# Shelf Life Estimation from Temperature Accelerated Studies

In temperature-accelerated studies, shelf life at a storage temperature T is estimated from the shelf life at an elevated temperature T2, according to

Shelf life is referred to as tMm) when the lower specification limit of content is 90%. Shelf life exhibits a log-linear relationship versus 1/T in a given temperature range when the activation energy is constant (Fig. 193). The latter condition usually is only met when the degradation mechanism is the same across the temperature range of exposure. For example, a shelf life of 6 months at 40°C corresponds to a shelf life of 3 years at 25°C when an activation energy of 22.1 kcal/mol is assumed.

4.4.2.1. Experimental Design ofAccelerated Testing

Experimental designs for accelerated temperature testing were proposed in the 1960s by Tootill,766 Kennon,767 and Lordi and Scott,767 as well as many others. A practical chart for estimating shelf life from the data from accelerated testing at 41.5 and 60°C, which was proposed by Lordi and Scott (Fig. 194), is introduced here because of its historical value in drug stability studies, although it is no longer useful because the advent of computers has made more complex numerical calculations trivial. Shelf lives at 25, 41.5, and 60°C , t90CT, t90(41,5), and t90(60), respectively, are represented by

In 190(60) = 2 In 190(41.5) — In 190(25) (4.9)

Figure 193. Temperature dependence of shelf life.

The bold, solid descending lines in Fig. 194 represent ¿90^ as a function of 4,(25) at a specific ts,o(4i.5) represented on the right-hand y-axis. The intersection of a horizontal line representing ¿90(41.5) and a bold, solid descending line representing ¿90(60) corresponds to ¿90(25). Thus, ¿90(25) can be estimated from ¿90(41,5) and ¿900(60). Activation energy can be determined from the

Figure 194. A Lordi chart for estimating shelf life from accelerated data. (Reproduced from Ref. 768 with permission.)

dashed lines. The measures shown on both sides of the figure are used to calculate ^90(60) and 4>(4i,5) from the percent of drug remaining, F, at time t for a first-order reaction.

Presently, shelf life is usually estimated by regression analysis according to Eq. (4.8) and its variants using computers. A simplified method for shelf-life estimation, regardless of reaction order, has been proposed,769-772 and various computer programs have been developed. However, it should be noted that the application of Eq. (4.8) is limited to a temperature range in which Ea can be regarded as constant (as discussed in Chapter 2).

Using data for the rate of production of degradants in addition to data for drug loss significantly enhances the precise estimation of shelf life.773 774 This is especially the case when the formation of degradation products can be measured with higher precision than the drug loss, and only degradation data at the initial stage are used for the estimation.

4.4.2.2. Estimation of Shelf Life Using Accelerated-Test Datu at a Single Level of Temperature

It is theoretically possible to estimate the shelf life of a product from a single measurement of drug degradation at a single time point and temperature if the activation energy for the degradation is known. Of course, the quality of the estimate is strongly affected by assay error. For example, when a pharmaceutical having a shelf life of t90CT is stored at a temperature T for a time span t, the probability that degradation percentage is determined to be x using an assay method having a standard deviation of o is represented by Eq. (4.10) (for zero-order degradation kinetics).775 Thus, when degradation percentage is determined to be x, the true value of shelf life is t90P5> at a probability represented by Eq. (4.11).

where

The distribution of the true value of the shelf life is shown as a function of the observed value for the degradation percentage in Fig. 195. As the observed value of degradation percentage decreases, the mean and range of the shelf-life estimate increase. Figure 196 shows the probability that the shelf life is longer than 2 years as a function of degradation percentage after a storage at 40°C for 6 months. The probability depends on activation energy and assay error.

Figure 197 shows the relationship between the observed degradation percentage and estimated shelf life at a probability of 95%. If the activation energy is known, the shelf life can be estimated from only the value of the degradation percentage observed after storage at 40°C for 6 months with the use of this figure. When the value of the activation energy is unknown, the shelf life should be estimated assuming a smaller value of Ea than expected in

Figure 195. Distribution of true shelf life predicted from degradation percentage observed at 40°C. x: Percent degraded during 6-month storage at 40°C. Assay error (standard deviation): 2%; activation energy: 20 kcal/mol. (Reproduced from Ref. 775 with permission.)

order to obtain a conservative value for the shelf life. The actual shelf life may, in fact, be longer.

The shelf life estimated from more than three values of degradation percentage observed at an elevated temperature exhibits a smaller distribution range than that estimated from a single value shown in Fig. 195.775 This is because the use of more than three observed data values provides information on the data variation and enables a Monte Carlo simulation of degradation data.776 This simulation method yields a longer shelf-life estimate because of the smaller distribution range.

As described in Chapter 6, stability testing guidelines recommend 40°C as the temperature for accelerated testing. Figure 198 shows the probability that all the observed values of degradation percentage after storage at 40°C for 2, 4, and 6 months are less than 10% (assuming a 90% specification) as a function of the true shelf life. This probability depends largely on activation energy and assay error. For example, for an energy of activation of 10

Figure 196. Relationship between degradation percentage observed at 40°C and probability of shelf life longer than 2 years. Assay error (standard deviation):-, 0.5; — —1;---, 1.5; ---, 2%. (Reproduced from Ref. 775

with permission.)

% remaining

Figure 196. Relationship between degradation percentage observed at 40°C and probability of shelf life longer than 2 years. Assay error (standard deviation):-, 0.5; — —1;---, 1.5; ---, 2%. (Reproduced from Ref. 775

with permission.)

Figure 197. Relationship between degradation percentage observed at 40°C and shelf life estimated at 95%

probability. Assay error (standard deviation):-, 2; — . —, 1.5;---, 1; ■■■, 0.5%. (Reproduced from Ref. 775

with permission.)

0-1-1-1-1

60 70 80 90 100

% remaining

Figure 197. Relationship between degradation percentage observed at 40°C and shelf life estimated at 95%

probability. Assay error (standard deviation):-, 2; — . —, 1.5;---, 1; ■■■, 0.5%. (Reproduced from Ref. 775

with permission.)

kcal/mol, even a product with a shelf life of less than 1 year exhibits degradation percentage data less than 10% at an increasing probability as assay error increases.777

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