Journal of Pharmacokinetics and Biopharmaceutics, Vol. 5, No. 5, 1977
`
`Bioavailability Assessment: Methods to Estimate
`Total Area (AUC 0-w) and Total Amount Excreted
`(Ae) and Importance of Blood and Urine Sampling
`Scheme with Application to Digoxin
`
`John G. Wagner 1'2 and James W. Ayres 1'3
`
`Received Jan. 18, 1977--Final Apr. 19, 1977
`
`Five methods are compared to estimate the total area .under the digoxin plasma or serum
`concentration-time curve (A UC 0-oo) after a single dose of drug. To obtain accurate estimates of
`A UC 0-oo, data required are concentrations at a sufficient number of sampling times to define
`adequately the concentration-time curve prior to the log-linear phase, and at least three, but
`preferably four or more equally spaced points in the terminal log-linear phase. One method
`(designated Method I) requires a digital computer; another (Method III) is the classical method
`(these two methods do not require equally spaced points in the log-linear phase). Method IIA is the
`accelerated convergence method of Amidon et al.; Methods IIB and llC are modifications of this
`method, but incorporate usual and orthogonal least squares, respectively, which make them more
`accurate with real (noisy) data. Methods I and llC gave very comparable estimates of A UC O-oo.
`Results indicate that digoxin administered orally in aqueous solution was completely (100%)
`absorbed when bioavailability estimates were based on oral and intravenous A UC 0--oo estimates
`and the actual doses, whereas formerly only about 80% absorption was reported, based on areas,
`under plasma concentration curves which were truncated at 96 hr. It is shown that the sampling
`scheme of blood can produce biased apparent bioavailability estimates when areas under
`truncated curves are employed, but an appropriate sampling scheme and application of method
`IIC yield accurate bioavailability estimates. This is important particularly in those bioavailability
`studies where one is attempting to determine the appropriate label dose for a new "fast-release"
`digoxin preparation relative to the label dose and bioavailability of currently marketed tablets. It is
`shown that the magnitudes and variability of apparent elimination rate constants and half-lives of
`digoxin, estimated ffom urinary excretion data by the tr- method, depend on which value of A~ ~ is
`used. The formerly reported greater interindividual variability of A UC data compared to Ar data
`for digoxin is explained in that the A UCs, but not the A~'s, involve the renal clearance, which
`exhibits considerable inter- and intraindividual variation.
`
`1College of Pharmacy and Upjohn Center for Clinical Pharmacology, The University of
`Michigan, Ann Arbor, Michigan 48109.
`2Address correspondence to Dr. John G. Wagner, Upjohn Center for Clinical Pharmacology,
`The University of Michigan Medical Center, Ann Arbor, Michigan 48109.
`P esent address: School of Pharmacy, Oregon State Umverslty, Corvallis, Oregon 97331.
`r
`"
`"
`
`533
`This journal is copyrighted by Plenum. Each article is available for $7.50 from Plenum Publishing Corporation, 227 West l 7th
`Street, New York, N.Y. 10011.
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`KEY WORDS: accelerated convergence method to estimate AUC 0-o0 and A~; bioavailabil-
`ity; estimation of total areas; estimation of total amounts excreted; blood sampling schemes for
`digoxin; elimination half-life of digoxin; intra- and interindividual variation of renal clearance
`of digoxin.
`
`INTRODUCTION
`Pharmacokinetic equations appropriate to estimation of absolute or
`relative bioavailability (used here with the connotation of absorption
`efficiency only, without the other component of rate of absorption) involve
`ratios of dose-corrected total areas under plasma or serum concentration-
`time curves (AUC 0-~), or total amounts of unchanged drug excreted in the
`urine in infinite time (A~) after a single dose of drug (la,2). The word
`"total" herein refers to AUC 0-oo or A ~ and not, as often erroneously used
`in the literature, to indicate AUC 0-T or Aft, where T is the investigator's
`last sampling time; hence AUC 0-T is a partial area and AT is a partial
`amount excreted in the urine. In this article, such partial areas and amounts
`excreted are simply designated by AUC'and Ae, respectively.
