`
`Logarithmic Transformation in Bioequlvalence: Application with
`Two Formulations of Perphenazine
`
`K. K. MIDHA*", E. D. ORMSBY*, J. W. HUBBARD", G. MCKAY*, E. M. HAWES*, L. GAVALAS§,
`AND I. J. MGGILVERAY‘
`
`Received July 20, 1990, from the “Colleges of Pharmacy and Medicine, University of Saskatchewan, Saskatoon
`Saskatchewan, Canada S7N 0W0, *Health Protection Branch, Tunney‘s Pasture, Ottawa, Ontario, Canada K1A 0L2, and
`Accepted for publication August 5,1992.
`§Zenith Laboratories Inc. Northvale, NJ 07647.
`
`
`Abstract DThe rationale for using the logarithmic transformation on
`concentration-dependent pharmacokinefic parameters a priori is pre-
`sented. This rationale is based on theoretical pharmacokinetic and
`statistical grounds, but is also applicable to the practice of physicians in
`dealing with variations of drug treatment within and between patients.
`The implications of the transformation on data analysis, specificwa
`analysis of variance, and estimation and inference from the analysis as
`it pertains to bioequivalence decisions are explored. implementation of
`the transformation is shown, with an example of two perphenazine
`formulations in a single-dose crossover study. It is concluded that the
`transformation has to be accepted on theoretical grounds because
`sample sizes are too small in bioequivalence studies and too susceptible
`to extreme values to state with any certainty the actual distribution of
`pharmacokinetic parameters or their differences within a subject.
`
`Over the past two decades there has been much discus-
`sion1-4 on the use of the logarithmic (log) transformation of
`concentration-dependent pharmacokinetic parameters before
`examination by analysis of variance (ANOVA) in bioequiv-
`alence studies. In several important papers5-8 in which
`decision rules for acceptance of bioequivalence are discussed,
`application ofthis transformation is alluded to, but its routine
`use has not been recommended. Recommendations to the
`_ Canadian Health Protection Branch9 have specified that
`bioequivalence decisions must be made on the basis of log
`transformed data. By contrast, the United'States Food and
`Drug Administration (FDA)10 requires that ANOVA be car-
`ried out on the raw scale unless there is strong evidence that
`supports the transformation. However, the Generic Drugs
`Advisory Committee11 of the FDA voted recently in favor of
`using the log transformation a priori. A guideline12 produced
`by the European Committee for Proprietary Medical Products
`has also recommended the transformation.
`In light of the recent ruling of the FDA advisory committee
`and the obvious trend towards the log transformation, the
`authors felt it was necessary to present the rationale and
`statistics concerning the analysis of a bioequivalence study
`based on the log scale. At a major bioavailability conference13
`in 1989, the point was raised that there was a need to educate
`both regulatory reviewers and the pharmaceutical industry
`on application of and inference from the log transformation.
`An example of the analysis is provided.
`
`Theoretical Section
`
`The purpose of a bioequivalence study is to show that the
`plots of concentration of drug in plasma versus time (plasma
`concentration—time profiles) of two drug products containing
`the same dosage of the same drug can be superimposed
`Because profiles from drug products even from the same lot,
`let alone two different formulations, will never be identical
`within a subject, the degree to which the two curves super-
`
`138 / Journal of Pharmaceutical Sciences
`Vol. 82, No. 2. February 1993
`
`impose must take account of this ”intrasubject” variation
`while still ensuring that the test formulation does not deliver,
`on the average, a different profile. Although this variation
`plays a significant role in the precision of the estimated
`relative bioavailability, the main interest lies in comparing a
`measure of central location of the pharmacokinetic parame-
`ters derived from the administration of test and reference .
`formulations.
`For immediate-release formulations, superimposition of
`the profiles following single dosings can usually be assessed
`by comparing extent and shape. These two characteristics of
`the profiles are estimated with the values of area under the
`curve (AUC) and the observed maximum concentration
`(CI-n“)! respectively. A crossover study is executed, and the
`probability that the relative average difference of the two 4
`products for each parameter falls within a prespecified range
`is calculated. This probability must not be greater than the
`consumer risk of a% (conventionally 5%) for both tails of the
`distribution. This prespecified range is called the bioequiva—
`lence interval (BI) and is the only link between clinical and
`blood concentration significance. It is generally agreed that,
`for most drugs, up to a 20% difference in AUC or C1mm between
`two formulations would have no clinical significance. The
`implication of accepting bioequivalence is that the two for-
`mulations would have the same clinical effect.
