statistical approaches for particle size distribution...

Statistical Approaches for

Particle Size Distribution Data

David ChristopherSchering-Plough Research Institute

PQRI and INFTG WorkshopDemonstrating Bioequivalence of Locally Acting Orally Inhaled

Drug Products

March 9-10, 2009

Hyatt Regency Bethesda

Bethesda, MD

David Christopher – PQRI BE Workshop, March 9-10, 2009 2

Acknowledgements

Walter Hauck

Ziqing Pan


Outline

• Brief overview of cascade impactor (CI) and particle size distribution (PSD) profiles

• Comparison of three statistical approaches:– Chi-square

– f2 Similarity Factor

– Multivariate Bioequivalence (MVBE)

• Discussion of how a statistical test may correctly meet some objectives (e.g., unbiasedness, scaled to reference variability, etc.) but fail to have enough power to detect differences of practical importance.

• Discussion of difficulty in establishing a "target" (i.e., consensus on profiles that are, or are not equivalent) against which the performance of a statistical test can be judged.


Cascade Impactor Particle Size

Distribution Profile

Deposition Site

Mass

Recovery

Actuator

Stem

Reference

Product


Cascade Impactor Particle Size

Distribution Profile

Deposition Site

Mass

Recovery Test

Product


Deposition Site

Mass

Recovery

Example Test / Reference PSD

Profiles

30 CI runs each for Test (Red)

and Reference (Blue) Products


PQRI Profile Comparisons WG

Realistic Scenarios

• WG developed an approach to simulate realistic

PSD profiles, including inter-site correlations

• Created 55 scenarios to cover a broad range of

PSD profile differences seen in real products

• Used these 55 realistic scenarios to evaluate the

performance characteristics of the Chi-square

Ratio Test

• f2 and multivariate PBE (MVBE) also evaluated

against these scenarios


Profile Examples from PQRI Profile Comparisons WG


Chi-Square Ratio Test

• FDA proposal requires 30 CI runs for Test and 30 CI runs for Reference

• Calculates Chi-square ratio as an overall measure of “distance” between Test and Reference PSD profiles, scaled to Reference product variability

• Uses re-sampling to create a distribution of these ratios

• Uses the 95th percentile of this distribution as the test statistic

• Smaller means more similar


Sample mean and confidence interval

1. Calculate Chi-sq Ratio of the kth triplet

R(Chi2)k =Ss Ws[ds(test, ref)]

2/es(test, ref)

2. Calculate mean Ch-sq Ratio of K triplets

R(Chi2) = (1/K) Si=1 to K (Chi-sq ratio(i)),

K = number of test/ref1/ref2 triplets (e.g. K=30)

3. Repeat the steps M times and calculate sample mean of Kratios of Chi-sq distance

^E(R) = (1/M) Sm=1 to MRm

4. The 95% upper confidence bound for the E(R), RU is the

empirical upper 95 percentile among the M Rm„s

5. Compare RU with q (pre-given). BE if R

U < q (e.g. =7.66).

Determining the equivalence limit,

q (=7.66) through simulation(using

Albuterol MDI)

1. Generate n = 1000 per product (10 lots, 100canisters/lot)

2. Real data mean and %CV used insimulations. Same between-lot and within-lot (between-canister) variability at eachstage. Two type of %CV (i.e. low and high)

Product ST & ACT Throat ST0 ST1 ST2 ST3 ST4 ST5 ST6 ST7

Low 20 10 30 20 20 20 10 20 20 20

High 10 5 15 10 10 10 5 10 10 10

3. Deposition in the stage was simulated fromlognormal dist.

4. Standardized to total =100.

Calculation of Chi-sq and Chi-sqRatio (based on Anderson CascadeImpactor)

