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Mean Kinetic Relative Humidity: A New Concept for Assessing the Impact of Variable Relative Humidity on Pharmaceuticals
It is well established that the rate of chemical degradation of solid-state pharmaceutical products, such as tablets and capsules, is highly dependent upon the environmental temperature and humidity. Indeed, this concept is the underpinning theoretical basis for the pharmaceutical industry guidelines that provide recommendations for long-term, intermediate, and accelerated storage conditions and for establishing expiration dates (1).
The concept of mean kinetic temperature (MKT) was first introduced in 1971 (2) and was further developed in the 1990s (3). MKT can be defined as the single calculated temperature at which the total amount of degradation over a particular period is equal to the sum of the individual degradations that would occur at various temperatures (4). Thus, MKT may be considered as an isothermal storage temperature that simulates the effects of variable storage temperature. The MKT has been found to be useful in a number of pharmaceutically relevant applications, such as providing a scientific means of summarizing variable climatic temperatures (5), handling storage and distribution of pharmaceuticals (6), and for handling temperature excursions (7). The MKT is distinct from the simple arithmetic mean temperature and is generally accepted as a more scientific basis for summarizing variable temperature than the arithmetic mean temperature.
For assessing the impact of variable humidity conditions, on the other hand, no analogous concept to MKT has yet been proposed, because there has been no established means of relating the rate of degradation with humidity. The simple arithmetic mean, therefore, is most commonly used currently for most pharmaceutical applications. As far back as 1977, however, a mathematical relationship between the relative humidity (RH) and the reaction rate in the solid state has been suggested (8), and work by Waterman et al. has since demonstrated that this relationship is widely applicable to pharmaceutical products (9–12). This relationship allows a mean kinetic relative humidity (MKRH) to be calculated analogously to MKT. This article proposes a method for calculating MKRH, and the author discusses the advantages of using MKRH over the simple arithmetic-mean relative humidity and the limitations of the MKRH approach.
where k = the rate constant of the degradation reaction, A = the 'pre-exponential factor', Ea = activation energy (KJ·mol-1), R = universal gas constant = 8.314 J·K-1·mol-1, and T = temperature (K).
where MKT = mean kinetic temperature (K), T 1 to T n = the variable temperature (K) measured at constant intervals, and n = the number of temperature measurements.
where B = the moisture-sensitivity term and RH = relative humidity (%).
where k1 is the rate constant of reaction at relative humidity RH1, and k2 is the rate constant of reaction at relative humidity RH2 at the same temperature.
The B term is typically in the range of 0 to 0.09. A B term of 0 indicates that RH has no effect on reaction rate, whereas a B term of 0.09 indicates a high dependency on RH: the degradation rate would double for every 8% increase in RH.
where MKRHisothermal = mean kinetic relative humidity (%), RH 1 to RH n = the variable relative humidity (%) measured at constant intervals, and n = the number of relative humidity measurements.
A complication that arises as a consequence of the humidity-corrected Arrhenius equation is that the established MKT equation holds true only for samples held at constant relative humidity, and Equation 5 for MKRH holds true for samples held at constant temperature.
Systems with both variable temperature and variable relative humidity
where MKRH = mean kinetic relative humidity (%) calculated for a given temperature, Tchoice, Tchoice = a chosen constant temperature (K), T 1 to T n = the variable temperature (K) measured at constant intervals, RH 1 to RH n = the variable relative humidity (%) measured at constant intervals, and n = the number of temperature and relative humidity measurements.
where MKT = mean kinetic temperature (K) calculated for a given relative humidity, RHchoice, and RHchoice = a chosen constant relative humidity (%).
The previous section shows how the MKRH can be calculated for variations in relative humidity, as long as the moisture-sensitivity parameter, B, is known for the product. The use of MKRH has advantages over the use of arithmetic mean relative humidity in that the MKRH provides a better estimate of the constant-storage relative humidity that simulates the effects of multiple, variable-relative humidity conditions for processes that are expected to follow the humidity-modified Arrhenius equation, such as chemical degradation. It also helps avoid underestimates which can mislead product developers.
