ASSESSING RELEASE LIMITS AND
MANUFACTURING RISK FROM A
BAYESIAN PERSPECTIVE
1
Areti Manola
amanola@its.jnj.com
OUTLINE
 Introduction
Review Q1E Stability Evaluation
 Definition of Release Limits



Allen, Dukes and Gerger Approach
Mixed Linear Model
A
Bayesian Approach to Manufacturing
Risk Estimation


Bayesian formulation of mixed model
Simulating future lots

Posterior predictive distribution
 Case
Studies
 Summary
2
ICH Q1E (2003) – STABILITY EVALUATION

Summary Points
A confidence level of 95% (one/two-sided at mean) is
recommended for shelf life calculation.
 Shelf lives for individual batches should first be
estimated using individual intercepts, individual
slopes and the pooled mean square error calculated
from all batches.
 Use shortest individual estimate for set(s) of batches
 Statistical test for batch poolability can be performed
using a level of significance of 0.25.


Comments


No definition or recommendations for release limits
calculations
Current technologies allow mixed models and
Bayesian approaches
3
DEFINITION OF RELEASE LIMITS
Specifications ensure that the identity, strength,
quality, and purity of a drug product are
maintained throughout its shelf life.
Release limits are the bounds of intervals on the
true lot mean formed on the basis of given
specifications and real time stability data so that a
future lot whose measured value at time of
manufacture falls within these limits has a high
level of assurance that its mean will remain within
specifications throughout shelf life.
4
RELEASE LIMITS
Internally derived and are the responsibility of
the manufacturer, lot acceptance limits
 Apply only at time of lot release
 Account for changes over time and uncertainties
due to process variability
 Intended to provide a high level of assurance that
a lot falling within release limits will conform to
quality requirements over the shelf life of the
product
 Important to the customer

Given Release Limits and
Specifications how can we
assess manufacturing risk?
5
ALLEN, DUKES, & GERGER (ADG)
- DETERMINATION OF RELEASE LIMITS: A GENERAL METHODOLOGY
(1991)
linear model:
yi  A  B  Ti   i
LRL  LSL  B  TSL  t1a ,k 
LRL
LSL
B
TSL
ST
S
t1-a,k
n
2
S
ST2 
for B 0
n
= lower release limit
= lower specification limit
= average slope for degradation
= shelf life
= standard error of average slope  shelf life
= assay standard deviation
= one-sided (1-a)%-ile critical t value with k
degrees of freedom
= number of replicate assays used for lot
release
105%
Percent of Label
Consider a fixed batch-specific
A recent poll of 8 companies found that the ADG
approach was used for either primary or secondary
calculations of release limits by all 8
100%
95%
Release
Release Assay +
Regression Loss Uncertainty
Regression Loss
90%
85%
0
6
12
18
Time (Months)
24
6
OTHER APPROACHES

Shao and Chow (1991)
Constructing Release Targets for Drug Products: a Bayesian
Decision Theory Approach
Various choices of release limits are viewed as part of an action
space; an action is chosen so as to minimize the cost through an
appropriately chosen loss function

Greg C. G. Wei (1998)
Simple Methods for Determination of the Release Limits for Drug
Products
“Conditional “ release limits (control the chance of failure for a given
lot - similar to Allen’s method) and “Unconditional” release limits
(control the chance of failure for all future lots); expected loss
function approach that minimizes cost due to lot failures at T0 & TSL

Murphy and Hofer (2002)
Establishing Shelf life, Expiry Limits and Release Limits
Conditions shelf life on control limits at time of release
7
MIXED LINEAR MODEL RANDOM INTERCEPT
yijk  A  ai  B j  Tijk  ijk
yijk = measurement of i-th batch at j-th condition and k-th time
point,
A = overall mean corresponding to process average at time 0,
αi = random i-th batch effect on intercept: αi ~ N(0 , α2),
Bj = fixed rate of change at j-th condition,
Tijk = k-th sampling time for batch i at j-th condition,
ijk = residual error: ijk ~ N(0, ε2).
Possible to decompose the residual error further into other variance
components if the design of the study permits, e.g. common
analytical runs for specific groups of lots
8
Note: Extend Allen’s approach to mixed modeling framework by
including additional variance terms
MANUFACTURING RISK ESTIMATION