`It is common practice in the digoxin (7-21), as well as the literature for
`many other drugs, to substitute the particular author's AUC or Ae figures
`for AUC 0-co or A e ~ in estimating bioavailability. Such estimates are herein
`called apparent bioavailabilities. For digoxin there have been almost as many
`blood sampling schemes as investigators. Apparent bioavailabili.ties depend
`on the sampling scheme employed, and they may be considerable underesti-
`mates of the true bioavailability. This will be very important in establishing
`the correct dose ratio of new "fast-release" digoxin formulations compared
`with currently marketed "slow-release" tablets, since an error of the order
`of 20% could have noticeable effects in the clinical use of digoxin. The
`shortcomings of reporting such apparent bioavailabilities have been pointed
`out before by other authors (4,6,16,24). We could find only one article (16)
`where estimates of A ~ for digoxin had been made, but the method was not
`given. No article could be found where AUC 0-oo had been estimated for
`digoxin after oral administration. Several authors (3,7,9,10,12,14-16,19-
`21) have collected either 0-6 or 0-10 day digoxin urinary excretion data in
`those cases when they were employing a radioimmunoassay method for
`digoxin in plasma or serum which had a sensitivity level of 0.2-0.5 ng/ml
`and which allowed them to follow digoxin in blood only for about 8 hr.
`However, as assay has been in the literature (22) since 1972 which allows
`measurement of digoxin down to 0.08 ng/ml of plasma or serum, and for
`96 hr after a single 0.5-mg dose of digoxin (5). This assay has been improved
`(23) and has a sensitivity limit of 0.05 ng digoxin/ml plasma when a 0.5-ml
`plasma sample is utilized. The authors have used this improved assay in
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`Bioavailability Assessment
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`several digoxin bioavailability studies where blood was sampled over a 96-hr
`period.
`The articles of Wagner et al. (2,5) and Lovering et al. (17) suggested that
`apparent bioavailabilities estimated from partial areas may be close to the
`true bioavailabilities under certain conditions. In a recent review on digoxin
`(18) it was stated: "However, if blood sampling is continued for an interval
`(T) after the dose which is sufficient to allow serum concentrations to
`become quite small, then AUC 0-T is a good approximation of AUC 0-oo."
`It was (18) further stated: "In our studies of comparative bioavailability, 4 hr
`of serum sampling gave results as reliable as 8 or 24 hr of sampling." The
`same authors (15) also stated: "Extending urine collections beyond 1 day or
`serum sampling beyond 4 or 8 hr does not necessarily reduce between
`subject variability or enhance the usefulness of the data." However,
`Beveridge et al. (16) stated: "Therefore, statements on bioavailabitity [of
`digoxin] based on areas under plasma curves up to 6 hr may differ from those
`based on cumulative urinary excretion'data, in this case by a factor of about 2
`[i.e., a 100% error] and could suggest that bioavailability was much worse
`than it actually was." This dichotomy of opinion prompted us to examine, in
`general, the assessment of bioavailability and, in particular, to study digoxin
`bioavailability. In the process, several new simple methods for estimating
`AUC 0-oo and A ~ were devised and applied.
`
`THEORETICAL
`
`Methods for Estimating AUC 0 - ~ and A ~
`All known methods and the new methods to be presented for estima-
`tion of AUC 0-oo and A ~ depend on accurate estimates of AUC or Ae at
`various times after administration of a single dose of drug. For AUC the
`trapezoidal rule (lb) is usually employed, and with sufficient sampling times
`(see rows 6, 7, and 8 of Table III) is accurate for digoxin. An even more
`accurate method would be that resulting from interpolation by the method
`of Fried and Zeitz (25), which has been computerized (lc), coupled with the
`trapezoidal rule using both observed and interpolated values. All methods
`(I, IIA, liB, IIC, and III) discussed below are applicable only to AUC or Ae
`data in the terminal, log-linear phase. Methods IIA, B, and C below also
`depend on having three or more blood or urine collections at equally spaced
`time intervals
`in the terminal log-linear phase of drug elimination; for
`accurate results, it is later shown that four such collections is the minimum
`necessary. The classical method (Method III) for estimating AUC 0-oo also
`depends on an accurate estimate of the apparent elimination rate constant,
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`536
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`
`which also requires at least three and preferably four or more plasma
`concentration-time points (i.e., Cp, t pairs).
