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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 |
5.9x10-5 |
thickness
of the stratum corneum |
9 |
|
Ls |
m |
uniform lower bound: 2x10-4 upper bound: 4x10-4 |
3.1x10-4 |
length
of the sweat duct |
1,10 |
|
Lf |
m |
uniform lower bound: 2x10-4 upper bound: 4x10-4 |
3.1x10-4 |
length
of hair follicle |
1,10 |
*The median value is
calculated from distributions generated by Mathcad®7.
1Michaels et al. (1975)
2Amsden and Goosen (1995)
3Scheuplein (1967)
4the aqueous-phase “pores” in the stratum corneum are assumed
to have tortuosity-coefficient bounds that are an order of magnitude less than
the oil-phase tortuosity coefficient in the stratum corneum
5 Reid et al. (1987); representative values were taken for
solutes diffusing through water
6The bounds for the molecular diffusion coefficient in the
oil phase are assumed to be an order of magnitude less than the bounds for the
aqueous phase
7Flynn (1990)
8water solubilities of various compounds were used from
Michaels et al. (1975)
9Scheuplein and Blank (1971)
10the distribution is assumed to be equal to the distribution of
thicknesses between the skin surface and capillary bed
All of the stochastic input parameters are assumed to be independent. Although one might intuitively expect that the octanol-water partition coefficient, Kow, and aqueous boundary-condition concentration (aqueous solubility), Cow, might be correlated, we do not assume any correlation between the input parameters (limited data from Michaels et al. (1975) support this assumption for the oil-water partition coefficient and the water solubility).
Mathcad®7 was used to perform a Monte Carlo analysis of the solutions provided in the previous section. Uncertainty distributions listed in Table 3.1 were developed in Mathcad®7 using a seed value of one, and the corresponding median values are reported in the table. Five hundred realizations were simulated to capture the uncertainty propagated by the sixteen stochastic input variables.
For brevity, only a subset of the solutions presented in the previous section is used to illustrate the probabilistic analysis. The time required to reach steady-state conditions and the mass flux for each of the penetration routes are examined in detail here. Figure 3.5 shows the distribution of times required to reach steady state for each of the three permeation routes. The median times to reach steady state are 4 minutes, 24 minutes, and 48 minutes for diffusion through the sweat duct, stratum corneum, and hair follicle, respectively. The distributions of steady-state times for transport through the sweat duct and hair follicle each span about two orders of magnitude. The difference between the distributions of steady-state times for the sweat duct and hair follicle is about an order of magnitude, where the sweat duct generally reaches steady state faster. The primary reason for the difference is that the molecular diffusion coefficient for the oil phase in the hair follicle was assumed to be about an order of magnitude less than the molecular diffusion coefficient for water, which comprises the sweat ducts. The results for the stratum corneum span nearly five orders of magnitude. The effective diffusion coefficient for the three-phase stratum corneum depends on a number of additional stochastic parameters that contribute additional uncertainty to the results.
Figure 3.5 Cumulative probability distribution of times required to reach steady state for chemical diffusion through three permeation routes of the skin using 500 realizations.
The mass flux into the bloodstream for each of the three permeation routes is calculated at two different times: 60 seconds and 1 hour. At 60 seconds (~0.02 hours), Figure 3.5 shows that most of the realizations for each permeation route are still in an early transient state; at 1 hour, most of the realizations (> 50%) for each of the permeation routes have reached a steady-state condition. By evaluating the mass flux into the bloodstream at these two times, we intend to glean information regarding the most important pathways and parameters during both early-time transient diffusion as well as long-term diffusion when the pathways are approaching steady state.
Figure 3.6 shows the cumulative distribution function for the
mass flux into the bloodstream, 
,
for each of the three permeation routes at 60 seconds. The units of mass flux used in the plot are
nanograms per square centimeter per hour [ng/cm2-h] where
1 ng = 10-9 g.
At this early transient time period, many of the realizations for
diffusion through the hair follicle and stratum corneum result in negligible
mass flux at the lower boundary (bloodstream) of the modeled domain; however,
the spread in results is significant, and values reach as high as 104
and 108 ng/cm2-h for the hair follicle and stratum
corneum, respectively. For most of the
realizations at this early time, the mass flux through the sweat duct is
greatest. The median mass flux into the
bloodstream from the sweat duct is 17 ng/cm2-h, whereas the median
mass fluxes from the hair follicle and stratum corneum are 10-9 and
0.02 ng/cm2-h, respectively.
Figure 3.6 Cumulative probability distribution of mass flux into the bloodstream at 60 seconds for chemical diffusion through three permeation routes of the skin using 500 realizations.
