A SSESSING R ELEASE L IMITS AND M ANUFACTURING R ISK FROM A B AYESIAN P ERSPECTIVE 1 Areti Manola...
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Transcript of A SSESSING R ELEASE L IMITS AND M ANUFACTURING R ISK FROM A B AYESIAN P ERSPECTIVE 1 Areti Manola...
ASSESSING RELEASE LIMITS AND MANUFACTURING RISK
FROM A BAYESIAN PERSPECTIVE
1Areti [email protected]
OUTLINE
Introduction Review Q1E Stability Evaluation Definition of Release Limits
Allen, Dukes and Gerger ApproachMixed Linear Model
A Bayesian Approach to Manufacturing Risk Estimation Bayesian formulation of mixed modelSimulating future lots
Posterior predictive distributionCase StudiesSummary
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)
LRL = lower release limit LSL = lower specification limit B = average slope for degradation TSL = shelf life ST = standard error of average slope shelf
life S = assay standard deviation t1-,k = one-sided (1-)%-ile critical t value with k
degrees of freedom n = number of replicate assays used for lot release
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
6
0for B 2
2,1 n
SStTBLSLLRL TkSL
Consider a fixed batch-specific linear model:
iii TBAy
85%
90%
95%
100%
105%
0 6 12 18 24Time (Months)
Per
cen
t o
f L
abel
Regression Loss
Release Assay + Regression Loss Uncertainty
Release
OTHER APPROACHES Shao and Chow (1991)
Constructing Release Targets for Drug Products: a Bayesian Decision Theory ApproachVarious 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 = 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
Note: Extend Allen’s approach to mixed modeling framework by including additional variance terms
ijkijkjiijk TBAy
8
MANUFACTURING RISK ESTIMATION Given Release Limits and Specifications,
manufacturing risk can be described through a 2x2 table given below:
Probabilities associated with the above 2x2 table can be estimated through a Bayesian posterior predictive distribution approach 9
cost to the company
End of Shelf Life
ReleasePass (%) Fail (%) Total (%)
Pass (%) C11 C12 R1
Fail (%) C21 C22 R2
Total (%) C1 C2 100
C12/R1= P(YSL<SpecSL|Y0≥RL)
A BAYESIAN FORMULATION OF THE MIXED MODEL
Without loss of generality, consider the mixed model:
y = vector of observations with mean E[y] = X = vector of fixed effects: βtr = [A , B]
= vector of random effects: tr = [α , ] with zero means
and variance-covariance matrix
= 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
Let θ be the vector of all parameters: θtr = [A , B , α , , 2 , 2 , 2 ]10
2
2
0
0
Var
εZXβy
A BAYESIAN FORMULATION OF THE MIXED MODEL
The likelihood : The prior distributions:
p(A), p(B) ~ Uniform Jeffreys’ prior:
The joint posterior distribution:
11
)|()()|( θyθyθ ppp
22
22
22 1
)(1
)(1
)(
ppp , ,
N
ii BTAy
NNN
ppppppBpApBAp
1
22
122
2
2
223
223
2
2222
22
22222222
2
1exp)(
22exp)()(
)I,ZXβ|y(111
),0|(),0|(
)θ|y()()()()|()|()()()y|,,,,,,(
prior
likelihood
)I,ZXβ(~θ|y 2N
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, i,ai
2, 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 (k(i)) to the estimated process mean value at T as follows:
k(i)T =Ai +i *T +ak(i) ,
where k(i) ~ N(0, i2). 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 .
