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

2. 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. 3 Dissolution profiles N=378 tablets from B=10 batches

4. 4 Dissolution Instability

5. 5 FDA Guidance “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.” Guidance for Industry Extended Release Oral Dosage Forms: Development, Evaluation, and Application of In Vitro/In Vivo Correlations CDER, Sept 1997

6. 6 Proposed dissolution limits 14 25 30 60 80

7. 7 USP Drug Release L-20 L-10 L U U+10 U+20 X 12 #(X i ) <3 XiXi L1 (n1=6) XiXi L2 (n2=n1+6) XiXi X 24 L3 (n3=n2+12)

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

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

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

11. 11 Mixed (static) modeling view N tablets ( i ) from B batches ( j ), testing at month x i

12. 12 Hierarchical (dynamic) Modeling view ibatch i xixi yiTyiT 1●●● ● ● 2●●........................ N●● j=1:B Random intercept & slope for each batch: i=1:N Dissolution result for each tablet: Data:

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

14. 14 PD Ve: Acceptable range of 

15. 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. 16 HAR1 or HCS 4 param UN 6 param Tablet residual covariance (V e ) Priors

17. 17 InvWishart Prior Component marginal prior distributions 0.4-31 0.8-54 1.4-98 2.4-164 4.4-299  ij c=1 c=3 c=10 c=30 c=100 ii 40,000 draws

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

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

20. 20 VeVuDIC HCSVC5476.17 HAR1VC5461.98 UN 5457.66 UNVC5456.27 UNVC Common Slope 5499.46 Effect of Covariance Choice: Deviance Information Criterion

21. 21 Parameter Estimates Proc MIXED vs WinBUGS VV VV Ve a b

22. 22 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  WARNING: Posterior sampling is not performed because the parameter transformation is not of full rank.

23. 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. 24 Shrinkage of Bayesian and mixed model batch intercept and slope estimates Intercept (dissolution near batch release %LC) Slope (rate of change in dissolution %LC/month)

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

26. 26 Predicting future results a (1) V  (1) b (1) V  (1) V e (1) ::::: a (d) V  (d) b (d) V  (d) V e (d) ::::: a (10000 ) V  (10000 ) b (10000) V  (10000) V e (10000)  fut (1)  fut (1) ::  fut (d)  fut (d) ::  fut (10000)  fut (10000) y fut,1 (1) …y fut,24 (1) ::: y fut,1 (d) …y fut,24 (d) ::: y fut,1 (10000 ) …y fut,24 (10000) Posterior sample Posterior predictive sample

27. 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. 28 y fut,1 (1) …y fut,24 (1) ::: y fut,1 (d) …y fut,24 (d) ::: y fut,1 (10000 ) …y fut,24 (10000) I(Pass @ L1)I(Pass @ L2)I(Pass @ L3)I(Fail) 0100 :::: 1000 :::: 0001 Pr(Pass @ L1)Pr(Pass @ L2)Pr(Pass @ L3)Pr(Fail) #(Pass @ L1)/ 10000 #(Pass @ L2)/ 10000 #(Pass @ L3)/ 10000 #(Fail)/ 10000 USP Estimate Probabilities Predicting testing results

29. 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. 30 Level testing and failure rate predictions

31. 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. 32 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. Acknowledgements David.LeBlond@abbott.com Thank you too!