# Allometry, Shallometry!

From the ^{*}P Less Than Corporation and the University of California San Francisco, San Francisco, California; and ^{†}Department of Anesthesiology, Perioperative and Pain Medicine, Stanford University, Stanford, California.

Accepted for publication February 4, 2016.

Funding: None.

The authors declare no conflicts of interest.

Reprints will not be available from the authors.

Address correspondence to Steven L. Shafer, MD, Department of Anesthesiology, Perioperative and Pain Medicine, Stanford University School of Medicine, 300 Pasteur Dr., MC-5640, Stanford, CA 94305. Address e-mail to steven.shafer@stanford.edu.

*Ctenocephalides canis*, the flea that infests your dog, can jump 15 cm.^{1} Fortunately, the miniscule body mass of a flea, about 1 mg,^{2} reduces the risk of injury to your dog when the flea lands. If the jumping height correlated with weight then, based on the performance of *C canis*, a 6000-kg (6,000,000,000 mg) African elephant^{a} could jump 900,000 km, more than twice the distance to the moon. That also poses little risk to your dog because an elephant returning from outer space would not survive reentry. However, it does illustrate that physiology does not scale linearly to weight.

The science of physiologic scaling, allometry, was first proposed by Snell in 1892.^{3} The principal may date to Galileo.^{b} In simple terms, allometry proposes that physiologic processes (e.g., oxygen consumption, blood flow, drug metabolism) are best scaled by weight raised to a power, typically 0.75 (or, in some instances, 0.66). Figure 1 shows the relationship between weight^{1.0} (usually just called “weight”) and weight^{0.75} for animals ranging from 0.0001 kg (0.1 gram) to 10,000 kg. The 0.1-g animal (or, more likely, organelle) physiologically scales 10 times more than predicted by weight, whereas the 10,000-kg animal physiologically scales one-tenth as much as predicted by weight. Numerous anatomic, physiologic, and mathematical explanations have been put forward for the empiric observation^{4} that physiology scales to weight to the 0.75 power.^{5–8}

The principles of allometric scaling have been useful in drug development, guiding the initial doses of drugs in human studies from the results of preclinical animal studies using animals of much smaller size (e.g., rats).^{9–11} However, what is the role of allometry in human pharmacokinetics, that is, when considering humans only, rather than scaling from other species? In particular, what is the role of allometric scaling in scaling the physiologic process of clearance in patients of different weights?

Figure 2A explores the calculation of a scaling parameter for clearance. The first relationship is as follows: Scalar = weight^{0.75}, represented by the 101 black dots from 0 to 100 kg. This is the allometric representation, where the “true” scalar for body mass is weight^{0.75}. The red line is a linear regression through the black dots that intersects the origin. The formula for the red line is as follows: Scalar = weight × 0.36. The red line is not very different from the black dots within the narrow range from 0 to 100 kg. This suggests that there will be very little difference in the accuracy of human pharmacokinetic models that use weight or weight^{0.75} as a scalar across the range from 0 to 100 kg. Figure 2B looks at the scalar between clearance and body mass in an adult population, ranging from 50 to 100 kg. This is a narrower range than in Figure 2A, and the linear regression better approximates weight^{0.75}. Figure 2C considers weights from 2 to 12 kg, as might be found in a study of infants. Again, the function Scalar = weight^{0.75} is well approximated by a linear model.

Some authors advocate modeling clearance in human pharmacokinetic models using weight^{0.75}, on the presumption that the models will better scale to children or to obese patients.^{12} Almost no data support the suggestion that human pharmacokinetic models better scale to weight^{0.75} than to weight, body surface area, or other common metrics.^{13},^{14} Indeed, as Figure 2A, 2B, and 2C shows, over the ranges typically found in human studies, weight^{0.75} can be well approximated by weight × constant. However, both models are *incorrect* because the physiology of clearance depends on more than weight alone.