`Consideration of the Log Transformation—It is com-
`monly believed that the distributions of many biological
`parameters have much longer right tails than would be
`expected had the parameters come from a normal distribu-
`tion. Kenny and Keeping“ state that if an outcome random
`variable is affected by many random causes, each of which
`produces a small proportional effect, the resulting distribu-
`tion can be represented by the log normal distribution.
`Applying this random process to pharmacokinetics, one can
`expect that a concentration measured at any time is a
`function of many random processes (absorption, metabolism,
`elimination) that act proportionally to the amount of drug
`present. Therefore, it can be envisaged that AUC or CmaXma)’
`take on a log normal distribution due to environmental and
`genetic influences on the many random processes from which
`the parameters arise. The values from this distribution may
`differ by a factor of < 10 to a factor as large as 100. Many drugs
`exhibit polymorphic metabolism, which can account for the
`wider ranges seen for some drugs. The distribution of the!)8
`parameters is difficult to ascertain because of the small
`number of subjects used in a bioequivalence study. However.
`the distribution of the concentration-dependent parameters
`for many drugs does show an extended right tail.15-17
`Although inter-subject or between-subject variation, 85 .
`described above, may have large clinical implications, it.is not
`of major importance in the crossover study because each.
`subject receives the drug from each formulation. The mot!t‘
`
`0022-3549/93/0200—0138302.5
`
`© 1993, American Pharmaceutical Associ
`
`
`ENDO - Ex. 2048
`
`Amneal v. Endo
`
`lPR2014-00360
`
`ENDO - Ex. 2048
`Amneal v. Endo
`IPR2014-00360
`
`

`

`x is
`
`'1
`
`
`
`t variation is the intrasubject or within-subject
`;
`oe» which is the variation exhibited by a single person
`'ven the same dose of drug (e.g., solution) over
`
`administrations (i.e., independent of formulation).
`
`;
`‘tude of the intrasubject variance depends greatly
`
`harmacokinetics of the drug. For instance, drugs with
`kinetics (e.g., little metabolism, well absorbed) will
`
`y have intrasubject coefficients of variation (CVs)
`
`‘
`- .
`to be <20%, whereas drugs with complicated
`
`cs (e.g., high first-pass effect, variable absorption) can
`estimated intrasubject CVs as large as 40% or more.
`
`in the above paragraph that we have switched from
`bject variance to intrasubject CV. This is the common
`
`‘7 professionals make when they wish to describe indi-
`in the population. The CV is used because it describes
`
`ation expected for individual patients from the pop-
`r. ., whether they exhibit large or small values. Implicit
`
`" use of the CV is the concept of relative variation. That
`jects with larger values will have larger variation in
`
`licated values than subjects with smaller values, but,
`' : tion relative to their individual means is the same.
`
`trast, ANOVA assumes constant variation, not relative
`
`on, and thus needs implimentation of the log transfor-
`n.
`
`ake18 suggested AUC and Cmax would have this
`
`e variation over subjects given the same drug. He
`ed, based on the common expression that AUC was
`
`to amount of drug absorbed over clearance, that vari-
`within a subject would be larger for subjects with larger
`
`(e.g., slower elimination) than for those with faster
`tion. Although this expression is satisfied for drugs
`
`,
`ery general conditions, we feel the same subject
`phcative effect holds true for most drugs. Further,
`
`'
`most analytical methods also show an increase in
`oe (but somewhat constant relative variance) in re-
`
`- as the response increases, we can expect
`larger
`
`on in larger AUCs, especially with higher Cmax values.
`
`main use of the log transformation is, therefore; to
`' e or make equal the intrasubject variation and not to
`
`ze the intersubject parameters, although the latter is
`also accomplished. Because ANOVA assumes that
`
`the model act in an additive way, the transformation
`
`the data from a proportional or relative scale into an
`ve one. The basic premise of the transformation is that
`
`.
`') = log (A) — log (B).
