10.1 Introduction

The skin is a primary route of exposure for chemicals in environmental and occupational settings; it also serves as a portal for systemic drug delivery using transdermal patches or is useful for local therapy of dermatological diseases using topical formulations. Exposure may also occur with the use of cosmetics and many personal care products. The skin is also a target for toxicity by these same agents owing to secondary or direct chemical action or as a result of immunological detection with amplification due to previous chemical sensitization. Because of this almost universal exposure of the skin to chemicals and drugs and because the toxicity is manifested in very visible and noticeable reactions, a great deal of attention has been focused on the effects of chemical exposure to this organ.

There are two major types of studies that comprise the field of dermatotoxicology: chemical absorption and irritation/sensitization. Computational approaches in this field have been focused on the former owing to its ability to generate quantitative data suitable for such an approach. This is also logical as a chemical must traverse the protective outer layer of the skin in order to gain access to its viable cells and exert a toxicological effect. Thus, computational approaches to predict toxicity are confounded by chemical properties that allow absorption to occur, focusing the best work at this point on models to predict dermal absorption. Additionally, absorption studies that investigate mechanisms and/or the toxicity involved have traditionally been conducted using single neat chemical exposure or binary mixtures, that is, a single drug/chemical delivered in a single vehicle. Such studies have demonstrated the significant effects that these vehicles may have on chemical absorption [1–4]. However, most topical chemical exposures, whether deliberate or accidental, are rarely in the form of a binary mixture - they are often as complex chemical mixtures consisting of penetrants, different vehicles, and/or chemical additives. Additional studies on vehicles with additives such as surfactants, alcohols, and solvents have confirmed that their presence may alter the barrier properties of the skin [5–8], yet limited work has been done to broadly incorporate the effect of complex mixture interactions [9, 10]. With the abundant presence of complex mixtures in cosmetic and pharmaceutical products, and through occupational and environmental scenarios, the ability to predict absorption is a highly desirable objective. This is the focus of the present chapter.

10.2 Principles of Dermal Absorption

The skin is composed of two primary layers, the epidermis which includes the outermost stratum corneum barrier and underlying viable keratinocytes, and the dermis [11] (Figure 10.1). The stratum corneum is relatively impermeable to most aqueous solutions and ions; however, it may be permeable to more lipophilic compounds. This layer is nonviable and considered to be the rate-limiting barrier in drug and chemical percutaneous absorption. It is axiomatic that a topically applied chemical must first traverse the stratum corneum barrier before it is capable of eliciting any toxicological or immunological effect on subsequent cell layers, making absorption both the primary factor in assessing the dermal effects of drugs and chemicals, as well as a confounding factor in all dermal computational toxicology models targeted toward intact skin.

Chemical absorption pathways can hypothetically involve both intercellular and intracellular passive diffusion across the epidermis and dermis and/or transappendageal routes, via hair follicles and sweat pores. Most available research has concentrated on the stratum corneum as the primary barrier to absorption, although the viable epidermis and dermis may also contribute resistance to the percutaneous penetration of specific chemical classes, for example, when the true barrier to absorption is not diffusional but is rather metabolic. Barrier is thus an operational definition and can be related to either a physical structure (e.g., the stratum corneum) or alternatively a biological process (e.g., diffusional resistance, metabolism, and vascular uptake) that retards absorption of topically applied chemicals.

The accepted hypothesis for dermal absorption is that the dominant pathway for chemicals to traverse the stratum corneum is through the intercellular lipids. Lipophilic compounds diffuse through this lipid milieu while polar molecules traverse the aqueous region of the intercellular lipids. This intercellular region, conceptualized as the mortar in the “brick and mortar” model of the stratum corneum [12], is now considered the most likely path for absorption of lipophilic drugs. Although this model is conceptually simple, the actual physical chemical environment of the intercellular lipids is complex. It is filled with neutral lipids (complex hydrocarbons, free sterols, sterol esters, free fatty acids, and triglycerides) that make up 75% of the total lipids, as well as other polar lipids [13, 14]. These intercellular lipids are also inextricably linked to the outer cellular membranes of the corneocytes, making a relatively complex and fluid structure that is often modeled as a simple homogeneous lipid pathway. Successive tape stripping, delipidization techniques, and use of epidermis through heat or chemical separation techniques, have been used by investigators to demonstrate the dominant influence that the stratum corneum and the lipid domain holds on penetration of hydrophilic and lipophilic chemicals. It is in this domain that many mixture and formulation components can modulate penetrating drugs.

