Interspecies allometric scaling: prediction of clearance in large animal species: Part II: mathematical considerations

Introduction

Within the published literature, there are numerous examples of studies that use published pharmacokinetic data to estimate the allometric relationship between a parameter such as clearance or volume of distribution vs. body weight (W). In this regard, among its uses, it is especially important for estimating an appropriate dose in large animal species for which there does not exist pharmacokinetic or clinical information to support dose estimation (Hunter & Isaza, 2002; Isaza & Hunter, 2004). It is also used to predict a terminal elimination half-life value, which may be applied by extension services such as the FARAD (http://www.farad.org) for estimating withdrawal times when drugs are administered to food-producing animals in an extralabel manner (Craigmill & Cortright, 2002).

Similar to its use for selecting the first time dose in humans (Boxenbaum & DiLea, 1995), interspecies extrapolation serves an important function within the framework of veterinary medicine. While it is now well-established that clearance can be extrapolated from small laboratory animals to man with a fair degree of accuracy, provided the rule of exponents is used (Mahmood & Balian, 1996), it is not known if the clearance from small animals can be extrapolated to large animals with the same degree of accuracy. Despite numerous examples of the successful application of allometric scaling (Mordenti, 1986; Hu & Hayton, 2001), there are also examples of its failure to accurately predict drug pharmacokinetics across species (Riviere et al., 1997).

The variable generally considered to be the most highly predictive for interspecies scaling is total body weight (W) raised to some power. This is because pharmacokinetic parameters are affected by the size and function of the eliminating organ, which in turn reflects an organism's metabolic demands (Boxenbaum, 1982). The general form of the allometric equation used in scaling pharmacokinetic parameters across animals is as follows (Mordenti et al., 1991):

(1)

Where, Y is the pharmacokinetic parameter of interest, W is body weight, and a and b are the coefficient and exponents of the allometry.

In Part I, we examined the accuracy of predicting clearance in large animal species. Unlike predictions for human clearance values, the inclusion of correction factors (brain weight and maximum life span potential) failed to improve predictions in large animal species. In addition, the error associated with the clearance predictions for large animal species tended to be substantially diminished when at least one large animal species was included in the regression analysis. Oftentimes, within the veterinary literature, there has been an assumption that if scaling is based upon an allometric equation associated with a high correlation coefficient (R²), there will also be a high degree of accuracy in the predicted pharmacokinetic parameter. However, the validity of this assumption is rarely tested because of the absence of data in the predicted animal species.

In this report, we examine the factors influencing the accuracy of our clearance predictions by addressing the following questions:

Inaccuracies in the allometric equation may lead to the selection of a sub-therapeutic dose if the dose is too low, a toxic dose if the prediction is too high, or withdrawal time predictions that lead to the presence of violative (potentially unsafe) residues in edible tissues. Therefore, it is important that the mathematical underpinnings associated with these regression procedures be appreciated and appropriate caution exercised. With this objective in mind, we manipulated datasets for which there was extensive species-specific information (i.e. organ weights) in an effort to illustrate the potential sources of error that can influence the accuracy of our allometric predictions. It is our hope that through these examples, investigators can be reminded of the basic principles underlying linear regression analysis and, therefore, allometric scaling.

Methods

Results

For completeness, we include Table 1, which contains the organ weight information provided in the Spector book chapter. Thus, the reader can conduct their own manipulation of these data. We also provide the allometric equations associated with the regression of body weight to organ weight across all 28 animal species in 1-5.

Examining the error structure influencing the accuracy of allometric extrapolations

Prediction errors based upon the degree of extrapolation

The dataset included 28 species whose weights ranged from 0.023 kg (mouse) to 6600 kg (elephant). As seen in 1-5, there is a highly correlated linear relationship between the log weights of the various organs (kidney, heart, lung and liver) and the values of log W of the various animal species (R² > 0.95). Interestingly, brain weight did not scale as well as did the other organ systems. The changes in slope (b) and intercept (a) for the entire dataset vs. the various data subsets are provided in Table 2. From these results, we see that the slopes and intercepts associated with the regression analysis performed on four large animal species are markedly different than that associated with the other data subsets. Particularly noteworthy is the absurdly large intercept seen both for the kidney and the heart (46.5 and 1805 g for the kidney and heart, respectively).

