dissolution stability of a modified release product 32 nd mbsw may 19, 2009 [email protected]
TRANSCRIPT
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OutlineOutline• 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
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3Hour
Me
an
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0 6
2 4 6 8
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18 24
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2 4 6 8
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Batch12345678910
Dissolution profilesN=378 tablets from B=10 batches
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4Month
Me
an
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0 10 20 30 40 50
1 2
0 10 20 30 40 50
3.5
5
0 10 20 30 40 50
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Batch12345678910
Dissolution Instability
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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 IndustryExtended Release Oral Dosage Forms:
Development, Evaluation, andApplication of In Vitro/In Vivo
CorrelationsCDER, Sept 1997
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2 4 6 8
20
40
60
80
10
0
Hours + jitter
% D
isso
lutio
n
Proposed dissolution limits
14
2530
60
80
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USP <724> Drug Release
L-20 L-10 L U U+10 U+20
X12
#(Xi)<3
XiL1 (n1=6)
Xi
L2 (n2=n1+6)
Xi
X24
L3 (n3=n2+12)
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2hr %LC20
2520 25
15
20
15 20
3.5hr %LC50
6050 60
30
40
30 40
8hr %LC100
105
110100 105 110
85
90
95
85 90 95
All p-values < 0.0001
Tablet residuals from fixed model:Correlation among time points
r = 0.79
r = 0.36
r = 0.54
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2hr Slope0.07
0.080.07 0.08
0.05
0.06
0.05 0.06
3.5hr Slope
0.20
0.250.20 0.25
0.10
0.15
0.10 0.15
8hr Slope0.1
0.20.1 0.2
-0.1
0.0
-0.1 0.0
Batch slopes:Correlations among time points
r = 0.21
p = 0.57
r = -0.37
p = 0.30
r = 0.76
p = 0.01
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2hr Initial19
20
21
19 20 21
16
17
18
16 17 18
3.5hr Initial44
46
4844 46 48
38
40
42
38 40 42
8hr Initial
96
98
10096 98 100
90
92
94
90 92 94
Batch intercepts:Correlations among time points
r = 0.92
p = 0.0002
r = 0.65
p = 0.04
r = 0.83
p = 0.003
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eZuXβy
Mixed (static) modeling viewN tablets (i) from B batches (j), testing at month xi
13
1
16
1
1
6333
33
313
16
633
3
31
3
3
3
13
1
0
0
0
0
0
0
0
0
0
0
0
0
NN
j
BB
B
j
j
BNN
i
NN
i
NN
i
b
a
ba
b
a
IxI
IxI
IxI
b
a
Ix
Ix
Ix
I
I
I
y
y
y
uB VI 0,MVN~u eN VI 0,MVN~e
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Hierarchical (dynamic) Modeling view
i batchi xi yiT
1 ● ● ● ● ●
2 ● ● ● ● ●
.
.
.
.
.
.
.
.
.
.
.
.