`in el and Assumptions—The general linear model for a
`
`riod,
`two-treatment crossover design,
`ignoring the
`, over efi‘ect,19 is:
`
`
`1“ Wijm1=Xim=u+n+Bjm+Wk+¢1+eijkl
`
`(1)
`
`'”
`
`and is used to describe the data after taking the natural logs
`of the individual parameters. The (SJ-(i) is the random effect of
`subject nested within sequence and is assumed to be normal
`(0, 023). The residual term is also assumed to be normal (0, 0%,)
`and independent of the random subject effects. The other
`terms are the fixed effects of sequence (7,), of period (ark), and
`of formulation (4),). As defined, the variance of a single
`observation is equal to 02 = 0% + 0%,, and the two observa—
`tions from the same subject have a covariance 0% or a
`correlation of 02,3/(0f3 + 0%,).
`The residual is the estimate of the intrasubject variation,
`0%,. As the name implies, it is the variation remaining that
`the model cannot explain. For this design, it is sometimes
`called the subject-by-formulation interaction term because it
`is mainly made up of variation due to a day-to-day variation
`in drug pharmacokinetics in a subject, the “true” intrasubject
`variation, and formulation variation. However, clean esti—
`mates of intrasubject and formulation variances can never be
`obtained because a tablet can only be given once and the
`pharmacokinetics of the drug in the subjects are unique for .75
`each dosing. The best that can be done is to estimate the
`subject-by-formulation interaction for each formulation when
`each formulation is repeated.20 Therefore, one assumption is
`that both the inter- and intra-variances from each formula-
`tion are equal. Also, contributing to the residual is analytical
`error and sample placement error. The analytical error is
`allowed to be as high as 15%21 but is more commonly ~5%.
`This error is more evident in the variation of a single
`concentration such as Cmax than with AUC, because AUC is
`an integral over many points with the same relative error.
`Recently, Gafl‘ney22 has studied the contributions of these
`variance components in bioequivalence studies. The last
`component due to placement of sampling times can have a
`great effect when input is fast and the frequency of sampling
`does not reflect the rate or the variation in time to reach C'max
`(than).
`The degrees of freedom, expected mean squares, and the
`appropriate error term to test for each source are given in
`Table I for the two-period, two-treatment, crossover model.
`Estimation and Inference from the Log Transforma-
`tion—We must consider the purpose of performing a bioequiv-
`alence study to decide what statistical parameters have to be
`estimated, but also the study design to account for design
`effects. What we need is an estimator of average relative
`bioavailability and its standard error such that a probability
`
`i—Expected Mean Squares for a Two-Period, Two-Treatment, Crossover Design
`
`roe
`
`d1a
`
`Expected Mean Squares
`
`02 + 202 +
`w
`B
`
`2nmnm(rm — Tm)2
`"Tn + "RT
`
`nTR + ”RT — 2
`
`05,, + 20‘?3
`
`2nrnnm'(77'm - Whrlz
`a5, +——"TR + nm
`
`02
`W
`
`+ .—.—_——__
`2nmnm'(¢m — <I>RT)2
`"TR + "RT
`
`”TR + nRT — 2
`
`03"
`
`F-Ratio
`Denominator
`
`Subject (SE0)
`
`Residual
`
`Residual
`
`Residual
`
`_b
`
`ass of freedom. " Not determined.
`
`
`
`Journal of Pharmaceutical Sciences/ 139
`Vol. 82, No. 2, February 1993
`
`

`

`statement or inference can be made. If the parent distribution
`of a parameter onr each formulation is log normal, then the
`resulting distributions after transformation of Y to X are
`normal (ubaf). There are two competing estimates of central
`tendency for Y, the median and the mean. The medians of the
`distributions can be estimated with eq 2:
`
`MEDIAN; = exp?!
`
`(2)
`
`For 1 = T,R (for the test and reference formulations, respec-
`tively), the least square means are X: (XTR, + XRT,)/2. A
`least squares mean is simply the mean1ofthe means from each
`of the sequence TR and RT. If the number of subjects in
`sequence TR (nTR) equals the number in sequence RT (nRT),
`then the least squares mean equals the overall mean for each
`formulation. Equation 2 also defines the geometric mean.
`The mean is obtained from eq 3:
`
`MEAN, = expi’l’sm
`
`(3)
`
`In eq 3, $123 is the estimated variance of the Xs. Note that the
`median will always be smaller than the mean for a skewed
`distribution. Land23 has given tables to calculate the exact
`confidence interval (CI) for the mean and they differ from the
`usual CI for the median because the exact CI must be wider
`to account for the estimation of 0123.