Table 2. Slope and intercept associated with the regression of log₁₀ weight vs. log₁₀ organ weight across the various data subsets

	All species	Small animal species	Large animal species	Cattle + 4 small species	Mouse + 4 large species
Kidney
b	0.85	0.77	0.54	0.82	0.83
a	7.41	6.09	46.5	6.60	6.76
Heart
b	0.98	0.91	0.10	0.93	0.92
a	5.76	5.62	1805*	5.72	8.33

• *This very large intercept value reflects the magnitude of error associated with inclusion of species with similar weights in the scaling procedure.

The data subsets were examined for differences in prediction accuracy as compared with that seen when using all 28 available data points (Tables 3 and 4). The largest prediction errors consistently occurred when allometric extrapolations were based upon a dataset containing solely large animal species. In most cases, of the four subsets examined, large animal organ weight prediction were the most accurate when the regression analysis included four small (mouse, rat, dog, and monkey) plus one large animal species (cattle). Inclusion of only one small animal species in a large animal dataset generally did not perform as well as the latter.

Table 3. Prediction error in kidney weights (observed/predicted values) based upon the allometric equations associated with the respective datasets*

	28 Animal dataset†	Small animal species	Large animal species	1 large + 4 small species	1 small + 4 large species
Dog	0.60		0.64		1.94
Rat	1.19		0.12		1.08
Mouse	1.16		0.06
Monkey	1.03		0.28		1.04
Cattle	0.85	0.46
Buffalo	0.87	0.54		0.89
Zebra	1.11	0.71		1.10
Giraffe	0.71	0.43		0.73
Elephant	1.38	0.79	3.29	1.50	1.77
Human	1.05	0.70	0.60	0.99	1.15

• *When more than one set of values were available on a particular species (e.g. there were three sets of data available for monkey), one set was selected for estimating prediction error under all conditions.
• ^†Error based upon interpolated values.

Table 4. Prediction error in heart weights (observed/predicted values) based upon the allometric equations associated with the respective datasets*

	28 Animal dataset†	Small animal species	Large animal species	1 large + 4 small species	1 small + 4 large species
Dog	1.56		0.04		1.37
Rat	0.88		0.0001		0.74
Mouse	1.09		0.0005
Monkey	0.60		0.01		0.52
Cattle	0.74	0.76
Buffalo	0.94	0.96		1.15
Zebra	2.79	2.85		3.34
Giraffe	0.83	0.85		1.03
Elephant	0.82	0.83	5.94	1.05	0.76
Human	0.80	0.82	0.08	0.93	0.71

• *When more than one set of values were available on a particular species (e.g. there were three sets of data available for monkey), one set was selected for estimating prediction error under all conditions.
• ^†Error based upon interpolated values.

To determine if the large animal species selected for inclusion in the ‘4 small + 1 large’ regression analysis had a substantial influence on the accuracy of the kidney or heart weight predictions, we recalculated the regression line, exchanging either elephant or buffalo for cattle. As seen in Table 5 (kidney) and Table 6 (heart), the results of the regression analysis generated using four small and one large animal species were similar, regardless of the species selected to represent the large animal.

Table 5. Prediction error (observed/predicted) in kidney weights: effect of using elephant, buffalo or cattle as the large species (4 small + 1 large) to generate the allometric equation

species	Obs/buffalo	Obs/elephant	Obs/cattle
Dog
Rat
Monkey
Mouse
Buffalo	0.81*	0.62	0.89
Zebra	1.01	0.80	2.42
Giraffe	0.66	0.50	0.73
Elephant	1.32	0.94*	1.50
Human	0.93	0.77	0.99
Cattle	0.70	0.53	0.77*

• *Back extrapolated values.

Table 6. Prediction error (observed/predicted) in heart weights: effect of using elephant, buffalo or cattle as the large species (4 small + 1 large) to generate the allometric equation

Large species	Obs/Pred
	Buffalo	Elephant	Cattle
Dog
Rat
Monkey
Mouse
Buffalo	0.99*	1.07	1.15
Zebra	2.92	3.13	3.34
Giraffe	0.87	0.95	1.03
Elephant	0.86	0.96*	1.05
Human	0.83	0.88	0.93
Cattle	0.78	0.85	0.92*

• *Back extrapolated values.