Anesthesiologists recognize that a massively obese person may be better given the appropriate dose for the “non-obese person” within. Anesthesiologists recognize that metabolic pathways in infants may be immature. We know that age, disease, and drugs may change metabolism. We also know that drugs that act rapidly on the brain require just one circulatory cycle. Because brain mass and cerebral blood flow vary little among adults, these drugs might be best given without any weight adjustment at all.^{11} Because weight is just one of many patient factors we consider in calculating the dose, it is best to calculate dose from pharmacokinetic models created from the population of interest rather than models adapted from other populations.

How should we select doses for children? Obviously, one would not administer the same dose to an 80-kg teenager and a 3-kg neonate. To address this, we need guidelines, that is, a formula to calculate dose as a function of some metric of body size (e.g., weight or height), age, or some other characteristic of the patient. In the absence of pediatric pharmacokinetic data, we may turn to Clark’s rule: the dose in a child is the adult dose divided by 68 multiplied by the child’s weight (in kg).^{c} Clark’s rule would work perfectly if the pharmacokinetics were exactly proportional to weight from children to adults. Because we recognize the limitations of dosing by weight alone, Clark’s rule seems to be a mindless calculation that assumes children are very small adults. Nevertheless, Clark’s rule performs about as well (or poorly) as any other scheme for calculating pediatric doses from adult doses.^{13}

The anesthesia community has benefitted from high-quality and innovative pharmacokinetic and pharmacodynamic research in adults and children. If pharmacokinetic and pharmacodynamic data are available, it makes sense that dosing should be guided by those data. For example, the present issue of *Anesthesia & Analgesia* includes a pharmacokinetic analysis of dexmedetomidine in neonates and infants by Su et al.^{15} The investigators evaluated 2 approaches to scaling for body size, a weight-proportional model and an allometric model. Before discussing the article by Su et al., we need to first review the difference between these models.

The proportional model scales volumes and clearances to weight. The appeal of this approach is its simplicity. When volumes and clearances are scaled to weight, pediatric doses can be expressed as dose per kilogram. Simplicity brings safety: there is relatively little chance for a calculation error when the dose is just weight × dose/kg. The allometric model scales volume of distribution to weight, but elimination clearance (and distributional clearance) to weight raised to the 0.75 power. Not only is taking weight to the three-fourth power not easy but also scaling volumes by weight^{1.0} and clearances by weight^{0.75}, as required by allometric models, introduces complexity in dose calculation, which will be shown below. As noted by Anderson and Meakin,^{16} “Models describing metabolic processes as nonlinear functions are difficult to calculate mentally at the workplace…. It is unlikely anesthetists will change from the traditional ‘mg per kg’ model.”

Let us look at the calculation of dose for a simple 1-compartment drug using weight-proportional scaling. We will consider 2 different situations: the bolus dose to start the case, and then the infusion to maintain targeted steady-state concentrations.

Let us assume we want to give a dose that will produce a concentration of 1 μg/mL 4 minutes after drug administration (e.g., at the time of tracheal intubation, as might be the case for a drug given to induce anesthesia). Let us also assume the volume of distribution is 20 mL/kg, and the elimination clearance is 2 mL/kg/min. If the child weighs W kg, then the volume of distribution is 20 (mL) × W (kg) and clearance is 2 (mL/min/kg) × W (kg). The elimination rate constant, k, is the clearance divided by the volume. Expressed mathematically,

. Because weight appears in the numerator and the denominator, it cancels out, that is, it does not affect k.

What is the dose that will result in a concentration of 1 μg/mL exactly 4 minutes after a bolus? The formula relating dose to concentration is as follows:

. Substituting the numbers for concentration (1 μg/mL), volume (20 (mL) × W (kg)), k (0.1 min^{−1}), and time (4 minutes) into this formula gives:

Solving this for dose gives:

It is OK if you skipped the math. As a clinician, all you need to know is the punchline: the dose is 30 μg/kg. Simple!