Percutaneous absorption through the intercellular pathway of the stratum corneum is driven by passive diffusion down a concentration gradient described at steady state by Fick's law of diffusion [15],

where coefficient, coefficient, and SA is the applied surface area, H is membrane thickness (or more precisely the convoluted intercellular path length) and Δx is the concentration gradient across the membrane. Since in vivo blood or in vitro perfusate concentrations after absorption are negligible compared to applied surface concentration, Δx reduces to the surface concentration (C) available for absorption. It is this relationship that allows the prediction of compound flux across the skin to be correlated to factors predictive of D and PC (e.g., octanol/water partition coefficients). Flux is expressed in terms of applied surface area, often normalized to cm². This is an oversimplification of the complex transport processes that occur in dermal diffusion [16, 17]; however, it forms the basis for most computational toxicology modeling efforts in this field.

The term is compound dependent and is termed the permeability coefficient (K_p), reducing the determination of flux to or , a first-order pharmacokinetic equation . Rearrangement of this equation yields the primary method used to experimentally calculate the value of K_p:

It must be stressed that both transdermal flux and K_p are not only chemical dependent but also tightly constrained by the membrane system studied, as well as the experimental design of the study used to estimate it (neat compound, vehicle, length of experiment, etc.). It is at this phase that additional mixture interactions may occur through altered solubility on the surface or in the stratum corneum lipids, which modulates the PC. The PC that is integral to K_p is the PC between the surface or applied vehicle and the stratum corneum lipids. Different vehicles will thus result in different PCs. Similarly, skin from different species may result in different PCs owing to differences in the stratum corneum lipids and surface characteristics (e.g., sweat, sebum). From the computational toxicology perspective, this translates into quantitative models whose parameters are very dependent upon experimental variables often not appreciated to be significant contributors to the process.

Dermal absorption data is often expressed as percentage of the dose absorbed. This is conceptually correct if one assumes that the permeability is unchanged across the dose as it represents a first-order pharmacokinetic process. It is also appropriate when comparing experimental treatments (e.g., temperature, vehicle) using the same applied dose. However, in many cases, topical dosing results in applying substantial amounts of chemical compared to what can be absorbed across the skin. In the case of soils, thick layering where most of the soil is not in contact with the skin, and thus not able to reach its surface to partition into suggests that most of an applied dose is not actually available for absorption. In such studies, only a monolayer of soil is in reality in contact with the skin. Unlike in fluid or gel matrices, compounds generally do not diffuse through distinct soil layers not in contact with the skin, unless water is added as another vehicle. In these cases, accounting for the applied dose as total dose in a multilayer system, overestimates available dose that artificially reduces the calculated percentage of the dose absorbed. Similarly, it overestimates the surface concentration used to estimate K_p. Caking of heavy dermal formulations results in a similar layering phenomenon.

The dose may also bind to the application device and not be available for absorption. In these cases, a large fraction of the dose may not be thermodynamically driving the diffusion process. Finally, when the dose is applied in solution, saturation may result in precipitation of chemical. Rapid evaporation of a volatile vehicle may precipitate it, thereby decreasing its availability for absorption. All of these factors lead to a phenomenon often seen in dermal absorption studies where the percentage of dose absorbed decreases with applied dose. Conducting a study at high applied doses may underestimate absorption of lower applied doses, and vice versa. Alternatively, for some chemicals dosed in volatile vehicles, evaporation increases thermodynamic activity of the penetrating solute through supersaturation and enhances absorption.

