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. .

Bootstrap re-sampling technique to evaluate the optimal formulation of theophylline tablets predicted by non-linear response surface method incorporating multivariate spline interpolation.

Optimal solutions of theophylline tablet formulations were derived from three types of experimental datasets, composed of different numbers of data-points using the response surface method incorporating multivariate spline interpolation (RSM(S)). The reliability of these optimal solutions was evaluated by a bootstrap re-sampling technique. Different levels of three causal factors were used as factors of response surface analysis: the lactose/cornstarch ratio (X(1)), the amount of carmellose calcium (X(2)), and the amount of hydroxypropylcellulose (X(3)). The target responses were the dissolution ratio of theophylline for the first 15 min (Y(1)) and the hardness (Y(2)) of each of the prepared tablets. Similar optimal solutions were estimated in three different sizes of datasets. A bootstrap re-sampling with replacements from the original dataset was applied, and optimal solutions for each bootstrap dataset were estimated. The frequency of the distribution of the optimal solution generated by the bootstrap re-sampling technique demonstrated almost normal distribution. The average and standard deviation of the optimal solution distribution were calculated as evaluation indices reflecting the accuracy and reproducibility of the optimal solution. It was confirmed that the accuracy was sufficiently high, irrespective of the dataset size; however, the reproducibility worsened with a decrease in the number of the experimental datasets. Consequently, it was considered that the novel evaluation method based on the bootstrap re-sampling technique was suitable for evaluating the reliability of the optimal solution.