Using Tolerance Intervals for Assessment of Pharmaceutical Quality

Variability and uncertainty: interpretation and usage of pharmacometric simulations and intervals

Variability and estimation uncertainty are important sources of variation in pharmacometric simulations. Different combinations of uncertainty and the variability components lead to a variety types of simulation intervals, and many realized and unrealized confusions exist among pharmacometricians on their interpretation and usage. This commentary aims to clarify some of the important underlying concepts and provide a convenient guideline on pharmacometric simulation conduct and interpretation.

Tolerance interval testing for assessing accuracy and precision simultaneously

Tolerance intervals have been recommended for simultaneously validating both the accuracy and precision of an analytical procedure. However, statistical inferences for the corresponding hypothesis testing are scarce. The aim of this study is to establish a whole statistical inference for tolerance interval testing, including sample size determination, power analysis, and calculation of p-value. More specifically, the proposed method considers the bounds of a tolerance interval as random variables so that a bivariate distribution can be derived. Simulations confirm the theoretical properties of the method. Furthermore, an example is used to illustrate the proposed method.

Use of a two-sided tolerance interval in the design and evaluation of biosimilarity in clinical studies

In assessing biosimilarity between two products, the question to ask is always "How similar is similar?" Traditionally, the equivalence of the means between products is the primary consideration in a clinical trial. This study suggests an alternative assessment for testing a certain percentage of the population of differences lying within a prespecified interval. In doing so, the accuracy and precision are assessed simultaneously by judging whether a two-sided tolerance interval falls within a prespecified acceptance range. We further derive an asymptotic distribution of the tolerance limits to determine the sample size for achieving a targeted level of power. Our numerical study shows that the proposed two-sided tolerance interval test controls the type I error rate and provides sufficient power. A real example is presented to illustrate our proposed approach.

Improving coverage probabilities for parametric tolerance intervals via bootstrap calibration

Statistical tolerance intervals are commonly employed in biomedical and pharmaceutical research, such as in lifetime analysis, the assessment of biosimilarity of branded and generic versions of biopharmaceutical drugs, and in quality control of drug products to ensure that a specified proportion of the products are covered within established acceptance limits. Exact two-sided parametric tolerance intervals are only available for the normal distribution, while exact one-sided parametric tolerance limits are available for a limited number of distributions. Approximations to two-sided parametric tolerance intervals often use the Bonferroni correction on the one-sided tolerance interval calculation; however, this often incurs a higher coverage probability than the nominal level. Recently, the usage of a bootstrap calibration has been demonstrated as a way to improve coverage probabilities of tolerance intervals for very specific and complex distributional settings. We present a focused treatment on using a single-layer bootstrap calibration to improve the coverage probabilities of two-sided parametric tolerance intervals. Simulation results clearly demonstrate the improved coverage probabilities towards the nominal level over the uncalibrated setting. Applications to medical data for various parametric distributions also highlight the utility of constructing these calibrated tolerance intervals.

Confidence, prediction, and tolerance in linear mixed models

The literature about Prediction Interval (PI) and Tolerance Interval (TI) in linear mixed models is usually developed for specific designs, which is a main limitation to their use. This paper proposes to reformulate the two‐sided PI to be generalizable under a wide variety of designs (one random factor, nested and crossed designs for multiple random factors, and balanced or unbalanced designs). This new methodology is based on the Hessian matrix, namely, the inverse of (observed) Fisher Information matrix, and is built with a cell mean model. The degrees of freedom for the total variance are calculated with the generalized Satterthwaite method and compared to the Kenward‐Roger's degrees of freedom for fixed effects. Construction of two‐sided TIs are also detailed with one random factor, and two nested and two crossed random variables. An extensive simulation study is carried out to compare the widths and coverage probabilities of Confidence Intervals (CI), PIs, and TIs to their nominal levels. It shows excellent coverage whatever the design and the sample size are. Finally, these CIs, PIs, and TIs are applied to two real data sets: one from orthopedic surgery study (intralesional resection risk) and the other from assay validation study during vaccine development.

Risk-Based Approach to Lot Release

In this work, a novel risk-based methodology for lot release is proposed. Its objective is to assess the risk that a lot declared to have passed truly meets product specifications. The methodology consists of 3 parts: adaptive sample size determination, estimation of the probability that the product was within specifications, and the lot-release decision. The methodology provides a probabilistic statement about the true quality of the batch. Having a probability estimate is the essential condition of risk-based decision-making. We demonstrate the proposed methodology on experimental data generated from 17 immediate-release solid oral drug products from a number of different manufacturers with 5 to 10 lots per manufacturer.

