When [D]o >> [P]o, Eqs. (2.46) and (2.47) can be used to describe the kinetics.

An example that closely follows this scheme is the hydrolysis of NSC-373965 (Scheme 69), a water-soluble prodrug of NSC-284356, where P, in this case, is formaldehyde.265 Initial hydrolysis of D in this example to generate formaldehyde was not dependent on formaldehyde being present in the starting solution.

2.2.3.2. Kinetic Models Describing Chemical Drug Degradation in the Solid State

The rate equations used to describe drug degradation in solution can be derived theoretically on the basis of the proposed degradation mechanisms. Data can then be tested to see if they conform to the proposed scheme. When the scheme is validated, the appropriate rate constant can be calculated and used to further refine the model. Similar theoretical rate equations for drug degradation in the solid state have been derived. Because drug degradation in the solid state generally occurs in a heterogeneous system where the physical state of the drug and other components varies with time, the rate equations describing solid-state degradation are much more complicated than those for degradation in solution.

A descriptor rate constant for solid-state degradation can be obtained once a theoretical rate equation has been derived and the data have been tested to see if they conform to the proposed model. However, for solid-state degradation in which the factors affecting the degradation mechanism have not been elucidated, because of the complexity involved, often an apparent constant (or constants) obtained by fitting the observed degradation curve to an empirical equation or equations is utilized. Such constants and the empirical relationships themselves can sometimes be used for stability prediction purposes. This section first discusses various theoretical equations used to describe the solid-state stability of drugs and introduces an empirical equation that can often describe the data adequately.

2.2.3.2.a. Diffusion-Controlled Reaction—The Jander Equation (Three-Dimensional

Diffusion). For a model in which a sphere of a reactant B exists in another reactant A (Fig. 18), a rate equation for reaction between A and B at the interface was derived by Jander in

the 1920s. When a reaction starts at the interface of a sphere B, with radius r, and proceeds inside the sphere forming a reaction product phase with a thickness y, the fractional decomposition, x, is given by Eq. (2.48), and the value of y is given by Eq. (2.49).

Assuming that the rate at which the thickness of the product phase, y, increases is proportional to the diffusion rate of A into B yields Eq. (2.50), which is integrated to give Eq. (2.51).

dt y

In these equations, [A]A is the concentration of A in the A phase, [A]B is the concentration of A in the interfacial area, and D is the diffusion constant. The Jander equation (Eq. 2.52) is derived from Eqs. (2.50) and (2.51).

Jander reported that the decarboxylation of inorganic carbonate can be described by Eq. (2.52), as shown in Fig. 19.266,267

CaCO3 + MoO3 ~ CaMoO4 + CO2 BaCO3 + SiO2 «> BaSiO2 + CO2 Assuming that the rate at which the thickness of the product phase, y, increases (Eq. 2.49), independent ofy, as represented by Eq. (2.53), yields Eq. (2.54). Prout and Tompkins reported that the decarboxylation of mercuric oxalate (Hg2C2O4) conforms to Eq. (2.54).268

10 20 30 40 SO 60 70 80 90 100

time (min)

10 20 30 40 SO 60 70 80 90 100

time (min)

Figure 19. Decarboxylation of inorganic carbonate in the solid state as described by the Jander equation. (Reproduced from Ref. 266 with permission.)

Figure 19. Decarboxylation of inorganic carbonate in the solid state as described by the Jander equation. (Reproduced from Ref. 266 with permission.)

The Jander equation has been applied to the degradation kinetics of various pharmaceuticals. For example, the degradation of freeze-dried thiamine diphosphate (Fig. 20) 269-270 and the degradation of propantheline bromide in the presence of aluminum hydroxide gel (Fig. 21)271 have been described by the Jander equation.

2.2.3.2.b. Autocatalytic Reactions Controlled by Formation and Growth of Reaction Nuclei. When a reaction proceeds with a growing number of nuclei or imperfection sites, it tends to exhibit an S-shaped degradation curve, where the rate of product formation depends on the rate of nuclei formation and growth, as represented generally by Eq. (2.55), where 1, m, and n are constants. And x is as defined in Eq. (2.48).

Equation Describing the Initial Stage of an Autocatalytic Reaction. The rate of an autocatalytic reaction is proportional to the number of nuclei (TV). Nuclei represent imperfection sites in the crystal where it is assumed that chemical reactions can take place. It is also assumed that as chemical reactions proceed, strain is placed on the crystal, resulting in more imperfections. The rate at which N increases at the initial stage of this autocatalytic process is described by Eq. (2.56). Equation (2.57) then describes the reaction rate.

Figure 20. Degradation of freeze-dried thiamine diphosphate plotted according to the Jander equation (11% RH). (Reproduced from Ref. 270 with permission.)

Figure 20. Degradation of freeze-dried thiamine diphosphate plotted according to the Jander equation (11% RH). (Reproduced from Ref. 270 with permission.)