Product ST &

ACT

Throat ST0 ST1 ST2 ST3 ST4 ST5 ST6 ST7

Test 14.28 38.26 1.83 2.06 2.24 7.56 17.97 13.11 1.59 0.51

Ref #1 18.56 46.32 2.15 0.43 0.92 9.11 12.00 7.11 2.44 0.82

Ref #2 19.22 47.51 2.03 0.82 0.83 9.06 10.21 7.15 2.01 0.94

Ref=(ref #1+ref #2)/2 18.89 46.91 2.09 0.63 0.87 9.08 11.10 7.13 2.23 0.88

D(test, ref) 4.61 8.65 0.26 1.43 1.37 1.52 6.87 5.97 0.64 0.37

Es =(test+ ref)/2 16.59 42.59 1.96 1.34 1.56 8.32 14.54 10.12 1.91 0.70

D2

/ Es

1.28 1.76 0.03 1.52 1.20 0.28 3.24 3.53 0.21 0.19

Chi-sq (test:ref) = 13.25

D(ref1, ref2) 0.66 1.19 0.12 0.39 0.08 0.05 1.79 0.04 0.43 0.12

d-sq/ave(ref1, ref2) .023 .030 .006 .249 .008 .0002 .288 .0002 .083 .016

Chi-sq(ref1,ref2) = 0.70

Chi-sq Ratio = 18.83 (the smaller the better)

Profile Analysis of Cascade Impactor Data: Proposed FDA Approach, Yi

Tsong, Ph.D. (2000)

http://www.fda.gov/ohrms/dockets/ac/00/slides/3609s1e/index.htm

Based on Albuterol MDI data and simulations

Chi-Square Ratio Test


f2 (or Similarity Factor)

• Developed for comparing dissolution profiles, but could potentially be applied to PSD profiles

• A population measure for assessing the similarity of two profiles

• Based on squared differences of cumulative distribution of Test and Reference

• Requires ordering of deposition sites– straightforward for inside impactor sites– how to treat outside impactor sites?

• No Reference product variability scaling

• In dissolution testing, similarity factor of 50 or greater indicates “similar” profiles

• Shah V, Tsong Y, Sathe P, Liu JP, In vitro Dissolution Profile Comparison – Statistics and Analysis of the Similarity Factor, f2, Pharmaceutical Research, Vol 15, No. 6, 1998.


Multivariate PBE (MVBE)

• Generalizes univariate PBE, including Reference product variability scaling

• Originally developed with in vitro bioequivalence in mind;e.g., treating four measures of spray pattern together in a single test rather than as four separate tests

• Shown statistically valid for dimensions up to 8 (not studied for >8)

• Would be better suited for cases where difference is on most stages rather than on just 1 or 2

• Chervoneva I, Hyslop T, Hauck WW. A multivariate test for population

bioequivalence. Statistics in Medicine 26:1208-23;2007.


How Do We Judge Performance?

• How consistently did the method agree

with the true answer?

• Must know what the true answer is

• Not simple…


Scenario 2a

“Minimal” differences in

impactor-sized profiles

between Reference and Test

R Total Mass = 113.43

R ISM = 58.85

T Total Mass = 112.54

T ISM = 56.42

CI Deposition Sites

ISM SitesBlue Line = Reference (R), Red Line = Test (T)

Realistic Scenarios:

Example Profiles

Proportion of WG

who judged profiles

equivalent

0.79


More pronounced differences in

impactor-sized profiles between

Reference and Test

ISM Sites


R ISM = 57.61


T ISM = 57.74Scenario 2b

CI Deposition Sites

Blue Line = Reference (R), Red Line = Test (T)


Example Profiles

Proportion of WG

who judged profiles

equivalent

0.50


Very visible differences in

impactor-sized profiles

between Reference and Test


R ISM = 57.42


T ISM = 56.04

ISM Sites

Scenario 2c

CI Deposition Sites

Blue Line = Reference (R), Red Line = Test (T)


Example Profiles

Proportion of WG

who judged profiles

equivalent

0.21


Box-Whisker Plot of 95th Percentile

95

thP

erc

en

tile

Scenarios

Chi-square Ratio Test

Lower Variability

Higher Variability



95

thP

erc

en

tile

Scenarios

Vcrit=7.66

Vcrit=2.75




95

thP

erc

en

tile

Scenarios

Vcrit=7.66

Vcrit=2.75


0.79

Equiv. 0.21

Equiv.



95

thP

erc

en

tile

Scenarios

Vcrit=7.66

Vcrit=2.75

Chi-square Ratio Test0.07

Equiv.

1.00

Equiv.


Vcrit= 50


Conclusions

• All three approaches generally agree in rank

order for the lower variability profiles

– MVBE may be more sensitive to differences in

variability

• No approach seems to be able to consistently

discriminate among differences likely to be of

practical importance

• Difficult to evaluate performance when there is

no clear consensus on “truth”

statistical approaches for particle size distribution...

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