(4.1)

Clearly, this model is limited by the size of the data set used. Nevertheless, this model is significant and interesting as it introduces nonlinearity (as log K

_{ow}^{2}) into the relationship between skin permeability and physicochemical parameters. It is also unique in that it presents log MW rather than MW as a significant parameter. Lien and Gao also commented that the correlations obtained were dependent not only on the biological system but also on the vehicle used. Indeed, in the absence of a lipophilic vehicle they determined that the ideal lipophilicity for optimal human skin permeability was achieved by a potential permeant with a logP value ranging from 2.5 to 6, the higher end of this range being an extrapolation based on the molecular space of their data set. The use of a lipophilic vehicle (in model studies featuring mouse skin as a membrane) saw this ideal range drop to approximately 0.4, potentially due to changes in penetrant solubility in the vehicle or potential effects of the solvent on the integrity of the stratum corneum barrier. They suggest that, while an increase in MW was characteristic of a decrease in permeability, increasing hydrogen bonding potential could have either a slightly positive or slightly negative effect.In a major development of, and from, the Potts and Guy approach, Cleek and Bunge (1993) described a range of models that estimated dermal absorption under a range of conditions. They commented that Potts and Guy’s (1992) model, given the inherent variation associated with skin permeability data, is a robust model which has been tested rigorously by the nature of its input data. The Potts and Guy equation averages most of the experimental variability and, as such, the weight of the experimental data strongly supports the validity of the correlation prediction. Thus, data which deviate significantly from the Potts and Guy equation (Eq. 3.36) should be evaluated with regard to the experimental procedure used in generating the data and data reproducibility before these data are accepted as being more representative than the correlation. Potts and Guy (1992) suggested that data varying from Eq. 3.36 by a factor of three or more would lie outside the 95 % confidence interval. These comments, from both Potts and Guy and Cleek and Bunge, imply that the equation can fit the data and that any erroneous data which are a poor fit are due to a poor experimental design and/or execution and not due to limitations of, or deficiencies in Eq. 3.36.

Cleek and Bunge went on to develop models based on three experimentally relevant scenarios: a single finite membrane, a semi-finite membrane and a finite two-membrane composite. All models have been, in various forms, reported previously (Crank 1975, Carslaw and Jaeger 1980; Ozisik 1980). All these models assumed that the membranes—either the stratum corneum, or the stratum corneum considered with the viable epidermis—are passive with regard to diffusivity, thickness and partition coefficients; that is, the vehicle and the absorbing chemical do not alter the nature or composition of the membrane. Cleek and Bunge consider the heterogeneous epidermal membranes as mathematically pseudo-homogeneous, resulting in the diffusivity, thickness and partition coefficient values as being effective properties. Thus, the stratum corneum can be considered as a single finite membrane:

where

(4.2)

A

is the surface area of chemical exposure

L

_{c }is the thickness of the stratum corneum

K

_{cv }is the equilibrium partition coefficient between the stratum corneum and the vehicle of the absorbing chemical

is the concentration of the absorbing chemical in the vehicle/formulation, which is assumed to remain constant during the exposure time, t

_{exp}is the normalised cumulative mass absorbed into the stratum corneum

with t

_{exp}being the exposure time normalised by the characteristic diffusion time () for a chemical in the stratum corneumD

_{c }is the effective diffusivity of the absorbing chemical in the stratum corneum

Thus, the normalised cumulative mass out of the stratum corneum is calculated by integrating the flux at L

where M

_{c }over the exposure time:(4.3)

_{out}is the normalised cumulative mass leaving the stratum corneum.Removal of the infinite series from 4.2 and 4.3 makes the equations more widely practical:

(4.4)

(4.5)

Cleek and Bunge commented that chemicals with different K

_{cv }values, indicated by different K_{ow}values, can still exhibit similar lag times as long as their D_{c }values were similar, suggesting that lag time is dependent on diffusivity and that lag time and diffusivity may not be directly related to log K_{ow}.They then proposed a semi-infinite membrane model, which was applied to the situation where the exposure time was short and where the penetrant will therefore only penetrate a short distance into the stratum corneum. In such cases, the stratum corneum is considered to be comparatively thick, or semi-infinite. In such cases, the normalised cumulative mass is given by:

(4.6)

They considered a finite two-membrane composite model which includes both the stratum corneum and viable epidermal layers, which provides a more realistic model to skin permeation than single-membrane systems:

where B is the permeability ratio between the stratum corneum and epidermis and measures the relative contribution of the stratum corneum and viable epidermis resistances

(4.7)

and λ_{n }, which are both eigenvalues.