`
`Method I
`Method I depends on nonlinear least-squares fitting to equation 1 of log
`linear AUC, t or Ae, t data using a suitable program and a digital computer:
`y = P(1) -P(3) e-P(2)'
`(1)
`
`For AUC data, y represents AUC at time t, P(1) represents AUC 0-w, P(2)
`represents h 1, and P(3) represents B1/~, 1, such that the plasma concentra-
`tion, Cp, in the log-linear phase is described by equation 2. The B1 in
`equation 2 (and later equation 13) is ~i complex function of model para-
`meters:
`
`(2)
`Cp = B 1 e-~ 1,
`For Ae data, y represents Ae at time t, P(1) represents A~, P(2)
`represents )tl, and P(3) is equivalent to C1RB1, where CIR is the renal
`clearance.
`To apply the method, we used the program NONLIN (26) and the
`Amdahl 470V/6 digital computer. P(1), P(2), and P(3) are the parameters
`estimated in the fittings. The method has two advantages: (a) the points may
`be equally or nonequally spaced and (b) one obtains the standard deviations
`of the estimated AUC 0 - ~ or A e ~ as well as measures of fit of predicted
`AUC or Ae to observed AUC or Ae values.
`
`Method IIA
`Method IIA is the accelerated convergence method of Amidon et al.
`(27). For a series of points, (ti, Y~), approaching an asymptote, Y~, and
`obeying first-order kinetics from time t', the general equation 3 applies,
`where A 1 is the first-order rate constant:
`(3)
`Yi = Yoo[1-e -A'('-c)]
`For three equally spaced points, at intervals, At, particular cases of equation
`3 may be written as
`
`(4)
`Y1 = Voo[1-e -x'a']
`(5)
`Y2 = Yoo[1-e -2x~a']
`(6)
`Y3 = Yoo[1-e -3a~at]
`Amidon et al. (27) used different symbolism such that Y1 = their X,,
`Y2 = their X,+I, I"3 = their X,+z, and Y~ = their X'. They also plotted the
`differences on the ordinate but ended up deriving the expression corres-
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`ponding to these differences being plotted on the abscissa. Hence we prefer
`to plot the differences on the abscissa in order to apply Methods lib and IIC
`discussed later.
`Consider a rectalinear plot of Y = Y~ (ordinate) vs. X = Y~+I-Y~
`(abscissa) with the equation of the straight line being Y = a + bX; two points
`on the line are (Y2- Y1), Y2 and (II3 - Y2), Y3; the line extrapolates back to
`give an ordinate intercept, a, equal to Yoo; the slope of the line, b, is given by
`equation 7, and the slope is negative since YI < Y2 < Y3 and (Y2- Ya) >
`(II3-- Y2).
`
`113- Y2
`(Y2- Y3)
`(y2_y1)_(y3_Y2) = Ya-2YE+Y~
`b
`When Y = II3, the equation of the line is given by
`
`(7)
`
`(8)
`
`(9)
`
`Yoo = I13
`
`]
`Y3- Y2
`y3 = yoo+[
`Y3-2Y2+ YI (Y3- Y2) = Yooq (Y3- I"2) 2
`Y3- 2Y2 + Y1
`Rearrangement of equation 8 gives equation 9, which is equivalent to
`the equation given by Amidon et al. (27):
`(Y3- Y2) 2
`Y3-2Y2+ Y1
`The validity of equation 9 with respect to first-order kinetics is readily
`checked by substituting for Y1, I"2, and Y3 in equation 9 from equations 4
`through 6 and showing that the right-hand side is equal to the left-hand side,
`As a simulation for AUC data after bolus intravenous administration of
`digoxin, let the values of the parameters of equation 1 be P(1)= 44.67,
`P(2) = 0.0122, and P(3) = 37.44. Substitution of these values and t = 24, 48,
`and 72 hr into equation 1 yielded the values below.
`t
`Y,
`24
`16.73 = Y1
`48
`23.82 =Y2
`72
`29.12=I"3
`Substitution of the above values into equation 9 and simplification gave
`Yoo = 44.81. The actual value of P(1) = Yoo = AUC 0-oo = 44.67, while the
`estimated value by this method is 44.81. Hence with these error-free data
`the method gave an answer with an error of +0.45%.