This brief review illustrates that many experimental variables can affect the determination of K_p, an essential metric in any topical computation toxicology exercise. Table 10.1 tabulates these relevant parameters that should be considered when designing a dermal penetration study or using literature date for a quantitative modeling study. Figure 10.2 illustrates both the anatomical basis where interactions may occur and its relationship to the computational modeling approach used.

10.3 Dermal Mixtures

Although topical administration offers several advantages over traditional routes [18], formulation development often requires overcoming the barrier function of the skin [19]. This is often accomplished by the inclusion of specific components that have been as carefully chosen as the drug/chemical itself. Such components have functions relative to the delivery, stability, or activity of the active ingredient [20]. Examples of commonly used components include surfactants and compounds to solubilize lipids within the stratum corneum [21–23] and penetration enhancers, which may increase the diffusion coefficient of drugs in the stratum corneum by disrupting the barrier of the stratum corneum and hence increasing the effective concentration of the drug in the vehicle [24–29].

The effects of formulation or vehicle on the rate and extent of absorption have been noted to be far greater with topical drug delivery rather than with any other route of administration [30]. This is exemplified by the broad potency range (I-V) of various marketed 0.5% betamethasone dipropionate products [31]. Cosmetic mixtures have additional aesthetic requirements of the vehicle and drug/chemical. Such criteria include visual appearance, odor and residual impression after application, all of which influence consumer acceptance and patient compliance [30]. In all topical formulations, other components may also be present for reasons totally unrelated to dermal penetration, yet may have effects on the stability or chemical partitioning of the formulations that would impact the penetration of the active ingredient.

In contrast to pharmaceutical formulations, occupational and environmental exposures are often to chemicals, which, from a dermal penetration perspective, are not functionally associated with the penetrant of interest; these include such compounds as contaminants, additives, and solvents [20]. Such chemicals are either present sequentially because they are applied to the skin independently at different times for unrelated purposes (e.g., cosmetics followed by sunscreen lotion), functionally for specific purposes (e.g., lubricants), or coincidentally because they are simultaneously associated together as waste or environmental contamination.

10.4 Model Systems

Assessment of percutaneous absorption for any topically applied drug or chemical, can be classified on the basis of either a model's level of biological complexity (in silico, in vitro, in vivo) or the specific species studied (human, laboratory rodent, monkey, pig). The goal of the research should also be taken into consideration. Is the work being conducted to study the mechanism of absorption (e.g., identify a specific mathematical model or assess the effect of a vehicle) or to quantitatively predict absorption in humans. Is the study designed to look at a local effect in the skin or a systemic effect after absorption? That is, are the skin concentrations the relevant metric or is flux of chemical across the skin important? Model systems and approaches in use today to assess dermal absorption have been extensively reviewed [20, 32].

The primary approach to assess dermal absorption in most computational toxicology studies is the in vitro diffusion cell. In this model, skin sections (full thickness, dermatomed to a specific thickness) are placed in a two-chambered diffusion cell where receptor fluid is placed in a reservoir (static cells) or perfused through a receiving chamber (flow-through cells) to simulate dermal blood flow. Chemical may either be dosed under ambient conditions neat or dissolved in a vehicle (vertical cells) or in water (side-by-side diffusion cells), resulting in finite versus infinite dosing conditions, respectively. This is a major variable in selecting the mathematical model to be used to derive the permeability constant. Selection of the receptor fluid (e.g., saline, albumin-based media) is also critical as absorption would only be detected if the penetrating compound were soluble in the receptor fluid. This is particularly important for hydrophobic penetrants. Many studies of pharmaceutical compounds use saline as the receptor fluid owing to the hydrophilic nature of many drugs, a choice that would falsely suggest minimal absorption for lipophilic chemicals as they would not be soluble in the receptor fluid, and thus could not be detected as absorbed. Steady-state flux is measured in these models and permeability calculated using the methods described in the following. In addition to the perfusate composition, temperature of the perfusate is also controlled, with pharmaceutical investigators suggesting that studies be conducted at 35 °C to mimic the surface temperature of the skin. These techniques have been exhaustively reviewed elsewhere) [20, 32–34].

