Pharmaceutical Research, Vol. 15, No. 6, 1998
`
`Research Paper
`
`In Vitro Dissolution Profile
`Comparison-Statistics and
`Analysis of the Similarity Factor, f2
`
`Vinod P. Shah,l.4 Yi Tsong,2 Pradeep Sathe/ and
`Jen-Pei Liu3
`
`Received December 4, 1997; accepted February 26, 1998
`
`Purpose. To describe the properties of the similarity factor (f2) as a
`measure for assessing the similarity of two dissolution profiles. Discuss
`the statistical properties of the estimate based on sample means.
`Methods. The f2 metrics and the decision rule is evaluated using exam(cid:173)
`ples of dissolution profiles. The confidence interval is calculated using
`bootstrapping method. The bias of the estimate using sample mean
`dissolution is evaluated.
`Results. 1. f2 values were found to be sensitive to number of sample
`points, after the dissolution plateau has been reached. 2. The statistical
`evaluation off2 could be made using 90% confidence interval approach.
`3. The statistical distribution of f2 metrics could be simulated using
`'Bootstrap' method. A relatively robust distribution could be obtained
`after more than 500 'Bootstraps'. 4. A statistical 'bias correction' was
`found to reduce the bias.
`Conclusions. The similarity factor f2 is a simple measure for the com(cid:173)
`parison of two dissolution profiles. But the commonly used similarity
`factor estimate f2 is a biased and conservative estimate of f2. The
`bootstrap approach is a useful tool to simulate the confidence interval.
`KEY WORDS: dissolution; similarity factor; estimation bias; boot(cid:173)
`strap confidence interval.
`
`INTRODUCTION
`
`For immediate release solid oral drug products, a single
`time-point dissolution
`specification has been
`routinely
`employed as a quality control release test. In general, a single
`point dissolution test does not characterize the dosage form
`completely, and therefore the dissolution profile and dissolution
`profile comparison is recommended in recently released guid(cid:173)
`ances by the Agency (1-4). For the post-approval changes such
`as (i) scale-up, (ii) manufacturing site, (iii) component and
`composition, (iv) equipment and process changes, a comparison
`of dissolution profiles between pre-change and post-change
`products is recommended in SUPAC-IR guidance (1) as it pro(cid:173)
`vides a more precise measurement of product similarity using
`
`dissolution characteristics. Dissolution profiles may be consid(cid:173)
`ered similar by virtue of (i) overall profile similarity and (ii)
`similarity at every dissolution sample time point. The dissolu(cid:173)
`tion profile comparison can be carried out using model indepen(cid:173)
`dent or model dependent methods.
`In the last decade, several methods for the comparison
`of dissolution profiles were proposed in the literature (5-9).
`However, a major problem has been the quantification of the
`comparison of dissolution profile. Shah et al. proposed a multi(cid:173)
`variate analysis of variance method to test for the difference
`between two dissolution profiles (5). Chow et al. proposed
`dissolution difference measurement and similarity testing based
`on parameters after fitting a one-degree autoregression time
`series model (6). Sathe et al. proposed dissolution difference
`measurement and similarity testing based on parameters of the
`profiles after fitting a selected mathematical model (7). Tsong et
`al. proposed dissolution difference measurement and similarity
`testing based on multivariate 'Mahalanobis' distance between
`two dissolution data sets (8). However, the statistical methods
`proposed in most of these examples involved the complicated
`estimation of covariance matrix.
`Recently, Moore and Flanner proposed a simple model inde(cid:173)
`pendent approach using mathematical indices to define difference
`factor, f~o and similarity factor, f2, to compare dissolution profiles
`(9). The f 1 and f2 factors are derived from Minkowski difference
`(average absolute differences) and mean-squared difference
`respectively. The similarity factor f2 and a similarity testing criteria
`based on f2 were therefore recommended for dissolution profile
`comparison in the FDA's Guidances for Industry (1-4). The sim(cid:173)
`plicity of similarity factor generated considerable interest. Subse(cid:173)
`quently, examples of the application of f2 appeared in the literature
`(I 0-12), and some statistical properties of f2 were also delineated
`in two papers ( 12, 13).
`The purpose of this work is to (i) describe f2 as a population
`measure for assessing the similarity of two dissolution profiles
`(ii) describe how a similarity criteria for f2 is defined for the two
`dissolution profiles (iii) discuss the statistical properties of f2, an
`estimate of population f2 based on sample means, (iv) discuss
`the estimation of the confidence interval of f2 based on f2 and
`calculation of the bias of f2, and (v) discuss the corresponding
`hypotheses for similarity testing based on f2 and f2. These discus(cid:173)
`sions will provide rational steps for the application of similarity
`factor f2 in dissolution profile comparison.
`
`SIMILARITY FACTOR
`
`A. Theoretical Considerations
`
`The profile comparison in general refers to the comparison
`of two dissolution profiles between (i) a reference batch and a
`test batch (ii) a pre-change batch and a post-change batch, and
`(iii) different strengths of products for biowaivers as discussed
`in various guidances. The principles can be applied at anytime
`when a profile comparison is needed.
`To illustrate the applications of similarity factor, f2, con(cid:173)
`sider the dissolution profiles of the two batches generated using
`P number of sample points. For comparison of the dissolution
`profiles of two batches, the dissolution measurements should
`be made under the same test conditions and the dissolution
`time points for both the profiles should be the same, e.g., for
`
`1 Office of Pharmaceutical Science, Center for Drug Evaluation and
`Research, Food and Drug Administration, HFD-350,1451 Rockville
`Pike, WOC 2, Rockville, Maryland 20852.
`2 Division of Biometrics III, Office of Epidemiology and Biometrics,
`Center for Drug Evaluation and Research, Food and Drug Administra(cid:173)
`tion, Rockville, Maryland 20857.
`3 Department of Statistics, National Cheng-Kung University, Tainan,
`Taiwan, ROC.
`4 To whom
`correspondence
`Shah vi @CDER.FDA.GOV)
`DISCLAIMER: The manuscript represents the personal opinions of
`the authors and does not necessarily represent the views or policies
`of FDA.
`
`addressed.
`
`(e-mail:
`
`should
`
`be
`
`889
`
`0724·8741/98/0600·0889$15.00/0 © 1998 Plenum Publishing Corporation
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 001
`
`

`

`890
`
`Shah, Tsong, Sathe, and Liu
`
`Table 1. Average Difference Between Two Dissolution Profiles of
`Reference Batches
`
`f 2 Limit
`
`2%
`
`83
`
`5%
`
`65
`
`10%
`
`50
`
`15%
`
`41
`
`20%
`
`36
`
`clear that once the average distance at any sample time point between
`any two reference batch is defined, the similarity limit based on f2
`can be defined independent to the test batch or the specific reference
`batch and independent to the number of sampling time point~ to
`be used in the assessment of dissolution similarity. Table I provides
`the f2 similarity limit~ for different average distances at multiple
`time points by appropriate substitution in Equation 1.
`
`B. Results and Discussions
`
`Example #1, One Reference Batch and Four Test Batches
`(Tables 2, 3 and Figures 1 and 2). To illustrate the concept of
`assessing similarity and dissolution profile comparison using
`
`%Dissolved
`
`90
`
`80
`
`70
`
`60
`
`50
`
`40
`
`30
`
`10
`
`15
`
`Test Batch 1
`
`Reference
`Batch
`
`~
`
`~
`
`W
`
`Time in Minutes
`Fig. 1. Actual mean data.
`
`H
`
`~
`
`f2Value
`00,------------------------------------------.
`
`immediate release products, 15, 30, 45, 60 minutes and for
`controlled release products, I, 3, 5 and 8 hours. Let (f.lr~> f.lrz,
`... , f.lrP) represent the dissolution measurements at P time
`points on the reference profile and (f.l11 , f.lt2> ... , f.l1p) be the
`corresponding P measurements on the test profile. The distances
`between the two profiles at these P time points are (I f.lri -
`f.l11 1,
`lf.lrz -
`f.l121, ... , lf.lrP -
`f.l1pl). The distances at the P time
`points may be combined into one measure, by either the sum,
`lf.lrl -
`f.ltll + lf.lr2 -
`f.ltzl + ... + lf.lrP -
`j..l,pl, or the
`Dl =
`s uare
`root
`of
`the
`sum
`of
`s uares, D2
`f.lt1) 2 + (f.lr2 -
`f.ltz) 2 + · · · + (f.lrP -
`f.ltP) 2
`[(f.lrl -
`].
`In 1996, Moore and Flanner proposed measurements of
`relative distance and similarity of two dissolution profiles as
`functions of D 1 and D 2 , as follows:
`
`and
`
`fz = 50•Iog{[ 1 + (1/P) i~ (f.lti -
`
`f.ln) 2 r112
`
`•100}
`
`= 50•log{[l +(IIP)D~r 1 '2•100}
`
`(I)
`
`where log is the logarithm based on 10. Note that f1 reflects
`the cumulative difference between the two curves at all time
`points, and is a measure of relative error between the two
`curves. Conceptually, f 1 which is a function of the average
`absolute difference between the two dissolution curves could
`be referred as a 'difference' factor. On the other hand, f2 metric
`is a function of the reciprocal of mean square-root transform
`of the sum of square distances at all points. Conceptually, f2
`which is a measure of the similarity in the percent dissolution
`between two curves, could be referred as a 'similarity' factor.
`When the two profiles are identical, f2 = 50•log(l 00) = 100,
`and when the dissolution of one batch is complete before the
`other begins, f2 = 50•log{[l + (l!P)~f= 1 (IOWr 112•100} =
`-.001, which can be rounded to 0. Thus the value of f2 ranges
`between 0 to 100 with a higher f2 value indicating more similar(cid:173)
`ity between the two profiles.
