Pharmaceutical Research; Vol. 13, No. 2, 1996
`
`Report
`
`Tmax: An Unconfounded Metric for
`Rate of Absorption in Single Dose
`Bioequivalence Studies
`
`RodneyP. Basson,!” Benito J. Cerimele,! Karl A.
`DeSante,' and Daniel C. Howey!
`
`Received August 24, 1995; accepted November 3, 1995
`
`Purpose. While peak drug concentration (Cmax) is recognized to be
`contaminated by the extent of absorption, it has long served as the
`indicator of change in absorption rate in bioequivalence studies. This
`concentration measure per se is a measure of extreme drug exposure,
`not absorption rate. This paper redirects attention to Tmax as the
`absorption rate variable.
`Methods. We showthat the time to peak measure (Tmax), if obtained
`from equally spaced sampling times during the suspected absorption
`phase, defines a count process which encapsulatesthe rate of absorp-
`tion. Furthermore such countdata appear to follow the single parameter
`Poisson distribution which characterizes the rate of many a discrete
`process, and which therefore supplies the proper theoretical basis to
`compare two or more formulationsfor differences in the rate of absorp-
`tion. This paper urgeslimiting the use of peak height measures based
`on Cmaxto evaluate only for dose-dumping,a legitimate safety concern
`with. any formulation. These principles and techniquesare illustrated
`by a bioequivalence study in which twotest suspensions are compared
`to a reference formulation.
`Results. Appropriate statistical evaluation of absorption rate via Tmax
`supports bioequivalence, whereas the customary analysis with Cmax
`leadsto rejection of bioequivalence. This suggests that the inappropriate
`use of Cmax as a surrogate metric for absorption rate contributes to
`the unpredictable and uncertain outcome in bioequivalence evalua-
`tion today.
`KEY WORDS:bioequivalence; absorption rate; Tmax; discrete count
`variable; Poisson distribution.
`
`INTRODUCTION
`
`schemesin vogue they are not easily amenableto properstatisti-
`cal evaluation. In the interim, the continuous variable Cmax,
`the extreme observed concentration, has annexed the function
`of Tmax and performsas a substitute measure for the rate of
`absorption. Figure 1 showsclearly that rate or speed of absorp-
`tion informationresides in the discrete time, or x-axis measure,
`not the continuous concentration or y-axis one. The paper ques-
`tions currentillogical BE practice, i.e., analyze the continuous,
`y-axis, output, Cmax variable and becauseit has greaterpreci-
`sion, assumeitis telling us something intelligent about Tmax,
`which is a discrete, x-axis, input variable.
`Such practice has nontrivial consequences for inferences
`or study conclusions. Rate analysis by Cmax mayreject bioequi-
`valence while the more appropriate analysis by Tmax finds
`differently. We illustrate this with an example.
`Cmaxis recognized as a extent-contaminated measure of
`rate. For example, the ratio Cmax/AUC has been proposed,
`and is being evaluated, because it ‘corrects for extent’ (2). If
`the unit of measure for Cmax is (say) g/ml, and the unit of
`measure for AUC& is (say) wg*hr/ml, the unit of measure for
`the ratio is hr~'. However, approaching rate via a ratio of
`two wrong axis variables (Cmax/AUC)imparts to the outcome
`variable a spurious continuous form, and this too can have
`nontrivial consequences on study conclusions.
`This paper redirects attention to Tmax as the absorption
`rate variable. Because absorption rate information resides
`therein we advocate 1) the routine use of equal spacing to
`collect samples, i.e., collect time data at a definite rate per hr
`during absorption and 2) simply analyze the corresponding
`count data (where count = Tmax times rate per hour). Such
`counts, multiples of the sampling times, are numeric integers.
`Feller (3) showed the ubiquitous single parameter Poisson distri-
`bution to be identified with numeric integer counts and process
`rates. The discrete Poisson distribution therefore provides a
`solid theoretical basis to compare two or more formulations
`for differences in the rate of absorption.
`In this paper we urge that y-axis peak height measures
`based on Cmax evaluate only for dose-dumping, a legitimate
`extent safety concern with any formulation.
`
`There’s no limit to how complicated things can get, on
`account of one thing always leading to another.
`—E.B. White.
