STATISTICAL ASSESSMENT OF MEAN DIFFERENCES BETWEEN TWO DISSOLUTION DATA SETS*

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1 Drug Information Journal, Vol. 30, pp , /96 Printed in the USA. All rights reserved. Copyright 1996 Drug Information Association Inc. STATISTICAL ASSESSMENT OF MEAN DIFFERENCES BETWEEN TWO DISSOLUTION DATA SETS* YI TSONG, PHD, AND THOMAS HAMMERSTROM, PHD Office of Epidemiology and Biometrics PRADEEP SATHE, PHD, AND VINOD P. SHAH, PHD Office of Pharmaceutical Sciences Center for Drug Evaluation and Research, Food and Drug Administration, Rockville, Maryland When comparing the dissolution data of a postapproval change product and a reference approval product, the goal is to assess the similarity between the mean dissolution values at the observed sample time points. The decision on accepting or rejecting the hypothesis that the two batches have similar dissolution is based on the evidence regarding whether the difference in mean dissolution values between the test and reference products is no larger than the maximum expected difference between any two batches of the approval product. When dissolution value is measured at a single time point, the confidence interval of the true difference between the two batches is compared with the prespecified similarity limits. When dissolution values are measured at multiple time points, a multivariate statistical procedure for difference assessment can be a generalized form of the t-statistic procedure. The proposed procedure is a modification and generalization of the regular bioequivalence test concept. The application of the proposed multivariate analysis procedure is illustrated using an example. Key Words: Dissolution test; Multivariate analysis of variance; Confidence region; Similarity test INTRODUCTION based on dissolution testing data are often accepted as adequate to assure dosage form DISSOLUTION DATA OF the drug product similarity and hence in vivo performance. are often used to assess the dosage form simi- The similarity of the product with respect to larity when the drug manufacturer is making dissolution means that the test (postapproval scale-up and postapproval changes, namely change) product has a dissolution performanufacturing site change, component and mance no different than that expected from compositional changes, and equipment and the reference (prechange) product except for process changes (1). Changes approved the potential batch-to-batch or lot-to-lot variation. Hence, in developing a statistical assess- *The views expressed in this paper are professional opinions of the authors and may not necessarily repredissolution data sets, the following need to ment of the difference between means of two sent the position of the U.S. Food and Drug Administration. be considered: Reprint address: Yi Tsong, Division of Biometrics III, HFD-720, Office of Epidemiology and Biostatistics, Center for Drug Evaluation and Research, FDA, A well-defined similarity limit of the pre- Fishers Lane, Rockville, MD change product is established before com- 1105

2 1106 Yi Tsong, Thomas Hammerstrom, Pradeep Sathe, and Vinod P. Shah paring the dissolution data of the test and proposes a statistical test for the hypothesis reference batches. The similarity limit is that the two batches are globally similar. set either by the knowledge of the characteristics of the product or by the empirical SINGLE TIME POINT experience on the batch-to-batch and the DISSOLUTION VALUE within-batch difference of the existing reference product. Similarity limits can be For most immediate release drug products, defined as global similarity or uniformly the dissolution is rapid and the quality control local similarity. Typically, a global similar- dissolution requirement is only a single time ity limit D g is defined as a tolerable differ- point measurement. By considering the disence between the test batch and the refer- solution measurement as a variable, the difence batch over all time points. The limit ference between the mean dissolution values D g is often defined as a given percentage of the test and reference product can be easily of dissolution, say, 10%. But with any measured and standardized with the regular measurement of difference, the similarity t-statistic. There is an expected difference limit is defined by the measurement corre- based on the empirical experience of the sponding to D g instead of D g itself. On batch-to-batch variation of mean dissolution the other hand, uniform local similarity is values of the reference product. Since the defined as maximum tolerable difference difference between mean dissolution values D t, at each sample time point. It is often of the test and reference batches is an estidefined as a given percentage of dissolu- mate based on the sampled tablets, a confition at time t, say, 12%. The similarity dence interval of the true difference can be limit is then defined by the difference mea- estimated and used when comparing with the surement at each time point and should expected difference. ref lect the similarity of the estimated batch For example, assume that the reference dissolution profiles, product dissolves rapidly and had one mea- 2. When the dissolution data of the reference surable dissolution value that is less than product vary significantly from batch to 100%. The average dissolution value based batch, similarity limits become large and on 12 tablets of a standard batch is 78.5% meaningless. In this case, dissolution data and maximum batch-to-batch difference of of test and reference batches may not be the standard batches is no more than 7.5%. comparable for similarity, With the manufacturing site change, the man- 3. The dissolution measurements of the test ufacturer did dissolution testing of 12 tablets and the reference batches are made under of the new batch after the site change (test identical conditions with an identical num- batch) and 12 tablets of a reference batch ber of units. The reference batch used manufactured recently at the old site. The should be the most recently manufactured dissolution values are given in the following prechange product, example. 4. The conclusion of similarity is made with the consideration of the adequacy of the Example 1 representativeness of the dissolution profiles (calculated with the sample tablets) Test batch Reference batch of the test and reference batches, and Dissolution values are sampled at the time points that properly and adequately repre sent the dissolution profile of the product Mean This paper proposes a statistical assessment of the difference between the mean dissolutions Std of the test and reference products and The difference, D, between test and reference

