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Bioanalytical Department, 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 November 14, 2005; accepted December 2, 2005)
). We are especially appreciative of Dr. Mahmood's thorough and thoughtful review of our paper and for the effort taken in preparing his commentary. Dr. Mahmood's commentary consists of essentially four statements (under Results and Discussion). Although they have not been numbered, we will respond to three of these bulleted paragraphs (as they express a difference of opinion) in the order of bullets 2, 4, and 1; the order of importance, as viewed by these authors.
Response to Point 2 Concerning the "Intrinsic Defects of Correction Factors." Dr. Mahmood is stating here his opinion that there is no "intrinsic defect" to the correction factors used in allometric analyses, as our paper has concluded. We believe that Dr. Mahmood has not understood our reasoning and arguments. In the Discussion section of the paper (page 1296), we tried to be clear in stating that what we have called an "intrinsic defect" in the correction factors would be apparent when good allometric relationships are examined. The good allometric relationship means that predictions from simple allometry would be the same or similar when different combinations of animal species are used. We believe that there are numerous examples with such good allometric relationships (e.g., r2 > 0.99; Hu and Hayton, 2001). However, from these examples, different combinations of animal species will yield substantially different predictions of human values when correction factors or the "rule of exponents" (ROE) is applied. We agree that simple allometry would also yield substantially different predictions in humans based on different combinations of animal species as illustrated in Table 1 of Mahmood's commentary (Mahmood, 2006), but this occurs only for compounds without a good allometric relationship. Please note that r2 > 0.90 or even 0.95 does not necessarily mean a good allometric relationship, since log-log transformation would minimize the deviations of data from the regression line. Although Dr. Mahmood did not provide the r2 in Table 1 (Mahmood, 2006), we speculate that the r2 would not be satisfactory, since different combinations yielded substantial differences in predictions in humans, indicating that the clearance values for those compounds were not on the regression lines. Therefore, the examples provided by Dr. Mahmood were not appropriate and would not undermine the "intrinsic defect" that correction factors or ROE are associated with, when they are applied to compounds with a good allometric relationship (in addition to compounds without a good allometric relationship).
Response to Point 4 Concerning the "Dilemma of Applying the Same Correction Factor to Two Allometric Relationships with Different Exponents". We proposed a dilemma when applying the same magnitude of correction to two allometric relationships having different exponents (0.71 versus 0.99; used as an example on page 1295 of our paper). Dr. Mahmood appears not to have followed our reasoning in presenting this dilemma. ROE was proposed by Mahmood and Balian (1996) based on their observation that when a relatively high allometric exponent is obtained, a correction using MLP or BrW to lower the human prediction from simple allometry may be necessary. Furthermore, the correction magnitude with use of BrW has been observed to be greater than that resulting from the application of MLP. We have now proven that point mathematically. The application of correction factors makes sense in that the allometric relationship for clearance, or more generally, metabolic rate, which has been extensively investigated in biology (West et al., 1997), follows a universal three-quarter power law. This is in line with the observation that when a higher exponent is obtained (e.g., greater than 1.0, or greater than 0.71), a greater downward correction needs to be applied (BrW correction is greater than MLP correction), as defined by ROE. This strongly suggested that the error in the prediction of human drug clearance obtained by simple allometry increases with increasing allometric exponent, although it may not increase "proportionally." Therefore, it is reasonable to question the soundness of applying the same correction magnitude for allometries with different exponents, e.g., 0.71 versus 0.99. A further thought, based on the above argument, is: can we adjust the exponents obtained from simple allometry with use of a fixed universal value (0.75, or other values, which could be determined statistically from large sets of data)? The latter proposal makes more sense than ROE. However, this has to be tested using large sets of data in order to be able to provide a practical value, because all of the allometric methods are empirical. This is why we maintain the utility of the ROE approach; its use in practice is certainly warranted at the current time.
Response to Point 1 Regarding the "Beauty of Correction Factors." We acknowledge that correction factors will vary with species—one of the important points of our paper. But we have a concern that it is being considered as an important characteristic of ROE. We have shown that the magnitude of correction by MLP and BrW varied, about 0.326 to 0.622 and 0.172 to 0.474, respectively, for the common combinations of species (Table 2, page 1295, Tang and Mayersohn, 2005). Given that approximately 50% of the predictions in humans by simple allometry were outliers (beyond the 0.5- to 2.0-fold range) (Nagilla and Ward, 2004; Tang and Mayersohn, 2006), we speculate that fixing the correction factor for MLP and BrW at their median values (0.455 and 0.290, respectively) will not cause important changes in prediction performance (neither better nor worse than that achieved per ROE).
In general, although we believe that ROE-based corrections remain valuable in practice, nonetheless, there are several interesting issues as those noted above, which should be taken into consideration, at least philosophically. There may be, and we fully expect there will be, better approaches than the currently available allometric methods, including the application of ROE. The disclosure of improved predictive allometry may need to await a greater insight into what has been an empirically based method.
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ABBREVIATIONS: ROE, rule of exponents; MLP, maximum life span potential; BrW, brain weight.
Address correspondence to: Huadong Tang, Bioanalytical Department, Wyeth Research, 401 N. Middletown Rd., Pearl River, NY 10965-1299. tangh3{at}wyeth.com
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