Equation (2.57) is equivalent to Eq. (2.55) when l and m equal zero. Equation (2.58), obtained by integrating Eq. (2.57), describes the initial hydrolysis rate of meclofenoxate hydrochloride in the solid state.272 273 Similarly, this equation describes the hydrolysis of propantheline bromide under high-humidity conditions, as shown in Fig. 22.274 Equation (2.58) has also been applied to the initial degradation of aspirin derivatives in the solid state (Fig. 23).275

The Prout-Tompkins Equation. It can be argued that, as more stress/nuclei appear, the rate of nuclei formation will eventually decrease. That is, when termination of the nuclei

Figure 21. Degradation of propantheline bromide in the presence of aluminum hydroxide gel plotted according to the Jander equation (75% RH). Drug: aluminum hydroxide gel = 50:4950 (weight ratio). (Reproduced from Ref. 271 with permission.)

Figure 21. Degradation of propantheline bromide in the presence of aluminum hydroxide gel plotted according to the Jander equation (75% RH). Drug: aluminum hydroxide gel = 50:4950 (weight ratio). (Reproduced from Ref. 271 with permission.)

occurs in addition to their propagation, the number of nuclei is determined by the probability of propagation (a) and the probability of termination (B), as described by Eq. (2.59). Equation (2.60) represents the reaction rate and is equivalent to Eq. (2.55) when 1, m, and n are equal to 1, 1, and 0, respectively.

Prout and Tompkins reported that the thermal degradation of potassium permanganate can be described by Eq. (2.61), which is obtained by integrating Eq. (2.60).276 Equation (2.60) adequately describes the degradation of solid aspirin in the presence of limited moisture, as shown in Fig. 24.277

The Kawakita Equation. Because the value of l in Eq. (2.55) depends on the catalytic effect of the degradation product, a more general form of Eq. (2.60) is Eq. (2.62). Kawakita reported that the reduction of ferric oxide can be described by Eq. (2.62) with various values of I, depending on temperature.278

The Avrami Equation. Equation (2.55) yields Eq. (2.63) when l = 0 and m = 1.

Equation (2.63) describes the reaction between ZnO and BaCC>3. It is also used to describe the rate of some polymorphic transitions279 280 described in Chapter 3.

2.2.3.2.c. Reaction Forming a Liquid Product-the Bawn Equation. A rate equation proposed by Bawn is applicable to reactions of solids forming gaseous and liquid products. In the latter case, the reaction rate is described by summing the reaction rates in the solid state and those in the solution formed by the liquid product:

where S is the solubility of the drug in the liquid formed, and ks and k, are the rate constants in the solid state and in solution, respectively. In Eq. (2.64), Sx, the product of the fraction degraded, x, and the solubility S, represents the molecular fraction of drug in solution, and (1 - x - Sx) represents the fraction in the solid state. Integrating Eq. (2.64) gives Eq. (2.65), which has been used to describe the decarboxylation of various benzoic acid derivatives.281 Decarboxylation of various alkoxyfuroic acids such as 5-(tetradecyloxy)-2-furoic acid282 and octyloxy furanoic acid283 284 was adequately described by Eq. (2.65), as shown in Figs. 25 and 26, respectively.

2.2.3.2.d. Reaction Controlled by an Adsorbed Moisture layer—The Leeson-Mattocks Equation. Leeson and Mattocks proposed a model in which drug degradation occurs in an adsorbed moisture layer. An S-shaped degradation curve observed for aspirin in the solid state (Fig. 27) was explained by this model.285 286 Aspirin was assumed to be dissolved rapidly in an adsorbed moisture layer so as to form a saturated solution in which decomposition occurs. It was assumed that the decomposition was catalyzed by hydronium ion, as described by Eq. (2.66), and that the rate increased as the amount of salicylic acid formed (x) increased, thus yielding an S-shaped curve.

60 120 180 240 300 time (h)

Figure 26. Decarboxylation of octyloxy furanoic acid plotted according to the Bawn equation. A = (Ski - Sk, - k,) / k,. (Reproduced from Ref. 283 with permission.)

60 120 180 240 300 time (h)

Figure 26. Decarboxylation of octyloxy furanoic acid plotted according to the Bawn equation. A = (Ski - Sk, - k,) / k,. (Reproduced from Ref. 283 with permission.)

0 80 160 240

time (day)

Figure 27. Time come of degradation of solid aspirin explained by the Leeson-Mattocks equation (60°C 80.6% RH). (Reproduced from Ref. 285 with permission of the American Pharmaceutical Association.)

In the above equation, V is the volume of the layer of adsorbed moisture, and K is the ionization constant of the degradant. Subsequent studies showed that the reaction was not affected by the formed salicylate, thus invalidating the model that led to Eq. (2.66).275

If it is assumed that the degradation rate of aspirin in the adsorbed moisture layer is determined by aspirin concentration, [D], the amount of moisture, [H2O], the volume of the adsorbed moisture layer, V, and a rate constant, k, then the reaction should be described by Eq. (2.67).

Each of these parameters was experimentally established by Carstensen and co-workers and utilized to predict the degradation curve.277 287 288 The predicted degradation curve diverged from the observed curve, indicating that the adsorbed moisture layer theory cannot fully explain the degradation of aspirin (Fig. 28).

For reactions in which the catalytic effect of degradation products is negligible and the volume of the adsorbed moisture layer and the drug solubility can be regarded as constant, the rate should be described by Eq. (2.68) according to the adsorbed moisture layer, and the degradation should conform to apparent zero-order kinetics.

