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The AAPS Journal

, Volume 15, Issue 2, pp 465–476 | Cite as

A Sensitivity Analysis of the Modified Chi-square Ratio Statistic for Equivalence Testing of Aerodynamic Particle Size Distribution

  • Benjamin Weber
  • Sau L. LeeEmail author
  • Robert Lionberger
  • Bing V. Li
  • Yi Tsong
  • Guenther Hochhaus
Research Article

Abstract

Demonstration of equivalence in aerodynamic particle size distribution (APSD) is one key component for establishing bioequivalence of orally inhaled drug products. We previously proposed a modified version of the Chi-square ratio statistic (mCSRS) for APSD equivalence testing and demonstrated that the median of the distribution of the mCSRS (MmCSRS) is a robust metric when test (T) and reference (R) cascade impactor (CI) profiles are identical. Here, we systematically evaluate the behavior of the MmCSRS when T and R CI profiles differ from each other in their mean deposition and variability on a single and multiple sites. All CI profiles were generated by Monte-Carlo simulations based upon modified actual CI data. Twenty thousand sets of 30 T and 30 R CI profiles were simulated for each scenario, and the behavior of the MmCSRS was correlated to metrics that characterize the difference between T and R product in mean deposition and variability. The two key findings were, first, that the MmCSRS is more sensitive to difference between T and R CI profiles on high deposition sites, and second, that a cut-off value for APSD equivalence testing based on the MmCSRS needs to be scaled on the variability of the R product. The former is considered as beneficial for equivalence testing of CI profiles as it decreases the likelihood of failing identical CI profiles by chance, in part, due to increasing analytical variability associated with lower deposition sites. The latter is expected to be important for consistently being able to discriminate equivalent from inequivalent CI profiles.

KEY WORDS

aerodynamic particle size distribution bioequivalence cascade impactor modified Chi-square ratio statistic orally inhaled drug products 

Supplementary material

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Fig. S3

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M1 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 34 kb)

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Fig. S4

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M3 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S5

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M4 (Fig. 2 without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20,% 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S6

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M5 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20,% 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S7

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M6 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20,% 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical, circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 34 kb)

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Fig. S8

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M7 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S9

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M8 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S10

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M9 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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Fig. S11

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M10 (Fig. 2) without inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM10_ESM.tiff (259 kb)
High-resolution image (TIFF 258 kb)
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Fig. S12

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M1 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM11_ESM.tiff (261 kb)
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Fig. S13

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M2 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM12_ESM.tiff (261 kb)
High-resolution image (TIFF 261 kb)
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Fig. S14

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M3 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM13_ESM.tiff (261 kb)
High-resolution image (TIFF 261 kb)
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Fig. S15

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M4 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM14_ESM.tiff (261 kb)
High-resolution image (TIFF 260 kb)
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Fig. S16

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M6 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 34 kb)

12248_2013_9453_MOESM15_ESM.tiff (262 kb)
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Fig. S17

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M7 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM16_ESM.tiff (261 kb)
High-resolution image (TIFF 261 kb)
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Fig. S18

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M8 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM17_ESM.tiff (261 kb)
High-resolution image (TIFF 261 kb)
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Fig. S19

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M9 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

12248_2013_9453_MOESM18_ESM.tiff (261 kb)
High-resolution image (TIFF 260 kb)
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Fig. S20

Behavior of the MmCSRS (displayed as average of the 20,000 samples (y-axis)) when T and R CI profiles differ from each other on multiple sites in their mean deposition and variability for CI profile M10 (Fig. 2) with inter-site correlation; x-axis, variability of the R CI profiles (displayed as the squared inverse of the coefficient of variation (CV%)), T and R CI profiles differed from each other by ±5%, 10%, 15%, 20%, 25%, or 30% on all deposition sites such that the cumulative total mass of all deposition sites was identical; circles, T and R CI profiles had the same variability on all deposition sites; triangles, the variability on the sites of T CI profiles was half of that of the R CI profiles; plus sign, the variability on the sites of T CI profiles was twice as large as that of the R CI profiles (JPEG 33 kb)

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12248_2013_9453_MOESM20_ESM.docx (15 kb)
Table S1 (DOCX 15 kb)

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Copyright information

© American Association of Pharmaceutical Scientists 2013

Authors and Affiliations

  • Benjamin Weber
    • 1
  • Sau L. Lee
    • 2
    • 4
    Email author
  • Robert Lionberger
    • 2
  • Bing V. Li
    • 2
  • Yi Tsong
    • 3
  • Guenther Hochhaus
    • 1
  1. 1.Department of Pharmaceutics, College of Pharmacy, Center of Pharmacometrics and Systems PharmacologyUniversity of FloridaGainesvilleUSA
  2. 2.Office of Generic DrugsCenter for Drug Evaluation and Research, Food and Drug AdministrationRockvilleUSA
  3. 3.Division of Biometrics VI, Office of BiostatisticsCenter for Drug Evaluation and Research, Food and Drug AdministrationSilver SpringUSA
  4. 4.MPN II, RM N-131, HFD-600RockvilleUSA

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