`
`METHODS
`
`A finding of bioequivalence (BE) serves as a surrogate
`for therapeutic identity (1). It is customary to evaluate bioequi-
`valence in vivo in healthy subjects, by comparing both rate
`and extent of drug absorptionofa test with a reference formula-
`tion. The area under the concentration time curve from time
`zero to time t (AUCt, wheret is the last measurabletime point)
`andsimilarly area underthe curve from timezeroto timeinfinity
`(AUC) are both recognized as uncontaminated measures of
`the extent of absorption.
`The situation for rate is in a state of flux. Time to peak
`data, Tmax,are collected but with the typical irregular sampling
`
`' Lilly Laboratory for Clinical Research, Eli Lilly and Company,India-
`napolis, Indiana 46285.
`2 To whom correspondence should be addressed.
`
`Revised Sampling Needed During the Absorption Phase
`
`The sampling times chosen to observe concentrations dur-
`ing a bioequivalence study attemptto balance conflicting objec-
`tives. First, an ethical imperative against unnecessary blood-
`letting translates into keeping sampling times to a practical
`minimum.Second,the need to sample densely enough through-
`out the suspected absorption phase so as not to miss the peak.
`Third, the desire to sample the time concentration curve for
`somethree or more half lives beyond the peak to supply a good
`measure ofthe full extent of absorption.
`We advocate equal spacing of the sampling times from
`time zero (or other suitable initial time) until approximately
`two or three times the expected peak concentration time to
`improvethe data and becausethis should have little impact on
`the total number of blood samples taken from a subject in the
`standard bioequivalence study. For example, a drug which has
`a Tmax of approximately half an hour in fasted subjects, and
`
`324
`
`(cid:36)(cid:81)(cid:71)(cid:85)(cid:91)(cid:3)(cid:21)(cid:19)(cid:19)(cid:26)
`Andrx 2007
`(cid:36)(cid:88)(cid:85)(cid:82)(cid:69)(cid:76)(cid:81)(cid:71)(cid:82)(cid:3)(cid:89)(cid:17)(cid:3)(cid:36)(cid:81)(cid:71)(cid:85)(cid:91)
`Aurobindo v. Andrx
`(cid:44)(cid:51)(cid:53)(cid:21)(cid:19)(cid:20)(cid:26)(cid:16)(cid:19)(cid:20)(cid:25)(cid:23)(cid:27)
`IPR2017-01648
`
`

`

`Tmax: An Unconfounded Metric for Rate
`
`325
`
`Var(X/Y) ~ Var(X)/Y? — 2 X Cov(X, YVY? + X?Var(Y)/Y*.
`
`{continuous y-axis)
`
`L~™
`
`Time (discrete x-axis)
`
`Fig. 1. Logical demonstration that information about the rate or speed
`of absorption resides in the discrete time or x-axis, not the continuous
`concentration or y-axis.
`
`WhenYis a constant, the two rightmost terms are zero and
`Concentration
`the ratio has variance proportional to Var(X). If X and Y do
`not correlate, the middle term becomes zero and the variance
`is proportional to Var(X) + X?Var(Y)/Y?. When X and Y do
`positively correlate, which we anticipate for Cmax and AUC,
`the result is likely to occur somewhere between these two
`extremes. In other words Cmax/AUC and Cmax havevariance
`approximately the same order of magnitude, and furthermore
`there is the problem of spurious attribution of continuity to a
`discrete time variable. Because ratios obscure the correspon-
`dence between theattribute being measured, andthe measure-
`ment chosen to do so,statisticians discourage forming ad hoc
`ratios of variables. The Cmax/AUC ratio exemplifies these
`disadvantages.
`
`a similar short half-life, is easily densely sampled every fifteen
`minutes for the first two hours (nine samples) and with diminish-
`ing frequency thereafter through eight hours. An equal sampling
`interval through the absorption phase ensures that a subject’s
`Tmax multiplied by the sampling frequency per hour is always
`a positive integer. These integer counts tell how long the absorp-
`tion process takes to reach maximum concentration for each
`subject and they encapsulate the processrate.
