1. Modeling Approaches to Multiple Isothermal Stability Studies for Estimating Shelf Life
Oscar Go, Areti Manola,
Jyh-Ming Shoung and Stan Altan
Non-Clinical Statistics

2. Contents Overview of Statistical Aspect of Stability Study
Accelerated Stability Study
Bayesian Methods
Case Study
Concluding Remarks

3. Purpose 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

4. Typical 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

5. Kinetic Models Orders 0, 1, 2 : where C0 is the assay value at initial
When k1 and k2 are small,

6. Estimation of Shelf Life
Lower
Specification
(LS)
Intersection of specification limit with lower 1-sided 95% confidence bound

7. Linear 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:

8. Shelf Life
If , 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)

9. Accelerated 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.

10. Arrhenius 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 Constant
Ea = Activation Energy
R = Universal Gas Constant (1.987)
T = Absolute Temperature

11. Assumptions Underlying Arrhenius 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

12. Nonlinear Parametrization (King-Kung-Fung Model)
Let T =298oK (25oC)

13. King-Kung-Fung Nonlinear Mixed Model
Indices
i = batch identifier
j = temperature level
l = time point
Parameters are :

14. King-Kung-Fung Model-Estimation of Shelf Life
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

15. Linearized Arrhenius Model
Take log on both sides of the Arrhenius equation
Assuming a zero order kinetic model

16. Linearized 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

17. Linearized Arrhenius Mixed Model
To include batch-to-batch effect in the model, we can add a random term to b0
Indices
i = batch identifier
j = temperature level
NB: log(C0 – 90 + v) – log(C0 – 90) is proportional to v.

18. Linearized Arrhenius Mixed Model
To summarize, the (Garrett, 1955) algorithm:
Fit a zero-order kinetic model by batch and and temperature level.
Estimate t90 and its standard error from each zero-order kinetic model.
Fit a linear (mixed) model to log(t90) on the reciprocal of Temperature(Kelvin scale).
Shelf life for a given temperature level is estimated from the model in step 3.

19. Comparison Between the Three Approaches
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

20. King-Kung-Fung Model: Bayesian Method
Indices
i = batch identifier
j = temperature level
l = time point
Parameters:
Additional Parameters:

21. Shelf Life
Consider

22. 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

23. Prior Distributions 23

24. Case Study

26. R/WinBUGS Simulation Parameters
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

27. Model Parameter Estimates
Parameters
True Value
Nonlinear Mixed Model
Bayesian Nonlinear Mixed Model
Estimate
95% Confidence Interval
Mean (Median)
95% Credible Interval C0
100.0
99.9
100.0 (100.0)
k298
0.26
0.24
0.24 (0.24)
Ea*
10.04
10.08
10.07 (10.07)
su2
0.20
0.32
0.24 (0.22)
se2
0.33
0.41
0.43 (0.42)
Bayesian method provides the ability to characterize the variability of parameter estimates, even when data are limited.

28. Linearized Arrhenius Model Bayesian Nonlinear Mixed Model
Shelf Life Estimates
Linear Mixed Model
Temperature
Estimate
Shelf Life
25C
41.3
32.9
30C
21.8
18.3
40C
6.1
5.2
Temperature
True Value
Nonlinear Mixed Model
Linearized Arrhenius Model
Bayesian Nonlinear Mixed Model
Estimate
90% Confidence Interval
Shelf Life
Mean (Median)
90% Credible Interval
25C
38.9
42.0
33.7
37.7
42.1 (41.8)
30C
20.5
21.6
18.3
20.4
21.7 (21.6)
40C
6.0
6.1
5.3
6.3
6.1 (6.1)

30. Variance Component

31. Degradation Rate

32. Shelf Life

33. Summary
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.