`In a real life situation, due to the batch-to-batch variation
`in dissolution profiles, it is not expected to have f2 value be
`anywhere near I 00 even when the two dissolution curves are
`generated from the same batch of tablets (or capsules). A test
`batch is therefore accepted as 'similar' to a reference batch if
`the dissolution profile difference between the two batches is
`no more than the dissolution profile difference between the two
`reference batches. Empirically, the experience in dissolution
`data analysis leads one to believe that an average difference of
`no more than 10% at any sample time point, of the batches of the
`same formulation may be acceptable. When this 10% average
`difference is substituted in the Equation I, f2 becomes:
`
`f2.10 = 50•log{[ I + (1/P) i~ 1101 2 r112
`
`•100}
`
`= 50olog{[lOir 112•100}
`= 50•1og(9.95037) = 49.89
`which may be rounded to 50 for simplicity. A test batch dissolu(cid:173)
`tion is therefore considered similar to that of the reference batch
`if the f2 value of the two true profiles is not less than 50. It is
`
`20+--------------------------------------------i
`
`tot-------------------------------------------1
`
`10
`15
`Global Average Difference
`Fig. 2. Actual profile comparison with similarity limits.
`
`20
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 002
`
`

`

`In Vitro Dissolution Profile Comparison
`
`%Dissolved
`
`1~,-------------------------------------------,
`
`Test Batch 2
`
`891
`
`similar to the reference batch even when one allows an average
`difference of 20% at all time points (Figure I). Using more time
`points after more than 85% dissolution, will invariably increase
`the f2 value leading to bias in the similarity assessment. For
`example, when using cumulative dissolutions up to 90 minutes,
`for the same four test batches, the f2 values increases in almost
`all test batches (Table 3). It is therefore important to limit the
`number of sample points to no more than one, once any of the
`batch (product) reaches 85% dissolution.
`
`C. Estimation of Similarity Factor
`
`30
`
`so
`
`90
`Time in Minutes
`
`120
`
`150
`
`180
`
`Fig. 3. Sample mean dissolution.
`
`f2, consider the following example. In Table 2 provides the
`actual cumulative dissolutions at 15, 30, 45, 60, 75 and 90
`minutes of a reference batch and four test batches. The f2 value
`for each of the four test batches compared to the reference
`batch is given in Table 3.
`Based on measurements up to 60 minutes (the time when
`the reference product is dissolved up to 87% ), it is clear that test
`batch #2 is similar to the reference batch with an average difference
`of 5% at the four time points. Test batch #3 can be claimed to
`be similar to the reference batch with an average difference of
`10% at the four time points. Test batch #1 can only be considered
`to be similar to the reference batch if the average difference
`between any two reference batches is 15%. Test batch #4 is not
`
`Table 2. Example #1: Dissolution Profile of One Reference and Four
`Test Batches
`
`Batch
`
`Reference
`Test batch #I
`Test batch #2
`Test batch #3
`Test batch #4
`
`15
`
`40
`28
`36
`43
`78
`
`30
`
`67
`51
`69
`78
`89
`
`% Drug dissolved in
`
`45
`
`80
`71
`84
`86
`91
`
`60
`
`87
`88
`89
`93
`93
`
`75
`
`89
`92
`93
`94
`95
`
`90 minutes
`
`91
`94
`95
`96
`98
`
`Table3
`
`When calculated up to
`60 minutes only
`When calculated up to
`90 minutes
`
`48
`
`52
`
`f2 Value for test batch
`
`2
`
`70
`
`71
`
`3
`
`54
`
`57
`
`4
`
`32
`
`36
`
`Note: f2 value calculated by using data presented for example #1, in
`Table 2.
`
`The properties illustrated in the last section are based on
`the f2 of the actual (population) dissolution profiles of the
`reference and test batches. In practice, dissolution testing is
`often carried out with no more than 12 units and the dissolution
`profile of each batch is an estimate based on dissolutions of
`the 12 units. Hence (x,~., x,2, ... , x,r) and (xt! , x1z., ... , xtP)
`are used to estimate (1-Lrl• !J-,2, ... , 1-Lrr) and (!J-,~> !-Lt2• ... , 1-Ltr)
`respectively, where xti.• Xri. are the mean dissolution value of
`the twelve tablets measured at the i-th time point of the test
`and the reference batch respectively. With these estimates, f2
`is estimated as follows
`I + (liP) i~ (x,i - x,)2
`
`f2 = 5Qelog{
`
`[
`
`-1/2
`•I 00}
`
`]
`
`D. Confidence Interval of Similarity Factor
`
`Because of the sampling variation for the estimate, dissolu(cid:173)
`tion similarity of the test and reference batches may not be
`assessed by direct comparison of f2 with the similarity limit, SL.