Given Release Limits and Specifications,
manufacturing risk can be described through a
2x2 table given below:
End of Shelf Life

Release
Pass (%) Fail (%) Total (%)
Pass (%)
C11
C12
R1
Fail (%)
C21
C22
R2
Total (%)
C1
C2
100
C12/R1=
P(YSL<SpecSL|Y0≥RL)
cost to the
company
Probabilities associated with the above 2x2 table
can be estimated through a Bayesian posterior
predictive distribution approach
9
A BAYESIAN FORMULATION OF THE MIXED MODEL
Without loss of generality, consider the mixed model:
y  Xβ  Zf  ε
y = vector of observations with mean E[y] = X
 = vector of fixed effects: βtr = [A , B]
f = vector of random effects: f tr = [α ,  ] with zero means
 a2 0 
and variance-covariance matrix
Varf   
2
 0   
 = vector of iid random error terms with mean zero and Var(ε ) = 2I
X , Z = matrices of regressors relating the observed y to β and f
Let θ be the vector of all parameters: θtr = [A , B , α ,  , a2 , 2 , 2 ]
10
A BAYESIAN FORMULATION OF THE MIXED MODEL
The likelihood : y | θ ~ N (Xβ  Zf ,  2 I)
 The prior distributions:

p(A), p(B) ~ Uniform
 Jeffreys’ prior:
2

p( a ) 
1
, p( 2 ) 
 a2
 The joint posterior distribution:
p(θ | y)  p(θ) p( y | θ )
1
 2
, p( 2 ) 
prior
1
 2
likelihood
p( A, B, a ,  ,  a2 ,  2 ,  2 | y)  p( A) p( B) p(a |  a2 ) p( |  2 ) p( a2 ) p( 2 ) p( 2 ) p( y | θ)
1 1 1
 N (a | 0,  a2 ) N ( | 0,  2 )
N ( y | Xβ  Zf ,  2 I)
 a2  2  2
3
3



2 2 2 2
 ( a ) (  ) exp
2 
 1

 2 1
2
 y  A  BT  a    

( ) exp
2
2 
2 i
 2  i 1

 2 a 2  
a
2
N
11
SIMULATION OF FUTURE LOTS (RANDOM INTERCEPT)


Generate a posterior sample representing a set of process parameters from
the posterior distribution of the parameters from the mixed model. This
represents a random process, indexed by i, with parameters: Ai, Bi,ai2, i2
For each posterior sample i, generate a mean value for a kth random lot at
time T (µk(i)T) by adding a lot effect (ak(i)) to the estimated process mean value
at T as follows:
mk(i)T =Ai +Bi *T +ak(i) ,
where ak(i) ~ N(0, ai2). Repeat this n times (k=1,2,…,n).

For each random lot with mean µk(i)T at time T, add measurement error as:
mk(i)T = µk(i)T + k(i)T, where k(i)T ~ N(0, i2).

Repeat above steps for N random processes.