Now, let us turn to the steady-state infusion rate for models in which the volumes and clearances are linearly proportional to weight. For the maintenance infusion, the math is even easier: the infusion rate is the target concentration times the clearance. Substituting in concentration (1 μg/mL) and clearance (2 (mL/kg/min) × W (kg)) gives:

That is also simple. To maintain a concentration of 1 μg/mL, give 2 μg/min/kg. Many infusion pumps are already set to administer drugs in units of μg/min/kg. The calculation for the weight-proportional model is easily performed at the bedside.

Let us repeat this exercise for the allometric model, where clearance is proportional to body weight^{0.75}. For this model, assume the volume of distribution is 20 mL/kg (as before), and clearance is 2 mL/min/kg^{0.75}. The child weighs W kg. Volume of distribution is therefore 20 (mL) × W (kg) and clearance (2 (mL/min/kg^{0.75}) × W (kg)^{0.75}. The elimination rate constant, k, is

Weight does not cancel out because it is raised to a different power in the numerator and the denominator. We again substitute concentration (1 μg/mL), volume (20 (mL) × W (kg)), k (

), and time (4 minutes) into the formula that relates concentration to dose and time:

. Solving this for dose gives:

Got that? The induction dose requires calculating

. This is what Anderson and Meakin^{16} were referring to as “difficult to calculate mentally at the workplace.”

Turning to the maintenance infusion, the dose is concentration time clearance:

Again, calculation of the infusion rate requires calculation of weight to the three-fourth power. Because our infusion pumps are not designed to infuse drugs based on weight^{0.75}, asking the anesthesiologists to make this calculation as part of the infusion setup is asking for trouble.

Table 1 summarizes the calculations required for the bolus dose and maintenance infusion rates to achieve and maintain a concentration of 1 μg/mL from 4 minutes based on these models. Which model is consistent with a safety culture? For a child of weight W, would you prefer to know that the bolus dose is 30 W, and the infusion rate is 2 W, as indicated by the proportional model. Or would you prefer to know that the bolus dose is 20W(

) and the infusion rate is

?

Obviously, the simple proportional model is the safer guideline. Asking physicians, nurses, or pharmacists to calculate dose as

is asking for trouble, particularly for commonly used anesthetic drugs for which the clinician is likely to calculate the dose in the moments before drug administration.

However, an even more important question is which dose is correct: 30 W or 20W(

)? If one method of dose calculation is clearly better than the other, then we should use our available tools (e.g., computers, infusion pumps) to give drugs optimally while avoiding calculation errors. There are 2 ways to determine which method is best. The first is to approach it from the data. If the weight-scaled model better describes the observed drug concentrations than the allometric model, then the weight-scaled model is the way to go. There is no “down” side to using a simple weight-scaled model if it describes the concentrations in the pharmacokinetic study as well as or better than the allometric model. However, if the allometric model better describes the concentrations in the pharmacokinetic study, then we have a dilemma. The weight-proportional model is simpler, hence safer, but would it be clinically consequential to use a weight-proportional model?

Returning to the article by Su et al. in this month’s issue of *Anesthesia & Analgesia*,^{15} the children ranged in weight from 2.3 to 11.9 kg. This is the weight range in Figure 2C. Within this range, weight^{0.75} is almost indistinguishable from weight × 0.60. Figure 3 shows that the differences between the two models are trivial compared with the unexplained intersubject variability in clearance. To obtain the “data” for Figure 3, we first evenly distributed patients in the numbers and weights as described by Su et al. in their Table 3.^{15} We calculated the clearance of each subject using the allometric model shown in their Table 4.^{15} We added intersubject variability by multiplying the estimated clearance by e^{η}, where η is a normal variable with a mean of 0 and a standard deviation of 28.58% (from their Table 4).^{15} These estimates of individual clearance appear as green dots. We plotted clearance predicted by the allometric model, (weight/70)^{0.75} × 0.657 L/min,^{15} and for the weight-proportional model, weight/70 × 1.17 L/min^{15} as the black dotted line and red line, respectively. The difference between the black dotted line and the red line is trivial compared with the unexplained intersubject variability (green dots). This is why Su et al. observed that the allometric model did not describe the data better (indeed, it was trivially worse) than the simple weight-normalized model. Of course! Because weight^{0.75} ≈ weight × 0.60 in this weight range, mathematically the results must be similar. The trivial differences between weight-proportional models and allometric models shown in Figure 2C and 3 are dwarfed by the effects of metabolic maturation in this age range, as well as by the unexplained between-subject variability common to all pharmacokinetic studies.

