Modeling Approaches to Modeling Approaches to Multiple Isothermal Stability Multiple Isothermal Stability Studies for Estimating Shelf Studies for Estimating Shelf LifeLife

Oscar Go, Areti Manola, Jyh-Ming Shoung and Stan AltanNon-Clinical Statistics

ContentsContents Overview of Statistical Aspect of Stability

Study Accelerated Stability Study Bayesian Methods Case Study Concluding Remarks

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Purpose of Stability TestingPurpose of Stability Testing To provide evidence on how the quality of a

drug substance or drug product varies with time under the influence of a variety of environmental factors (such as temperature, humidity, light, package)

To establish a re-test period for the drug substance or an expiration date (shelf life) for the drug product

To recommend storage conditions

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Typical DesignTypical Design Randomly select containers/dosage units at

time of manufacture, minimum of 3 batches, stored at specified conditions.

At specified times 0, 1, 3, 6, 9, 12, 18, 24, 36, 48, 60 months, randomly select dosage units and perform assay on composite samples

Basic Factors : Batch, Strength, Storage Condition, Time, Package

Additional Factors: Position, Drug Substance Lot, Supplier, Manufacturing Site

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Kinetic ModelsKinetic Models Orders 0, 1, 2 :

where C0 is the assay value at initial

When k1 and k2 are small,

1

20

)2(

0)1(

00)0(

1)(

)(

)(1

tkC

tC

eCtC

tkCtCtk

tkCCtC

tkCCtC

2200

)2(

100)1(

)(

)(

5

Estimation of Shelf LifeEstimation of Shelf Life

Intersection of specification limit with lower 1-sided 95% confidence bound

LowerSpecification(LS)

6

Data Plot with Regression Line and Lower Confidence Limit Assay (%Label)

Time (months)0 3 6 9 12 18 24 30 36

85

90

95

100

Linear Mixed ModelLinear Mixed Model

where

yijk = assay of ith batch at jth temperature and kth time point,

= process mean at time 0 (intercept),

i = random effect due to ith batch at time 0:

Bj = fixed average rate of change,

Tijk = kth sampling time for batch i at jth temperature,

ijk = residual error:

7

ijkijkjiijk TBy

),0(~ 2 Ni

),0(~ 2 Nijk

Shelf LifeShelf LifeIf , the expiration date ( TSL ) at condition

i is the solution to the quadratic equation

LSL = 90% = lower specification limit, q = (1-)th quantile, (=0.05 and z-quantile was used for the case study)

8

2ˆ)ˆˆ(ˆˆ SLiSLi TBVarqTBLSL

0iB

Accelerated Stability TestingAccelerated Stability Testing Product is subjected to stress conditions. Temperature and humidity are the most

common stress factors. Purpose is to predict long term stability and

shelf life. Arrhenius equation captures the kinetic

relationship between rates and temperature. The usual fixed and mixed models ignore any relationship between rate and temperature.

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Arrhenius EquationArrhenius Equation Named for Svante Arrhenius (1903 Nobel

Laureate in Chemistry) who established a relationship between temperature and the rates of chemical reaction

where kT = Degradation Rate

A = Non-thermal ConstantEa = Activation EnergyR = Universal Gas Constant (1.987)T = Absolute Temperature

TR

E

T

a

AeTkk

)(

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Assumptions Underlying Assumptions Underlying Arrhenius ApproachArrhenius Approach

The kinetic model is valid and applies to the molecule under study

Homogeneity in analytical error

NB: Humidity is not acknowledged in the equation

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Nonlinear Parametrization Nonlinear Parametrization (King-Kung-Fung Model)(King-Kung-Fung Model)

Let T =298oK (25oC)

TR

E

T

a

Aek

R

Ea

ekA 298298

TR

E

T

a

ekk1

298

1

298

tekCtC TR

E

T

a

1

298

1

2980)(

Tk12

King-Kung-Fung King-Kung-Fung Nonlinear Mixed ModelNonlinear Mixed Model

ijlijl

TR

e

iijl tekuCC j

aE

1

298

1

2980

*

),0(~

),0(~2

2

N

Nu

ijl

ui

Indices i = batch identifier j = temperature level l = time point

Parameters are :22*

2980 ,),ln(,, uaa EEkC

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King-Kung-Fung Model-King-Kung-Fung Model-Estimation of Shelf LifeEstimation of Shelf Life

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Shelf life at a given temperature Tj = T is the solution tSL in the following equation

where

t0.95,df is the Student’s 95th t-quantile with df degrees

)))((ˆ())((ˆ,95.0 SLijldfSLijl tCarVttCLSL

tekCtCE TR

e

ijl

aE

1

298

1

2980

*

))((

Linearized Arrhenius ModelLinearized Arrhenius Model

Take log on both sides of the Arrhenius equation

Assuming a zero order kinetic model

TR

EAkeAk a

TTR

E

T

a

loglog

ttCCktkCtC TTTT logloglog 00

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Linearized Arrhenius ModelLinearized Arrhenius Model Combining the two equations and solving for

log t

Set t to t90 , time to achieve 90% potency for each temperature level (CT ( t90 )=90 )

Expressed as linear regression problem

TR

EAtCCt a

T log))(log(log 0

TR

EAC90t a 1

log)90log(log 0

01

T

90t1

log 10

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Linearized Arrhenius Mixed ModelLinearized Arrhenius Mixed Model

To include batch-to-batch effect in the model, we can add a random term to

ijij

iij Tv90t

1log 10

),0(~

),0(~2

2

N

Nv

ij

vi

Indices i = batch identifier j = temperature level

17

Linearized Arrhenius Mixed ModelLinearized Arrhenius Mixed Model

To summarize, the (Garrett, 1955) algorithm:

1)Fit a zero-order kinetic model by batch and and temperature level.