`Method IIB
`Method IIB is a modification of Method IIA, but three or more points
`may be utilized (see Fig. 3 as an example). Equation 10 is a generalization of
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`538
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`equation 8:
`
`Wagner and Ayres
`
`Y/= Yoo-(slope)(Y~+l- Y/)
`
`(10)
`
`Equation 10 indicates that a plot of Y~ vs. (Y/+I - Y/) will have an ordinate
`intercept equal to Yoo and a negative slope, when the method of least squares
`is applied to the data and first-order kinetics is obeyed. To illustrate the
`method, the data used to illustrate Method IIA were extended by adding a
`I"4 value for 96 hr and these data are shown below:
`
`t
`24
`48
`
`72
`96
`
`Y = Y~ X = Y~+I- Y~
`16.73= Y1--.~7.09
`23.82 = Y i ~
`5.30
`
`29.12= Y3~.._y4 ~ > 3 " 9 4
`
`
`33.06 =
`
`Using ordinary least-squares linear regression and the pairs of values
`X = 7 . 0 9 and Y=16.73, X=5.30, and Y=23.82, and X = 3 . 9 4 and
`Y = 29.12, equation 11 was obtained with a correlation coefficient of 1.000.
`The intercept, 44.64, is
`
`Y = 44.64- 3.935X
`
`(11)
`
`the estimate of AUC 0 - ~ and is within -0.07% of the actual value of 44.67.
`Methods IIA and liB give the same answer when there are only three Y~
`values and the line is based on only two points.
`
`Method IIC
`Method IIC is the same as Method liB, with the exception that
`orthogonal least squares (28) is used in place of ordinary least squares. In
`applying both methods, trapezoidal areas should not be rounded off before
`calculating the parameters of the least-squares line, but final intercept values
`should be rounded off to the same number of places as the original blood
`level or urinary excretion data. The equations used to obtain the slope and
`intercept of the orthogonal least-squares line are shown in the Appendix.
`With the same data as used to illustrate Method liB, Method IIC gave the
`same intercept, 44.64, with an error of -0.07%. However, with other data
`sets orthogonal least squares and ordinary least squares do not give the same
`answer. Since both YI and Y~+I - Y~ contain errors, orthogonal least squares
`is preferred from a statistical point of view. Hence Method IIC is preferred
`to Method liB.
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`Method III
`Method II is the classical method (la) for estimating AUC 0 - ~ using
`
`AUC 0-c~ = trapezoidal area 0 - T + Cr/A 1
`
`(12)
`
`In equation 12, T is the last sampling time in the log-linear phase and
`(~T is the estimated concentration at that time obtained with equation 14.
`The value of h l is obtained by least-squares regression based on equation
`13.
`
`In Cp = In B1 -h it
`d T : e (lnBl-xlT)
`
`(13)
`(14)
`
`As indicated formerly, one should have a minimum of three and preferably
`four or more Cp, t or Ae,t pairs in the log-linear phase and adequate sampling
`in the early time period with blood data to apply equation 12.
`
`Other Methods
`When the Guggenheim method (ld,29) was applied to the data which
`were used to illustrate Methods liB and IIC, the AUC 0-oo estimate was
`37.50, with a large error of -16.1%. When the "rate method" (le) was
`applied to the same data, the AUC 0-az estimate was 37'.64, with a large
`error of -15.3 %. Since these two methods gave such large errors even with
`"error-free" data, they were not considered further for application to
`digoxin AUC,t data. The Results and Discussion section indicates that the
`same conclusion was reached with respect to digoxin urinary excretion data.
`
`Variability of AUC and Ae Data
`There have been several reports (9,15,18) that interindividual AUC
`data are more variable than interindividual Ae data, and the conclusion was
`reached that Ae data provide a more reliable estimate of apparent bioavaila-
`bility than AUC data. There is a simple pharmacokinetic explanation for
`such differences in variability. AUC data are influenced by an additional
`variable, namely renal clearance, Cln, which does not influence Ae data. If
`the system obeys linear pharmacokinetics, then equations 15-17 apply:
`
`Ae = Cln (cid:12)9 AUC
`Ae ~ = fFF*D
`AUC 0-00 = fFF*D/CIn
`
`(15)
`(16)
`(17)
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`
`In equations 16 and 17, f is the fraction of the drug which reaches the
`circulation which is excreted in the urine, F is the bioavailability factor
`concerned with incomplete absorption ( 0 - F-< 1), F* is the bioavailability
`factor concerned with the "first-pass effect" (0 <-F*- < 1), and D is the dose
`administered orally. For the intravenous route, F = 1 and the same equa-
`tions apply. F* has meaning only when both oral and intravenous data are
`considered together.
`Koup et al. (14) reported renal clearances of digoxin in eight subjects
`following both bolus intravenous injections and intravenous infusions.
`Interindividual coefficients of variation of the C1R values were 40.8% and
`37.8% for bolus and infusion, respectively. Intrasubject variability of CIR is
`reflected by the coefficient of variation calculated from the differences in the
`pairs of CIR values for the eight subjects; this coefficient of variation was
`43.8%. This suggests that intraindividual variation and interindividual
`variation of the C1R of digoxin are very similar. Similar calculations made by
`the authors from CIR data collected in two recent digoxin bioavailability
`studies, one involving 12 and the other involving 15 subjects, also gave little
`differences between inter- and intraindividual variation of C1R. Thus one can
`readily see with such an additional variable (CIR) being "contained in" AUC
`data, but not in Ae data, why AUC data are more variable than Ae data.
`Equations for both AUC and Ae will be analogous to equations 16 and 17,
`but will also ~contain exponential terms; those equations for data in the
`log-linear phase will contain a term with e -~1' (like equation 1); those for
`data in the postabsorptive, distribution phase will contain two terms, one
`with e -~1' and one with e -x2'. Therefore, on a theoretical basis one would
`expect AUC to a given time to be more variable than AUC 0-oo, and Ae to a
`given time to be more variable than A e ~.
`
`Estimation o| AUC 0 - ~ from Data ot Wagner et al. (5)
`Wagner et al. (5) reported digoxin plasma concentration-time data for
`two subjects administered labeled doses of 0.5 mg of digoxin by 1-hr
`constant-rate intravenous infusion, as a solution orally, as two B & W
`tablets, and as two Fougera tablets (both 0.25 mg/tablet) orally. The blood
`sampling scheme employed is shown in the sixth row of Table III. For each
`subject and each treatment, AUC 0-oo was estimated by Methods I, IIA,
`IIB, IIC, and III. The needed AUC,t data were obtained by application of
`the trapezoidal rule (lb) to each set of Cp,t data. Bioavailability estimates,
`based on ratios of dose-corrected AUC 0-oo's, were compared with appar-
`ent bioavailabilities, calculated from dose-corrected AUCs at various times
`after administration, using the 1-hr intravenous infusion data as the "stan-
`dard" in both cases.
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`Bioavailability Assessment
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`541
`
`Importance of Blood Sampling Scheme
`Digoxin plasma concentration-time data were simulated for a "fast-
`release" (A) and a "slow-release" (B) digoxin formulation. Equations used
`are shown in the Appendix. The advantage of such a procedure is that the
`exact answers are known against which "experimental answers" may be
`compared. Eight different blood sampling schemes were compared: six of
`these were taken from the literature; a seventh scheme was used by the
`authors in a recent unpublished study; the eighth scheme led to the points
`shown in Fig. 2. The equations used to generate these data were chosen so
`that the simulated plasma concentrations from 3 hr to infinity were identical
`to the third decimal place for A and B. This provided a minimum (not a
`maximum) test to show the effect of the blood sampling scheme on apparent
`bioavailabilities. It also provided a means to show that the reasoning of
`Lovering et al. (17) is faulty, when applied to digoxin.
`
`Estimation of A~ from Data of Juhl et al. (20)
`The individual subject/treatment sets of A~,t data of Juhl et al. (20)
`were employed to obtain estimates of A~ ~ by Methods I, IIA, IIB, and IIC.
`Ae S, were compared with Bioavailability estimates, based on ratios of the ~
`
`apparent bioavailabilities, based on ratios of Ae's to various times. Appar-
`ent elimination rate constants, A 1, were calculated, using the method of least
`squares, the different A~ estimates, and equation 18, in which In I is the
`intercept and its value depends on the particular model which applies.