Our laboratory recently conducted a study assessing the effect of experimental factors including static versus flow through diffusion cells, finite versus infinite dose, albumin versus saline as receptor fluid, and saturated versus unsaturated doses on observed flux and calculated K_p across six compounds: caffeine, cortisone, diclofenac sodium, mannitol, salicylic acid, and testosterone [35]. The important factors identified were dose volume, saturation level, and vehicle. Notably, for these compounds, the experimental system used or receptor fluid composition had minimal effect. In a later study using the same compounds and using different 96-well PAMPA (parallel artificial membrane permeability assays) artificial membrane systems, it was found that the correlation between diffusion cells and PAPMA systems for these same compounds were best with saturated solutions and improved when skin retention was compared to membrane retention [36].

The second major approach used to assess dermal absorption is in vivo. This is the primary approach used to assess drug absorption by all routes of administration. It is also the approach used in many toxicology disposition studies where full mass balance is attempted. The chemical is dosed on the surface of an animal and total excreta (urine, feces, expired air) collected and analyzed for parent compound or metabolites. Radiolabeled compounds are often employed in these studies. These data are usually expressed as percentage dose absorbed per unit of surface area exposed, and is well adapted to laboratory rodent models. However, the resulting metrics are not optimal for computational toxicology approaches. The dose may be applied occluded (evaporation of dose prevented) or non-occluded (dose site open to ambient environment). In calculating the absorbed dose, the chemical at the dose site must be segregated from other tissues that would reflect absorbed chemical. This usually involves gently washing the non-absorbed chemical with a soapy solution. When larger animals (e.g., pigs, primates) or humans are studied and total mass balance is not possible (e.g., feces and expired air cannot be collected), the fraction of a systemically absorbed compound excreted in the urine must first be determined using parenteral dosing. In some classic studies, this parenteral route correction factor was conducted in monkeys [37] under the assumption that systemic distribution, metabolism, and elimination of these pesticides are similar in man and primate. In pigs, separate parenteral injections have been similarly made to determine fractional excretion by other routes [38].

For many pharmaceutical compounds administered as transdermal drug delivery systems, absorption may be assessed by determining the area under the curve (AUC) of the plasma concentration-time profile, the peak plasma flux and time of peak flux, much as it is for determining bioavailability from oral and other routes of administration. These are classical metrics of biopharmaceutical bioequivalence studies and are extensively covered in other texts [39, 40]. If plasma concentration-time data are available from intravenous or immediate release formulations of the same compound, then various deconvolution methods can be employed to infer the absorption profile over time from the transdermal route. This gives a more complete picture of the rate of absorption and can be useful for analyzing time-dependent variations in the absorption rate [41, 42].

There are several perfused skin preparations with an intact functional microvasculature. The major advantage of such a perfused system is that subsequent systemic influences on absorbed chemical are not present, yet the tissue is fully functional with an intact microcirculation unlike simpler in vitro models. The perfused rabbit ear model, perfused pig ear model, in situ sandwich skin flap in athymic rats, and the hybrid rat-human sandwich flap have been developed [43], but each intuitively has severe limitations. The isolated perfused porcine skin flap (IPPSF) developed in our laboratory is a unique ex vivo skin preparation that has an intact functional cutaneous microcirculation. Predictions from IPPSF studies have correlated well with in vivo absorption data for several drugs and insecticides [44–46]. IPPSFs are physiologically and biochemically viable and therefore can be used to assess cutaneous toxicity of topically applied chemicals [47]. The latter is most important as cutaneous toxicity as well as dermal absorption of various pesticide formulations can be assessed simultaneously.

