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

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- 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

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.”

80

60

30

25

14

L1 (n1=6)

Xi

X12

X24

L2 (n2=n1+6)

Xi

Xi

L3 (n3=n2+12)

#(Xi)<3

L-20 L-10 L U U+10 U+20

r = 0.36

r = 0.79

r = 0.54

All p-values < 0.0001

r = 0.76

p = 0.01

r = 0.21

p = 0.57

r = -0.37

p = 0.30

r = 0.92

p = 0.0002

r = 0.83

p = 0.003

r = 0.65

p = 0.04

Random intercept & slope for each batch:

j=1:B

Dissolution result for each tablet:

i=1:N

Data:

UN

6 param

HAR1

4 param

HCS

4 param

- 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

UN

6 param

HAR1 or HCS

4 param

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

40,000 draws

rij

UN

12 params

VC

6 params

Process mean

6 param

VC

6 param

VC Common slope

3 param

UN

12 param

a

b

Va

Vb

Ve

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

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

# 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[ , ])}

Intercept (dissolution near batch release %LC)

Slope (rate of change in dissolution %LC/month)

Intercepts

Slopes

Posterior predictive sample

Posterior sample

# 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]

Estimate Probabilities

USP <724>

- “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

- 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

Thank

you too!

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.