Oxidative degradation of solid sulpyrine in the presence of moisture was described by Eq. (2.68).289 This equation was also used to describe decarboxylation of 4-aminosalicylic acid.290,291

0 100 200 300 400

Figure 28. Aspirin degradation curve (62.5°C,water content: 10%). Observed; ♦, predicted from Eq. (2.67). (Reproduced from Ref. 277 with permission.)

0 100 200 300 400

Figure 28. Aspirin degradation curve (62.5°C,water content: 10%). Observed; ♦, predicted from Eq. (2.67). (Reproduced from Ref. 277 with permission.)

2.2.3.2.e. Use of Empirical Expressions Such as the Weibull Equation. It is often difficult to derive the rate equation for degradation of drugs in the solid state because the physical state of the system varies with time in complex ways. This is especially true for degradation in the presence of moisture, such that derivation of adequate rate equations is often impossible. Empirical equations such as the Weibull equation (Eq. 2.69) have been used to describe this kind of solid drug degradation. Degradation curves for various drugs have been fitted to the Weibull equation without significant deviations (Fig. 29).292 Weighted least-squares analysis of data fit to the Weibull equation has resulted in smaller deviations.293

1n1n

1n1n

2.2.3.3. Calculation of Rate Constants by Fitting to Kinetic Models

A rate constant for drug degradation can be obtained by fitting an observed degradation curve to a suitable kinetic model chosen on the basis of a proposed mechanism. Since any experimental data include errors, least-squares regression analysis is usually carried out to test the validity of the model and to calculate the apparent rate constant(s).

The concentration of remaining drug, [D], and the fraction of degradation, x, are generally represented by a linear equation for zero-order kinetics or linearized forms of various equations, e.g., log [D], versus time for first-order kinetics. Recently, with the advent of computers, nonlinear regression analysis has become popular. Various methods for obtaining accurate estimates utilizing linear regression analysis were developed,294 and some of these are still used as a method of analysis or to obtain initial values needed in nonlinear regression analysis. Various programs are available for the preliminary estimation of parameters for drug degradation kinetics of various orders.295 Computer programs such as MULTI,296 NONLIN, and other commercial software programs are also used, especially spreadsheet programs such as EXCEL, which allow one to perform nonlinear regression analyses very easily.

In the estimation of rate constants through the fitting of degradation data to a kinetic model, the validity of the model and the reliability of the estimated rate constant should be evaluated, taking into account experimental errors. Additional data are sometimes required to obtain accurate estimates. For example, in the case of consecutive reactions, the time courses for both the parent drug and the intermediate are required to estimate the pseudofirst-order rate constant for the formation and loss of the intermediate (see Section 2.2.3.7.f), especially when k/k, is larger than 0.5.297

2.2.4. Temperature

Temperature is one of the primary factors affecting drug stability. The rate constant/ temperature relationship has traditionally been described by the Arrhenius equation,

where Ea is the activation energy and A is the frequency factor. This equation is a variant of the equation describing the effect of temperature on equilibrium processes that was developed by van't Hoff in 1887. Arrhenius applied his equation to various reaction processes.298 Comparison of the empirical Arrhenius equation to the Eyring equation, Eq. (2.7), shows some similarities and some differences. The frequency factor A in the Arrhenius equation corresponds to the product of the universal collision and entropy terms in Eq. (2.7) while the Ea term in Eq. (2.70) is related to the enthalpy term in Eq. (2.7).299 Because aplot of the logarithm of k against the reciprocal of absolute temperature generally yields a linear relationship (Arrhenius plots), the frequency factor and activation energy are regarded as independent of temperature, and the activation energy, Ea, is used as a measure of the temperature dependence of the rate constant. However, the Eyring equation suggests that both A and Ea should be temperature-dependent. Ea can be shown to be related to AH as indicated in Eq. (2.71). Observed linear Arrhenius plots can be explained by the much larger temperature dependency of the exponential term in Eq. (2.70) as compared to that of A. In theory, however, Arrhenius plots should not be linear.

Nevertheless, Arrhenius plots have been traditionally used to describe the temperature dependency for various chemical reactions by regarding A and Ea as independent of temperature. A prerequisite for the application of Eq. (2.70) [and Eq. (2.7)] is that the degradation mechanism does not change in the temperature range of interest. Ea values of about 10-30 kca/mol (40-130 kJ/mol) are generally observed in the degradation of drug substances. Table 4 shows Ea values for degradation of representative drug substances. The values of Ea are presented in units of calories per mole rather than kilojoules per mole because those were the units reported in the original reference sources (1 kcal/mol = 4.18 kJ/mol).

As an alternative to Arrhenius plots, the data can be fitted to the Eyring equations: k

A plot of In k/T versus 1/T is linear. Thus, plots of either k versus 1/T or k/T versus 1/T are usually plotted by taking either the natural logarithm (In) or the logarithm to the base 10 (log) of k. The slopes of plots of log k or log k/T versus 1/T are -Ea/2.303RT and -AHt/2.303 RT, respectively.