10.5 Local Skin Versus Systemic Endpoints

Before a quantitative model is developed, it is important to have some knowledge of precisely what endpoint is being modeled. If a kinetic study is being conducted, is the focus on predicting transdermal flux and serum concentration-time profiles or is the goal to determine how much chemical will reach target sites within the skin? The same experimental model systems may be used for both endpoints; however, different data may have to be collected. For transdermal flux, the perfusate is collected and either the K_p or AUC determined. However, if local skin deposition is the endpoint, then skin samples or biopsies must be collected to assess the amount of chemical retained in skin, not just the amount which traversed skin which would be estimated if only flux were monitored. As discussed in the methods section above, very lipophilic molecules may not partition into perfusate, further underestimating the quantity of chemicals exposed to the skin. Unlike perfusate flux parameters, there are no well-established metrics for dose deposition into skin.

Finally, cell culture studies are often used to assess direct cutaneous toxicity to skin cells such as keratinocytes. As discussed above, this can be used to define whether a potential topical chemical that penetrates the skin can cause epidermal cell dysfunction. However, systemically administered chemicals may also distribute to the skin and modify keratinocyte cell function. Damage to skin, especially if the mechanism of action is immunological, does not require topical exposure.

10.6 QSAR Approaches to Model Dermal Absorption

Since passive diffusion is the primary driving force behind dermal absorption, physicochemical factors such as molecular weight and structure, lipophilicity, pK_a, ionization, solubility, partition coefficients, and diffusivity can influence the dermal absorption of various classes of chemicals. In addition, penetration of acidic and basic compounds will be influenced by the skin surface, which is weakly acidic (pH 4.2–5.6), as only the uncharged moiety of weak acids and bases is capable of diffusing through the lipid pathway. Several of these factors (e.g., molecular weight and partition coefficients) have been used to predict absorption of various drug classes [48–50].

The first such relationship which is still widely used to assess chemical absorption is that of Potts and Guy [51]:

where MW is the molecular weight. This equation was subsequently modified (Potts and Guy [48]) to relate K_p to molecular properties of the penetrants as follows:

where MV is molecular volume, is the hydrogen-bond donor acidity, and is the hydrogen-bond acceptor basicity.

The most promising approach is to further extend this rationale using linear free energy relationships (LFER) to relate permeability to the physical properties of the penetrant under defined experimental conditions (dose, membrane selection, vehicle). Geinoz et al. [52] critically reviewed most of such quantitative structure-permeability relationships (QSPeR) applied to dermal absorption and their study should be consulted. Abraham's LFER model is representative of the dermal QSPeR approaches presently available [53]. This model was selected as it is broadly accepted by the scientific community as being descriptive of the key molecular/physiochemical parameters relevant to solute absorption across the skin. This basic model can be written as follows:

where is the dipolarity/polarizability, R₂ represents the excess molar refractivity, V_x is the McGowan volume and the other parameters are as described earlier. The variables c, a, b, s, r, and v are strength coefficients coupling the molecular descriptors to skin permeability in the specific experimental system studied.

Our laboratory has focused significant research on the effects of chemical mixtures on dermal absorption of penetrant compounds. In order to incorporate mixture effects, our laboratory has been exploring using an additional term operationally called the mixture factor (MF) yielding the following:

The nature of the MF is determined by examining the residual plot (actual–predicted log k_p) generated from the base LFER equation based on molecular descriptors of the permeants, against a function of the physical chemical properties of the mixture/solvents in which they were dosed [20, 54]. Figure 10.3 depicts the improvement in prediction of k_p when this approach is used. This depiction also illustrates the great impact that a formulation has on k_p (seen by the heights of the columns) around a specific penetrant when only the penetrant's molecular properties are considered (inset a) versus when both penetrant and mixture properties are included (inset b). When similar analyses were conducted across both diffusion cell as well as the perfused IPPSF, different mixture factors were needed for each model system [55]. Also noted was that the QSPR model could be reduced to four terms in each system; however, the four terms differed between the two biological systems. This suggests that the rate limiting process for a mixture's effect on absorption differs between the two biological systems studied. These findings are encouraging and imply that a QSPR model is possible for estimating dermal absorption as a function of chemical mixture composition.

