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3.0 A Geoscience Approach to Modeling Chemical Transport through Skin Clifford K. Ho |
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Ho, C.K., 2004, Probabilistic Modeling of Percutaneous Absorption for Risk-Based Exposure Assessments and Transdermal Drug Delivery, Statistical Methodology, Vol. 1/1-2, 47-69. Uncorrected proof (PDF 620 KB)
Modeling chemical transport through human skin (percutaneous absorption) serves an important role in two primary arenas: (1) hazardous chemical-exposure assessments and (2) transdermal drug delivery. In the former, models are used to understand relevant features and processes of percutaneous absorption so that protective measures can be designed and implemented that minimize the risk of dermal absorption of toxic chemicals (Stewart and Dodd, 1964; Bird, 1981; Flynn, 1990; EPA, 1992; Ness, 1994; Poet et al., 2000; McDougal and Boeniger, 2002; Poet and McDougal, 2002). In the latter arena, researchers are striving to enhance the viability of transdermal delivery of drugs such as analgesics, insulin, and more recently, peptides and proteins (Amsden and Goosen, 1995; Potts and Guy, 1995; Kalia and Guy, 2001). Transdermal delivery of drugs that require low dosages for long periods can be more effective, less costly, and less painful than traditional alternatives such as injection, intravenous infusion, or oral ingestion.
Developing accurate and reliable models of chemical transport through the skin can yield information regarding the important features and processes that contribute to the retardation or enhancement of chemical permeation. Many of the models that have been considered previously have focused on steady-state Fickian diffusion through the skin (Michaels et al., 1975; Flynn, 1990). Transient models have been developed (Scheuplein, 1967; Kalia and Guy, 2001), but simplifying assumptions were made so that analytical solutions could be obtained. In addition, boundary conditions and properties were stylized according to the field of application (either for exposure assessment or drug delivery). For example, models used for exposure assessments typically focus on industrial solvents and other hydrocarbons (e.g., trichloroethylene), which are generally lipophilic and hydrophobic. On the other hand, models used in drug-delivery studies often focus on hydrophilic solutes (i.e., drugs that dissolve in water). These considerations impact the boundary conditions of the models, as well as the choice of partitioning coefficients and the layers of skin that are included in the models. The application-specific models make it difficult to draw general conclusions regarding the features and processes that most affect the permeability of particular solutes and solvents. In addition, models are often used with deterministic property values, which do not consider the large uncertainty inherent in biological systems and properties.
A number of similarities exist between models of percutaneous absorption and models of contaminant transport in geologic media. Both systems involve complex, multiphase, heterogeneous structures with transient diffusion and sorption of chemical species. Models of contaminant transport in fractured rock also simulate features (e.g., fractures) that may create “fast-flow” paths within a larger continuum. This is analogous to hair follicles and sweat glands that penetrate the skin, potentially creating shunts for chemical transport through the skin. In addition, probabilistic methods have been developed for performance assessments of complex transport processes in geologic structures to quantify risk and uncertainty and to perform sensitivity analyses (DOE, 1998). These methods can be applied to models of percutaneous absorption to perform similar assessments.
The purpose of this work is to review previous models of percutaneous transport and to identify important assumptions and issues relevant to each model. A stochastic analysis adopted from contaminant transport models of geologic media is then applied to models of percutaneous absorption. In particular, a mechanistic model of transient percutaneous absorption will be developed that is used in conjunction with probabilistic methods to estimate probability distributions for permeation and chemical dose. Uncertainty distributions for model-input parameters are developed, and a Monte Carlo analysis is performed using the mechanistic model to quantify the impacts of the uncertainties on the simulated results. Sensitivity analyses are also performed to identify the parameters that are most important to the simulated results. A review of the anatomy of the skin and the factors that are likely to impact chemical permeation are provided first, followed by a description and discussion of the models and results.
