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The Posterior Probability of Dissolution Equivalence. David J LeBlond 1 , John J Peterson 2 and Stan Altan 3 ) 1 Exploratory Statistics, Abbott, [email protected] 2 Research Statistics Unit, GlaxoSmithKline Pharmaceuticals 3 Pharmaceutical R&D, Johnson & Johnson .

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The posterior probability of dissolution equivalence l.jpg

The Posterior Probability of Dissolution Equivalence

David J LeBlond 1 , John J Peterson 2 and Stan Altan 3)

1 Exploratory Statistics, Abbott,[email protected]

2 Research Statistics Unit, GlaxoSmithKline Pharmaceuticals

3 Pharmaceutical R&D, Johnson & Johnson

Midwest Biopharmaceutical Statistical Workshop

Muncie, Indiana

May 25, 2011

Outline l.jpg

  • Objective

  • Background

    • Why dissolution?

    • Equivalence defined

    • Current practice

  • Why a Bayesian approach?

  • Posterior probability defined

  • MCMC

  • Examples

    • Equivalence of 2 lots

    • Equivalence of 2 processes (multiple lots)

    • Model dependent comparisons

  • Summary

MBSW May 25, 2011

Objective l.jpg

  • Make this tool available to you so you can use it if you want to.

    • Statistical Modeling

    • Software (R, WinBUGS)

    • Example Data & Code

    • [email protected]

MBSW May 25, 2011

Importance of in vitro dissolution l.jpg
Importance of in-vitro dissolution

  • Surrogate measure of in-vivo dissolution

  • In-vivo dissolution rate affects drug bio-availability

  • Bio-availability may affect PK (blood levels)

  • Blood levels may affect safety and efficacy

  • Compendial requirement for most solid oral dosage forms

  • Need to show “equivalence” for process/ method change or transfer to obtain a bio-waiver.

  • Need to show “non-equivalence” to prove in-vitro method can detect formulation / process differences.

MBSW May 25, 2011

Measurement of in vitro dissolution l.jpg


% Dissolved



Measurement of in-vitro dissolution

  • 1 tablet/ stirred vessel

  • 1 run usually = 6 tablets

  • solution sampled at fixed intervals

  • samples assayed

  • cumulative concentration

  • expressed as % of dosage form Label Content (%LC)

  • Are and “equivalent”?

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Equivalence defined l.jpg
Equivalence defined

  • Identify parameter space based on

    • Difference in Dissolution at multiple time points

    • Difference in profile model parameters

    • Condensed univariate distance measure

  • Establish similarity region

    • Constraints on parameter space

    • Based on “customer requirements”

  • Obtain distance estimate from data

    • Conforms to parameter space

  • Equivalence: distance estimate is “sufficiently contained within” the similarity region.


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Example f2 similarity metric see reference 9 l.jpg





Example: f2 similarity metric(see reference 9)

  • parameter space: Dissolution differences, Di, at p time points.

  • similarity region:

  • distance estimate = (point estimate)

  • Equivalence: (no measure of uncertainty)

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The confidence set approach l.jpg
The confidence set approach

TOST (one dimensional)







“MOST” (multi-dimensional)




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Confidence set approach considerations

  • Must choose similarity region shape.

  • Must choose confidence region shape.

  • The number of shapes increases with number of dimensions.

  • Lack of conformance between similarity and confidence region shapes  conservative test

  • Conclusion depends on shape choices.

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Confidence set approach considerations10 l.jpg
Confidence set approach considerations

  • The confidence level is not the probability of equivalence.

  • It is the probability of covering the “true” difference in repeated trials.

  • What if you really want to know the probability of equivalence?

    • risk based decision making (ICH Q9)

MBSW May 25, 2011

Proposed bayesian approach l.jpg
Proposed Bayesian Approach

distance estimate: Joint Posterior of

Distance measures

Measure of Equivalence

= Integrated density

= Posterior Probability of Equivalence

Obtained by counting from MCMC

Similarity region

(“customer requirement”)

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Bayesian equivalency in a nutshell l.jpg
Bayesian equivalency in a nutshell

Probability Model


Dissolution Data

(Test and Reference)

Prior Information



Draws from the joint posterior distribution of distance parameters

(10-100 thousand)

Count the fraction of draws within the similarity region

Conclude equivalency if fraction exceeds some limit (e.g. 95%)

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Example 1 multivariate dissolution model l.jpg
Example 1: Multivariate Dissolution Model

% Dissolution vector, Y, for the ith tablet from the kth lot …

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Example 1 non informative prior information17 l.jpg
Example 1: Non-informative Prior Information



can be shown (see appendix) to have the distribution

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Example 1 definition of equivalence l.jpg
Example 1: Definition of Equivalence

Define a rectangular similarity region, S, as

and require that

to conclude equivalence.

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Example 1 results l.jpg
Example 1: Results

500 of 10,000 draws plotted

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Example 2 hierarchical model l.jpg
Example 2: Hierarchical Model

% Dissolution vector, y, for the ith tablet from the kth run …

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Example 2 non informative prior information l.jpg
Example 2: Non-informative prior information

elements of Vtablet

elements of Vrun

Max = 40

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Example 2 definition of equivalence same as example 1 l.jpg
Example 2:Definition of Equivalence(same as Example 1)

Define a rectangular similarity region, S, as

and require that

to conclude equivalence.

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Example 2 results l.jpg
Example 2: Results

1000 of ~2,000 draws plotted

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Example 3 a model dependent comparison l.jpg
Example 3: A model dependent comparison

  • Data from reference 12

  • 3 lots: 1 reference and 2 post-change lots

  • A minor change and a major change lot

  • 12 tablets per Lot

  • Pre-change and Test Lots have different time points

  • Comparison requires a parametric dissolution profile model

  • Similarity region defined on the model parameter space

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Dissolution profile models l.jpg
Dissolution profile models





( 1st order kinetics )


…and some others (Higuchi, Gompertz, Hixson-Crowell,…)

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Weibull parameters l.jpg
Weibull parameters

M measures content

T is time to 63.2% Dissol.

beta measures delay



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Weibull parameterization in winbugs l.jpg
Weibull parameterization in WinBUGS

  • The following seemed to reduce colinearity and improve convergence.

    • Replace T with lna = -b lnT

    • Replace b with lnb

    • transform time (t) from minutes to hours

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Nonlinear mixed model in winbugs l.jpg
Nonlinear mixed model in WinBUGS

% Dissolution, Y, for the ith tablet from the kth lot at the jth time (t) point…

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Weibull example judging similarity by confidence set approach l.jpg
Weibull Example: Judging similarity by confidence set approach

“…At present, some issues are unresolved such as

(i) how many standard deviations (2 or 3) should be

used for a similarity criterion,

(ii) what to do if the ellipse is

only marginally out of the similarity region …”

from Sathe, Tsong, Shah (1996) Pharm Res 13(12) 1799-1803

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Weibull example posterior probability of dissolution equivalence l.jpg
Weibull Example: Posterior Probability of Dissolution Equivalence

Prob = 0

2SD Similarity Region

Prob = 0.949

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Pros and cons of a bayesian approach l.jpg
Pros and Cons of a Bayesian Approach Equivalence

  • Pros

    • Based on simple counting exercise (MCMC)

    • Probability estimate for risk assessment

    • Exact conformity between the similarity region and the estimate (integrated posterior)

    • Incorporation of prior information (or not) as appropriate

    • True equivalence (not significance) test

    • Rewards high data information content

  • Cons

    • Requires (usually) MCMC

    • Coverage properties require calibration studies.

    • Regulatory acceptance?

MBSW May 25, 2011

References l.jpg
References Equivalence

  • Schuirmann DJ (1981) On hypothesis testing to determine of the mean of a normal distribution is contained in a known interval, Biometrics 37:617

  • Berger RL (1982) Multiparameter hypothesis testing and acceptance sampling, Technometrics 24(4) 295-300

  • Schuirmann DJ (1987) Comparions of two one-sided procedures and power approach of rassessing the equivalence of average bioavailability, Journal of Pharmokinetics and Biopharmaceutics 15:657-680.

  • Shah VP, Yamamoto LA Schirmann D, Elkins J and Skelly JP (1987) Analysis of in vitro dissolution of whole versus half controlled release theophilline tablets, Pharm Res 4: 416-419

  • Food and Drug Administration. Guidance for Industry: Immediate Release Solid Oral Dosage Forms. Scale-Up and Postapproval Changes (SUPAC-IR): Chemistry, Manufacturing and Controls, In Vitro Dissolution Testing and In Vivo BE. 1995

  • Tsong Y, Sathe P, an dShah VP (1996) Compariong 2 dissolution data sets fro similarity ASA Proceedings of the Biopharmaceutical Section 129-134

  • Berger RL and Hsu JC (1996) Bioequivalence trials, intersection-union tests and equivalence confidence sets, Statistical Science 11(4) 283-319

  • J.W.Moore and H.H.Flanner, Mathematical Comparison of curves with an emphasis on in vitro dissolution profiles. Pharm. Tech. 20(6), : 64-74, 1996

  • Moore JW and Flanner HH (1996) Mathematical comparison of dissolution profiles, Pharmaceutical Technology 24:46-54

  • Tsong Y, Hammerstrom T, Sathe P, and Shah VP (1996) Statistical assessment of mean differences between two dissolution data sets, Drug Information Journal 30: 1105-1112

  • Polli JE, Rekhi GS, and Shah V (1996) Methods to compare dissolution profiles, Drug Information Journal 30: 1113-1120.

  • Sathe PM, Tsong Y, Shah VP (1996) In vitro dissolution profile comparion: statistics and analysis, model dependent approach, Pharmaceutical research 13(12): 1799-1803.

  • Polli JE, Rekhi GS, an dShah VP (1996) Methods to compare dissoltuion profiles and a rationale for wide dissoltuion specifications for metroprolol tartrate tablets j pharm Sci 86:690-700

  • FDA (1997) Guidance for industry: extended release oral dosage forms: development, evaluation, and application of in vitro/ in vivo correlations

  • Food and Drug Administration. Guidance for Industry: Dissolution Testing of Immediate Release Solid Oral Dosage Forms, 1997

  • Food and Drug Administration. Guidance for Industry: SUPAC-MR: Modified Release Solid Oral Dosage Forms. 1997

  • Chow SD and Ki FYC (1997) Statistical comparison between dissoltuion profiles of drug products, Journal of Biopharmaceutical statistics, 7(30): 241-258

  • Tsong Y, Hammerstrom T, an Chen JJ (1997) Multipoint dissoltuion specification and acceptance sampling rule based on profile modeling an dprincipal component analysis, Journal of biopharmaceutical statistics 7(3) 423-439.

  • Liu JP, Ma MC, Chow SC (1997) Statistical evaluation of similarity factor f2 as a criterion for assessment of similarity etween dissoltuion profiles Drug Info J 31: 1255-1271

  • Ju HL and Liaw S (1997) On the assessment of similarity of drug dissolution profile – a simulation study Drug Info J 31 1273-1289

MBSW May 25, 2011

References35 l.jpg
References Equivalence

  • Shah VP, Tsong Y, Sathe P, and Liu J-P (1998) In vitro dissolution profile comparisons – statistics and analysis of the similarity factor f2, Pharm. Res. 15: 889-896, 1998

  • FDA (2000) Guidance for Industry Waiver of In Vivo Bioavailability and Bioequivalence Studies for Immediate- Release Solid Oral Dosage Forms Based on Biopharmaceutics Classification System

  • FDA (2000) Guidance for industry: bioavailability and Bioequivalence studies for orally administered drug products – general considerations

  • Ma M-C, Wang BC, Liu J-P, and Tsong Y (2000) Assessment of similarity between dissolution profiles, Journal of Biopharmaceutical statistics 10(2) 229-249

  • Gohel MC and Panchal MK (2000) Comparison of in vitro dissolution profiles using a novel, model independent approach, Pharmaceutical technology, March, 2000, pp 92-102

  • Gudrun F (2001) Clinical Data Management - Guidelines for the Registration of Pharmaceuticals -- Notes for Guidance, Points to Consider and Related Documents for Drug Approval with Biostatistical Methodology - Guidelines on Dissolution Profile Comparison, Drug Information Journal, Vol. 35(3), pp 865-874

  • FDA (2001) Guidance for industry: statistical approaches to bioequivalence.

  • Eaton ML, Muirhead RJ, Steeno GS (2003) Aspects of the dissolution profile testing problem, Biopharmaceutical Report 11(2) 2-7

  • Senn S (2001) Statistical issues in bioequivalence, Statistics in Medicine 20: 2785-2799

  • Paulo Costa*, Jose´ Manuel Sousa Lobo (2001) Modeling and comparison of dissolution profiles, European Journal of Pharmaceutical Sciences 13, 123–133

  • Chow S-C, Shao j, and Wang H (2003) In vitro bioequivalence testing, Statistics in Medicine 22:55-68

  • Saranadasa H (2001) Defining similarity of dissolution profiles through Hotelling’s T2 statistic, Pharmaceutical Technology Februrary 2001, pp 46-54

  • Tsong Y, Sathe PM, and Shah VP (2003) In vitro dissoltuion profile comparison, pp 456-462, in Encyclopedia of Biopharmaceutical statistics, Marcel Dekker

  • Yi Tsong, Meiyu Shen, Vinod P Shah 2004 Three-stage sequential statistical dissolution testing rules. J Biopharm Stat Vol. 14, Issue 3, Pages 757-79

  • Saranadasa H and Krishnamoorthy K (2005) A multivariate test for similarity of two dissolution profiles, Journal of Biopharmaceutical Statistics 15, 265-278

  • EMEA guidance

  • WHO guidance

  • J. Siepmann∗, F. Siepmann (2008) Mathematical modeling of drug delivery, International Journal of Pharmaceutics 364 (2008) 328–343

  • Selen Arzu; Cruañes Maria T; Müllertz Anette; Dickinson Paul A; Cook Jack A; Polli James E; Kesisoglou Filippos; Crison John; Johnson Kevin C; Muirhead Gordon T; Schofield Timothy; Tsong Yi (Profiled Author: Polli, James E.) 2010Meeting report: applied biopharmaceutics and quality by design for dissolution/release specification setting: product quality for patient benefit. Food and Drug Administration, Silver Spring, Maryland, USA The AAPS journal;12(3):465-72

  • Yong Zhang, Meirong Huo, Jianping Zhou, Aifeng Zou, Weize Li, Chengli Yao, and Shaofei Xie (2010) DDSolver: An Add-In Program for Modeling and Comparison of Drug Dissolution ProfilesThe AAPS Journal, Vol. 12, No. 3, 263-271

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Appendix derivation of prior distribution of s i shown on slide 17 l.jpg
Appendix EquivalenceDerivation of prior distribution of si shown on slide 17

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