Temperature is obviously an important parameter because most reactions proceed faster at elevated temperatures than at lower temperatures. The terms Ea and AH are a measure of how sensitive the degradation rate of a drug is to temperature changes. Table 5 shows the effect of a 10°C change in temperature on the rate constant. If the Ea for a degradation process is only 10 kcal/mol, this temperature change results in only a 1.76-fold change in drug reactivity. However, if the Ea is 30 kcal/mol, a 10°C increase in temperature results in about a 5.5-fold increase in the degradation rate.

2.2.4.2. Quantitation of the Temperature Dependency of Degradation Rate Constants

Estimation of an appropriate rate or rate constant for drug degradation is an important step in predicting the stability of pharmaceuticals. Knowing how such a rate or rate constant changes with temperature in a quantitative way may allow one to predict the stability at other temperatures. Even if a rate or rate constant cannot be estimated by fitting the data to a theoretical or empirical equation, constants such as time required for 10% degradation (¿,0) can be utilized instead of rate constants. Stability prediction is possible, for example, from the relationship between the reciprocal of t9„ and temperature.

In the previous section, the Arrhenius equation was described. The Arrhenius equation was applied to the prediction of drug degradation in the 1,40s and 1,50s. Taking the logarithm of both sides of Eq. (2.70) yields

Drughydrolysis |
Ea (kcal/mol) |
PH° |
Reference |

Hydrolysis | |||

Ampicillin |
9.2b |
9.78 |
65 |

18.3b |
4.93 |
65 | |

Cefotaxime |
9.4 |
8.94 |
73 |

24.7 |
5.52 |
73 | |

Echothiophateiodide |
10 |
OH- |
49 |

23 |
H+ |
49 | |

Indomethacin |
10.1 |
11 |
58 |

Methylphenidate |
12.35 |
OH- |
19 |

15.98 |
H+ |
19 | |

Oxazolam |
12.4 |
8.0 |
93 |

Succinylcholine chloride |
13.09 |
OH- |
32 |

17.23 |
H+ |
32 | |

Haloxazolam |
15.2 |
1.3 |
94 |

Mitomycin C |
16.2bc |
5.20 |
105 |

Procaine |
16.8 |
H+ |
10 |

Hydrocortisone sodium phosphate |
17.0 |
H+ |
47 |

Diazepam |
17.2 |
10.18 |
87 |

Atropine |
17.2 |
H+ |
16 |

20.9 |
H+ |
26 | |

Acetaminophen |
17.42 |
6 |
53 |

Phenethicillin |
17.6 |
6.7 |
68 |

Amobarbital |
18.6b |
10.12 |
80 |

Benzocaine |
18.6 |
H+ |
10 |

Ethylparaben |
18.7 |
9.16 |
22 |

Meperidine |
18.77 |
6 |
28 |

Nitrofurantoin |
18.9 |
H+ |
109 |

Rifampicin |
19.2 |
H+ |
111 |

Oxazepam |
20.0 |
3.24 |
87 |

26.1 |
8.52 |
87 | |

Benzylpenicillin |
20.3 |
2.7 |
64 |

Thiamine hydrochloride |
21 |
9.90 |
102 |

29 |
1.70 |
102 | |

Cefadroxil |
21.4 |
7.00 |
71 |

Carmethizole |
21.5 |
9.90 |
44 |

Chlorphenesin carbamate |
21.8 |
H+ |
43 |

Cephalothin |
22.6 |
5.00 |
70 |

28.5 |
10.00 |
70 | |

Sulfacetamide |
22.9d |
7.40 |
60 |

Carbenicillin |
23.4 |
4.45 |
67 |

27.45 |
10.47 |
67 | |

Furosemide |
23.5 |
H+ |
100 |

Chloramphenicol |
24.0 |
6.00 |
15 |

Chlorambucil |
24.4= |
7.00 |
120 |

Pilocarpine |
25.02 |
OH- |
37 |

Cocaine |
26.2 |
6.25 |
34 |

Moricizine |
27.5 |
6.0 |
61 |

Clindamycin |
38.0 |
1.10 |
(continued) |

Table 4. Continued

Drug hydrolysis

Reference

Racemization and epimerization

Etoposide

Hetacillin

Epinephrine

Cefsulodin

Pilocarpine

Dehydration Prostaglandin E1

Isomerization Amphotericin B Prostaglandin E1

Decarboxylation Etodolac

Oxidation Ascorbic acid

Morphine Procaterol

18 24

16.4

34.5

156 163 158 166 37

142 142

148 142

119 179 202 184

°H+, Hydroniumion-catalyzedreaction; OH-,hydroxideion-catalyzedreaction. ^ Activation enthalpy AH

cValue reported in the reference (68 kJ/mol) has been converted to kilocalories per mole. 'Value reportedinthe reference (95.9 kJ/mol) has been converted tokilocalories permole. eValue reportedinthe reference (102 kJ/mol) has been converted to kilocalories permole. riValue reported in the reference (76 kJ/mol) has been converted to kilocalories per mole.