Figure 3.7 shows the cumulative distribution function for the
mass flux into the bloodstream, 
,
at 1 hour. At 1 hour, many of the
realizations have reached steady state (see Figure 3.5), and the spread in
mass-flux values is significantly reduced for each of the three permeation
routes. The distribution of results for
the three-phase stratum corneum exhibits the most uncertainty, primarily
because the model relies on the largest number of stochastic variables compared
to the other permeation routes. In general, the mass flux into the bloodstream
at 1 hour is greatest in the stratum corneum, followed by the mass flux from
the hair follicle and sweat duct. The
median mass fluxes into the bloodstream from the sweat duct, hair follicle, and
stratum corneum are 190, 560, and 8900 ng/cm2-h, respectively. Recall that the mass flux into the
bloodstream from the hair follicle at early times was significantly less than
the mass flux from the sweat duct because of the lower molecular diffusion
coefficient in the oil phase of the hair follicle. However, the fractional area of hair follicles per unit area of
skin is significantly greater than the fractional area of sweat ducts (see
Table 3.1). Therefore, after an hour
(when sufficient time had elapsed for steady-state conditions to be approached),
the larger fractional area of hair follicles allowed more mass to diffuse into
the bloodstream per unit area of skin. Similarly, as steady-state diffusion was
approached in the stratum corneum at 1 hour, the large surface area allowed
relatively more mass to diffuse into the bloodstream. At later times, when more of the realizations would achieve
steady state in the stratum corneum, we would expect that the mass flux in the
bloodstream would be dominated by the stratum corneum.
Figure 3.7 Cumulative probability distribution of mass flux into the bloodstream at 1 hour for chemical diffusion through three permeation routes of the skin using 500 realizations.
Finally, the distributions of total mass flux into the bloodstream from the three permeation routes at 60 seconds and 1 hour are plotted in Figure 3.8. The median mass fluxes at 60 seconds and 1 hour are 1.3x103 and 2.4x104 ng/cm2-h, respectively. The uncertainty is reduced at 1 hour relative to at 60 seconds because as time progresses and steady-state conditions are approached, the solutions depend on fewer stochastic parameters. It is interesting to compare the resulting distributions with required dosages of various drugs reported in the literature. Amsden and Goosen (1995) report that the required adult dosage for various peptides can range from 2-4 micrograms per day for vasopressin (an antidiuretic hormone that regulates the excretion of body water through urine) to ~3,000 micrograms per day for insulin (a hormone used to convert sugar to energy). Assuming that a transdermal patch covers approximately 10 cm2 of skin, the required average mass flux into the bloodstream would be approximately 13 ng/cm2-h for vasopressin and 1.3x104 ng/cm2-h for insulin. Figure 3.8 shows that the probabilities of obtaining the required mass fluxes for vasopressin and insulin at 60 seconds after application of the transdermal patch is about 75% and 35%, respectively, using the assumed input distributions in this model. At 1 hour, the probabilities increase to 95% and 55%, respectively. It is important to note, however, that the values derived in this model are based on general input values that are intended to capture a large range of uncertainty for percutaneous absorption. Using properties specific to these two drugs (e.g., water solubility, partition coefficients, etc.) in the model will likely change the results.
Figure 3.8 Cumulative probability distribution of total mass flux into the bloodstream (sum of stratum corneum, sweat duct, and hair follicle) at 60 seconds and 1 hour using 500 realizations.
Analyses can be performed to determine the sensitivity of the dependent variables (e.g., mass flux into the bloodstream) to the stochastic independent variables (e.g., thickness of the stratum corneum, length of the sweat duct, etc.). A stepwise linear regression is a modified version of multiple regression that selectively adds input parameters (independent variables) to the regression model in successive steps. The stepwise process continues until no more variables with a significant effect on the dependent variable are found. The order of parameter selection for incorporation into the regression model gives an indication of their relative importance. The change in the coefficient of determination (DR2) for a given step indicates the fraction of the variance in the model output explained by the input parameter added in that step.
In order to implement a linear regression, a rank transformation (assigning the smallest value of a given variable a value of 1, the next largest value of a given variable a value of 2, and so on) of independent and dependent variables is required and is generally used to compensate for potential non-linear relationships in the stepwise linear regression method for complex model results. Rank transformation essentially allows regression on the strength of the monotonic relationship, instead of the strength of the linear relationship between independent and dependent variables. This stepwise linear regression method provides insight into the relationship between uncertainty in input parameters and the uncertainty in modeling results for complex probabilistic models. In addition, this method provides a quantitative basis for prioritizing the importance of relevant input parameters and processes.
