November 1, 2003

Robert T. Magari, Ph.D.**BioPharm International**, BioPharm International-11-01-2003, Volume 16, Issue 11

*Biopharmaceutical products in storage change as they age, but they are considered to be stable as long as their characteristics remain within the manufacturer's specifications. The number of days that the product remains stable at the recommended storage conditions is referred to as the shelf life.*

Biopharmaceutical products in storage change as they age, but they are considered to be stable as long as their characteristics remain within the manufacturer's specifications. The number of days that the product remains stable at the recommended storage conditions is referred to as the shelf life. The experimental protocols commonly used for data collection that serve as the basis for estimation of shelf life are called stability tests.

Shelf life is commonly estimated using two types of stability testing: real-time stability tests and accelerated stability tests. In real-time stability testing, a product is stored at recommended storage conditions and monitored until it fails the specification. In accelerated stability tests, a product is stored at elevated stress conditions (such as temperature, humidity, and pH). Degradation at the recommended storage conditions can be predicted using known relationships between the acceleration factor and the degradation rate.

Temperature is the most common acceleration factor used for chemicals, pharmaceuticals, and biological products because its relationship with the degradation rate is characterized by the Arrhenius equation. Several methods of predicting shelf life based on accelerated stability testing are described in the article. Humidity and pH also have acceleration effects but, because they are complex, they will not be discussed in detail here. Also, details on statistical modeling and estimation are outside the scope of the article, but we provide references to computer routines.

Figure 1. A simulated set of stability results also showing the estimated degradation and 95% confidence limits.

The assessment of shelf life has evolved from examining the data and making an educated guess, through plotting, to the application of rigorous physical-chemical laws and statistical techniques. Regulators now insist that adequate stability testing be conducted to provide evidence of the performance of a drug or a biopharmaceutical product at different environmental conditions and to establish the recommended storage conditions and shelf life.

^{1-3}

Recently, Tsong reviewed the latest approaches to statistical modeling of stability tests,

^{4}

and ICH has published some guidelines for advanced testing design and data analysis.

^{5,6}

Modeling has become easier due to availability of standard statistical software that can perform the calculations. However, an understanding of the general principles of stability testing is necessary to apply these programs correctly and obtain appropriate results. Thus, the purpose of this paper is to provide an outline of the basic approaches to stability testing, as well as to create a foundation for advanced statistical modeling and shelf life prediction.

Table 1. Estimates of the degradation model and Table 2. Estimates of degradation rates, days of stability and 95% confidence limits.

Since degradation is usually defined in terms of loss of activity or performance, a product is considered to be degrading when any characteristic of interest (for example potency or performance) decreases. Degradation usually follows a specific pattern depending on the kinetics of the chemical reaction. The degradation pattern can follow zero-, first-, and second-order reaction mechanisms.

^{6}

In zero-order reactions, degradation is independent of the concentration of remaining intact molecules; in first-order reactions, degradation is proportional to that concentration.

^{6,7}

Zero- and first-order reactions involve only one kind of molecule, and can be described with linear or exponential relationships. Second- and higher-order reactions involve multiple interactions of two or more kinds of molecules and are characteristic of most biological materials that consist of large and complex molecular structures. Although it is common to approximate these reactions with an exponential relationship, sometimes their degradation pattern needs to be modeled more precisely, and no shortcuts will suffice.

The degradation rate depends on the activation energy for the chemical reaction and is product specific. We don't always have to deal with higher-order equations; in many cases, the observed responses of different orders of reactions are indistinguishable for products that degrade slowly.

The degradation rate depends on the conditions where the chemical reaction takes place. Products degrade faster when subjected to acceleration factors such as temperature, humidity, pH, and radiation. Modeling of the degradation pattern and estimation of the degradation rate are important for assessing shelf life. Experimental protocols used for data collection are called stability tests. In practice, evaluators use both real-time stability tests and accelerated stability tests. The real-time stability test is preferable to regulators. However, since it can take up to two years to complete, the accelerated tests are often used as temporary measures to expedite drug introduction.

