Effective diffusivity in membranes with tetrakaidekahedral cells and implications for the permeability of human stratum corneum

Mechanisms of penetration and diffusion of drugs and cosmetic actives across the human Stratum corneum

Based on the structure of the Stratum corneum (SC) the potential penetration/diffusion pathways of drugs and cosmetic actives through the SC are presented and discussed. The well-known lipophilic pathway across the SC is presented and relevant examples are used to show that highly lipophilic molecules such as glucocorticoids, coenzyme Q10 etc. are accumulated in the SC and penetrate into the inner liquid like layer of the SC lipid bilayer by lateral diffusion. The diffusion into and across the SC of highly hydrophilic drugs and active substances such as urea, amino acids and peptides is still under discussion. Another diffusion pathway for the highly hydrophilic molecules via the corneocytes and the corneodesmosomes is presented and discussed, the corneocytary diffusion pathway.

Transdermal and lateral effective diffusivities for drug transport in stratum corneum from a microscopic anisotropic diffusion model

This paper presents a computational model of molecular diffusion through the interfollicular stratum corneum. Specifically, it extends an earlier two-dimensional microscopic model for the permeability in two ways: (1) a microporous leakage pathway through the intercellular lipid lamellae allows slow permeation of highly hydrophilic permeants through the tissue; and (2) the model yields explicit predictions of both lateral (D‾_‖^sc) and transdermal (D‾_⊥^sc) effective (average, homogenized) diffusivities of solutes within the tissue. We present here the mathematical framework for the analysis and a comparison of the predictions with experimental data on desorption of both hydrophilic and lipophilic solutes from human stratum corneum in vitro. Diffusion in the lipid lamellae is found to make the effective diffusivity highly anisotropic, with the predicted ratio D‾_‖^sc/D‾_⊥^sc ranging from 34-39 for fully hydrated skin and 150 to more than 1000 for partially hydrated skin. The diffusivities and their ratio are in accord with both experimental data and the results of mathematical analyses performed by others.

Predictive Design and Analysis of Drug Transport by Multiscale Computational Models Under Uncertainty

Computational modeling of drug delivery is becoming an indispensable tool for advancing drug development pipeline, particularly in nanomedicine where a rational design strategy is ultimately sought. While numerous in silico models have been developed that can accurately describe nanoparticle interactions with the bioenvironment within prescribed length and time scales, predictive design of these drug carriers, dosages and treatment schemes will require advanced models that can simulate transport processes across multiple length and time scales from genomic to population levels. In order to address this problem, multiscale modeling efforts that integrate existing discrete and continuum modeling strategies have recently emerged. These multiscale approaches provide a promising direction for bottom-up in silico pipelines of drug design for delivery. However, there are remaining challenges in terms of model parametrization and validation in the presence of variability, introduced by multiple levels of heterogeneities in disease state. Parametrization based on physiologically relevant in vitro data from microphysiological systems as well as widespread adoption of uncertainty quantification and sensitivity analysis will help address these challenges.

Compartmental modeling of skin absorption and desorption kinetics: Donor solvent evaporation, variable diffusion/partition coefficients, and slow equilibration process within stratum corneum

This work expands the recently developed compartmental model for skin transport to model variable diffusion and/or partition coefficients, and the presence of slow equilibration/slow binding kinetics within stratum corneum. The model was validated by comparing it with the diffusion model which was solved numerically using the finite element method. It was found that the new compartmental model predictions agreed well with that of the diffusion model, providing a sufficient number of compartments was used. The compartmental model was applied to two previously published experimental data sets: water penetration and desorption data and the finite dose dermal penetration of testosterone. Significant improvement of the fitting quality for all these data sets was achieved using the compartmental model.

Surging footprints of mathematical modeling for prediction of transdermal permeability

In vivo skin permeation studies are considered gold standard but are difficult to perform and evaluate due to ethical issues and complexity of process involved. In recent past, a useful tool has been developed by combining the computational modeling and experimental data for expounding biological complexity. Modeling of percutaneous permeation studies provides an ethical and viable alternative to laboratory experimentation. Scientists are exploring complex models in magnificent details with advancement in computational power and technology. Mathematical models of skin permeability are highly relevant with respect to transdermal drug delivery, assessment of dermal exposure to industrial and environmental hazards as well as in developing fundamental understanding of biotransport processes. Present review focuses on various mathematical models developed till now for the transdermal drug delivery along with their applications.

