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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

Midwest Biopharmaceutical Statistical Workshop

Muncie, Indiana

May 25, 2011

Outline

- 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

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Objective

- 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]

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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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% Dissolved

0

Time

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

- 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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0

D1

0

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

TOST (one dimensional)

5%

5%

8%

2%

Yes

No

“MOST” (multi-dimensional)

No

Maybe

Yes

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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 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

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

Probability Model

(Likelihood)

Dissolution Data

(Test and Reference)

Prior Information

(non-informative)

MCMC

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

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

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Example 1: Non-informative Prior Information

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Example 1: Non-informative Prior Information

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Example 1: Non-informative Prior Information

and

Since

can be shown (see appendix) to have the distribution

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Example 1: Non-informative Prior Information

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Example 1: Definition of Equivalence

Define a rectangular similarity region, S, as

and require that

to conclude equivalence.

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Example 2: Equivalence of 2 processes

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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

elements of Vtablet

elements of Vrun

Max = 40

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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 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

Probit:

Logistic:

Weibull:

Exponential:

( 1st order kinetics )

Quadratic:

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

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Weibull parameters

M measures content

T is time to 63.2% Dissol.

beta measures delay

0.5

2.0

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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

% 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

“…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

Prob = 0

2SD Similarity Region

Prob = 0.949

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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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References

- 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

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References

- 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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AppendixDerivation of prior distribution of si shown on slide 17

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