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

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

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

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

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

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

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:

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)

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.

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

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

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

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

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

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

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

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.

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.

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

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

Shelf Life Consider

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

Prior Distributions 23

Case Study

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

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 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 su2 0.20 0.32   0.24 (0.22) 0.13 - 0.43 se2 0.33 0.41 0.43 (0.42) 0.29 - 0.64 Bayesian method provides the ability to characterize the variability of parameter estimates, even when data are limited.

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 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

Variance Component

Degradation Rate

Shelf Life

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.