**Date Published:** February 11, 2019

**Publisher:** Springer International Publishing

**Author(s):** Piotr J. Rudzki, Przemysław Biecek, Michał Kaza.

http://doi.org/10.1208/s12248-019-0293-2

**Abstract**

**Reliable results of pharmacokinetic and toxicokinetic studies are vital for correct decision making during drug discovery and development. Thus, ensuring high quality of bioanalytical methods is of critical importance. Incurred sample reanalysis (ISR)—one of the tools used to validate a method—is included in the bioanalytical regulatory recommendations. The methodology of this test is well established, but the estimation of the sample size is still commented on and contested. We have applied the hypergeometric distribution to evaluate ISR test passing rates in different clinical study sizes. We have tested both fixed rates of the clinical samples—as currently recommended by FDA and EMA—and a fixed number of ISRs. Our study revealed that the passing rate using the current sample size calculation is related to the clinical study size. However, the passing rate is much less dependent on the clinical study size when a fixed number of ISRs is used. Thus, we suggest using a fixed number of ISRs, e.g., 30 samples, for all studies. We found the hypergeometric distribution to be an adequate model for the assessment of similarities in original and repeated data. This model may be further used to optimize the sample size needed for the ISR test as well as to bridge data from different methods. This paper provides a basis to re-consider current ISR recommendations and implement a more statistically rationalized and risk-controlled approach.**

**Partial Text**

Reliable results of pharmacokinetic and toxicokinetic studies are vital for correct decision making during drug discovery and development. Thus, assuring high quality of bioanalytical methods is of critical importance. The American Association of Pharmaceutical Scientists (AAPS) and the US Food and Drug Administration (FDA) were the driving forces behind discussions on the bioanalytical method validation in the 1990s (1). Both AAPS and FDA are constantly involved in the evolution of bioanalytical requirements, including incurred samples reanalysis (ISR) (2,3). Professional organizations like the European Bioanalysis Forum have also presented their opinions on the test (4). Finally, the ISR was included in the bioanalytical regulatory recommendations by the European Medicines Agency (EMA) (5), the Health Canada (6), and the FDA (7). Although the ISR is now part of the regulatory documents, the topic is still under much discussion in the bioanalytical and pharmaceutical community (8–16).

The symbols and terms used in this manuscript are defined in Table I. Calculations and Figs. 1, 2, 5, and 6 have been generated in the R version 3.5.0 (25). For Figs. 1 and 2, we have selected a red-yellow-blue color scheme for seven diverging percentage difference classes in a colorblind-safe mode (26). Figures 3 and 4 have been generated in Microsoft Office Excel 2007.Table ISymbols and Terms UsedSymbol or termHypergeometric distributionISR testValues tested and/or calculation methodaNSize of the populationStudy sample size—number of unique biological samples in a clinical study(20), (50), 100, (200), (250), 500, 1000, 1500, 2500, and 5000nNumber of the experimentsNumber of ISRsFixed number: 10, 20, (30), 50, and 100 or fixed ratio: 5% · N (9), 7% · N (27), 10% · N, 10% · N for the studies with N ≤ 1000 and 100 + 5% · (N-1000) for the studies with N > 1000 (5, 7)KNumber of successes in the populationNumber of ISR pairs meeting %difference criteria if all samples from the clinical study have been analyzedK = p · NK ∈ [0,1, …, N]kNumber of successes in n experimentsNumber of ISR pairs meeting %difference criteria observed in the reanalyzed samplesk ∈ [0,1, …, n]p = %ISRSuccess ratetrue percentage of ISR pairs meeting %difference criteria (when all samples have been reanalyzed)p ∈ [0; 100%]documentclass[12pt]{minimal}

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begin{document}$$ widehat{p} $$end{document}p^ = %isrEstimated success rateThe estimated percentage of ISR pairs meeting %difference criteria (when a portion of the samples has been reanalyzed)documentclass[12pt]{minimal}

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begin{document}$$ widehat{p}=frac{mathrm{k}}{n}cdotp 100% $$end{document}p^=kn·100%Passing rate–Probability of passing the ISR testCalculated using the hypergeometric distribution passing rate ∈ [0; 1]%difference–Percentage difference between the original concentration and the concentration measured during the repeat analysisdocumentclass[12pt]{minimal}

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begin{document}$$ %mathrm{difference}=frac{mathrm{repeat}-mathrm{original}}{mathrm{mean}}cdotp 100%kern0.5em $$end{document}%difference=repeat−originalmean·100%(7)aValues in brackets were used in selected tests onlyFig. 1Cumulative distribution function for different study sample sizes (N), when the ratio of the number of ISRs to study the sample size (n/N) is fixed at 5% (a), 7% (b), 10% up to 1000 samples, and then 5% (c) and 10% (d)Fig. 2Cumulative distribution function for different study sample sizes (N), when the number of ISRs (n) is fixed at 10 (a), 20 (b), 50 (c), and 100 (d)Fig. 3Differentiation of non-reproducible (red) and reproducible (green) methods for different study sample sizes (N) using the current regulatory ISR sample size (5,7)Fig. 4Differentiation of non-reproducible (red) and reproducible (green) methods for different study sample sizes (N), when the number of ISRs (n) is fixed at 10 (a), 20 (b), 30 (c), and 50 (d)

The goal of the ISR test is to confirm that a bioanalytical method is reliable (5,7). Thus, the probability of meeting the acceptance criteria should depend mainly—or even solely—on the bioanalytical method performance. Our new approach shows that this is not the case when the sample size is based on a fixed n/N ratio. The hypergeometric distribution revealed that the passing rate for a particular method performance (%ISR) is related to N. Surprisingly, this dependence is hard to observe for a fixed n.

The hypergeometric distribution is an appropriate model to understand and optimize ISR sample size better. Our study revealed that the passing rates are currently related to clinical study size. Interestingly, the passing rates are much less dependent on clinical study size when a fixed number of ISR samples are used; therefore, we propose to use a constant number of samples, e.g., 30, for ISR for all studies. This paper provides a basis to re-consider ISR methodology and implement a more statistically rationalized and risk-controlled approach.

Source:

http://doi.org/10.1208/s12248-019-0293-2