`
`Statistical Methodology for Counts
`
`This section merely alerts the reader to a large and scientifi-
`cally substantiated statistical methodology for analyzing dis-
`crete variables, including integer counts. For example, since the
`underlying distribution for a positive countis not the ubiquitous
`normaldistribution many readers are likely familiar with, nor-
`mal (Gaussian) theory methods should not be unthinkingly
`applied to counts. In a wide array of applications counts like
`those generated above have been found to be well-modeled by
`the pure random event Poisson distribution (3). The summary
`statistic for such counts across subjects for a formulation is
`supplied by the estimated distribution mean. Furthermore, com-
`parison of the summary statistics from the counts for twodiffer-
`ent formulations determines whether formulations have the
`samerates of absorption. Modern computers and software make
`the proper analysis of counts very much more convenient than
`it used to be. Analysis of count data for subjects that arise from
`a crossover study may be analyzed by numerous procedures.
`Performing the computations for counts in particular within the
`context of a generalized linear model is a recent innovation
`comprehensively described in McCullagh and Nelder (4). SAS
`Institute (5) provides software that can perform the necessary
`computations. Cyrus Mehta and Nitin Patel (Cytel Software
`Corp., 6) have developed software to perform exact nonpara-
`metric inference for count data.
`
`Reasons for Avoiding Ratios When We Can
`
`Cmax,the extreme value of a concentration-time profile,
`is a single variable, whereas AUC is a composite variable
`similar to an average. Accordingly, AUCis better behaved and
`exhibits lower intrasubject variability than does Cmax. The
`ratio Cmax/AUC appears to cancel out the extent effect, but it
`retains unattractive aspects of any ratio. The varianceofa ratio,
`X/Y is a relatively complex function of the variances of both
`X and Y and their covariance:
`
`A Metric to Evaluate Dose-dumping
`
`Whether Cmax is actually so extremeas to bepotentially
`unsafe depends uponthe other contributors to the concentration
`profile. The peak excursionstatistic: Cex = Cmax- average of
`all other concentrations of the profile up to time t could signal
`whether formulation ‘dumping’ has occurred. This statistic, like
`AUC,is a composite measure statistic, but will behave like
`Cmax in applications.It too will exhibit substantial intrasubject
`variability. Cex, an alwayspositive continuous random variable
`like Cmax,is log-normally distributed. It offers little that Cmax
`does not offer more directly. Both Cex and Cmaxrelate primar-
`ily to the question of extent while their connection to the ques-
`tion of rate is tenuous.
`Westlake (7) first suggested log transforms of Cmax and
`AUCtbe analyzed, and a confidence interval method similar
`to that given by Schuirmann (8) could compare Cex or Cmax
`values between two formulations. The continuousdata for sub-
`jects arising in a crossover study can be analyzed by generalized
`linear model procedures. A comprehensive approach to per-
`forming the computations is described in SAS Institute (9).
`
`RESULTS
`
`An Example
`
`A single dose three way crossover study was conducted
`by Lilly Research Laboratories to assess bioequivalence of two
`test suspension formulations, A and B, against a reference
`formulation, C. Sixteen healthy male volunteers were random-
`ized to receive antibiotic drug formulations in either one of
`three sequences: ABC, BCA or CAB, and began treatment,
`only fifteen subjects who completed the study contributed to
`the analysis reported. Formulation periods were separated by
`a 3 day washout period. Since food is known to delay the
`absorption of someantibiotics, blood samples were drawn from
`subjects after overnight fast followed by two additional hours
`of fasting at times 0 hrs, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75,
`2.0, 2.5, 3.0, 4.0, 5.0, 6.0, and 8.0 hrs after dosing in each
`period of formulation administration. Table 1 lists selected phar-
`macokinetic data for the study.
