Allometric scaling is one of the most widely used approaches in predicting human pharmacokinetic (PK) parameters [e.g., clearance (CL), V_d, t_1/2] based on values in animals, The basic 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 observed power function is empirical, although there is some possible underlying physiological rationale (Boxenbaum, 1982; Mordenti, 1986). For example, the “3/4 power law” of metabolic rate was theoretically derived from the hydrodynamics and fractal geometry of nutrition-supply network of the organisms (West et al., 1997). However, there are numerous examples of substantial differences between predicted and observed values in humans. Great effort has been focused on how to improve the accuracy of allometric scaling. The various modifications include in vitro correction (Lave et al., 1997), a two-term power function approach (Boxenbaum and Fertig, 1984), maximum life-span potential or brain weight correction (Mahmood and Balian, 1996b), “rule of exponents” (Mahmood and Balian, 1996a), and unbound CL approach (Feng et al., 2000). Unfortunately, none of these modifications is completely satisfactory since there are always deviations from prediction. Basically there are two reasons leading to such deviations. One is the deviations of the values of PK parameters in certain species (animal or human) from the assumed power function. The other is the measurement errors in the reported PK parameters. The former could be considered as a “position error” (though it is not an error), and such an error is fixed for each species. The latter measurement error is a random one.

Power functions are well known for creating substantial errors in data fitting. The log-log transformation of the data will visually minimize the deviations from a regression line. A high R², greater than 0.90 or even 0.95, does not guarantee that all the data points will be close to the regression line. The extrapolation of this regression line to obtain a predicted human value, which is obtained from fitting data based on a limited number of animal species, may have great uncertainty associated with it. It is also well known that the regression process does not treat the weight of each animal species comparably. The measurement errors in a given parameter from an animal species could lead to significant prediction error in humans as a result of fitting this power function. Therefore, it is necessary and desirable to know how quantitatively a measurement error or a position error in a given parameter in animals affects the regression analysis and the ultimate quantitative prediction in humans.

In this report, we have derived a general equation, which describes the mathematical relationship between predicted PK parameters in humans and the body weights of selected animal species and the values of the corresponding measured PK parameters. Using CL as an example, simulations were performed to examine the dependence of the variability of predicted human CL on the variability of animal CL. Finally, data from the literature were examined to demonstrate the species dependence as derived from theory. Based on these results, some suggestions are proposed for the optimal selection of animal species and the application of animal data in allometric scaling.

Materials and Methods

Theory. The function relating predicted PK parameters in humans (P_predicted) to animal body weights (W) and observed animal PK parameters (P_i) is derived in the following sections.

The log-log transformation of P = a · W^b gives

Let Y = log P; X = log W; a = 10^α; b = β. Then, eq. 1 can be simplified to 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, α and β can be calculated as Substituting Y = log P; X = log W into eqs. 3 and 4, and further substituting α and β into a = 10^α; b = β, expressions of a and b are obtained as where By assuming a human body weight of 70 kg, the predicted P in humans is obtained from Direct fitting of power functions with incorporation of a weighting strategy has been shown not to improve the prediction performance by allometric scaling (work not shown). The above log-log transformation and linear regression appears to be the best approach in allometry. In reality, this method is inevitably applied in allometric scaling.