So-called Arrhenius plots in which the logarithm of rate (rate constant) is plotted against the reciprocal of absolute temperature (degrees kelvin) have been used to validate the conformity of the degradation rates of various drugs to this equation. Linear Arrhenius plots were reported for the degradation of penicillin G (Fig. 30),300 the discoloration of a liquid

(cal/mol) |
k k T2/ T1 (T1= 20°C, T2 = 30°C) |

10,000 |
1.76 |

15,000 |
2.34 |

20,000 |
3.11 |

25,000 |
4.12 |

30,000 |
5.48 |

Figure 30. Arrhenius plots for the degradation of penicillin G at pH 1.2 and pH 4.56. Note: The authors plotted the values of 1/T from high to low values (rather than from low to high values as is normally done). (Reproduced from Ref. 300 with permission.)

Figure 30. Arrhenius plots for the degradation of penicillin G at pH 1.2 and pH 4.56. Note: The authors plotted the values of 1/T from high to low values (rather than from low to high values as is normally done). (Reproduced from Ref. 300 with permission.)

multisulfa preparation (Fig. 31),301 and the degradation of liquid multivitamin preparations (Fig. 32),302303 to give a few examples. A linear Arrhenius plot indicates that, once degradation rates are obtained at several temperature levels, the degradation rate at some other specific temperature can be estimated. Thus, the Arrhenius equation has been successfully applied to the prediction of the stability of various pharmaceuticals: the degradation of vitamin A dosage forms,304 the change in appearance of tablets and powders,305,306 the degradation of multivitamin tablets,307 and many other examples too numerous to cite here.

The Arrhenius equation is valid in the temperature range where both A and Ea in Eq. (2.70) can be regarded as constant. Changes in the degradation mechanism with temperature may result in nonlinear Arrhenius plots. A very low degradation rate of phenylbutazone tablets was observed at 50°C in contrast to a high rate at 60°C, suggesting a change in the degradation mechanism.308 The Arrhenius plots for the hydrolysis of soybean phosphatidyl-choline exhibited different slopes at temperatures above and below its phase-transition temperature (Fig. 33).309 The slopes also changed with changes in the pH of the solution.

Figure 31. An Arrhenius plot for the discoloration of a liquid multisulfa preparation (Reproduced from Ref. 301 with permission of the American Pharmaceutical Association.)

Figure 31. An Arrhenius plot for the discoloration of a liquid multisulfa preparation (Reproduced from Ref. 301 with permission of the American Pharmaceutical Association.)

2.9 3.0 3.1 3.2 3.3 3.4 (xlO""3«"1)

Figure 32. An Arrhenius plot for the degradation of thiamine hydrochloride in a liquid multivitamin preparation (Reproduced from Ref. 303 with permission of the American Pharmaceutical Association.)

2.9 3.0 3.1 3.2 3.3 3.4 (xlO""3«"1)

Figure 32. An Arrhenius plot for the degradation of thiamine hydrochloride in a liquid multivitamin preparation (Reproduced from Ref. 303 with permission of the American Pharmaceutical Association.)

On the other hand, the degradation rate of amorphous indomethacin at a temperature slightly lower than its melting point yielded the same linear Arrhenius plots as obtained for the molten substance (Fig. 34).310

A recent paper proposed that the hydrolysis rate of aspirin could not be predicted on the basis of the Arrhenius plots, owing to changes in the activation energy in the temperature range 30-70°C.311 The authors proposed that the structures of icebergs formed near the hydrophobic groups in the activated complex changed between the temperature ranges below 42°C, 42-58°C, and above 58°C. However, this interpretation has been questioned based on the statistical uncertainty of the observed differences in the activation energy.312313

Figure 33. Arrhenius plots describing the degradation of soybean phosphatidylcholine at three pH values. Phosphatidylcholine concentration: 0.05M(Reproduced from Ref. 309 with permission.)

Figure 33. Arrhenius plots describing the degradation of soybean phosphatidylcholine at three pH values. Phosphatidylcholine concentration: 0.05M(Reproduced from Ref. 309 with permission.)

Figure 34. An Arrhenius plot for the degradation of indomethacin in amorphous (o) and molten (A) states. (Reproduced from Ref. 310 with permission.)

2.2.4.2.a. Prediction of Degradation Rate by Linear Regression Analysis of the Arrhenius Equation. Drug degradation rates at room temperature can be estimated by using Arrhenius plots of the rate constants calculated at each of several temperatures, as described above. This regression method using multiple rate constants calculated separately at different temperatures has been called "classical Arrhenius analysis" and is still used as a basic method for the prediction of drug stability. However, this classical analysis may result in large errors in the estimated rate constant. Slater et al.3 14 determined the degradation rate of vitamin A in multivitamin tablets at 40, 45, 50, and 55°C and obtained the Arrhenius plot shown in Fig. 35. Estimation of the rate at 25°C according to the regression equation derived from this plot resulted in a calculated degradation rate curve that deviated significantly from that observed experimentally, as shown in Fig. 36.314

0.0641 0.0481 0.0321

0.0641 0.0481 0.0321

0.0161

Figure 35. An Arrhenius plot for the degradation of vitamin A in multivitamin tablets, obtained by classical analysis. The upper and lower curves represent 95% significance limits. (Reproduced from Ref. 314 with permission.)