`However, in bioequivalence, the paramount estimation is
`for the relative medians or means. Estimates for the ratio of
`medians (RELBIOMED) are obtained with eq 4:
`
`RELBIOMED = eprrT 5‘11
`
`(4)
`
`The standard error of the difference (STEdifl) in the exponent
`is:
`
`1
`1
`STEdjfi‘: (_+—)Sw
`nTR
`nRT
`
`(5)
`
`In eq 4, S3,, is the residual from the crossover ANOVA.
`RELBIOMED is a consistent maximum likelihood estimator
`but also has a slight bias. Recently, Liu and Wen24 examined
`the properties of RELBIOMED and recommended a correction
`factor. Figure 1 gives a plot of the correction factors for
`intrasubject CVs from 10 to 50% and crossover sequence sizes
`from 6 to 50. Although the estimator consistently underesti-
`mates the true ratio of medians, this bias is not likely to be of
`any consequence (< 1%) given the required sample sizes to
`
`1.000
`
`0.975
`
`FACTOR
`CORRECTION
`
`0.050
`
`o
`
`'10
`
`20
`
`so
`
`40
`
`so
`
`SAMPLE SIZE
`
`Figure 1—Bias correction factors for balanced crossover studies. Key to
`CVW: (~--—) 10%; (- - - -) 20%; (—) 30%; (— — —) 40%; (— —) 50%.
`
`140 / Journal of Pharmaceutical Sciences
`Vol. 82, No. 2, February 1993
`
`achieve the power to make a bioequivalence decision for largei
`intrasubject CVs.
`;?
`If one is unwilling to assume the distributional structure:
`noted above, a nonparametric method8 can be used. However, .
`this method is not without distributional assumptions. It
`assumes that the differences calculated for each subject come
`from a distribution that is symmetric and that the differences
`are independent.25 Thus,
`the log transformation is also ,
`recommended to stabilize the variation.
`Two other important parameters that are used in summa.
`rizing results of a bioequivalence study are measures of
`variation. As mentioned earlier, two main sources of varia.
`tion can be estimated. From the “expected mean squares of
`subject (sequence)” term from Table I, one can estimate the
`intersubject variance component 0%. This is a measure of the
`variance of the distribution of subject means if subjects were {
`sampled an infinite number of times. The estimate 8% is.
`obtained by using the expected mean squares (EMS) for
`subject (sequence) and residual terms and solving for 0323. It is ‘
`usually more convenient to express this variation as a CV.
`The CVB, or the intersubject CV, is estimated from the log;
`scale from eq 6:
`
`-
`
`CVB = V expSi —1 x 100%
`
`(6):
`
`A very good approximation (ACVB) for CVBs, up to 40%, is
`obtained by simply taking the square root of Sf; and multi-
`plying by 100%.
`The estimate of the intrasubject variance is simply the
`residual mean square from the ANOVA, and its CVs can be
`obtained from eq 6 or the approximation by substituting $2
`for S2. CVW or intrasubject CV estimates the relative van- =1
`ationBone expects from an individual when given the drug on 7
`different occasions.
`,
`Another CV estimate could also be calculated. This is the‘
`CV of the observed Ys. The model assumes that the variance .
`of an individual observation is the sum of the inter- and
`intrasubject variances. Thus, an estimate of this pooled CV '
`(CVP) is CVP =
`expsg + Sl' — 1 x 100%. CVP or its approx- ‘
`imation can be used to compare CVs from parallel studies.
`What needs to be defined now is the criterion for accepting.
`bioequivalence. This has produced a profusion of papers from
`all corners of statistical thought from Baysianz“,27 to multi- ‘-
`variate28 to robust statistics,29 from nonparametrict3 to clas- :
`sical parametric techniques,56 and to the latest swing to- _-
`wards individual bioequivalence.30 Common to all these .
`methods are the a priori defining of limits that have clinical}
`significance. These are the bioequivalence limits introduced'
`earlier in the paper and they give meaning to the whole}?