N ● ● ● ● ●
j=1:B VaMVNj ,~ 3
VbMVNj ,~ 3 660
0
V
VVu
Random intercept & slope for each batch:
iibatchbatchi exyii
ei VMVNe ,0~ 3i=1:N
Dissolution result for each tablet:
Data:
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3
2
1
3
2
1
00
00
00
1
1
1
00
00
00
HCSHCS
4 param4 param
3
2
1
2
2
3
2
1
00
00
00
1
1
1
00
00
00
HAR1HAR14 param
3
2
1
2313
2312
1312
3
2
1
232313
122212
231221
00
00
00
1
1
1
00
00
00
UNUN
6 param
Tablet residual covariance (Ve)
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PD Ve: Acceptable range of
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
rho
det
erm
inan
t
HCS
HAR1
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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
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HAR1for 999.0,999.0
HCSfor )999.0,499.0(~
3,2,1),001.0,001.0(~2
Unif
Unif
kInvGammak
HAR1 or HCS
4 param
3,30~ 3232313
2212
21
IInvWishart
sym
UN
6 param
Tablet residual covariance (Ve) Priors
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InvWishart PriorComponent marginal prior distributions
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
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
0 4 8 12 16 20 24 28 32 36 40
0 4 8 12 16 20 24 28 32 36 40
0.4-31
0.8-54
1.4-98
2.4-164
4.4-299
3,~ 32332233113
222112
21
IcInvWishart
sym
ij
c=1
c=3
c=10
c=30
c=100
i40,000 draws
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232313
232212
131221
232313
232212
131221
0
0
bbb
bbb
bbb
aaa
aaa
aaa
UN12 params
23
22
21
23
22
21
00
00
00
0
0
00
00
00
b
b
b
a
a
a
VC6 params
Batch intercept & slope covariance (Vu)
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Batch intercept & slope Priors
3,~,3,~ 3232313
2212
21
3232313
2212
21
IcInvWishart
sym
IcInvWishart
sym
bbb
bb
b
aaa
aa
a
UN12 param
3,2,1),001.0,001.0(~
)001.0,001.0(~2
2
kInvGamma
InvGamma
bk
ka
VC6 param
3,2,1),001.0,001.0(~2 kInvGammakaVC Common slope
3 param
33
334
3 10,0~,10,
90
50
20
~ IMVNbIMVNa
Process mean
6 param
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Ve Vu DICHCS VC 5476.17
HAR1 VC 5461.98
UN UN 5457.66
UN VC 5456.27
UN VCCommon
Slope
5499.46
Effect of Covariance Choice:Deviance Information Criterion
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Parameter Estimates Proc MIXED vs WinBUGS
)4.3(0.600
0)7.5(3.110
00)1.1(2.2
)9.2(1.400
0)1.9(6.140
00)2.2(3.3
V
310
)5.8(4.1300
0)9.1(2.10
00)4.0(3.0
310
)8.17(7.2300
0)6.4(6.40
00)8.0(0.1
V
)3.1(4.16)1.1(0.10)4.0(0.3
0)4.1(6.18)4.0(5.4
00)2.0(7.2
)2.1(4.16)1.1(8.9)4.0(0.3
0)4.1(5.18)4.0(5.4
00)2.0(8.2
Ve
)9.0(0.94
)1.1(3.41
)5.0(4.17
)9.0(0.94
)3.1(3.41
)6.0(4.17
a
210
)2.4(5.7
)3.2(7.16
)9.0(1.7
210
)2.5(2.8
)9.2(9.16
)3.1(2.7
b
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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.
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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[ , ])}
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Shrinkage of Bayesian and mixed Shrinkage of Bayesian and mixed model batch intercept and slope model batch intercept and slope estimatesestimates
2h
r
15
16
17
18
19
20
21
Bayesian Fixed Model Mixed Model
Estimation Method
3.5
h36
39
42
45
48
Bayesian Fixed Model Mixed Model
Estimation Method
8h
r
85
90
95
100
105
Bayesian Fixed Model Mixed Model
Estimation Method
Intercept (dissolution near batch release %LC)
2h
r
0
0.02
0.04
0.06
0.08
0.1
0.12
Bayesian Fixed Model Mixed Model
Estimation Method
3.5
h
0
0.05
0.1
0.15
0.2
0.25
0.3
Bayesian Fixed Model Mixed Model
Estimation Method
8h
r
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Bayesian Fixed Model Mixed Model
Estimation Method
Slope (rate of change in dissolution %LC/month)
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WinBUGS Batch intercept and slope WinBUGS Batch intercept and slope estimates: Bayesian “BLUPs”estimates: Bayesian “BLUPs”
[1,1]
[2,1][3,1] [4,1]
[5,1]
[6,1]
[7,1]
[8,1]
[9,1]
[10,1]
box plot: Init[,1]
14.0
16.0
18.0
20.0
22.0[1,2]
[2,2]
[3,2] [4,2]
[5,2]
[6,2]
[7,2]
[8,2]
[9,2]
[10,2]
box plot: Init[,2]
35.0
40.0
45.0
50.0
[1,3]
[2,3] [3,3]
[4,3]
[5,3]
[6,3]
[7,3]
[8,3]
[9,3] [10,3]
box plot: Init[,3]
85.0
90.0
95.0
100.0
105.0
Inte
rcepts
[1,1]
[2,1]
[3,1]
[4,1]
[5,1][6,1]
[7,1]
[8,1]
[9,1]
[10,1]
box plot: slope[,1]
0.0
0.05
0.1
0.15
[1,2] [2,2]
[3,2] [4,2] [5,2]
[6,2][7,2]
[8,2]
[9,2]
[10,2]
box plot: slope[,2]
-0.1
0.0
0.1
0.2
0.3
0.4
[1,3]
[2,3]
[3,3]
[4,3]
[5,3]
[6,3]
[7,3]
[8,3]
[9,3] [10,3]
box plot: slope[,3]
-0.4
-0.2
0.0
0.2
0.4
Slo
pes
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Predicting future resultsPredicting future results
a(1) V(1) b(1) V
(1) Ve(1)
: : : : :
a(d) V(d) b(d) V
(d) Ve(d)
: : : : :
a(10000) V(10000) b(10000) V
(10000) Ve(10000)
fut(1) fut (1)
: :
fut (d) fut (d)
: :
fut (10000) fut (10000)
yfut,1(1) … yfut,24
(1)
: : :
yfut,1(d) … yfut,24
(d)
: : :
yfut,1(10000) … yfut,24
(10000)
Posterior sample Posterior predictive sample
)()()(3
)(, ,~ d
edfut
dfut
difut VxMVNy
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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]
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yfut,1(1) … yfut,24
(1)
: : :
yfut,1(d) … yfut,24
(d)
: : :
yfut,1(10000) … yfut,24
(10000)
I(Pass @ L1) I(Pass @ L2) I(Pass @ L3) I(Fail)
0 1 0 0
: : : :
1 0 0 0
: : : :
0 0 0 1
Pr(Pass @ L1) Pr(Pass @ L2) Pr(Pass @ L3) Pr(Fail)
#(Pass @ L1)/
10000
#(Pass @ L2)/
10000
#(Pass @ L3)/
10000
#(Fail)/
10000
USP <724>Estimate Probabilities
Predicting testing results
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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
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Level testing and failure rate predictionsLevel testing and failure rate predictions
60
65
70
75
80
85
90
95
100
0 6 12 18 24 30 36 42 48
Months of Storage
Pro
ba
bil
ity
of
Pa
ss
ing
at
Le
ve
l 1
(%
)
Mixed Model
Fixed Model
Bayesian
0
5
10
15
20
25
30
35
0 6 12 18 24 30 36 42 48
Months of Storage
Pro
bab
ilit
y o
f P
assi
ng
at
Lev
el 2
(%
)
Mixed Model
Fixed Model
Bayesian
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
0 6 12 18 24 30 36 42 48
Months of Storage
Pro
bab
ilit
y o
f P
assi
ng
at
Lev
el 3
(%
)
Mixed Model
Fixed Model
Bayesian
0
2
4
6
8
10
12
14
16
0 6 12 18 24 30 36 42 48
Months of Storage
Pro
bab
ilit
y o
f F
aili
ng
Dis
solu
tio
n
Tes
tin
g
Mixed Model
Fixed Model
Bayesian
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SummarySummary
• 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
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The invaluable suggestions of, encouragement from, and helpful discussions with
John Peterson, GSKOscar Go, J&JJyh-Ming Shoung, J&JStan Altan, J&J
are greatly appreciated.
AcknowledgementsAcknowledgements
Thankyou too!