Time of interest: T=0 (at release) and T=Shelf Life (e.g. 24 mos.).
Note: Independence Chain Metropolis-Hastings algorithm used in SAS Proc Mixed
procedure
12
MANUFACTURING RISK ESTIMATION
Using the simulated lot data at T0 and TSL, calculate the
probabilities of future lots falling into each of the 4 possible
outcomes in relation to pass and fail at Release and end of
shelf life .
P(YSL<SpecSL|Y0≥RL)
P(YSL>SpecSL|Y0 ≥ RL)
P(YSL<SpecSL|Y0<RL)
Yj = Lot mean at j-th time point, SL=shelf life
13
P(YSL>SpecSL|Y0<RL)
OC CURVES CORRESPONDING TO THE 2X2 TABLE
14
CASE STUDIES
15
EXAMPLES 1 – ASSAY FOR IMPURITY B
Stability data (up to 18 mos.) for the assay of
Impurity B; 9 lots stored at 25C/60%RH
temperature condition; 24 months shelf life
 Specification: ≤ 2.3
o
16
EXAMPLE 1: MIXED EFFECTS MODEL
yij  A  ai  B  Tij  ij
yij = assay for ith lot at jth stability time point
A = overall process mean at time of manufacture
ai = random effect of the ith lot: ~ N(0, a2)
B = rate of change per month
Tij = jth stability time point for ith lot
ij = Residual Variability ~ N(0, 2)
a ’s, and  ’s are mutually independent
17
17
EXAMPLE 1 – MIXED EFFECTS MODEL RESULTS
Example 1 – Maximum Likelihood Compared
to Posterior Estimates
Fixed
Effect
A
MLE (se)
1.56 (0.02)
B
0.18 (0.01)
0.18 (0.15 – 0.20)
MLE
Posterior Median
(95% interval)
0.0017
0.0019 (0.0006 – 0.0071)
0.0016
0.0016 (0.0010 – 0.0025)
Variance
Terms
 a2 (lot)
 2 (resid)
Posterior Mean
(95% interval)
Overall
Mean at T0
1.56 (1.52-1.60)
Monthly
Rate
Lot Variability
Residual
18
EXAMPLE 1 – ADG RELEASE LIMITS CALCULATION
RLU  2.3  b T  z0.95  Var(b T )  Var(Re sid )  1.87
Example 1 - % of simulated lots in
categories of pass/fail for a
specification=2.3 at 24 mos. and RL
=1.87 (ADG method)
End of Shelf Life
Release
Pass
Fail
Total
Pass
99.99%
0
99.99%
Fail
0.01%
0
0.01%
19
EXAMPLE 2: DISSOLUTION OF IR TABLET
Stability data (up to 18 mos.) for 30 minutes dissolution;
7 lots stored at 25C/60%RH temperature condition
 Q-specification = 80% at 30 minutes; 24 months shelf life
o
20
EXAMPLE 2: MIXED EFFECTS MODEL
yk (ij)  A  ai  B Tij   ij   k (ij)
yk(ij) = dissolution of kth vessel for ith lot at jth stability time point
A = overall process mean at time of manufacture
ai = random effect of the ith lot: ~ N(0, a2)
B = rate of change per month
Tij = jth stability time point for ith lot
ij = error II (Run-to-Run and unknown source of variability):
~ N (0, 2 )
k(ij) = error I (Vessel-to-Vessel variability): ~ N(0, 2)
a ’s, ’s and  ’s are mutually independent
21
21
EXAMPLE 2: MIXED EFFECTS MODEL RESULTS
Example 2 – Maximum Likelihood Compared
to Posterior Estimates
Fixed
Effect
A
MLE (se)
87.1 (1.0)
B
-1.0 (1.3)
-1.1 (-3.7 – 1.3)
MLE
Posterior Median
(95% interval)
0.8
1.9 (0.1 – 12.3)
10.9
10.5 (5.7 – 18.6)
12.6
12.7(10.5 – 15.6)
Variance
Terms
 a2 (lot)
 2 (run)
 2 (resid)
Overall
Mean at T0
Posterior Mean
(95% interval)
87.1 (84.8-89.4)
Lot Variability
Monthly Rate
Run Variability
Residual
22
EXAMPLE 2 - ADG RELEASE LIMITS CALCULATION
RL  80  b  T  z0.95  Var(b  T )  Var( Run)  Var(Vessel ) / 6
 89.29
Example 2 - % of simulated lots in
categories of pass/fail for a Q= 80% at 24
mos. and RL criterion= 89.29(ADG method)
End of Shelf Life
Release
Pass
Pass
Fail
Total
30% (97%)*
1% (3%)*
31%
Fail
65%
4%
69%
* probabilities conditional to row total (how many
passed or failed shelf life specification from23those
that passed the RL criterion)
EXAMPLE 3: DISSOLUTION TRANSDERMAL SYSTEM
o
Stability data (up to 24 mos.) for the 30 minutes dissolution, 8 lots
stored at 25C/60%RH temperature condition; additional Release
data from 25 lots; 24 months shelf life.
Specification (% label claim)
Release
13 - 19
TSL (24 months)
9 - 16
o 3 Level Criterion for Testing
Transdermal Delivery
Systems – USP <724>
24
EXAMPLE 3: MIXED EFFECTS MODEL RESULTS
 Mixed
effects modeling with fixed intercept and
slope and random intercept, run and vessel
effects (similar to Example 2)
Example 3 – Maximum Likelihood Compared to Posterior
Estimates
Fixed
Effect
MLE (se)
Posterior Mean
(95% interval)
A
15.5 (0.1)
15.5 (15.2-15.9)
B
-2.4 (0.1)
-2.4 (-2.7 – 2.2)
MLE
Posterior Median
(95% interval)
Variance
Terms
 a2
(lot)
0.47
0.48 (0.24 – 0.95)
 2
(run)
0.22
0.22 (0.12 – 0.39)
 2
(resid)
0.42
0.42 (0.37 – 0.48)
25
EXAMPLE 3 - ADG RELEASE LIMITS CALCULATION
RLL  9  b  T  z0.95  Var(b  T )  Var( Run)  Var(Vessel ) / 6
 14.8
Example 3 - % of simulated lots in
categories of pass/fail for a Q= 9 - 16%
at 24 mos. and RL criterion= 14.8 - 19
(ADG method)
End of Shelf Life 9 - 16
Release
14.8- 19
Pass
Fail
Total
0.2%
79.4%
79.6%
(0.2%)
Pass
(99.8%)
Fail
18.9%
1.5%
20.4%
* probabilities conditional to row total
26
SUMMARY