`In accordance with FDA bioequivalence guidelines (1),
`analyses were applied to natural logarithmic transformations
`of three variables: Cmax, AUCt and AUC. Sequences were
`tested for significance. No significant differences between
`
`

`

`Table 1. Selected Pharmacokinetic Data for a Bioequivalence Study
`
`Tmax
`half-life
`Cmax
`AUCt
`AUC
`
`Subject
`Sequence
`Treatment
`(hrs)
`(hrs)
`(ug/ml)
`(wg*hr/ml)
`(g*hr/ml)
`
`Basson, Cerimele, DeSante, and Howey
`
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`A
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`B
`
`12.9
`12.8
`10.7
`0.582
`0.75
`CAB
`1
`13.7
`13.4
`16.0
`0.533
`0.50
`ABC
`2
`10.4
`10.1
`12.1
`0.567
`0.50
`BCA
`3
`14.2
`13.8
`15.7
`0.523
`0.75
`CAB
`4
`11.6
`11.4
`10.6
`0.528
`0.50
`CAB
`5
`12.8
`12.3
`13.5
`0.594
`0.50
`BCA
`6
`14.7
`14.4
`14.0
`0.644
`0.75
`ABC
`7
`13.1
`12.9
`15.2
`0.463
`0.50
`ABC
`8
`9.6
`9.4
`12.5
`0.527
`0.50
`BCA
`9
`16.0
`15.8
`13.9
`0.573
`0.75
`BCA
`10
`14.4
`14.1
`19.1
`0.538
`0.50
`CAB
`11
`12.8
`12.5
`13.2
`0.566
`0.25
`ABC
`12
`11.7
`11.5
`12.8
`0.467
`0.25
`CAB
`13
`15.4
`15.2
`14.1
`0.579
`0.50
`ABC
`14
`18.2
`18.1
`16.2
`0.524
`0.50
`BCA
`15
`12.4
`12.1
`6.3
`0.827
`0.50
`ABC
`16
`152
`14.9
`21.6
`0.561
`0.25
`CAB
`1
`14.8
`14.7
`11.7
`0.554
`0.75
`ABC
`2
`11.2
`10.8
`73
`0.486
`1.25
`BCA
`3
`15.7
`15.4
`17.8
`0.481
`0.50
`CAB
`4
`11.1
`10.8
`10.7
`0.525
`0.50
`CAB
`5
`16.4
`16.3
`14.4
`0.527
`0.75
`BCA
`6
`13.3
`13.2
`117
`0.542
`0.75
`ABC
`7
`13.2
`13.1
`17.4
`0.488
`0.25
`ABC
`8
`8.5
`8.4
`8.7
`0.433
`0.75
`BCA
`9
`16.3
`15.9
`8.6
`0.629
`0.50
`BCA
`10
`13.7
`13.6
`7.8
`0.460
`L715
`CAB
`11
`12.9
`12.5
`12.6
`0.506
`0.50
`ABC
`12
`12.6
`12.4
`16.2
`0.473
`0.50
`CAB
`13
`14.6
`14.1
`TA
`0.849
`1.25
`ABC
`14
`17.2
`16.8
`16.2
`0.486
`0.50
`BCA
`15
`12.5
`12.3
`12.7
`0.413
`0.75
`ABC
`16
`12.2
`11.9
`10.6
`0.491
`0.75
`CAB
`1
`14.9
`14.7
`19.4
`0.480
`0.25
`ABC
`2
`12:1
`11.9
`9.1
`0.586
`0.50
`BCA
`3
`12.9
`12.8
`15.9
`0.526
`1.00
`CAB
`4
`12.5
`12.2
`15.2
`0.545
`0.50
`CAB
`5
`14.7
`14.5
`16.7
`0.488
`0.50
`BCA
`6
`15.0
`14.6
`14.3
`0.779
`0.50
`ABC
`7
`11.0
`10.8
`127
`0.445
`0.75
`ABC
`8
`10.5
`10.3
`12.1
`0.464
`0.75
`BCA
`9
`13.6
`13.4
`9.9
`0.628
`0.50
`BCA
`10
`15.1
`14.9
`17.3
`0.477
`0.25
`CAB
`11
`14.6
`14.3
`19.6
`0.494
`0.50
`ABC
`12
`11.8
`11.6
`19.9
`0.570
`0.25
`CAB
`13
`15.9
`15.6
`11.6
`0.604
`0.50
`C
`ABC
`14
`
`
`
`
`
`
`
`BCA Cc 0.50 0.520 19.8 16.015 16.3
`
`Cc
`Cc
`Cc
`Cc
`Cc
`Cc
`Cc
`c
`Cc
`C
`Cc
`
`sequences implied the basic validity assumption underlying the
`crossover design had not been violated. Single degree of free-
`dom contrasts constructed from the sequence X period means
`served to compare both the new formulations against the refer-
`ence. Confidence intervals (90%) were constructed for the dif-
`ference in the means of both new formulations versus the
`reference formulation using the 1n transformed data. Theanti-
`logarithms of each set of confidence limits supply 90% confi-
`dencelimits for the ratio of test and reference product averages.