A "Large-N" Content Uniformity Process Analytical Technology (PAT) Method for Phenytoin Sodium Tablets

Accurate assessment of tablet content uniformity is critical for narrow therapeutic index drugs such as phenytoin sodium. This work presents a near-infrared (NIR)-based analytical method for rapid prediction of content uniformity based on a large number of phenytoin sodium formulation tablets. Calibration tablets were generated through an integrated experimental design by varying formulation and process parameters, and scale of manufacturing. A partial least squares model for individual tablet content was developed based on tablet NIR spectra. The tablet content was obtained from a modified United States Pharmacopeia phenytoin sodium high-performance liquid chromatography assay method. The partial least squares model with 4 latent variables explained 92% of the composition variability and yielded a root mean square error of prediction of 0.48% w/w. The resultant NIR model successfully assayed the composition of tablets manufactured at the pilot scale. For one such batch, bootstrapping was applied to calculate the confidence intervals on the mean, acceptance value, and relative SD for different sample sizes, n = 10, 30, and 100. As the bootstrap sample size increased, the confidence interval on the mean, acceptance value, and relative SD became narrower and symmetric. Such a 'large N' NIR-based process analytical technology method can increase reliability of quality assessments in solid dosage manufacturing.

Content uniformity testing for stratified samples via parametric tolerance interval testing

Historically in the biopharmaceutical setting, USP<905> has been used to establish that a batch of drug product has acceptable content uniformity. More recently, alternative approaches such as the two one-sided parametric tolerance interval test (PTI-TOST) have been proposed to establish content uniformity. Traditionally, the PTI-TOST is implemented as a sequential, two-tiered test, under the generally accepted assumption that the data are independently and identically distributed. Since the material is sequenced through the manufacturing process over a period of time, there are conceptually arguable locations within each batch, for instance: beginning, middle, and end. In such a situation, a practitioner may wish to evaluate potential effects of these batch locations, for example, during process validation. If location (stratified) differences exist within the batch and if multiple samples are taken from each location, significant within-location correlations may be induced in the data. In such a case, the traditional PTI-TOST underestimates the total variability, thereby improperly boosting the power of the test method. When there is reason to believe that location variances exist, the batch may be evaluated using stratified sampling, and the location effect may be modeled. In this paper, a two-tiered PTI-TOST that accounts for both between-location and within-location variance components is introduced. Operating characteristic curves and practical advice are given to aid the practitioner's uptake of the proposed method.

Wald tests for variance-adjusted equivalence assessment with normal endpoints

Equivalence tests may be tested with mean difference against a margin adjusted for variance. The justification of using variance adjusted non-inferiority or equivalence margin is for the consideration that a larger margin should be used with large measurement variability. However, under the null hypothesis, the test statistic does not follow a t-distribution or any well-known distribution even when the measurement is normally distributed. In this study, we investigate asymptotic tests for testing the equivalence hypothesis. We apply the Wald test statistic and construct three Wald tests that differ in their estimates of variances. These estimates of variances include the maximum likelihood estimate (MLE), the uniformly minimum variance unbiased estimate (UMVUE), and the constrained maximum likelihood estimate (CMLE). We evaluate the performance of these three tests in terms of type I error rate control and power using simulations under a variety of settings. Our empirical results show that the asymptotic normalized tests are conservative in most settings, while the Wald tests based on ML- and UMVU-method could produce inflated significance levels when group sizes are unequal. However, the Wald test based on CML-method provides an improvement in power over the other two Wald tests for medium and small sample size studies.

Quality assurance test of delivered dose uniformity of multiple-dose inhaler and dry powder inhaler drug products

The delivered dose uniformity is one of the most critical requirements for dry powder inhaler (DPI) and metered dose inhaler products. In 1999, the Food and Drug Administration (FDA) issued a Draft Guidance entitled Nasal Spray and Inhalation Solution, Suspension, and Spray Drug Products-Chemistry, Manufacturing and Controls Documentation and recommended a two-tier acceptance sampling plan that is a modification of the United States Pharmacopeia (USP) sampling plan of dose content uniformity (USP34<601>). This sampling acceptance plan is also applied to metered dose inhaler (MDI) and DPI drug products in general. The FDA Draft Guidance method is shown to have a near-zero probability of acceptance at the second tier. In 2000, under the request of The International Pharmaceutical Aerosol Consortium, the FDA developed a two-tier sampling acceptance plan based on two one-sided tolerance intervals (TOSTIs) for a small sample. The procedure was presented in the 2005 Advisory Committee Meeting of Pharmaceutical Science and later published in the Journal of Biopharmaceutical Statistics (Tsong et al., 2008). This proposed procedure controls the probability of the product delivering below a pre-specified effective dose and the probability of the product delivering over a pre-specified safety dose. In this article, we further propose an extension of the TOSTI procedure to single-tier procedure with any number of canisters.