Imidazole and amine are indicator variables for the imidazole and aliphatic amine groups.

Consideration of Chen’s initial QSPR in the context of maximum flux shows that the mole fraction term in Eq. 5.1 is related to the solubility (C _s ) term in this expression and all other terms are related to membrane permeability. They commented that the partition coefficient and the diffusion coefficient both depend on the solute–solvent–membrane interaction, a finding in common with the findings of Hadgraft and colleagues, discussed above.

In their second such study, Chen et al. (1996) examined a larger data set and refined Eq. 5.1:
where

Chen et al. reported that Eq. 5.2 gave better predictions than their previous model (Eq. 5.1; Chen et al. 1993). Thus, they applied Eq. 5.2 to predict the flux of 171 new compounds which were not included in their previous study. This analysis yielded a simplified model in which the imidazole descriptor is not included:

While Chen’s studies examined in detail the various subclasses in their data sets, they did not apply this analysis to the whole data set. Although the models are statistically highly relevant, they require the measurement of specific properties, such as the solubility of permeants in isopropyl alcohol as a method does not currently exist to compute this value. Therefore, Cronin et al. (1998) reanalysed the data published by Chen, with the aim of developing QSAR models based on readily calculable descriptors and with greater mechanistic insight for the whole data set. Thus, using the data from Chen’s two studies, they analysed a data set of the flux for 256 compounds. Five of Chen’s original data were omitted due to ambiguities in their structures, and the thirteen compounds common to both studies were only included once. Cronin et al. calculated 43 descriptors for each member of the data set including the octanol–water partition coefficient (as log P if available, c log P otherwise, which may have the potential to introduce variance in the study as calculations and predictions of log P often differ—see Chap. 9), topological indices and various measures of hydrogen bonding. Stepwise regression and the removal of outliers considering their residuals produced the following relationship between flux and significant descriptors:
where HA and HD are, respectively, the number of hydrogen bond acceptor and donor groups present on a penetrant, and ⁶ χ is the sixth-order path molecular connectivity.

Thus, the highly significant model describes permeability across the PDMS membrane in terms of hydrogen bonding and, to a lesser extent, molecular topology. The flux is inversely related to the simple count of hydrogen bonding groups available on a molecule, and the topological expression, ⁶ χ, is based on a count of the number of paths of six atoms, irrespective of the presence of heteroatoms and therefore described molecular volume, or molecular bulk. It is, in Eq. 5.4, associated with a decrease in flux as ⁶ χ increases. Cronin et al. commented that the significance of such a specific descriptor may encode more subtle information on the relative importance of six-membered rings compared to, for example, five-membered rings and their comparative significance in influencing permeation across the PDMS membrane—in a general mechanistic sense, larger or bulkier molecules are less likely to pass across the membrane. In comparing Cronin’s model with those developed by Chen, it is clear that Chen’s are statistically more significant, which may be due to their analysis of subsets rather than the complete data set. Nevertheless, the models from all three studies do find commonality in that Chen’s use of parameters describing molecular charge was rationalised as describing hydrogen bonding, a phenomenon of high significance in Cronin’s model. They also found molar solubility in isopropyl alcohol to be significant, and which Cronin also suggested could be related to hydrogen bonding. Cronin also compared their model to the Potts and Guy (1992) algorithm for human skin permeability, highlighting the differences in both models. Nevertheless, solvent selection, particularly after the mechanistic work of Hadgraft, highlighted above, may play a role in producing very different models, as does the comparative simplicity of the PDMS membrane compared to the multilayered and significantly more complex human skin. However, one issue to additionally consider is the limited number of descriptors employed in early QSAR-type studies of human skin, such as Potts and Guy (1992) and Flynn (1990) where permeability was quantified in terms of a small range of descriptors whose significance was determined by reference to experimental studies; the analysis of PDMS might therefore reflect the methodology of analysing a wider range of descriptors; this might also be considered in the significance of ⁶ χ in Cronin’s model, as topological parameters were not calculated by Chen. While this might also speak to the ease with which such parameters can be calculated, particularly by non-experts, it does suggest a limited value in making such comparisons particularly when later QSAR studies of human skin examine a wider range of parameters (e.g. Patel et al. 2002). Further, the composite and possibly covariate nature of parameters such as log P may also lend itself to a more empirical and less mechanistic approach to algorithm development. Thus, studies such as those by Chen et al. (1993, 1996) and Cronin et al. (1998) suggest that more complex methods may be required to discern specific mechanistic information and that the dual purpose of such models—predictive ability and the provision of mechanistic insight—might not always be a relevant outcome for all analyses.

A novel approach was taken to address this issue by applying artificial neural networks (ANNs) (Agatonovic-Kustrin et al. 2001). They used the data originally published by Chen et al. (1993, 1996) and modified by Cronin et al. (1998) for their analysis. They optimised and analysed their neural network model, which was based on a wide range of descriptors similar in type and range to those examined by Cronin et al. They generated a 12-parameter nonlinear QSAR model, based on descriptors that characterise dielectric energy, –OH and –NH₂– groups present on a molecule, the count of ring structures present in a molecule, the lowest unoccupied molecular orbital, EL affinity, molecular weight, total energy, dipole and descriptors of connectivity and molecular bulk. The model they developed indicated that intermolecular interactions (dipole interaction, electron affinity), hydrogen bonding ability (the presence of amino and hydroxyl group) and molecular shape and size (topological shape indices, molecular connectivity indices, ring count) were important for drug penetration through PDMS membranes. log P was not found to be a significant descriptor in their analysis, which they suggested was due to the inability of this parameter to account for intramolecular interactions, including intramolecular hydrogen bonding.

As with Cronin’s study, Agatonovic-Kustrin et al. found that topological indices were significant. They commented that their inclusion was significant as they could be calculated for any structure, real or hypothetical, and their inclusion was significant for drug discovery and new drug development. Their model included as significant descriptors topological shape indices of the first order (κ ¹) and connectivity indices of the first and second order (χ ¹ and χ ², respectively) which allowed specific quantification of molecular shape and bulk properties, describing similarity or dissimilarity of molecules based on the comparative values of the significant topological indices for molecules being compared. Topological shape indices encoded information on structural features such as size, shape, branching pattern, cyclicity and symmetry of molecular graphs. κ values are derived from fragments of one-bond, two-bond and three-bond fragments, with each count being made relative to fragment counts in reference structures. The first-order shape index, κ ¹, encodes molecular cycles, with κ ² and κ ³ encoding linearity and branching, respectively. Thus, the model proposed by Agatonovic-Kustrin et al. shows that an increase in κ ¹ decreased membrane permeation due to an increase in molecular size and lipid solubility. χ values indicate the extent of branching present in a molecule, which is the sum of the carbon atoms in a molecule linked to neighbouring carbons atoms, forming the χ index from which specific information on the number of bond fragments can be determined. Such values can be used to quantify aspects of a molecular structure; χ ⁰, or zero-order connectivity indices, provides information on the number of atoms in a molecule; χ ¹, or the first-order connectivity index, encodes the properties of single bonds, being a weighted count of bonds and is related to the types and position of branching in the molecule; and χ ², the second-order connectivity indices, is derived from fragments of two bond lengths, providing information about types and positioning of branching, indicating structural flexibility. Thus, Agatonovic-Kustrin et al. found that an increase in branching, based on the significance of the χ ¹ and χ ² descriptors in their model, suggested an increase in surface area and molecular volume, resulting in an increased solubility and reduced partition coefficient. Their analysis suggested that the increase in the χ ¹ and χ ² descriptors was consistent with a decrease in membrane penetration and that the χ ¹ and χ ² descriptors were covariant to an extent, although sufficiently different to each encode different, specific characteristics of the penetrating molecules; for example, χ ² can differentiate between structural isomers, whereas χ ¹ values for isomers are identical. Lower values of χ ¹ and χ ² are associated with comparatively more elongated molecules or those with only a single branching atom. They commented that an increase in molecular topology, characterised by the significance of the κ ¹, χ ¹ and χ ² descriptors, and an increase in ring count and molecular mass result in a decrease in flux across the PDMS membrane. Thus, mechanistically, a more bulky molecule is less likely to pass through the membrane. Overall, however, the most significant term in their 12-descriptor nonlinear QSAR was dielectric energy—essentially, the change in charge rearrangement of a molecule, which accompanies the change in hydrogen bonding strength. The model proposed by Agatonovic-Kustrin et al. suggested that an increase in dielectric energy is associated with an increase in membrane permeation.