosmetics industry. Part I of this Ph.D. thesis addresses the transport of chemicals into and near a hair follicle. A comprehensive 2-D axisymmetric finite difference model has been developed to solve the transient diffusion equation describing transport of an unionized chemical species through stratum corneum, viable epidermis, dermis and hypodermis in a large cylindrical volume surrounding a realistic geometrical representation of the hair shaft and encircling layers of the follicular sheath. The numerical technique introduces a novel non-orthogonal coordinate system fitted to the follicle's outer boundary. The barrier properties of the follicular infundibulum are modeled as a stratum corneum-like epithelium with a thickness diminishing with depth. Inputs to the calculation are the partition and diffusion coefficients of the applied drug in each skin layer, the permeability of the outer root sheath (ORS), taken as the hair follicle's outer boundary, and volumetric rate coefficients describing clearance into the systemic circulation via the dermal vasculature. Concentration profiles for a model permeant representing the lipophilic dye Bodipy® FL C 5 (BFL) were obtained for upper and lower bound estimates on the permeability of the ORS and the concentration of BFL in the sebum. The results are compared to experimentally obtained published in vitro (no clearance via the vasculature) dye distributions around hair follicles, and show the infundibulum to be the structure via which the hair follicle increases permeation into dermal tissue. The contribution of the sebaceous duct to the permeability of the hair follicle is shown to further increase the delivery of BFL to the deeper dermis, in qualitative agreement with published experimental results. Simulations of the in vivo permeation of BFL show that typical levels of clearance by the vasculature yield maxima in the average concentration profiles calculated at the hair follicle/skin tissue boundary, and in the viable epidermis and the dermis near the follicle. Average dermal concentrations near the hair follicle are up to 90 % smaller than in the in vitro case. Furthermore we use the model to simulate the diffusion of BFL into the skin from a small patch-like device. Delivery to the skin is nearly non-existent for the region of the tissue does not lying directly below the patch. Part II presents the mathematical structure for a 3-D model of the steady-state diffusion of an unionized chemical substance through and near the duct of an eccrine sweat gland. We develop the structure of the solution to the model equations with reference to physico-chemical parameter values for the diffusion of BFL.