This approach was further validated where melting point of the vehicle constituents described mixture effects across both saturated and unsaturated doses when all experimental variables were also accounted for [56]. A similar approach was useful to predict absorption of a wide variety of 56 agrochemicals across a diverse set of 150 formulations [57].

When the diffusion cell data of Riviere and Brooks [54] was revisited, using a pharmacokinetic model that accounted for transient processes in the absorption profiles, the MF approach resulted in a concise QSPR containing terms for log P and MW of the penetrant and an MF for refractive index of the mixture [58]. This was consistent with previous results while improving the model fit, largely due to revised permeability estimates that account for transient vehicle effects. Figure 10.4 shows the QSPR developed which was expressed as the product of diffusivity and partitioning terms proportional to K_p:

This relationship included molecular weight MW and log P, similar to the Potts and Guy [51] model, but also includes an MF term for refractivity (MF_Refrac) to adjust for the properties of the dosing vehicle.

In another effort, application of a compartmental dermatopharmacokinetic model to the IPPSF data of Riviere and Brooks [55] allowed for estimation of permeability constants (as opposed to the previously used AUC) and regression to a QSPR. A relationship relating K_p to penetrant properties log P and MW and mixture factors for the Connolly molecular area (CMA) and total polar surface area difference (TPSAd) [60] was developed :

The consecutive improvement in model fit as each MF term is added to the QSPR is shown in Figure 10.5. The QSPR in IPPSF contained similar properties to that in diffusion cells and suggests that it may be possible to combine the data and develop a single QSPR that would account for penetrant and vehicle effects on permeability across the different membrane systems.

The literature on QSPR is exhaustive and rapidly growing. The limitation of applying these approaches to chemical absorption is the lack of large and comparable databases of chemical dermal absorption, as well as the lack of availability of molecular descriptors for many compounds. As will be discussed in the following, data suitable for large-scale analyses must be rigorously controlled relative to the species studied, the nature of the experiments (in vitro vs in vivo), dose, surface area, vehicle, and method of sample collection and analyses.

10.7 Pharmacokinetic Models

The final area of computational toxicology to be discussed in this chapter applied to dermal absorption is the general area of pharmacokinetic models. These are usually developed as extensions of whole-animal-based models using classic compartmental [55, 61] or physiologically based [62] approaches. These texts should be consulted for an overview of the basic assumptions inherent in each genre of pharmacokinetic modeling, as these are carried forward when applied to the skin. There are an enormous variety of approaches taken to develop dermatopharmacokinetic models. These applications to the skin have recently been well reviewed elsewhere [17, 63].

The primary purpose for most dermatopharmacokinetic models is to quantitate the linkage between anatomical and physiological properties of skin that play rate-limiting roles in absorption with target sites for which concentration-time profiles are needed, such as plasma. The complexity of the models is a function of the chemical being studied as well as the level of precision (concentration, time frame) required for the prediction. For example, models created to estimate total percentage of the dose absorbed are much simpler than those designed to predict the time and magnitude of peak plasma concentrations in a subject.

Numerous other factors play a role in the structure and complexity of models employed. In general, the stratum corneum is assumed to be a homogeneous membrane into which the compound partitions from the surface. Some models attempt to take into account the tortuous nature of the intercellular pathway when defining actual diffusion path length. For volatile compounds, models may specifically include evaporative loss from the surface of the skin or binding to other surface sites (e.g., application device, hair). The next stage of potential complexity is whether metabolism occurs in the epidermis, a process that may dramatically increase model complexity. The drug then enters the dermis or is assumed to directly enter the blood. If a compound is vasoactive, dermal blood flow may be specifically modeled. At all three tissue levels, irreversible or very slow tissue binding may also be included to model the formation of so-called depots in the stratum corneum or fat. An example of a pharmacokinetic model used in our laboratory [64–66] is depicted in Figure 10.6 which also illustrates the concept of a fixed skin depot as well as how it can be used to link to systemic pharmacokinetic models for making in vivo predictions.

