# Factorial Analysis

In formulation studies, all the factors affecting the stability of the pharmaceutical product have to be considered. Because the stability of pharmaceuticals is generally affected by numerous and complex factors, quantitative analysis of the role of each would involve a very large and complex series of experiments. The effect of each individual factor would have to be tested under conditions in which all other factors are maintained constant. Factorial analysis attempts to minimize the number of experiments needed to get meaningful results and, therefore, to save time and labor. Figure 159. Remission function of ascorbic acid-lactose mixture as a function of colored sample content. •,

Powder; o, tablet. (Reproduced from Ref. 646 with permission.)

Figure 159. Remission function of ascorbic acid-lactose mixture as a function of colored sample content. •,

Powder; o, tablet. (Reproduced from Ref. 646 with permission.) Figure 160. Changes in dielectric constant of gelatin (a) and methylcellulose microcapsules (b) during aging at 45°C. (Reproduced from Ref. 650 with permission.)

Consider the following example. The dependence of degradation rate on reactant concentrations ([A], [B], [C], . . .) is described by additive terms for each degradation pathway, as represented by the rate equation (Eq. 2.4).657 Therefore, k^ [the terms inside brackets in Eq. 2.41 is described by Eq. (4.2) when, for example, hydroxide ion (B) and phosphate ion (C) catalyze the degradation independently.

k,bs = kb[B] + kc[C] + ■

On the other hand, a factor such as ionic strength that directly affects the rate constant k [Eqs. (2.4), (2.6), and (2.7)] enters into the rate equation as a product term:

kobs=/(A) {kb [B] + kc [c]+. . . }

To obtain kb, kc, . . ., and/(A), experiments are performed at two levels for each factor, such that 2n experiments are required when the number of factors is n. For example, 23 experiments are performed at low and high levels of each factor, as shown in Fig. 161, when three factors Figure 161. 23-factorial analysis. Upper-case letters (A, B, C) represent the high level of each factor; and lower-case letters (a, b, c) represent the low level. (Reproduced from Ref. 657 with permission.)

(A, B, and C) affect the degradation rate. Unless factor A interacts with the additive factors (B and C), kb, kc and/(A) can be obtained from the experimental results.657 The Plackett-Burman method (n factors are analyzed by performing n + 1 experiments)658 and the randomized block method are employed to perform factorial analysis experiments less than 2n.

The application of factorial analysis in stability studies of various pharmaceuticals has been reported. For example, the effects of temperature, ionic strength, buffer concentrations, and other factors on complex formation between polyvinylpyrrolidone and drug substances such as salicylic acid were analyzed.659 Factorial analysis was performed to elucidate the effects of excipients, light, humidity, temperature, and other factors on the stability of chlorpromazine hydrochloride,660 aspirin,661 662 and amphotericin B.663 Factorial analysis has also been performed in investigations of the physical stability of dosage forms. The effects of humidity and packaging on the dissolution rate of controlled-release theophylline tablets were analyzed using such a design.664

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