Using estimated E a and B values to calculate MKT and MKRH. One inconvenience of the MKRH approach is that a moisture-sensitivity parameter, B, is required for the calculation, in the same way that Ea is required for the calculation of MKT. In the calculation of MKT, it is common practice to use an estimated value for Ea. USP <1150> (4) advocates the use of 83.144 KJ·mol-1 (19.87 KCal·mol-1) based partly on the average of the Ea values cited in the literature between 1950 and 1980 (13, 14) and partly due to the convenience that Ea/R equates to almost exactly 10,000 K-1. The MKT is always higher than the arithmetic mean temperature; the magnitude of the difference increases with higher Ea and with more extreme differences in the variable temperature measurements. Analogously, the (isothermal) MKRH is typically higher than the arithmetic mean relative humidity, and the magnitude of the difference increases with higher B terms and with more extreme differences in the variable relative humidity measurements.
The average Ea determined here is higher than the commonly cited literature value. Although the reason for this is uncertain, it may reflect that the chemical reactions in this experiment occurred in the solid state, and the literature value was, instead, mainly obtained from solution-state reactions. In any case, caution should be exercised if a generic average value for the Ea or B term is used to calculate MKT or MKRH, because a wide range of Ea and B terms appear to be observed in solid-state chemical reactions. In some cases, depending on the magnitude of temperature or relative humidity variability, the choice of Ea or B term used in the calculation may have a negligible effect on the MKT or MKRH. This can be misleading, however, because processes with high Ea terms (or B terms) are sensitive to changes in temperature (or relative humidity), and so even small differences in mean temperature (or relative humidity) can significantly change the amount of degradation. Therefore, we would recommend determining the Ea and B term on a case-by-case basis by means of a short, accelerated stability study as described by Waterman et al. (9–12). Alternatively, the Ea and B term can be estimated by careful analysis of existing long-term, intermediate, and accelerated stability data; a manuscript detailing a method for achieving this is in preparation.
Limitations of the MKRH approach. The MKRH approach has some limitations that need to be considered when assessing the impact of relative humidity on the stability of a pharmaceutical substance or product. The most important consideration is that MKRH covers only chemical degradation. There are a number of other factors that may be affected by humidity but would not be expected to obey the humidity-corrected Arrhenius equation (e.g., dissolution performance, morphology changes of the active or an excipient, depression of glass transition temperatures, hydration or deliquescence of the active or an excipient). The risk of humidity variability on each of these factors needs to be considered separately to the calculation of MKRH; similar limitations also apply to the use of MKT.
An equation for the calculation of MKRH is based upon the humidity-corrected Arrhenius equation, which has been shown to be applicable to a wide variety of solid-state pharmaceutical products and active substances. The calculation of the MKRH requires a moisture sensitivity parameter, B, to be known, analogously to the way Ea is required for the calculation of MKT. In many situations, such as those involving chemical degradation, the use of MKRH is more appropriate than the use of the arithmetic mean relative humidity, in much the same way that MKT is more appropriate than the use of arithmetic mean temperature.
It is relatively simple to calculate the MKRH for constant temperature situations or to calculate the MKT for constant humidity conditions using established methods, such as those described in USP <1150> (4). In situations that have both variable temperatures and humidity conditions, it may be necessary to use a combined temperature and relative humidity calculation as described above, which provides a continuum of constant temperature and constant relative humidity combinations that would age the product to the same degree as the set of variable temperature and variable relative humidity conditions. However, in most situations, this combined calculation is not likely to be necessary, because either the relative humidity can be considered to be effectively constant (e.g., when considering packaged products over short time periods, such as with temperature excursions that occur during the distribution of products), or the temperature can be considered constant (e.g., during stability testing of packaged products in which the relative humidity inside the packaging changes dynamically over time and testing occurs at specific conditions, such as 25 °C/60% RH or 40 °C/75% RH). Despite its limitations and its apparent complexity, the MKRH approach should prove useful in a number of pharmaceutically relevant situations, as has the analogous concept of MKT.
The author would like to thank Ken Waterman for his inspirational scientific guidance and Bill Porter for stimulating technical discussions. The author would also like to thank Tim Lukas, Bruno Hancock, and Pierre Barratt for their useful comments and feedback.
Garry Scrivens, PhD, is in Pharmaceutical Sciences at Pfizer Global Research and Development, Ramsgate Road, Sandwich, Kent, UK, CT13 9NJ, tel.
+ 44 0 1304 649578, email@example.com
Submitted: Mar. 2, 2012; Accepted Mar. 28, 2012.
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