This issue of *Anesthesia & Analgesia* also includes an allometric model by Sitsen et al.^{17} to describe the influence of epidural blockade on propofol pharmacokinetics. During peer review, the authors were asked to consider a weight-proportional model. In response, the authors determined (as expected) that the weight-proportional model was indistinguishable from their allometric model. They justified not including the weight-proportional model because their intent was assessing physiology, not guiding drug dosing. Although this is a weak argument for the additional complexity of the allometric model, the article was accepted without including the weight-proportional model. Readers should accept the authors’ recommendations to the reviewers that the model not be used to guide drug dosing, unless implemented by a computer.

The principle of allometry arose out of empiric observations of organisms ranging from fleas to elephants. It appears to be nearly universal in biology. Investigators are still exploring the underlying principles revealed by these observations. However, allometry was developed to describe the average for species of varying sizes not the subject-to-subject differences among individuals within the same species. Allometry was not developed to apply to modest weight scales, as done by Su et al. looking at children ranging from 2.3 to 11.9 kg. It was not developed to predict pediatric doses from adult doses and performs poorly at doing so.^{13} There is no scientific justification for the view that human clearances are better described using weight^{0.75}. These models should only be used when weight^{0.75} rather than weight^{1.0} provides a sufficient improvement in the pharmacokinetic model and dose calculation to justify the risk of calculation error. Given that the allometric and simple models are not very different (Fig. 2, A–C), this will rarely be the case.

The data reported by Su et al. are important. We applaud the authors characterizing the pharmacokinetics of anesthetic drugs in infants and children, a population too often neglected in clinical research. Submissions such as these are enthusiastically welcomed by reviewers and editors of all journals, including *Anesthesia & Analgesia*.

However, investigators of human pharmacokinetics should follow the advice of our mentor, the late Lewis Sheiner, to “let the data speak.” Unless it is clearly and convincingly demonstrated to be better than a simple weight-proportional model, we consider allometry shallometry.

## DISCLOSURES

**Name:** Dennis M. Fisher, MD.

**Contribution:** This author helped write the manuscript.

**Attestation:** Dennis M. Fisher approved the final version of the manuscript.

**Name:** Steven L. Shafer, MD.

**Contribution:** This author helped write the manuscript.

**Attestation:** Steven L. Shafer approved the final version of the manuscript.

## RECUSE NOTE

Steven L. Shafer is the Editor-in-Chief for *Anesthesia & Analgesia*. This manuscript was handled by James G. Bovill, Guest Editor-in-Chief, and Dr. Shafer was not involved in any way with the editorial process or decision.

**FOOTNOTES**

a http://animals.nationalgeographic.com/animals/mammals/african-elephant/. Accessed February 2, 2016.

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b Galileo Galilei, Dialogues Concerning Two New Sciences, 1638. Available at: http://oll.libertyfund.org/titles/753. Accessed February 2, 2016.

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c Available at: https://en.wikipedia.org/wiki/Clark’s_rule. Accessed February 2, 2016.

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## REFERENCES

*Ctenocephalides canis*(Curtis, 1826) and the cat flea, Ctenocephalides felis felis (Bouché, 1835). Vet Parasitol. 2000;92:239–41