BIOPHARMACEUTICS AND DRUG DISPOSITION, VOL. 16, 37-58 (1995)
`
`AN EVALUATION OF NUMERICAL INTEGRATION
`ALGORITHMS FOR THE ESTIMATION OF
`THE AREA UNDER THE CURVE (AUC)
`IN PHARMACOKINETIC STUDIES
`
`ZHILING YU AND FRANCIS L. S. TSE*
`
`Department of Drug Metabolism, Sandoz Research institute, East Hanover. NJ 07936, U.S.A.
`
`ABSTRACT
`Six numerical integration algorithms based on linear and log trapezoidal methods as
`well as four cubic-spline methods were proposed for estimation of area under the curve
`(AUC). These six different algorithms were implemented using IMSL/IDLTM command
`language and evaluated using data simulated under five different dosing conditions and
`two different sampling conditions. Comparisons between AUC estimations using these
`six different algorithms and the theoretical results were made in terms of both overall
`AUC values and the superimposability of the concentration-time profiles. In well designed
`studies with ample data points, the algorithm based on IMSL/IDLTM function
`CSSHAPE with concavity preservation gave the best performance. In contrast, when the
`frequency of blood collection was limited, the algorithm based on the log trapezoidal
`rule proved to be stable with reasonable accuracy, and is recommended as the practical
`method for numerical interpolation and integration in pharmacokinetic studies.
`Algorithms based on the combination of the log trapezoidal rule and cubic-spline
`methods using IMSL/IDLTM function CSSHAPE can be developed to enhance overall
`performance.
`
`KEY WORDS Area under the curve Numerical integration Cubic spline Trapezoidal method
`Log trapezoidal method
`
`INTRODUCTION
`
`The area under the concentration-time curve (AUC) is one of the most important
`parameters in pharmacokinetic analysis. A function of dose and clearance,
`it is often used as a direct indicator of the extent of bioavailability of a drug.
`The value of AUC may be determined by fitting an analytic function C(t) based
`on a compartmental model to experimental concentration (C) versus time ( t )
`data, and integrating C(t) analytically. Alternatively, it may be estimated
`by direct numerical integration of the data.' Although a variety of direct
`numerical integration algorithms for AUC estimation have been r e p ~ r t e d , ~ - ~
`
`*To whom reprint requests should be sent.
`
`CCC 0142-2782/95/010037-22
`0 1995 by John Wiley & Sons, Ltd.
`
`Received 23 June 1994
`Accepted 2 September 1994
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`38
`
`Z. YU AND F. L. S. TSE
`the linear trapezoidal method remains the most commonly used despite its
`tendency to produce systematic biases reflecting the concavity and spacing of
`the data.' The popularity of the linear trapezoidal method is due largely to its
`simplicity, compared with other algorithms based on relatively complicated
`interpolating methods, the computer software for which has been developed
`mainly for local use and is not generally
`Recent advancements in computer technology have resulted in a number of
`sophisticated spline interpolation methods that are suitable for pharmacokinetic
`applications. These programs are commercially available,6-8 and can be readily
`used to estimate the AUC values for any given set of concentration-time
`observations. In the present study, algorithms based on these spline methods
`as well as the classical trapezoidal methods were proposed and applied to data
`simulated under various dosing and sampling conditions. Comparisons between
`estimations using these different algorithms and the theoretical results were
`made in terms of both overall AUC values and the superimposability of the
`concentration-time profiles.
`
`SYSTEM AND METHODS
`
`These algorithms were implemented using the IMSL/IDLTM command
`language on a VAXstation 4000-60 under the VMS operating system.
`IMSL/IDLTM is a complete computing environment for the interactive analysis
`and visualization of scientific and engineering data. The C/Math/Library,
`designed by IMSL, has been integrated into the structure of the Interactive Data
`Language (IDLTM) by RSI.7
`In order to evaluate these numerical integration algorithms, simulated data
`without noise were generated under various dosing and sampling conditions.
`The superimposability of the concentration-time profiles is judged by the sum
`of absolute differences between the theoretical and the calculated AUC value
`at each time interval. Comparisons between these numerical integration methods
`were made in terms of both the percentage differences between the theoretical
`and the calculated total AUC values and the superimposability of the
`concentration-time profiles.
`
`Dosing condition I: intravenous bolus injection
`In a linear time-invariant system, the plasma concentration of drug, Cs(t),
`at time t following a unit single intravenous bolus dose can be approximated
`by a polyexponential e q ~ a t i o n : ~
`
`Cs(t) = c (Cke-k.')