Some dermatopharmacokinetic models attempt to develop linkages between chemical concentrations in a specific compartment with a toxicological effect based on mechanisms of action [63]. This work is in its infancy but holds promise should sufficient data to conduct these modeling exercises become available.

Models of aggregated data, containing measurements for multiple compounds, vehicles, and/or experimental systems, present additional challenges. Often the goal of such models is to provide comparable estimates of permeability parameters for inclusion in a QSPR or other secondary analyses. This can be difficult when different treatments have different underlying mechanisms, and thus would necessitate using different models. The traditional approach is to model each experimental unit (e.g., flux profile) independently, but often the non-identifiability of parameters describing mechanistic effects, such as binding or evaporative loss, degrade the estimates of interest (e.g., permeability). In such cases, it may be beneficial to model the data simultaneously. A simultaneous model allows for individual experimental units to estimate shared parameters, for instance, the time for an ethanol vehicle to evaporate.

One particularly powerful method for combining data is through mixed effect modeling. In a mixed effects model each of the treatments share some parameters (fixed effects) and individual variations from those shared parameters are treated as random variables. The estimation technique finds the fixed effect values (typical values for the dataset) as well as the random effect values (specific variations) for each experimental unit. Additional fixed effects to describe relationships between chemical and physical properties can be added to the model. The structure of the model greatly reduces the number of parameters as compared to estimating parameters for individual experimental units, and hence increases the power of the analysis.

A mixed model approach was used to evaluate models for 12 compounds in 24 vehicle mixtures in porcine skin and silastic membrane diffusion cells, where the random effects accounted not only for differences in diffusivity by treatment but also for time-varying dynamics of vehicle effects [58]. Further refinement of the models for the porcine skin diffusion cells allowed for incorporation of an embedded QSPR in the model [59]. The approach was also extended to model a QSPR in the ex vivo IPPSF system for 12 compounds in 10 vehicle mixtures [60].

In each of these cases, the aggregated data allows for system-wide parameters to be influenced by data from multiple experimental units, which can help prevent statistical problems when the parameter is not readily identifiable in some of the experimental units. An example of this would be if ethanol evaporation is modeled. In a case where the penetrant diffuses slowly, the evaporation rate of ethanol would be visible in the measured flux. But if the penetration rate is high enough, the dose is absorbed before the ethanol evaporates and the effect of the evaporation, and the ability to estimate the rate of evaporation, is not attainable from the data.

Many transdermal flux profiles exhibit time-dependent variation in model parameters, such as diffusivity or partitioning. This is particularly apparent when comparing between different dosing vehicles, where modulation of the membrane or evaporation of the vehicle produce transient effects. It is often difficult to directly model such dynamics because the mechanism is either uncertain or not estimable from available data. Stochastic differential equations have been used in pharmacokinetic models to account for unpredictable, time-dependent deviations from deterministic predictions [67–71]. Incorporation of such a random process in the model may help to account for these transient effects and improve the precision of estimates of parameters of interest. In one analysis, a random process was used to account for transient changes to apparent diffusivity to allow for improved estimates of permeability coefficients in various penetrant-vehicle mixtures, where mechanistic models of the vehicle dynamics were either unknown or computationally intensive [59]. The ability of the random process model to compensate for the transient dynamics is illustrated in Figure 10.7, which compares the model fit for a few example profiles with and without the random process. The development of models and tools for random processes in pharmacokinetics, and applications to dermatological models, is an ongoing area of research.

10.8 Conclusions

The percutaneous absorption and dermatotoxicity of topically applied drugs and chemicals is a major concern in pharmacology, pharmaceutical and toxicological sciences. As research in these areas continue, and regulatory oversight is applied to more classes of substances, efforts have been made to develop predictive models to quantitate chemical exposure to the skin and systemic circulations. Model complexity increases with greater anatomical or physiological details. Before realistic predictive models can be constructed, experimental data must be collected of sufficient quality to make such efforts worthwhile.

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