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`0090-9556/05/3309-1297–1303$20.00
`DRUG METABOLISM AND DISPOSITION
`Copyright © 2005 by The American Society for Pharmacology and Experimental Therapeutics
`DMD 33:1297–1303, 2005
`
`Vol. 33, No. 9
`4143/3046936
`Printed in U.S.A.
`
`A NOVEL MODEL FOR PREDICTION OF HUMAN DRUG CLEARANCE BY
`ALLOMETRIC SCALING
`
`Huadong Tang1 and Michael Mayersohn
`
`Department of Pharmaceutical Sciences, College of Pharmacy, The University of Arizona, Tucson, Arizona
`
`Received February 9, 2005; accepted June 2, 2005
`
`ABSTRACT:
`
`Sixty-one sets of clearance (CL) values in animal species were
`allometrically scaled for predicting human clearance. Unbound
`fractions (fu) of drug in plasma in rats and humans were obtained
`from the literature. A model was developed to predict human CL:
`CL ⴝ 33.35 ml/min ⴛ (a/Rfu)0.770, where Rfu is the fu ratio between
`rats and humans and a is the coefficient obtained from allometric
`
`scaling. The new model was compared with simple allometric
`scaling and the “rule of exponents” (ROE). Results indicated that
`the new model provided better predictability for human values of
`CL than did ROE. It is especially significant that for the first time
`the proposed model improves the prediction of CL for drugs illus-
`trating large vertical allometry.
`
`Allometric scaling is widely used in predicting human clearance
`(CL) based on animal data. Since prediction errors are commonly
`observed in the practical application of this approach, various modi-
`fications to allometric scaling have been proposed. These modifica-
`tions include in vitro metabolic data (Lave et al., 1997), correction by
`either maximum life-span potential (MLP) or brain weight (BrW)
`(Mahmood and Balian, 1996b), the “rule of exponents” (ROE) (Mah-
`mood and Balian, 1996a), and scaling unbound CL (Feng et al., 2000).
`Correction by in vitro metabolic data was successful in predicting
`human CL of 10 extensively metabolized drugs (Lave et al., 1997).
`Based on a data analysis of 16 drugs, however, Mahmood (2002)
`concluded that the use of in vitro data obtained from liver microsomes
`to predict hepatic CL in humans did not provide reliable predictions.
`In addition, in vitro metabolic corrections cannot be applied to com-
`pounds eliminated by excretion. Scaling unbound CL across animal
`species improved the prediction for certain compounds (Feng et al.,
`2000); however, it failed to predict well for a few compounds with
`large vertical allometry such as diazepam and valproate. Recently,
`Mahmood (2000) suggested that unbound CL cannot be predicted any
`better than total clearance. Corrections either with MLP or BrW have
`been shown to be inappropriate if they are used indiscriminately,
`which led to the idea of ROE. This rule provides selection criteria for
`use of MLP or BrW, based on the values of the exponents obtained
`from simple allometry (Mahmood and Balian, 1996a). Although ROE
`has been shown to improve the prediction significantly compared with
`simple allometry, this method is still not satisfactory in predicting
`large vertical allometry. More recent studies (Nagilla and Ward, 2004)
`
`This work was presented at the American Association of Pharmaceutical
`Scientists Annual Meeting, Salt Lake City, Utah, October 26, 2003.
`1 Current address: Bioanalytical Department, Wyeth Research, Pearl River,
`New York.
`Article, publication date, and citation information can be found at
`http://dmd.aspetjournals.org.
`doi:10.1124/dmd.105.004143.
`
`found that the corrections using MLP or BrW or the rule of exponents
`in allometric scaling did not result in significant improvements in
`predictions of human CL. Furthermore, they proposed that the mon-
`key liver blood flow approach was superior to the rule of exponents.
`This controversy is currently not resolved (Mahmood, 2005; Nagilla
`and Ward, 2005).
`The coefficients (a) of the power function have been considered
`important in determining the magnitude of CL, because the exponents (b)
`have been shown to be relatively constant, with a typical value close to
`0.75 (Boxenbaum, 1982). Based upon analysis of more than 60 drugs, we
`have observed that the water-octanol partition coefficient (log P) and the
`ratio of unbound fraction (fu) in plasma between rats and humans (Rfu)
`may provide simple rules for anticipating the occurrence of large vertical
`allometry. Based upon these findings, therefore, we attempted to develop
`a new model for predicting human CL.