ADG method does not address risk in a statistically
derived probability sense, more a heuristic
calculation than statistical.
 Applies to individual lots as manufactured
 More decision rule rather than risk control strategy
ADG approach can be extended to the mixed effects
model.
 Allows for more flexible description of a
manufacturing process and relevant variance
components
 Sets the stage for a hierarchical Bayesian approach
Current technology allows the application of a
Bayesian approach in a fairly direct and
uncomplicated way.
27
SUMMARY

Bayesian
posterior predictive approach addresses
manufacturing risk by allocating measured outcomes into
categories of acceptable and unacceptable lots at both release
and end of shelf life given specifications and release limits






Predictive posterior distribution of future lots can be easily
generated  a natural interpretation of manufacturing risk as a
probability.
The risks associated with the manufacturing process are expressed
via 2x2 tables displaying joint release and end of shelf life outcomes
as probabilities.
Release limits as a control strategy can be assessed by calculating
the OC curve corresponding to the 2x2 table outcomes generated
across a range of release point values or intervals.
Costs to the company associated with the risks can be calculated.
Provides elements of a comprehensive risk control strategy missing
in the ADG method
28
Expert opinions, historical data from diverse sources and
prior knowledge may be integrated into a prior distribution.
REFERENCES

Allen, Dukes, & Gerger (1991). Determination of Release
Limits: A General Methodology . Pharmaceutical Research, Vol. 8,
No. 9, pp.1210-1213.

Shao and Chow (1991). Constructing Release Targets for
Drug Products: a Bayesian Decision Theory Approach. JRSS,
Series C (Applied Statistics) 1991, 40, No. 3, pp. 381-390.

Greg C. G. Wei (1998). Simple Methods for Determination of
the Release Limits for Drug Products. Journal of Biopharmaceutical
Statistics, 1998, 8(1), 103-114.

Murphy and Hofer (2002). Establishing Shelf life, Expiry
Limits and Release Limits. Drug Information Journal, 2002, vol. 36,
pp. 769-781.

Andrew Gelman, John B. Carlin, Hal S. Stern and Donald B.
Rubin (2004). Bayesian Data Analysis. 2nd ed. Chapman &
Hall/CRC.
29