Powder
Technology,
42 (1985)
.
285 - 286
-
_
285
_
Letter to the Editor Reply to Comments on ‘Ordered Mixtures Interactive Mixtures’ H. EGERMANN Pharmaceutical Technoloey Division, Institute of Organic and Pharmaceufical Chemistry, University of Innsbruck, A-6020 Innsbruck, Innrain S2a (Austria) (Received
November
20,1984)
The recent discussions [1 - 71 on the nomenclature of powder mixtures have developed general lines of a consistent and uniform terminology. Agreement was achieved amongst the contributing authors [1 - 71 that two different types of terms are necessary, one to define the degree of mixing (ordered, random, incomplete), and one to distinguish between mixtures of interacting powders and of free-flowing constituents (interactive, noninteractive). Along these lines, unambiguous terms of powder mixtures may be derived which allow reference to the degree of mixing and/or to the type of the mixture, as shown by Egermann and Orr [6], and by Thiel C7] in his recent comments. Some misinterpretations may have happened in the course of the discussions with respect to the quality and to the quality control of interactive mixtures. To produce ordered mixtures (a C on), the basic requirement is to achieve ordered adhesion [S]_ Ideally ordered adhesion implies a constant number of the ingredient particles to adhere to each square unit of the total carriers’ surface area. Whether a perfectly ordered mixture ((T= 0) may be formed, depends further on the particle size distribution of the components [9] _ With monosized constituents only, the quantity of the fines per carrier particle will become constant by number and by weight, and o = 0, independent of the arrangement of the interactive units (or adhesion units [71) in the mixture. With polydisperse carriers, by virtue of their differences in particle surface area, the quantity of the fines per interactive unit is not identical_ Then randomization of the different sized carriers 0032-5910/85/$3.30
will produce (T> 0. From this, Egermann [3] questioned the validity of a previous statement of Thiel et aI_ [Z] that in a perfectly ordered mixture the carriers should be fully randomized. To achieve a pseudorandom mixture (or interactive random mixture [S] ) with a polydisperse carrier requires not only random adhesion of the ingredient but also randomization of the carriers_ Interactive unit segregation will increase the variance of the mixture 17 J_ Thiel [7 3 appears to have misund2P,Tstood the type of mixture being considered when he assumed Egermann [3] to have questioned this latter statement. On the basis of sampling, a mixture will be ordered with a probability 1 - 01,if the upper confidence limit of (T, as estimated from the standard deviation s of the assay sample (n individual samples), is below an, provided that there is normal distribution of the sample content. Applying the x*-statistics [lo], 52 -= cJ*
x2,-I.
Tbiel
[4, 73 has shown that this condition
I--P
(1)
n--l
is
met if n-1 s * <1 (2) ( @R > x*n-1. 1-a Thiel is correct in his view [d, 71 that the presence of agglomerates in incomplete mixtures (o > on) may skew the content dis’ribution and invalidate eqn. (2). However, this does not imply that the existence of the hypothetical ordered mix could not be established [4 J , as becomes obvious by considering the statistical properties of interactive mixtures. In the equilibrium situation, random adhesion will produce a pseudorandom mixture with sample mean content x and standistribution of dard deviation en_ Normal the adherent ingredient may be provided, and then s does not meet eqn. (2). If deagglomeration remains incomplete (a > on), two different types of distribution may coexist in the mixture [ 111 _The ingredient proportion dispersed as individual particles, with mean 8, 0 Elsevier Sequoia/Printed
in The Netherlands
260
and standard deviation err = (Tn,, conforms to normality again. The agglomerated proportion, with parameters xZ = z-z, and cZ > may exhibit skewed distribution of ORI, Poissonian type and skew the distribution of the total mis, where D = ((zR,Z 1- 022)i’2 With the assay sample, and s = (s*l + +2)1’2
(3) the mean is T= 4 -t _, (4)
As a consequence of differences in weight and of a small number of the agglomerates as compared with the number of samples of the mixture, the estimates Zand s may not be representative of xand CJ[ll - 131. Experimentally, Egermann 1131 has shown that, in extreme situations, it is possible that no agglomerate may be found in the samples, even if the agglomerated ingredient proportion of the mix is high. Then s may be found equal to si, which is representative of on1 < cR only, and thus may meet the condition of eqn. (2) erroneously. However, the fact that an ordered mix has not been formed is immediately evident from the low mean 2, of the assay sample being representative of ;k’, < x_ In contrast to the standard deviation o, the coefficient of variation CV is inversely related to the mean z_ In terms of CV, (5) C,, of the incomplete mix (CV > C,) is above C, unless equilibrium is approached (C,, = Ca). Likewise with the assay samples, SI -2,
=cv,>-
S
s
=cv
(6)
cv,. being the lowest estimate of the coefficient of variation of the incomplete mix, is higiier than the coefficient of variation of the pseudorandom mix (CV = C,), and cannot assume a value significantly below CR with a probability > 1 -CY_ Accordingly, the coefficient of variation is superior to establish the boundary between ordered and pseudorandom mixes on the basis of sampling. Substitution to eqn. (2) gives eqn. (7):
cv *
n-l
(c,1X21-a.
n-
(7)
<1 1
which is not invalidated by the presence of agglomerates skewing the content distribution of incomplete interactive mixtures. Irrespective of these statistical considerations, in principle one must agree with Thiel et aZ_ [2] that the proof of an ordered mixture will be difficult in practice, as a consequence of the high degree of fineness necessary to accomplish complete adherence of the ingredient [9]. Using a micronized drug powder with the representative mean particle size & [14] amounting to about 10 pm, CR is only I%, even if Xis as low as 10 I.rgper sample, which is at the smallest dosage level of active ingredients in pharmaceutical preparations. The analytical error may be of the same magnitude as C,, thus making impossible the judgement of a lower than random content variation_ With the high quality of the random systems in mind, not only the evidence but also the practical relevance of ordered mixtures appears to be questionable, as demonstrated by Egermann [9] in 1976. Nevertheless, the theoretical and practical problems involved in ordered mixtures [S, 91 to not affect the need of a consistent terminology which appears to have been estabiished from the discussions.
REFERENCES 235. 26 (1980) 1 H. Egermann, Powder Technol., 2 W. J. Thiel, F_ Lai and J_ A. Hersey, Powder TechnoI. 28 (1981) 117. 3 H. Egennann,.Powder TechnoZ , 30 (1981) 289. 4 W_ J. Thiel. Powder Technol.. 33 (1982) 287. 5 H. Egermann, Powder Technol, Sk (1983) 135. 6 H. Egermann and N. A. Orr, Powder TechnoZ.. 36 (1983) 117. 7 W. J. Tbiel, Powder Technol , 39 (1984) 147. 8 H. Egermann, Powder TechnoZ , 27 (1980) 203. 9 H. Egermann, Acta Pharm. TechnoL. 22 (1976) 207.
10 11
H. Harnby, Powder TechnoL. 5 (1971/72) H. Egerman, Inf. J. Pharm. Tech. & Prod. 3 (1982)
81. Mfr..
59.
12
H. Egermann,
13 14
H. Egermann, Pharm H. Egermann, Powder
Pharm.
Irzd., 40 (1978) 1377. Ind.. 41 (1979) 285. Technol., 31 (1982) 231.