Mass release curves as the constitutive curves for modeling diffusive transport within biological tissue

In diffusion governed by Fick's law, the diffusion coefficient represents the phenomenological material parameter and is, in general, a constant. In certain cases of diffusion through porous media, the diffusion coefficient can be variable (i.e. non-constant) due to the complex process of solute displacements within microstructure, since these displacements depend on porosity, internal microstructural geometry, size of the transported particles, chemical nature, and physical interactions between the diffusing substance and the microstructural surroundings. In order to provide a simple and general approach of determining the diffusion coefficient for diffusion through porous media, we have introduced mass release curves as the constitutive curves of diffusion. The mass release curve for a selected direction represents cumulative mass (per surface area) passed in that direction through a small reference volume, in terms of time. We have developed a methodology, based on numerical Finite Element (FE) and Molecular Dynamics (MD) methods, to determine simple mass release curves of solutes through complex media from which we calculate the diffusion coefficient. The diffusion models take into account interactions between solute particles and microstructural surfaces, as well as hydrophobicity (partitioning). We illustrate the effectiveness of our approach on several examples of complex composite media, including an imaging-based analysis of diffusion through pancreatic cancer tissue. The presented work offers an insight into the role of mass release curves in describing diffusion through porous media in general, and further in case of complex composite media such as biological tissue.

Anatomically Detailed and Large-Scale Simulations Studying Synapse Loss and Synchrony Using NeuroBox

The morphology of neurons and networks plays an important role in processing electrical and biochemical signals. Based on neuronal reconstructions, which are becoming abundantly available through databases such as NeuroMorpho.org, numerical simulations of Hodgkin-Huxley-type equations, coupled to biochemical models, can be performed in order to systematically investigate the influence of cellular morphology and the connectivity pattern in networks on the underlying function. Development in the area of synthetic neural network generation and morphology reconstruction from microscopy data has brought forth the software tool NeuGen. Coupling this morphology data (either from databases, synthetic, or reconstruction) to the simulation platform UG 4 (which harbors a neuroscientific portfolio) and VRL-Studio, has brought forth the extendible toolbox NeuroBox. NeuroBox allows users to perform numerical simulations on hybrid-dimensional morphology representations. The code basis is designed in a modular way, such that e.g., new channel or synapse types can be added to the library. Workflows can be specified through scripts or through the VRL-Studio graphical workflow representation. Third-party tools, such as ImageJ, can be added to NeuroBox workflows. In this paper, NeuroBox is used to study the electrical and biochemical effects of synapse loss vs. synchrony in neurons, to investigate large morphology data sets within detailed biophysical simulations, and used to demonstrate the capability of utilizing high-performance computing infrastructure for large scale network simulations. Using new synapse distribution methods and Finite Volume based numerical solvers for compartment-type models, our results demonstrate how an increase in synaptic synchronization can compensate synapse loss at the electrical and calcium level, and how detailed neuronal morphology can be integrated in large-scale network simulations.

Mathematical models for dermal drug absorption

INTRODUCTION

Mathematical models of dermal transport offer the advantages of being much faster and less expensive than in vitro or in vivo studies. The number of methods used to create such models has been increasing rapidly, probably due to the steady rise in computational power. Although each of the various approaches has its own virtues and limitations, it may be difficult to decide which approach is best suited to address a given problem.

AREAS COVERED

Here we outline the basic ideas, drawbacks and advantages of compartmental and quantitative structure-activity relationship models, as well as of analytical and numerical approaches for solving the diffusion equation. Examples of special applications of the different approaches are given.

EXPERT OPINION

Although some models are sophisticated and might be used in future to predict transport through damaged or diseased skin, the comparatively low availability of suitable and accurate experimental data limits extensive usage of these models and their predictive accuracy. Due to the lack of experimental data, the possibility of validating mathematical models is limited.

