Role of individual test samples in optimal solutions in pharmaceuticals predicted using a nonlinear response surface method

Establishing a method to evaluate the reliability of an optimal solution is an exciting challenge for the nonlinear response surface method. We reported previously that the bootstrap (BS) resampling technique and Kohonen's self-organizing map are promising tools for meeting this challenge. To understand the usefulness of these techniques further, we employed a formulation optimization study of photocrosslinked polyacrylic acid (PAA) hydrogel as a case study. In a series of experiments, a large number of optimal solutions were generated with BS resampling and they were classified into three distinct clusters with SOM clustering. Using analysis of Bayesian estimation, we clarified the mode of generating clusters; e.g., cluster 2 was distinguished by the difference in features between the BS optimal solutions and the original optimal solution, whereas cluster 3 was distinguished by the substantial change in the shape of the response surfaces. We concluded that cluster 1 represents the global optimal solution, and then estimated 95% confidence intervals of the optimal solutions using the BS optimal solutions. These findings prove that our method is a valid approach for evaluating nonlinear optimal solutions. This method has applications for establishing a science-based rationale for, and a design space in, pharmaceutical formulation development.