`
`m
`
`k = 1
`
`and for a single dose of DIv
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`39
`
`If plasma drug concentration values Ci are measured at times ti (i= 1, . . . , n),
`the area under the curve AUC at each time interval is:
`
`The total AUC from 0 to tn is
`
`The following set of parameters was used: m = 2 , DIv=50mg, c1 = O - 1 L-l,
`~2=0*025L-l, X1=2.0h-l, X2=0*2h-'.
`
`Dosing condition 2: constunt-rate intravenous in fusion
`If a drug is administered intravenously at a constant rate ko for a time period
`to, the plasma concentration, Co(t), of unchanged drug at time t can be
`approximated by the following equation:
`
`where u(t) is the unit step function:
`
`u(t)=O
`u(t)=l
`
`for t<O
`for t>O
`
`The following equations for AUC were derived:
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`40
`
`Z . YU AND F. L. S. TSE
`-ko
`k I l [ :
`
`1
`
`-(t,-to)u(t,-to)
`
`1
`
`The following set of parameters was used: m = 2, ko = 50 mg h- I , to = 4 . 0 h,
`cl=O.l L-I, ~2=0'025L-', XI =2*0h-', X2=0'2h-',
`
`Dosing condition 3: first-order absorption
`Assuming a first-order drug-absorption rate k, from a drug dosage
`formulation of dose DA, the plasma concentration, CA(t), of drug at time t
`can be approximated by the following equation:
`
`The following equations for AUC were derived:
`
`The following set of parameters was used: m = 2, D, = 100 mg, k, = 3 -0 h- l,
`~ 1 = 0 . 1 L-', C2=O*O25L-', X1=2*0h-', h2=0*2h-'.
`
`Dosing condition 4: consecutive first-order drug release and
`first-order absorption
`Assuming a first-order drug-release rate k, from an oral dosage form of DRA,
`and assuming a first-order drug-absorption rate k,, the plasma concentration,
`CRA(t), of drug at time t can be approximated by the following equation:
`
`where k, # k, # Xk.
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`41
`
`The following equations for AUC were derived:
`
`The following set of parameters was used: rn = 2, DRA = 200 mg, kr = 1.0 h-l,
`ka=3.0h-', cl=O.l L-', ~Z=0*025L-', X1=2*0h-', Xz=0*2h-'.
`
`Dosing condition 5: consecutive zero-order drug release and
`first-order absorption
`Assuming a zero-order drug-release rate k, from the controlled-release oral
`dosage form of dose DZA , and assuming a first-order drug-absorption rate k,,
`the plasma concentration, CZA(t), of drug at time t can be approximated by
`the following equation:
`
`where to = DzA/kz, and Xk # k, .
`The following equations for AUC were derived:
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`42
`
`Z . YU AND F. L. S. TSE
`
`+ k 2 [
`
`k = 1
`
`ckka
`
`t o ) I)
`
`The following set of parameters was used: rn = 2, DzA = 300 mg, k, = 25 mg h- I,
`ka=3.0h-', ~ 1 = 0 * 1 L-I, ~ 2 = 0 * 0 2 5 L - ' , X1=2*0h-', X2=0.2h-l.
`Two different sampling conditions were tested for each dosing case, one
`with frequent blood collection for analysis and the other with relatively few
`blood samples.
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
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`43
`
`ALGORITHMS
`
`Pharmacokinetic experiments typically provide the value of the function C(ti)
`at a set of time points ti (i= 1, . . . , n). The purpose of using an approximating
`function is to provide an analytic expression for C ( t ) so that its value at any
`arbitrary time t can be calculated. Interpolation schemes must model the function
`between the known points C ( t i ) by some plausible functional form, which
`should be sufficiently general so as to be able to approximate the many classes
`of functions that might arise in practice.
`
`Algorithm I : the linear trapezoidal rule
`The linear trapezoidal method is based on linear interpolation between two
`adjacent data points. Assuming tj< t < tj+ , where j = 1, . . . , n - 1, the
`algorithm is described by the following equations:
`
`Algorithm 2: the log trapezoidal rule
`The log trapezoidal method is based on the assumption that C ( t ) values vary
`exponentially between two adjacent data points. Assuming ti< t < ti+ , where
`j = 1, . . . , n - 1, defining E as the function tolerance, the algorithm is
`summarized as follows.
`If C(tj) < E then
`
`else if C(tj+ 1 ) < E then
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`44
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`Z. YU AND F. L. S. TSE
`else if IC(tj+l)-C(tj)I < E then
`
`C(t) = C(tj)
`
`AUC5" = (tj, - t,)C(tj)
`
`' j
`
`else
`
`where
`
`end if.
`
`Cubic spline methods
`When a smooth function is to be approximated locally, polynomials are the
`approximating functions of choice because they can be evaluated, differentiated,
`and integrated easily, but if a function is to be approximated on a larger interval
`with many data points, a high-degree approximating polynomial may have to
`be chosen, whose oscillatory nature can produce an inaccurate approximation.
