Veterinary Anaesthesia and Analgesia, 2000, 27, 2–5

COMMISSIONED ARTICLE

Anaesthesia in the fourth dimension. Is biological

scaling relevant to veterinary anaesthesia?

Tim Morris, BVetMed, PhD, CertLAS, DipACLAM, MRCVS

SmithKline Beecham Pharmaceuticals, New Frontiers Science Park, Harlow, Essex, UK

Correspondence: Dr Tim Morris, SmithKline Beecham Pharmaceuticals, New Frontiers Science Park, Third Avenue, Harlow, Essex CM19

5AW, UK.

Keywords Allometric equation, anaesthesia, biology,scaling.

Fundamental laws, such as gravity and thermo-dynamics, are in common use in the physical sci-ences but are less applicable to biology. However,although there are fewer biological laws relative toother disciplines, the ones that exist are of greatimportance, e.g. natural selection. Another examplemay be a new law of biological network design whichexplains observations on how metabolic rate chan-ges as organisms change size. It has been known forsome time that as animal size increases its metabolicrate falls (Rubner 1883; Kleiber 1932; Adolph 1949;Schmidt-Nielsen 1984; Calabrese 1991). This hasdeveloped from a physiological curiosity to a toolthat is used extensively to predict drug effects acrossspecies (Lave et al. 1999). The function of this articleis to review the development of this concept, and toexamine its application in veterinary anaesthesia.

Inter-species scaling (‘the mathematical expres-sion of methods used to scale up or down, size orcapacity’) developed in biology from methods usedin chemical engineering (Boxenbaum 1984). Fourgeneral types of scaling are described:1 Changing the number of units working in parallel(a biological example would be more glomeruli inlarger animals).2 Maintaining design and function whilst changingsize, e.g. larger animals have larger livers.3 Altering the flow scheme of the basic system, forexample, employing different metabolic rates in ani-mals of different sizes.

2

4 Choosing another type of equipment, e.g. develop-ing different metabolic pathways in differently sizedorganisms.

More specifically allometric scaling concentrateson scaling factors related to size, while ignoring nonsize-related factors such as metabolic rate. The equa-tion that describes this allometric relationship is:

Log p=Log a+b. Log W

where p is the parameter of interest, e.g. oxygenconsumption, W is the bodyweight, a the interceptquantifying the value of p when bodyweight equals1 kg, and b the exponent, i.e. the gradient of the line.The equation can be simplified to p = aWb (Morris1995; Morris 1999). Under ‘real-life’ conditions, thevalue of the exponent depends on the variable beingexamined, which complicates calculations.However, in 1981 Lindstedt and Calder provided auseful classification, which simplified interpretationof the expression. The value of the exponent for vol-umes of organs (heart, lung, etc.) is approximately1, because relative to each other and the body as awhole they are indispensable, thus they increase indirect proportion to increased size. In contrast, theskeleton is required to be disproportionately strongerin larger animals and so the exponent is greater than1. Of major importance is the fact that the value ofthe exponent for metabolic rate – which will deter-mine the rate of drug metabolism – is 0.75 (Bartels1982; Riviere et al. 1997).

The historical approach to understanding howthis relationship developed begins with first accept-ing the generalization that anatomical features andbiochemical reactions are similar across the same

Veterinary Anaesthesia and Analgesia, 2000, 27, 2–5

Biological scaling in veterinary anaesthesia T Morris

biological order. Second, it must then be acceptedthat there are inevitable physical consequences asorganisms increase in size. A well understood exam-ple is represented by body surface area to bodyweightratio, which falls, as animals get larger. Thus theirability to lose heat is also reduced. Because metabolic,transport and waste processes are optimized for aparticular temperature, it follows that without adap-tation these processes would become inefficient asorganisms grew in size. It is believed that withincreasing animal size, evolutionary pressures selec-ted for a reduction in the metabolic rate as a meansof limiting the inevitable rise in body temperature.This selected adaptation is given as the explanationfor the observations that, in species spanning a wideweight range, physiological parameters, such as oxy-gen consumption, ventilation rate, renal clearanceand nitrogen output only correlate linearly whenplotted across bodyweight on a log : log scale withan exponent value of about 0.75. Consequently, asbody size increases the values of these physiologicalparameters are reduced proportionately, e.g. onegram of shrew tissue has a metabolic rate 1000 timesgreater than a gram of blue whale tissue.

It is probably less well appreciated that temporalparameters, e.g. the duration of the cardiac cycle, lifespan, drug half-life, etc., also correlate linearly whenplotted against bodyweight on a log : log scale. Thevalue of the exponent is 0.25. Consequently, theduration of a given parameter increases with bodysize, and may be usefully exemplified by comparingthe lifespan of the shrew (short-lived) with that ofthe blue whale.