In real-time stability tests, a product is stored at recommended storage conditions and monitored for a period of time (t

_{test}

). Product will degrade below its specification, at some time, denoted t

_{s}

, and we must also assure that t

_{s}

is less than or equal to t

_{test}

. The estimated value of t

_{s}

can be obtained by modeling the degradation pattern. Good experimental design and practices are needed to minimize the risk of biases and reduce the amount of random error during data collection. Testing should be performed at time intervals that encompass the target shelf life and must be continued for a period after the product degrades below specification. It is also required that at least three lots of material be used in stability testing to capture lot-to-lot variation, an important source of product variability.

^{1,2}

The true degradation pattern of a certain product, assuming that it degrades via a first-order reaction, can be described as follows:

The observed result (Y) of each lot has a random component φ associated with that lot, as well as a random experimental error, ε.

Both α and δ represent the fixed parameters of the model that need to be estimated from the data, while φ and ε are assumed to be normally distributed with mean = 0, and standard deviations of σ_{φ} and σ._{ε} respectively. Equation 2 is a nonlinear mixed model. Details on the estimation process are outside the scope of this paper.^{8,9}

Let C represent a critical level where the essential performance characteristics of the product are within the specification. A product is considered to be stable when Y ≥ C. Product is not stable when Y < C, while Y < C occurs at t_{s}. The manufacturer determines the value of C. The estimated time that the product is stable is calculated as

Here, a and d are the estimated values of the intercept and the degredation rate. The standard error of the estimated time can be obtained from the Taylor series approximation method and is used to calculate confidence limits. The labeled shelf life of the product is the lower confidence limit of the estimated time.^{8} Public safety is paramount, that is why we use the lower confidence limit. Lots should be modeled separately when lot-to-lot variability is large. More details on this issue are found in references 9 and 10.

We simulated data for three lots tested for a total period of 600 days (Table 1 and Figure 1). The product loses its activity as it ages, but it is considered to be performing within the specification until it reaches 80% of its activity (C = 0.8). The estimated lot-to-lot standard deviation is 0.000104, and the estimate of experimental error is 0.000262. Therefore, the shelf life of the product was determined to be 498 days. This represents the lower 95% confidence limit corresponding to the estimated time of 541 days.

Figure 2. A set of simulated data showing degradation of product at four different temperatures.

In accelerated stability testing, a product is stored at elevated stress conditions. Degradation at recommended storage conditions could be predicted based on the degradation at each stress condition and known relationships between the acceleration factor and the degradation rate. A product may be released based on accelerated stability data, but the real-time testing must be done in parallel to confirm the shelf-life prediction.

^{1}

Sometimes the amount of error of the predicted stability is so large that the prediction itself is not useful. Design your experiments carefully to reduce this error. It is recommended that several production lots should be stored at various acceleration levels to reduce prediction error. Increasing the number of levels is a good strategy for reducing error.

Table 3. Predictions of parameters at 25ÃÂ°C based on the Arrhenius equation.

Temperature is probably the most common acceleration factor used for chemicals, pharmaceuticals, and biological products since its relationship with the degradation rate is well characterized by the Arrhenius equation. This equation describes a relationship between temperature and the degradation rate as in Equation 4.

This relationship can be used in accelerated stability studies when the following conditions are met:

- A zero- or first-order kinetics reaction takes place at each elevated temperature as well as at the recommended storage temperature.
^{7}

- The same model is used to fit the degradation patterns at each temperature.
^{7,8}

These requirements do not fully guarantee that the Arrhenius equation can be used to predict the degradation rate at storage temperature, but they are a good start. Do not compromise the analytical accuracy during the course of the study to distinguish between the degradation rates at each temperature.

Select temperature levels based on the nature of the product and the recommended storage temperature. The selected temperatures should stimulate relatively fast degradation and quick testing but not destroy the fundamental characteristics of the product. It is not reasonable to test at very high temperatures for a very short period of time, since the mechanisms of degradation at high temperatures may be very different than those at the recommended storage temperature. Choose the adjacent levels appropriately so that degradation trends are larger than experimental variability. Choosing levels depends on the nature of the product and analytical accuracy, but other practical implications may be considered. Testing should be performed at time intervals that encompass the target stability at each elevated temperature. Acquire some data below C so that the degradation trend can be determined.