The skin is a complex organ that serves to protect humans from chemical, physical, and biological intrusion, while retaining moisture and providing thermal regulation. It consists of three primary regions: the epidermis, the dermis, and the hypodermis (see Figure 3.1). The epidermis is the outermost layer of the skin in contact with the environment, ranging between 0.075 and 0.20 mm thick in most regions and between 0.4 and 0.6 mm thick in the palms and soles (Amsden and Goosen, 1995; Flynn, 1990). It consists of the stratum corneum, which forms the outermost layer of the epidermis, and the viable epidermis, which consists of the granular, spinous, and basal layers. The epidermis does not contain any capillary vasculature, so chemicals that transport through the epidermis must also transport partially through the dermis to reach the bloodstream. The cells in the epidermis are continually shed to the surface and replaced from the basal layer. These cells are replaced completely on the average of once every two weeks.
Figure 3.1 Skin features relevant to percutaneous absorption of chemicals.
The outermost layer of the epidermis, the stratum corneum, is the primary barrier to permeation of most drugs and chemicals (Scheuplein, 1976; Michaels et al., 1975). The stratum corneum is between 10 and 50 mm thick (15-20 cell layers thick) and contains dead keratinized cells (keratinocytes) with lipid lamellae filling the intercellular regions. It is composed of a very heterogeneous structure containing approximately 20-40% water, 20% lipids, and 40% keratinized protein. The keratinocytes, connected together in a planar array by desmosomes, are thin platelets filled with polar protein strands woven into compact and dense keratin fibers. The lipids form a continuous, albeit extremely tortuous, intercellular network between the keratinocytes. The compactness of the keratinocytes and the limited amount of intercellular lipid results in the low permeability of the stratum corneum.
The underlying dermis contains the vasculature (blood vessels and lymph vessels) that can uptake chemicals diffusing through the skin. The vasculature can reach to within a few microns of the undersurface of the epidermis. The dermis consists of a moderately dense network of connective tissue composed of collagen fibers and elastic fibers. It varies in thickness from 1 to 4 mm depending on the location of the body. Diffusion through this layer is analogous to diffusion through hydrogels (Amsden and Goosen, 1995).
Hair follicles and sweat glands, called skin appendages, break the continuity of the epidermal and dermal layers throughout most of the surface of the body. On average, 40 to 100 hair follicles and 210 to 220 sweat ducts exist per square centimeter of skin, occupying about 0.1% of the total surface area (Scheuplein, 1967). Hair follicles extend through the epidermis into the dermis, where the base of the follicle is well vascularized. Sebaceous glands attached to the sides of the follicles secrete sebum, a lipid mixture, into the region between the hair and the sheath. The sweat glands consist of tubes extending from the dermis, where the tube is coiled and vascularized, to the skin surface where a watery mixture (sweat) is excreted to provide thermal regulation.
3.3 Skin Permeation Routes and Previous Models
Based on the physiology of the skin, three possible pathways exist for passive transport of chemicals through the skin to the vascular network (Scheuplein, 1965; Scheuplein and Blank, 1971): (1) intercellular diffusion through the lipid lamellae; (2) transcellular diffusion through both the keratinocytes and lipid lamellae; and (3) diffusion through appendages (hair follicles and sweat ducts). Figure 3.2 illustrates these potential pathways.
Figure 3.2 Skin permeation routes through the stratum corneum: (1) intercellular diffusion through the lipid lamellae; (2) transcellular diffusion through both the keratinocytes and lipid lamellae; and (3) diffusion through appendages (hair follicles and sweat ducts).
A number of models have been developed that simulate one or more of these pathways. Michaels et al. (1975) considered the first two modes of transport by modeling the steady-state behavior of the stratum corneum as a two-phase “brick and mortar” region (the aqueous protein phase in the keratinocytes was modeled as the bricks and the intercellular lipid phase was modeled as a continuous mortar). They assumed that the transport was the sum of steady diffusion (1) through the lipid and protein in series and (2) through the lipid phase via a tortuous path. They estimated tortuosities (~10:1) and diffusion coefficients through the lipid and protein. Experiments were conducted using cadaver skins and several different drug chemicals. Results showed a permeation dependence on pH (higher pH gave a higher flux for the same concentration) and mineral oil/water partition coefficient (larger partition coefficients yielded greater fluxes). They concluded that the ratio of the lipid diffusivity to the protein diffusivity (one of the two important parameters in their model) was 10-2 to 10-3, meaning that the diffusion coefficient for the lipid phase was about 500 times less than the diffusion coefficient for the protein phase, which they estimated to be about 2x10-7 cm2/s (from Michaels et al., 1975).