`The alternative is to subdivide the interval of approximation into sufficiently
`small intervals so that, on each such interval, a polynomial of relatively low
`degree can provide a good approximation to the function. Cubic splines are
`smooth, fourth-order piecewise polynomial functions. The goal of cubic-spline
`interpolation is to obtain an interpolation formula that is smooth in the first
`derivative, and continuous in the second derivative, both within an interval and
`its boundaries. The cubic-spline function is given by
`
`or
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`45
`
`where &!Rn
`is its break-point sequence, cii are its local polynomial coefficients,
`and i = 1, . . . , n. Two IMSL/IDLTM cubic-spline interpolation functions,
`CSINTERP and CSSHAPE, are evaluated in this study.
`
`Algorithm 3: cubic-spline interpolation with the not-a-knot condition
`In this algorithm, the IMSL/IDLTM function CSINTERP is applied using
`the default settings, and the interpolant is determined by the not-a-knot
`condition,1° i.e., the first and the last interior knots are not active. This
`algorithm is useful when one knows nothing about end-point derivatives.
`
`Algorithm 4: complete cubic-spline interpolation
`The function CSINTERP allows the user to specify various end-point
`conditions, such as the value of the first or second derivative at the right and
`left points. When the first derivatives at both the right and left end points are
`specified, the resulting spline interpolant is called the complete cubic-spline
`interpolant.'O In this algorithm, the IMSL/IDLTM function CSINTERP is
`used; the first derivatives at both the right and left end points are specified using
`the keywords ILEFT, LEFT, IRIGHT, RIGHT:
`
`Algorithm 5: shape-preserving cubic-spline interpolation
`The function CSSHAPE is designed so that the shape of the curve matches
`the shape of the data. In this algorithm, the IMSL/IDLTM function CSSHAPE
`is used using the default settings. This computation is based on a method by
`Akima" to reduce wiggles in the interpolant and the end-point conditions are
`automatically determined by the prograrn.'OJ1
`
`Algorithm 6: concavity-preserving cubic-spline interpolation
`This algorithm is based on the function CSSHAPE using the keyword
`CONCAVE. The interpolation scheme is designed to preserve the convex and
`concave regions implied by the data. 12,13
`The numerical integrations for AUC in algorithms 3-6 were calculated using
`IMSL/IDLTM function SPINTEG, with double precision throughout the
`computing.
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`46
`
`Z. YU AND F. L. S. TSE
`
`1
`
`2
`
`4
`
`5
`
`~
`
`0.00
`0.10
`0-25
`0.50
`1.00
`1.50
`2.00
`3.00
`4.00
`6.00
`8-00
`12.00
`18.00
`24 - 00
`
`Table 1. Simulated concentration-time data for sampling condition 1
`Time (h)
`Dosing condition
`3
`Concentration (mg L-I)
`0.000 OOO
`0*000 OOO
`0.000 OOO
`0.311 138
`2.978 574
`0-576 931
`1.477031
`5 * 307 592
`1.288489
`3 * 787 682
`6.168 480
`2-175068
`6.576 939
`4.626 117
`3.294595
`6-938 698
`3.114 920
`30995418
`6.262 104
`2.263 967
`4.514711
`1 - 540 361
`4- 557 244
`5-313730
`3 * 366 732
`1-213423
`5.940 855
`2.067 682
`0.806 954
`2-352 810
`1 -358 898
`0.540 798
`1.547 291
`0-607 613
`0-242 995
`0.694866
`0.182 972
`0.073 189
`0-209290
`0 * 022 044
`0.055 110
`0.063 037
`
`6.250 000
`5-318 902
`4.221690
`2.970444
`1.700090
`1.174958
`0.929478
`0.698408
`0.563 339
`0.376 523
`0.252 371
`0.113397
`0.034155
`0.010 287
`
`0~ooo000
`0.040 251
`0.201 945
`0.573 494
`1.261 788
`1 *738 133
`2.068 691
`2.528 502
`2-869 309
`3-366 514
`3.699007
`4-071 257
`0.917 000
`0-276 188
`
`IMPLEMENTATION AND DISCUSSION
`
`Concentration-time data simulated under the frequent and infrequent sampling
`conditions are shown in Table 1 and Table 2, respectively. The exact AUC
`values at each time interval and the total AUC values for five different tested
`dosing conditions under sampling condition 1 are presented in Table 3.
`The sums of absolute differences between the theoretical and the calculated
`AUC values at each time interval for each of the six different algorithms
`under sampling condition 1 for five different tested dosing conditions are
`summarized in Table 4, while the percentage differences between the theoretical
`and the calculated total AUC values are summarized in Table 5 . Rankings
`
`(h) data for sampling condition 2
`Table 2. Simulated concentration (mg L-')-time
`Dosine condition
`3
`
`1
`
`2
`
`4
`
`5
`
`t
`t
`C
`C
`t
`C
`0.00 0~000OOo 0.00 0.000000
`0.00 5.318 902"
`1.00 3.294595 0.25 5.307592
`0.10 5.318 902
`2.00 4.514711 0.50 6.168480
`0-25 4.221 690
`0.50 2.970444
`3.00 5.313730 1.00 4.626 117
`4-00 5.940855 2.00 2.263967
`1.00 1-700090
`6.00 2.352810 4.00 1.213423
`4.00 0.563339
`8.00 1.547291 8.00 0.540798
`8.00 0.252371
`24.00 0.063037 24.00 0.022044
`24.00 0.010287
`aConcentration value at time zero was set as the same value as the first measurement.
`
`t
`t
`C
`0.00 0*000OOo 0.00
`0-25 1.477031 1.00
`0-50 3.787682 3.00
`1-00 6.576939 6.00
`2.00 6.262104 8-00
`4.00 3.366732 12.00
`8.00 1.358898 18.00
`24.00 0.055 110 24.00
`
`C
`0.000 OOO
`1.261 788
`2.528 502
`3.366 514
`3.699007
`4-071 257
`0.917 OOO
`0.276 188
`
`~
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
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`47
`
`Table 3. Exact AUC values (mg h L-I) for sampling condition 1
`Dosing condition
`3
`0.161 005
`0.646 777
`1 *486 193
`2.753 175
`1.905 380
`1.322235
`1.839 242
`1 .363 229
`1.988 821
`1.329 970
`1 -489 000
`0.849 030
`0.255 723
`17.389780
`
`Time (h)
`
`0.0-0.1
`0.1-0.25
`0.25-0.5
`0.5-1 *o
`1 '0-1 ' 5
`1.5-2.0
`2.0-3.0
`3.0-4.0
`4.0-6.0
`6-0-8.0
`8 * 0- 12.0
`12.0-18.0
`18.0-24.0
`Total
`
`1
`0.576 931
`0-711 558
`0-886 578
`1 * 119 527
`0.700 824
`0.519 292
`0-799 020
`0-627 125
`0 * 926 665
`0 * 620 626
`0 * 694 866
`0.396 214
`0.119 337
`8.698564
`
`2
`0 * 029 622
`0.141 961
`0.439 436
`1.393 487
`1 .833 298
`2.132 593
`4.933 066
`5.638 487
`6.899 981
`3.825 386
`4.258 353
`2.427 882
`0.731 264
`34.684816
`
`4
`5
`0.001 394
`0-010 872
`0.017 082
`0- 127 661
`0.095 868
`0.661 735
`0.467 312
`2.717 094
`0.757 867
`3-449 001
`0-956 110
`3.321 064
`2-313 263
`5.383 344
`2 - 705 641
`3.916 970
`6.269 454
`5 * 276 692
`7.087 639
`3.368 724
`15.638 751
`3.729285
`2.122 701
`11.604 721
`3.203 956
`0-639 308
`34-724450 51-119059
`
`Table 4. The sums of absolute differences (mg h L-I) between the theoretical and the
`calculated AUC values at each time interval under sampling condition 1
`Algorithm
`3
`2
`4
`0.019040
`0.016950
`0.048851
`1.009292 4-266334 30654290
`0.248722
`0.058419
`00086458
`0.452 012
`0.094 399
`0.099 944
`1.370 362
`8.503 419
`4.870 534
`
`Dosing
`condition
`1
`2
`3
`4
`5
`
`1
`0.228 039
`2-145 950
`0-529 968
`1 a046 329
`3.934 579
`
`6
`5
`0.011733
`0.036434
`10930124 2.665773
`0-175728 0.062724
`0.202 004
`0.063 779
`5.087 574
`5.667 770
`
`Table 5. The percentage differences between the theoretical and the calculated total AUC
`values under sampling condition 1
`Algorithm
`Dosing
`condition
`3
`4
`2
`1
`0.561 593 0.015 819 0.009524
`1
`2.621 573
`2.089683 6.076276 5.009921
`2
`5.718281
`0.074832 0.051952
`1'396026 -0.657009
`3
`-0.811 195 0.059466 0.052822
`4
`1.743006
`1.593731 3.063319 7.516405
`6.929438
`5
`
`6
`5
`0.153 819 -0.035 082
`5.102510
`4.812269
`-0.058742
`0*010504
`0.146019 -0.006654
`60897057
`6.476168
`
`based on the sums of absolute differences and the percentage differences are
`shown in Table 6 and Table 7 respectively. Figure 1 shows on a semilogarithmic
`scale the simulated concentration-time profiles for five different tested dosing
`conditions. Figures 2-6 show the interpolation features of six different
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`48
`
`2. YU AND F. L. S . TSE
`
`Table 6. Rankings based on the sums of absolute differences under
`sampling condition 1
`Algorithm
`3
`4
`2
`3
`6
`5
`1
`3
`2
`3
`3
`6
`17
`17
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`6
`3
`6
`6
`2
`23
`
`2
`5
`1
`5
`5
`1
`17
`
`5
`4
`2
`4
`4
`4
`18
`
`6
`1
`4
`2
`1
`5
`13
`
`Table 7. Rankings based on the percentage differences under
`sampling condition 1
`Algorithm
`3
`4
`2
`1
`6
`3
`2
`4
`3
`2
`2
`6
`17
`14
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`6
`5
`6
`6
`5
`28
`
`2
`5
`1
`5
`5
`1
`17
`
`5
`4
`4
`3
`4
`4
`19
`
`6
`3
`2
`1
`1
`3
`10
`
`Bolus IV
`
`First-order Absor tion
`
`E
`
`1.00
`
`D
`: 0.10
`
`0 10
`
`0 01
`0
`
`5
`
`15
`10
`Time (hr)
`
`20
`
`25
`
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`’~~~
`
`0.01 0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`0.01 w
`5
`15 20 25
`10 15 20 25
`5
`10
`0
`0
`Figure 1. Simulated concentration-time profiles on a semilogarithmic scale for five different
`dosing conditions
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`49
`
`0
`
`5
`
`10 15 20
`Time (hr)
`
`25
`
`0
`
`5
`
`10
`
`15 20
`
`25
`
`0
`
`5
`
`10 15 20 25
`
`10.00
`
`1 .oo
`
`0.10
`
`10 00
`
`1 00
`
`0 10
`
`0 01
`0.01
`5 10
`25
`15
`0
`20
`20 25
`10 15
`5
`0
`10 15
`20 25
`5
`0
`- .
`Figure2. Interpolation features of six different interpolation methods on a semilogarithmic Gale
`for dosing condition 1 under sampling condition 1
`
`10 00
`
`- J 2 1.00
`- C - 0
`
`: 0 10
`c
`c 0 U
`
`5
`
`10
`15 20
`Time (hr)
`
`25
`
`0
`
`5
`
`10
`
`15 20 25
`
`0 01
`0
`
`10.00
`
`100
`
`0 10
`
`0.01 - 0
`
`10 15 20 25
`
`5
`
`0.01 L
`0.01
`5
`15 20 25
`5
`0
`10
`15 20
`0
`10
`25
`25
`5
`20
`0
`10
`15
`Figure 3. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 2 under sampling condition 1
`
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`50
`
`Z . YU AND F. L. S. TSE
`
`10.00
`
`2 -
`
`J
`E 1.00
`c
`0
`I e
`: 0.10
`c
`
`0 U
`
`0.01
`0
`
`10.00
`
`1 .oo
`
`0.10
`
`5
`
`15 20
`10
`Time (hr)
`
`25
`
`0.01
`0
`
`5
`
`10
`
`15 20 25
`
`Not-o-knot Condition
`10.00
`
`1 .oo
`
`0.10
`
`0.01
`0
`
`5
`
`10
`
`15 20 25
`
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`51
`
`interpolation methods for these dosing conditions under sampling condition 1.
`The results shown in Table 6 and Table 7 indicate that with relatively frequent
`blood collection the concavity-preserving cubic-spline interpolation method
`performed the best, the log-trapezoidal-rule method performed satisfactorily,
`while the linear-trapezoidal-rule method was the least acceptable in terms
`of both the percentage differences between the theoretical and the calculated
`total AUC values and the superimposability of the concentration-time
`profiles. The exact AUC values at each time interval and the total AUC
`values for five different tested dosing conditions under sampling condition 2
`are presented in Table 8. The sums of the absolute differences between
`the theoretical and the calculated AUC values at each time interval for
`each of the six different algorithms under sampling condition 2 for five
`different tested dosing conditions are summarized in Table 9, while the
`percentage differencces between the theoretical and the calculated total
`AUC values are summarized in Table 10. Rankings based on the sums
`of the absolute differences and the percentage differences are shown in
`Table 11 and Table 12 respectively.
`
`Linear Trapezoidal Rule
`' " " '"'7
`lO.OOf---rCp'
`
`10.00
`
`u
`
`- 3 1.00
`c 0 - e
`: 0.10
`
`5 10 15 20
`Time (hr)
`
`25
`
`0 V
`
`0.01
`0
`
`10.00
`
`1 .oo
`
`0.10
`
`Not-a-knot Condition
`10.00
`
`1 .oo
`
`0.10
`
`5
`
`10 15 20 25
`
`0.01
`0 5 10 15 20 25
`
`10.00
`
`1 .oo
`
`0.10
`
`100
`
`0 10
`
`0 01
`0
`
`10.00
`
`1 .DO
`
`0.10
`
`0.01 ,....,....I
`0 5 10 15 20 25
`
`0.01
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`0.01
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`Figure 6. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 5 under sampling condition 1
`
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`51 * 119 059
`
`Total
`
`34.724 450
`
`Total
`
`17.389 780
`
`Total
`
`34-684 816
`
`Total
`
`8-698 564
`
`Total
`
`3.203 956 8
`11.604721
`15-638 751
`7.087639 r
`8.975095 7
`4.027240
`0.581 657 9
`2
`t (h)
`AUC
`N
`Dosing condition 5
`
`18.0-24.0
`12.0-18.0
`8.00-12.0
`6-00-8-00
`3-00-6-00
`1-00-3.00
`0-00-1.00
`
`6.491 294
`8.645 416
`9.300 314
`6.770065
`2.717094
`0.661 735
`0.138 533
`
`8.00-24-0
`4-00-8*00
`2.00-4.00
`1.00-2.00
`0*50-1*00
`0.25-0.50
`0-00-0.25
`
`8*00-24*0 2.593 753
`4*00-8-00
`3.318 791
`2-00-4-00
`30202471
`1-00-2*00 3.227615
`0-50-1*00
`2.753 175
`1.486 193
`0.25-0.50
`0.00-0.25
`0.807 782
`
`7.417499
`3 * 825 386
`6.899 981
`5.638 487
`4-933 066
`3.965 891
`2.004 505
`
`8 '00-24 '0
`6.00-8.00
`4-00-6.00
`3*00-4-00
`2-00-3-00
`1.00-2.00
`0.00- 1 -00
`
`1 -210 418
`1.547 291
`2.646 261
`1 .I19 527
`0.886 578
`0.711 558
`0.576 931
`
`8 * OO-24 * 0
`4.00-8 -00
`1 40-4-00
`0.50- 1 -00
`0-25-0-50
`0 * 10-0 * 25
`0.00-0.10
`
`t (h)
`AUC
`Dosing condition 4
`
`t (h)
`AUC
`Dosing condition 3
`
`t (h)
`AUC
`Dosing condition 2
`
`t (h)
`AUC
`Dosing condition 1
`
`Table 8. Exact AUC (mg h L-') values for sampling condition 2
`
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`53
`Table 9. The sums of the absolute differences (mg h L- l ) between the theoretical and
`the calculated AUC values at each time interval under sampling condition 2
`Algorithm
`3
`2
`1
`36.563279
`0.510838
`1.833432
`7.382011
`1.365817 251.795755
`2-841519 0.526476
`10.050050
`6.481359
`0.891031
`430567994
`4.274760
`1.853597
`8.669344
`
`6
`4
`5
`1.543015
`4.509087 0.424415
`8.005442
`13.845556 2.881438
`1.952117 0-599369 2.801483
`3.155182 2-857008 1-742771
`4-934047 5.262025 5.761025
`
`Dosing
`condition
`1
`2
`3
`4
`5
`
`Table 10. The percentage differences between the theoretical and the calculated total
`AUC values under sampling condition 2
`Algorithm
`6
`4
`5
`3.474208 -15.078697
`-36.022054
`7.133887
`19.705576
`35.052609
`-13.116950
`-1.912744
`-6.356301
`6.788858 -6.655332
`2.844657
`6-621915
`7.647732
`6.947657
`
`Dosing
`condition 1
`3
`2
`4.837070 -400.490639
`1 20.041813
`1.071 170
`720.559478
`2 18.697052
`3 13.458668 -0.097365
`-54.278646
`4 15.899756 -0.831601
`121*710407
`6.445799
`0.838635
`3-231076
`5
`
`Table 11. Rankings based on the sums of absolute differences under
`sampling condition 2
`Algorithm
`3
`4
`6
`5
`6
`5
`6
`3
`6
`4
`6
`3
`30
`20
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`4
`3
`5
`5
`2
`19
`
`2
`2
`1
`1
`1
`1
`6
`
`5
`1
`2
`2
`3
`4
`12
`
`6
`3
`4
`4
`2
`5
`18
`
`Table 12. Rankings based on the percentage differences under
`sampling condition 2
`Algorithm
`3
`4
`6
`5
`6
`5
`6
`3
`6
`4
`2
`6
`26
`23
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`4
`3
`5
`5
`3
`20
`
`2
`2
`1
`1
`1
`1
`6
`
`5
`1
`2
`2
`3
`5
`13
`
`6
`3
`4
`4
`2
`4
`17
`
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`54
`
`Z . YU AND F. L. S. TSE
`
`.~
`
`Time (hr)
`
`Shope-preserviny-
`1 0 . 0 o r '
`
`'
`
`' ~ '
`
`'
`
`' " '
`
`'
`
`'
`
`' 1
`
`0 01
`25
`5
`15 20
`0
`10
`o
`25
`5
`10
`20
`15
`0
`5
`10
`25
`15 20
`Figure 7. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 1 under sampling condition 2
`Not-a-knot Condition
`
`10.00
`
`1 .oo
`
`0.10
`
`10.00
`
`1 .oo
`
`0.10
`
`10 00
`
`v
`
`h 2 2 1 0 0
`c 0 - e
`: 0 1 0
`
`0 V
`
`001
`0
`
`10 00
`
`1 .oo
`
`0 to
`
`5
`
`10
`
`15
`
`20
`
`25
`
`0.01 w
`
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`0.01
`0
`
`5
`
`10
`
`15
`
`20
`
`25
`
`10.00
`
`1 .oo
`
`0.10
`
`0.01
`001
`o
`15
`5
`25
`20
`10
`25
`20
`5
`10
`15
`0
`5
`25
`20
`10
`15
`0
`Figure 8. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 2 under sampling condition 2
`
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`55
`
`10.oi)
`
`h
`
`v
`
`J p 1.00
`c 0 -
`0
`c
`8
`0 U
`
`-0.10
`
`0.01
`0
`
`5
`
`10 15
`Time (hr)
`
`20
`
`25
`
`10
`
`15
`
`20
`
`25
`
`0.01 0 1 5
`
`0.01 0
`
`5
`
`10
`
`15 20
`
`25
`
`10 00
`
`1 00
`
`0 10
`
`0.01
`0
`20
`25
`15
`10 15 20 25
`5
`0
`10 15
`20 25
`5
`5
`0
`10
`Figure 10. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 4 under sampling condition 2
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`56
`
`Z. YU AND F. L. S. TSE
`
`10.00
`
`- 2 2 100
`c 0 - e
`c
`: 0.10
`
`v
`
`0 U
`
`0 01
`0
`
`5
`
`15
`10
`Time (hr)
`
`20
`
`25
`
`0 5
`
`10
`
`15 20 2 5
`
`0.01 u 0
`
`5
`
`10
`
`15 20 25
`
`Complete Cubic Spline
`
`1000[]
`
`10 00
`
`100
`
`0.10
`
`10 00
`
`100
`
`0 10
`
`0.01 u 3
`
`0.01
`0
`
`5
`
`10
`
`15
`
`20
`
`5
`
`10
`
`15 20 25
`
`25
`
`0 01
`0 5
`
`10
`
`15
`
`20
`
`25
`
`Figure 11. Interpolation features of six different interpolation methods on a semilogarithmic scale
`for dosing condition 5 under sampling condition 2
`
`Figures 7- 11 show the interpolation features of six different algorithms for
`the five tested dosing conditions under sampling condition 2. The results shown
`in Table 11 and Table 12 indicate that with relatively few data points the
`log-trapezoidal-rule method performed very satisfactorily. This is because
`of the inherent stability of the method. Unsurprisingly, the cubic-spline
`interpolation methods performed unsatisfactorily, mainly because of insufficient
`number of data points and the large time interval before the last data point.
`In order to gain more insight into the performance of these numerical integration
`
`Table 13. The sums of absolute differences (mg h L-') between the theoretical and the
`calculated AUC values at each time interval under sampling condition 2, based on data
`without the last time interval
`Algorithm
`4
`3
`1.664497
`1.761 073
`2.843 900
`2.954 775
`0.761504
`0.452073
`0.793 138 0.565 990
`4.443029
`5.223748
`
`6
`5
`0.434025
`0.417517
`2.540 317
`1.927 932
`0.492401
`0.263032
`0.531 859
`0.432044
`4-501937 4.603410
`
`Dosing
`condition
`-_
`1
`2
`3
`4
`5
`
`1
`0.942 582
`1 *916 886
`0.932 539
`1 *660 592
`3.899 151
`
`2
`0-510837
`1-363 300
`0.526466
`0.873 758
`1.853561
`
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`NUMERICAL INTEGRATION ALGORITHMS FOR AUC
`
`57
`
`Table 14. The percentage differences between the theoretical and the calculated total
`AUC values under samulina condition 2, based on data without the last time interval
`Algorithm
`
`Dosing
`condition
`2
`1
`5.618934
`1
`11.384660
`1.353320
`2
`3.740415
`3
`2.915985 -0.114520
`4
`2.480530 -1.083990
`6-092914
`0-894641
`5
`
`3
`-0.463770
`3.974520
`1.074863
`-1.812010
`10.638 180
`
`6
`4
`5
`-3.856900 4.127898 -2.706160
`4.240516 5-577618 5.023288
`0.576249 0.025095
`0.189720
`-0.821020
`0.403540 -0.790290
`9.183892 8.998558
`9.480684
`
`Table 15. Rankings based on the sums of absolute differences under
`sampling condition 2, based on data without the last time interval
`Algorithm
`3
`4
`6
`5
`6
`5
`2
`5
`4
`3
`6
`3
`24
`21
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`4
`2
`6
`6
`2
`20
`
`2
`3
`1
`4
`5
`1
`14
`
`6
`2
`4
`3
`2
`5
`16
`
`5
`1
`3
`1
`1
`4
`10
`
`5
`4
`6
`1
`1
`3
`15
`
`Table 16. Rankings based on the percentage differences under
`sampling condition 2, based on data without the last time interval
`Algorithm
`3
`4
`1
`3
`3
`4
`4
`5
`3
`5
`6
`4
`20
`18
`
`Dosing
`condition
`1
`2
`3
`4
`5
`Total
`
`1
`6
`2
`6
`6
`2
`22
`
`2
`5
`1
`2
`4
`1
`13
`
`6
`2
`5
`3
`2
`5
`17
`
`methods, data were reanalyzed without consideration of the last time interval.
`The sums of the absolute differences between the theoretical and the calculated
`AUC values at each time interval are summarized in Table 13 and the percentage
`differences between the theoretical and the calculated total AUC values are
`summarized in Table 14. Rankings based on the sums of the absolute differences
`and the percentage differences, shown in Table 15 and Table 16 respectively,
`indicate that the log-trapezoidal-rule method consistently yielded the best
`performance when applied to pharmacokinetic studies with relatively few blood
`collections, such as those typical in toxicologic trials.
`
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`58
`
`2. YU AND F. L. S. TSE
`
`CONCLUSIONS
`
`The log-trapezoidal-rule method is simple and stable with reasonable accuracy,
`and is recommended as the practical method for numerical interpolation and
`integration in pharmacokinetic studies, especially when the frequency of blood
`sampling is limited. In well designed studies with frequent blood collection,
`however, improved results can be obtained from the concavity-preserving
`cubic-spline interpolation method using the IMSL/IDLTM function CSSHAPE.
`Algorithms based on the combination of the log trapezoidal rule and cubic-spline
`methods based on IMSL/IDLTM function CSSHAPE can be developed to
`enhance overall performance in pharmacokinetic studies.
`
`REFERENCES
`
`1. K. Yamaoka, T. Nakagawa and T. Uno, Statistical moments in pharmacokinetics. J
`Pharmacokinet. Biopharm., 6 , 547-558 (1978).
`2. K. C. Yeh and K. C. Kwan, A comparison of numerical integrating algorithms by trapezoidal,
`Lagrange, and spline approximation. J. Pharmacokinet. Biopharm., 6 , 79-98 (1978).
`3. K. C. Yeh and R. D. Small, Pharmacokinetic evaluation of stable piecewise cubic polynomials
`as numerical integration functions. J. Phurmacokinet. Biopharm., 17, 721-740 (1989).
`4. R. D. Purves, Optimum numerical integration methods for estimation of area-under-the-curve
`(AUC) and area-under-the-moment-curve (AUMC). J. Pharmacokinet. Biopharm., 20,211-226
`(1 992).
`5. W. L. Chiou, Critical evaluation of the potential error in pharmacokinetic studies of using
`the linear trapezoidal rule method for the calculation of the area under the plasma level-time
`curve. J. Pharmacokinet. Biopharm., 6 , 539-546 (1978).
`6. IMSL, IMSL Math/Library, IMSL Inc., Houston, TX, 1991.
`7. IMSL, IMSLADLTM, Visual Numerics Inc., Houston, TX, 1992.
`8. C. de Boor, Spline toolbox for use with MATLAB, The Mathworks, Inc., Natick, MA,
`1992.
`9. P. Veng-Pedersen, Model-independent method of analyzing input in linear pharmacokinetic
`systems having polyexponential impulse response I: theoretical analysis. J. Pharm. Sci., 69,
`298-304 (1980).
`10. C . de Boor, A Practical Guide to Splines, Springer, New York, 1978.
`11. H. Akima, A new method of interpolation and smooth curve fitting based on local procedures.
`J. ACM, 17, 589-602 (1970).
`12