`The SL proposed in the guidances is 50 (1-4). Assuming the
`expected value of f2 equals f2, i.e., E(f2) = f2, for an assurance
`of 95% correct decision, one should compare the 90% lower
`confidence limit of E(f2) with SL instead. In order to have a
`mathematical form of the confidence interval, one needs to derive
`the sampling distribution of f2. Each component of the mean
`vector x, = (X,~., xr2.• ... , x,r) and x, = (Xti , x,2_, ... , x,p) is a
`random variable with standard error se(xki), where k=r,t, and the
`elements in xk are correlated. In order to have a standard (or
`asymptotically standard) distribution for f2, one needs to standard(cid:173)
`ize f2 by its covariance matrix. If there is a known standardized
`form offz, it would be a complicated function of the 'Mahalanobis'
`distance as described by Tsong eta!. (7,8). Alternatively, the 90%
`confidence· interval can be simulated through bootstrap method
`as given by Tsong eta!. (14).
`A bootstrap sample of f2 can be generated by random
`sample with replacement twelve times from x,i =(xrlj.• x,2i, ... ,
`Xrrj) and x,i = ( x, 1i, x,2i.• ... , X1pi_), wherej=1 to 12. Let x',i
`=( Xrlj'.• Xr2j', ... , XrPj'_) and x',i = ( XtJj.'• X1zj.'• .. , X1pj-.), j'= I
`to 12, be the twelve dissolution vectors re-sampled from the
`12 tablets of the test and reference batches respectively. Note
`that some of the vectors of dissolution values may be identical
`because of the replacement in the sampling. Let f2 denote the
`estimated f2 value of the bootstrap sample. Considering that M
`sets of sample are generated using the bootstrap mt;_thod, the
`90% percent confidence interval is defined by [Lfz, Uf2], where
`Lf2 and Uf2 are the 5th and 95th percentiles of the Ef2 values.
`Since distribution of f2 is skewed, an adjustment may be neces(cid:173)
`sary. The adjusted confidence interval, BC" of E(f2) of bias
`correction (f2("!), f2("2l) is defined with
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 003
`
`

`

`892
`
`Shah, Tsong, Sathe, and Liu
`
`a! = <I>(Z0 + (Z0 + z<"'l)f[l - a(Z0 + z<o:l)]
`a2 = <I>(Z0 + (Z0 + z(l-o:l)f[ I - a(Zo + z(l-o:J)
`zo = <P- 1(#(f2(m) < f2)/M)
`
`a-~ (f20 -
`- " ' '
`
`f2(iJ) /{6[ ~ (f2(·J- f2(il)] }
`' 2 3 / 2
`' 3 " ' '
`
`where a is the level of type I error, f2(iJ is the i-th jackknife statistic,
`f2(·J is the mean of jackknife statistics, f2(m) is the bootstrap
`estimate of the m-th bootstrap sample, f2 is the original sample
`mean, z<"'J is the a-th percentile of standard normal distribution.
`
`Example #2, One Reference Batch and Five Test Batches
`(Tables 4, 5 and Figure 2). To illustrate the application of
`bootstrap method in confidence interval estimation and assess(cid:173)
`ment of dissolution similarity, consider the cumulative percent
`of dissolution at 30, 60, 90 and 180 minutes of five test batches
`and one reference batch with 12 tablets each as shown in Table
`4. Table 4 also provides the sample means of each batch at
`every time point. The covariance and correlations among time
`points are given in Table 5. From Table 5, the correlation
`between two time points can be as high as 0.93 and some times,
`the cumulative percent dissolved at different time points may
`be negatively correlated. The mean dissolution values of test
`batch #I differ from the reference batch by no more than 8%.
`Test batch #2 dissolved 15% more than the reference batch at
`30 minutes, but the differences between the test and reference
`
`batches are less than 8% at any time point after 30 minutes.
`Test batch #3 is more than 12% different compared with the
`reference batch at 90 minutes and less than I 0% at any other
`time points. Test batch #4 differs with the reference batch by
`more than 19% at 30 minutes and shows no difference at any
`other time point. Test batch #5 differs with the reference batch
`by more than 17% at 60 minutes, but Jess than 10% at any
`other time point. The f2 of the five test batches are 60.04 for
`test batch #I, 51.08 for test batch #2, 51.19 for test batch #3,
`50.07 for test batch #4 and 48.05 for test batch #5. When taking
`f2 as f2 for dissolution similarity assessment, one would consider
`that all test batches except batch #5 have dissolution profile
`similar to the reference batch, when the similarity criterion
`value, 50 (computed based on an I 0% average distance at all
`key time points) is used. However, the bootstrap confidence
`intervals of Ecf2) give the lower 90% confidence limits lower
`than similarity criterion in this example, using either the per(cid:173)
`centage confidence interval (PI) or the BC"' confidence interval
`which are given in Table 6 with 100, 200, 400, 500 and 1 ,000
`bootstrap samples. The 90% lower confidence limits BC"' based
`on the 500 samples are 52.79 for test batch #1, 48.39 for test
`batch #2, 48.59 for test batch #3, 48.38 for test batch #4 and
`46.11 for test batch #5. It indicates that all test batches except
`test batch #I fail to show dissolution similarity to the reference
`batch when the f2 value of 50 is used as a cutoff point for
`accepting similarity between two dissolution profiles.
`
`Table 4. Example #2 Dissolution Data of Reference and Five Test Batches
`
`Reference batch
`
`Test batch l
`
`T batch 2
`
`30
`
`60
`
`90
`
`180
`
`30
`
`60
`
`90
`
`180
`
`30
`
`60
`
`90
`
`180
`
`36.1
`33
`35.7
`32.1
`36.1
`34.1
`32.4
`39.6
`34.5
`38
`32.2
`35.2
`34.92
`
`28.7
`26.4
`25.4
`23.2
`25.1
`28.7
`23.5
`26.2
`25
`24.9
`30.4
`22
`25.80
`
`58.6
`59.5
`62.3
`62.3
`53.6
`63.2
`61.3
`61.8
`58
`59.2
`56.2
`58
`59.5
`
`80
`80.8
`83
`81.3
`72.6
`83
`80
`80.4
`76.9
`79.3
`77.2
`76.7
`79.27
`
`Test batch 3
`
`48.2
`53.1
`52.4
`49.5
`50.7
`54.1
`50.3
`50.6
`49.1
`49.5
`53.9
`46.3
`50.64
`
`63.8
`68.3
`70
`65.5
`68
`70.8
`66.1
`67.7
`63.6
`66.7
`70.4
`63
`67.00
`
`93.3
`95.7
`97.1
`92.8
`88.8
`97.4
`96.8
`98.6
`93.3
`94
`96.3
`96.8
`95.08
`
`85.6
`90.6
`89.5
`92.2
`87.6
`93.6
`85.1
`88
`85.8
`86.6
`89.9
`88.7
`88.6
`
`38.75
`36.16
`38.49
`37.27
`48.12
`48.45
`41.08
`39.64
`36.06
`36.69
`39.95
`43.41
`40.34
`
`17.1
`16
`12.7
`15.1
`14.1
`12.1
`14.4
`19.6
`14.5
`14
`18.2
`13.2
`15.08
`
`61.79
`61.21
`63.89
`62.52
`77.18
`80.62
`67.62
`63.68
`61.59
`63.6
`67.98
`74.07
`67.15
`
`85.14
`84.25
`84.94
`85.65
`95.32
`95.05
`84.94
`80.73
`82.22
`84.5
`87.4
`93.95
`87.01
`
`Test batch 4
`
`58.6
`59.5
`62.3
`62.3
`53.6
`63.2
`61.3
`61.8
`58
`59.2
`56.2
`58
`59.5
`
`80
`80.8
`83
`81.3
`72.6
`83
`80
`80.4
`76.9
`79.3
`77.2
`76.7
`79.27
`
`100.2
`97.3
`96.39
`95.47
`99.3
`98.94
`99.03
`95.63
`96.12
`98.42
`98.1
`97.8
`97.73
`
`93.3
`95.7
`97.1
`92.8
`88.8
`97.4
`96.8
`98.6
`93.3
`94
`96.3
`96.8
`95.08
`
`48
`52
`48
`53
`45
`48
`51
`49
`44
`53
`49
`52
`49.33
`
`41.5
`43.7
`46.3
`44
`42.6
`44.4
`43
`44.4
`44.8
`41.7
`42.3
`42
`43.39
`
`60
`75
`60
`70
`60
`66
`71
`63
`60
`68
`63
`68
`65.33
`
`84
`89
`83
`93
`84
`90
`91
`89
`84
`81
`86
`87
`86.75
`
`Test batch 5
`
`78
`78.3
`78.3
`79.9
`73.2
`78.4
`79
`79.6
`78.7
`76.9
`77
`78.2
`77.96
`
`86.4
`85.9
`86.9
`88.6
`81.4
`86.2
`87.5
`87.3
`86.9
`84.5
`81.9
`92.4
`86.33
`
`103
`99
`101
`103
`105
`103
`100
`104
`103
`104
`105
`104
`102.83
`
`98.3
`102.9
`96.4
`96
`95.5
`98.4
`99.5
`99.9
`97.8
`100
`97.9
`100.3
`98.58
`
`Time
`
`Tablet
`I
`2
`3
`4
`5
`6
`7
`8
`9
`10
`]]
`12
`Mean
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`II
`12
`Mean
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 004
`
`

`

`In Vitro Dissolution Profile Comparison
`
`Table 5. Covariance and Correlation Matrices of the Six Batches
`
`Covariance
`
`Correlation
`
`Std
`
`2.36
`2.84
`2.98
`2.73
`4.28
`6.62
`4.97
`1.55
`2.96
`5.10
`3.67
`1.90
`2.47
`2.37
`2.68
`2.68
`2.56
`2.84
`2.98
`2.73
`5.81
`1.74
`2.90
`2.10
`
`030
`
`5.55
`-0.21
`-0.57
`0.19
`18.30
`27.39
`18.73
`3.29
`8.79
`12.33
`4.18
`-1.48
`6.10
`3.74
`3.70
`1.40
`5.10
`-0.98
`-0.56
`0.81
`2.18
`1.02
`0.67
`-0.72
`
`060
`
`-0.21
`8.09
`7.89
`4.92
`27.39
`43.86
`30.52
`4.67
`12.33
`26.06
`11.18
`-5.21
`3.74
`5.60
`5.95
`3.51
`-0.98
`8.09
`7.89
`4.92
`1.02
`3.02
`3.52
`1.13
`
`090
`
`-0.58
`7.88
`8.88
`5.05
`18.73
`30.53
`24.75
`3.85
`4.18
`11.18
`13.48
`-2.23
`3.70
`5.95
`7.19
`4.03
`-0.56
`7.89
`8.88
`5.05
`0.67
`3.52
`8.40
`1.73
`
`0180
`
`0.19
`4.92
`5.05
`7.43
`3.28
`4.67
`3.85
`2.40
`-1.48
`-5.21
`-2.23
`3.61
`1.40
`3.51
`4.03
`7.17
`0.81
`4.92
`5.05
`7.43
`-0.72
`1.13
`1.73