Humidity and pH can be used along with temperature to accelerate degradation, but modeling of multi-factor degradation is very complex. A model for parameter estimation and prediction of shelf life when temperature and pH are used as acceleration factors is given by Some et al.^{11}

Assuming that the degradation pattern follows a first-order reaction as described in Equation 2, the Arrhenius equation (Equation 4) can be used to predict the degradation rate at recommended storage temperature. First, an acceleration factor, λ, is calculated as the ratio of the degradation rate at elevated temperature to the degradation rate at storage temperature.

^{9}

This ratio, which can be worked out easily from Equation 4, can be expressed as

The true degradation pattern at storage temperature can be expressed as

(Here, λ indicates this was evaluated from accelerated tests.) The testing result (Y) will include random components representing lot-to-lot variability and experimental error. Once the estimates of α and δ are obtained, stability time is calculated in a similar fashion as in real-time stability testing. Shelf life is the lower confidence limit of the estimated time.

**An example.** Simulated data for three lots, each aged at four elevated temperatures for 300 days, are shown in Table 2. The performance of each lot at each time point is measured in three replicates. A critical level of C = 0.8 is the criterion. Data and trends are presented in Figure 2, with the estimates of degradation rates and days given in Table 2. The estimated degradation rate is observed to increase with temperature. The number of days that product performs within the specifications (C = 0.8) is 217 days when stored at 35Â°C. Stability time will drop down to 24.8 days when product is stored at 65Â°C. The estimated activation energy is 15.4 kcal/mol. Predictions at 25Â°C based on the Arrhenius equation are presented in Table 3. The product will perform within specification for an estimated 572 days. Using the lower limit, the recommended shelf life is 561 days.

Nomenclature

Activation energy is usually estimated from the accelerated stability data. However, when the activation energy is known, the degradation rate at storage temperature may be predicted from data collected at only one elevated temperature. This practice is sometimes preferred in industry since it reduces the size and time of accelerated stability tests. Experience indicates that some pharmaceutical analytes have activation energy in the range of 10 to 20 kcal/mol, but it is unlikely you will have precise information or be able to make assumptions about the activation energy of a certain product.

^{12}

Unstable Weather Conditions and Shelf-Life

The bracket method is a straightforward application of the Arrhenius equation that can be used if the value of the activation energy is known.^{12} Assuming that stability of a product at 50Â°C is 32 days, and it will be stored at 25Â°C, then, t_{e} = 32 days, T_{e} = 273 + 50Â°C = 323K, and T_{s} = 273 + 25Â°C = 298K. We know that activation energy is E_{a} = 10 kcal/mol. Stability at recommended storage temperature is calculated with a modified version of Equation 5 as:

Calculated stability is highly dependent on the value of activation energy. A stability of 435 days results when E_{a} = 20 kcal/mol.

The bracket method should not be confused with bracketing, which is an experimental design that allows you to test a minimum number of samples at the extremes of certain factors, such as strength, container size, and container fill.^{3,4,6} Bracketing assumes that the stability of any intermediate levels is represented by the stability of the extremes and testing at those extremes is performed at all time points.^{4}

The Q-Rule states that the degradation rate decreases by a constant factor when temperature is lowered by certain degrees. The value of Q is typically set at 2, 3, or 4. This factor is proportional to the temperature change as Q

^{n}

, where n equals the temperature change in Â°C divided by 10Â°C. Since 10Â°C is the baseline temperature, the Q-Rule is sometimes referred to as Q

_{10}

.

To illustrate the application of the Q-Rule, let us assume that the stability of a product at 50Â°C is 32 days. The recommended storage temperature is 25Â°C and n = (50 - 25)/10 = 2.5. Let us set an intermediate value of Q = 3. Thus, Q^{n} = (3)^{2.5} = 15.6. The predicted shelf life is 32 days 3 15.6 = 500 days. This approach is more conservative when lower values of Q are used.^{12} Both Q-Rule and the bracket methods are rough approximations of stability. They can be effectively used to plan elevated temperature levels as well as the duration of testing in the accelerated stability testing protocol.

Theoretically, the Arrhenius equation does not apply when more than one kind of molecule is involved in reactions. However, if the degradation rate and temperature are linearly related, the prediction of shelf life can be approximated by the Arrhenius equation. Statistics that test the appropriateness of this approximation are presented in literature.