`In ( A ~ - A e ) = In l-A~t
`(18)
`
`RESULTS AND DISCUSSION
`
`Estimation of AUC 0-oo from Data of Wagner et aL (5)
`The averages of duplicate assay values reported by Wagner et al. (5)
`were used as the Cv,t data. The AUCs were obtained by trapezoidal rule.
`AUCs in the log-linear phase, corresponding to 24, 48, 72, and 96 hr for oral
`treatments and 25, 49, 73, and 97 hr for the 1 hr constant-rate intravenous
`infusion, were employed to estimate AUC 0-oo values, which are shown in
`Table I. Plotting of data according to equation 10 (see Fig. 3 for example
`with urinary excretion data) indicated that elimination was not apparent first
`order at 12 hr, but was at 24 hr, when plasma data were evaluated. Data for
`the Fougera tablet in subject 2 were anomalous in that the data did not obey
`apparent first-order kinetics (see footnotes to Table I). For the other seven
`sets of data, the AUC 0-o's estimated by Method IIC agreed extremely well
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`Table I. Summary of AUC 0--oo for Digoxin Estimated from the Plasma Concentration-Time
`Data of Wagner et al. (5)
`
`Subject
`
`Method
`
`I
`
`IIA
`IIB
`IIC
`III
`I
`
`IIA
`liB
`IIC
`III
`
`1 hr i.v.
`infusion
`
`43.2 ~
`(0.82) b
`41.5 c
`43.1 a
`43.2 a
`44.8 a
`44.7 ~
`(4.97)
`37.3 c
`42.8 ~
`44.3 ~
`47.5 ~
`
`AUC 0-o0 [(ng/mi) x hr]
`
`Solution
`orally
`
`B & W tablet Fougera tablet
`orally
`orally
`
`40.1 a
`(3.42)
`51.2 ~
`39.2 a
`39.3 ~
`37.6 a
`42.0 ~
`(0.02)
`42.1 c
`42.0 ~
`42.0 ~
`49. I a
`
`20.1 a
`(0.08)
`20.3 c
`20.1 a
`20.1 ~
`20.0 a
`21.7 a
`(1.06)
`24.9 c
`21.5 ~
`21.7 ~
`20.2 a
`
`9.03 a
`(0.20)
`9.50 c
`9.00 a
`9.02 ~
`8.53 ~
`15.2 ~
`(8.56)
`e
`
`.e
`
`e
`23.4 ~
`
`'*Four partial areas to 25, 49, 73, and 97 hr were used for the intravenous data and those to 24,
`48, 72, and 96 hr were used for the oral data.
`bNurnbers in parentheses are standard deviations of the estimated areas.
`CThree partial areas to 25, 49, and 73 hr were used for the intravenous data and those to 24, 48,
`and 72 hr were used for the oral data.
`dThree partial areas to 48, 72, and 96 hr were used.
`eMethods gave ridiculous answers, indicating that data were not obeying first-order kinetics;
`this was also indicated by the very large standard deviation of 8.56 for an estimated area of
`15.2 by Method I and the discrepancy of the answers obtained by Methods I and III.
`
`with the A U C 0-o0's estimated by Method I, the m e a n absolute deviation
`being 0.17 (ng/ml) x hr (0.5% of m e a n by Method I), and in four out of the
`seven sets the estimates were identical. For Methods I and IIB the m e a n
`absolute deviation was 0.45 (ng/ml) • hr (0.7% of m e a n by Method I), for
`Methods I and III it was 2.3 (ng/ml) x hr (7.3% of m e a n by Method I), and
`for Methods I and I I A it was 3.45 (ng/ml) • hr (11% of m e a n by Method I).
`Figure 1 illustrates the variation in apparent digoxin bioavailability as a
`function of time. The " t r u e " bioavailabilities, based on dose-corrected
`A U C 0-oo ratios, are shown above the infinity signs at the far right of the two
`graphs. These " t r u e " bioavailabilities, based on A U C 0-oo's obtained by
`Method I, are very similar to those estimated by Method IIC. This new
`interpretation of the data of W a g n e r et al. (5) indicates that digoxin,
`administered as an aqueous solution orally in those studies, was completely
`absorbed. This agrees with similar estimates made from 10-day urinary
`excretion data (3) and disagrees with data summarized by G r e e n b l a t t et al.