A stepwise linear-regression analysis was performed using Statistica 6.0 on the results of the total mass flux into the bloodstream at 60 seconds and 1 hour (Figure 3.8 shows the results of the simulated total mass flux). Table 3.2 lists the key parameters that were identified at 60 seconds and 1 hour and their corresponding incremental contributions (DR2) to the coefficients of determination (R2). Also shown are the semi-partial correlations for each parameter. The semi-partial correlation is a measure of the proportion of (unique) variance accounted for by the parameter relative to the total variance of the output variable after controlling for the other input parameters. The semi-partial correlation is similar to the incremental coefficients of determination, but the sign of the semi-partial correlation indicates whether the correlation is positive or negative. Parameters with incremental coefficients of determination greater than 0.005 are presented.
Table
3.2 Summary of parameters important
to the simulated total mass flux into the bloodstream at two different times
based on stepwise linear-regression analysis.
See Table 3.1 for a list of all input parameters and their
distributions.
|
Step |
Variable |
R2 |
DR2 |
Semi-Partial
Correlation |
|
60
Seconds |
||||
|
1 |
aqueous solubility limit |
0.382 |
0.382 |
0.608 |
|
2 |
thickness of stratum corneum |
0.701 |
0.319 |
-0.581 |
|
3 |
aqueous molecular diff. coefficient |
0.727 |
0.026 |
0.172 |
|
4 |
sweat-duct porosity |
0.751 |
0.024 |
0.151 |
|
5 |
sweat-duct tortuosity coefficient |
0.770 |
0.019 |
0.131 |
|
6 |
oil molecular diffusion coefficient |
0.786 |
0.016 |
0.121 |
|
7 |
octanol-water partition coefficient |
0.800 |
0.014 |
0.113 |
|
8 |
oil tortuosity coefficient |
0.807 |
0.007 |
0.087 |
|
1 Hour |
||||
|
1 |
aqueous solubility limit |
0.747 |
0.747 |
0.846 |
|
2 |
thickness of stratum corneum |
0.875 |
0.128 |
-0.376 |
|
3 |
octanol-water partition coefficient |
0.919 |
0.045 |
0.203 |
|
4 |
oil molecular diffusion coefficient |
0.939 |
0.019 |
0.131 |
|
5 |
oil tortuosity coefficient |
0.948 |
0.009 |
0.103 |
At 60 seconds, the total mass flux is most sensitive to the aqueous solubility limit, Cow, which was used as the upper boundary condition for the concentration in the skin. Based on the incremental coefficient of determination, this parameter accounts for 38% of the variability in the results, with larger values resulting in larger mass fluxes. The thickness of the stratum corneum is also important, with nearly 32% of the variability in the simulated mass flux explained by this parameter. As indicated by the negative sign of the semi-partial correlation for this parameter, there is an inverse relationship between the thickness of the stratum corneum and the simulated total mass flux into the bloodstream (i.e., the thicker the stratum corneum, the lower the mass flux). The aqueous molecular diffusion coefficient and parameters associated with the sweat duct are the next most important parameters, followed by the oil-phase molecular diffusion coefficient, the octanol-water partition coefficient, and the oil-phase tortuosity coefficient. In total, 81% of the variability in the output is explained using the multiple regression model with these 8 key parameters (the top five or six parameters could be used with nearly the same confidence).
At 1 hour, nearly 95% of the total variability in the simulated mass flux into the bloodstream can be explained by five key parameters. The two most important parameters are the aqueous solubility and the thickness of the stratum corneum, similar to the results at 60 seconds. However, the next three most important parameters are associated with transport through the oil phase in the hair follicle. As explained earlier, at 1 hour, more realizations have reached steady state, and the larger number of hair follicles (follicle porosity) relative to the number of sweat ducts (sweat duct porosity) allows a greater simulated mass flux into the bloodstream when compared to earlier transient times when many of the realizations had not yet allowed mass to reach the bottom boundary.
Figure 3.9 provides a graphical interpretation of the results of the stepwise linear regression using the simulated total mass flux into the bloodstream as the dependent variable. It should be noted that although the results show the strongest sensitivity to aqueous solubility limit, which was used as the aqueous-phase skin concentration at the upper boundary of the simulated domains, the distribution assigned to this parameter in Table 3.1 spans over five orders of magnitude. The range was taken from a variety of chemicals that were reported in the literature to capture the full uncertainty distribution for various chemicals. If the chemical of interest is prescribed or known, this parameter (or an analogous form of it, i.e., the vehicle-tissue partition coefficient per Scheuplein, 1976) will likely exhibit a much smaller range of uncertainty. The resulting sensitivity to this parameter, while still significant, may therefore be reduced if the chemical is prescribed a priori.
