Dissolution stability of a modified release product 32 nd MBSW May 19, 2009

1. Dissolution stability of a modified release product 32nd MBSW May 19, 2009 David.LeBlond@abbott.com

2. Outline • Multivariate data set • Mixed model (static view) • Hierarchical model (dynamic view) • Why a Bayesian approach? • Selecting priors • Model selection • Parameter estimates • Latent parameter (“BLUP”) estimates • Posterior prediction • Estimating future batch failure and level testing rates

3. Dissolution profilesN=378 tablets from B=10 batches

4. Dissolution Instability

5. FDA Guidance Guidance for Industry Extended Release Oral Dosage Forms: Development, Evaluation, and Application of In Vitro/In Vivo Correlations CDER, Sept 1997 • “VII.B. Setting Dissolution Specifications • A minimum of three time points … • … should cover the early, middle, and late stages of the dissolution profile. • The last time point … at least 80% of drug has dissolved …. [or] … when the plateau of the dissolution profile has been reached.”

6. Proposed dissolution limits 80 60 30 25 14

7. L1 (n1=6) Xi X12 X24 L2 (n2=n1+6) Xi Xi L3 (n3=n2+12) #(Xi)<3 USP <724> Drug Release L-20 L-10 L U U+10 U+20

8. Tablet residuals from fixed model:Correlation among time points r = 0.36 r = 0.79 r = 0.54 All p-values < 0.0001

9. Batch slopes:Correlations among time points r = 0.76 p = 0.01 r = 0.21 p = 0.57 r = -0.37 p = 0.30

10. Batch intercepts:Correlations among time points r = 0.92 p = 0.0002 r = 0.83 p = 0.003 r = 0.65 p = 0.04

11. Mixed (static) modeling viewN tablets (i) from B batches (j), testing at month xi

12. Random intercept & slope for each batch: j=1:B Dissolution result for each tablet: i=1:N Hierarchical (dynamic) Modeling view Data:

13. UN 6 param HAR1 4 param HCS 4 param Tablet residual covariance (Ve)

14. PD Ve: Acceptable range of r

15. Why a Bayesian approach? • Asymptotic approximations may not be valid • Allows quantification of prior information • Properly accounts for estimation uncertainty • Lends itself to dynamic modeling viewpoint • Requires fewer mathematical distractions • Estimates quantities of interest easily • Provides distributional estimates • Fewer embarrassments (e.g., negative variance estimates) • Is a good complement to likelihood (only) methods • WinBUGS is fun to use

16. UN 6 param HAR1 or HCS 4 param Tablet residual covariance (Ve) Priors

17. c=1 c=3 c=10 c=30 c=100 0.4-31 0.8-54 1.4-98 2.4-164 4.4-299 si InvWishart PriorComponent marginal prior distributions 40,000 draws rij

18. UN 12 params VC 6 params Batch intercept & slope covariance (Vu)

19. Process mean 6 param VC 6 param VC Common slope 3 param Batch intercept & slope Priors UN 12 param

20. Effect of Covariance Choice:Deviance Information Criterion

21. a b Va Vb Ve Parameter EstimatesProc MIXED vs WinBUGS

22.  WARNING: Posterior sampling is not performed because the parameter transformation is not of full rank. Posterior from Proc Mixed(SAS 8.2) 391 proc mixed covtest; 392 class batch tablet time; 393 model y= time time*month/ noint s; 394 random time time*month/ type=un(1) subject=batch G s; 395 repeated / type=un subject=tablet R; 396 prior /out=posterior nsample=1000; NOTE: Convergence criteria met. • Runs in SAS 9.2, however… • SAS only strictly “supports” the posterior if • random type=VC with no repeated, or • random and repeated types both = VC

23. WinBUGS dynamic modeling # Prior InvVe[1:T,1:3]~dwish(R[,],3) acent[1]~dnorm(0.0,0.0001) acent[2]~dnorm(50,0.0001) acent[3]~dnorm(100,0.0001) for ( j in 1:3) { b[ j ]~dnorm(0.0,0.001) gacent[ j ]~dgamma(0.001,0.001) gb[ j ]~dgamma(0.001,0.001) } # Likelihood # Draw the T intercepts and slopes for each batch for ( i in 1:B) { for ( j in 1:3) { alpha[i, j] ~ dnorm(acent[ j ], gacent[ j ]) beta[i, j] ~ dnorm(b[ j ], gb[ j ]) } } # Draw vector of results from each tablet for (obs in 1:N){ for ( j in 1:3){ mu[obs,j]<-alpha[Batch[obs],j]+beta[Batch[obs],j]*(Month[obs]-xbar)} y[obs,1:T ]~dmnorm(mu[obs, ], InvVe[ , ])}

24. Intercept (dissolution near batch release %LC) Slope (rate of change in dissolution %LC/month) Shrinkage of Bayesian and mixed model batch intercept and slope estimates

25. Intercepts Slopes WinBUGS Batch intercept and slope estimates: Bayesian “BLUPs”

26. Predicting future results Posterior predictive sample Posterior sample

27. WinBUGS posterior predictions # Predict int & slope for future batches for (j in 1:3){ b_star[ j ]~dnorm(b[ j ], gb[ j ]) acent_pred[ j ]~dnorm(acent[ j ], gacent[ j ]) a_star[ j ]<-acent[ j ] - b[ j ]*xbar} # Obtain the Ve components Ve[1:3,1:3] <- invVe[ , ]) for (j in 1:3){ sigma[ j ] <- sqrt(Ve[j,j])} rho12 <- Ve[1,2]/sigma[1]/sigma[2] rho13 <- Ve[1,3]/sigma[1]/sigma[3] rho23 <- Ve[2,3]/sigma[2]/sigma[3]

28. Predicting testing results Estimate Probabilities USP <724>

29. Semi-parametric bootstrap prediction • “Fixed model” prediction (no shrinkage) • 10 intercept and 10 slope vectors via SLR • 378 tablet residual vectors • -or- • “Mixed model” prediction (shrinkage) • 10 intercept vector BLUPs • 10 slope vector BLUPs • 378 tablet residual vectors • Sample with replacement to construct future results

30. Level testing and failure rate predictions

31. Summary • A multivariate, hierarchical, Bayesian approach to dissolution stability illustrated • Some options for specifying the covariance priors • Estimation and shrinkage of the latent batch slope and intercept parameters • Posterior prediction of future data • Prediction of future failure and level testing rates • “Fixed” most pessimistic… (no shrinkage?) • “Mixed” lowest failure rate… (non-asymptotic?) • Give WinBUGS a try

32. Thank you too! David.LeBlond@abbott.com Acknowledgements The invaluable suggestions of, encouragement from, and helpful discussions with John Peterson, GSK Oscar Go, J&J Jyh-Ming Shoung, J&J Stan Altan, J&J are greatly appreciated.