Yj = Lot mean at j-th time point, SL=shelf life
13
P(YSL<SpecSL|Y0≥RL)
P(YSL<SpecSL|Y0<RL)
P(YSL>SpecSL|Y0<RL)
P(YSL>SpecSL|Y0 ≥ RL)
OC CURVES CORRESPONDING TO THE 2X2 TABLE
14
CASE STUDIES
15
EXAMPLES 1 – ASSAY FOR IMPURITY B
o 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
16
17
EXAMPLE 1: MIXED EFFECTS MODEL
yij = assay for ith lot at jth stability time point
A = overall process mean at time of manufacturei = random effect of the ith lot: ~ N(0, 2)
B = rate of change per monthTij = jth stability time point for ith lot
ij = Residual Variability ~ N(0, 2) ’s, and ’s are mutually independent
ijijiij TBAy
17
EXAMPLE 1 – MIXED EFFECTS MODEL RESULTS
18
Example 1 – Maximum Likelihood Compared to Posterior Estimates Fixed Effect MLE (se)
Posterior Mean (95% interval)
A 1.56 (0.02) 1.56 (1.52-1.60)B 0.18 (0.01) 0.18 (0.15 – 0.20)Variance Terms MLE
Posterior Median(95% interval)
(lot)0.0017
0.0019 (0.0006 – 0.0071)
(resid)0.0016
0.0016 (0.0010 – 0.0025)
22
Overall Mean at T0
Monthly Rate
Residual
Lot Variability
EXAMPLE 1 – ADG RELEASE LIMITS CALCULATION
19
87.1)(Re)(3.2 95.0 sidVarTbVarzTbRLU
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%
EXAMPLE 2: DISSOLUTION OF IR TABLETo 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
20
21
EXAMPLE 2: MIXED EFFECTS MODEL
yk(ij) = dissolution of kth vessel for ith lot at jth stability time point
A = overall process mean at time of manufacturei = random effect of the ith lot: ~ N(0, 2)
B = rate of change per monthTij = 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) ’s, ’s and ’s are mutually independent
)()( ijkijijiijk TBAy
21
EXAMPLE 2: MIXED EFFECTS MODEL RESULTS
22
Example 2 – Maximum Likelihood Compared to Posterior Estimates
Fixed Effect MLE (se)
Posterior Mean (95% interval)
A 87.1 (1.0) 87.1 (84.8-89.4)
B -1.0 (1.3) -1.1 (-3.7 – 1.3)
Variance Terms MLE
Posterior Median(95% interval)
(lot) 0.8 1.9 (0.1 – 12.3)
(run) 10.9 10.5 (5.7 – 18.6)
(resid) 12.6 12.7(10.5 – 15.6)
222
Overall Mean at T0
Monthly Rate
Residual
Run VariabilityLot Variability
EXAMPLE 2 - ADG RELEASE LIMITS CALCULATION
23
29.89
6/)()()(80 95.0
VesselVarRunVarTbVarzTbRL
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 LifeRelease Pass Fail Total
Pass30%
(97%)* 1% (3%)* 31% Fail 65% 4% 69%* probabilities conditional to row total (how many passed or failed shelf life specification from those that passed the RL criterion)
EXAMPLE 3: DISSOLUTION TRANSDERMAL SYSTEM
24
Specification (% label claim)Release 13 - 19
TSL (24 months) 9 - 16
EXAMPLE 3: MIXED EFFECTS MODEL RESULTS
25
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)
Variance Terms MLE
Posterior Median(95% interval)
(lot) 0.47 0.48 (0.24 – 0.95)
(run) 0.22 0.22 (0.12 – 0.39)
(resid) 0.42 0.42 (0.37 – 0.48)
222
Mixed effects modeling with fixed intercept and slope and random intercept, run and vessel effects (similar to Example 2)
EXAMPLE 3 - ADG RELEASE LIMITS CALCULATION
26
8.14
6/)()()(9 95.0
VesselVarRunVarTbVarzTbRLL
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 - 16Release14.8- 19 Pass Fail Total
Pass79.4%
(99.8%)0.2%
(0.2%)79.6%
Fail 18.9% 1.5% 20.4%* probabilities conditional to row total
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 Expert opinions, historical data from diverse sources and
prior knowledge may be integrated into a prior distribution.
28
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