Kleiber Rules!

Have you ever wondered why clinicians (and even some physiologists) relate a number of measurements to body surface area? For example, they talk about cardiac index (cardiac output divided by, yes, body surface area), and urinary protein losses in nephrotic syndrome as 3.5g/1.73m2. Why relate these measurements to area? Does this make sense?

Such an approach did make sense (at least on 23 July 1839) to the mathematician Sarrus and the (heh) doctor of medicine Rameaux, who proposed this "surface law" to the French Royal Academy. Theirs was a first primitive attempt at scaling - trying to derive "laws" that apply whatever the size of the organism. Clinicians, who usually have to deal with variations in size over about two orders of magnitude (say 1 kilogram to about 100kg), have been quite happy to accept the surface law and in so doing lag a hundred years behind biologists, who cannot make do with sloppy approximation, dealing as they do with organisms ranging in size from a mycoplasma (so small that at physiological pH the mycoplasma contains perhaps two hydrogen ions), to a hundred tons of blue whale. Biologists consider weights ranging over twenty one orders of magnitude! It seems inconceivable that there are scaling laws that apply across this vast spectrum of life. But there are such "laws", and they have far reaching implications for such diverse areas as ageing, athletic performance, and weight-adjusted dosing.

What is the basis of Sarrus and Rameaux's argument? At first glance, it seems quite sensible to argue that:

QED? Not quite. Nature is often more subtle than nineteenth century mathematicians. It is quite clear that small animals do live fast, short, metabolically hectic existences compared to say elephants, but unfortunately for Sarrus and Rameaux the rule isn't what they predict! We are about to dip into the complex world of scaling.

Scaling Basics

The boundary where biology intersects mathematics is littered with the intellectual corpses of frustrated clinicians who have tried unsuccessfully to "crack into" the maths. A physician, I will not try to preach any mathematical gospel and will stick to simple explanations. (Mirth-filled mathematicians may suppress their laughter long enough to point out where I've erred). We do however need some basic 'scalingspeak':

Allometric Scaling Equations

The magnitude of many body processes changes in a regular fashion as the size of the organism changes. A surprising number of such processes can be described in a very simple fashion by:

where M is the body process in question (for example metabolic rate), x is a measure of the size of the organism, and a and k are constants. For example, a large number of measurements suggest that the relationship between metabolic rate (let's call this Pmet) and weight (mb) is:

.. where P is measured in kcal/day and mb is the weight in kilograms. This equation, first proposed in 1932 by the brilliant animal researcher Max Kleiber has been amply justified, and is good for men, mice and elephants! There is a small amount of uncertainty about the exact exponent (his original equation actually had an exponent of 0.74) but the value is certainly extremely close to 3/4. There are a whole host of other similar relationships, and many of them depend on similar "fractions involving a quarter" for example 3/4 and 1/4. But before we look at these, let's recall that if we take the logarithm of both sides of an equation, our original equation becomes:

You can see that this equation (basically y = c + k*z) is the equation of a straight line. What does this tell us? Simply, that if we want to visualise what's going on in one of the above equations, we can plot log(M) against log(x), and we should get a straight line! We can also indulge in other mathematical meddling - draw a regression line through this log-log plot, draw in the 95% confidence intervals, and the slope of the line will be k, and the point where it meets the y axis will be log(a) (that is where log(x)=0, in other words where x=1).

Picture of Max Kleiber
Max Kleiber

A list of allometries

In the past fifty years, many people have come up with interesting allometric equations. Let's have a look at some of these! (Most data come from Schmidt-Nielsen, and apply to mammals). The lists look intimidating, but it's worthwhile browsing through them quickly, for we refer back to them in later discussion. While looking through, continually ask yourself "Why should this be?".