`4.39
`
`030
`
`1.00
`-0.03
`-0.08
`0.03
`1.00
`0.97
`0.88
`0.50
`1.00
`0.81
`0.38
`-0.26
`1.00
`0.64
`0.56
`0.21
`1.00
`-0.15
`-0.08
`0.13
`1.00
`0.40
`0.16
`-0.24
`
`060
`
`-0.03
`1.00
`0.93
`0.63
`0.97
`1.00
`0.93
`0.46
`0.81
`1.00
`0.60
`-0.54
`0.64
`1.00
`0.94
`0.55
`-0.15
`1.00
`0.93
`0.63
`0.40
`1.00
`0.70
`0.31
`
`090
`
`-0.08
`0.93
`1.00
`0.62
`0.88
`0.93
`1.00
`0.50
`0.38
`0.60
`1.00
`-0.31
`0.56
`0.94
`1.00
`0.56
`-0.08
`0.93
`1.00
`0.62
`0.16
`0.70
`1.00
`0.28
`
`Batch
`
`2
`
`3
`
`4
`
`5
`
`6
`
`Time
`
`030
`060
`090
`0180
`030
`060
`090
`0180
`030
`060
`090
`0180
`030
`060
`080
`0180
`030
`060
`090
`0180
`030
`060
`090
`0180
`
`893
`
`0180
`
`0.03
`0.63
`0.62
`1.00
`0.50
`0.46
`0.50
`1.00
`-0.26
`-0.54
`-0.32
`1.00
`0.21
`0.55
`0.56
`1.00
`0.13
`0.63
`0.62
`1.00
`-0.24
`0.31
`0.28
`1.00
`
`Efron and Tibshrani (15) indicated that in general a boot(cid:173)
`strap of 400 sample sets give precise estimate. However, the
`rate of convergence of the bootstrap confidence limits is data
`dependent, and it is recommended to calculate a few bootstrap
`estimates in order to make sure that the estimate is stable. Table
`6 shows that the confidence intervals are quite stable with 500
`sample sets for both Percent interval and BC" estimate.
`
`E. Bias of the Estimate of Similarity Factor
`
`The confidence interval estimated using bootstrap method
`is for the expected value of f2, E(f2). The assessment of dissolu-
`
`tion similarity using the confidence interval as in the last section
`is unbiased only if f2 is an unbiased estimate of f2, which means
`E(f2) = f2. Assuming that there are n tablets in both the test
`the expected value of
`and
`reference batches, consider
`[(liP) LF=dLf=I (xtij - Xrij)/n) 2],
`
`E[ (liP) i~ t~ (xtij - Xri)/n rJ
`= E( (liP{~ [ ( t (xtij - Xri)/n -
`
`(f.lti - 1-lri) n
`
`Type Test Sample
`of CI batch mean Mean
`
`I 00 Bootstraps
`
`Table 6. Bootstrap Confidence Intervals
`
`200 Bootstraps
`
`400 Bootstraps
`
`500 Bootstraps
`
`I ,000 Bootstraps
`
`CI
`
`Mean
`
`CI
`
`Mean
`
`CI
`
`Mean
`
`CI
`
`Mean
`
`CI
`
`PI"
`Bcah
`PI
`Bca
`PI
`Bca
`PI
`Bca
`PI
`Bca
`
`2
`
`3
`
`4
`
`5
`
`60.03
`60.08c
`51.08
`51.01
`51.19
`51.19
`50.07
`50.06
`48.05
`48.05
`
`50.96
`
`61.16 (54.26, 70.28) 60.57
`(53.73, 70.13) 60.17
`(52.79, 68.15) 60.22
`(52.84, 68.69) 60.11
`(54.34, 70.73)
`(54.18, 70.24)
`(54.07, 70.35)
`(54.19, 70.73)
`(48.36, 53.63) 50.97
`(48.23, 53.32) 51.03
`(48.33, 53.68) 51.01
`(48.25, 53.71) 50.98
`(48.37, 53.68)
`(48.37, 53.46)
`(48.39, 53.74)
`(48.35, 53.77)
`(48.52, 54.05) 51.27
`51.22 (48.47, 54.11) 51.16
`(48.59, 54.05) 51.29
`(48.59, 54.10) 51.29
`(48.49, 53.94)
`(48.14, 53.95)
`(48.41, 53.91)
`(48.47, 53.87)
`( 48.49, 51.50) 49.96
`(48.51, 51.42) 49.93
`(48.41, 51.55) 49.% (48.39, 51.55) 49.99
`(48.96, 51.90)
`(48.75, 51.69)
`( 48.63, 51.85)
`(48.60, 51.74)
`(46.52, 49.91) 48.14 (46.35, 49.89) 48.00 (46.08, 49.91) 48.01
`( 46.11' 50.09) 48.01
`(46.01, 49.78)
`( 46.41' 49 .89)
`(46.32, 50.33)
`( 46.33, 50.33)
`
`49.86
`
`48.17
`
`(53.01, 68.34)
`(53.89, 70.24)
`(48.25, 53.69)
`(48.37, 53.74)
`(48.54, 54.56)
`( 48.41' 54.22)
`(48.38, 51.59)
`(48.47, 51.73)
`(46.05, 50.04)
`(46.15, 50.17)
`
`" Percent confidence interval.
