Introduction
The use of models in science and engineering is long established and has played an invaluable role in both understanding the science underlying many processes and in analysing the engineering that has produced results. Modelling dermal exposures has as its ultimate objective the development of expert systems capable of reliably predicting the extent to which a molecule will be percutaneously absorbed without the need to make experimental measurements. The ideal model of dermal exposure should take account of the processes all the way from the details of an exposure to a specific substance to an assessment of the risk resulting from that exposure. The risk assessment practitioner requires a realistic and flexible model compatible with available computational resources and capable of making discriminatory predictions. There are three principal steps and a complete model should allow the parameters that control each of these to be altered to reflect the conditions under which the exposure occurs. The first of these is the exposure itself for which the variables may include: the formulation and mode of application, the duration and number of exposures, the use of protective clothing and personal hygiene. The second is the percutaneous penetration process in which the complex nature of the skin, the reservoir effect, and the influence of penetration enhancers and of damage to the skin should be considered. Finally, the representation of the disposition of the substance within the systemic system should include consideration of its half-life, of the metabolic processes that it can undergo, of its elimination kinetics and of its toxicity. This presentation will be concerned with models of percutaneous absorption, which have as their primary objective the prediction of the dermal permeability coefficient, K_p, of a substance through skin. Kp is the quantity that is measured in a steady-state experiment and is given by the relationship:

K_p = K_m . D_m / h

where D_m is the permeant diffusivity in the membrane, K_m is its partition coefficient between the stratum corneum and the vehicle (very often substituted for by the octanol-water partition coefficient K_ow), and h is the thickness of the stratum corneum. For more simple substances the modelling of transport processes through solid and liquid media is regularly carried out at an atomistic level using techniques such as molecular dynamics and, recently, first principles electronic calculations. These techniques cannot be used for a complex biological membrane such as skin and here predictive work has been almost completely confined to two techniques. The first of these is the use of quantitative structure activity relationships (QSARs) that relate statistically the measured biological activity of a series of compounds to their physicochemical and /or structural properties. The second is based on the available theories for the selected process with the resulting mathematical equations being solved either analytically or using numerical methods. For percutaneous penetration, equations originating in the partition of the diffusing substance between layers and macroscopic theories for its transport along the resulting pathways are utilised.

QSAR Models for Percutaneous Penetration
The numerous QSARs that have been developed to model percutaneous penetration have been recently reviewed (Moss et al, 2002). The most frequently used physicochemical properties on which the predictions of K_p are based are the octanol-water partition coefficient of the substance and some measure of its molecular bulk e.g., molecular weight or molecular volume. Perhaps the best known of these was developed by Potts and Guy (1992), who were among the earliest to demonstrate the use of the logarithm of K_ow and molecular size. They used a collection of data published earlier (Flynn, 1990) and argued that their model provided a simple mechanistic description of SC permeability that made it unnecessary to invoke the idea of aqueous pores to explain the values of K_p for small polar compounds – their behaviour was accounted for by their small size. They achieved a value for r² of 0.67 for 93 compounds with the algorithm:

logK_P = – 6.3 + 0.71 logK_ow – 0.0061 MW

Other QSARs subsequently published have, for the most part, relied on the same set of data, originating with Flynn (1990). They have differed in that they have sometimes treated smaller groups of related compounds and have also tended to introduce additional physicochemical properties, for example to rationalise the role of hydrogen bonding in the process. Most recently, Patel et al (2002) published a new QSAR based on data for 158 compounds, including those used by Potts and Guy. With the algorithm:

log K_P = 0.652 log K_OW – 0.00603 MW – 0.623 ABSQon
– 0.313 SsssCH – 2.30

they achieved an r² value 0.76. ABSQon is the sum of absolute charges on oxygen and nitrogen atoms and SsssCH is the sum of E-state indices for all methyl groups. These were selected on the basis of statistical analyses as being the two most significant of the 169 physicochemical descriptors calculated for each of 158 compounds. On elimination of 15 outliers the value of r² was optimised at 0.90 for the remaining 143 compounds.

We have applied the QSAR model of Potts and Guy (1992) to the data for the 158 compounds listed by Patel et al and found that it gave an r² value of 0.66 (Fitzpatrick and Corish, 2002). Removal of the same 15 outliers as had been discarded by Patel et al increased this value of r² to 0.74. We then used standard statistical techniques to select the most significant 15 outliers to the line given by the Potts and Guy equation (these were not the same as those chosen by Patel et al) and applied that QSAR to the remaining 143 compounds. The resulting value of r² was 0.88, which is not significantly different from best value obtained by Patel et al.

As has been pointed out before now, it is important to realise that the data used in these QSARs are not ideal for this purpose. They were originally measured using a variety of techniques and protocols each for its own particular reason and were subsequently assembled into sets for use in the development of the QSARs. Adding complexity to the model, in the form of additional physicochemical properties, will be expected to improve the predictive capacity of a model. It is also evident that very significant improvements can be made through the removal of judiciously chosen outliers – and this without any reference to the quality of the relevant or remaining measurements. These data have probably now been pushed to their limit. Rather than continuing to extend the number of compounds on the list, the route to better QSARs may lie in the collection of purpose-measured and validated data for selected compounds, and in the development of a range of QSARs each specific to a group of related compounds.

Mathematical Modelling of Percutaneous Penetration
As has been done for a number of analogous matter transport processes, the relevant standard differential equations have been adopted and solved for the movement of molecules through the stratum corneum. The best-known and most frequently used model of this kind is that developed by Cleek and Bunge (1993) and later extended by the same authors (Bunge and Cleek, 1995a,b). The essence of their model is a set of algebraic equations that accurately represents dermal absorption – including non steady state contributions – within certain simplifying assumptions. The model adequately represents the exact solution to the unsteady state diffusion equations for a two-membrane composite and the viable epidermis contribution to highly lipophilic molecules and also accounts for larger absorption rates during initial exposure. Three cases are considered in detail: single finite membrane; semi-infinite membrane and a finite two-membrane composite and the equation can be applied to predict absorption or to analyse experimental data to provide values for permeability. Again this model uses Potts and Guy (1992) correlation for steady-state permeability from water for parameter estimation and so ultimately reverts back to the experimental data sets described in the previous section.

More recently, a new mathematical model has been developed by Kruse (2002). This is a more versatile advanced simulation programme that can model and estimate uptake via the skin, after specified dermal exposures. It employs a physiological description of the skin with both stratum corneum and viable epidermis layers. A thin water layer usually covers the surface of the stratum corneum but contact with another solvent vehicle or a vapour or the application of a solid diffusant can also be modelled. In addition to passage through the two-layer route (stratum corneum and viable epidermis), a parallel route that circumvents this barrier can be invoked. Finally, clearance of the penetrant by blood perfusion can also be modelled. The differential equations describing the diffusion through the layers are numerically integrated using Advanced Continuous Simulation Language (ACSL, Mitchell and Gauthier Associates, Cambridge, MA, USA) software package. The model uses physicochemical data of the compound and QSARs (e.g., Guy and Potts, 1992; Cleek and Bunge, 1993) to derive the necessary information for its implementation. As with other mathematical models it is possible, once the equations have been solved for the specified conditions, to calculate and output all the information of interest. This can include the rate of absorption, the cumulative absorption, the quantity retained in the skin reservoir, the permeability coefficient and the lag time (theoretical values and values derived from the simulation). In tests this model has been shown to give results that agree very well with analogous results using the Bunge and Cleek model. It is also being successfully used within the EDETOX¹ project in parallel with some of the series of experimental measurements that are in progress specifically to provide data suitable to test models.

The Future
The development of an integrated model of dermal exposure and absorption, capable of providing realistic predictions on which risk assessments could be based, will require that the limitations currently in evidence, and outlined above, be overcome. These limitations include the quality, reliability and consistency of the data on which QSAR and other models depend. In particular, accurate reproducible data must be measured under conditions that reflect typical real occupational exposures. Such data would enable decisions on the choice of optimum QSAR models and of the ranges of chemicals for which such models might be suitable. More reliable and flexible models will then be needed to encompass variables such as formulation/vehicle/phase and the detailed exposure regime, including non-steady-state events. Simulations based on these models will then need to integrate the specific types of information relating to the different process of exposure, adsorption and disposition to realistically assess the risk. The EDETOX project is already addressing some of the current deficiencies.

A fuller understanding of these exposure and adsorption processes and a predictive capacity based on this understanding will eventually provide the knowledge base necessary to develop new formulations and application techniques for beneficial but potentially toxic substances to increase the efficacy and safety of their use.

Acknowledgement
The authors are grateful for support from the European Commission through the EDETOX Project and from their colleagues in the Consortium.

¹ Contract QLRT-2000-00196, Fifth Framework Programme of the European Commission

References
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Bunge, A.L. Cleek, L.K. and Vecchia, B.E., 1995b, Pharm. Res. 12, 972.
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Fitzpatrick, D. and Corish, J., 2002, to be published.
Flynn, G.L., 1990, in ‘Principles of Route-to-Route Extrapolation for Risk Assessmnet’, eds. T.R.Gerrity and C.J.Henry, (Elsevier, New York) p.93.
Kruse, J., 2002, private communication.
Moss, G.P., Dearden, J.C., Patel, H. and Cronin, M.T.D., 2002, Tox. in Vitro,16, 299.
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