Arne Naegel

Detailed modeling of skin penetration—An overview

In recent years, the combination of computational modeling and experiments has become a useful tool that is proving increasingly powerful for explaining biological complexity. As computational power is increasing, scientists are able to explore ever more complex models in finer detail and to explain very complex real world data. This work provides an overview of one-, two- and three-dimensional diffusion models for penetration into mammalian skin. Besides diffusive transport this includes also binding of substances to skin proteins and metabolism. These models are based on partial differential equations that describe the spatial evolution of the transport process through the biological barrier skin. Furthermore, the work focuses on analytical and numerical techniques for this type of equations such as discretization schemes or homogenization (upscaling) techniques. Finally, the work compares different geometry models with respect to the permeability.

The Role of Corneocytes in Skin Transport Revised—A Combined Computational and Experimental Approach

PurposeTo investigate mechanisms of compound–corneocyte interactions in a combined experimental and theoretical approach.Materials and MethodsExperimental methods are presented to investigate compound–corneocyte interactions in terms of dissolution within water of hydration and protein binding and to quantify the extent of the concurrent mechanisms. Results are presented for three compounds: caffeine, flufenamic acid, and testosterone. Two compartmental stratum corneum models M1 and M2 are formulated based on experimentally determined input parameters describing the affinity to lipid, proteins and water. M1 features a homogeneous protein compartment and considers protein interactions only via intra-corneocyte water. In M2 the protein compartment is sub-divided into a cornified envelope compartment interacting with inter-cellular lipids and a keratin compartment interacting with water.ResultsFor the non-protein binding caffeine the impact of the aqueous compartment on stratum corneum partitioning is overestimated but is successfully modeled after introducing a bound water fraction that is non-accessible for compound dissolution. For lipophilic, keratin binding compounds (flufenamic acid, testosterone) only M2 correctly predicts a concentration dependence of stratum corneum partition coefficients.ConclusionsLipophilic and hydrophilic compounds interact with corneocytes. Interactions of lipophilic compounds are probably confined to the corneocyte surface. Interactions with intracellular keratin may be limited by their low aqueous solubility.

A comparison of two- and three-dimensional models for the simulation of the permeability of human stratum corneum.

The stratum corneum is the outermost layer of cells in mammalian epidermis. It is widely believed to play the key role for the barrier function of the skin. This study characterises how the cell geometry influences the permeability of the membrane. It is based on a diffusion model, which is evaluated using numerical simulation. Three different geometry concepts, i.e., ribbon, cuboid and tetrakaidekahedral type, in two and three space dimensions are compared. The results confirm that tetrakaidekahedral cells with an almost optimal surface-to-volume ratio provide a barrier, in which a minimal amount of mass is used very effectively. Additionally, the study supplies tools to quantify this and links the results to the theory of homogenization.

Finite dose skin mass balance including the lateral part: comparison between experiment, pharmacokinetic modeling and diffusion models.

This work investigates in vitro finite dose skin absorption of the model compounds flufenamic acid and caffeine experimentally and mathematically. The mass balance in different skin compartments (donor, stratum corneum (SC), deeper skin layers (DSL), lateral skin parts and acceptor) is analyzed as a function of time. For both substances high amounts were found in the lateral skin compartment after 6h of incubation, which emphasizes not to elide these parts in the modeling. Here, three different mathematical models were investigated and tested with the experimental data: a pharmacokinetic model (PK), a detailed microscopic two-dimensional diffusion model (MICRO) and a macroscopic homogenized diffusion model (MACRO). While the PK model was fitted to the experimental data, the MICRO and the MACRO models employed input parameters derived from infinite dose studies to predict the underlying diffusion process. All models could satisfyingly predict or describe the experimental data. The PK model and MACRO model also feature the lateral parts.

