The application and limitations of mathematical modelling in the prediction of permeability across mammalian skin and polydimethylsiloxane membranes

Abstract

Predicting the rate of percutaneous absorption of a drug is an important issue with the increasing use of the skin as a means of moderating and controlling drug delivery. One key feature of this problem domain is that human skin permeability (as K(p)) has been shown to be inherently non-linear when mathematically related to the physicochemical parameters of penetrants. As such, the aims of this study were to apply and evaluate Gaussian process (GP) regression methods to datasets for membranes other than human skin, and to explore how the nature of the dataset may influence its analysis. Permeability data for absorption across rodent and pig skin, and artificial membranes (polydimethylsiloxane, PDMS, i.e. Silastic) membranes was collected from the literature. Two quantitative structure-permeability relationship (QSPR) models were used to compare with the GP models. Further performance metrics were computed in terms of all predictions, and a range of covariance functions were examined: the squared exponential (SE), neural network (NNone) and rational quadratic (QR) covariance functions, along with two simple cases of Matern covariance function (Matern3 and Matern5) where the polynomial order is set to 1 and 2, respectively. As measures of performance, the correlation coefficient (CORR), negative log estimated predictive density (NLL, or negative log loss) and mean squared error (MSE) were employed. The results demonstrated that GP models with different covariance functions outperform QSPR models for human, pig and rodent datasets. For the artificial membranes, GPs perform better in one instance, and give similar results in other experiments (where different covariance parameters produce similar results). In some cases, the GP predictions for some of the artificial membrane dataset are poorly correlated, suggesting that the physicochemical parameters employed in this study might not be appropriate for developing models that represent this membrane. While the results of this study indicate that permeation across rodent (mouse and rat) and pig skin is, in a statistical sense, similar, and that the artificial membranes are poor replacements of human or animal skin, the overriding issue raised in this study is the nature of the dataset and how it can influence the results, and subsequent interpretation, of any model produced for particular membranes. The size of the datasets, in both absolute and comparative senses, appears to influence model quality. Ideally, to generate viable cross-comparisons the datasets for different mammalian membranes should, wherever possible, exhibit as much commonality as possible.