Little change with weight

Changes in (almost) direct proportion to weight

Note how none of the above has an exponent that substantially deviates from one - as the weight increases, so does the blood volume, the size of the heart and the size of the lungs. Why should this be, if the metabolic rate is only increasing with the three-quarter power of the weight? Note that we don't have an equation for heart stroke volume (do you?) but it's reasonable to assume that it's directly related to heart weight.

Scaling to the quarter power

Aha! Now we're getting somewhere. If metabolic rate is increasing as stated, and heart stroke volume (and lung tidal volume) are increasing in proportion to mass, then something has to give, and thus respiratory rate and heart rate decrease with increasing size. (You might wish to ask the question "So why not keep the heart rate the same, and have a proportionally smaller heart?")

Scaling to the three quarter power

Note that the units of VO2max and Pmet are different. If we recalculate using the same units, we find that the VO2max is consistently about ten times the Pmet.

The original 
log-log plot from Kleiber's 1947 paper

A log-log plot of metabolic heat production against body weight, taken from Kleiber's 1947 article. It is clear that most points (green) closely fit the red line with a slope of three quarters, rather than the weight line (slope of 1) or the surface line (slope of two thirds). Note that although the differences between the lines seem small, this is a log-log plot so the deviations are actually vast!
Do you wonder why the whale is where it is?

Other allometries

Hmm, okay we just put these in for fun, and to show that not everything is based on quarter powers. Note that Schmidt-Nielsen gives a good explanation for the approximately 2/3 exponent in the above allometric equation for the metabolic cost of running (Others have found somewhat different exponents).

Using allometric equations

"The above is all very well" you say "but what is the practical use of all this bumf?" Some of the immediate practical implications are:
  1. Using 3/4 power scaling
    Have you noticed that for some drugs (an example is neuromuscular blocking agents) per kilogram dosing seems to be wholly appropriate, while for others, giving a dose based on the body weight is inadequate at low body weights (for example, aminophyllin and many antibiotics, administered to children)! Careful consideration may lead you to the conclusion that the initial dose of a neuromuscular blocker may depend on blood volume (which scales with weight) but if the metabolic removal of a drug is important (for example, aminophyllin) then three-quarter power scaling will apply.

    It thus seems reasonable to relate certain drug doses to the 3/4 power of the weight, and not to either weight or surface area. This isn't as difficult as it sounds. The cheapest pocket calculator usually has a square root function and a memory, so if we know the dose for a 70kg person, let's call it D70 and our patient weighs say 32kg, we could work out the relevant dose D32 as follows:

    This looks formidable but we can easily find the three quarter power of 32/70 as follows:

    1. Cube 32/70:
      • Clear memory and then work out 32/70 ( 32 70 0.457 )
      • Store the result in memory ( )
      • Multiply the result by itself, and then by the value retrieved from memory.
        ( 0.457 0.457 0.209 0.0955 )
    2. Find the fourth root of the result - press square root twice. ( giving the result we want: 0.556)

    You should not go out and apply this willy-nilly for all drugs, and for all people. Body composition and behaviour vary remarkably within the human species, especially at the extremes of size and age. Three quarter power scaling goes a long way towards explaining why we encounter statements in the medical literature about the peculiar doses we need to give to children, and very small or very large adults. If you're dealing with a metabolic variable (such as a drug dose, a clearance, cardiac output, or whatever) think Kleiber!

    Now that you know this, how about taking the per kilogram doses you use for children and comparing them with the per kilogram doses recommended for adults and seeing whether they square up? A really keen person might like to compare veterinary doses of drugs (for example, fentanyl) with human doses.

    The justification for the above formula is that if D70 = a*703/4 and D32 = a*323/4 then D32 / D70 = (323/4) / (703/4) which gives us D32 = D70 * (32/70)3/4.

  2. Read "the literature" with caution
    We have already encountered our dimensionally incorrect friends Harris and Benedict. Similar circumspection could perhaps be applied to all reports where for example a metabolic parameter is related to either weight or to surface area. The likelihood of a major difference is probably small when dealing with humans (except at the extreme ranges of body mass) but you should at least feel uncomfortable with information where an empiric index is used.