1D-3D hybrid modeling-from multi-compartment models to full resolution models in space and time

Investigation of cellular and network dynamics in the brain by means of modeling and simulation has evolved into a highly interdisciplinary field, that uses sophisticated modeling and simulation approaches to understand distinct areas of brain function. Depending on the underlying complexity, these models vary in their level of detail, in order to cope with the attached computational cost. Hence for large network simulations, single neurons are typically reduced to time-dependent signal processors, dismissing the spatial aspect of each cell. For single cell or networks with relatively small numbers of neurons, general purpose simulators allow for space and time-dependent simulations of electrical signal processing, based on the cable equation theory. An emerging field in Computational Neuroscience encompasses a new level of detail by incorporating the full three-dimensional morphology of cells and organelles into three-dimensional, space and time-dependent, simulations. While every approach has its advantages and limitations, such as computational cost, integrated and methods-spanning simulation approaches, depending on the network size could establish new ways to investigate the brain. In this paper we present a hybrid simulation approach, that makes use of reduced 1D-models using e.g., the NEURON simulator—which couples to fully resolved models for simulating cellular and sub-cellular dynamics, including the detailed three-dimensional morphology of neurons and organelles. In order to couple 1D- and 3D-simulations, we present a geometry-, membrane potential- and intracellular concentration mapping framework, with which graph- based morphologies, e.g., in the swc- or hoc-format, are mapped to full surface and volume representations of the neuron and computational data from 1D-simulations can be used as boundary conditions for full 3D simulations and vice versa. Thus, established models and data, based on general purpose 1D-simulators, can be directly coupled to the emerging field of fully resolved, highly detailed 3D-modeling approaches. We present the developed general framework for 1D/3D hybrid modeling and apply it to investigate electrically active neurons and their intracellular spatio-temporal calcium dynamics.

A multiscale MD-FE model of diffusion in composite media with internal surface interaction based on numerical homogenization procedure

Mass transport by diffusion within composite materials may depend not only on internal microstructural geometry, but also on the chemical interactions between the transported substance and the material of the microstructure. Retrospectively, there is a gap in methods and theory to connect material microstructure properties with macroscale continuum diffusion characteristics. Here we present a new hierarchical multiscale model for diffusion within composite materials that couples material microstructural geometry and interactions between diffusing particles and the material matrix. This model, which bridges molecular dynamics (MD) and the finite element (FE) method, is employed to construct a continuum diffusion model based on a novel numerical homogenization procedure. The procedure is general and robust for evaluating constitutive material parameters of the continuum model. These parameters include the traditional bulk diffusion coefficients and, additionally, the distances from the solid surface accounting for surface interaction effects. We implemented our models to glucose diffusion through the following two geometrical/material configurations: tightly packed silica nanospheres, and a complex fibrous structure surrounding nanospheres. Then, rhodamine 6G diffusion analysis through an aga-rose gel network was performed, followed by a model validation using our experimental results. The microstructural model, numerical homogenization and continuum model offer a new platform for modeling and predicting mass diffusion through complex biological environment and within composite materials that are used in a wide range of applications, like drug delivery and nanoporous catalysts.

A microscopic multiphase diffusion model of viable epidermis permeability

A microscopic model of passive transverse mass transport of small solutes in the viable epidermal layer of human skin is formulated on the basis of a hexagonal array of cells (i.e., keratinocytes) bounded by 4-nm-thick, anisotropic lipid bilayers and separated by 1-μm layers of extracellular fluid. Gap junctions and tight junctions with adjustable permeabilities are included to modulate the transport of solutes with low membrane permeabilities. Two keratinocyte aspect ratios are considered to represent basal and spinous cells (longer) and granular cells (more flattened). The diffusion problem is solved in a unit cell using a coordinate system conforming to the hexagonal cross section, and an efficient two-dimensional treatment is applied to describe transport in both the cell membranes and intercellular spaces, given their thinness. Results are presented in terms of an effective diffusion coefficient, D¯(epi), and partition coefficient, K¯(epi/w), for a homogenized representation of the microtransport problem. Representative calculations are carried out for three small solutes-water, L-glucose, and hydrocortisone-covering a wide range of membrane permeability. The effective transport parameters and their microscopic interpretation can be employed within the context of existing three-layer models of skin transport to provide more realistic estimates of the epidermal concentrations of topically applied solutes.

Application of numerical methods for diffusion-based modeling of skin permeation

The application of numerical methods for mechanistic, diffusion-based modeling of skin permeation is reviewed. Methods considered here are finite difference, method of lines, finite element, finite volume, random walk, cellular automata, and smoothed particle hydrodynamics. First the methods are briefly explained with rudimentary mathematical underpinnings. Current state of the art numerical models are described, and then a chronological overview of published models is provided. Key findings and insights of reviewed models are highlighted. Model results support a primarily transcellular pathway with anisotropic lipid transport. Future endeavors would benefit from a fundamental analysis of drug/vehicle/skin interactions.

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.

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.

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.