2)Estimate t90 and its standard error from each zero-order kinetic model.

3)Fit a linear (mixed) model to log(t90) on the reciprocal of Temperature(Kelvin scale).

4)Shelf life for a given temperature level is estimated from the model in step 3.

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Comparison Between the Three Comparison Between the Three ApproachesApproaches Linear Mixed Model

Loses information contained in the Arrhenius relationship when it is valid

Linearized Arrhenius Model (Garrett) Simple and does not require specialized

software Not clear how to estimate shelf life in relation

to ICH guideline Ignores heteroscedasticity in the error terms Difficult to interpret the random effect

Nonlinear Model (King-Kung-Fung) Computationally intensive Computing convergence issues

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King-Kung-Fung Model: King-Kung-Fung Model: Bayesian Method

),0(~

),0(~2

2

N

Nu

ijl

ui

Indices i = batch identifier j = temperature level l = time point

Parameters:22*

2980 ,),ln(,, uaa EEkC

Additional Parameters: 303303298 90,,90 TTT tkt

20

ijlijl

TR

e

iijl tekuCC j

aE

1

298

1

2980

*

Shelf LifeShelf Life Consider

21

ZtkCLSL TT 9090 0

TR

E

T

a

ekk1

298

1

298

.account into take toadded wasZ

).,0(~,eg

0,about lsymmetrica and dataoftindependenis

2

2

u

uNZ

Z

King-Kung-Fung Model: King-Kung-Fung Model: Bayesian Method-Prior Distributions

Provides a flexible framework for incorporating scientific and expert judgment, incorporating past experience with similar products and processes

Expert opinions Process mean at time 0 is between 99%

and 101% No information regarding degradation rate No information regarding activation energy Batch variability is between 0.1 and 0.5

with 99% probability Analytical variability is between 0.1 to 1.0

with 99% probability

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Prior DistributionsPrior Distributions

)2,6(~

)2,10(~

),(~

),(~

)1.0,100(~

12

12

*

298

0

u

a IE

Ik

NC

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Case StudyCase Study

24

25

R/WinBUGS Simulation R/WinBUGS Simulation ParametersParameters 3 chains 500,000 iterations/chain Discard 1st 100,000 simulated values in each

chain Retain every 100th simulation draw A total of 27,000 simulated values for each

parameter

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Model Parameter EstimatesModel Parameter Estimates

Bayesian method provides the ability to characterize the variability of parameter estimates, even when data are limited.

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ParametersTrue Value

Nonlinear Mixed Model Bayesian Nonlinear Mixed Model

Estimate95%

Confidence Interval

Mean (Median)95% Credible

Interval

C0 100.0 99.9 98.3 - 101.5 100.0 (100.0) 99.5 - 100.4

k298 0.26 0.24 0.15 - 0.32 0.24 (0.24) 0.20 - 0.28

Ea* 10.04 10.08 9.89 - 10.27 10.07 (10.07) 9.99 - 10.16

u 0.32 

0.24 (0.22) 0.13 - 0.43

0.41 0.43 (0.42) 0.29 - 0.64

Shelf Life EstimatesShelf Life Estimates

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Linear Mixed ModelTemperature Estimate Shelf Life

25C 41.3 32.930C 21.8 18.340C 6.1 5.2

TemperatureTrue Value

Nonlinear Mixed ModelLinearized Arrhenius

ModelBayesian Nonlinear

Mixed Model

Estimate90%

Confidence Interval

Shelf Life

Estimate90%

Confidence Interval

Mean (Median)

90% Credible Interval

25C 38.9 42.0 31.9 - 52.1 33.7 37.7 31.1 - 44.4 42.1 (41.8) 35.9 - 49.1

30C 20.5 21.6 17.8 - 25.4 18.3 20.4 17.2 - 23.6 21.7 (21.6) 19.1 - 24.5

40C 6.0 6.1 5.2 - 6.9 5.3 6.3 5.1 - 7.5 6.1 (6.1) 5.5 - 6.8

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Variance ComponentVariance Component

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Degradation RateDegradation Rate

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Shelf LifeShelf Life

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SummarySummary King-Kung-Fung model is a practical way to

characterize multiple isothermal stability profiles and has been shown to be extended easily to a nonlinear mixed model context.

Bayesian method permits integration of expert scientific judgment in characterizing the stability property of a pharmaceutical compound.

The Bayesian credible interval can be interpreted in a probabilistic way and provides a more natural meaning to shelf life compared with the frequentist repeated sampling definition.

The problem of determining the appropriate degrees of freedom in mixed modeling is eliminated by Bayesian method.

Bayesian method is flexible and can be easily applied to a wide family of distributions.

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