    For the record, Kleiber reformatted the Harris-Benedict equation (using their data) in a less dimensionally-challenged way:

    Clearly, it's easier to use the incorrect version, but with modern calculators and computers, we have little excuse.

  3. The really good stuff
    The real applicability of scaling will probably only become apparent when we understand what it's all about - why for example as size increases the heart rate slows, in the place of decreasing cardiac size relative to weight. Similar slowing may be intimately associated with the lifespan of the organism, which increases with mass. The lifespan of captive mammals has been found to approximately fit the allometric equation:

    The corresponding equation for birds is

    Obviously these are broad rules, with several exceptions. Most animals seem to survive for about 800 million to a billion heartbeats - humans are exceptional in this regard, as you can easily calculate. We don't know what causes ageing, but it's tantalising that the exponent in the above equations is so close to a quarter. It's not unreasonable to assume that faster metabolisms cause faster metabolic wear-and-tear on cellular structures (mitochondria, telomeres and so on), and that this may give us fundamental insights into ageing.

    Quarter-power scaling has even been reported in plants and microbes! (References: Niklas KJ Am J Bot 1994 81 134 and Hemmingsen AM Rep Steno Mem Hosp 1950 4 1, ex West 1997).

Explanations Please!

There have been many attempts to explain all, or at least some of the allometric equations we presented above. If we re-visit our friends Sarrus and Rameaux, we see that their argument was that as weight increased so metabolic rate would increase at a lesser rate, depending on surface area. In other words, they effectively proposed an allometric equation with an exponent of 2/3 relating metabolic rate and weight. Although this is not the case, it's not intuitively obvious why they are wrong. Some still argue that the interspecific rule has an exponent of 3/4 but that within the species the exponent is 2/3. There is little evidence for this, and the primitive surface area rule also doesn't take into account things such as fur (which tends to be longer and a more effective insulator in larger animals, excluding really large tropical animals), and the thermal conductance of animals, which decreases with increasing size. ( The practical assessment of surface area is in any case a nightmare - do you include the area of the ears (about 20% of a rabbit's surface area), the skin area between the legs and the body and so on?)

In some simple circumstances, we have reasonable explantions for allometric equations. Take the case of pores in the egg of a bird - as the size of the egg varies, so does the thickness of the egshell, and total pore area is accurately adjusted to ensure that evaporative water loss is constant for all eggs (about 15% of weight). Once we examine more complex cases, our logic tends to break down.

Some attractive explanations for allometric equations must be rejected out of hand. For example, if we were to predict that the primary determinant of bone mass was the static load placed on the animal, then the exponent should be 1.33 and not the 1.08 we noted above. It is clear that static loads do not determine bone mass - the important consideration appears to be stresses on the skeleton during movement! It is often easier to reject an explanation based on allometric equations, than to accept it - Karl Popper would be delighted!

"Why three quarters?"

There have been several attempts to explain the 3/4 exponent that seems to govern metabolism when related to weight. None appears entirely convincing. Here are a few:


  1. Kleiber M. Physiological Reviews 1947 27 511-541. A brilliant review. Read it!
  2. Taylor and Weibel. Respiratory Physiology 1981 44 1-10.
  3. Schmidt-Nielsen K, Scaling. Why is animal size so important? Cambridge University Press, 1984. ISBN 0 521 31987 0, ISBN 0 521 26657 2. Get the book. A fascinating read! Much of the above is based on this book.
  4. McMahon TA Science 1973 179 1201-4
  5. West GB, Brown JH and Enquist BJ. Science 1997 276 122-6.
  6. Banavar JR, Maritan A and Rinaldo A. Nature 1999 299 130-2 and Supplementary information at nature.com
  7. Heusner. Respiratory Physiology 1982 48 1-12. Arguments in favour of a 2/3 exponent within a species.

Web References
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