3 Assessing Differences Between Two Dissolution Data Sets 1107 batch is 2.32%. The standard error of difference, stderr(d), is: ( )/11 = Then the 90% confidence interval of D would be the interval that contains all the differences µ 2 µ 1 such that: [(µ 2 µ 1 ) D]/stderr(D) t 22,.95.(1) where S pooled = (S 1 + S 2 )/2 is the sample vari- ance-covariance matrix pooled across both batches, x 1 = (x 11,x 12,...,x 1p ) is the sample mean dissolution of the reference batch, and x 2. is the sample mean dissolution of the test batch. For the mean difference of two batches in dissolution measurements at multiple time points, the confidence region (CR), is defined as: Where t 22,.95 is the 95th percentile of t distribution with degrees of freedom = 22, that is, 2n 2. The lower and upper limits of the interval L90 and U90 can be calculated by (2): L90 = 2.32 t 22,.95 stderr(d) = U90 = t 22,.95 stderr(d) = Since both the lower and upper 90% confidence limits are within the inter-batch variation, ±7.5% ( 7.5% 0.639% and 4.001% 7.5%), the product after change is accepted as a product with a dissolution value similar to that of the prechange product batch. MULTIPLE TIME POINT DISSOLUTION ment at each time point may be considered a variable. Variation in dissolution changes with time. These variables represent dissolution of one tablet at different time points, which are correlated. The use of the single time point approach then becomes erroneous, necessitating a new difference measurement. A statistical distance often used to mea- sure the difference between two multivariate means is the Mahalanobis distance (M-distance) (3): D M = [(x 2 x 1 ) S 1 pooled(x 2 x 1 )], (2) CR = K[(y (x 2 x 1 )) S 1 pooled(y (x 2 x 1 )] F P, 2n P 1,.90, (3) When the dissolution values are measured at where y = (µ 21 µ 11, µ 22 µ 12,...,µ 2P µ 1P ) multiple time points, dissolution measure- is the difference within CR, K = [(n 2 )/(2n)] TABLE 1 Dissolution Data of a Reference Batch and a Test Batch % Dissolution Batch Tablet 5-min 10-min 15-min 20-min 30-min 60-min 90-min 120-min REF REF REF REF REF REF MEAN TEST TEST TEST TEST TEST TEST MEAN

4 1108 Yi Tsong, Thomas Hammerstrom, Pradeep Sathe, and Vinod P. Shah FIGURE 1. Mean dissolution of the test and reference batches. (2n P 1)/[(2n 2)P], and F P, 2n P 1,.90 is the respectively, the two M-distances of the two 90th percentile of F-distribution with degrees values in CR that give the minimum and of freedoms P and 2n P 1. The multivari- maximum M-distance to the original (the ate confidence region is compared with the l point of no dissolution difference). D M = 0if overall similarity limits for overall similarity. l CR contains the origin. Otherwise, D M and It is to be noted that formula (1) is a special D u M can be calculated using Lagrange Multicase of formula (3) when P = 1. plier method (4). The global similarity can Formula (3) gives a p-variate 90% confidence be verified if the 90% confidence interval: region for the possible true differences. l Let D M be the lower 90% limit, and D u M be the (D l M,D u M ) upper 90% limit of the confidence interval of l true M-distance D M..D M and D u M represent, is imbedded in:

5 Assessing Differences Between Two Dissolution Data Sets 1109 TABLE 2 Variance-Covariance Matrices of the Reference and Test Batches Time Point 5-min 10-min 15-min 20-min 30-min 60-min 90-min 120-min REF 5-min S1 10-min min min min min min min TEST 5-min S2 10-min min min min min min min ( {[D g ] S 1 pooled[d g ]], {[D g ] S 1 pooled[d g ]}), where [D g ] is the 1 P vector with all entries (x 2 x 1 ) =( 17.54, 3.39) K = [6 6/(6 + 6)][( )/ [( )2] = 1.35 equals to D g, the difference specified as the global similar limit. The application of the n 1 + n 2 p 1 = 9, F 2,9,.90 = 3.01 procedure is illustrated in the following two examples (Table 1). CR = {1.35 (µ 21 µ 11 ) (µ 22 µ 12 ) 3.39 Data The dissolution data and the corresponding mean dissolution of the test batch and the reference batch are given in Table 2 and Fig- ((µ 21 µ 11 ) , ure 1. Let S 1 and S 2 denote the sample covari- (µ 22 µ 12 ) 3.39) 3.01} ance matrices of dissolutions at the eight time points for the test and reference batches, rewhere (µ 21 µ 11 ) and (µ 22 µ 12 ) are the possispectively, as shown in Table 3. The mean profiles of test and reference batches will ble value of difference at 15- and 90-minutes be compared for similarity. Let D g be the respectively. The confidence region is shown empirically defined limit, which is obtained in Figure 2. from the standard batches of the reference product. D M = DATA1 data (comparing the 15- and 90-minute sample time points only): For comparing test and reference batches P = 2, n = 6, S pooled = (S 1 + S 2 )/2, The points ((µ 21 µ 11 ), (µ 22 µ 12 )) on the boundary of CR with the minimum and the maximum M-distance from (0, 0), the origin, are ( 15.03, 2.90) and ( 20.05, 3.87), respectively. The 90% confidence interval of D (D l M,

6 1110 Yi Tsong, Thomas Hammerstrom, Pradeep Sathe, and Vinod P. Shah FIGURE 2. The 90% confidence region of difference in percentage dissolved between batches at 15- and 90-minutes. D u M ), is (8.95, 11.93). This is to be compared P = 8, n = 6, S pooled = (S 1 + S 2 )/2, with: (x 2 x 1 ) =( 23.69, 20.52, 17.54, 14.80, RD = {[15] S 1 pooled[15]} = , 3.39, 5.00) T 2 = Since is larger than 9.63, it is concluded that the two batches are not globally K = [6 6/(6 + 6)[( )/ similar. [( )8] =.1125 DATA2 data (Comparing all eight time D M = points): (D l M,D u M )=(19.65, 33.32)

7 Assessing Differences Between Two Dissolution Data Sets 1111 TABLE 3 Dissolution Data of Two Approved Standard Batches % Dissolution Batch Tablet 5-min 10-min 15-min 20-min 30-min 60-min 90-min 120-min STD STD STD STD STD STD MEAN STD STD STD STD STD STD MEAN Since D u M is greater than RD, it is concluded percentile value. With a fixed number of tablets, that the two batches are not globally similar. more measurements (time points) may lead to a better representation of dissolution DISCUSSION information. Since the second degrees of freedom of F is 2n P 1, however, the When dissolution is measured at multiple maximum number of time points that can be time points, the statistic of Mahalanobis disdramatically used is 2n 2. The F percentile increases tance is used to assess the difference between with the decrease of the second the means of two sets of data with adjustment degrees of freedom. It leads to a large confi- for difference in measurement variation at dence region and a large confidence interval different time points and for the correlation of M-distance. A model-dependent proce- among the measurements at multiple time dure may also be used in such cases. points. Under the assumption that the dissotributed When the data are neither normally dis- lution measurement is multivariate normally or lognormally distributed, one distributed and that the two batches have the needs to consider the nonparametric or bootidentical variance-covariance structure, the strapping procedures. With within-batch dis- 90% confidence interval of the M-distance solution variation being small, decisions can be estimated. When the 90% confidence made with point estimation are sometimes interval is imbedded within the similarity used. They should, however, be used cau- limits, the true difference between the test tiously and conservatively. For example, for and reference batches is no more than the 12 tablets in each batch and with a coefficient expected true difference between two batches of variation of less than 0.10, instead of using of the same product as the reference batch. 15% difference as similarity limit, one may When the data are not normally distributed use 10% instead. one may consider using a lognormal distribution assumption; similarity interpretation is Acknowledgment The authors thank Drs. Roger Wilon the means of the log transformed distribuliams, Lawrence Lesko, and William Fairweather for tion values. their support and encouragement, and CDER s Immedi- As shown in the formula, the confidence ate Release Dissolution Working Group for their suggestions region is the ellipsoid restricted by the F and discussions.

8 1112 Yi Tsong, Thomas Hammerstrom, Pradeep Sathe, and Vinod P. Shah REFERENCES 2. Snedecor GW, Cochran WG. Statistical Methods. Seventh Edition. Ames, IO: The Iowa State University 1. Guidance for Industry, Immediate Release Solid Oral Press; Dosage Forms Scale-Up and Post-Approval changes, 3. Johnson RA, Wichern DW. Applied Multivariate Chemistry, Manufacturing and Controls. In Vitro Dissolution Analysis. Englewood Cliffs, NJ: Prentice-Hall, Inc.; Testing, and In Vivo Bioequivalence Docu mentation. Rockville, MD; Center for Drug Evaluation 4. Freund JE. Mathematical Statistics. Englewood and Research, FDA; November Cliffs, NJ: Prentice-Hall, Inc., 1962.

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