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Transcript presentation_David5-26-2009-7-34-37. ppt
Dissolution stability
of a modified release product
32nd MBSW
May 19, 2009
[email protected]
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
2
Dissolution profiles
N=378 tablets from B=10 batches
2
4
6
36
48
18
24
8
100
80
60
40
20
Batch
30
100
Mean
80
60
40
20
0
6
12
100
1
2
3
4
5
6
7
8
9
10
80
60
40
20
2
4
6
8
2
Hour
4
6
8
3
Dissolution Instability
0
5
10 20
30 40
50
8
100
80
60
Batch
40
1
2
3
4
5
6
7
8
9
10
Mean
20
1
2
3.5
100
80
60
40
20
0
10
20 30
40 50
0
Month
10 20
30 40
50
4
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.”
5
100
Proposed dissolution limits
40
60
60
25
20
% Dissolution
80
80
30
14
2
4
Hours + jitter
6
8
6
USP <724> Drug Release
L-20
L1 (n1=6)
L2 (n2=n1+6)
L-10
L
U
U+10
U+20
Xi
X12
Xi
X24
L3 (n3=n2+12)
Xi
#(Xi)
<3
7
Tablet residuals from fixed model:
Correlation among time points
110
100 105 110
105
r = 0.36
100
8hr %LC
95
90
85 90 95
50
85
60
60
r = 0.79
50
r = 0.54
3.5hr %LC
40
30
25
20
15
20
2hr %LC
20
40
30
25
20
15
All p-values < 0.0001
8
Batch slopes:
Correlations among time points
0.1
0.2
0.2
0.1
8hr Slope
0.0
-0.1
-0.1
0.25
0.20
0.25
0.20
r = 0.21
p = 0.57
3.5hr Slope
0.10
0.07
0.08
0.08
0.07
2hr Slope
0.15
0.0
0.15
r = 0.76
p = 0.01
0.10
r = -0.37
p = 0.30
0.06
0.05
0.06
0.05
9
Batch intercepts:
Correlations among time points
100
96
98
98 100
96
8hr Initial
94
92
90 92
48
r = 0.92
p = 0.0002
44
46
48
46
44
3.5hr Initial
42
94
90
r = 0.83
p = 0.003
40
38
40
42
38
21
19
20
20
19
2hr Initial
18
21
r = 0.65
p = 0.04
17
16
17
18
16
10
Mixed (static) modeling view
N tablets (i) from B batches (j), testing at month xi
y Xβ Zu e
y1
I3
y
I3
i
y
N 3 N 1 I 3
x1 I 3
I3
a
0
xi I 3
b 61
0
x N I 3 3 N 6
x1 I 3 0
0 I3
0
0
0
xi I 3
0
0
0
I3
a1
b1
0
1
a
j
0
j
b
j
x N I 3 3 N 6 B
N 3 N 1
aB
b
B 6 B1
u ~ MVN 0,I B Vu e ~ MVN 0, I N Ve
11
Hierarchical (dynamic) Modeling view
Data:
i
batchi
xi
yi T
1
●
●
●●●
2
●
●
●●●
.
.
.
.
.
.
.
.
.
.
.
.
