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
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.
Figure 2 summarizes the Ea and B terms of approximately 40 different chemical degradation processes that occur in solid-state pharmaceutical products
(e.g., tablets), as determined by the author. The average Ea for the sample of products is approximately 120 KJ·mol-1, ranging from 74 to 172 KJ·mol-1 and the average B term is approximately 0.04, ranging from 0 to 0.09.
Figure 2: The range of activation energy (Ea) and B terms observed in the laboratory for a sample of 40 degradation reactions that occur in solid oral-pharmaceutical
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.
Application of the MKRH approach to packaged pharmaceuticals.
Manufacturers are usually concerned with the stability of packaged products and substances, but the relative humidity external
to the packaging is not the same as that inside an air-tight container, such as a capped bottle or blister pack. One potentially
useful application of MKRH is for providing a convenient, single, constant, relative-humidity value that can be used to summarize
and assess the impact of the variable relative humidity that occurs inside pharmaceutical packaging. The relative humidity
inside a pharmaceutical pack can be accurately simulated as a function of the external conditions using the methods described
by Waterman et al. if the moisture vapor transmission rate (MVTR) is known for the packaging and the moisture vapor sorption isotherm is known
for the product or its constituents (15). For example, Figure 3 shows how the relative humidity is expected to vary inside a capped, 40-cc polyethylene terephthalate bottle containing 100
tablets with an initial water content of 2.0%, which, according to the product's moisture vapor sorption isotherm, equates
to a water activity of 0.15; the bottle is stored at 30 °C/75% RH for 6 months. The relative humidity inside the bottle starts
at approximately 15% RH (i.e., equilibrated to the initial water activity of the product), and, due to the water transmission
through the bottle, slowly equilibrates towards the external relative humidity of 75% RH. The MKRH approach enables this variable,
dynamically changing internal relative humidity to be expressed conveniently as a scientifically relevant mean value. For
comparison, Figure 3 shows the MKRH calculated using B terms of 0.04 and 0.09 and the arithmetic mean relative humidity.
Figure 3: The variable relative humidity inside a capped bottle containing product of initial water content 2.0%, which, according
to the product’s moisture vapour sorption isotherm equates to a water activity of 0.15; stored at 30 °C/75% RH for six months.
The mean kinetic relative humidity (MKRH) over the six-month period is shown for the product with a B term of 0.04 and 0.09,
along with the arithmetic mean relative humidity.
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.