`(18) and with the original estimates of W a g n e r et al. (5), based on apparent
`bioavailabilities. The graphs in Fig. 1 illustrate how impossible it is to
`compare apparent bioavailabilities of different investigators who sample
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`Bioavailability Assessment
`
`543
`
`J
`
`B
`
`C
`
`J
`
`i
`20
`
`J._...L.__J
`40
`TiME
`
`.
`
`J
`
`.
`J
`80
`60
`IN HOURS
`
`~
`
`
`400
`
`a~
`
`/
`
`110
`
`1OO
`~ 90
`
`>-
`
`8 0
`
`_a ~ 70
`J ~ 6o
`
`~-
`
`~:
`
`<
`
`50
`
`40
`
`30
`
`20
`10
`
`.o [-
`loo{-
`
`so
`
`Z,
`
`~
`so
`~ 50
`,o
`~ ~o
`
`O
`
`20
`
`1OO
`
`~:~
`
`80
`60
`40
`TiME IN HOURS
`Fig. 1. Apparent bioavailabilities as a function of
`time for two subjects based on the digoxin plasma
`concentrations of Wagner et al. (5). Apparent
`bioavailabilities were calculated from the dose-
`corrected ratio of areas under
`the plasma
`concentration-time
`curves.
`The
`"true"
`bioavailabilities are given by the points above the
`infinity sign and are based on total areas esti-
`mated by Method [. Key: A, oral solution relative
`to 1-hr intravenous infusion; B, B & W (Lanoxin)
`tablet relative to 1-hr intravenous infusion; C,
`Fougera tablet relative to I-hr intravenous infu-
`sion. Top, subject 1; botton, subject 2.
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`544
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`Wagner and Ayres
`
`Table II. Comparison of Results Based on Partial Areas with Those Based on Total Areas
`Obtained by Method I
`
`Apparent bioavailability of digoxin in the
`Fougera tablet relative to that in the B & W tablet
`Subject 1
`Subject 2
`57.7 Decrease
`41.3 Increase
`48.5 [ with
`60.7 I with
`44.7 ~
`time
`69.4 ~ time
`
`Average
`49.5
`54.6
`57.1
`
`Area utilized
`0-5hr
`0-96 hr
`0-oo
`
`blood for different times, such as 3, 4, 5, 6, 8, 12, 24, 48 and 96 hr (7-21). It
`also should be noted that in three out of four examples the "true" bioavaila-
`bility estimates for the tablets were lower than the values estimated from
`AUC 0-96 hr data. Also, throughout the 12-96 hr period, the trend lines
`rise for subject 1 (curves A and B) and subject 2 (curves A and C), but the
`trend line falls in the same period for subject 2 (curve B). It is also obvious
`from the figure that a 6-hr sampling scheme, as recommended by the Food
`and Drug Administration for digoxin (31), cannot provide "true" bioavaila-
`bility estimates for digoxin.
`Table II compares results based on partial areas with those obtained by
`total areas, when the Fougera tablet is compared to the B & W tablet as a
`"standard." For subject 1 there is a decrease with increase in time, and for
`subject 2 there is an increase with increase in time. Klink et al. (13) also
`reported data on the apparent bioavailabilities of digoxin from the B & W
`tablet and a Towne-Paulsen tablet, relative to digoxin elixir. For the B & W
`tablet the values were 71%, 83%, 96%, and 106%, and for the Towne-
`Paulsen tablet they were 65%, 74%, 85%, and 101%, based on ratios of
`AUC values to 5, 12, 24, and 48 hr, respectively. These results make
`questionable the conclusions of Greenblatt et al. (15,18) and Lovering et al.
`(17) that areas under truncated plasma or serum concentration-time curves
`are satisfactory to estimate digoxin bioavailability.
`
`Importance o| Blood Sampling Scheme
`Figure 2 shows simulated digoxin plasma concentrations for a "fast-
`release" (A) and a "slow-release" (B) digoxin formulation. Table III
`summarizes apparent bioavailabilities as a function of time, estimated from
`AUCs obtained by trapezoidal rule from the data shown in Fig. 2. Care must
`be taken to read Table III correctly. The "true" bioavailability is 97.4%.
