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SHORT COMMUNICATION

Utility of the Coefficient of Determination (r²) in Assessing the Accuracy of Interspecies Allometric Predictions: Illumination or Illusion?

Bioanalytical R&D, Drug Safety and Metabolism, Wyeth Research, Pearl River, New York (H.T.); and Department of Pharmaceutical Sciences, College of Pharmacy, The University of Arizona, Tucson, Arizona (M.M.)

(Received May 14, 2007; Accepted August 28, 2007)

	Abstract

Top Abstract Materials and Methods Results and Discussion Appendix References

 
The appropriateness of relying on the coefficient of determination (r²) as a statistical metric for judging the predictability of human clearance (CL) based on interspecies animal data was assessed. An explicit mathematical expression was derived for r² as a function of species body weight and the corresponding measured value of CL. The derived mathematical function demonstrated that r² is numerically large in most instances. Simulations using random CL generated from a common combination of species of mouse, rat, and monkey resulted in an r² of 0.75 as the minimum, and 0.95 and 0.98 at 50th and 75th percentiles, respectively, given that total CL values increase with increasing species body weight. Analysis of literature data also indicated that the prediction accuracy of human CL was not correlated with values of r². Therefore, it is concluded that r² is a limited statistical measure when assessing allometric scaling for the purpose of predicting human CL.

	Materials and Methods

Top Abstract Materials and Methods Results and Discussion Appendix References

 
Theory. Expression of predicted PK parameters among species. The function relating predicted PK parameters () in humans or animal species to animal body weights (W_i, i = 1 to n, where n is the number of animal species) and observed animal PK parameters (P_i) has been described previously (Tang and Mayersohn, 2005). The following highlights the major mathematical functions needed in the subsequent derivations.

(1)

(2)

(3)

where

(4)

(5)

The predicted PK parameter value, _, in the species of interest is obtained from

(6)

Expression for r². The log-log transformation of P = a · W^b gives

(7)

Let

Then, eq. 7 can be simplified to

(8)

r², by definition, is expressed as

(9)

where

(10)

(11)

(12)

Although r² can be explicitly expressed when eqs. 10 to 12 are placed into eq. 9, a visually clearer form is not readily available. Therefore, a common combination of animal species (mouse, rat, and monkey, with body weights assigned as 0.03, 0.3, and 3 kg, respectively), was used for illustration purposes.

(13)

(14)

(15)

(16)

Substituting eqs. 13 to 16 into eq. 9 results in

(17)

(18)

Note that, and this is true most of the time, values for total CL in mouse, rat, and monkey follow the corresponding order of body weight. Let CL_monkey = L · CL_rat, CL_rat = M · CL_mouse, where L, M > 1, r² will be equal to

(19)

Simulating r² values. Although a wide range of CL values for each species is considered here for simulation purposes, in reality CL values usually do not exceed certain limits in each species. The values ranged from 0.001 times liver blood flow (LBF) at the low end to 5 times LBF at the high end, for each species. In total, 10,000 random values of CL from each species were generated from a uniform distribution of [0.001x LBF to 5x LBF], in each species, where the LBF for mouse, rat, and monkey was 5.40, 3.31, and 2.62 l/h · kg, respectively (Davies and Morris, 1993). Thus, 10,000 r² values were computed. In reality, the magnitude of CL follows the order of species body weight; therefore, the r² values obtained were further constrained under the expected order of CL: CL_monkey > CL_rat > CL_mouse. All calculations and simulations were performed using MATLAB, version 6.5 (MathWorks Inc., Natick, MA).

Literature Data Evaluation. A large set of allometric data including CL values in rat, monkey, dog, and human is available (Jolivette and Ward, 2006). The combination of animal species in this data set was different from the combination of species in the above-mentioned example developed under Theory. Due to the close body weights of monkey and dog, it is expected that some CL values in those species will not strictly follow the order of body weights. Therefore, lower r² values are expected from that combination than from simulations obtained from the combination of mouse, rat, and monkey. Nevertheless, the correlation between the prediction performance and the r² values can still be assessed.

	Results and Discussion

Top Abstract Materials and Methods Results and Discussion Appendix References

 
Values for r² obtained from the species combination mouse, rat, and monkey are derived from eq. 19. Notice that 2, r² is, therefore, always greater than 0.75 (when ), and equal to 1 (when = 2, or log L = log M). Furthermore, due to the existence of an approximate allometric relationship, that is,

(20)

(21)

the value of L is usually close to that of M. Therefore, one expects that for most of the cases, r² is high and close to 1. Even in some extreme situations, for example, when L = 10 and M = 1000, that is, the CL in rat is 1000-fold higher than that in mouse, whereas the CL in monkey is only 10-fold higher than that in rat, given the same 10-fold of difference in body weights between rat and mouse, or monkey and rat, the resulting r² is still as high as 0.92.