Flynn (1990), however, stated that the density and compactness of the intracellular protein in the keratinocytes of the stratum corneum presents a thermodynamically and kinetically impossible passageway for chemical transport. Other recent investigators also supported the belief that when comparing evidence for intercellular versus transcellular diffusion, intercellular diffusion through the lipid lamellae is the predominant mode of transport (Amsden and Goosen, 1995). As a result, Flynn (1990) proposed an alternate “aqueous pore pathway” in parallel (as opposed to in series) with the lipid pathways through the stratum corneum to represent the limited intercellular aqueous phase. Although the location of these aqueous pathways was uncertain, Flynn included these pathways to accommodate the diffusion of polar compounds. Other researchers have argued a similar phenomenon by considering both a polar and non-polar pathway through the stratum corneum. Elias (1981) described the polar pathway in terms of aqueous pores in the small aqueous phase between the lipid lamellae. Supporting this theory, Cooper (1984) found that polar molecules such as water and small ions permeated the skin and that the flux was independent of the oil/water partition coefficient. As the polarity of the molecules decreased, the flux became a function of the partition coefficient (Scheuplein and Blank, 1971).
Results of Flynn (1990) showed that diffusion was a direct function of the octanol/water partition coefficient, Kow, and molecular weight, MW (for a given Kow, chemicals with larger molecular weights exhibited lower diffusion; for a given MW, chemicals with greater Kow yielded more diffusion). The study showed that larger molecular-weight chemicals permeated slower in general, but the phase of the vehicle (water or oil) delivering the chemical was not specified. Scheuplein (1976) and Potts and Guy (1995) point out that the permeability of a chemical depends on the phase of the vehicle. If the molecular weight is high, indicating a more lipophilic compound, then the permeability will be greater if the vehicle is a water than an oil since the compound will want to partition out of the water and into the tissue. Flynn (1990) also presented a simple exposure-assessment equation using the results of his modeling that expressed the cumulative mass entering the skin. The permeability coefficient was determined from simple equations that were correlated to experimental results for different Kow and MW values. The equation assumed that the cumulative mass entering the skin took place after the lag time, which Flynn estimated could be approximately 10 minutes for MW<150 and 1 hour for MW>150.
Scheuplein (1967) developed analytical transient models of percutaneous absorption considering transport via appendages. He compared transport through appendages with transport through the intact stratum corneum, which he modeled as two single-phase regions: (1) the stratum corneum with a thickness of 10 mm and (2) the aqueous viable epidermis and papillary dermis with a combined thickness of 200 mm. To determine the cumulative amount of chemical transported, he used a composite slab solution (using resistances in series). From this he concluded that the appendages (follicles and sweat ducts), which had several orders of magnitude higher diffusion coefficients, allowed greater transport at early times, but that the bulk stratum corneum would allow greater diffusion at longer times. To determine concentrations profiles, he used a steady-state solution to determine the steady concentrations in the two slabs and a semi-infinite solution to determine the transient concentrations in the two slabs. The semi-infinite solution does not yield a concentration of zero at the boundary of the basal layer, which was inconsistent with his general formulation, but it did provide some relative comparisons. He also showed that the partitioning coefficient between the lipid and aqueous regions could also impact the concentration gradient.
Kalia and Guy (2001) developed a number of analytical solutions for transient diffusion of drugs through the skin. They treated the skin as a homogeneous slab, but they considered different scenarios for the delivery vehicle (e.g., patch with a reservoir on top, patch with drug dispersed, drug in an ointment, etc.). They concluded that a unified model that could consider the effects of molecular weight and partition coefficients was necessary.
Other models have been developed that do not consider the specific routes of transport through the skin but attempt to describe the general rate of chemical transport through the skin and/or into the circulatory system using empirical observations and lumped-capacitance models. These models are generally described as physiologically-based pharmaco-kinetic models (PBPK). The general method is to correlate existing data to simple compartment models that represent the skin and various processes and regions associated with uptake into the body. Potts and Guy (1995) and Poet et al. (2000) have developed PBPK models that can predict chemical diffusion and uptake through the skin using physical properties of the chemical. Potts and Guy (1995) developed a model that provides an algorithm to predict permeability from the drug’s physical properties. Multiple regression analyses were performed using previous data of the permeability coefficient for different chemicals, and the molecular volume and the hydrogen bond activity parameters were determined to be important. However, this model is only valid when the stratum corneum is the rate-limiting barrier to percutaneous absorption (i.e., for polar compounds). Poet et al. (2000) used a PBPK model to estimate skin permeability values and to predict exhaled concentrations of trichloroethylene (TCE) when subjects were exposed to TCE. Good agreement was obtained between predicted and observed TCE concentrations, but the relative importance of the various features and processes were not elucidated. In general, specific routes of permeation that contribute to the overall rate of transport through skin are not considered in PBPK models.
3.4 Model Development
Most of the models of percutaneous absorption that have been developed previously treat the skin as a homogeneous medium with an effective (average) permeability coefficient. These include many of the transient analyses (e.g., Kalia and Guy, 2001) and the PBPK analyses (e.g., Poet et al., 2000). A few models that do consider multiphase heterogeneous transport through the various layers and pathways of the skin often assume steady-state conditions (e.g., Michaels et al., 1975; Flynn, 1990). Scheuplein (1967) developed models of transient diffusion through different pathways in the skin, but deterministic models were used. In the following sections we develop a probabilistic, transient, multiphase model of chemical transport through various routes in the skin to address the inherent uncertainties in the processes and parameters associated with percutaneous absorption.
In particular, we consider the following possible pathways: (1) intercellular diffusion through the lipids and aqueous “pores” in the stratum corneum (pathway #1 in Figure 3.2); and (2) diffusion through appendages (hair follicles and sweat ducts) (pathway #3 in Figure 3.2). We do not consider the transcellular pathway across keratinocytes and lipids (pathway #2 in Figure 3.2) because the evidence presented earlier suggests that diffusion through the keratinocytes would be extremely small.
3.4.1 Intercellular Diffusion through the Stratum Corneum
Intercellular diffusion through the stratum corneum is modeled as a three-phase continuum. The keratinized cells in the stratum corneum are considered to be an immobile protein phase, which can provide reversible interactions (adsorption and desorption) with chemicals in the mobile phases. The mobile phases include the lipid (or oil) and aqueous (water) phases in between the keratinocytes. A differential control volume consisting of these three phases is shown in Figure 3.3.
Figure 3.3 Control volume for intercellular chemical diffusion through a three-phase region in the stratum corneum: p = immobile protein phase (keratinocytes), o = mobile oil (lipid) phase, w = mobile water (aqueous) phase.
Assuming that Fick’s Law governs the diffusive mass transport through the mobile regions, a one-dimensional mass balance of a chemical diffusing through this three-phase region results in the following partial differential equation for the concentration as a function of time, t [s], and penetration distance into the skin, x [m]:
(3.1)
where C is the concentration of the chemical present in the phase [kg/m3-phase]; Do, Dw, and Dp, are the molecular diffusion coefficients for the oil, water, and protein phases, respectively [m2/s]; f is the porosity of a given phase [m3-phase/m3-total]; t is the tortuosity coefficient (inverse of tortuosity) that expresses the ratio of the linear path length to actual path length; subscript o denotes the oil (or lipid) phase; subscript w denotes the water (or aqueous) phase; and subscript p denotes the protein (or keratinized cell) phase.
We assume that local equilibrium exists and that partitioning between the three phases can be expressed using the following linear relationships:
Co = KowCw (3.2)
Cw = KwpCp (3.3)
Co = KopCp (3.4)
where K is the partitioning coefficient of the phases denoted by the subscripts. Because the protein phase is hydrophilic (Michaels et al., 1975), we also assume that the water-protein partition coefficient is near unity (Kwp = 1). Therefore, the water and protein concentrations are equivalent (i.e., Cw = Cp) and the oil-protein partition coefficient, Kop, is equivalent to the oil-water partition coefficient, Kow (often referred to as the octanol-water partition coefficient). The octanol-water partition coefficient is used widely as a measure of polarity in organic chemistry (Flynn, 1990). Using these assumptions, we can then re-write Equation (3.1) in terms of the water concentration as follows:
(3.5)
where 
(3.6)
(3.7)
Dsc is the effective diffusion coefficient of the three-phase stratum-corneum continuum and Rsc is the retardation factor of the three-phase stratum-corneum continuum. The boundary and initial conditions for Equation (3.5) are written as follows:
(3.8)
(3.9)
(3.10)
Equation (3.8)
assumes that the surface of the stratum corneum is maintained at a constant
concentration, 
,
in the aqueous phase. Equation (3.9) assumes that at a distance Lsc
from the surface, capillaries are present that have effectively zero
concentration due to a continuous advective flow in the bloodstream (this also
assumes that the aqueous region just beneath the stratum corneum in the viable
epidermis and dermis does not contribute significantly to the overall
resistance of chemical transport).
Finally, Equation (3.10)
assumes that the initial concentration in the stratum corneum is zero.
The solution to Equations (3.5) to (3.10), which yields the aqueous concentration as a function of time and location in the skin, is presented in Crank (1975, pp. 49-51) and can be written in non-dimensional form as follows:
(3.11)
The mass flux into the blood stream (dose), 
[kg/m2-s], can be calculated using
Fick’s Law at the downstream boundary of the stratum corneum (i.e., x = Lsc):
(3.12)
In addition, the cumulative amount of mass (per unit area) diffusing into the blood stream (cumulative dose), Q [kg/m2], can be expressed as follows:
(3.13)
The expressions for both the mass flux, 
,
and cumulative dose, Qsc,
can be readily non-dimensionalized.
Finally, the time required for the system to reach steady-state
conditions, 
[s],
can be approximated by the following expression adapted from Crank (1975, p. 51):
(3.14)
3.4.2 Diffusion through Sweat Ducts
Chemical permeation through sweat ducts is modeled as a single-phase aqueous diffusion process. A control volume consisting of a sweat duct (or hair follicle) in a larger continuum is shown in Figure 3.4. For simplicity, the region around the sweat duct is assumed to be impermeable (no interactions), and the sweat duct is assumed to be filled with water. The region around the sweat duct is included in the control volume to represent a larger unit area of skin for normalization with the other transport pathways.
Figure 3.4 Control volume for diffusion through an appendage (sweat duct or hair follicle). Diffusion is assumed to occur through a single-phase fluid in the appendage.
A mass balance of a chemical species diffusing through this control volume can be written as follows:
(3.15)
where 
(3.16)
(3.17)
where the subscript s
denotes the sweat duct. Note that the
porosity of the sweat duct, fs, represents the fractional
area that the sweat ducts occupy per unit area of skin. It depends on both the density of sweat
ducts and the area available for diffusion within each sweat duct. The boundary and initial conditions for Equation (3.15) are the same as those
expressed in Equations (3.8) to (3.10)
with Cs replacing Cw. The constant aqueous concentration on the surface of the sweat
duct is assumed to be the same as the constant aqueous concentration applied to
the surface of the skin, 
,
in Equation (3.8). In addition, the distance between the
surface of the sweat duct and the location where the chemical is carried into
the blood stream is denoted as Ls. The solutions for the normalized
concentration, the mass flux into the bloodstream, the cumulative mass, and the
time to reach steady state are expressed as follows for chemical transport in
the sweat duct:
(3.18)
(3.19)
(3.20)
(3.21)
3.4.3 Diffusion through Hair Follicles
Diffusion through hair follicles is modeled in a similar fashion as diffusion through sweat ducts. The primary difference is that the follicle is assumed to be filled with an oil phase instead of an aqueous phase. Using the control volume shown in Figure 3.4, a mass balance on a chemical species diffusing through the oil phase in the follicle can then be expressed as follows:
(3.22)
where 
(3.23)
(3.24)
where the subscript f denotes the hair follicle. The porosity of the hair follicle, ff, represents the fractional area that the follicles occupy per unit area of skin. It depends on both the density of follicles and the available area for diffusion within each follicle. The boundary condition at the surface of the hair follicle is slightly different than the corresponding boundary conditions of the stratum corneum and sweat duct because the follicle concentration is written in terms of the oil phase. Assuming local equilibrium between the aqueous concentration at the surface boundary of the skin and the concentration in the oil phase at the surface of the follicle, the boundary and initial conditions for the follicle can be written as follows:
(3.25)
(3.26)
(3.27)
The distance between the surface of the follicle and the location where the chemical is carried into the bloodstream is denoted as Lf. The solutions for the normalized concentration, the mass flux into the bloodstream, the cumulative mass, and the time to reach steady state are expressed as follows for chemical transport in the hair follicle:
(3.28)
(3.29)
(3.30)
(3.31)
It is important to note that the expressions for the mass flux into the bloodstream in the above solutions use an effective diffusion coefficient preceding the concentration gradient term. The effective diffusion coefficient accounts for the reduced area of the appendages per unit area of skin, as well as the reduced area for diffusion caused by phase interference in the three-phase stratum corneum. However, the coefficient preceding the second derivative in the diffusion equation is expressed by the ratio of the effective diffusion coefficient and the retardation factor, which yields the “diffusivity” for the diffusion equation that appears in the exponent term of the solutions.