, the ratio of the lag times for the stratum corneum and viable epidermis.

Equations 4.3 and 4.6 yield identical results during shorter exposure times as the chemical does not penetrate far enough into the stratum corneum to make the finite nature of this layer relevant. Equation 4.6 therefore describes absorption into the stratum corneum during the non-steady-state period of the diffusion process.

This means the diffusion process no longer depends on G, and Cleek and Bunge commented that this expression is accurate for large (>10) values of G which reflects the lack of a compromised stratum corneum barrier. Further, in the context of the partition between the viable epidermis and the stratum corneum, and the role of the former in influencing skin permeability, Cleek and Bunge also derived an expression for k

where k

_{p }that sets an upper limit for permeation of highly lipophilic species:(4.9)

_{p }is the predicted permeability coefficient calculated from a QSAR equation, such as that presented by Potts and Guy (1992).The overall power of this study is that it includes the effects of chemical characteristics (i.e. log P and molecular weight) and exposure—particularly exposure time—into models of permeant uptake in skin in both steady-state and non-steady-state conditions. It also considers the skin “compartments” and provides a more balanced model that considers the molecular requirements for permeability in the context of partitioning between the skin (epidermal) layers. Thus, the cumulative mass absorbed into the stratum corneum can be used to assess dermal exposure risks.

Bunge, Cleek and co-workers developed the above study with two subsequent publications. They next examined the effect of molecular weight and lipophilicity on dermal absorption (Bunge and Cleek 1995). They emphasised the influence of partitioning between the stratum corneum and the viable epidermis (defined as B) in the context of key physicochemical properties and considered the balance of properties required for effective skin permeation rather than simply considering partitioning into the lipid stratum corneum. They considered four main models which could be used to estimate B. These models demonstrated, for example, how the value of B varied with log K

_{ow}across the four models for a molecule with a constant molecular weight (300). Their methods vary in how they consider the hydrophilic–hydrophobic balance between the viable epidermis and stratum corneum and how this manifests itself in the determination of B. Their first method considers that the hydrophilic–hydrophobic balance between the viable epidermis and stratum corneum varies according to log K_{ow}, specifically that , which assumes that molecular weight dependence is the same in both the viable epidermis and the stratum corneum. Correcting this estimate for steady-state permeability of the stratum corneum suggests that the effect attributed to K_{ow}by Model 1 is too strong. Model 2 makes a similar assumption but considers B to be a function of and also suggests that B does not change with molecular weight. Model three suggests that molecular weight dependence is exponential for the stratum corneum and that there is no molecular weight dependence for the viable epidermis, and is represented by . Model 4 assumes that B is dependent exponentially on molecular weight for the stratum corneum and inversely related to the square root of molecular weight for the viable epidermis, being represented by . Models 3 and 4 predict that B decreases as molecular weight increases. Model 3 predicts a more rapid decrease in B, compared to Model 4, as it assumes that permeability in the viable epidermis is unaffected by molecular weight, whereas stratum corneum permeability decreases exponentially. Thus, with the caveat of considering very small molecules (i.e. MW < 36), they suggest that their fourth model is the most realistic, as it suggests a more moderate decrease in permeability across the viable epidermis decreases with increasing molecular weight but more moderately than the permeability across the stratum corneum. Thus, in most cases, estimates of cumulative mass absorbed can be made from D_{c }, L_{c }and K_{cv }(as defined above) and B, the relative size of stratum corneum permeability to the viable epidermis permeability. Their modelling studies suggested that B is only important for highly lipophilic, small molecular weight permeants. When B is considered important Bunge and Cleek recommend the application of the Potts and Guy (1992) algorithm to determine skin permeability.Finally, in this series, Bunge et al. (1995) compared models of steady-state and non-steady-state absorption for estimating dermal absorption. The non-steady-state methods were considered in the cases of short exposure times and larger absorption, as well as considering the contribution of the hydrophilic barrier of the viable epidermis to the passage of lipophilic chemicals. They found that, for example cases, the steady-state modelling approach significantly underestimated the dermal absorption under non-steady-state conditions, meaning that permeability values calculated from data sets which include non-steady-state data will be incorrect. They further determined that calculating permeability values from cumulative absorption data measured for exposure times less than 18 times the stratum corneum lag time overestimates permeability. Models of non-steady-state diffusion, including those in the context of finite dosing, are discussed in greater detail in Chaps. 6 and 8.

