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## The Posterior Probability of Dissolution Equivalence

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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,david.leblond@abbott.com 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 MBSW May 25, 2011**Objective**• Make this tool available to you so you can use it if you want to. • Statistical Modeling • Software (R, WinBUGS) • Example Data & Code • david.leblond@abbott.com MBSW May 25, 2011**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**100**% 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”? MBSW May 25, 2011**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. ? MBSW May 25, 2011**D2**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) MBSW May 25, 2011**The confidence set approach**TOST (one dimensional) 5% 5% 8% 2% Yes No “MOST” (multi-dimensional) No Maybe Yes MBSW May 25, 2011**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. MBSW May 25, 2011**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**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”) MBSW May 25, 2011**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%) MBSW May 25, 2011**Example 1: Is the Reference lot equivalent to the Test lot?**6 tablets per lot MBSW May 25, 2011**Example 1: Multivariate Dissolution Model**% Dissolution vector, Y, for the ith tablet from the kth lot … MBSW May 25, 2011**Example 1: Non-informative Prior Information**MBSW May 25, 2011**Example 1: Non-informative Prior Information**MBSW May 25, 2011**Example 1: Non-informative Prior Information**and Since can be shown (see appendix) to have the distribution MBSW May 25, 2011**Example 1: Non-informative Prior Information**MBSW May 25, 2011**Example 1: Definition of Equivalence**Define a rectangular similarity region, S, as and require that to conclude equivalence. MBSW May 25, 2011**Example 1: Results**500 of 10,000 draws plotted MBSW May 25, 2011**Example 2: Equivalence of 2 processes**MBSW May 25, 2011**Example 2: Hierarchical Model**% Dissolution vector, y, for the ith tablet from the kth run … MBSW May 25, 2011**Example 2: Non-informative prior information**elements of Vtablet elements of Vrun Max = 40 MBSW May 25, 2011**Example 2:Definition of Equivalence(same as Example 1)**Define a rectangular similarity region, S, as and require that to conclude equivalence. MBSW May 25, 2011**Example 2: Results**1000 of ~2,000 draws plotted MBSW May 25, 2011**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 MBSW May 25, 2011**Dissolution profile models**Probit: Logistic: Weibull: Exponential: ( 1st order kinetics ) Quadratic: …and some others (Higuchi, Gompertz, Hixson-Crowell,…) MBSW May 25, 2011**Weibull parameters**M measures content T is time to 63.2% Dissol. beta measures delay 0.5 2.0 MBSW May 25, 2011**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 MBSW May 25, 2011**Nonlinear mixed model in WinBUGS**% Dissolution, Y, for the ith tablet from the kth lot at the jth time (t) point… MBSW May 25, 2011**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 MBSW May 25, 2011**Weibull Example: Posterior Probability of Dissolution**Equivalence Prob = 0 2SD Similarity Region Prob = 0.949 MBSW May 25, 2011**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? MBSW May 25, 2011**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 MBSW May 25, 2011**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 MBSW May 25, 2011**AppendixDerivation of prior distribution of si shown on**slide 17 MBSW May 25, 2011

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