`
`Materials and Methods
`
`A literature search was performed to obtain animal data for allometric
`scaling of systemic CL (CL used in this article refers to systemic CL) and fu
`ratio in rats and humans. Only data sets including at least three animal species
`were used for scaling. Coefficients and exponents were obtained by fitting
`body weight and CL, CL ⫻ MLP, or CL ⫻ BrW on a log-log scale according
`to the allometric equation: CL or CL ⫻ MLP or CL ⫻ BrW ⫽ a⫻ Wb. CL in
`humans was calculated by using the coefficients and exponents obtained and
`human body weight reported, or by assuming 70 kg (if weight was not reported
`in the publication). MLP was calculated by using MLP ⫽ 10.839 䡠 W0.636 䡠
`BrW⫺0.225 (Boxenbaum, 1982). The rule of exponents was applied as de-
`scribed by Mahmood and Balian (1996a): 1) if the exponent from simple
`allometry is between 0.55 and 0.70, simple allometry is applied; 2) if the
`exponent is between 0.70 and 1.0, CL ⫻ MLP approach is applied; 3) if the
`exponent is greater than 1.0, CL ⫻ BrW approach is applied; 4) if the exponent
`is less than 0.50, simple allometry is applied since none of the approaches
`could improve the prediction. Predictability was assessed by percentage error
`⫺ CLobs)/CLobs] ⫻ 100% for over-prediction and,
`(PE), which is [(CLpred
`⫺ CLpred)/CLpred] ⫻ 100% for under-prediction. A power model is
`[(CLobs
`proposed,
`
`ABBREVIATIONS: CL, clearance; Rfu, ratio of unbound fraction in plasma between rats and humans; MLP, maximum life-span potential; BrW,
`brain weight; ROE, rule of exponents; PE, percentage error; GV150526A, sodium 4,6-dichloro-3-[(E)-3-(N-phenyl)propenamide]indole-2-carbox-
`ylate; PK, pharmacokinetic.
`
`1297
`
`Apotex v. Novartis
`IPR2017-00854
`NOVARTIS 2106
`
`

`

`Downloaded from
`
`dmd.aspetjournals.org
`
` at ASPET Journals on February 28, 2018
`
`1298
`
`TANG AND MAYERSOHN
`
`TABLE 1
`Comparison of predictability of human clearance obtained from simple allometry, the new model equations, and the rule of exponents
`The order of drugs is arranged according to the ascending values of exponent b obtained from simple allometry.
`
`Compounds
`
`References
`
`Rfu
`
`a
`
`b
`
`CLobs
`
`Simple Allometry
`
`Equation 5
`
`ROE
`
`CLpred
`
`PE
`
`CLpred
`
`PE
`
`CLpred
`
`PE
`
`Indinavir
`Remoxipride
`CI-921
`Ofloxacin
`Dofetilide
`Nilvadipine
`Enprofylline
`Ceftizoxime
`Talsaclidine
`Cefoperazone
`Bosentan
`Antipyrine
`Moxifloxacin
`Tamsulosin
`Recainam
`Nicardipine
`Cefmetazole
`Ketamine
`Cefotetan
`Tolcapone
`Moxalactam
`Propranolol
`Sildenafil
`Tirilazad
`Ro 24-6173
`Sematilide
`Cefazolin
`Mofarotene
`Diazepam
`Caffeine
`Cefpiramide
`FCE22101
`NS-105
`Felbamate
`Midazolam
`Dolasetron
`Mibefradil
`Quinidine
`
`Lin et al., 1996
`Widman et al., 1993
`Paxton et al., 1990
`Nakamura et al., 1983; Okazaki et al., 1992; Kawakami et al., 1994
`Smith et al., 1992
`Terakawa et al., 1987; Tokuma et al., 1987; Naritomi et al., 2001
`Tsunekawa et al., 1992
`Murakawa et al., 1980
`Leusch et al., 2000
`Sawada et al., 1984
`Lave, et al., 1996a; Ubeaud et al., 1995
`Boxenbaum and Fertig, 1984; Chiou and Hsu, 1988
`Siefert et al., 1999
`van Hoogdalem et al., 1997; Matsushima et al., 1998
`Scatina et al., 1990
`Higuchi et al., 1980; Naritomi et al., 2001
`Murakawa et al., 1980; Komiya et al., 1981
`Bjorkman and Redke, 2000
`Komiya et al., 1981; Matsushita et al., 1990
`Lave et al., 1996b
`Sawada et al., 1984; Mahmood, 1999
`Chiou and Hsu, 1988; McNamara et al., 1988
`Walker et al., 1999
`Bombardt et al., 1994
`Lave et al., 1997
`Hinderling et al., 1993
`Lee et al., 1980; Sawada et al., 1984
`Lave et al., 1997
`Laznicek et al., 1982; Mahmood and Balian, 1996a
`Lave et al., 1997; Bonati et al., 1984
`Murakawa et al., 1980; Ohshima et al., 1991
`Efthymiopoulos et al., 1991
`Kumagai et al., 1999; Mukai et al., 1999
`Adusumalli et al., 1991; Palmer and McTavish, 1993
`Lave et al., 1997
`Sanwald-Ducray and Dow, 1997
`Lave et al., 1997
`Belpaire et al., 1977; Chiou and Hsu, 1988; Mahmood and Balian,
`1996a
`Cosson et al., 1997
`Sumatriptan
`Izumi et al., 1996, 1997; Mahmood, 1999
`Troglitazone
`Gaspari and Bonati, 1990; Lave et al., 1997
`Theophylline
`Stopher et al., 1988
`Amlodipine
`Kim et al., 1998a,b
`DA-1131
`Bjorkman and Redke, 2000
`Alfentanil
`Nakamura et al., 1983; Mahmood and Balian, 1996a
`Norfloxacin
`Busch et al., 1998
`Meloxicam
`Bjorkman and Redke, 2000
`Methohexitone
`Kaul et al., 1999
`Stavudine
`Amphotericin B Hutchaleelaha et al., 1997; Robbie and Chiou, 1998
`Fentanyl
`Bjorkman and Redke, 2000
`Propafenone
`Puigdemont et al., 1991
`SU 5416
`Sukbuntherng et al., 2001
`Ciprofloxacin
`Siefert et al., 1986; Mahmood, 1999
`Valproate
`Loscher, 1978; Loscher and Esenwein, 1978; Chiou and Hsu, 1988
`ACNU
`Mitsuhashi et al., 1990
`Ethosuximide
`Battino et al., 1995; Mahmood and Balian, 1996a
`Thiopentone
`Bjorkman and Redke, 2000
`AL01576
`McNamara et al., 1988; Park et al., 1988; Brazzell et al., 1990
`Warfarin
`Nagashima and Levy, 1969; von Oettingen et al., 1975
`Ro25-6833
`Richter et al., 1998
`GV150526A
`Iavarone et al., 1999
`APE
`S.D.
`
`ml/min
`1.15 198.00 0.349
`2.94
`36.50 0.362
`9.73
`13.74 0.439
`1.04
`7.58 0.457
`0.98
`19.20 0.462
`0.90
`32.07 0.514
`0.46
`6.34 0.526
`0.99
`11.24 0.563
`0.99
`29.23 0.564
`4.23
`6.77 0.577
`1.00
`17.19 0.578
`1.00
`4.52 0.589
`1.15
`19.34 0.589
`20.00
`61.00 0.594
`1.02
`2.20 0.601
`1.46
`72.34 0.630
`3.73
`12.80 0.633
`0.93 120.50 0.635
`7.78
`7.13 0.639
`1.00
`7.23 0.646
`1.28
`4.97 0.651
`1.15
`49.68 0.662
`1.25
`28.95 0.679
`2.17
`26.50 0.693
`1.30
`68.82 0.716
`0.95
`19.67 0.727
`0.56
`4.79 0.733
`1.00
`11.50 0.733
`5.00
`37.57 0.737
`0.94
`6.36 0.750
`14.59
`4.70 0.755
`0.69
`11.18 0.756
`1.00
`7.90 0.759
`1.19
`1.50 0.766
`1.00
`52.30 0.785
`0.90
`57.44 0.793
`2.00
`66.88 0.804
`1.41
`47.51 0.805
`
`ml/min ml/min
`⫺52
`1325
`872
`43
`119
`170
`89 ⫺113
`188
`53 ⫺178
`146
`105
`136
`30
`⫺96
`560
`285
`59 ⫺426
`315
`⫺2
`126
`123
`⫺82
`588
`321
`74
`79
`6
`140
`200
`43
`43
`55
`28
`154
`236
`53
`48
`761
`1485
`⫺6
`30
`28
`472
`944
`100
`129
`188
`46
`1170
`1787
`53
`30
`108
`256
`⫺5
`118
`113
`⫺18
`93
`79
`⫺27
`1050
`827
`420
`518
`23
`⫺15
`580
`503
`840
`1440
`71
`313
`431
`38
`53
`108
`104
`259 ⫺194
`770
`27
`860
`3087
`137
`154
`12
`19
`6
`553
`⫺79
`494
`278
`141
`199
`41
`30
`39
`30
`798
`1465
`84
`1232
`1670
`36
`532
`2032
`282
`330
`1452
`340
`
`1.01
`0.98
`0.69
`3.00
`1.00
`1.23
`1.02
`0.60
`0.88
`1.00
`2.12
`1.06
`0.33
`0.88
`1.10
`7.04
`1.87
`1.00
`0.57
`0.98
`15.00
`0.58
`13.50
`
`31.71 0.808
`12.44 0.810
`1.89 0.817
`29.00 0.821
`11.58 0.825
`24.85 0.834
`90.02 0.836
`0.35 0.855
`72.75 0.857
`18.80 0.870
`1.03 0.870
`59.66 0.882
`71.07 0.890
`56.00 0.908
`17.65 0.927
`3.66 0.944
`50.71 0.957
`0.60 1.012
`3.67 1.059
`0.35 1.104
`0.37 1.126
`1.10 1.180
`2.00 1.196
`
`1333
`411
`51
`490
`353
`448
`1360
`12
`1000
`572
`30
`730
`1104
`949
`423
`7
`805
`13
`215
`28
`4
`27
`6
`
`982
`383
`61
`949
`385
`859
`3139
`13
`2777
`758
`41
`2525
`3117
`2652
`1085
`202
`2950
`44
`330
`38
`44
`165
`322
`
`⫺35
`0
`19
`94
`9
`92
`131
`13
`178
`32
`38
`246
`182
`179
`157
`2786
`266
`240
`53
`35
`1006
`513
`5266
`323
`850
`
`ml/min
`33
`1757
`95
`232
`44 ⫺332
`154
`5
`330
`214
`⫺7
`522
`251 ⫺25
`217
`72
`452 ⫺30
`48 ⫺54
`298
`113
`107
`148
`293
`90
`79
`64
`60
`101
`673
`43
`86 ⫺50
`1412
`21
`31
`4
`153
`30
`95
`2
`606 ⫺73
`375 ⫺12
`229 ⫺153
`709 ⫺19
`344
`10
`174
`229
`219 ⫺252
`159
`489
`145
`6
`14 ⫺36
`285 ⫺73
`164
`16
`40
`33
`702 ⫺14
`818 ⫺51
`⫺7
`497
`500
`52
`
`474 ⫺181
`236 ⫺74
`72
`42
`191 ⫺156
`220 ⫺61
`337 ⫺33
`1050 ⫺29
`22
`84
`999
`0
`319 ⫺79
`19 ⫺57
`743
`2
`2088
`89
`816 ⫺16
`283 ⫺50
`20
`188
`423 ⫺90
`23
`73
`140 ⫺54
`15 ⫺86
`2 ⫺108
`55
`102
`8
`28
`78
`86
`
`ml/min
`872 ⫺52
`170
`43
`89 ⫺112
`53 ⫺175
`136
`30
`285 ⫺96
`59 ⫺434
`⫺2
`123
`321 ⫺83
`79
`7
`200
`43
`55
`28
`236
`53
`761
`1485
`⫺7
`28
`944
`100
`188
`46
`1787
`53
`108
`260
`⫺4
`113
`79 ⫺18
`827 ⫺27
`518
`23
`503 ⫺15
`477 ⫺76
`135 ⫺132
`47 ⫺13
`84 ⫺817
`466
`1626
`71 ⫺93
`20
`5
`128 ⫺286
`155
`10
`21 ⫺43
`290 ⫺175
`907 ⫺36
`642
`21
`285 ⫺16
`
`319 ⫺318
`178 ⫺131
`42 ⫺21
`324 ⫺51
`104 ⫺239
`253 ⫺77
`912 ⫺49
`8 ⫺50
`⫺2
`980
`466 ⫺23
`17 ⫺76
`384 ⫺90
`550 ⫺101
`971
`2
`270 ⫺57
`60
`757
`⫺3
`785
`8 ⫺63
`37 ⫺481
`30
`7
`4
`0
`39
`44
`132
`2100
`185
`395
`
`CI-921, 9-关关2-methoxy-4-关methylsulphonylamino兴-phenyl兴amino兴-N,5-dimethyl-4-acridinecarboxamide; NS-105, (⫹)-5-oxo-d-prolinepiperidinamide monohydrate; SU 5416, semaxanib; ACNU,
`1-(4-amino-2-methyl-5-pyrimidinyl)methyl-3-(2-chloroethyl)-3-nitrosourea hydrochloride; Ro 24-6173, an N-methyl-D-aspartate receptor antagonist; Ro25-6833, a cephalosporin; FCE22101, a
`penem antibiotic; DA-1131, a carbapenem antibiotic; AL01576, 2,7-difluoro-spiro-9H-fluorene-9,4⬘-imidazolidine)-2⬘,5⬘-dione; APE, average of absolute percentage error.
`
`

`

`and transformed into
`
`CL ⫽ ␣写 Pi
`Log CL ⫽ Log ␣ ⫹ 冘 ␤i 䡠 Log Pi
`
`␤i
`
`MODEL FOR PREDICTING HUMAN DRUG CLEARANCE
`
`1299
`
`(1)
`
`(2)
`
`TABLE 2
`A summary of outliers for predictions of human clearance (PEs greater than
`200%) based on simple allometry, new model equation, and rule of exponents
`
`Methods
`
`APE
`
`N (PE ⬎ 200%)
`
`N (PE ⬎ 500%)
`
`N (PE ⬎1000%)
`
`Downloaded from
`
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`
` at ASPET Journals on February 28, 2018
`
`Simple allometry
`Rule of exponents
`Equation 5
`
`%
`323
`185
`78
`
`11
`11
`6
`
`APE, average of absolute percentage error.
`
`6
`6
`1
`
`5
`3
`0
`
`Second, using the new model (e.g., eq. 5), only six compounds had
`percentage errors over 200%, with 548% for diazepam and 200 to
`300% for the other five. In contrast, 11 compounds using the ROE
`method had prediction percentage errors greater than 200%, with
`2100% for GV150526A, 1626% for diazepam, 1485% for tamsulosin,
`and 200 to 1000% for the other eight compounds (Table 2). Therefore,
`the new model predicted the large vertical allometry with greater
`success compared with ROE.
`Comparisons of the predictability of human CL from simple allom-
`etry with the new model (eq. 5) and ROE may be visualized in Fig. 1
`
`where Pi is the variable for a, b, Rfu, ore ClogP (exponential values of water-
`octanol-water partition coefficient, ClogP). The transformed model was
`screened by a backward step-wise procedure (P value entrance criterion at 0.1
`and P value removal criteria at 0.2) to obtain parameters of statistical signif-
`icance (Intercooled Stata 7.0, Stata Corporation, College Station, TX).
`
`Results
`The interest and rationale for developing a new allometric model
`equation was based on our previous findings that Rfu, combined with
`ClogP, could be used to formulate rules to predict qualitatively the
`occurrence of large vertical allometry in predicting human CL (Tang
`and Mayersohn, 2005, in press). The current study was undertaken to
`create and test a model in which parameters such as Rfu and ClogP, as
`well as coefficient a and exponent b from simple allometry, could
`potentially be useful to quantitatively predict human CL. ClogP was
`removed from the model since it did not add any statistical improve-
`ment. Coefficient a, exponent b, and Rfu were found to be statistically
`significant with P values of ⬍0.001, ⬍0.05, and ⬍0.001, respectively.
`The model equation incorporating these three variables was:
`
`CL ⫽ 36.6 䡠 (ml/min) 䡠 a0.82 䡠 b0.71 䡠 Rfu
`
`⫺0.70共R2 ⫽ 0.82兲
`
`(3)
`
`The exponential value of b (0.71) is close to that of a (0.82) and Rfu
`(0.70). b is relatively constant and varies over a much narrower range
`(⬃ 0.35–1.20) than a (0.31–200) or Rfu (0.33–20); therefore, b was
`not considered to be an important variable. Thus, a and Rfu were used
`as the only variables to redevelop the model, which resulted in the
`simplified eq. 4,
`
`CL ⫽ 33.35 䡠 (ml/min) 䡠 a0.77 䡠 Rfu
`
`⫺0.71
`
`(4)
`
`which retained an R2 of 0.81, indicating that the three-variable model
`does not improve the prediction performance. Values for CL increase
`with a, indicating that the coefficient a from simple allometry is a
`primary determinant of CL. In contrast, CL decreases when Rfu
`increases due to the negative power of Rfu. This inverse relationship
`makes sense in that a higher value for fu in animals compared with
`humans may lead to an over-prediction of CL by simple allometry.