Monte Carlo Simulations. CL was used as an illustrative example of a PK parameter. Typical body weights of animals commonly used in allometry are listed in Table 1. Assuming that there is a perfect allometric relationship, CL = a · W^b, between CL and W, and setting a = 30 ml/min, b = 0.75, “seed values” of CL for each species for Monte Carlo simulations can be obtained (Table 1). Log-normal distribution, N(ln CL_seed + CV), is assumed for CL, where the coefficient of variation (CV) is 30% or 100%. Thirty percent and 100% CV are used to assess the effect of the experimental measurement errors and the overall “errors” (including both measurement errors and position errors) on the prediction performance, respectively. The magnitude of the position errors is considered to be much greater than that of the measurement errors. The arithmetic mean and median values of CL_predicted in humans from these simulations are listed for comparison with the theoretically perfect value (726.0 ml/min) predicted from the power function. For some combinations of commonly used animal species, in addition to the simulations where all the P_i values were variables, simulations were also performed by assuming that only one P_i was variable, whereas the others were held constant (at the seed values). This was done to assess the contribution of each species to the prediction performance. Different combinations of animal species from five species to two species were used for simulation purposes to select optimal combinations of animal species. Percentage errors (PEs), which are [(CL_predicted – CL_observed)/CL_observed] × 100%, for over-prediction and [(CL_observed – CL_predicted)/CL_predicted] × 100%, for under-prediction, were used to assess the prediction performance. All the calculations and simulations were performed with MATLAB 6.5 (MathWorks, Inc., Natick, MA).

Literature Data Experimentation. Twenty-six sets of allometric scaling data for CL from at least three species, including mouse, rat, and dog, were randomly collected from the literature. CL in humans was predicted by allometry. Percentage change is defined as [(CL_{remove-ith-species} – CL_all)/CL_all] × 100% for an increase and [(CL_all – CL_{remove-ith-species})/CL_all] × 100% for a decrease, where CL_all is the CL predicted using all animal species and CL_{remove-ith-species} is the CL predicted by removing the ith species.

Results

Theoretical experimentation. The equation, not only directly depicts the relationship between the predicted PK values in humans and those observed in animals, but more importantly it indicates the dependence of the prediction variability on animal body weights, which determine the value noted in the exponent of P_i. Each animal P_i is raised to a specific power, A_i + 1.845B_i. Prediction of CL is used as an example of a parameter, P, in the following discussion, although the principle can be applied to any PK parameter, such as volume of distribution and half-life. A typical example of an animal species combination, mouse, rat, rabbit, and dog, was used to illustrate the body weight dependence suggested by eq. 9. Substituting body weights from these species into A_i + 1.845B_i gives the equation The exponent (–0.005357) for rat CL is close to 0, indicating that the CL in the rat would have little effect on the prediction of human CL when this specific set of animals is chosen. In contrast, the CL in the dog would be expected to have a large effect on the predicted CL in humans. For example, a doubling of the values of dog CL results in a 0.75-fold increase in the predicted CL in humans, whereas there is little prediction effect on the predicted value in humans even with a 100-fold change in rat CL (Fig. 1).

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Fig. 1.

Fold-range in predicted human clearance as a function of the fold-range in clearance in selected animal species. A hypothetical combination of animal species was considered: mouse (long-dashed line), rat (dotted line), rabbit (short-dashed line), and dog (solid line). Note that log-scales are used.

The roles that different species play in their contribution to the prediction accuracy and variability were also examined by simulations allowing only one P_i to vary at a time. Commonly used species combinations were examined: mouse, rat, rabbit, monkey, dog; mouse, rat, monkey, dog; and mouse, rat, dog (Table 2). The results demonstrate that the prediction of a value in humans is most sensitive to the value in dogs, whereas the rat made essentially no contribution to the prediction. For example, the 30% CV random error in rat CL only generates a mean of 0.01% PE. The mouse also showed a small contribution to the prediction variability. These results are consistent with the results derived from the model equation, which shows that rat CL has a trivial effect on the variability in predicted human CL, whereas a significant effect is observed using the value of CL from the dog.

Literature Data Experimentation. The mouse, rat, rabbit, or dog was individually removed from allometric scaling to examine the effect of deleting one species on the prediction in humans (by using the log-log transformation and linear regression method), since the majority of the 26 allometry data sets used these four species. The rat contributed virtually nothing to the prediction, because all the values of percentage change were very close to 0 after removal of the rat. In contrast, removal of the dog from the allometry resulted in a significant change in predicted CL values in humans (Table 3, Fig. 2).