The role of tight junctions in skin barrier function and dermal absorption

The skin protects our body from external assaults like pathogens, xenobiotics or UV irradiation. In addition, it prevents the loss of water and solutes. To fulfill these important tasks, a complex barrier system has developed which comprises the stratum corneum, tight junctions, the microbiome, the chemical barrier and the immunological barrier. These barriers do not act separately, but influence each other e.g. after external manipulation or in skin diseases. Especially the two mechanical barriers, i.e. stratum corneum and tight junctions, are of great interest for drug delivery, because they are the first interaction partners of drug delivery systems and play the major role in skin absorption. Tight junctions are of special interest, as they are centrally localized in this complex barrier system in the outermost viable layer - the stratum granulosum of the interfollicular epidermis and the companion cell layer of the hair follicle - and because they can react very quickly to stimuli. We summarize here our current knowledge about tight junction barrier function in mammalian interfollicular epidermis and hair follicles, and the interaction of tight junctions with other skin barrier components in health and disease. Furthermore, we discuss their relevance for drug delivery and provide examples for tight junction modulators.

Numerical simulation of skin transport using Parareal

In silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination’s performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal’s convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.

Computational modeling of the skin barrier.

A simulation environment for the numerical calculation of permeation processes through human skin has been developed. In geometry models that represent the actual cell morphology of stratum corneum (SC) and deeper skin layers, the diffusive transport is simulated by a finite volume method. As reference elements for the corneocyte cells and lipid matrix, both three-dimensional tetrakaidecahedra and cuboids as well as two-dimensional brick-and-mortar models have been investigated. The central finding is that permeability and lag time of the different membranes can be represented in a closed form depending on model parameters and geometry. This allows a comparison of the models in terms of their barrier effectiveness at comparable cell sizes. The influence of the cell shape on the barrier properties has been numerically demonstrated and quantified. It is shown that tetrakaidecahedra in addition to an almost optimal surface-to-volume ratio also has a very favorable barrier-to-volume ratio. A simulation experiment was successfully validated with two representative test substances, the hydrophilic caffeine and the lipophilic flufenamic acid, which were applied in an aqueous vehicle with a constant dose. The input parameters for the simulation were determined in a companion study by experimental collaborators.

Efficient multigrid solver for the 3D random walker algorithm

The random walker algorithm is a graph-based segmentation method that has become popular over the past few years. The basis of the algorithm is a large, sparsely occupied system of linear equations, whose size corresponds to the number of voxels in the image. To solve these systems, typically comprised of millions of equations, the computational performance of conventional numerical solution methods (e.g. Gauss-Seidel) is no longer satisfactory. An alternative method that has been described previously for solving 2D random walker problems is the geometrical multigrid method. In this paper, we present a geometrical multigrid approach for the 3D random walker problem. Our approach features an optimized calculation of the required Galerkin product and a robust smoothing using the ILUβ method. To reach better convergence rates, the multigrid solver is used as a preconditioner for the conjugate gradient solver. We compared the performance of our new multigrid approach with the conjugate gradient solver on five MRI lung images with a resolution of 96 x 128 x 52 voxels. Initial results show an increasing in speed of up to four times, reducing the average computation time from six minutes to less than two minutes when using our proposed approach. Employing a multigrid solver for the random walker algorithm thus permits accurate interactive segmentation with fewer delays.

Mathematical modelling of the viable epidermis: impact of cell shape and vertical arrangement:

In-silico methods are valuable tools for understanding the barrier function of the skin. The key benefit is that mathematical modelling allows the interplay between cell shape and function to be elucidated. This study focuses on the viable (living) epidermis. For this region, previous works suggested a diffusion model and an approximation of the cells by hexagonal prisms. The work at hand extends this in three ways. First, the extracellular space is treated with full spatial resolution. This induces a decrease of permeability by about 10%. Second, cells of tetrakaidecahedral shape are considered, in addition to the original hexagonal prisms. For both cell types, the resulting membrane permeabilities are compared. Third, for the first time, the influence of cell stacking in the vertical direction is considered. This is particularly important for the stratum granulosum, where tight junctions are present.