Supplementary material
An introduction to the approaches used by DDSolver
Yong Zhang,1 Meirong Huo,1 Jianping Zhou,1Aifeng Zou,1Weize Li,1Chengli Yao,1 and Shaofei Xie2
1 Department of Pharmaceutics, ChinaPharmaceuticalUniversity,No.24,Tongjiaxiang, 210009, Nanjing, China.
2Center for Instrumental Analysis, ChinaPharmaceuticalUniversity (Key Laboratory of Drug Quality Control and Pharmacovigilance, Ministry of Education),No.24,Tongjiaxiang, 210009, Nanjing, China.
This supplementary material provides an overview of the approaches used to assess the similarity between drug dissolution profiles used by DDSolver program. The advantages and disadvantages of each method are also discussedin detail.
- Contents -
1. Exploratory data analysis
2. ANOVA-based approaches
3. Ratio test procedures
4. Pairwise procedures
4.1. Difference factor and similarity factor
4.2. Rescigno index
5. Multivariate confidence region procedures
5.1. 90% CI of difference method
5.2. Multivariate statistical distance method
6. Model-dependent approaches
7. Bootstrap f2 method
8. Chow and Ki’s Method
8.1. Global similarity
8.2. Local similarity
9. Release rate comparison
REFERENCES
1. Exploratory data analysis
Exploratory data analysis is usually considered as the first step in obtaining an improved understanding of dissolution data both graphically and numerically (1).By plotting the average dissolution data with error bars extending to one standard deviation (SD) at each sampling time point, the dissolution data for both the reference and the test formulations can be overlaid and illustrated in one chart. Then the dissolution data are summarized numerically, 95% confidence intervals for the mean data at each sampling time point are calculated, and a chart of mean dissolution profiles with 95% confidence regions is plotted which can be used for preliminary comparison of the two profiles. If the confidence regions for the two formulations at a given time point do not overlap, then the mean dissolution profiles at that time point may be considered to differ significantly from each other.
2. ANOVA-based approaches
Analysis of variance is a powerful and commonly used statistical procedure for testing the equality of multiple population means. Several types of ANOVA-based approaches have been used to compare dissolution data, including univariate ANOVA (2,3), multivariate ANOVA (MANOVA) (4), analysis of covariance (ANCOVA) (5) and level and shape approaches(6). However, these methods are not appropriate because dissolution-testing results over time are not independent due to the nature of the dissolution test. As an alternative, repeated-measures ANOVA (split-plot approach) appears more suitable for comparing multipoint dissolution profiles(5,7,8) because it is specially designed to compare repeated data measurements. However, in practical application, none of these ANOVA-based approaches is widely accepted or recommended by the FDA. This is probably because: (1) some degrees of freedom may be lost when considering time, which is not of primary pharmaceutical interest, as a class variable in an ANOVA model, and if a formulation-by-time interaction effect does exist, the interpretation becomes much more complicated(9),and (2) ANOVA-based approaches address the question of statistical sameness rather than pharmaceutical sameness. In other words, from a statistical viewpoint, these methods may be satisfactory, but from a pharmaceutical and practical viewpoint, theyare overly discriminating and not useful(4). Therefore, DDSolver does not place much emphasis on these methods, and only the simplest form of univariate ANOVA method is implemented. This method is used to compare separately the mean dissolution data at each time point. It is equivalent to the Student’s t-test when only two formulations are being compared and can be considered as a statistical interpretation of the results of the exploratory data analysis.
3. Ratio test procedures
A ratio test procedure is a model-independent approach for comparing dissolution profiles;these can be further classified into ratio tests of percent dissolved (PD), area under the dissolution curve (AUC), and mean dissolution time (MDT). The method can be used to calculate the ratio of mean PD, mean AUC, or average MDT of the test formulation to the reference formulation at each time point, followed by an estimation of its standard error (SE) and 90% confidence interval (CI)(2,4). Here the ratio test of PD is taken as an example.Because the two compared formulations are independent of each other, the 90%CI of the ratio of two independent groups of PD values can be approximately estimated according to:
in which RatioT/Rand SERatio can be calculated using
and ,
where nT and nR are the numbers of units of the test and reference formulations respectively, is the t-value with degrees of freedom and a confidence limit of 90%, SERatio is the standard error of the mean PD ratio, SET and SER are the standard errors of the test and reference formulations, and and are the mean PD of the test and reference formulations at the time point under consideration.
Compared with the AUC and MDT ratio tests, which require as a first step the calculation of AUC or MDT, and which can be considered meaningful only when most of the drug has been dissolved, the PD ratio test may be preferable because of its simplicity of calculation and its ability to construct a similarity criterion at any time point. When the PD ratio test is used, a ratio value of approximately 1.0 with a narrow 90% CI may indicate a “local” sameness at that time point. However, in most cases, the ratio value, as well as its 90%CI, varies throughout the dissolution time, and therefore it is difficult to establish specification limits for overall similarity. This disadvantage of the ratio test procedure has greatly limited its application. Hence, it wasnecessary to implement more widely accepted approaches in the DDSolver program.