Magari et al. used a polynomial model to fit the degradation of a reagent (HmX PAK) for the Coulter HmX Analyzer.^{13} The following degradation pattern was consistent at all elevated temperatures:

β_{0}, β_{1}, and β_{2} are the parameters of the second-degree polynomial and t is time. The degradation rate is a function of time, which is not constant in this case.

Degradation at storage temperature can be predicted from the degradation at elevated temperatures as

The acceleration factor, λ, is based on the Arrhenius equation. Statistical tests indicated that the use of this equation was appropriate in this case. Shelf-life predictions were also verified by real-time stability testing results.

When most of the assumptions required to use the Arrhenius equation are not satisfied, comparisons to a product with a known stability is performed to assess shelf life. This approach requires having a similar product with a known shelf life to be used as a control. The new or test product is expected to demonstrate similar behavior to the control since they belong to the same family and have the same kinetics of degradation. Side-by-side testing of the control and test products at different elevated temperatures is then performed. It is necessary to assume that the same model can represent the degradation pattern at each elevated and storage temperature.

If the degradation patterns of the test and control samples at the same elevated temperatures are not statistically different, it can be assumed that they will degrade similarly at the storage temperature. The closer the elevated temperatures are to the storage temperature, the more confident we can be in making this statement. The experimental protocols used are similar to the protocols used with the Arrhenius equation. Degradation patterns of a family of products at certain elevated temperatures can be modeled and used to check the behavior of a new product that belongs to that family.

The complication that was alluded to with Equation 1 is that degradation models are usually nonlinear mixed models, where lot-to-lot variability is the random component. Estimation of the parameters of the models is important for the accuracy of shelf-life predictions. We recommend using the maximum likelihood (ML) approach to estimate these parameters.

Since no closed-form solutions for ML estimates exist, an iterative procedure is performed, starting with some initial values for the parameters and updating them until differences between consecutive iterations are minimal and the estimates converge to their final value. Initial values are usually chosen by experience. The closer these values are to the final values, the faster the model will converge. We used PROC NLMIXED of SAS for data analysis.^{14} Values of the real-time stability model (Equation 2) converged relatively quickly, while several initial values for the parameters of the accelerated model (Equation 6) were tried before they converged. Statistical theory and the applicability of ML estimation are common in the literature, and many computer routines are available to facilitate data analysis. However, experience with the modeling and estimation processes is necessary, since any unexpected results must be appropriately interpreted. It is quite easy to get useless numbers from a computer run.

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(2) ICH. Stability testing for new drug substances and products. Q1A(R). Geneva; 1994.

(3) ICH. Stability testing for new drug substances and products. Q1A(R2). Geneva; 2003.

(4) Tsong Y. Recent issues in stability testing, *J Biopharm Stat* 2003; 13:vii-ix.

(5) ICH. Stability data evaluation. Q1E. Geneva; 2003.

(6) ICH. Bracketing and matrixing designs for stability testing of new drug substances and products. Q1D. Geneva; 2003.

(7) King SP, et al. Statistical prediction of drug stability based on nonlinear parameter estimation. *J Pharm Sci* 1984; 73:2332-2344.

(8) Tydeman MS, Kirkwood TBL. Design and analysis of accelerated degradation tests for the stability of biological standards. 1. Properties of maximum likelihood estimators. *J Biol Stand* 1984; 12:195-206.

(9) Magari R. Estimating degradation in real time and accelerated stability tests with random lot-to-lot variation: a simulation study. *J Pharm Sci* 2002; 91:893-899.

(10) Tsong Y, et al. Shelf life determination based on equivalence assessment. *J Biopharm Stat* 2003; 13:431-449.

(11) Some I, et al. Stability parameter estimation at ambient temperature from studies at elevated temperatures. *J Pharm Sci* 2001; 90:1759-1766.

(12) Anderson G, Scott M. Determination of product shelf life and activation energy for five drugs of abuse. *Clin Chem* 1991; 37:398-429.

(13) Magari R, et al. Accelerated stability model for predicting shelf-life. *J Clin Lab Anal* 2002; 16: 221-226.

(14) SAS Institute. SAS/stat user's guide, vers. 8. Cary (NC); 2000. **BPI**