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Modeling Approaches to Multiple Isothermal Stability Studies for Estimating Shelf Life

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

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Modeling Approaches to Multiple Isothermal Stability Studies for Estimating Shelf Life

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  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 C0is 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 = kthsampling 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-)thquantile, (=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= Tis the solution tSL in the following equation where t0.95,df is the Student’s 95tht-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 tto 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

  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

  25. 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

  26. Model Parameter Estimates • Bayesian method provides the ability to characterize the variability of parameter estimates, even when data are limited.

  27. Shelf Life Estimates

  28. Variance Component

  29. Degradation Rate

  30. Shelf Life

  31. 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.

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