The underlying principles and the mathematicsused to predict drug dosage by extrapolation andinterpolation between different species have beenexplored in detail by Morris (1999). In brief, becausetemporal parameters are linked with bodyweight byan exponent whose value is 0.25, while that linkingbodyweight and volume has an approximate valueof 1.0, then variables described in terms of volume/time, e.g. cardiac output, are related to weight by aexponent with a value of about 0.75:

Volume

Time�

M1.0

M0.25=M0.75

(Lindstedt & Calder 1981) where M is body weight.This linking of physiological variables to body

weight raised to the power 0.75 is the origin of termssuch as ‘the 3

4power scaling law’.

This phenomenon has influenced dose extra-

3

polation of many drugs (Riviere et al. 1997) andthere is ample evidence that it probably applies toanaesthetics. For example, allometrically predictedpentobarbitone sleeping time was found to be similarto actual sleep times in 30-day-old and adult rabbits(Weatherall 1960). Glen (1980) found that althoughthe absolute utilization rate of propofol was 2.22 mgkg−1 min−1 in mice and 0.28 mg kg−1 min−1 in pigs,the allometric calculation based on a bodyweightraised to the power of 0.75 yielded much closervalues of 0.88 (mice) and 0.70 mg kg−1 min−1 inpigs (Glen 1980). More recently, the plasma levelof the muscle relaxant metocurine has been scaledallometrically from the rat, through the cat, the dog,in man and to the horse (Gronert et al. 1995).

Therefore, it is important to appreciate that whenusing a given drug in a population of animals whosesize differs markedly, that dosage should not, in thefirst instance, be extrapolated on a mg kg−1 basis,but on a 3

4power scaling. Worksheets and computer

programs to help in this extrapolation have beenpublished (Morris 1999; Hawk 1999). This has par-ticular importance in the field of laboratory animalmedicine and science, where anaesthetists may beexpected to assist laboratory animal veterinariansrequiring specialist advice (Schatzmann 1997).

The principal general reservation with the accu-racy of allometric scaling is that its dependency onthe link between size and other physiological par-ameters is prone to invalidation when different meta-bolic pathways develop (Lave et al. 1999) althoughaccuracy may be improved by incorporating meta-bolic data (Lave et al. 1995). For example, pento-barbitone sleeping time could not be predictedallometrically in 1-day-old rabbits, compared with30-day-old and adult animals (see above) becausethese have poorly developed metabolic pathways(Weatherall 1960). Furthermore, the inability of thecat to conjugate phenolic compounds resulted inabnormally low propofol utilization rates – whenpredicted allometrically – in comparison to mice andpigs (Glen 1980).

Care is also required when applying allometricscaling to the behaviour of inhaled anaestheticsbecause the alveolar minute volume is linked withsize across various species by the exponent value of0.75 (Fiserova-Bergerova & Hughes 1983). Res-piratory rate falls with increasing size, and so therate of induction and recovery will scale to 3

4power,

with induction being faster in smaller, rather thanlarger animals (Kohler et al. 1999). At steady state,MAC values across animals of differing size are simi-

Biological scaling in veterinary anaesthesia T Morris

Veterinary Anaesthesia and Analgesia, 2000, 27, 2–5

lar because alveolar gas concentration is not influ-enced by size.

It is worth examining how a ‘34

scaling law’became established because conventional Euclideangeometry predicts that 2

3scaling would occur (Rubner

1883). This prediction is based on the fact that heatloss is proportional to surface area (which is relatedto a linear dimension by a factor raised to the 2ndpower) while mass is proportional to volume (whichis related to length raised to the 3rd power). Variousfactors, including gravity (Calabrese 1991) havebeen used to account for the difference between pre-dicted 2

3scaling and the more accurate working value

of 3

4. Recently, more fundamental explanations have

been proposed, based on the characteristics of ‘net-works’ (West et al. 1997; West et al. 1999; Banavaret al. 1999). The mathematical proofs are relativelycomplex but the concept is presented in a moreaccessible way in some accompanying commentaries(Willis 1997; Mackenzie 1999). These propose thatnatural selection has developed ‘networks’ that max-imize metabolic capacity with maximal efficiency byminimizing transport distances and times. Oneapproach (West et al. 1997; West et al. 1999) fav-ours fractally designed networks that scale to bodymass × 0.75. (Fractals are a curve, or geometricalfigure, each part of which has the same statisticalcharacter as the whole. Fractals are useful in descri-bing partly random natural phenomena, especiallynonuniform structures in which similar patternsrecur at progressively smaller scales). Anotherapproach proposes that allometric scaling to bodymass × 0.75 is a feature of distribution networksirrespective of assumptions such as fractal geometricdesign (Banavar et al. 1999). These new approachesare controversial and have been criticized (Miller &Austad 1999). Nevertheless, it is the current prop-osition that the constraints on metabolic rate andlifespan which dictate an organism’s physiology andanatomy are imposed by network design factors ondimensions within the organism, rather than themore obvious and quantifiable external dimensionssuch as surface area, body volume or weight.

This is the hidden fourth dimension that dictatesthe law of 3

4power scaling. Anaesthetists working

with species of markedly differing body sizes shouldbe aware that this is also the level at which drugsexert their effect.

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