`Test and reference formulations are declared bioequivalent
`whenthe calculated limits are contained within the conventional
`
`bioequivalence range .80 to 1.25. Results of the bioequivalence
`evaluation are summarized in Table 2: The pointestimate differ-
`ence between B and C for the Cmax/AUC& ratio was —.18.
`The arithmetic 90% CI was —28.6% to —4.7% so this ratio
`variable here fails the bioequivalence interval
`(—20% to
`20%) specification.
`Since subjects were sampled at a constantrate of four per
`hour for two hours during the suspected absorption phase and
`all subjects peaked in concentration within two hours,the alter-
`native procedure waseasily implemented. Tmax could assume
`at most eight distinct values, and Tmax multiplied by four is
`
`€
`€
`

`

`Tmax: An Unconfounded Metric for Rate
`
`327
`
`Table 2. 90% Confidence Limits on Separation of Means
`
`
`Contrast
`
`Variable
`
`Ratio
`
`Lower
`Limit
`
`Upper
`Limit
`
`Pass/Fail
`
`Avs. C
`
`Pass
`1.12
`82
`95
`Cmax
`Pass
`1.04
`.93
`99
`AUCt
`Pass
`1.04
`94
`99
`AUC
`Fail
`97
`70
`82
`Cmax
`Pass
`1.07
`.96
`1.01
`AUCt
`
`
`
`
`1.01 .96 1.07AUC Pass
`
`B vs. C
`
`DISCUSSION
`
`an integer, or count. Table 3 summarizes the answer to the
`range, B andCaresignificantly different at approximately the
`question “How many quarters to reach Cmax?”for each subject
`20% level in yet another two sidedtest.
`and period on the study.
`Analyses for these count data by stratified linear rank test
`(6) and within the crossover context by Poisson regression (5)
`are summarized in Table 4. Contrasts of sequence X period
`means similar to the ones used in the crossover analysis for
`continuous variables provided the two tests of the two Poisson
`regression mean parameter estimates given in Table 4. The
`Poisson regression deviance of 13.5053 is below its asymptotic
`chi-square value (24) for 24 degrees of freedom. The observed
`chi-square value correspondsto a p-value >.95 which usually
`is an indication the specified model fits the data very well. We
`have examined several examples and in all of them ‘under
`dispersion’ seemsto be the norm.In general this should alert
`to the possibility of an incorrectly specified model (unlikely)
`or outliers in the data (much morelikely). The phenomenon
`needs further investigation.
`Table 5 showshowthe Poisson regression computerprint-
`leads easily to meaningful rate estimates.
`In the upper
`out
`part of the table, with three treatmenteffects aliased with four
`sequence X period means, the program supplied estimates for
`the four sequence X period means (Table 5, column 3) need
`rectification into three treatment effects that sum to zero (col-
`umn 4); treatment means are derived from these effects by
`adding each in tur to the overall mean (intercept). Column 5
`showscountscale estimates obtained by applying the exponen-
`tial transform to column 4 entries. Entries in parentheses are
`the square root of the entry above.
`Haight (10) provides several methods for comparing Pois-
`son means. Underthe null hypothesis the three observed means
`(31.4, 43.4 and 31.4) are the same. The binomial probability
`law can evaluate them forsignificance. For example, to compare
`p. (B) and w (C) we evaluate the appropriate confidenceinterval
`for 8 = 0.5 in 30 binomial trials, and observe whether p/(w
`+ w) = 0.58 lies inside or outside the CI. If we use the software
`package (5) to evaluate a 60% CI for 15 successes in 30trials
`we obtain .41—.59. Since .58 is barely contained within the
`
`A B vs.C ratio estimate for Cmax below 1.0 implies the
`B formula has the lower extreme concentration value, so clearly
`dumping and attendant safety issues do not arise for B (Table
`2). The example shows a suspected common occurrence in
`bioequivalence testing. The usurper variable Cmax declares
`that B and C are not bioequivalent in absorption rate under the
`standard approach, while the appropriate analysis (Table 5)
`applied to the logical estimator available for rate of absorption,
`i.e., Tmax, finds insufficient evidenceto reject the null hypothe-
`sis. We should infer or conclude that B, while numerically lower
`in absorption rate by chance, has the samerate of absorption
`as C. Appropriate analysis therefore finds the two formulas, B
`and C,
`to be bioequivalent in both rate and extent!. Three
`outlier individuals here lead the two competing procedures to
`different conclusions.