Another note of interest regarding the sensitivity analysis is that the molecular diffusion coefficients for water and oil depend on a number of additional parameters such as the molecular weight of the diffusing species, temperature, viscosity, etc. The molecular weight (or molecular volume) of a chemical has been identified as an important parameter impacting percutaneous absorption (Flynn, 1990; Potts and Guy, 1995). If a functional relationship between the molecular diffusion coefficient and these parameters had been used (see, for example, Reid et al., 1987), uncertainty distributions could have been assigned to these additional parameters in lieu of the diffusion coefficient to identify the relative importance of these parameters on percutaneous absorption.
Figure 3.9 Results of stepwise linear regression analysis showing the dependence of the total mass flux (sum of mass flux through stratum corneum, sweat duct, and hair follicle) on key independent stochastic parameters at two different times.
The foregoing analyses and results have illustrated significant features and benefits of a probabilistic assessment of percutaneous absorption. In particular, the ability to quantify the uncertainty associated with a specific model and to identify the parameters most important to the simulated results can benefit studies ranging from exposure assessments to transdermal drug delivery.
3.5.3.1 Implications for Exposure Assessment
The United States Environmental Protection Agency produced a comprehensive report describing the principals and applications of dermal exposure assessment (EPA, 1992). The report summarizes the mechanisms of dermal absorption, techniques for measuring dermal absorption, and mathematical models available at the time for dermal absorption and risk assessment. The purpose of this report was to provide exposure assessors with the knowledge and tools required to evaluate dose and subsequent health risks associated with dermal exposure at waste disposal sites or contaminated soils where contaminants could reside in soil, air, and water. The report stated that exposure and risk assessment associated with dermal contact remains the least well understood of the major exposure routes (relative to ingestion and inhalation) because considerable uncertainty exists in parameters and processes associated with dermal absorption. Strangely, all of the mechanistic models presented in the report are deterministic in nature; none of them provide a quantification of uncertainty or sensitivity analyses of the mechanisms and parameters associated with percutaneous absorption. We believe that the use of probabilistic analyses that incorporate inherent uncertainty in the model parameters and processes will yield more useful and meaningful results when determining and reporting health risks associated with dermal absorption.
3.5.3.2 Implications for Transdermal Drug Delivery
One of the significant problems of transdermal drug delivery is the ability to deliver sufficient doses of a particular drug through the skin. Several methods have been developed to augment the passive diffusion of water-soluble drugs such as peptides and proteins through the skin (Amsden and Goosen, 1995), and these are briefly summarized below:
· Prodrugs: Lipophilic groups are covalently bonded onto functional groups of the drug to improve partitioning into the intercellular lipid lamellae of the stratum corneum. Enzymes detach the lipophilic groups in vivo, rendering them free and active. However, prodrugs have molecular size restrictions and require synthesis.
· Chemical permeation enhancers: Compounds exist that alter the skin as a permeability barrier. Known permeation enhancers include solvents and surfactants; however, the physical basis for the method of enhancement is still unknown. No general theory of chemical enhancement has been provided.
· Iontophoresis: An iontophoretic device consists of two electrodes immersed in an electrolyte solution and placed on the skin. When an electric current is applied across the electrodes, an electric field is created across the stratum corneum that drives the delivery of ionized drugs. The primary route of ion transport appears to be through hair follicles or sweat glands, although additional uncertain pathways may be created. This method is restricted to short-term delivery.
· Electroporation: Electroporation involves the application of high-voltage electric pulses to increase the permeation through lipid bilayers. This differs from iontophoresis in the duration and intensity of the application of electrical current (iontophoresis uses a relatively constant low-voltage electric field). The high-voltage electric pulse of electroporation is believed to induce a reversible formation of hydrophilic pores in the lipid lamellae membranes that can provide a high degree of permeation enhancement, but the physics and dynamics of this process are not completely understood. This method is restricted to short-term delivery.
· Ultrasound: Ultrasound applies sound waves having a frequency greater than 16 kHz to the skin, which causes compression and expansion of the tissue through which the sound waves travel. The resulting pressure variations cause a number of processes (e.g., cavitation, mixing, increase in temperature) that may increase the permeation of drugs. Again, the exact processes and physics have not yet been determined, and this method is restricted to short-term delivery.
In all of these methods, significant uncertainty exists regarding the processes and parameters that truly cause enhanced permeation. Probabilistic simulations and sensitivity analyses coupled with mechanistic models of each of these processes can help to identify the most likely processes and parameters that are significant to drug-delivery enhancement. While models of empirical correlations are intended to provide similar information, they are not inherently based on physical processes or mechanisms. If processes or parameters are introduced that have not been used in these correlations, the results are likely to be erroneous. Therefore, mechanistic models combined with probabilistic analyses are needed to better understand and improve these methods.
