Particle size statistical diameters

The following sections detail existing in vitro/in vivo correlations for the major aerosol modalities and, where appropriate, comparisons are made with lung model predictions. Measures of particle diameter, particle size statistics, and aerosol test methods are also discussed. Aerosol test methodologies are included in the discussion because, as described above, sizing results are highly dependent upon the method and apparatus used. The correlations that have been developed and any predictions that can be made from them are therefore specific to the use of particular experimental methods, and it is important that the applicability of the different instruments/methods be understood. 

Equation (2.28), being statistical in nature, requires a large number of particles to be measured, especially if the spread of particle size is wide. The possibility of error from this source is stressed by Arnell and Henneberry who found that in a particular sample of finely ground quartz, two particles in a total of 335 had a diameter about twenty times the most probable diameter, and that if these were overlooked the calculated value of A would be nearly doubled. 

Particle size distribution is usually plotted on a log-probabiHty scale, which allows for quick evaluation of statistical parameters. Many naturally occurring and synthetic powders foUow a normal distribution, which gives a straight line when the log of the diameter is plotted against the percent occurrence. However, bimodal or other nonnormal distributions are also encountered in practice. 

Avera.ge Particle Size. Average particle size refers to a statistical diameter, the value of which depends to a certain extent on the method of deterniiaation. The average particle size can be calculated from the particle-size distribution (see Size measurement of particles). 

When a distribufion of particle sizes which must be collected is present, the aclual size distribution must be converted to a mass distribution by aerodynamic size. Frequently the distribution can be represented or approximated by a log-normal distribution (a straight line on a log-log plot of cumulative mass percent of particles versus diameter) wmich can be characterized by the mass median particle diameter dp5o and the standard statistical deviation of particles from the median [Pg.1428]

In Nature, however, we always have a contiiinous distribution of particles. This means that we have all sizes, even those of fractional parentage, i.e.-18.56n, 18.57p, 18.58 p, etc. (supposing that we can measure 0.01 p differences). The reason for this is that the mecheuiisms for particle formation, i.e.- precipitation, embryo and nucleation growth, Ostwald ripening, and sintering, are random processes. Thus, while we may speak of the "statistical variation of diameters", and while we use whole numbers for the particle diameters, the actuality is that the diameters are fractional in nature. Very few particle-size" specialists seem to recognize this fact. Since the processes are random in nature, we can use statistics to describe the... 

Particle shape plays an important role in particle size determination. The simplest definition of particle size diameter is based on a sphere, which has a unique diameter. In reality, however, many particles are not well represented by this model. Figure 1 illustrates the variety of shapes that may be found in particle samples [1]. As the size of a particle increases, so does its tendency to have an irregular shape [2], complicating statistical analysis. Particle shape coefficients have been derived for different geometries [3], and various equivalent diame-... 

Quite different site densities are obtained if these assumptions are changed. Perez et al.13 have calculated the surface site statistics using a computer model which can simulate incomplete layers by removing atoms from complete shells. The atoms removed are those which have the smallest number of first and second nearest neighbours. Many more types of site are considered in the models used by Perez et al. However, one of the most interesting results of their calculations is to demonstrate that for all sites, apart from B2 sites, there are very pronounced oscillations in number as the particle size is increased. Figure 2 shows the variation in the number of B2, B3, and B4 sites, and Figure 3 shows the ratio of B3/B4 sites as a function of particle size. Any reaction which is controlled by this ratio will show activity maxima for particle diameters of 0.8 and 2.0 nm. On the other hand B2 and B2 sites are the ones most likely to catalyse structure insensitive reactions. 

In the case of low-dose drug products, content uniformity of the dosage form can be negatively affected if the particle size of the API is either too large or the size distribution is too broad. This could result in the need to deliver API below a certain size to ensure content uniformity. Recently, Rohr et al.4 expanded upon previous work to develop a statistical model predicting the volume mean diameter of API necessary to meet USP stage 1 criteria on content uniformity for tablets with a 99% confidence level [Eq. (8.1)] ... 

With uniform particles the difference between the statistical diameters diminishes. When all particles are the same size, the statistical diameters are the same. 

Fairs (1943) has criticized the method of linear measurements above described. He points out that the diameter so measured does not correspond with the Stokes or effective diameter det but is usually greater. The importance of avoiding some shape factor to convert d to de is obvious but over and above this lies the fact that a linear measure is of statistical interest only, as already inferred. A diameter to be useful must be related to measures of mass or surface. Schweyer (1942) in his comprehensive analysis of particle size techniques (about which more will be said later) has dealt with this subject in detail. 

Compared to other methods, TEM is more reliable, accurate and intuitive. Meanwhile it can observe two-dimensional and three-dimensional forms of the particles. However, it only observes the particles in limited range, so it does not have statistical meaning. Based on centrifugal subsidency method, the particle size analyzer can get all kinds of the particles and the differential and cumulative diameter distribution curves of them. In order to dominate or avoid the particles agglomerate, the key is the suspension method in the test. In summary, the three methods are compensated for each other. 

For an assembly of particles, each linear measurement quantifies the particle size in only one direction. If the particles are in random orientation, and if sufficient particles are counted, the size distribution of these measurements reflects the size distribution of the particles perpendicular to the viewing direction. Because of the need to count a large number of particles in order to generate meaningful data these diameters are called statistical diameters. 

NIR has been used to determine the mass median diameter of a micronized active compound contained in a lactose monohydrate at a concentration of 4% by weight and a size between 8 and 20 pm [223]. Multivariate statistical analysis was applied to zero order NIR spectra using particle size distributions by low angle laser light scattering as a reference technique. Due to its speed, simplicity and low operating costs it was demonstrated that this is a viable alternative to other methods used to carry out this type of analysis. 

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