`In retrospect some will claim that appropriate analysis of
`the rate data in this study merely confirms that 15 subjects
`cannotdefinitively answer the rate question. In our experience
`a sample in the vicinity of 30 is needed to begin to compare
`discrete variables with reasonable power. The insensitivity of
`absorption rate as measured in small bioequivalencestudies is
`an issue beyond the scope of this paper. The bothersomeevi-
`dence, illustrated by the study,
`is that appropriate statistical
`evaluation with Tmax supports bioequivalencein rate of absorp-
`tion for the test suspension (B), whereas the customary analysis
`with illogical Cmax leadsto rejection of bioequivalence.
`The unpredictable and uncertain outcome arising from
`using Cmaxas a metric for absorption rate hasled to a search for
`surrogate metrics, and alsoto calls to modify the bioequivalence
`interval. We saw in the study example that the Cmax/AUCo
`ratio for B vs. C failed the bioequivalence interval specification
`when Cmax did. This was not surprising, it merely confirmed
`
`Table 4. Two Analyses of Counts Formed from Tmax Observations
`
`Sided
`p-
`Table 3. Frequency Tabulation of Counts for Three FormulationsLLEEaTTTEaNES
`Test
`Contrast
`Inference form
`test
`value
`Formulation
`1
`2
`3
`4
`5
`6
`7
`Total
`
`A
`B
`c
`
`2
`2
`3
`
`9
`6
`8
`
`4
`4
`3
`
`0
`0
`1
`
`0
`2
`0
`
`0
`0
`0
`
`0
`1
`0
`
`15
`15
`15
`
`1.0
`Two
`Stratified linear Avs.C Exact test
`0.18
`Two
`rank test
`Bvs.C Permutation
`1.0
`Two
`Poisson
`Avs.C Asymptotic test
`
`
`Bvs.C Likelihood ratio Tworegression 0.20
`
`
`

`

`328
`
`Basson, Cerimele, DeSante, and Howey
`
`Table 5. Study Poisson Regression Absorption Parameter Estimates
`
`Actual In
`
`Parameter
`
`Program
`Estimate
`(In scale)
`
`Estimate
`(sum
`terms)
`
`exp(estimate)
`(count scale)
`
`Estimate
`in minutes
`(= count X 15)
`
`35.0
`2.333
`0.8473
`Intercept
`31.4
`2.095
`0.2412
`A mean
`r
`43.4
`2.894
`5776; 5513
`B mean
`p
`31.4
`2.095
`0.2412
`C mean
`@
`26.3
`1.7554
`0.5627
`deviance
`o?
`o
`scale (\/dev)
`(0.7501)
`(1.3249)
`19.9
`
`
`0.8473
`—0.1078
`0.2155
`—0.1078
`
`expected behavior for an ad hoc ratio. There are many proposals
`in the literature suggesting something be done about the Cmax
`interval. Forexample, Schultz and Steinijans (11) want to widen
`:
`es
`3
`the bioequivalence range for Cmax from .8 — 1.25 to .7 —
`1.43. Clearly a wider interval would help, but we contend a
`superior solution to the unpredictability problem lies in first
`using a refined sampling scheme so as to empower Tmax to
`fariction.as ie meitie for absorviion fate
`Tp
`.
`
`REFERENCES
`.
`.
`.
`.
`oo.
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`2. L. Endrenyi, S. Fritsch and W.Yan. Cmax/AUCis aclearer measure
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`
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`SAS
`Institute:
`SAS/STAT™
`. The G
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`pp. 329-352.
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`—_9_ SAS Institute: SAS/STAT™ User's Guide, Release 6.3 Edition.
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`
`