# Physics of the Body Mass Index

**S. P. Apell, O. Wahlsten and H. Gawlitza**

The individual and societal costs of obesity are well known: excess cardiovascular stress, chronic and cancerous diseases, and early mortality ^{[1]}. For this reason, there is considerable medical and popular interest in having a simple, reliable, and cheap way to monitor health risks by reducing physiological features of obesity to a single number. Such a number could also facilitate comparisons of health status between different regions, populations, ages and genders, and it could help an individual monitor his or her own body. Perhaps the best-known such number is the *Body Mass Index* (BMI). In this article we develop a straightforward, physically-motivated model to cast the definition of the BMI into some interesting equivalent forms, and show that it has a fundamental physical justification in terms of bodily metabolic rate and heat loss.

**A Brief History**

The BMI is not the only index of general health and weight status. Several alternative and complementary indices such as the Ponderal Index, Body Volume Index, Skin Fold Method, Waist-to-Hip Ratio, and Sagittal Abdominal Diameter have been developed ^{[1]}. Through a comparative study of obesity indices, Ancel Keys of the University of Minnesota and his collaborators introduced in 1972 the notion of *body mass index* as the best performing such measurement ^{[2]}. This was a renaming of the *Quetelet index* proposed in 1832 by the Belgian mathematician, astronomer and statistician Adolphe Quetelet, who was also known as the "founder of social physics." Quetelet did not intend his index to be used to characterize obesity or general health status, but rather to help him define a "normal man" by fitting a Gaussian curve to the distribution he found for the index, since using mass alone did not work ^{[3]}.

For a person of weight M (kilograms) and height H *(*meters), BMI is defined as ^{[4, 5]}:

BMI = M/H^{2} =*p*V/H^{2},

where we assume that an individual has an average body density *p* = (M/V) where V is the body volume. The Centers for Disease Control cautions that while the correlation between BMI and body fat is fairly strong, there are variations by race, sex, and age ^{[6]}. For adults, CDC guidelines classify a person with a BMI of __<__ 18.5 as underweight, 18.5 to 24.9 as normal weight, 25 to 29.9 as overweight, and __>__ 30 as obese. If M and H are measured in pounds and inches, then M/H2 must be multiplied by a factor of 703 to give the BMI on this scale. For aspects of BMI related to children, see ^{[7]}.

**A Prolate Spheroidal Model Human**

Quetelet constructed his index based on the empirical finding that the weight of adult humans scales with the square of their height. If we grew equally in all directions our weight would grow as the cube of our height, a model often used in many physics papers on animal scaling (see, e.g., ^{[8]}), but this is the case only during our first year of growth. A physicist's common first-approximation approach of modeling a system as a sphere is not applicable to human beings: if it were, our weight would increase by a factor of 27 in the time that we grew from a height of two feet to six feet! The next simple model one might think of, a right circular cylinder, would also not do as it would not capture the notion that most people are wider in their middles than at their ends. To represent adult proportions, we need a more realistic model intermediate between these two extremes. Also, since most people are very concerned with their waistline circumference W, it would be handy to cast the formulation of BMI in terms of that measurement.

Consider modeling a person as a prolate spheroid, that is, as an ellipse rotated about its major axis. With the person's height as the major axis and their waistline diameter D as the minor axis, their volume will be proportional to HD^{2}, or, in terms of their waistline circumference, V ~ HW^{2}. Hence, dropping unnecessary constants,

BMI ~ *p* HW^{2}/H^{2} ~ *p*W^{2}/H.

Average body density does not vary much among the population, even between very obese and very muscular people, so *p* can be treated as essentially constant ^{[9]}. Your height H will not vary much once you have reached adulthood, so you can conclude that your BMI depends on the square of your waistline measurement. Adding an inch to a 36-inch waist at a BMI of 22 will push your BMI to just over 23.

Another perspective is provided by eliminating height in favor of weight via H ~ M/*p*W^{2}, which gives

BMI ~ *p*^{2} W^{4}/M.

For a group of people *of the same mass*, BMI varies as the fourth power of waistline, a result which strikingly emphasizes the role of an individual's waist/height "aspect ratio" in computing their BMI. As a side comment, it is interesting to note that resistance to laminar flow in arteries and veins also scales with the fourth power of their circumference; one is led to wonder what effect these fourth-power dependencies have on human evolution.

**Energy Considerations**

The most fundamental aspect of many physical systems is their energy budget. If we wish to construct an index which in some sense measures the accumulation of unnecessary fat reserves, we should compare energy input versus energy loss. For most animals, biologists measure the energy input via the so-called basal metabolic rate, which scales as about the three-fourths power of body mass, BMR ~ M^{3/4}. There are slight variations in the exponent when one starts to go into details of various animal groups as well as interpreting data for one single group, but our purpose here is to develop a simple physical argument ^{[10,11]}. As for energy loss, for an organism that is simply sitting around, we consider heat leaving the body as the major energy output. This will be proportional to its surface area S, hence

(energy input/energy output) ~ W^{3/4}/S.