Barratt (1995) also analysed the data set published by Flynn (1990), using the log P and log of permeability coefficients (listed by Barratt as log PC) published by Flynn. He calculated the molecular volumes for each member of the data set and obtained melting point values from literature sources. A series of cross-validated QSARs were derived from these data, and the data were also subjected to principal component analysis. The chemicals in the data set were categorised as (a) steroids, (b) other pharmacologically active chemicals and (c) small organic molecules without pharmacological activity. In categorising the data in this way he noted that the subset of hydrocortisone derivatives were consistently modelled poorly by these methods (this is an issue that will be discussed in detail later). QSARs were derived for each of these subdivisions of Flynn’s data set.

Barratt’s use of melting point is not explained in detail, other than to comment on its use as a parameter that reflects solubility. It is difficult to elaborate on the significance of the use of this parameter in this study. Melting point, however, is dependent on hydrogen bonding and as such may infer indirectly the importance of that phenomenon in skin permeation. Its inclusion resulted in a slight increase in the amount of variance explained by the model (compared to Potts and Guy (1992)) from 71.1 to 76.6 %. In the context of Potts and Guy’s comments on the variability of data (that up to 30 % variance is common in skin permeability experiments) and the additional context that Flynn’s data set is constructed from a number of different literature sources and therefore a number of different laboratory studies, Barratt concluded that models of this nature could describe fully the variance in the data.

Barratt further commented that principal component analysis demonstrated the existence of distinct groupings within the data—this, again, is interesting in the context of subsequent Gaussian process studies of skin absorption, which will be discussed later, as well as the issues raised by Johnson et al. (1995), and later quantified by Moss and Cronin (2002) regarding the quality of some of the original data from the Flynn data set. The most significant QSAR reported by Barratt (units cm/h) was derived from a subset of 60 “small molecules and steroids”, but excluding the hydrocortisone derivatives, taken from Flynn’s data set:

(4.10)

The hydrocortisone derivatives were poorly modelled by this expression and this was attributed to their analysis within the same laboratory environment (Anderson et al. 1988; Raykar et al. 1988) and, by Barratt, to a particular experimental technique associated with this laboratory. Similar conclusions were drawn for the underestimation of permeability by the model for ethylbenzene, styrene and toluene. Despite the use of a linear model Barratt is clear that the overall trend, extrapolated beyond the molecular space of this data set, is not linear as increases in lipophilicity will increase permeability only to the point where aqueous solubility in skin layers beneath the stratum corneum permits further permeation. Barratt’s final conclusion is that while the model explains 90 % of the variance in the data and is expected to yield accurate predictions, it has not been tested experimentally.

Another study around the same time—in what could be described as another analysis of subsets of Flynn’s data set—was reported by Potts and Guy (1995). They investigated the role of hydrogen bonding by developing a predictive model based on a subset of 37 non-electrolytes. The data they used was again taken from Flynn’s data set. The resulting QSAR highlighted the significance of hydrogen bonding but did not feature a hydrophobicity term. The QSAR derived was as follows:

where

(4.11)

MV

is molecular volume

is the solute hydrogen bond acidity

is the solute hydrogen bond basicity.

Equation 4.11 is interesting in that it highlights a significant issue with the QSAR (and related) analyses of skin permeability. By altering the input to the model (the membership of the data set, including its size), it substantially alters the output—the QSAR algorithm. This study, and its QSAR, indicates that the factors (molecular descriptors) influencing the skin permeability of non-electrolytes are significantly different to those which influence permeability for the whole data set. In particular, it provides an understanding that hydrogen bond activity is inversely related to skin permeability and that the hydrogen bond acceptor ability of a potential permeant appears to be more significant than its hydrogen bond donor ability, linking potential permeation directly to permeant structure. However, a direct comparison with Potts and Guy (1992) is not possible as that model focused on a model of permeability based on lipophilicity and molecular volume/weight, and did not specifically explore effects of hydrogen bonding on permeability. Further, the reduced size of the data set lessens the power of the model as a viable predictive model compared to their earlier study.

At a similar time, a substantial analysis, resulting in a range of QSAR models, of skin permeability was published by Abraham et al. (1995). Their study was based on earlier work by this group (Abraham 1993) and relates to their application of a general solvation equation:

where

(4.12)

SP

is the property of a series of solutes in a system; in the context of skin permeability, this is most likely k

_{p }, but could also be steady-state flux.R

_{2}is an excess molar refraction and is determined from knowledge of the compound refractive index, according to an earlier method by this group (Abraham et al. 1990). This term describes the tendency of a solute to interact with a phase through π or n electron pairs.

represents solute-effective, or overall hydrogen bond acidity

represents solute-effective, or overall hydrogen bond basicity

V

_{x }is the McGowan characteristic volume.

Their data sets were examined using multiple linear regression analysis. A series of subsets of the larger data set were examined. For example, they examined the data subsets which were previously investigated by El Tayar et al. (1991). Abraham reports that “very poor” correlations were often obtained in this analysis, attributing this to the values of alcohols and steroids lying on two distinct lines in a plot of log P

_{oct}against −log k_{p }, although the utilisation of small data sets and the extrapolation of results from them might also contribute to poor models with limited statistical value. Further, El Tayar commented that the poor correlations were due to the Δlog P term, which reflects solute hydrogen bond acidity, which has a retarding effect on skin permeability due to hydrogen bond formation between permeants and hydroxyl groups in the alkyl side chains of ceramides during permeation via the intercellular route. Abraham and co-workers proposed an alternative equation to that presented by El Tayar, using the same data but based on their generalised model:(4.13)

They then added three further permeability data—values for diethyl ether, 2-ethoxyethanol and butanone, taken from Scheuplein and Blank (1971), which resulted in the following expression:

(4.14)

Thus, the models of Abraham et al. are significantly different in their mechanistic insight than those produced by El Tayar et al., with the former study focusing on the role of hydrogen bonding and polarisability, which means it sits close to Potts and Guy’s algorithm for skin permeability (Potts and Guy 1992). Similar findings were, however, found by Abraham and colleagues for other subsets of the Flynn data set, including analysis of the Roberts et al. (1977) subset, where Abraham did not find evidence to support Roberts’ proposal that phenols permeated the skin by two different routes based on their lipophilicities. They showed that, for phenols, there is a significant contribution from solute acidity which was not relevant for alcohols and steroids:

(4.15)

By combining the two data subsets—that is, combining the phenol, alcohol and steroid data which were collated by Flynn, they produced another model of skin permeability:

(4.16)

Given the statistical quality of this model, they recommend that it is a good generalised model for skin permeability. Further, as the model retains its statistical quality, compared with the models of subsets (Eqs. 4.12–4.15) they suggest that there is no separate mechanism of permeation for each of the same chemical classes and that their permeability can be described by a single equation. They further analysed a series of models based on what they describe as the “log P

_{oct}model”. In general, they found poorly correlated models, compared to their hydrogen bond-based approach, described above, resulted. Nevertheless, they did find that “better” models (in terms of the correlation coefficient) were obtained when larger data sets were used. They also briefly examined the process of back-diffusion upward through skin layers, suggesting that the regression equation for the ingress of exogenous chemicals (inward rate constant) and log K_{m }(equilibrium constant) are similar, whereas the rate constant for back-diffusion, log k_{pback}, is very different. This, they suggest, means that log k_{pback}is wholly or partially diffusion controlled. They base this on the comparative size of coefficients in Eqs. 4.13 and 4.17, which represents back-diffusion, below:(4.17)

This implies that molecular structural effects have a smaller effect on log k

_{pback}than on the k_{p }.Abraham and colleagues followed up this comprehensive study by later considering the nature and issue of outliers within the data sets, by examining issues with potentially erroneous values for some of the steroid data (Abraham et al. 1997, 1999). They stated, in the first of these papers, that although algorithms based on logP (or logK

_{ow}) and molecular weight (or volume) appeared to have utility, particularly in the prediction of logk_{p }, these algorithms [those published by Potts and Guy (1992, 1995) and others, related models using the same, or similar data sets, such as Brown and Rossi (1989), Fiserova-Bergerova et al. (1990), McKone and Howd (1992), Guy and Potts (1993) and Wilschut et al. (1995)] were empirical in nature and yielded little information on the structural features of solutes that influenced water–skin permeability. Thus, they continued their solvation-based approach which focused on the use of hydrogen bond descriptors. They initially (Abraham et al. 1997) focused on steroids and included data from Johnson et al. (1995) in their analysis which appeared to be at odds with earlier published work by Scheuplein et al. (1969). Interestingly, they omitted water from their data analysis as, on partitioning from bulk water to skin, they commented that it is not acting as a solute at all. Therefore, using the data they had previously collated from Flynn (1990), and reduced into a smaller, more chemically specific data set, they obtained the following expression: