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Drug Metabolism and Disposition Fast Forward
First published on May 26, 2005; DOI: 10.1124/dmd.105.004135

0090-9556/05/3309-1294-1296$20.00
DMD 33:1294-1296, 2005

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A MATHEMATICAL DESCRIPTION OF THE FUNCTIONALITY OF CORRECTION FACTORS USED IN ALLOMETRY FOR PREDICTING HUMAN DRUG CLEARANCE

Huadong Tang1, and Michael Mayersohn

Department of Pharmaceutical Sciences, College of Pharmacy, The University of Arizona, Tucson, Arizona

(Received February 9, 2005; Accepted May 20, 2005)


   Abstract
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 Abstract
Materials and Methods
 Results
 Discussion
 References
 
The functionality of the correction factors, maximum life-span potential (MLP), and brain weight (BrW) used in allometry is mathematically described. Correction by MLP or BrW is equivalent to a multiplication of some constants by the predicted values in humans from simple allometry, but they have no relationship to measured pharmacokinetic parameters in the animal species. The values of these constants (FMLP or FBrW) were calculated for some commonly used combinations of animal species. For all combinations of animal species, the value of FBrW is always greater than that of FMLP with a fold-increase of about 1.3 to 1.9. Different combinations of species give different values of FBrW and FMLP. In addition, the role of correction factors (MLP and BrW) or the "rule of exponents" (ROE) was evaluated. An intrinsic defect in using correction factors or ROE was revealed; different study designs will produce significantly different prediction results. However, ROE may still serve as a useful practical approach in predicting human CL since it was derived from real observations and has been applied to many examples.

Allometric scaling is one of the most widely used approaches in predicting human pharmacokinetic (PK) parameters (CL, Vd, t1/2) from animals. The allometric relationship has been observed to follow the power function: parameter = a (body weight) b, where a and b are a coefficient and an exponent, respectively. The power function is empirical, and it produces prediction errors that are observed in practice. Therefore, various modifications have been proposed to improve these predictions, including in vitro correction (Lave et al., 1997Go), two-term power function approach (Boxenbaum and Fertig, 1984Go), maximum life-span potential (MLP) or brain weight (BrW) correction (Mahmood and Balian, 1996bGo), rule of exponents (ROE) (Mahmood and Balian, 1996aGo), and unbound CL approach (Feng et al., 2000Go). Among these modifications, ROE appears to be widely used in industry. ROE states: 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, the CL x MLP approach is applied; 3) if the exponent is greater than 1.0, CL x BrW is applied. ROE is based on empirical observations, and it suggests that when a large exponent is observed (greater than the generally observed exponent, 0.75), it is likely that CL will be over-predicted. A MLP or BrW correction is thus used to lower this potential over-predicted value. Interestingly, some recent studies by Ward and coworkers (Nagilla and Ward, 2004Go; Ward and Smith, 2004Go) indicated that ROE did not show any advantage over approaches with or without other correction factors. Furthermore, Ward and coworkers showed that the prediction of CL based on the monkey liver blood flow method was superior to ROE (Nagilla and Ward, 2004Go; Ward and Smith, 2004Go). This controversy remains unresolved.

The current work was not designed to solve this controversy or judge which approach is superior. In a previous study (Tang and Mayersohn, 2005Go), a general allometric equation was derived, which is equivalent to the approach used to obtain predicted CL in humans by performing a log-log transformation of the power function followed by linear regression. Based on the general allometric equation, we recognized the functional relationship for the aforementioned correction factors (MLP and BrW) used in allometry. In addition, methods of MLP correction, BrW correction, and ROE used in allometric scaling were evaluated based on those observations.


   Materials and Methods
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 Abstract
 Materials and Methods
Results
 Discussion
 References
 
Theory. The function relating predicted PK parameters in humans (Ppredicted) to animal body weights (W) and observed animal PK parameters (P) was derived from our previous study (Tang and Mayersohn, 2005Go) and is presented in eq. 1,

(1)
where n different animal species are used in allometric scaling, and Pi is the value of the PK parameter in the ith animal species


W is the body weight of a specified animal species.

The scaling of MLP-corrected CL, thus, is equivalent to

(2)
which is further equivalent to

(3)
where

(4)
The scaling of BrW-corrected CL, thus, is equivalent to

(5)
which is further equivalent to

(6)
where

(7)
The common values for W, MLP, and BrW of animal species are listed in Table 1. MLP and BrW were calculated from body weight according to the literature (Boxenbaum, 1982Go). All the mathematical equations were programmed in MATLAB 6.5 (The MathWorks, Inc., Natick, MA).


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  		TABLE 1 Values for animal body weights, MLP, and BrW

 


   Results
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It can be seen from the first part of eqs. 3 and 6 that FMLP and FBrW, are only dependent on the combinations of animal species selected, since body weight, MLP, and BrW in each animal species are fixed constants. The second part of the equation, , is, in fact, the predicted value in humans obtained from simple allometry. The above analyses clearly indicate that the functionality of the correction factors, MLP and BrW, as used in allometry, is actually a multiplication of some constants by the predicted values in humans obtained from simple allometry and has no bearing on measured PK parameters in the animal species. Those values of FMLP and FBrW for some common combinations of animal species are calculated and presented in Table 2.


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  		TABLE 2 The values for FMLP and FBrW for some common combinations of animal species

 

Several conclusions may be reached on the basis of this analysis. First, the values of FMLP vary from 0.33 to 0.62, which means that the predicted value based upon a MLP correction is about 1/3 to 2/3 of the predicted value from simple allometry, depending on the combination of species examined. The values of FBrW vary from 0.17 to 0.47, which means that the predicted value based upon a BrW correction is about 1/5 to 1/2 of the predicted value from simple allometry, depending on the combination of species.

Second, for all combinations of animal species, the value of FBrW is always greater than that of FMLP with a fold-increase of about 1.3 to 1.9. In another words, the magnitude of the correction by BrW is always greater than that by MLP. This phenomenon, that is, the observed larger magnitude of correction by the BrW method than by the MLP method, is theoretically proven here.

Finally, different combinations of species give different values for FBrW or FMLP (Table 2). The most significant difference is that between the combination of (mouse, rat, dog) and (mouse, rat, monkey), which represents almost a 2-fold difference by replacing the dog with the monkey.


   Discussion
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 Abstract
 Materials and Methods
 Results
 Discussion
References
 
The functionality of the correction factors, MLP and BrW, is described here mathematically for the first time. The correction with MLP and BrW in allometry is actually equivalent to multiplying some fixed constants by the values predicted from simple allometry. This statement also reveals an important fact: that prior to the experiments to obtain PK parameters in each animal species, the magnitude of correction by MLP or BrW has already been fixed once the combination of animal species has been chosen. Based on the above statements, we can evaluate the role of correction factors (MLP and BrW) or ROE.

First, we can ask whether it is reasonable that the magnitude of correction by MLP or BrW is fixed once the combination of animal species is chosen, no matter what experimental values are measured in animals and no matter what exponents are obtained from simple allometry. For example, assuming that using the same combination of animal species, the exponents from simple allometry for drug A and drug B are 0.71 and 0.99, respectively. As illustrated above, the magnitude of correction for both drugs using MLP correction is the same. Is this reasonable?

Second, we can ask whether it is reasonable that the magnitude of correction is only dependent on the animal species selected. There are numerous drugs whose PK parameters follow a good allometric relationship across animal species. The prediction of the PK parameters for those drugs, according to the MLP, BrW, or ROE correction methods, would be totally different if other animal species were used. For example, assume that the mouse, rat, monkey, and dog are available species for experimentation with those drugs. One group of investigators chooses mouse, rat, and monkey and obtains exponents between 0.71 and 0.99 or greater than 1.00. The other group chooses the mouse, rat, and dog, and this should result in the same exponents. The first group would correct the predicted value from simple allometry by 0.62, if the exponents are between 0.71 and 0.99 (therefore, corrected by the MLP method) or by 0.47 if the exponents are greater than 1.00 (therefore, corrected by the BrW method). In contrast, the second group would correct the predicted value from simple allometry by 0.32 if the exponents are between 0.71 and 0.99 (therefore, corrected by the MLP method) or by 0.17 if the exponents are greater than 1.00 (therefore, corrected by the BrW method). A 2- to 3-fold difference in prediction values would occur between the two groups only because of choice of animal species. This phenomenon is inevitable and can be considered to be an intrinsic defect associated with use of correction factors or ROE in allometry.

The main purpose of this study was to illustrate the mathematical description of MLP or BrW or ROE correction and to point out the intrinsic defect with use of those correction methods. This intrinsic defect may partially explain the prediction errors made by ROE as noted in many examples (Nagilla and Ward, 2004Go; Ward and Smith, 2004Go). However, this is not to say that the ROE method should be discarded. On the contrary, because ROE was proposed based on real observations, it has been applied to many examples and has shown an improved predictability over that of simple allometry. Since there are no other affirmatively proven methods at this time, which can significantly improve the predictability of CL in humans, the empirical ROE method may still serve as an optional approach in predicting human CL.


   Footnotes
 
Article, publication date, and citation information can be found at http://dmd.aspetjournals.org.

doi:10.1124/dmd.105.004135.

ABBREVIATIONS: PK, pharmacokinetics; CL, clearance; MLP, maximum life-span potential; BrW, brain weight; ROE, rule of exponents.

1 Current address: Bioanalytical Department, Wyeth Research, Pearl River, New York. Back

Address correspondence to: Michael Mayersohn, College of Pharmacy, The University of Arizona, 1703 East Mabel Street, Tucson, AZ 85721. E-mail: mayersohn{at}pharmacy.arizona.edu


   References
Top
 Abstract
 Materials and Methods
 Results
 Discussion
 References
 


Boxenbaum H (1982) Interspecies scaling, allometry, physiological time and the ground plan of pharmacokinetics. J Pharmacokinet Biopharm 10: 201–227.[CrossRef][Medline]

Boxenbaum H and Fertig JB (1984) Scaling of antipyrine intrinsic clearance of unbound drug in 15 mammalian species. Eur J Drug Metab Pharmacokinet 9: 177–183.[Medline]

Feng MR, Lou X, Brown RR, and Hutchaleelaha A (2000) Allometric pharmacokinetic scaling: towards the prediction of human oral pharmacokinetics. Pharm Res (NY) 17: 410–418.

Lave T, Dupin S, Schmitt C, Chou RC, Jaeck D, and Coassolo P (1997) Integration of in vitro data into allometric scaling to predict hepatic metabolic clearance in man: application to 10 extensively metabolized drugs. J Pharm Sci 86: 584–590.[CrossRef][Medline]

Mahmood I and Balian JD (1996a) Interspecies scaling: predicting clearance of drugs in humans. Three different approaches. Xenobiotica 26: 887–895.[Medline]

Mahmood I and Balian JD (1996b) Interspecies scaling: predicting pharmacokinetic parameters of antiepileptic drugs in humans from animals with special emphasis on clearance. J Pharm Sci 85: 411–414.[CrossRef][Medline]

Nagilla R and Ward KW (2004) A comprehensive analysis of the role of correction factors in the allometric predictivity of clearance from rat, dog and monkey to humans. J Pharm Sci 93: 2522–2534.[CrossRef][Medline]

Tang H and Mayersohn M (2005) Accuracy of allometrically predicted pharmacokinetic parameters in humans: role of species selection. Drug Metab Dispos 33: 1288–1293.[Abstract/Free Full Text]

Ward KW and Smith BR (2004) A comprehensive quantitative and qualitative evaluation of extrapolation of intravenous pharmacokinetic parameters from rat, dog, and monkey to humans. I. Clearance. Drug Metab Dispos 32: 603–611.[Abstract/Free Full Text]


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This Article
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