0.0161

0.0001

Figure 35. An Arrhenius plot for the degradation of vitamin A in multivitamin tablets, obtained by classical analysis. The upper and lower curves represent 95% significance limits. (Reproduced from Ref. 314 with permission.)

Figure 36. Time course of degradation of vitamin A at 25°C. —, Estimated using upper and lower predicted k values from the Arrhenius plot, — experimentally observed (upper and lower curves represent 95% significance limits. (Reproduced from Ref. 314 with permission.)

In classical Arrhenius analysis, the rate constants are calculated by fitting normal degradation versus time data, which have inherent experimental errors, and then fitting the calculated rate constants again to the Arrhenius equation. The errors included in the original data points are thus not directly reflected in the Arrhenius plots. Therefore, the weight given in the analysis to a single data point varies if the number of data points differs among the different temperature levels. Furthermore, the variation of estimates of rate constant obtained from the Arrhenius plots becomes larger owing to the reduced number of degrees of freedom. This occurs because the number of data points used in the Arrhenius regression analysis (the number of calculated rate constants) is smaller than the number of original data points used to estimate the original experimental rate constants.

The Arrhenius regression analysis can also be performed by using the multiple rate constants calculated from each of the drug concentration/time data points observed at a single temperature level. As shown in Fig. 37, the data at several temperature levels are fitted to Eq. (2.75) (for first-order degradation kinetics), which is obtained by replacing the rate constant k of Eq. (2.70) with the concentration of remaining drug, [D]:

where [D], is initial drug content. This "modified Arrhenius analysis"314-316 uses the same weight for each data point and provides an estimate of the rate constant at room temperature with a smaller 95% confidence interval owing to the larger number of degrees of freedom (the larger number of data points used). Nevertheless, in the analysis of the degradation of vitamin A, this method provided a regression curve similar to that obtained by classical Arrhenius analysis, resulting in a calculated room-temperature degradation curve which deviated significantly from that observed experimentally.

The large deviation between the estimated rate constants for a given temperature extrapolated from classical (Fig. 35) and modified Arrhenius regression analysis (Fig. 37) and the experimental rate constants determined at the actual temperature is due to the linear regression analysis method used. That is, the values of the logarithms of the rate constants do not reflect directly the errors in the experimental data. As Bentley pointed out,317 when the error term e is added to the logarithm of k in accordance with Eq. (2.76), in the linear

Figure 36. Time course of degradation of vitamin A at 25°C. —, Estimated using upper and lower predicted k values from the Arrhenius plot, — experimentally observed (upper and lower curves represent 95% significance limits. (Reproduced from Ref. 314 with permission.)

least-squares regression analysis of the rate constant k at a temperature T, this results in a larger contribution from the data obtained at lower temperatures than from those obtained at higher temperatures (Fig. 38). To avoid this problem, Bentley proposed a weighted least-squares analysis in which the error term e is added to k as described by Eq. (2.77) rather than Eq. (2.76).

Weighted least-squares analysis provides the variance of experimental errors on the basis of which the linearity of the Arrhenius plots can be assessed.

2.2.4.2.b. Prediction of Degradation Rate by Nonlinear Regression Analysis of the Arrhenius Equation. Although linear regression analysis provides biased results and weighted least-squares analysis is required to improve the estimates, nonlinear regression analysis does not suffer from the same problem. Representing Eq. (2.75) in a nonlinear manner yields Eq. (2.78) for first-order degradation kinetics.318 Similarly, Eq. (2.79) is obtained for zero-order degradation kinetics.

30 n

30 n

2.9 3.0 3.1 3.2 3.3 3.4 3.5 2,9 3.0 3.1 3.2 3.3 3.4 3,5

Figure 38. Linear Arrhenius regression by weighted least-squares analysis (a) and least-squares analysis (b). _y-axis: (1) logarithmic scale, (2) arithmetical scale. (Reproduced from Ref. 317 with permission.)

2.9 3.0 3.1 3.2 3.3 3.4 3.5 2,9 3.0 3.1 3.2 3.3 3.4 3,5

Figure 38. Linear Arrhenius regression by weighted least-squares analysis (a) and least-squares analysis (b). _y-axis: (1) logarithmic scale, (2) arithmetical scale. (Reproduced from Ref. 317 with permission.)

The frequency factor A corresponds to the rate constant at infinite temperature. Replacing A with ¿298, which has a more practical meaning than A [rate constant of degradation at 25°C represented by Eq. (2.80)], yields Eq. (2.81).

Again, replacing ^ in Eq. (2.81) with the shelf life of a pharmaceutical, t90 (time required for 10% degradation), yields Eq. (2.82). Similarly, Eq. (2.83) is obtained for zero-order degradation kinetics.