`Values listed in rows 1 to 8 under "Apparent bioavailability" are those
`which would be estimated with the given sampling scheme and the areas
`under the truncated curves up to the last sampling time. Thus with the 3-hr
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`
`2.0
`
`1.8
`
`1.6
`
`1.4
`
`Z
`
`z
`g
`<
`
`1.0
`
`~" 0.8
`_z
`
`N
`
`0.6
`
`~
`
`0.4
`
`0.2
`
`)•,,.TWO SIMULATED FORMULATIONS
`
`iDENTICAL PLASMA LEVELS
`GIVE
`k FROM 3 HOURS TO
`INFINITY
`
`3
`
`4
`
`5
`
`12 24 48 72 96
`
`0
`
`1
`
`2
`
`7
`6
`t0 11
`9
`8
`TIME
`IN HOURS
`Fig. 2. Simulated digoxin plasma concentrations for "fast-release" and
`"slow-release" digoxin formulations.
`
`sampling scheme in row 1 one underestimates the "true" bioavailability by
`100 (97.4-84.3)/97.4 = 13.4%. With the 6-hr sampling scheme in row 2 one
`underestimates the "true" bioavailability by 14.7%. For this simulation the
`96-hr sampling schemes of rows 6-8 give AUC 0-96 hr estimates between
`99.3% and 99.8% of the "true" bioavailability, while Method IIC (last
`column of Table III) gives estimates from 100.1% to 100.4% of the "true"
`bioavailability. Values listed in the last row of Table III are those obtained
`with equations 23 and 25 of the Appendix. Discrepancies between the
`numbers when one reads vertically in the table--for example, comparing
`84.3 in row 1 with 76.9 in row 9--are caused by the sampling scheme's not
`truly defining the actual curves. This is a m i n i m u m (not a maximum) test and
`does show that sampling schemes only up to 8 hr do introduce appreciable
`error in bioavailability estimates. As stated in the Introduction, Beveridge et
`al. (16) claimed that with real data areas under digoxin plasma concentra-
`tions up to 6 hr can lead to a 100% underestimate of bioavailability.
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`gh
`
`97.4 I
`97.5"
`97.5"
`97.8"
`___a
`a
`___a
`d
`__a
`oo
`
`x 100
`
`AUC 0-0o for A
`AUC 0-0o for B
`Bioavailability
`
`fActual bioavailability for the simulation.
`"Values obtained from AUC 0-o0's estimated by Method IIC from AUCs to 24, 48, 72, and 96 hr.
`aBioavailability estimate cannot be made by Methods 1 through IIl since data are not in log-linear phase.
`CSampling scheme for points shown in Fig. 2.
`bSampling scheme used in recent (unpublished) study of K. S. Albert, J. W. Ayres, J. G. Wagner, et al.
`Reference for the sampling scheme shown, which was used in an actual digoxin bioavailability study.
`
`96.6
`96.8
`97.7
`97.2
`
`94.2
`
`86.1
`
`86.2 88.9 92.7
`
`76.9 84.1
`
`96
`
`24
`
`12
`
`8
`
`=
`
`x 100
`
`AUC for A
`AUC for B
`
`Apparent bioavailability (%)
`
`88.1
`83.1
`
`6
`
`-
`
`84.3
`
`3
`
`True values from equations
`0, 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 6, 8, 12, 24, 48, 72, 96
`0, 0.25, 0.5, 0.75, 1, 1.5, 3, 5, 7, 9, 12, 24, 48, 72, 96
`0, 0.25, 0.5, 0.75, 1, 1.5, 3, 5, 12, 24, 48, 72, 96
`0, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 4, 7, 24
`0,0.167,0.333,0.5,0.75,1,1.5,2,3,4,8
`0, 0.5, 1, 1.5, 2, 3, 4, 6
`0, 0.5, 1, 3, 6
`0, 0.5, 1.0, 1.5, 2, 3
`
`c
`b
`(5)
`(21)
`(10)
`(19)
`(8)
`(6)
`
`Blood sampling scheme (hr)
`
`Reference"
`
`Table ill Importance of the Blood Sampling Scheme in Estimating Bioavailability Using Simulated Data for "Fast-Release" (A) and
`
`"Slow-Release" (B) Digoxin Formulations (Fig. 2)
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`
`Lovering et al. (17) stated: "The AUC ratios at t = 2T are, in most
`cases, within a fe