`h Bca adjusted confidence interval.
`c Jackknife mean.
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 005
`
`

`

`894
`
`Shah, Tsong, Sathe, and Liu
`
`where al; and a~ are the variances of percent dissolution mea(cid:173)
`sured at the i-th time point of the test and reference batches
`respectively.
`When n becomes large, the expected value of the mean
`squared differences between sample means E[(l/P)LF=dLf= 1
`(x1;j-xri)/n} 2
`] becomes very close to (1/P)[LF=,(IJ.-1;-IJ.-,;)2
`], the
`mean squared difference between population means. Hence, f2
`is an asymptotically unbiased estimator of f2.
`In contrast,
`
`E(fz) = E(50•log{[ I + (1/P) ;~ (x,; - x,J 2 r112
`
`•100})
`
`= E{ 100 - 25•log[ I + (1/P) ;~ (x,; - x,;) 2
`
`]}
`
`= 100 - 25•log( I + E[ (1/P) ;~ (x,; - x,;) 2
`
`] )
`
`100 - 25•log( I + (liP{~ (IJ.-1; -
`
`ILrY
`
`+ ;~ (al; + a~;)/n ])
`
`(2)
`
`with Taylor's expansion
`
`= 100 - 25{ log( 1 + (1/P{~ (~J.-1;
`
`-
`
`1Ln)2])
`
`+[(liP);~ (al; + a~;)/n]/[ 1 +(liP{~ (~J.-1;- ILri
`
`+ ;~ (al; + a;;)/n]]
`( [ (1/P) ;~ (a?; + a~;)/n] I [I + (liP{~ (~J.-1;
`+ ;~ (af; + a~;)/n ]])
`
`2
`/2 + ... }
`
`-
`
`IJ.-,;)2
`
`-
`
`which in term
`
`< 100 - 25•log( I +(1/P) ;~ (IJ.,; -
`
`IJ.-,;)2])
`
`IJ.-,;) 2 r112
`
`= 5Qolog{[ 1 + (1/P) ;~ (~J.-1; -
`
`= fz.
`
`With bias
`
`•100}
`
`-25[(1/P) t (af; + a;;)/n]/[1 + (1/P)[t (IJ.,;- ILri
`
`1-1
`
`J-1
`
`- ([ (1/P) ;~ (a?; + a~;)/n ]/[I + (1/P{~ (~J.,; -
`
`ILri)2
`
`+ i~ (al; + a~;)/n JJr/2 + ... }
`
`This implies that the use off2 is conservative in assessment
`of dissolution similarity for the criterion defined for actual
`dissolution profiles.
`
`F. Bias Correction
`
`As shown in last equation, the estimation bias is contrib(cid:173)
`uted by the term LF= 1 ( al; + a;;)/n within the log function. An
`intuitive bias correction would lead to subtracting the unbiased
`estimate of LF= 1 (a~ + a~;)/n within f2 and we have the follow(cid:173)
`ing unbiased estimate
`
`- ;~ (s~ + s~;)/n
`
`] }
`
`-112
`•100)
`
`where sf; and s~; are the unbiased estimates of variance at the
`i-th time point of the test and reference batches respectively.
`The confidence interval is then adjusted accordingly. For the
`five test batches in example #2 the bias adjusted estimates are
`given in Table 7. It is shown in Table 7 that for the five
`comparisons, the bias adjusted estimates of f2 are not much
`different to the biased estimate f2 because of the small dissolu(cid:173)
`tion variance of the six (one reference and five test) batches.
`However, the adjustment is not valid when LF=, (x1; -
`x,;) 2 < Lr= I (s;; + s~)/n.
