All You Need to Know About Allometric Scaling: An Integrative Review on the Theoretical Basis, Empirical Evidence, and Application in Human Pharmacology

Abstract

Scaling approaches are used to describe or predict clearance for paediatric or obese populations from normal-weight adult values. Theoretical allometry assumes the existence of a universal bodyweight-based scaling relationship. Although theoretical allometry is highly disputed, it is commonly applied in pharmacological data analyses and clinical practice. The aim of the current review is to (1) increase pharmacologists' understanding of theoretical allometry to better understand the (implicit) assumptions and (dis)advantages and (2) highlight important methodological considerations with the application of this methodology. Theoretical allometry originated in an empirical, and later debated, observation by Kleiber of a scaling exponent of 0.75 between basal metabolic rate and body mass of mammals. The mathematical framework of West, Brown, and Enquist provides one possible explanation for this value. To date, multiple key assumptions of this framework have been disputed or disproven, and an increasing body of evidence is emerging against the existence of one universal allometric exponent. The promise of ease and universality of use that comes with theoretical approaches may be the reason they are so strongly sought after and defended. However, ecologists have suggested that the theory should move from a 'Newtonian approach', in which physical explanations are sought for a universal law and variability is of minor importance, to a 'Darwinian approach', in which variability is considered of primary importance for which evolutionary explanations can be found. No scientific support was found for the application of allometry for within-species scaling, so the application of basal metabolic rate-based scaling principles to clearance scaling remains unsubstantiated. Recent insights from physiologically based modelling approaches emphasise the interplay between drugs with different properties and physiological variables that underlie drug clearance, which drives the variability in the allometric scaling exponent in the field of pharmacology. To deal with this variability, drug-specific or patient-specific adaptations to theoretical allometric scaling are proposed, that introduce empiric elements and reduce the universality of the theory. The use of allometric scaling with an exponent of 0.75 may hold empirical merit for paediatric populations, except for the youngest individuals (aged ≤ 5 years). Nevertheless, biological interpretations and extrapolation potential attributed to models based on 0.75 allometric scaling are theoretically unfounded, and merits of the empirical application of this function should, as for all models, always be supported by appropriate model validation procedures. In this respect, it is not the value of the allometric exponent but the description and prediction of individual clearance values and drug concentrations that are of primary interest.

Fig. 1

Excerpt from Quantitative Relations in the Physiological Constitutions of Mammals by EF Adolph, published in 1949, highlighting the fragmentation of the scientific consensus on the scaling exponent and alluding to the existence of a distribution of exponents rather than a single universal value

Fig. 2

Visualisation of the concepts and parameters used in the framework of West, Brown, and Enquist. Shown are a side view (top) and cross-sectional view (bottom) of the vascular system, which branches from the aorta (left) to the capillaries (right)

Fig. 3

Number of research publications on allometric scaling in drug pharmacokinetics over the past five decades as derived from PubMed. Results are based on a search on 13 May 2024, with the query “(pharmacokinetics OR PK) AND (allometric scaling OR allometry OR theoretical allometry)”

Fig. 4

Application of theoretical allometry in human pharmacology can be visualised as a temple (left). It is founded on two principles: Kleiber’s observational ‘law’ and the framework of West, Brown, and Enquist (WBE) for scaling of basal metabolic rate (BMR). Additionally, three assumptions are required to support the use of theoretical allometry (AS0.75) within human pharmacology. In reality, the basis of the temple is poorly supported, and the assumptions are shown to generally not hold (right). CL clearance

Fig. 5

The relationship between clearance (arbitrary values, expressed in unit volume/unit time) and bodyweight (kg) on a (A) linear scale and (B) double-log scale. The blue line represents the allometric relationship for clearance, and the solid line section represents the broadly accepted application range (25–90 kg). The green lines represent clinically observed patterns, with (1) in the very young clearance values that are below the allometric line, (2) in children between approximately 2 and 5 years clearance values above the allometric line, and (3) a wide range of patterns in the obese population with clearance values increasing, decreasing, or remaining constant with increasing bodyweight. The text indicates how these observed patterns are accounted for with extensions to the allometric framework