Chemical transport through skin is similar in many ways to chemical transport through geologic media. Both processes involve diffusion and absorption in heterogeneous multiphase media. Figure 3.3 shows that the intercellular pathway for chemical diffusion through the stratum corneum is analogous to diffusion through a porous media containing three phases: (1) an immobile solid phase, (2) a mobile water phase, and (3) a mobile oil phase. Figure 3.4 shows that the permeation route through appendages (sweat ducts and hair follicles) is analogous to transport through fractures or other fast-flow-path features (e.g., root holes, animal burrows, etc.) that exist in a relatively less permeable surrounding geologic media. Complex models of these systems have been developed in the geoscience arena to better understand processes relevant to applications such as enhanced oil recovery, geothermal energy production, subsurface contaminant migration, and environmental remediation.
Numerical simulators used in the oil industry to model three-phase movement of hydrocarbons in the subsurface can probably be applied to models of percutaneous absorption. Advantages of these codes is that they can be applied to transient, multidimensional, non-isothermal, heterogeneous systems. Most of the models used in studies of percutaneous absorption have assumed steady-state, one-dimensional, isothermal, homogeneous systems.
Numerical codes developed to model flow and transport through fractured rock can also be applied to simulations of percutaneous transport through appendages, which act as shunts through the bulk of the skin. Formulations have been developed to include the interactions between fast-flow features and a surrounding low-permeability media (Ho, 1997). These formulations were implemented in “dual-permeability” models that were developed to model flow and chemical transport in fractured rock where interaction between two different permeability materials was important. These dual-permeability models can be used to model the simultaneous transport of chemicals through the bulk stratum corneum as well as through the appendages (hair follicles and sweat ducts). Current models assume that the transport through the appendages and through the stratum corneum are independent.
Finally, constitutive relations developed in the geosciences may be applicable to models of percutaneous transport. For example, numerous studies have been performed to relate parameters such as the effective diffusion coefficient in porous media to parameters such as the tortuosity coefficient and mobile-phase porosity. Processes such as diffusion in porous media under a thermal gradient have also been investigated in great detail. The use of these studies in trying to better understand processes associated with percutaneous absorption may be extremely beneficial.
Percutaneous absorption plays an important role in applications dealing with exposure assessment and transdermal drug delivery. Unfortunately, previous models have focused on deterministic, steady-state, homogeneous systems when evaluating chemical transport through the skin. In this study, a transient, three-phase, mechanistic model of percutaneous absorption has been developed. Penetration routes through the skin that were modeled include the following: (1) intercellular diffusion through the stratum corneum comprised of an immobile protein phase and mobile aqueous (water) and oil (lipid) phases; (2) diffusion through aqueous-phase sweat ducts; and (3) diffusion through oil-phase hair follicles. Uncertainty distributions were assigned to model parameters and a probabilistic Monte Carlo analysis was performed to simulate a distribution of mass fluxes through each of the routes. Results indicated that at early times, before steady-state conditions had been established, transport through the sweat ducts provided a significant amount of mass flux into the bloodstream. Because of the uncertainty in the input parameters, a large range of mass fluxes was simulated through each of the three routes at this early time. At longer times (1 hour), when many of the realizations had reached steady state, the uncertainty was reduced, and the relative importance of the pathways had changed. Diffusion through the stratum corneum became important because of the relatively large surface area. Similarly, despite the lower oil-phase molecular diffusion coefficient of the hair follicles, diffusion through the hair follicles was more significant than diffusion through the sweat ducts at later times because of the larger simulated porosity of hair follicles.
Sensitivity analyses were also performed using a stepwise linear regression analysis. Parameters that were most important to the simulated mass flux were identified, and the relative importance of each parameter was quantified through the incremental coefficients of determination and semi-partial correlations. These analyses were found to be extremely useful in not only quantifying the uncertainty in the simulated output variables, but also in identifying the input parameters that were most important to the simulated results. These probabilistic methods can provide more meaningful interpretations of exposure assessments and risk regarding dermal uptake of contaminants. In addition, new methods of enhancing transdermal drug delivery (e.g., ultrasound, electroporation, etc.) have a great deal of uncertainty surrounding the physical processes associated with these methods. Mechanistic models of multiphase, heterogeneous transport through the skin coupled with probabilistic analysis can shed additional insight into how these methods can be improved through identification and refinement of important parameters and processes.
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