3.4.4 Uncertainty Distributions of Input Parameters
The parameters that are used in the solutions presented above can be highly uncertain. As a result, distributions of values are assigned to each of the input parameters using parameter values available in the literature to capture the inherent uncertainty. If insufficient data existed to define a distribution for a parameter, professional judgment was used to assign a distribution for that parameter based on the available values. A Monte Carlo analysis is then performed to obtain a probabilistic distribution of results using the derived solutions. Table 3.1 summarizes the stochastic variables and associated uncertainty distributions that are used in this study.
Table 3.1 Stochastic variables and their uncertainty distributions.
|
Stochastic Variable |
Units |
Distribution |
Median Value* |
Description |
Reference |
|
fo |
- |
uniform lower bound: 0.15 upper bound: 0.20 |
0.18 |
oil-phase
porosity in the stratum corneum |
1 |
|
fw |
- |
uniform lower bound: 0.15 upper bound: 0.40 |
0.27 |
aqueous-phase
porosity in the stratum corneum |
1,2 |
|
fp |
- |
uniform lower bound: 0.35 upper bound: 0.40 |
0.38 |
protein-phase
porosity in the stratum corneum |
1,2 |
|
fs |
- |
log uniform lower bound: 3.6x10-5 upper bound: 8.4x10-3 |
4.5x10-4 |
fractional
area of sweat ducts per unit area of skin (sweat duct porosity) |
3 |
|
ff |
- |
uniform lower bound: 1.5x10-3 upper bound: 3.8x10-3 |
2.8x10-3 |
fractional
area of hair follicles per unit area of skin (follicle porosity) |
3 |
|
to |
- |
log uniform lower bound: 0.01 upper bound: 0.1 |
0.034 |
oil-phase
tortuosity coefficient in stratum corneum |
1 |
|
tw |
- |
log uniform lower bound: 0.001 upper bound:0.01 |
3.1x10-3 |
aqueous-phase
tortuosity coefficient in stratum corneum |
4 |
|
ts |
- |
uniform lower bound: 0.1 upper bound: 1.0 |
0.56 |
sweat
duct tortuosity coefficient |
N/A |
|
tf |
- |
uniform lower bound: 0.1 upper bound: 1.0 |
0.52 |
hair
follicle tortuosity coefficient |
N/A |
|
Dw |
m2/s |
log uniform lower bound: 10-10 upper bound: 10-9 |
3.2x10-10 |
molecular
diffusion coefficient in aqueous phase |
5 |
|
Do |
m2/s |
log uniform lower bound: 10-11 upper bound: 10-10 |
3.2x10-11 |
molecular
diffusion coefficient in oil phase |
6 |
|
Kow |
- |
log normal mean log(Kow):
2.0 st. dev. log(Kow): 1.4 |
8.8 |
octanol-water
partition coefficient |
7 |
|
Cow |
kg/m3 |
log uniform lower bound: 0.003 upper bound: 800 |
2.0 |
fixed
aqueous concentration at the skin surface (aqueous solubility limit) |
1,8 |
|
Lsc |
m |
log uniform lower bound: 5x10-6 upper bound: 6x10-4 |