-0.1054i

X |
f 1 l) |
1] | ||||||||||||||||||||

R |
298 T | |||||||||||||||||||||

L |
\ ^ |
The estimates of tM and E, can be obtained directly by the nonlinear regression analysis of the observed degradation versus time data according to Eq. (2.82) or Eq. (2.83). A Monte Carlo simulation study performed by King et al.319 suggested that nonlinear regression analysis could provide more reliable estimates, with smaller deviations and biases, than does linear analysis. As shown in Fig. 39, nonlinear analysis produced an estimate of t90 closer to the theoretical value with smaller and more symmetrical 95% confidence limits than did linear analysis. 2.2.4.2.c. Nonisothermul Prediction of Degradation Rate Rate Equation for Nonisothermal Prediction. In the previous sections, the estimation of rate constants at room temperature by linear and nonlinear Arrhenius regression of the degradation data determined at several fixed temperatures was described. The method of nonisothermal prediction of degradation rate was developed to reduce the experimental effort required by allowing kinetic parameters to be estimated from a single set of drug concentration versus time data obtained while the temperature is changed as a function of time according to some algorithm. For the nonisothermal prediction of degradation rate, the rate constant k is represented by Eq. (2.84), in which the term, T, or temperature, in the Arrhenius equation is replaced by a temperature time function, G( t). The basic theory for nonisothermal prediction of degradation rate was established in the 1950s.320 321 Its application to the stability prediction of pharmaceuticals was reported by Rogers322 and extended by Eriksen and Stelmach.323 Initial temperature programs or algorithms used relationships that could easily be integrated when inserted into Eq. (2.84). For example, the following relationships were employed by Rogers322 and Eriksen and Stelmach,323 respectively: In these equations, ? 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 Ea and kj> (rate constant at ?) can be obtained by fitting the drug concentration versus time data to the following equation: 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 co-workers326-328 proposed a method for obtaining optimal kinetic parameters. They described the degradation versus time curve as a function of Ea 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 Owen332 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. 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? at a temperature T. Using Eq. (2.90), k? can be represented by Eq. (2.91) in the case of first-order degradation kinetics. Inserting the coefficients a>, ab . . . , an, [which are obtained by fitting the drug concentration versus time data to Eq. (2.90)] into Eq. (2.91) yields individual kT values at a series of temperatures, T, which are then used to estimate kinetic parameters according to the Arrhenius equation (Eq. 2.70). Yoshioka et al.334 reported a method for obtaining the estimates of t90 and E, directly from the concentration versus time data using an equation obtained by replacing the frequency factor A with t90 Equation (2.92) was used to describe the process for first-order degradation kinetics. where In addition, many papers have reported various methods for nonisothermal estimation of kinetic parameters and its application to stability prediction of pharmaceuticals.335-347 In addition, many papers have reported various methods for nonisothermal estimation of kinetic parameters and its application to stability prediction of pharmaceuticals.335-347 Reliability of Kinetic Parameters Estimated by Nonisotheml Prediction. The reliability of the kinetic parameters estimated by fitting degradation data observed under varying temperature conditions to a nonisothermal rate equation depends largely on the temperature program used. The accuracy and precision of kinetic parameter estimation using various nonisothermal temperature programs were studied by using a series of Monte Carlo simu-lations.334 348 Three hundred sets of experimental data with specified temperature and assay errors were generated using the 12 linear temperature programs shown in Fig. 40. These data sets were then analyzed according to Eq. (2.92), resulting in estimates of t«, and Ea values. All of the temperature programs used provided estimates with similar means, which were close to the theoretical values, but with significantly different variations. For example, a SO 100 150 100 20 25 30 20 25 30 Figure 41. Effect of linear temperature programs on tx (a) and Ea (b) estimates and variations. Theoretical value of t90, 100 weeks: theoretical value of Ea, 25.00 kcal/mol; assay error (standard deviation): 2%. (Reproduced from Ref. 334 with permission.) SO 100 150 100 20 25 30 20 25 30 Figure 41. Effect of linear temperature programs on tx (a) and Ea (b) estimates and variations. Theoretical value of t90, 100 weeks: theoretical value of Ea, 25.00 kcal/mol; assay error (standard deviation): 2%. (Reproduced from Ref. 334 with permission.) marked difference in the variations of the tM, and Ea estimates was observed between temperature programs 1 and 5 when the theoretical values of tM, and Ea were 100 weeks and 25 kcal/mol, respectively (see Fig. 41). The variation of the estimates depended on the temperature variation range and the final percentage of degraded drug, so that estimates with smaller variations were obtained by using a program providing a larger temperature range and a higher percentage degraded (see Fig. 42). The t90, estimate is affected by the number of observations at temperatures near room temperature; thus, estimates with smaller variations are obtained by using a program including temperatures near room temperature. For example, when an Ea value of 10 kcal/mol and programs 3 and 6 were used, estimates with enormous variations were obtained. Although temperature programs providing a higher final percentage degraded and a larger temperature range and those including temperatures close to room temperature are needed