Because MMAD is the most widely used metric, it provides a good starting point for discussions about data analysis. To find
MMAD, it is necessary to process the raw analytical data to ascertain as accurately as possible the particle size below which
50% of the particle population lies. Figure 2 shows cascade impaction data presented in two different ways. The first is a
straightforward linear plot of cumulative sized mass, derived by summing the percentage masses collected at each stage, against
particle size, the upper cut off diameter of each stage (see Figure 2a). The second is a log-probit plot of the same data
(see Figure 2b) (5).
Figure 2: Converting a linear plot of cascade impaction data (percentage cumulative mass against upper cut-off diameter) to
a Log-Probit plot makes it easier to derive the particle-size metrics of interest.
The design of multistage cascade impactors is such that the cut-off diameters are typically not linearly spaced. This can
result in five or more stages of separation in the size range of most interest (e.g., less than 5 μm for inhalation). The
x axis is therefore often presented using a logarithmic scale, as in the log-probit plot, allowing data point spacing to be
more equal, rather than grouping measurements at the finer and coarser ends of the size distribution.
Using the probit function to express cumulative percentage mass also makes it easier to accurately derive metrics of interest.
Probit 4 equates to a cumulative percentage of 15.8%, Probit 5 to 50% and Probit 6 to 84.1%. This type of representation more
precisely scrutinizes the tail ends of the mass distribution, as required for GSD definition.
With either plot, the question remains of how to determine the particle size that equates to a cumulative mass of 50% as this
almost always lies between two data points. Several different approaches are possible and, though consistency may exist within
individual companies and organisations, no single method dominates across the community.
Implicit within the USP guidance for calculating MMAD is the suggestion that one approach is to assume that the distribution
of the mass-weighted raw data is log-normal and to perform a linear regression over the whole data set, thereby giving equal
weighting both to the tails of the distribution and the central area (6). For data sets where measurements are equally valid
at all points, this is an efficient way of using all the data to guide a calculation to a more accurate outcome. However,
with cascade impaction, only relatively small amounts of material collect on stages at either end of the distribution, so
measurement errors are therefore at their highest, making such an approach potentially disadvantageous.
The adoption of this approach provokes two questions for those selecting an MMAD calculation method:
- How many OINDPs comply with the stipulation of log-normality?
- How accurate is the method for those that don't?
A recent Stimuli paper to the Revision Process from the Pharmacopoeial Forum considers both of these issues (1). It reviews three alternative techniques for calculating MMAD and cross-compares all four
methods. Two of the three alternatives involve sigmoid curve fitting using the Mercer-Morgan-Flodin (MMF) and the Chapman-Richards
(CR) models respectively, the third is simple linear interpolation between the two points on either side of the MMAD.
This article draws two key conclusions. The first is that, on the basis of the cross-section of orally inhaled products investigated,
log-normality is the exception rather than the norm. Furthermore, it is impossible to easily predict which products will exhibit
such behavior and which will not. The second is that while an assumption of log-normality can produce significantly different
answers to those provided by the curve fitting techniques, for formulations that deviate from this distribution, the simple
technique of two point interpolation does not. Close agreement is observed between interpolated data and that obtained from
curve fitting for all formulations (see Figure 3).
Figure 3: Comparing four curve fitting techniques for multistage cascade impactor data for a device metered dry powder inhaler
(a) and pressurized metered dose inhaler (b). (REPRINTED WITH PERMISSION FROM PHARMACOPEIAL FORUM 36 (3). COPYRIGHT 2010 US
Mathematically, interpolation is the most straightforward approach and already well-used (7, 8). This article reinforces the
validity of interpolation confirming it as a robust method for more routinely encountered distributions.