N
●
●
●●●
Random intercept & slope for each batch:
j ~ MVN 3 a,V
j=1:B
j ~ MVN 3 b,V
V
Vu
0
Dissolution result for each tablet:
i=1:N
ei ~ MVN3 0,Ve
yi batch batch xi ei
i
i
12
0
V
66
Tablet residual covariance (Ve)
UN
6 param
HAR1
4 param
HCS
4 param
12 12 23 1 0 0 1
2
0
0
12
12
2
12
2
2
13
23
3
0 0 3 13
23
1 0 0 1
0 2 0
0 0 2
3
0
0
3
0 1
1 0
0 2 0
0 0
3
1
2 1 0
0 2
0
1 0
1
1 0
0 2
0
1 0
12
1
13 1 0
23 0 2
1 0
0
0
0
3
13
0
0
3
PD Ve: Acceptable range of
determinant
1
0.8
0.6
HCS
HAR1
0.4
0.2
0
-1
-0.8 -0.6 -0.4 -0.2
0
0.2
0.4
0.6
0.8
1
rho
14
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
15
Tablet residual covariance (Ve) Priors
UN
6 param
HAR1 or HCS
4 param
12
12 22
13 23
sym
~ InvWishart30 I 3 ,3
32
k2 ~ InvGam m a(0.001,0.001), k 1,2,3
Unif (0.499,0.999) for HCS
~
Unif 0.999,0.999 for HAR1
16
InvWishart Prior
Component marginal prior distributions
12
12 1 2
13 1 3
22
23 2 3
sym
~ InvWishartc I 3 ,3
32
40,000 draws
i
-0.8 -0.6 -0.4 -0.2
c=3
c=10
0.4-31
0
4
8
12 16 20 24 28 32 36 40
0
4
8
12 16 20 24 28 32 36 40
0
4
8
12 16 20 24 28 32 36 40
c=30
ij
-1
c=1
4
8
c=100
0.2
0.4
0.6
0.8
1.4-98
2.4-164
0
0
0.8-54
12 16 20 24 28 32 36 40
4.4-299
1
0
4
8
12 16 20 24 28 32 36 40
17
Batch intercept & slope covariance (Vu)
UN
12 params
a21 a12 a13
2
0
a12 a 2 a 23
2
a 23
a3
a13
2
b1 b12 b13
0
b12 b22 b 23
2
b13 b 23 b 3
VC
6 params
a21 0
0
2
0
0 a2 0
0
2
0
a3
2
b1 0
0
0
0 b22 0
2
0
0 b3
18
Batch intercept & slope Priors
Process mean
6 param
UN
12 param
a21
a12 a22
a13 a 23
VC
6 param
VC Common slope
3 param
20
4
a ~ MVN 3 50,10 I 3 , b ~ MVN 3 0,103 I 3
90
b21
sym
~ InvWishartc I 3 ,3, b12 b22
a23
b13 b 23
sym
~ InvWishartc I 3 ,3
b23
a2k ~ InvGam m a(0.001,0.001)
2bk ~ InvGam m a(0.001,0.001), k 1,2,3
a2k ~ InvGamma(0.001,0.001), k 1,2,3
19
Effect of Covariance Choice:
Deviance Information Criterion
Ve
Vu
DIC
HCS
VC
5476.17
HAR1
VC
5461.98
UN
UN
5457.66
UN
VC
5456.27
UN
VC
5499.46
Common Slope
20
Parameter Estimates
Proc MIXED vs WinBUGS
a
17.4(0.5)
41.3(1.1)
94.0(0.9)
17.4(0.6)
41.3(1.3)
94.0(0.9)
b
7.1(0.9)
2
16
.
7
(
2
.
3
)
10
7.5(4.2)
7.2(1.3)
2
16.9(2.9) 10
8.2(5.2)
V
0
0
2.2(1.1)
0
11
.
3
(
5
.
7
)
0
0
0
6.0(3.4)
0
0
3.3(2.2)
0
14
.
6
(
9
.
1
)
0
0
0
4.1(2.9)
V
0
0
0.3(0.4)
3
0
1
.
2
(
1
.
9
)
0
10
0
0
13.4(8.5)
0
0
1.0(0.8)
3
0
4
.
6
(
4
.
6
)
0
10
0
0
23.7(17.8)
Ve
0
0
2.7(0.2)
4
.
5
(
0
.
4
)
18
.
6
(
1
.
4
)
0
3.0(0.4) 10.0(1.1) 16.4(1.3)
0
0
2.8(0.2)
4
.
5
(
0
.
4
)
18
.
5
(
1
.
4
)
0
3.0(0.4) 9.8(1.1) 16.4(1.2)
21
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.
WARNING: Posterior sampling is not performed because the
parameter transformation is not of full rank.