Mosteller ^{[12]} found that body surface area scales as the square root of the product of a person's weight and height, a result supported by later findings ^{[13, 14]}. This gives

(energy input/energy output) ~ W^{3/4}/(WH)^{1/2} W^{1/4}/H^{1/2} ~ (BMI)^{1/4}.

Physically, Quetelet's empirical index introduced nearly 200 years ago can be interpreted as the ratio of the basal metabolic rate to the body's heat loss rate to the *fourth* power. Increasing your caloric intake without a corresponding increase in energy burn-off will thus have a serious effect on your BMI. The fourth-power dependence shows why it is difficult to improve a BMI once it has reached an unhealthy number. Since the possible amount of stored energy in the form of fat tissues should be related to the metabolic rate and the amount of lost energy is related to surface area, a large storage capacity coupled to a relatively small area will give a large index - the case for "spherical" individuals. The best advice for us humans is to strive for a relatively large eccentricity and watch out for changes in our waistline.

It is always instructive to examine the physical bases for everyday phenomena. Following in the tradition of Quetelet's social physics, we have explored here how a common measure of human health relates to basic geometrical and energy-balance quantities.

Finally, we want to point out that Quetelet's observations also imply that the metabolic rate is beautifully fine-tuned to be directly proportional to the body surface area. In contrast, a simplified physical "spherical cow" argument leads to the suggestion that there is a lower limit to the size of animals based on the false assumption that energy generation is proportional to volume and losses (true) to the area, and that eventually the latter wins for small enough an animal. In fact, as pointed out in ^{[15]}, the lower limit is related to when cells begin behaving independently from each other rather being part of an individual (in vivo versus in vitro).

We encourage readers to consider how this and other socially-relevant measures might be similarly as well as further analyzed.

**References**

1. R. Huxley, S. Mendis, E. Zheleznyakov, S. Reddy and J. Chan, , *Body mass index, waist circumference and waist:hip ratio as predictors of cardiovascular risk – a review of the literature*, European Journal of Clinical Nutrition **64**(1),16-22 (2010).

2. A. Keys, F. Fidanza, M.J. Karvonen, N. Kimura and H.L. Taylor, *Indices of Relative Weight and Obesity*, J. Chron. Dis. **25**(6-7), 329-343 (1972).

3. G. Eknoyan, *Adolphe Quetelet (1796-1874) – the average man and indices of obesity,* Nephrology Dialysis Transplantation **23**(1), 47-51 (2008).

4. A. Quetelet, *Recherches sur le poids de l'homme aux different ages.* Nouveaux Memoire de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles. 1832, t. VII.

5. A. Quetelet, *A Treatise on Man and the Development of his Faculties*. Originally published in 1842. Reprinted in 1968 by Burt Franklin, New York.

6. http://www.cdc.gov/healthyweight/assessing/bmi/adult_bmi/index.html (Sept 14 – 2011)

7. N. J. MacKay, *Scaling of human body mass with height; the Body Mass Index revisited*, J. Biomech. **43**(4)*,* 764-66 (2010).

8. H. Lin, *Fundamentals of biological scaling*, American Journal of Physics **50**(1), 72-81 (1982).

9. For an extreme being of only fat or only muscles there is a density change of less than 20%.

10. C.R. White and R.S. Seymour, *Mammalian basal metabolic rate is proportional to body mass**2/3*, PNAS **100**(7), 4046-49 (2003).

11. G.B. West, W.H. Woodruff and J. H. Brown, *Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals,* PNAS **99**(Supplement 1), 2473-78 (2002).

12. R.D. Mosteller, *Simplified calculation of body-surface area.* N Engl J Med **317**(17), 1098 (1987).

13. T.K. Lam and D.T. Leung, *More on simplified calculation of body-surface area*. N Engl J Med **318**(17), 1130 (1988).

14. B.J.R. Bailey and G.L. Briars, *Estimating the surface area of the human body*, Statistics in medicine **15**(13), 1325-32 (1996).

15. G. B. West, W.H. Woodruff, and J. H. Brown, Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals, PNAS **99**, 2473-2478 (2002).

S. P. Apell, O. Wahlsten and H. Gawlitza

*Department of Applied Physics*

*Chalmers University of Technology*

*SE-412 96 Göteborg, Sweden*

These contributions have not been peer-refereed. They represent solely the view(s) of the author(s) and not necessarily the view of APS.