Evaluation of the reliability of nonlinear optimal solutions in pharmaceuticals using a bootstrap resampling technique in combination with Kohonen's self-organizing maps.

The response surface method incorporating multivariate spline interpolation (RSM-S) is a powerful technique for the formulation optimization of pharmaceuticals. However, no satisfactory method has been developed to evaluate the reliability of the optimal solution. We integrated bootstrap (BS) resampling and Kohonen's self-organizing maps (SOM) into RSM-S using the formulation optimization of theophylline tablets as the model experiment. The hardness and the 63.2% drug-release times of the tablets were measured as response variables. Based on the data set obtained, the simultaneous optimal solution was estimated using RSM-S. Leave-one-out cross-validation showed the optimal solution to be reliable. Concurrently, a large number of BS samples were generated from the original data set using BS resampling, and simultaneous optimal solutions for each BS sample (BS optimal solutions) were estimated. The distribution of the BS optimal solutions was far from a normal distribution, suggesting a mixture of global and local optimal solutions. SOM clustering was used to identify the set of global optimal solutions. SOM clustering divided the BS optimal solutions into several clusters, and the reliability of the optimal solution was evaluated from the cluster that included the optimal solution. This study offers a promising method for evaluating the reliability of nonlinear optimal solutions. .