Predicting Viable Skin Concentration: Modelling the Subpapillary Plexus

Abstract

The skin concentration of a substance after a topical application or exposure determines both local treatment outcomes and the dermal toxicity assessment of various products. However, quantifying the time course of those concentrations at skin effect sites, such as the viable epidermal, superficial dermis and appendages in humans is especially problematic in vivo, making physiologically based mathematical modelling an essential tool to meet this need. This work further develops our published physiologically based pharmacokinetic and COMSOL based dermal transport modelling by considering the impact of the superficial subpapillary dermal plexus, which we represent as two well stirred compartments. The work also studied the impact on dermal concentrations of subpapillary plexus size, depth, blood velocity and density of subpapillary plexus vessels. Sensitivity analyses are used to define the most important transport determinants of skin concentrations after topical application of a substance, with previously published results used to validate the resulting analyses. This resulting model describes the available experimental data better than previous models, especially at deeper dermal depths.

Keywords: COMSOL; Computational modelling; Subpapillary plexus; Viable skin concentration.

Fig. 1

Schematic of blood circulation in the skin

Fig. 2

An illustration for the composition of the model developed for the subpapillary plexus. The l_ca represents the length of the computational area, h_ca represents the thickness, J_in is the flux of drug entering the capillary loops from the arteriole plexus, J_out is the flux of drug entering the venule plexus from the capillary loop, D_p represents the diffusion coefficient between the venules and arterioles in the plexus and D_vs is the diffusion coefficient in the viable skin

Fig. 3

Concentration of a solute at different depths within the viable skin with different sizes for the total plexus. Each of the concentrations is recorded adjacent to the arterial branch of the capillary loop. The blue colour signifies the standard which has a total plexus size of 50μm. The red has total plexus size of 70μm, the green has a sizes of 90μm, and the black has a total plexus size of 30μm

Fig. 4

The figure shows the concentration at different depths within the viable epidermis and dermis. The gaps in concentration are related to the area of the subpapillary plexus. The dotted lines represent the concentration near the arteriole, while the solid lines show the concentration near the venule. The red colour represents a subpapillary plexus depth of 450μm, the blue colour is the standard depth of 400μm, and the black colour is the deepest plexus with a depth of 500μm}

Fig. 5

The concentration of solute present at each depth when different velocities of blood in the capillary loops were imposed. The concentrations were all taken adjacent to the arteriole. The blue colour was the standard and had the velocity stated in the method (0.65 ± 0.3 mm/s). The red line considers when the velocity was tenfold smaller than the standard velocity. Meanwhile, the green line considered ¼ of the standard velocity, the black line implemented ½ of the standard velocity, and the cyan line used ¾ of the standard velocity

Fig. 6

The figure shows the concentration when the density of the pores in the subpapillary plexus was changed. The dashed lines represent the concentration adjacent to the venular branch of the capillary loop while the solid line represents the concentration near the arteriole branch. The black colour represents the standard density of 11%. The red colour represents a pore density of 19% while the blue represents a pore density of 25%

Fig. 7

A comparison of mathematical models which aim to explain the impact of the blood vessels. The green line represents the concentration profile for the subpapillary plexus model introduced in this paper. The black line represents the enhanced capillary model in [1], while the yellow line represents the first capillary model in [2]. The simple linear model is described by the red colour while the previously used distributed-elimination model is shown in blue