Reliability assessment for the optimal formulations of pharmaceutical products predicted by a nonlinear response surface method

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Abstract

A novel method of evaluating the reliability of the optimal formulations of pharmaceutical products was developed based on statistical techniques. Hydrogel ointments, PEGylated emulsions, and solid dispersions were chosen as the model data for the pharmaceutical products, and the formulations of these models were optimized using a nonlinear response surface method incorporating multivariate spline interpolation. A bootstrap resampling method combined with a Kohonen's self-organizing map, was used to estimate the confidence intervals of the optimal formulations. To understand the factors significantly affecting the optimal formulations, a leave-one-factor-out (LOFO) method and a random number technique were introduced as sensitivity analyses. Our results suggest that the random number technique is a better approach than the LOFO method. To determine the design space and control space based on a scientific rationale that can satisfy a number of specifications of the pharmaceutical responses, a novel approach that takes advantage of the random number technique was investigated. The control space was successfully defined as a super-cubic area inscribed in a super-spherical area in the design space of the factors.

Introduction

In many pharmaceutical formulations, there are complicated relationships between the formulation factors and the response variables relating to the effectiveness, usefulness, stability, and safety of the product. Therefore, the causal relationships must be understood in designing a pharmaceutical formulation. In recent years, ICH Q8 guidance (Yu, 2008) has propounded the establishment of a science-based rationale. The concept of quality by design described in the ICH Q8 guidance states that “quality cannot be tested into products, i.e., quality should be built in by design.”

Previously, we developed an ingenious response surface method incorporating multivariate spline interpolation (RSM-S), which has been used to determine acceptable formulations of pharmaceuticals (Takayama et al., 2004). The basic concept of multivariate spline interpolation (MSI) involves a boundary element method. Green functions are used for the minimum curvature interpolation of multidimensional data points (Sandwell, 1987). MSI estimates multidimensional data using a thin-plate spline that represents the sum of the interpolations made with a Green function and a linear polynomial equation (Wahba, 1990). This can naturally interpolate observational data, including experimental errors. Using RSM-S, we can easily understand nonlinear relationships between causal factors and response variables, and estimate a stable and reproducible simultaneous optimal solution. Furthermore, this method does not require any complicated procedures, such as an artificial neural network, and it has been applied to pragmatic cases to optimize pharmaceutical formulations (Onuki et al., 2004, Onuki et al., 2005). However, no proper method of evaluating the effects of individual factors on the response variables has yet been established for RSM-S. The aim of this study was to evaluate the factors affecting the optimal solutions estimated by RSM-S, using a bootstrap (BS) resampling method, Kohonen's self-organizing map (SOM), and a leave-one-factor-out (LOFO) method or a random number technique.

The ICH Q8 guideline suggests that the design space must be identified in a scientific sense in the development of a pharmaceutical formulation. The design space is defined as the multidimensional combination and interaction of factors that have been demonstrated to provide an assurance of quality. However, few tangible methods have been suggested with which to establish a science-based design space and control space (ICH Draft Consensus Guideline, Pharmaceutical Development, Annex to Q8, 2007; MacGregor and Bruwer, 2008). In addition to the reliability assessment presented in this study, a novel method of determining the design space and control space is proposed that can satisfy various specifications.

Section snippets

Sensitivity analysis based on a leave-one-factor-out method

A leave-one-factor-out method was newly developed to understand the causal factors affecting optimal solutions. The procedure of the LOFO method is shown in Fig. 1. LOFO samples, in which the factor to be considered (Xi) was removed from the original dataset (X1, X2, …, Xi−1, Xi+1, …, Xn), were prepared. A number of duplicated samples with replacement (BS-LOFO samples) were generated using the BS resampling method. Details of the BS method have been reported elsewhere (Efron and Tibshirani, 1993

Model data

Three cases of formulation optimization were chosen as the model data from previously published articles, as summarized in Table 1 (Takayama et al., 1985, Wu et al., 2001, Fan et al., 2004). Ketoprofen hydrogels, containing 1-O-3-n-butylcyclohexanol (OEBC) and diisopropyl adipate (DIA) as chemical enhancers and isopropanol (IPA) as a solvent, were used as case A (Wu et al., 2001). The amounts of OEBC, DIA, and IPA were selected as the causal factors. All other components in the hydrogels were

Original optimal solutions and BS-optimal solutions

The individual datasets summarized in Table 1 were simultaneously optimized using RSM-S and the original optimal solutions were estimated. The BS resampling method was applied to each case and the BS-optimal means and their standard deviations were calculated. The results are shown in Table 2. Further details about estimating the original optimal and BS-optimal solutions have been fully described in previous papers (Arai et al., 2007, Onuki et al., 2008). In cases A and B, the original optimal

Conclusions

The simultaneous optimization of a model dataset was performed using RSM-S. Reliability and reproducibility were estimated with the BS resampling method and a sufficient number of model formulations were required to estimate high-precision optimal solutions. To analyze the sensitivities of the factors on the optimal formulations, the random number technique was a better approach than the LOFO method. To establish the design space and control space that can satisfy the specifications of the

Acknowledgment

This study was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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