4. Pairwise procedures
Pairwise proceduresarethe most widely used method for assessing the similarity between a pair of dissolution data. The distinction of this method is that the similarity can be evaluated usinga single statistical index estimated from the individual raw data (or mean data) of two profiles. These indices include the difference factor f1, the similarity factor f2(10) and the two Rescigno indices(11).
4.1. Difference factor and similarity factor
The difference factor f1 is a measure of the relative error between two curves, while the similarity factor f2 is a measure of the similarity in the percent of dissolution between two curves.These two factors can be respectively defined by:
where Rt, Tt are the percentage dissolved of the reference and test profile respectively at time point tand n is the number of sampling points. For the profiles to be considered “similar”, f1 should be close to 0, and f2 should be close to 100. Current FDA guidelines(12-14)suggest that two profiles can be considered similar if f1 is less than 15 (0–15) and f2 is greater than 50 (50–100), which is equivalent to an average difference of 10% at all sampling time points.Because thef1 and f2 methods are recommended by the FDA and are used by many pharmaceutical researchers, it is necessary to provide here a detailed discussion of the issues concerning the use and the statistical properties of these two factors.
(1) When using the f1 method to assess the difference between a single pair of formulations, the f1 factor will change if test and reference are interchanged, yet the differences between the two mean profiles remain the same. To avoid this problem, a modified formula has been proposed as follows(15):
in which the denominator is the sum of the average values of the two formulations at each time point, instead of that of the reference formulation only.
(2) Because f1is a measure of the overall relative error, it is not sensitive to individual relative error at each time point, the average of individual relative errors, f3, has beenproposed as another alternative measurement for evaluating the difference between two dissolution profiles (16):
However, thef3factor still requires further study before its general acceptance as a measure of the similarity.
(3) f2is sensitive to the number of time points, especially after the dissolution plateau has been reached (>85%) (17). That is to say, if a large number of sampling points are selected after the asymptote is achieved, then the overall average difference will be small, and f2 will tend to be larger than 50, which will result in a determination of similarity although the two profiles could be very different. Therefore, the FDA recommends a limit of one sampling time point after 85% dissolution. In cases where both test and reference products dissolve more than 85% within 15 minutes, the comparison with an f2 test is unnecessary.
(4) Becausef2does not take into account the variability within the test and reference batches (17,18), when using the mean values from both curves at each time point to calculate f1and f2, it is suggested that the within-batch coefficient of variation (CV) at the earlier time points (e.g., 15 minutes) should not be more than 20%, and at other time points should not be more than 10%.
(5) Because of the invariant property of f2 with respect to location change, it is insensitive to the shape of the curves and cannot take into account unequal spacing between successive sampling times(19). Therefore, the FDA requires that adequate sampling be performed until the asymptote is reached, for example, at 15, 30, 45, 60, and 120 minutes, until either 80% of the drug is released or an asymptote is reached for delayed-release dosage forms(13).
(6) f2 is proposed for evaluating the degree of sameness of two curves, and therefore it is not appropriate for situations where the average difference at any sampling time point is greater than 15% between two batches. In instances where within-batch variation is more than 15% CV, a multivariatestatisticaldistance approach, which will be introduced in the next section, is more suitable.
(7) Becausef2 is usually calculated on the basis of mean dissolution data without considering the percentage coefficient of variation at each time point, a slightly conservative estimate of the similarity between the two formulations may be obtained. To reduce this possible bias, a modified formula has been proposed to calculate the similarity factor:
where and are the variances of the observed percentage dissolved at the i-th time point for the reference and test formulations respectively. However, a previous study demonstrated that when the dissolution variance is small, the values might not be very different from the f2 values(20). More recently, to take into account the variability in dissolution data at each time point, several other methods have been proposed for calculating the weighting factor in Moore and Flanner’s f2equation(21-23).
(8) Although the FDA has recommended the calculation of f1 andf2 using mean dissolution data, alternatively f1 andf2 can be estimated using individual dissolution data. It has been reported that the values of f1 andf2 were not statistically different when calculated using mean or individual dissolution data(24), but other research has shown that the difference between two formulations was less when mean rather than individual dissolution data points were used to calculate f1,f2, or other similarity factors(25).
4.2. Rescigno index
The Rescigno index, which was originally proposed for evaluating the bioequivalence of two formulations based on plasma-concentration versus time curves(11), has also been used to compare dissolution profiles(4,26).It is defined as:
where Ri and Ti are the percentage of drug dissolved at the i-th time point for the reference and test formulations respectively, and j is1 and 2 for the first- (ξ1) and second-order (ξ2) Rescigno index respectively. The Rescigno index, which can be calculated using the trapezoidal rule, takes on values from zero (which indicates no difference between the reference and test formulations) to one (which indicates complete dissolution of one formulation before the otherbegins to dissolve).
A major difference between f1, f2 and the Rescigno index is that the first two take into account only the n sampling times when determining the profile differences, whereas the Rescigno indexalso takes into account the spacing between successive sampling times by evaluating integrals over time(27).One major disadvantage of the Rescigno index is that it is impossible to establish a critical value (cutoff point value, criteria) for concluding that two profiles are similar.Therefore, this index has been used only in relative comparative studies and seldom as a point estimate of similarity. Moreover, the statistical properties of this index have yet to be investigated.
5. Multivariate confidence region procedures
Multivariate confidence region procedures areamongthe model-independent methods which are recommended by the FDA for comparing dissolution profiles in instances where within-batch variation is greater than 15%(12,28). According to this approach, for two profiles to be considered similar, the difference between the test and reference profiles should be less than or equal to the maximum expected difference between any two batches of approved products. Here the maximum expected difference, also called the similarity limit or tolerance limit, is often defined as a given percentage dissolution, say, 10% or 15%. Alternatively, the similarity limits can also be set on a product basis, rather than on the empirical basis of an average difference of 10%. Because the procedure is constructed to determine whether the difference between two products is greater than the tolerance limit, it can be considered as a modification and generalization of the bioequivalencetest concept. This procedure can be further classified into 90% CIofdifference methods (for single-time-point dissolution values) and multivariatestatisticaldistance methods (for multiple-time-point dissolution profiles).
5.1. 90% CI of difference method
For immediate-release drug formulations, only a single-time-point dissolution value is required for the evaluation of drug dissolution properties. Therefore, it is easy to test whether the difference between the mean dissolution values of the test and reference formulations is significantly different from zero using a simple Student’s t-test. However, whenthe concern is whether the difference between two products is greater than a tolerance limit (e.g., 10%), it is essential to estimate the confidence interval of the true difference and to compare the upper limit of the confidence interval with the maximum tolerance. By considering the dissolution value of each formulation as an independent variable, the difference between the two variables is only a single estimate based on the two sample means, and the 90%CI of the difference can be calculated as follows:
in whichDiffT-R and SEDiff can be calculated using
and
where nT and nR are the numbers of units of the test and reference formulations respectively, is the t-value with degrees of freedom and a confidence limit of 90%, SEDiff is the standard error of the difference, SDT and SDR are the standard deviations of the test and reference formulations, and and are the mean percentage dissolved of the test and reference formulations. Then the two profiles can be considered similar if the upper limit of the 90%CI is less than the specified maximum tolerance.
5.2. Multivariate statistical distance method
In situations where within-batch variation is greater than 15%, FDA guidelines recommend use of a multivariateconfidenceinterval method to assess the similarity between two drug products.This can be done using the following stepwise procedure:
(1) Establish a similarity limit in terms of multivariate statistical distance (MSD) based on interbatch differences in dissolution from reference (standard approved) formulations.
(2) Calculate the MSD between the test and reference mean dissolutions.
(3) Estimate the 90% CI of the true MSD as determined in step (2).
(4) Compare the upper limit of the 90%CI with the similarity limit determined in step (1). The test formulation is declared to be similar to the reference formulation if the upper limit of the 90%CI is less than or equal to the similarity limit.
However, no exact procedure for calculating the MSD is stated explicitly by the FDA;therefore, it may be more complex to use this method than to use the f2 method. However, in practice, the procedure proposed by Tsong et al. can be considered a well-accepted method and is actually recommended by the FDA in this situation(28).According to this method, a multivariate statistical distance, called the Mahalanobis distance, is used to measure the difference between two multivariate means.This distance measure can be calculated as
where is the sample variance-covariance matrix pooled across both batches, XT and XR are the vectors of the sample meansfor the test and reference profiles,and ST and SR are thevariance-covariance matrices of the test and reference profiles.
To determine the similarity limits in terms of the MSD, it is proposed to use the following equation
where [Dg] is a 1×p vector with all elements equal to an empirically defined limit Dg (e.g., 15%) for the maximum tolerable average difference at all time points andp is the number of sampling points.By assuming that the data follow a multivariate normal distribution, the 90% confidence region (CR) forthe true difference between the mean vectors, , can be computed for the resultantvector to satisfy the following condition:
where K is the scaling factor () and is the 90th percentile of the F-distribution with degrees of freedoms p and . It is obvious that must be greater than (p+1). Then the 90% confidence region of , (, ), can be easily calculated, and similarity can be concluded if is less than or equal to the similarity limit DM_max.