The prediction of the dissolution rate constant by mixing rules: the study of acetaminophen batches

The purpose of this article is to promote two simple and scalable methods to accelerate the formulation development of formulated granules using acetaminophen as a model system. In method I, formulated granules made from the batch of small particle-sized acetaminophen (1) by ball milling the batch of large particle-sized acetaminophen (2), and the mixture of the two batches at equal weights (mix) gave the dissolution rate constants (k) of k(1) = 0.43 +/- 0.15 minutes(-1), k(2) = 0.18 +/- 0.01 minutes(-1), and k(mix) = 0.30 +/- 0.03 minutes(-1) for 75 wt percent formulation; k(1) = 0.75 +/- 0.01 minutes(-1), k(2) = 0.18 +/- 0.01 minutes(-1), and k(mix) = 0.34 +/- 0.03 minutes(-1) for 62 wt percent formulation; and k(1) = 0.28 +/- 0.01 minutes(-1), k(2) = 0.16 +/- 0.01 minutes(-1), and k(mix) = 0.22 +/- 0.02 minutes(-1) for 30 wt percent formulation. In method II, the mixture of the formulated granules produced by mixing the formulated granules from the two batches at equal weights gave dissolution rate constants of k(mix) = 0.30 +/- 0.03 minutes(-1), 0.30 +/- 0.02 minutes(-1), and 0.22 +/- 0.01 minutes(-1) for 75 wt percent, 62 wt percent, and 30 wt percent formulations, respectively. After fitting the three data points of k(1), k(2), and k(mix) to the 10 mixing rules in materials science--series mixing rule, Hashin and Shtrikman upper bound, logarithmic mixing, Looyenga mixing rule, effective media approximation (EMA), three-point lower bound, Torquato approximation, three-point upper bound, Maxwell mixing rule, and parallel mixing rule--we found that the selection of the best suited mixing rules based on k(1), k(2), and k(mix) was solely dependent on the formulations under a given operating condition and regardless of whether the system was a powder mixture or a granular mixture. The values of k(1), k(2), and k(mix) in both the 75 wt percent and 30 wt percent formulations were enveloped by the parallel mixing rule and Maxwell mixing rule, whereas the values of k(1), k(2), and k(mix) for the 62 wt percent formulation were encompassed by the logarithmic mixing rule, Hashin and Shtrikman upper bound, and the series mixing rule. Apparently, the best suited mixing rules could be used to predict the right proportions of either the powder mixture (Method I) or the granular mixture (Method II) for obtaining any other desired dissolution rate constant, k(mix), whose value fell in between the values of k(1) and k(2).