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
22
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[ , ])}
23
Shrinkage of Bayesian and mixed model
batch intercept and slope estimates
Intercept (dissolution near batch release %LC)
105
20
48
19
45
18
17
8hr
100
3.5h
2hr
21
42
39
95
90
16
36
15
85
Bayesian Fixed Model Mixed Model
Bayesian Fixed Model Mixed Model
Bayesian Fixed Model Mixed Model
Estimation Method
Estimation Method
Estimation Method
0.3
0.4
0.1
0.25
0.3
0.08
0.2
0.2
0.06
0.15
0.04
0.1
0.02
0.05
0
8hr
0.12
3.5h
2hr
Slope (rate of change in dissolution %LC/month)
0.1
0
-0.1
-0.2
0
Bayesian Fixed Model Mixed Model
Estimation Method
Bayesian Fixed Model Mixed Model
Estimation Method
Bayesian Fixed Model Mixed Model
Estimation Method
24
WinBUGS Batch intercept and slope
estimates: Bayesian “BLUPs”
Intercepts
box plot: Init[,1]
box plot: Init[,2]
22.0
50.0
box plot: Init[,3]
[1,2]
105.0
[8,2]
[8,1]
[1,3]
20.0
[1,1]
100.0
[6,2]
45.0
[6,1]
[7,1]
18.0
[2,1]
[3,1]
[4,1]
[3,2]
[5,1]
[4,2]
[9,2]
[5,3]
40.0
16.0
90.0
14.0
35.0
box plot: slope[,1]
85.0
box plot: slope[,3]
box plot: slope[,2]
[8,1]
Slopes
0.15
0.4
0.4
[3,2]
[3,1]
[1,1]
[7,1]
[9,1]
[4,2]
[6,2]
[5,1]
[1,2]
[6,1]
[2,1]
[10,1]
0.2
[2,2]
[5,3]
[7,3]
[3,3]
[9,2]
[5,2]
[9,3]
0.3
[4,1]
0.1
[4,3]
0.2
[7,2]
[6,3]
[2,3]
[8,2]
[10,2]
[1,3]
0.0
[8,3]
0.1
0.05
-0.2
0.0
0.0
[10,3]
[4,3]
[7,3]
[7,2]
[5,2]
[2,2]
[9,3]
[3,3]
95.0
[10,1]
[9,1]
[8,3]
[6,3]
[2,3]
[10,2]
-0.1
-0.4
25
[10,3]
Predicting future results
Posterior predictive sample
Posterior sample
a(1)
V(1)
b(1)
V(1)
Ve(1)
fut(1)
fut (1)
yfut,1(1)
…
yfut,24(1)
:
:
:
:
:
:
:
:
:
:
a(d)
V(d)
b(d)
V(d)
Ve(d)
fut (d)
fut (d)
yfut,1(d)
…
yfut,24(d)
:
:
:
:
:
:
:
:
:
:
a(10000
V(10000
b(10000)
V(10000)
Ve(10000)
fut (10000)
fut (10000)
)
)
yfut,1(10000 … yfut,24(10000)
)
y(futd ),i ~ MVN3 (futd ) x (futd ) ,Ve(d )
26
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]
27
Predicting testing results
I(Pass @ L1)
I(Pass @ L2)
I(Pass @ L3)
I(Fail)
yfut,1(1)
…
yfut,24(1)
0
1
0
0
:
:
:
:
:
:
:
yfut,1(d)
…
yfut,24(d)
1
0
0
0
:
:
:
:
:
:
:
0
0
0
1
yfut,1(10000 … yfut,24(10000)
)
USP <724>
Estimate Probabilities
Pr(Pass @ L1)
Pr(Pass @ L2)
Pr(Pass @ L3)
Pr(Fail)
#(Pass @ L1)/
10000
#(Pass @ L2)/
10000
#(Pass @ L3)/
10000
#(Fail)/
10000
28
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
29
Probability of Passing at Level 2 (%)
Level testing and failure rate predictions
Probability of Passing at Level 1
(%)
100
95
90
85
80
Mixed Model
75
Fixed Model
70
Bayesian
65
60
0
6
12
18
24
30
36
42
35
Mixed Model
30
Fixed Model
25
Bayesian
20
15
10
5
0
0
48
6
12
Probability of Failing Dissolution
Testing
Probability of Passing at Level 3 (%)
2.5
2.25
2
Mixed Model
1.5
Fixed Model
Bayesian
1.25
1
0.75
0.5
0.25
0
0
6
12
18
24
30
Months of Storage
24
30
36
42
48
Months of Storage
Months of Storage
1.75
18
36
42
48
16
14
12
Mixed Model
10
Fixed Model
8
Bayesian
6
4
2
0
0
6
12
18
24
30
36
Months of Storage
30
42
48
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
31
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.
Thank
you too!
[email protected]
32