ASSESSING RELEASE LIMITS AND MANUFACTURING RISK

FROM A BAYESIAN PERSPECTIVE

1Areti Manolaamanola@its.jnj.com

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

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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

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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.

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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?

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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

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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

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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

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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:

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)|()()|( 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

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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

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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

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CASE STUDIES

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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

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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

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EXAMPLE 1 – MIXED EFFECTS MODEL RESULTS

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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)

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Overall Mean at T0

Monthly Rate

Residual

Lot Variability

EXAMPLE 1 – ADG RELEASE LIMITS CALCULATION

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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

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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

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EXAMPLE 2: MIXED EFFECTS MODEL RESULTS

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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)

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Overall Mean at T0

Monthly Rate

Residual

Run VariabilityLot Variability

EXAMPLE 2 - ADG RELEASE LIMITS CALCULATION

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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

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Specification (% label claim)Release 13 - 19

TSL (24 months) 9 - 16

EXAMPLE 3: MIXED EFFECTS MODEL RESULTS

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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

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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.

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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.

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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