`The inverse functional relationship between fu and CL predicted in
`humans, therefore, may correct the over-predictions caused by signif-
`icant differences in fu between animals and humans.
`The exponents of a and Rfu have very similar absolute values.
`Changing ⫺0.71 to ⫺0.77 for the exponent of the fu ratio only slightly
`affects CL. For example, an Rfu of 10 raised to the power ⫺0.71 is
`0.19, whereas 10 raised to the power ⫺0.77 is 0.17. Most fu ratios are
`smaller than 10; therefore, the equation was further simplified to
`
`CL ⫽ 33.35 䡠 (ml/min) 䡠冉 a
`
`Rfu
`
`冊0.77
`
`(5)
`
`The term, a/Rfu, could be referred as an “fu-corrected a.” The predict-
`ability of CL estimations for eq. 5, as well as for simple allometry and
`ROE, are given in Table 1. The significant improvement in prediction
`performance by the proposed model, compared with ROE, could be
`judged from three perspectives.
`First, the average absolute values of percentage error by eq. 5,
`ROE, and simple allometry were 78%, 185%, and 323%, respectively.
`The significant
`improvement
`in prediction by the new model
`is
`apparent.
`
`FIG. 1. Predicted human clearance as a function of observed human clearance. Pre-
`dicted values are based upon simple allometry (top), the new model equation derived
`here (eq. 5; middle) and the rule of exponents (bottom). The solid lines are the lines of
`identity and the dashed lines represent a range associated with 200% error.
`
`

`

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`
`dmd.aspetjournals.org
`
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`
`FIG. 3. Unbound fraction of drug in plasma (fu) for the average in all animal species
`(top) and in humans (bottom) as a function of fu in rats for 61 compounds. The
`average fu values in animals are based on at least two animal species including rats.
`The solid lines indicate the lines of 5-fold or 0.2-fold. The dotted lines indicate the
`lines of 2-fold or 0.5-fold. The dashed lines indicate the lines of identity.
`
`exponents greater than 0.70. In contrast, the new model equation
`proposed here results in more accurate predictions of human CL and
`a more random pattern of errors.
`
`Discussion
`The use of fu ratio between rats and humans, rather than between all
`animals and humans, was based on our observation that the fu in rats
`is representative of the average fu in animals (Fig.3). In contrast, many
`significant differences between fu in rats and fu in humans were
`observed (Fig. 3). One question could be raised concerning why
`scaling by the unbound CL approach did not provide stable and good
`predictability, because it appears that correcting CL by fu in each
`animal species would be more favorable than just considering only
`rats and humans. One possible explanation could be attributed to the
`serious error underlying data fitting to the power function (Smith,
`1984) and the considerable measurement error of fu, especially for
`highly plasma-bound compounds. When three or more animal species
`are included for scaling unbound CL, the same number of fu variables
`with errors is also introduced into the data fitting, and may generate
`
`FIG. 2. Percentage error in prediction of human clearance as a function of observed
`human clearance. Percentage errors are from predictions based upon simple allom-
`etry (top), the new model eq. 5 (middle), and the rule of exponents (bottom). The
`inset plots are limited to 400% error, which encompasses most of the error range.
`The solid lines indicate 0% error. The dashed lines indicate the range associated
`with 200% error. Symbols: simple allometric slope values less than 0.7 (circle), less
`than 0.7–1.0 (triangle), or greater than 1.0 (rectangle).
`
`and Fig. 2. The dashed line in the graphs represents a 200% error
`range. Simple allometry results in substantial over-prediction of hu-
`man CL for many compounds (especially those with low CL). The
`ROE method considerably reduces that error, whereas it still retains a
`few large over-predictions and leads to biased under-predictions. The
`under-predictions by the ROE method are primarily the result of
`applying MLP or BrW corrections to compounds having allometric
`
`

`

`MODEL FOR PREDICTING HUMAN DRUG CLEARANCE
`
`1301
`
`Let
`
`Y ⫽ log P; X ⫽ log W; a ⫽ 10␣; b ⫽ ␤
`
`Then, eq. A1 can be simplified to
`
`Y ⫽ ␣ ⫹ ␤ 䡠 X
`
`(A2)
`
`Suppose n different animal species are used for allometric scaling.