`concept ofaccepting a new formulation on bioavailability data‘l
`alone, in lieu of clinical data It18 generally agreed that a safe
`limitis 20%; that1s, if the test formulation on the average
`produced a concentration—time profile within 20% of the
`reference formulation, the test formulation could assume all},
`clinical support from the reference formulation for its mar-5:
`keting. Therefore, the BI is commonly referred to as being;
`80—120% when an additive scale is used. However, a syma
`metric interval on an asymmetric distribution seems inap-i
`
`propriate and, ifthe log scale13 used, the BI should be set atj
`
`80—125% because the chance of passing the criterion is
`optimal when the formulations have identical means Note,
`‘i
`that the log of 0.801s equal, except for sign, to log 1.25. ThuS; '
`a BI symmetric on the log scale expresses itself asymmetfl'
`
`cally on the original scale [i.e., a “20%” increase of the 178513
`
`formulation on log (additive) scale1s a 25% increase on thQ
`original (relative) scale]. Of course, narrower or wider limit9
`may be chosen a priori for safety or efficacy concerns of clini
`.,
`
`
`
`

`

`
`cance with particular groups of drugs.
`alluded to earlier, there have been many papers written
`- manipulation of data to state bioequivalence. However,
`of them do not take into account the design in which the
`
`“in. eters are collected. Therefore, only three methods seem
`g:acceptable to the regulatory authorities. These are (1)
`
`the 90% CI for RELBIOMED must fall entirely within the
`1’) that both of the two one-sided hypothesis6 tests should
`
`'ficant at the 5% level, and (3) that the nonparametric
`
`018 of RELBIOMED should fall within the BI.
`r log-transformed data, the 90% CI for RELBIOMED is
`
`ted as follows:
`
`ifLOWER,UPPER = expo—(T - in) : tones“ + m — stEm (7)
`
`
`
`_ (ET—ink In (0.80)
`STE
`
`2 t0.05,nTR + nRT — 2
`
`(8)
`
`" convenient to multiply the estimate (eq 4) and its
`Hence limits (eq 7) by 100 to express them as percents.
`{‘e two one-sided hypothesis tests decisions are based on
`
`and 9:
`
`.
`
`'1'
`
`ln(1.25) — (ET — ER)
`tu= —-— 2 t
`STE
`
`0.05m“ + nRT — 2
`
`(9)
`
`
`
`E‘I-th tests produce p values of <0. 05, equivalence is
`‘Ja.ted
`
`vée nonparametric 90% CI is formed by taking all the
`,'u 1 minus period 2 differences for sequence TR and
`g acting each from all the period 1 minus period 2
`
`C nces from sequence RT. These differences are ranked
`
`'ieach divided by 2 and then exponentiated. There will be
`11 'r differences. The estimate of RELBIOMED is the
`
`difference, and the lower and upper limits are the
`. priate observed differences depending on the sample
`
`1 of the two sequences. Tables identifying the appropriate
`ed difference for each limit for crossovers up to 12
`3 per sequence are given in Hauschke et al.,7 and a
`1"» : equation is given in Conover.25
`
`
`
`..
`
`
`
`
`Experimental Section
`1 —Perphenazine is a potent antipsychotic drug that undergoes
`- .ve first-pass metabohsm.31 The drugin the population may
`
`at least a bimodal distribution for AUC and C1mm according to
`mine slow and rapid hydroxylators.32
`Methodology—A modification to a published radioimmu—
`
`(RIA) procedures-‘1 was used to assay perphenazine1n plasma.
`
`_ uodification was an extraction step that guarded against inter—
`i v from metabolites The lower limit of quantitation was 39
`
`'
`1
`, which allows for good measure of the elimination phase of the
`Sup to 48 h or longer after the administration of a single oral dose
`
`4 ). Standard curves were linear over the range of 39 to 2.5
`
`ng/mL, with an average accuracy and precision of 0.9 ng/mL and 8%,
`respectively. Samples were subject to single extraction, and aliquots
`(150 ML) of the extracts were reconstituted in 7% bovine serum
`albumin and analyzed in triplicate. The test samples were always
`analyzed in parallel with the standards and quality control standards
`that were made in blank plasma obtained from each subject before
`dosing. The highest mean difference from the nominal (added) value
`for the quality control samples was 13 (:12)% at the lowest concen-
`tration (0.18 ng/mL).
`Subjects—Twenty-seven subjects were selected from a cohort of
`healthy male volunteers, aged 18—45 years, and deviating by no more
`than 10% from the ideal weight for height according to the Metro-
`politan Life Insurance Company Bulletin, 1983. Each subject was
`required to be free of cardiovascular, hepatic, renal, or gastrointes-
`tinal disease, or drug abuse or alcohol dependence, as assessed by
`physical examination and review of medical history by the supervis‘
`ing physician. In addition, the blood pressure, electrocardiogram, and
`results of clinical laboratory tests (blood chemistry, hematology, and
`urinalysis) were required to be within normal ranges. Smokers
`(15/27) were permitted to continue smoking at their normal rate of
`consumption throughout the study. All subjects were required to
`abstain from the use of all other drugs (including nonprescription
`drugs) for at least 1 week prior to until after completion of the study.
`The subjects were also required to refrain from consuming alcoholic
`or cafieine—containing beverages from 48 h prior to dosing until after
`the collection of the last blood sample. Physical examination and
`laboratory tests were also carried out during and after the study to
`monitor the health of participating subjects.
`Study Implementation—Each volunteer gave his informed con-
`sent before participating in the study. The study protocol was
`approved by the local institutional review board. Of the 27 subjects
`who started the study, 24 subjects completed both legs ofthe crossover
`study. Three subjects dropped out for personal reasons. Each subject
`received each of the 16-mg reference formulations (Trilafon, Schering
`Corporation, lot #3ADM3) and the test formulation (Zenith Labora-
`tories, lot #107-183) according to their randomized sequence with a
`2-week washout period. After an overnight fast and drawing a
`zero-time blood sample, the dose was given and 20-mL blood samples
`were collected at 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 6.0, 8.0, 12.0, 16.0, 24.0,
`28.0, 36.0, and 48.0 h. Plasma was separated by centrifugation within
`30 min of harvesting and stored at —23 °C until analysis. Standard-
`ized meals were provided at 4.0 and 9.0 h post-dosing.
`Parameters—Pharmacokinetic parameters for perphenazine were
`determined by standard techniques. The Cmax and tmax values were
`obtained directly from the plasma concentration—time curve. The
`AUL up to the last time showing a measurable concentration of
`perphenazine (AU03) was determined by the linear trapezoidal rule
`for ascending and the log trapezoidal rule for descending portions of
`the curve.34 A rate constant for the disappearance of drug from
`plasma (k) was calculated by least squares regression of the natural
`logs of the last three plasma concentrations. AUCE,n values were
`determined by adding to the appropriate AUG}, the quotient of the
`estimated last plasma concentration and the appropriate value of k.
`A half-life value (t1,2) for the disappearance of the drug from plasma
`was calculated from the quotient of In 2 and k.
`
`Results and Discussion
`
`Table II lists estimates for the medians and, when appro-
`priate, means and CVs of some pharmacokinetic parameters
`for the perphenazine study. Each parameter listed in Table II
`
`I ll—Summary Statistics for Perphenazlne Pharmacoklnetlc Parameters for Test and Reference Formulations
`Reference Formulation
`.-'
`Test Formulation
`arameter
`‘
`‘
`
`Median
`12.92
`14.58
`1.17
`—"
`—
`—
`
`Mean
`14.32
`16.59
`1.29
`2.38
`0.076
`13.40
`
`CVP
`47.9
`54.3
`45.6
`43.6
`49.3
`91.3
`
`ACVP
`45.4
`50.8
`43.5
`——
`—
`—
`
`Median
`13.39
`15.81
`1.18
`—
`—
`—
`
`Mean
`14.79
`17.50
`1.28
`2.54
`0.078
`12.84
`
`CV...
`46.9
`47.5
`41.4
`29.5
`55.9
`76.4
`
`ACVP
`44.6
`45.1
`39.8
`—
`—
`—
`
`
`
`
`‘1‘" possible to calculate for all subjects (see text). b —: N°i determined.
