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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

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

  • 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

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

MBSW May 25, 2011

importance of in vitro dissolution
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.

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measurement of in vitro dissolution


% 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
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





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
The confidence set approach

TOST (one dimensional)







“MOST” (multi-dimensional)




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confidence set approach considerations
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
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)

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proposed bayesian approach
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
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
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
Example 1: Non-informative Prior Information



can be shown (see appendix) to have the distribution

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example 1 definition of equivalence
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
Example 1: Results

500 of 10,000 draws plotted

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example 2 hierarchical model
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
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
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
Example 2: Results

1000 of ~2,000 draws plotted

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example 3 a model dependent comparison
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
Dissolution profile models





( 1st order kinetics )


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

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weibull parameters
Weibull parameters

M measures content

T is time to 63.2% Dissol.

beta measures delay



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weibull parameterization in winbugs
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
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
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
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
Pros and Cons of a Bayesian Approach
  • 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?

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  • 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.
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  • Polli JE, Rekhi GS, and Shah V (1996) Methods to compare dissolution profiles, Drug Information Journal 30: 1113-1120.
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  • 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
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  • FDA (2001) Guidance for industry: statistical approaches to bioequivalence.
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