`Therefore, there are n sets of (X, Y) data to fit using linear regression.
`Based on the method of least squares for linear regression, ␣ and ␤
`can be calculated as
`
`(A3)
`
`(A4)
`
`共Xi ⫺ X៮ 兲共Yi ⫺ Y៮ 兲
`
`(Xi ⫺ X៮ )2
`
`冘 i
`
`⫽1
`
`n
`
`冘 i
`
`⫽1
`
`n
`
`␣ ⫽ Y៮ ⫺ ␤ 䡠 X៮
`
`Substituting Y ⫽ log P, X ⫽ log W into eqs. A3 and A4, and further
`substituting ␣ and ␤ into a ⫽ 10␣, b⫽ ␤, expressions of a and b are
`obtained as
`
`CLpredicted ⫽ B 䡠 共 fu
`
`rat)⫺0.77 䡠 共 fu
`
`human)0.77
`
`(7)
`
`␤ ⫽
`
`n
`
`i⫽1
`
`Ai
`
`Pi
`
`n
`
`i⫽1
`
`Bi 䡠 log Pi
`
`a ⫽写
`b ⫽ 冘
`冉1 ⫺ Bi 䡠 log写
`
`1 n
`
`Ai ⫽
`
`where
`
`Wj冊
`
`n
`
`j⫽1
`
`n⫺1
`
`Wk
`
`log写
`
`n
`
`l⫽1
`n
`
`冣2
`
`Wl
`
`log
`
`Wi
`n
`
`写 k
`
`⫽1
`k⫽i
`
`n 冢log Wk ⫺
`
`冘 k
`
`⫽1
`
`䡠
`
`1 n
`
`Bi ⫽
`
`Downloaded from
`
`dmd.aspetjournals.org
`
` at ASPET Journals on February 28, 2018
`
`(A5)
`
`(A6)
`
`(A7)
`
`(A8)
`
`By assuming a human body weight of 70 kg, the predicted P in
`humans is obtained from
`
`Ppredicted ⫽ a 䡠 70b ⫽写
`
`n
`
`i⫽1
`
`(Ai⫹1.845Bi)
`
`Pi
`
`(A9)
`
`where Ppredicted is the predicted PK parameter in humans and Pi is the
`measured PK parameter in an animal species,
`
`Wj冊
`
`(A10)
`
`冉1 ⫺ Bi 䡠 log写
`
`n
`
`j⫽1
`
`1 n
`
`Ai ⫽
`
`greater error in predicting human values than what is generated from
`the error noted in only one species, the rat, in the new proposed model.
`Here is an example to visualize this concept. Suppose three species,
`mouse (0.03 kg), rat (0.25 kg), and dog (15 kg), are used for allometric
`scaling of unbound CL. The final predicted CL in humans by allom-
`etry can be expressed as:
`
`CLpredicted ⫽ A 䡠 共 fu
`
`mouse)0.36 䡠 共 fu
`
`rat)⫺0.17 䡠 共 fu
`
`dog)⫺1.19 䡠 共 fu
`
`human)1.0
`
`(6)
`
`where A is a function of CL observed in each animal species and the
`body weight of animals (derivation under Appendix). The new model
`can be expressed as:
`
`where B is not equal to A, but is also a function of CL observed in each
`animal species and the body weight of animals. It is obvious that the
`correction of fu in each species incorporates more variance by intro-
`ducing more fu variables compared with both simple allometry and the
`new model.
`Certainly, the new model is empirical, just as are all of the other
`approaches. No solid physiological or biochemical basis could be offered
`at this time. The model proposed here does not consider many other
`potential types of useful information such as in vitro metabolic differ-
`ences across species, which may account for deviations in predictions.
`Therefore, the empirical model that has been proposed should be ex-
`pected, in practice, to result in errors in prediction, such as when a
`significant metabolic/elimination difference is seen across the species
`examined. Nevertheless, the new model was shown to be simple, rea-
`sonable, and more predictive than the currently available approaches. In
`particular, the new model significantly improves for the first time the
`prediction of the occurrence of large vertical allometry noted in humans.
`In summary, a novel and simple model, incorporating a and the fu ratio
`between rats and humans, has been proposed and shown to provide a
`better predictability than the currently available allometric techniques in
`estimating values of CL in humans. Most important, it significantly
`improves the prediction of large vertical allometry.2
`
`Acknowledgments. We thank Dr. Harold Boxenbaum for useful
`suggestions in the development of the new model equations, and Drs.