Discussion

The equation that has been derived here offers insight into allometric analysis in that it describes quantitatively the dependence of prediction variability on each animal species (i.e., species weight). An immediate practical significance of these findings will allow investigators to recognize such animal species, or body weight dependence, on the predicted human value and permit design of better or more optimal experiments. For example, large variability in the data for dog PK parameters would have a high potential risk for producing large deviations in predicted human values, whereas variability in rat data can essentially be ignored. Having such a quantitative equation available and realizing the magnitude of the species weight dependence, investigators may increase sample size for the species having the most significant effect on the predicted value to improve its accuracy, and appropriately reduce or eliminate completely the sample size for the species with the least effect. As demonstrated by both theoretical and literature experimentation, rats had no significance in predicting human PK parameters as long as the body weight of the rat is not the smallest in the species used in the allometric relationship. Why, then, have rats been widely, almost inevitably, included in allometric scaling? One reason is that the rat is relatively inexpensive and readily available. The other reason may be that investigators rely on the concept “the more, the better,” without recognizing the magnitude of improvement in prediction brought by adding more species; the rat, in this discussion. In fairness, however, the role of species weight in allometric prediction has not been recognized until now.

It is apparent and not surprising that the more species used in establishing an allometric relationship, the better the human prediction will be. However, it is costly, time-consuming, and not realistic to design allometric experimentation to include five or more species. A practical and economical approach to minimize the number of animal species is to recognize the role of species weight, while still achieving the desired prediction goal. Monte Carlo simulations using different combinations of animal species were performed to select the “best” or optimal combination of animal species. The results showed, not surprisingly, that, in general, the more species used, the better becomes prediction performance (Tables 4 and 5). By comparing the percentage errors among different combinations of animal species, the following observations, in terms of prediction performance, could be obtained.

1. The five-species combination is the best, having the smallest PE mean; however, it offers no significant improvement over the four-species combinations.

2. Certain three-species combinations, such as mouse/rat, monkey, and dog, showed a prediction performance similar to that of the four species combinations, whereas some three-species combinations, such as rabbit, monkey, and dog, showed a significantly worse prediction performance.

3. The two-species combination showed the worst prediction performance, especially for the combinations of mouse and rat, rabbit and monkey, or monkey and dog, which should be forbidden combinations in allometric scaling.

4. Comparison of any combination having more than three species including mouse and rat showed that removal of the rat has little effect on the prediction performance, which is consistent with the previous findings.

5. A three-species combination, mouse/rat, rabbit, and dog, or mouse/rat, monkey and dog, may be economically desirable without sacrificing the predictability compared to a five-species combination.

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Fig. 2.

Percentage difference in the predicted human value for clearance (“box and whisker” plots) when one individual species is removed from a combination of three or more species. The inset graph illustrates a limited, but the major, range of percentage difference. Analysis is based on real data taken from the literature (Table 3).

However, investigators should keep in mind that all of the above observations are purely mathematical. The differences or similarities between animal species and humans in terms of anatomy, physiology, biochemistry, etc. are not considered. These differences or similarities would account for different CVs used in the above analyses. Unfortunately, there are still no significant findings, or agreement, regarding these differences/similarities as to what animal species should be used to provide the best prediction in humans.

To summarize, an equation has been derived that relates the accuracy of predicted PK parameters in humans to species weight used in allometric scaling, and the Monte Carlo simulations provided a quantitative approach to appreciate the prediction variability in a species-dependent way (or more generally, body weight-dependent). The awareness that such dependence exists may be helpful in selecting animal species and designing experiments, such as increasing the sample size for species having the greatest effect on the prediction, and reducing or even deleting the species having the smallest effect on the prediction. It is especially noted that rats were found to have no significance in predicting human PK parameters as long as the body weight of rats is not the smallest among the species used in the allometric experimentation. In addition, some economical combinations of three animal species, mouse/rat, rabbit, and dog, or mouse/rat, monkey, and dog, which allow a theoretically reasonable predictability, are proposed.