`
`Journal of Pharmaceutical Sciences] 141
`Vol. 82, No. 2, February 1993
`
`

`

`shows CVs of >40%, which is characteristic of phenothiazine
`antip'sychotics.35—37 These estimates of relative variation are
`mixtures of intersubject and intrasubject variance compo-
`nents. Note that fluctuations in post-Cmax plasma concentra-
`tions prevented the calculation of a value for k in some
`individuals. It is important to recognize that variations in the
`time to the last concentration were characteristic of the
`subject, not the formulation. After reaching Cm“, plasma
`concentrations showed evidence of secondary peaking in
`many subjects, as has been reported with other phenothiazine
`antipsychotic drugs.16v35’36 This phenomenon may be a result
`of biliary recycling. In any event, the difficulty of obtaining a
`reasonable estimate of k (and therefore the AUCf,/AUCE,n
`ratio) should not invalidate studies on such drugs. Therefore,
`various mean values for k, tm, and AUC3 were calculated
`from 17/24 subjects for the reference product, and 18/24 values
`for the test product. The inability to estimate these parame-
`ters in so many subjects effectively ruled them out of conten-
`tion as bioequivalence-determining variables.
`Figure 2 presents the average profiles for both the reference
`and test formulations. Table III lists the estimates Of AUCE,
`and Cm” on the log scale for each subject and formulation by
`sequence. The means presented in this table are used in the
`calculation of the two least squares means. The similar
`magnitudes of the standard deviation for each formulation
`and period indicate similar sequence groups. It is recom-
`mended that transformed values be reported to at least three
`decimal places to maintain the accuracy when summary
`statistics are transformed back to the original scale.
`Interaction diagrams for AUCf, and Cmax are given in
`Figures 3 and 4, respectively. The spreads of the values for
`each formulation for both parameters are similar, indicating
`that the formulation variance is similar. The steepness and
`criss-crossing of the lines joining subjects is an indication of
`how variable the subject-by—formulation interaction is across
`subjects.
`Table IV presents some intersubject distribution statistics
`for AUCI', and C,mam and their logs. The Shapiro—Wilk test
`indicated a lack of normality, probably due to the positive
`skewness of the distributions (W test, p < 0.01) of both AUC
`and Cmax on the raw scale. The distributions of log-
`
`1.2
`
`—L O
`
`(ng/ml) 99:5ca Pn
`CONCENTRATION
`
`4 812162024283236
`
`48
`
`Figure 2—Mean plasma concentration—time profiles following test (0)
`reference (A) formulations of perphenazine.
`
`TIME (Hrs)
`
`142/ Journal of Pharmaceutical Sciences
`Vol. 82, No. 2, February 1993
`
`.-
`
`Table Ill—Estlmates of lndivldual Perphenazlne AUC}, and Cmax
`Values on Log Scale
`AUC‘
`C
`
`Sequence
`Subject
`0
`max
`Test
`Ref
`Test
`Ref
`
`
`TR
`
`A
`B
`G
`I
`K
`M
`O
`R
`U
`V
`W
`
`1.894
`2.980
`2.717
`2.319
`2.085
`2.756
`2.665
`2.661
`2.576
`2.446
`1.664
`
`2.129
`2.794
`2.929
`2.651
`2.001
`2.813
`2.747
`2.460
`2.498
`2.649
`1.635
`
`—0.333
`0.116
`0.258
`0.053
`—0.580
`0.397
`0.664
`0.481
`0.226
`—0.090
`—0.71 5
`
`—0.190
`0.545
`0.550
`0.126
`—0.523
`0.436
`0.490
`—0.051
`0.097
`0.083
`-—0.358
`
`
`
`4..A___-__..'
`
`_
`XTR
`STD
`RT
`
`0.963
`0.944
`3.551
`3.406
`X
`0.181
`0.118
`2.571
`2.514
`—3
`0.430
`0.491
`0.491
`0.476
`—
`—0.248
`0.137
`2.335
`2.386
`C
`—0.094
`0.100
`2.256
`2.306
`D
`0.105
`0.887
`2.790
`3.328
`E
`—0.31 5
`—0.383
`1.998
`1.945
`F
`0.128
`0.061
`2.669
`2.449
`H
`0.239
`0.083
`2.625
`2.428
`J
`0.061
`~0.020
`2.851
`2.747
`L
`—0.057
`—0.01 2
`2.682
`2.684
`N
`0.428
`0.095
`2.501
`2.339
`O
`1 .152
`0.951
`3.597
`3.345
`P
`0.241
`0.494
`2.898
`3.103
`S
`0.248
`0.007
`2.211
`2.201
`T
`_
`0.157
`0.200
`2.618
`2.603
`—
`Xm
`
`
`
`
`
`— 0.447 0.416 0.387STD 0.381
`
`‘ Not applicable.