`Stacey Tannenbaum (Novartis Pharmaceutical Co.) and Iftekkar Mah-
`mood (U.S. Food and Drug Administration) for providing part of the
`allometric data used in the analyses.
`
`Appendix: Derivation of Equation 5
`Part I: Derivation of the Function Relating Predicted PK Parameters in
`Humans (Ppredicted) to Animal Body Weights (W) and Observed Animal
`PK Parameters (Pi)
`The log-log transformation of P ⫽ a 䡠 Wb gives
`
`log P ⫽ log a ⫹ b 䡠 log W
`
`(A1)
`
`2 The proposed model (eq. 5) was tested using one example of large vertical
`allometry (reboxetine), whose data were available to the authors during the
`revision of the manuscript. We predicted an Rfu greater than 5 for reboxetine. The
`data kindly provided by one of the reviewers (courtesy of Pfizer, Inc.) showed fu
`values of 0.17 and 0.02 in rats and humans, respectively, which translate to an Rfu
`of 8.5. Prediction of human CL based upon eq. 5 resulted in a PE of 104%,
`compared with 1395% and 804% based upon simple allometry and the ROE
`method, respectively.
`
`

`

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`
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`
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`
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`
`n⫺1
`
`Wk
`
`log写
`
`n
`
`l⫽1
`n
`
`冣2
`
`Wl
`
`(A11)
`
`log
`
`Wi
`n
`
`写 k
`
`⫽1
`k⫽i
`
`n 冢log Wk ⫺
`
`冘 k
`
`⫽1
`
`䡠
`
`1 n
`
`Bi ⫽
`
`where W is the animal body weight.
`
`Part II: Derivation of Equation 5 Based on Equation A9
`Based on eq. A9, predicted human CL (CLpredicted) using simple
`allometry from a combination of animals such as the mouse (0.03 kg),
`rat (0.25 kg), and dog (15 kg) gives
`
`CLpredicted ⫽ (CLmouse)⫺0.36 䡠 (CLrat)0.17 䡠 (CLdog)1.19
`
`(A12)
`
`Since
`
`CLu ⫽
`
`CL
`fu
`
`(A13)
`
`scaling of unbound CL can be done by substituting CLu for CL,
`resulting in
`
`
`
`冉CLpredictedhuman 冊1.0
`
`fu
`
`
`
`⫽冉CLmousemouse冊⫺0.36
`
`fu
`
`
`
`䡠冉CLratrat冊0.17
`
`fu
`
`
`
`䡠冉CLdogdog冊1.19
`
`fu
`
`(A14)
`
`Therefore, CL predicted in humans by scaling unbound CL can be
`obtained,
`
`CLpredicted ⫽ A 䡠 共 fu
`
`mouse)0.36 䡠 共 fu
`
`rat)⫺0.17 䡠 共 fu
`
`dog)⫺1.19 䡠 共 fu
`
`human)1.0
`
`(A15)
`
`where A is the CL value predicted in humans using simple allometry
`and is equal to
`
`A ⫽ (CLmouse)⫺0.36 䡠 (CLrat)0.17 䡠 (CLdog)1.19
`
`(A16)
`
`By substituting a from eq. A5, the new model equation (eq. 5 in text)
`is
`
`CL ⫽ 33.35 䡠 (ml/min) 䡠冉 a
`CL ⫽ 33.35 䡠 (ml/min) 䡠冉写
`Ai冊0.77
`
`and can be expressed as
`
`Rfu
`
`n
`
`i⫽1
`
`Pi
`
`冊0.77
`䡠冉fu
`
`rat冊0.77
`
`human
`
`fu
`
`(A17)
`
`(A18)
`
`That is,
`
`where
`
`CLpredicted ⫽ B 䡠 共 fu
`
`rat)⫺0.77 䡠 共 fu
`
`human)0.77
`
`(A19)
`
`B ⫽ 33.35 䡠 (ml/min) 䡠冉写
`
`n
`
`Ai冊0.77
`
`Pi
`
`i⫽1
`
`References
`
`(A20)
`
`Adusumalli VE, Yang JT, Wong KK, Kucharczyk N, and Sofia RD (1991) Felbamate pharma-
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`in paediatric patients. Part I: Phenobarbital, primidone, valproic acid, ethosuximide,