Systems with both variable temperature and variable relative humidity
For systems that experience both temperature and relative-humidity variability, the calculations become more complex, but
still can be dealt with by MKT and MKRH approaches. To illustrate one complicating factor, two similar, hypothetical scenarios
are considered. In both scenarios, the same hypothetical solid-state pharmaceutical product is exposed to identical temperatures
and relative humidity conditions, but in Scenario 1, the high temperature coincides with high relative humidity, and in Scenario
2, the high temperature coincides with low relative humidity, as shown in Table I.
Table I: Hypothetical scenarios for systems with both temperature and relative-humidity variability.
As a consequence of the exponential nature of the humidity-corrected Arrhenius equation, Scenario 1 is more stressful to the
product than Scenario 2. For instance, if the hypothetical product had an Ea of 120 KJ·mol-1 and a B term of 0.04 (see later for a discussion on average typical values for Ea and B), Scenario 1 would result in three-fold more degradation than in Scenario 2, and yet both scenarios have identical
average temperature and humidity values (i.e., identical MKT, MKRH, arithmetic mean temperature, and arithmetic mean humidity).
This illustrates how, in situations in which both temperature and humidity are varying, it can be misleading to calculate
the MKT or MKRH in isolation of the other. In order to express situations with both varying temperature and humidity as a
single, constant temperature and humidity condition, it is first important to recognize that there is a continuum of constant
temperature and humidity conditions that would result in the same amount of degradation as the varying conditions, because
any increase in the constant "average" temperature can be compensated by a decrease in the constant "average" relative humidity.
Any temperature can be chosen, therefore, within the variable temperature range, and a corresponding MKRH can be calculated
using Equation 6:
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.
Alternatively, any relative humidity can be chosen (within the variable relative humidity range), and a corresponding MKT
can be calculated using Equation 7:
where MKT = mean kinetic temperature (K) calculated for a given relative humidity, RHchoice, and RHchoice = a chosen constant relative humidity (%).
Despite their apparent complexity, these equations are relatively easy to apply using spreadsheet software. One potential
problem with this approach is that there is a degree of arbitrariness when choosing Tchoice (or RHchoice). If this approach is used, then the use of the MKT calculated without consideration of the variable relative humidity (e.g.,
according to USP <1150>) should be a good choice for Tchoice in most situations, and the associated MKRH can be calculated using Equation 7 (4). Similarly, the use of the MKRH calculated without consideration of the variable temperature should be a good choice for
RHchoice, and the associated MKT can be calculated using Equation 7. Figure 1 shows the MKT and MKRH combinations calculated for Scenario 1 and Scenario 2 using this 'combined' approach to temperature
and relative humidity variations. As shown in Figure 1, the use of MKT and MKRH calculated in isolation of each other overestimates the degradation for Scenario 2 but underestimates
the degradation for Scenario 1, and the use of arithmetic mean temperature and humidity significantly underestimates the degradation
for Scenario 1 and even underestimates Scenario 2.
Figure 1: The solid lines represent the temperature-relative humidity combinations that are calculated to result in the same
amount of degradation as the variable temperature and relative humidity in Scenarios 1 and 2. These lines were calculated
for a product with an activation energy (Ea) of 120 KJ·mol-1 and a B term of 0.04, which represent average values for solid-state pharmaceutical products. (ALL FIGURES
ARE COURTESY OF THE AUTHOR)