`
`G. Corresponding Hypotheses for Similarity Testing
`When f2 is used as an estimate of f2, the application of f2
`for the assessment of dissolution similarity, can be interpreted
`in two ways.
`
`I. When the similarity limit SLrz is set independent of the
`data of existing reference batches, for example a fixed SLrz =
`
`Table 7. Bias Adjusted Estimate of f2
`
`Testing
`batch
`
`2
`3
`4
`5
`
`fz
`
`60.03
`51.08
`51.19
`50.07
`48.05
`
`2: (S~; + S~;)/12"
`
`2: (X,; ~ X,;Jl h
`
`9.94
`6.82
`4.67
`4.95
`3.99
`
`154.81
`358.96
`354.34
`393.36
`474.61
`
`f2*
`
`60.73
`51.29
`51.34
`50.21
`48.14
`
`a Average of sum of within batch variances.
`b Sum of between batch mean squares.
`
`Par Pharm., Inc.
`Exhibit 1043
`Page 006
`
`

`

`In Vitro Dissolution Profile Comparison
`
`895
`
`50 for all products, the similarity limit is set for f2 instead of
`Ecf2). The similarity comparison using the lower 90% limit of
`the bootstrap confidence interval of E(f2) as proposed earlier
`is an approximation test for the following hypotheses,
`HQ: 100 - 25 log( 1 + (liP{~ (f.L1;
`
`f.L,;) 2
`
`-
`
`+ ;~ (rr~ + rr 2 ,;)In]) :S SLrz
`
`versus
`
`H~: I 00 - 25 log( I + (liP{~ (f.L1; -
`
`f.L,;) 2
`
`similarity measurement f2 is not an exception. Even the loga(cid:173)
`rithm transformation of f2 complicates the known sampling
`distribution of mean squared differences. As pointed out by
`Liu et al. (12) there is no mathematical formula for the sampling
`distribution either in exact or asymptotic form. It is therefore
`difficult to assess the type I (consumer's risk) and type II
`(manufacturer's risk) error rates. Without these error rates, it
`is difficult to evaluate the power, sample size, magnitude of
`the bias, validity of the approximation, and sensitivity of the
`f2 test.
`The similarity factor f2 is a function of the mean differences
`and does not take into account the differences in dissolutions
`within the test and reference batches. Hence careful interpreta(cid:173)
`tion is warranted when f2 is used as a similarity factor for
`batches with large difference in variance.
`
`+ ;~ (rr~ + rr2,;)/n ]) > SLr2
`
`CONCLUSIONS
`
`H0 will be rejected only when both Ll=l (f.L1; -
`f.Lri and
`Ll= 1 ( rr~ + rr~;)/n are small. Hence it may be considered to be
`a conservative test for the following hypotheses regarding f2,
`Ho: 100-25 log (I + (1/P) [Ll=l (f.L1; -
`
`]) :S SLrz
`
`f.Lr;) 2
`
`versus
`H.: 100-25 log (1 + (liP) [Ll=l (f.L1; - f.L,i]) > SLrz
`
`In this case, the proposed procedure can be taken either
`as a conservative test for H0 or it is considered as an approximate
`test for Hb when LF= 1 (rr~ + rr~;)/nP is very small (i.e., if n is
`large and within batch variances, rri\ and rr;; are small at every
`sampling time point). Correction for estimation bias may not
`exist when batches are similar but within-batch variances are
`large.
`2. However, if SLrz is product specified and determined
`by experience with the observed means of the reference batches,
`the similarity limit is actually set for E(f2) instead of f2 • In
`another word, it is actually SLE(fzl instead of SLrz. Correspond(cid:173)
`ingly, the hypotheses are defined instead as
`
`H0: 100-25 log(! + (1/P)LL~=l (f.L,;- f.L,i+ L~=l (rr~
`
`versus
`
`H~: 100-25 log(! + (1/P)[L~=I (f.L,;- f.L,;) 2
`+ 2:~= 1 ( rr~ + rr~;)/n]) > SLE(f2l
`
`(3)
`
`and the test would be considered unbiased.
`
`H. Limitations of Similarity Factor
`
`Measurements of difference or similarity of the profiles
`are often based on combining the differences at all time points
`into one measurement. Such measurements are often estimated
`by substituting sample means for the actual means. However,
`with the dissolutions correlated at the sample time points, such
`estimates are often complicated in that the variation of the
`estimate is difficult to calculate and the estimate itself may be
`biased, and the statistical properties are difficult to derive. The
`
`Through mathematical scaling, the f2 measurement takes
`the values ranging from 0 to I 00. A convenient critical value
`of 50 is derived for similarity of dissolution profiles based on
`average difference of I 0% at all sampling time points. Since
`the f2 is sensitive to the measurements obtained after either test
`or reference batch has dissolved more than 85%, it leads to the
`recommendation of limiting to no more than one sampling time
`point after 85% dissolution.
`In conclusion, similarity factor f2 provides a simple
`measure for the comparison of two dissolution profiles. The
`analysis and discussion suggest that the commonly used
`estimate f2 has complicated statistical prope