Accelerated Stability Modeling

Understanding the factors that impact stability and then generating product stability knowledge are key considerations detailed in the International Conference of Harmonization (ICH) Q8 (1) development standard as well as the Q1E standard. Accelerated stability analysis is a strategy used to quickly evaluate alternative formulations, packaging and processes; however, it is not always clear how to estimate expiry from the accelerated data.  

Understanding the product and process is becoming more of an expectation based on FDA guidance and technical presentations. Building representative product and process models for stability is now considered an essential part of product and process development. Once a product/process model has been established and verified it can be used to support and/or justify potential formulation, process, site, and supplier changes.

Linear and non-linear regression models are recommended in guidance from the health authorities for stability determination. Specifically, ICH Q1E 2.6 General Statistical Approaches states (2):

“Regression analysis is considered an appropriate approach to evaluating the stability data for a quantitative attribute and establishing a retest period or shelf life. The nature of the relationship between an attribute and time will determine whether data should be transformed for linear regression analysis. The relationship can be represented by a linear or non-linear function on an arithmetic or logarithmic scale. In some cases, a non-linear regression can better reflect the true relationship.”

Building a product or process stability model typically follows the following steps:

• State the purpose
• Perform a risk assessment to identify the key factors driving the responses of interest (3)
• Develop a stability study design and associated power analysis (4)
• Collect the data
• Fit the model (linear and or non-linear)
• Evaluate the model’s usefulness, accuracy, and associated errors
• Evaluate the model’s predictive potential and associated errors
• Determine rate of degradation and expiry at nominal conditions and at accelerated conditions
• Verify any early accelerated predictions with long-term stability studies at nominal storage conditions.

Traditionally there has often been an attempt to use the Arrhenius transformation for all accelerated conditions. This approach is good when the assumptions associated with the Arrhenius equation are valid; however, often the Arrhenius transformation becomes the erroneous equation as it does not fit the data and the product or process assumptions cannot be satisfied. The health authorities have issued warning letters to companies that use Arrhenius transforms when the data does not support this method of stability estimation (5). This paper will outline an approach to model and predict linear and non-linear stability data under accelerated and nominal storage conditions. In general, the approach presented in this paper is to model the measured data, understand the scientific rational associated with the degradation pathway (what makes it degrade?), and then generalize the model for multiple accelerated conditions and at the nominal condition.

There are many factors other than environmental that may impact product stability.  These factors should be considered as well when developing accelerated and long-term stability and self-life claims:

• Temperature
• Humidity
• API concentration
• Water content
• Amount of an excipient
• Processing conditions and/or set points
• Packaging materials/method.

Study Design
Proper design of experiments for data collection and factor effect isolation is crucial for building linear and non-linear stability models.  All factors should be orthogonal relative to each other and have zero correlation (or near zero). For this simple example, there are multiple time points at multiple temperatures. There are an equal number of samples at each time and temperature condition. Replicates are suggested at each condition to reduce the analytical method variation and to improve the precision of the estimates. In this study, the number of replicates was three at each time point. A power analysis (Figure 1) should be performed to assure the stability study has good statistical power (>0.95). This study is a 3*3*9=81 runs, three temperatures with three replicates and nine time points. Simulation can also be used to evaluate study plans and estimate their impact on stability.

Analysis Method
The following is the step-by-step procedure for accelerated stability modeling and expiry determination:

1.Measure the data at multiple time periods and at multiple temperatures. Generate a plot of the data (Figure 2) to visualize the relationship of the curves over time.  Save the slopes from the curve and place them in a table.  Examination of the slopes will indicate if the environmental factor accelerates the rate of change/degradation.

2. Use a multiple factor analysis of covariance (ANCOVA) model to fit the data. Examine the effects test (Figure 3) to make sure all terms in the model are significant. The inverse prediction (Figure 4) will provide the expiry and the 95% lower confidence interval. Also examine the whole model to determine the RSquare and the quality of the ANCOVA model. Check the residuals to make sure no hidden factor has entered into the study.

3. Save the expiry and the 95% predictions (Figure 5) at each temperature into a table.  Acceleration rate is the ratio of each slope at temperature to the nominal storage condition.

4. To build a generalized model (Figure 6) of how temperature accelerates the rate of degradation and expiry, fit a model of temperature to the coefficients (slopes), expiry, and 95% confidence interval (CI) and acceleration rate. Models may be linear or non-linear in their fitting parameters. Make sure the models selected makes good scientific sense and can be generalized.  Modern software programs make this easy to do.

5. Save the equations for the fit for the 95% CI, expiry, acceleration rate, and the slope:

Predicted Slope= (-0.0000009387967452312) + 0.0000001815600541022 * :Temperature + 0.0000000218633389171 * :Temperature ^ 2, Quadratic Model

Predicted 95% CI= (-27.8898211558579) + 23.9294245426114 * Exp(0.0376760939674008 * :Temperature), 3 Parameter Exponential Model

Predicted Expiry= 33.5848316003024 + 7328.12608635937 * Exp(-0.196329985274204 * :Temperature), 3 Parameter Exponential Model

Predicted Accelaration Rate= 25.9675472550865 + 1008.32460629621 * Exp(-0.116629176337002 * :Temperature), 3 Parameter Exponential Model

6. Check the model to make sure it matches the actual data. Correct any modeling errors.  
7. Create a profiler (Figure 7) from the equation. This can be done using a modern statistical package such as SAS/JMP.

8. Predict expiry at any temperature using the profiler. A temperature of 4 C was not part of the study design; however, it now can be modeled using the prediction profiler.
9. 95% CIs are often not helpful in estimation of long-term stability due to the limited sample size.  The sample size directly impacts the 95% CI but not the slope as much. Expiry based on the slope is typically the primary focus.  In this example, it is estimated at 4 C the product will be stable for 3375 days or 9.2 years.
10. Finally, long-term stability evaluation at nominal storage conditions will be used to confirm the early model prediction and will provide an independent secondary determination of stability and changes in dissolution.  Understanding rates of degradation should factor into shelf life and release specification limits (6).

Summary
Reliable accelerated stability modeling and estimation has long been a problem in a variety of process and product modeling and prediction situations. The novel procedure discussed in this paper for the characterization of stability performance provides a clear, statistically defendable method for determining accelerated stability. Four primary tools are used in the generation of the analysis: design of experiments for the design of the study, ANCOVA model fitting, linear and non-linear model fitting of the coefficients, and profilers to integrate the equation and improve visualization and prediction. Long-term verification of accelerated conditions should always follow early determinations of expiry, acceleration rates and rates of degradation

References
1. ICH Q8 (R2) Pharmaceutical Development (ICH, 2009).
2. ICH Q1E, Evaluation for Stability Data (ICH, 2003).
3. ICH Q9 Quality Risk Management (ICH, 2006).
4. ICH Q1A(R2) Stability Testing of New Drug Substances and Products (2003).
5. FDA, FDA Warning Letter to ACell (April 26, 2013), www.fda.gov/ICECI/EnforcementActions/WarningLetters/2013/ucm352061.htm
6. ICH Q6B Specifications: Test Procedures and Acceptance Criteria for Biotechnological/Biological Products (ICH,1999).

About the Author
Thomas A. Little, PhD is president, Thomas A. Little Consulting,12401 North Wildflower Drive, Highland, UT  84003, [email protected]