Influence of error in Y variable estimate: relationship to W

To ascertain the degree to which one value can influence the regression equations, the kidney weight of the dog, mouse or elephant was sequentially changed (one at a time) by introducing a 100% error (doubling the weight of the kidney). The other species included in the regression analysis (rat and cattle) were not modified. The question being addressed was whether the ratio of kidney weight predicted without error (Y_{no error}) vs. with error varied as a function of where the error was introduced along the Y-axis. The resulting slope and intercept values are provided in Table 7.

Table 7. Relationship between species to which error has been added (introducing a 100% error to kidney weight) and the impact of including that data (with error) in the points used for generating the allometric equation

	b	a	R 2
No error	0.848	5.47	1.0
Elephant error	0.888	5.70	0.996
Dog error	0.849	6.30	0.995
Mouse error	0.808	6.96	0.997

Introducing error into the mouse data and into the elephant data had equal but opposite effects on the slope of the line. However, the error in the mouse data had a greater impact on the value of the intercept, thereby compensating for the change in slope when predicting kidney weights for the large animal species. Conversely, the impact of mouse error on the kidney weight predictions of small animal species was comparable with the impact of elephant error on the predictions in large animal species (Table 8), Thus, when there is a limited number of species associated with the regression analysis, each datapoint has the greatest impact on the predictions of Y for animals whose values of W are closest to the deviant observation. While this outcome may be intuitively obvious, this basic point is sometimes overlooked by investigators when performing allometric scaling.

Table 8. Prediction error (ratio of Y_{no error} to Y with error) when error is introduced into the ‘observed’ dataset

Species with error		Mouse	Dog	Elephant
	kg
Dog	13	0.87	0.87*	0.87
Elephant	6600	1.12	0.87	0.68*
Mouse	0.023	0.68*	0.87	1.12
	0.1	0.72	0.87	1.05
	10	0.86	0.87	0.88
	100	0.95	0.87	0.80
	1000	1.04	0.87	0.73
	10 000	1.14	0.87	0.67

• *Back extrapolated values.

If we compare the error occurring in predictions for large animal species, the greatest error risk was associated with the dataset containing error in the elephant organ weight (e.g. 27% over-prediction for a 1000 kg animal). In contrast, the introduction of error in the mouse data resulted in only a 4% under-prediction for the 1000 kg animal. When a midpoint species (dog) is the source of the error, the change is primarily in the intercept rather than the slope. Consequently, the resulting magnitude of prediction error (Y_{no error}/Y_{dog error}) is comparable throughout the range of W values examined. It is also of interest to note that the dog predictions were consistently robust to the changes, exhibiting the least fluctuation in predicted values. This is consistent with its location at the central point on the curve. It should be noted that in real life situations, this ‘data error’ can appear both in either W and/or in the pharmacokinetic parameter.

Real example: difference in pharmacokinetic predictions when using two different equations reported for same drug

To examine how clearance predictions can change as a function of the data used to generate an allometric equation, we considered equations previously published in two separate manuscripts that explored allometric relationships for the veterinary fluoroquinolone, enrofloxacin. As the scaling provided by Cox et al. (2004) was based upon values corrected for protein binding, we needed to correct the Bregante et al. (1999) estimates for interspecies differences in protein binding. Therefore, to compare the two sets of equations, we re-calculated the allometric equation of Bregante et al. The resulting equations were very similar, differing primarily in the respective values of the allometric coefficient:

(2)

(3)

Using these two equations, we predicted clearance values for hypothetical species ranging in weight from 1 to 1000 kg. When comparing the ratios of the two sets of estimates (Table 9), we see that the magnitude of the divergence in predictions increases only slightly as values of W increase from 1 to 1000 kg. Nevertheless, this slight deviation would result in more than a 50% difference in predicted clearance values in a large animal species (i.e. a doubling or halving of the predicted dose).

Table 9. Comparison of predicted clearance values based upon equations of Cox et al., and Bregante et al.

Hypothetical W	Bregante et al., prediction of total clearance	Cox et al., prediction of total clearance	Total clearance predictions Bregante/Cox
1	23	16	1.44
10	136	92	1.47
100	805	536	1.50
1000	4769	3115	1.53

Discussion

Despite the potential for extrapolation error, the reality is that allometric predictions are needed across many veterinary practice situations. For this reason, it is important to consider mechanisms for minimizing these errors. In Part I (Mahmood et al., 2006), we reported that when generating allometric predictions of clearance for large veterinary species, the inclusion of at least large animal in the scaling, with or without human data, can improve the accuracy of the predictions. In this current report (Part II), we clearly see that in addition to the physiological issues discussed in Part I, there is a simple mathematical basis for this observation.

