# Comparing drug dissolution profiles: proposals based on tolerance intervals

Dr. Yi Huang, Dept of Math and Stat, UMBC

Dr. Thomas Mathew, Dept of Math and Stat, UMBC

Shuyan Zhai, Dept of Math and Stat, UMBC

Drug dissolution is defined as the amount of drug substance that goes into solution, per unit time, under standardized conditions. In other words, dissolution measures the in-vitro drug release, as a function of time. Availability of a drug in solution is usually essential for drug absorption and appearance in the blood, in order to eventually provide therapeutic benefits. The sample dissolution profile is the recorded percentage amount dissolved at prespecified time points. The corresponding population dissolution profile is the curve representing the mean dissolution rate over time.

Dissolution tests are performed to compare prototype formulations of a drug product. For example, after a drug product has been approved, for post-approval changes such as equipement changes, process changes, changes in component and composition, scale up etc., the U.S. FDA requires a comparison of dissolution profiles between pre-change and

post-change products, and establish similarity between the two profiles based on suitable criteria.

An easy to use measure for checking the similarity of two dissolution profiles is a similarity factor (denotedby f2), recommended by the FDA. The similarity factor f2 is an inverse function of the sum of the squares of a suitably scaled difference between the percentage amount dissolved at prespecified time points. The factor takes values between 0 and 100, with the value 100 indicating perfect similarity. According to FDA guidance, values between 50 and 100 are taken to be evidence for similarity.

The use of factors such as f2 has been criticized by various researchers, since the decision based on such a similarity factor does not take into account the variability among dissolution profiles, or the correlation among the percentage amount dissolved at different time points. In other words, even though f2 is a random variable, this fact is not taken into account in the decision making process. Motivated by this, researchers have suggested various alternative criteria, and have developed statistical inference methodology based on them. However, the factor f2 continues to be a popular criterion, and a statistically rigorous approach based on f2 is currently unavailable.

In our research, we propose to develop a lower tolerance limit for the distribution of f2. A lower tolerance limit is such that a specified proportion (say, 0.90) or more of the distribution will be above the limit, with a specified confidence level (say, 0.95). The confidence level reflects sampling variability, since the limit will be computed using a sample of dissolution profiles. If such a lower tolerance limit is large (say, more than 50), we are 95% confident that at least 90% of the f2 distribution is above 50. This could be taken as strong indication of similarity between the profiles. Note that such a tolerance limit does take into account the variability among the dissolution profiles. A major challenge in developing such a tolerance limit is the lack of a tractable analytic expression for the distribution of f2, even if we assume multivariate normality for the profile data.Nevertheless, it is possible to use non-parametric methods to derive the tolerance limit; however, a bootstrap calibration is necessary to obtain an accurate limit. An added complication is that the mean of the multivariate normal distribution is usually a non-linear function of unknown parameters, and closed form expressions are not available for the maximum likelihood estimates. In the area of dissolution profile testing, a two-parameter Weibull model is very often used for the mean. All these features make the tolerance limit computation, and the Monte Carlo estimation of its coverage probability, very computationally intensive. We could obtain only very limited numerical results, and they indicate excellent accuracy. Our request for an account on tara is to speed up this computation.

We also plan to investigate the tolerance limit approach to various other criteria suggested in the literature.While we won’t be able to recommend one specific criterion for dissolution profile comparison (this could as well be a regulatory issue), our work will put the existing criteria on a firm statistical footing.

The work described above is the doctoral work of the graduate student

Shuyan Zhai.