Statistical and Regulatory Considerations for Multiple Measures in Bioequivalence Testing

Equivalence tests for ratio of means in bioequivalence studies under crossover design

There are several problems concerning the statistical definition of average bioequivalence provided by the U.S. Food and Drug Administration. We proposed a ratio of means based on the original bioavailability measure as the definition for average bioequivalence. Under the log-normal distribution assumption, we proposed a hypothesis testing-based method and a confidence interval-based method to answer the question of whether the ratio of means falls into a predetermined interval. For the hypothesis testing-based method, we decomposed the null two-sided hypothesis of the ratio of means into two one-sided hypotheses. With the intersection-union theorem for asymptotic tests, we constructed two asymptotic size-α tests for the original null hypothesis. The method of variance estimation recovery was adopted to develop the confidence interval-based method. Simulation studies showed that the proposed methods can maintain the empirical type I error rate closely at the nominal level and is as powerful as the two one-sided t-test for testing the ratio of means under different settings. The application of the proposed methods was illustrated through six datasets in real-world examples.

Multivariate Assessment for Bioequivalence Based on the Correlation of Random Effect

Background and Objective

Bioequivalence tests are fundamental step in assessing the equivalence in bioavailability between a test and reference product. In practice, two separate linear mixed models (LMMs) with random subject effects, which have an area under the concentration-time curve (AUC) and the peak concentration (C_max) as the responses, have become the gold standard for evaluating bioequivalence. Recently, Lee et al developed a multivariate hierarchical generalized linear model (HGLM) for several responses that modeled correlations among multivariate responses via correlated random effects. The objective of this study was to apply this multivariate analysis to the bioequivalence test in practice and to compare the performance of multivariate HGLM and separate LMMs.

Methods

Three pharmacokinetic datasets, fixed-dose combination (naproxen and esomeprazole), tramadol and fimasartan data were analyzed. We compared the 90% confidence interval (CI) for the geometric mean ratio (GMR) of a test product to a reference product using the multivariate HGLM and two conventional separate LMMs.

Results

We found that the 90% CIs for the GMRs of both AUC and C_max from the multivariate HGLM were narrower than those from the separate LMMs: (0.843, 1.152) vs (0.825, 1.177) for C_max of esomeprazole in fixed-dose combination data; (0.805, 0.931) vs (0.797, 0.941) for C_max in tramadol data; (0.801, 1.501) vs (0.762, 1.578) for C_max and (1.163, 1.332) vs (1.009, 1.341) for AUC in fimasartan data, consistent with the random subject effects from two separate LMMs being highly correlated in the three datasets (correlation coefficient r = 0.883; r = 0.966; r = 0.832).

Conclusion

This multivariate HGLM had good performance in the bioequivalence test with multiple endpoints. This method would provide a more reasonable option to reduce the 90% CI by adding correlation parameters and thus an advantage especially in evaluating the bioequivalence of highly variable drugs with broad 90% CIs.

The Subject-by-Formulation Interaction in Multivariate Bioequivalence

This paper addresses hypothesis testing problems concerning the subject-by-formulation interaction matrix for the assessment of multivariate bioequivalence. Two problems are addressed: (a) the problem of testing if the subject-by-formulation interaction matrix itself is zero, and (b) the problem of testing if suitable scalar valued functions of the subject-by-formulation interaction matrix is below a threshold. Approximate tests are developed in both cases and the accuracy of the approximation is numerically investigated. The results are illustrated with an example. Even though the literature on univariate bioequivalence testing addresses average bioequivalence, variance bioequivalence and subject-by-formulation interaction, the literature on multivariate bioequivalence deals only with the problem of average bioequivalence. This work appears to be the first attempt to address tests for the subject-by-formulation interaction matrix for testing multivariate bioequivalence.

Kullback-Leibler divergence for evaluating bioequivalence

In this paper we propose a methodology for evaluating the bioequivalence of two formulations of a drug that encompasses not only average bioequivalence (ABE), but also the more recently introduced measures of population bioequivalence (PBE) and individual bioequivalence (IBE). The latter two measures are concerned with prescribability (PBE) and switchability (IBE). The main idea is to use the Kullback-Leibler divergence (KLD) as a measure of discrepancy between the distributions of the two formulations. Two formulations are declared bioequivalent if the upper bound of a level-alpha confidence interval for the KLD is less than a given goalpost to be set by a regulator. This new methodology overcomes many of the disadvantages of the corresponding measures recommended by the FDA. In particular the KLD: (i) possesses the natural hierarchical property that IBE => PBE => ABE; (ii) satisfies the properties of a true distance metric; (iii) is invariant to monotonic transformations of the data; (iv) generalizes easily to the multivariate case where equivalence on more than one parameter (for example, AUC, C(max) and T(max)) is required; and (v) is applicable over a wide range of distributions of the response variable (for example, those in the exponential family). The performance of the KLD relative to the metric proposed in guidance by the FDA for the evaluation of individual bioequivalence is evaluated using a simulation study. Previously published retrospective analyses using the FDA-proposed metric are contrasted with those based on the KLD. It is concluded that the KLD is a viable alternative to the FDA-proposed metric and that its mathematical and statistical properties make it a readily interpretable measure of the differences between formulations.

Effects of variability in hepatic clearance on the bioequivalence parameters of a drug and its metabolite: simulations using a pharmacostatistical model

A semi-physiological pharmacostatistical model was developed and used to study the manner in which intra-individual variability in hepatic clearance is transmitted to the area-under the plasma concentration-time curve from zero to infinity (AUC) of a drug and its metabolite. In order to clarify the effects, the model contained no other sources of variability. As the drug's hepatic extraction ratio increased, the coefficient of variation (CV) of the AUC of the drug increased, whereas that of the metabolite decreased. The AUC of the metabolite increased as the drug's non-hepatic clearance increased. At high extraction ratios, the CV of the AUC of the drug was insensitive to the contributions of other forms of clearance. The results suggest that under certain circumstances, smaller samples sizes could be used in bioequivalence studies for highly variable drugs if the test were based on the metabolite rather than the parent drug.

Use of the repeated cross-over design in assessing bioequivalence: (within and between subjects variability - Schuirmann Confidence Intervals estimation)

In 1992, the Division of Bioequivalence in the Office of Generic Drugs published a guide to Statistical procedures for bioequivalence studies using a standard two-treatments cross-over design (1). This paper describes the application of the guidelines to a practical protocol and the recent Proc MIXED (SAS) will be shown to be much more convenient than the traditional Proc GLM for theoretical and practical reasons (correct estimation of residuals, analysis of the within-subjects variation, direct calculation of the Schuirmann 90% Confidence Intervals). This new procedure was applied to a study protocol on riluzole (Rilutek) including a replicate design with the within-subject and between-subject variances being estimated on Cmax and AUC biopharmaceutic parameters.