`
`3.6
`
`3.1
`
`1.6
`
`
`
`LnAUCI(ng.hr/ml) !° 0)
`
`2.1
`
`Reference
`
`in
`FORMULATION
`5
`Flgure 3—Comparison of In AUG; of each subject by formulationl
`(formulation—subject interaction).
`"
`
`Test
`
`transformed data were less skewed and less platykurtic thall-
`the distributions of natural data. The log transformation not.
`only reduced skewness and kurtosis but, in each case, Ten‘}
`
`
`
`
`

`

`1.2
`
`0.8
`
`0.4
`
`0.0
`
`
`
`-——1———————I—
`Reference
`Test
`
`FORMU LATION
`_
`yre 4—Comparison of in Cmax values of each subject by formulation
`, ulation—subject interaction).
`
`; e W—Intersublect Distribution Characteristics
`
`Skewness
`
`1.44
`0.15
`1.52
`0.31
`
`Kurt sis
`o
`2.28
`0.34
`2.75
`0.60
`
`Shapiro-Wilk (W) Test
`p values
`0.003
`0.782
`0.003
`0.490
`
`
`
`
`
`" :
`. 2‘-
`
`eter
`
`f. t,
`" UC},
`'
`,
`'r max
`
`,
`
`
`4.:- the distribution approximately normal (W test, p >
`
`_
`). It is realized that any test on distributional assump-
`1: : with small sample sizes has very low power and is
`uenced greatly by extreme values. Therefore, it should not
`
`weft to the observed data set to decide which scale should be
`for analysis.
`
`examination of the intrasubject error for AUCE, was
`ed out by correlating the positive studentized residuals
`
`the ANOVA with the mean AUCf, values for each
`,
`
`ject. The studentized residuals are simply the residuals
`) from the model fit divided by their stande error. Each
`_'ect produces two residuals that are equal but opposite in
`
`. ; thus, only the positive ones were used. The usefulness of
`: standardization is that residuals from different scales can
`
`"compared. The correlation coefficient was significant for
`
`- raw data (p < 0.001) but was not significant in the log
`
`,1" ysis (p = 0.11). These data indicate that there was a
`rtionship between the size of the residual and the magni-
`
`fs e ofAUCf, that disappeared when the analysis was done on
`’3 log scale. Thus, in a theoretical intrasubject comparison,
`
`- given difference between test and reference has a greater
`‘ lute magnitude at higher parametric values. The impli-
`
`n of this result is that the use of the raw scale places
`I weight on the higher values and correspondingly
`
`r-weights the smaller values.
`,
`.' 81' results were obtained for analysis of Cmax data,
`Srough the correlation coefficient for the raw data (p =
`
`
`
`0.053) was slightly higher than the nominal 0.05. There was
`no significant correlation in the corresponding log trans-
`formed data (p = 0.291).
`The results of ANOVA on the log transformed AUCf,
`(Tables V and VI) or Cmax (Tables VII and VIII) revealed that
`neither pharmacokinetic parameter showed any effect of
`formulation, period, or sequence of administration. In each
`case, there was a significant effect of subject nested within
`sequence that showed the usefulness of the crossover design.
`The intrasubject CVW is estimated to be 14 and 20% for
`AUC and CW, respectively, which indicates reasonably
`consistent values were obtained within a subject. One feature
`of using the log transformation is that CVW is expected to be
`constant regardless of the magnitudes or units of the param-
`eters. Thus, a drug with a CVW of 20% would be expected to
`have a residual mean square of ~0.04 over many studies. For
`example, over three studies of chlorpromazine,3s the CVW
`varied by only 5% even when different analytical methods and
`subjects were used.
`The intersubject CVB was obtained by partitioning the
`subject (seq) mean square. This partitioning yields estimates
`of CVB of 46 and 39% for AUCE, and Cm”, respectively. Also
`calculated are the pooled CVP estimates, which agree well
`with the CVps for AUCB and C,mm for each formulation (Table
`II).
`Estimates of bioequivalence from the three methods are
`included in Tables V and VI. For these sequence sample sizes,
`the nonparametric estimate of RELBIOMED is the average of
`the 72n and 73rd difference and the lower and upper limits are
`the 43rd and 102nd difference, respectively. Note a