to obtain estimates with small variations, optimization of temperature programs is not easily achieved due to the interrelationship of these factors. As shown in Fig. 43, nonlinear temperature programs with different patterns (Fig. 44) provided estimates with different variations even when the final degradation ratios were the same. The Monte Carlo simulation study suggested that estimates with relatively small variations for a pharmaceutical having a tM, of approximately 3 years can be obtained by using the program T(°C) = 25 + 1.56 x 10-5 /■'(week) for a 40-week experiment and the program T(°C) = 25 + 4t (week) for a 10-week experiment. Advantages of Nonisothemal Prediction. In both isothermal and nonisothermal predictions of degradation rate at room temperature from data obtained at elevated temperatures, degradation data that include higher overall degradation yield more reliable estimates. Therefore, the difference in the precision of the estimates between isothermal and Figure 42. Effects of final degradation ratio and temperature variation range on Ea (a) and f90 (b) estimates. Temperature variation range: —, 15°C; -, 30°C;---, 45°C. o, Ea =25 kcal/mol; a, Ea = 10 kcal/mol. (Reproduced from Ref. 334 with permission.) Figure 42. Effects of final degradation ratio and temperature variation range on Ea (a) and f90 (b) estimates. Temperature variation range: —, 15°C; -, 30°C;---, 45°C. o, Ea =25 kcal/mol; a, Ea = 10 kcal/mol. (Reproduced from Ref. 334 with permission.) nonisothermal predictions is due to the difference in the final percentage degradation achieved. In the prediction of degradation rate according to the Arrhenius equation for degradations exhibiting slightly curved Arrhenius plots, nonisothemal prediction may provide an Figure 43. Effect of nonlinear temperature programs on the variance of t90 estimate (effect of n in the nonlinear temperature program, T (°C) = 25 + kf). Assay error (standard deviation): 2%; temperature error (standard deviation), 0.5°C. Theoretical value of t90, 156 weeks; Ea, 25 kcal/mol. (Reproduced from Ref. 348 with permission.) Figure 43. Effect of nonlinear temperature programs on the variance of t90 estimate (effect of n in the nonlinear temperature program, T (°C) = 25 + kf). Assay error (standard deviation): 2%; temperature error (standard deviation), 0.5°C. Theoretical value of t90, 156 weeks; Ea, 25 kcal/mol. (Reproduced from Ref. 348 with permission.) estimate with a smaller bias than does isothermal prediction. Monte Carlo simulation studies showed that a tm estimate close to the theoretical value was obtained for degradation exhibiting the slightly curved Arrhenius plots shown in Fig. 45 when nonisothermal prediction was used [temperature program: T(°C) = 25 + 4 t (week)] rather than isothermal prediction using four levels of constant temperature (50, 60, 70, and 80°C); see Table 6.348 Figure 45. Slightly curved Arrhenius plots used for Monte Carlo simulation studies. (1, 2) Concave plots; (3) linear plots; (4, 5) convex plots. (Reproduced from Ref. 348 with permission.) Figure 45. Slightly curved Arrhenius plots used for Monte Carlo simulation studies. (1, 2) Concave plots; (3) linear plots; (4, 5) convex plots. (Reproduced from Ref. 348 with permission.) Table 6. Estimated t90 Values from Slightly Curved Arrhenius Plots«6 Duration of experiment Number in Fig. 45 (weeks) Nonisothermal Isothermal Duration of experiment Number in Fig. 45 (weeks) Nonisothermal Isothermal Table 6. Estimated t90 Values from Slightly Curved Arrhenius Plots«6
"Reference 348. È Theoretical i90(25> 100(weeks):Ea, 25kcal/mol; assayerror, 2%; temperatureerror, 0.5°C; temperature program, T(°C) = 25 + 42. "Reference 348. È Theoretical i90(25> 100(weeks):Ea, 25kcal/mol; assayerror, 2%; temperatureerror, 0.5°C; temperature program, T(°C) = 25 + 42. Nonisothermal prediction using a temperature program including temperatures near room temperature reduces the effect of the deviation from the Arrhenius equation. Thus, nonisothermal prediction appears to be more suitable for t90 estimation when Ea might vary with temperature owing to possible changes in degradation mechanisms, an observation quite common in some drug degradation studies. The advantage of nonisothermal prediction was shown in the estimation of £,o values of a vitamin A syrup.349 Nonisothermal prediction from data observed at temperatures ranging from 25 to 54.7°C using the program T(°C) = 25 + 0.004t4 (week) (Fig. 46) provided a t,o, estimate of 83.0 days, which was close to the value observed at 25°C (82.7 days), whereas isothermal prediction from data observed at 40, 50, 60, and 70°C provided an estimate of 60.9 days. Relatively accurate t,o estimates can be obtained by nonisothermal prediction using a program including temperatures near room temperature even if the Arrhenius plots are not strictly linear. 2.2.4.3. Stability in Frozen Solutions Freezing is often assumed to slow chemical degradation. The assumption is correct in the majority of cases; however, freezing of aqueous solutions of drugs may increase drug degradation when the degradation occurs via bimolecular or higher orders of reaction. At temperatures just below freezing, solutes, including the drug and its reactants, become concentrated in the space between the ice crystals, thus effectively concentrating the drug and reactants. Freezing has also been proposed to enhance drug degradation by forming ice structures, resulting in an arrangement of the drug molucules that is suitable for degradation. The epimerization rate of moxalactam in frozen solutions, which occurs in the unfrozen aqueous phase after initial ice formation, exhibits linear Arrhenius plots as shown in Fig. 47, indicating that the effect of concentration change caused by freezing is negligible in this case.350 However, the bimolecular hydrolysis rate of n-propyl4-hydroxybenzoate (propylparaben) in alkaline medium increased at temperatures between -4 and -14°C (Fig. 48). This cannot be fully explained by the increase in dr |

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