Stability of Drugs and Dosage Forms Sumie Yoshioka

72 Chapter 2 • Chemical Stability of Drug Substances
For example, the following relationships were employed by Rogers 322
Stelmach,323 respectively:
and Eriksen and
(2.85)
(2.86)
In these equations, T 0 and T are the temperatures at time zero and time t, respectively, and
b and a are constants. Assuming that the temperature changes according to Eq. (2.86), a rate
equation obtained by combining the first-order rate equation, Eq. (2.12), with Eq. (2.84) can
be integrated to give Eq. (2.87). The estimates of E a and k T0 (rate constant at T 0 ) can be
obtained by fitting the drug concentration versus time data to the following equation:
(2.87)
Estimations using Eq. (2.87) and variants of this equation were performed manually with limited
temperature programs and were not generally applicable to drug degradation studies. 324,325
New analysis methods using flexible temperature programs have been reported with the
increasing availability of computers to facilitate subsequent calculations. Zoglio and coworkers
326-328 proposed a method for obtaining optimal kinetic parameters. They described
the degradation versus time curve as a function of E a by using the arithmetic mean of an
individual rate constant at time t as the mean rate constant and by representing temperature
change in terms of a linear or polynomial expression. 326-328 Kay and Simon performed this
estimation using an analog computer. 329 Edel and Baltzer applied a stepped heating program
to this method. 330
In contrast to these approximate methods, the nonlinear regression methods reported
by Madsen et al. 331 and Tucker and Owen 332 utilized numerical integration of Eq. (2.88),
which is obtained from the general rate equation (Eq. 2.11) and Eq. (2.84). Equation (2.88)
becomes Eq. (2.89) for first-order degradation kinetics.
(2.88)
(2.89)
Hempenstall et al. 333 reported a calculation method that can be performed by simple
computers, whereby the degradation curve is represented by a polynomial equation in order
to easily obtain a rate constant k T at a temperature T. Using Eq. (2.90), k T can be represented
by Eq. (2.91) in the case of first-order degradation kinetics. Inserting the coefficients a 0 , a 1 ,
. . . , a n , [which are obtained by fitting the drug concentration versus time data to Eq. (2.90)]