Interpretation of relaxation time constants for amorphous pharmaceutical systems

The molecular mobility of amorphous pharmaceutical materials is known to be a key factor in determining their stability, reactivity, and physicochemical properties. Usually such molecular mobility is quantified using relaxation time constants. Typically relaxation processes in amorphous systems are non-exponential and relaxation time constants are usually obtained from experimental data using a curve fitting procedure involving the empirical Kohlrausch-Williams-Watts (KWW) equation. In this article we explore the possible relationship between the KWW curve fitting parameters (tau(KWW), beta(KWW)) and common statistical measures of the average and the distribution (e.g., median, standard deviation) of the relaxation time values. This analysis is performed for several common statistical distributions (e.g., normal, lognormal, and Lorentzian), and the results are compared and analyzed in the context of pharmaceutical product stability predictions. The KWW function is able to describe relaxation processes stemming from several different statistical distribution functions. Under some circumstances the "average" relaxation time constant of the KWW equation (tau(KWW)) is significantly different from common statistical measures of the central value of a distribution (e.g., median). Simply knowing the relaxation time constants from the fit of the KWW equation is not sufficient to completely characterize and quantify the molecular mobility of amorphous pharmaceutical materials. An appreciation of the distribution of relaxation times and the resulting effects upon the KWW constants should be considered to be essential when working with amorphous pharmaceutical materials, especially when attempting to use relaxation time constants for predicting their physical or chemical stability.