The results from the simulations also indicated that r² values were high in the majority of cases based on the random values for CL in each species. The r² values were highly right-skewed toward larger values (Fig. 1). The r² values for the 50th and 75th percentiles are 0.95 and 0.98, respectively. These results clearly demonstrate that the common use of r², whose values are often considered to be "good" if they are greater than 0.90 or 0.95, is misleading.

View larger version (15K): [in this window] [in a new window]   FIG. 1. Distribution of r2 values computed by random CL generated from a uniform distribution of [0.001x LBF to 5x LBF] in mouse, rat, and monkey.

View larger version (12K): [in this window] [in a new window]   FIG. 2. Correlation between the r2 values and the prediction -fold value for human CL (CLpredicted divided by CLobserved). Based on the allometric data from rat, monkey, and dog (n = 103; Jolivette and Ward, 2006). Values for r2 greater than 0.5 (n = 94) are shown in A, and values greater than 0.90 (n = 64) are shown in B.

In summary, r² has been shown to not be an appropriate statistical measure for allometric scaling. We are not aware of another simple statistical measure that would serve in its place. The purpose of this communication was not to discourage reporting of r² values for allometric relationships. However, we caution that use of r² as a measure for the quality of an allometric relationship is not appropriate. Claiming a good allometric relationship based on r² values greater than 0.90 or 0.95 is not appropriate, and more practically important is the degree of confidence that one has in the predicted human value.

	Appendix

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Symbols Used in Derivations under Materials and Methods a: The coefficient of the allometric power function, P = a · W^b

A_i: Equal to

and used in

which calculates the coefficient (a) of the allometric power function. Note, the PK parameter, P_i, observed in one animal species (i) is raised to its specific exponent, A_i, which is only dependent on the body weights across animal species and bears no relation to observed P_i.

b: The exponent of the allometric power function, P = a · W^b

B_i: Equal to

and used in

which calculates the exponent (b) of the allometric power function. Note, the log P_i is multiplied by its specific scalar, B_i, which is only dependent on the body weights across animal species and bears no relation to observed P_i.

L: Equal to

M: Equal to

n: The number of animal species

P_i: The PK parameter observed in species i

_i: The PK parameter predicted in species i

W_i: The body weight of species i

X_i: Equal to log W_i, and used to transform the allometric power function, P = a · W^b, to linear function, Y = + ß · X

Y_i: Equal to log P_i, and used to transform the allometric power function, P = a · W^b, to linear function, Y = + ß · X

: Mean of Y_i

:Predicted Y_i

: Equal to log a, and used to transform the allometric power function, P = a · W^b, to linear function, Y = + ß · X

ß: Equal to b, and used to transform the allometric power function, P = a · W^b, to linear function, Y = + ß · X

	Footnotes

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

doi:10.1124/dmd.107.016444.

ABBREVIATIONS: PK, pharmacokinetics; CL, clearance; LBF, liver blood flow.

Address correspondence to: Dr. Huadong Tang, Bioanalytical R&D, Drug Safety and Metabolism, Wyeth Research, 401 N. Middletown Rd., Pearl River, NY 10965. E-mail: tangh3{at}wyeth.com

	References

Top Abstract Materials and Methods Results and Discussion Appendix References

 


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

Davies B and Morris T (1993) Physiological parameters in laboratory animals and humans. Pharmacol Res 10: 1093–1095.[CrossRef]

Hu TM and Hayton WL (2001) Allometric scaling of xenobiotic clearance: uncertainty versus university. AAPS PharmSci 3: 1–14.[CrossRef][Medline]

Jolivette LJ and Ward KW (2006) Extrapolation of human pharmacokinetic parameters from rat, dog, and monkey data: molecular properties associated with extrapolative success or failure. J Pharm Sci 94: 467–483.

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

Smith RJ (1984) Allometric scaling in comparative biology: problems of concept and method. Am J Physiol 246: R152–R160.[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]

Tang H and Mayersohn M (2006) A global examination of allometric scaling for predicting human drug clearance and the prediction of large vertical allometry. J Pharm Sci 95: 1783–1799.[CrossRef][Medline]

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