Interspecies scaling is conducted across species with values of W that generally vary by several orders of magnitude. Even when the smallest species is the laboratory mouse (0.02 kg) and the largest is the dog (10–12 kg), this amounts to almost a 500- to 600-fold difference in W. When a pharmacokinetic parameter is predicted in humans, the human W is outside the regression line. Although, this practice belies the general notion that the prediction of a given parameter should only be made within the range of the independent (X-axis) parameter, the W of man is only about sevenfold greater than that of the dog. On the other hand, large animals like horse, camel, donkey, and mule are almost 30- to 50-fold heavier than the canine. This wide margin of difference between weights may be one of the reasons that the predicted pharmacokinetic values in large animals are in error.

Looking at the results of the extrapolation exercises using heart and kidney weights, a ‘rule of thumb’ is that slightly better predictions in large animal species are generated when the regression analysis includes several small plus one large animal species. However, as seen in our analysis, it is difficult to obtain an a priori determination of which large animal species will produce the least biased allometric prediction. Nevertheless, we did observe that there are certain factors that can be considered for minimizing the risk of prediction error.

Based upon the conditions examined in this study, we can see that deviation in even one animal species can markedly skew our predictions. In fact, when we selectively added error to the ‘actual’ values of any particular animal species, we found that the greatest bias occurs in predictions for species of similar weight. Therefore, it is important to base extrapolations on as many species of similar weights as is technically feasible.

In the comparison of clearance values estimated on the basis of the equations provided by Bregante et al. vs. Cox et al., we find that despite the similarity in these equations, there was a 50% difference in the clearance values predicted in large animal species. This observation underscores the importance of recognizing the magnification of error that occurs when converting from logarithmic back to a linear scale. Most importantly, even in the presence of a high correlation coefficient, extrapolation error can be substantial and therefore, a high R² should interpreted as a criterion for assuring a high level of accuracy in our interspecies predictions.

As we attempted to illustrate in this set of exercises, it is important to keep sight of the fundamental principles underlying simple linear regression (Draper & Smith, 1981). When we use log–log transformation to plot the relationship between clearance and W, we are assuming that there is a constant % CV about the values of clearance associated with W being considered. If this is not a correct assumption, the use of linear regression analysis is not appropriate. Simply translated, allometry functions under an assumption that the variability in clearance (expressed relative to a species population mean) is identical across target animal species (becomes linearized by using a log–log plot). This point is rarely addressed in published allometry study reports.

We are also relying upon an assumption that the function describing the relationship between log₁₀ clearance and log₁₀W is best described by a single slope and intercept. This assumption raises one very important stumbling block: the reliance on R² as an indicator of prediction accuracy. As seen in the organ datasets and in our exercises in Tables 7 and 8, a high R² value does not necessarily indicate that particular segments within the total range of W values would not have been better fit if we used some nonlinear regression method. A minor deviation in linearity is particularly critical when we recognize that our clearance estimates must ultimately be back-transformed to the linear scale.

Even if the relationship is indeed linear, we need to appreciate the factors that may influence the variance about the slope of the line or about the intercept. As we saw with our organ weight example, a substantial change in the slope of the line can occur by altering the weight range of the animal species (even though R² values typically remained very high). We also found that the nature of the change in the regression equation resulting from a modification in a single species parameter value depended upon where the animal was located within the range of Ws. The value of the intercept was most greatly influenced by the mouse parameter value. Mouse error also influenced the slope. Introducing error in the midrange value (dog) had little to no effect on slope but some effect on intercept. Elephant values had negligible impact on the value of the intercept but had a substantial effect on slope. These findings are consistent with the simple principle that the prediction error in a regression equation is at its minimum at the midpoint of the line and increases as we move in either direction from that midpoint. Therefore, our best predictions are most likely to occur when we interpolate a parameter value for species that fall within the midpoint of our observed values of W.

In conclusion, the following points need to be considered when predicting doses for large animal species on the basis of allometric equations: