# Bodyfat Analysis - 20-Compartment Percentage Formula

There have been a variety of formulas presented to convert values of Average Body Density to Bodyfat Percentages. In order for ease of use, they have all been relatively simple formulas, with few terms to calculate.

The first was the Siri 2-Compartment model (W.E. Siri, 1956). It is based on the assumption that the human body is made up of just two types of material: fat-mass (FM); and fat-free-mass (FFM); and that the two are assumed to have average densities of 0.900 kg/cc and 1.100 kg/cc respectively. Further, it is assumed that the relative amounts of the three major components of the FFM (aqueous, mineral and protein) are also known as defined, and that they are constant and the same in all individuals. The Siri 2-comp model formula is Bodyfat % = (4.95/Db - 4.50) *100

Siri made an improved 3-Compartment model formula a few years later (W.E. Siri, 1961) which corrected for the total body water (Siri 3-Comp). The Siri 3-comp model formula is Bodyfat % = (2.11/Db) - (0.78 * W - 1.354) * 100

Brozek made a different 2-compartment model shortly after (J. Brozek, 1963), by revising some quantitative assumptions. The Brozek 2-comp model formula is Bodyfat % = (4.57/(Db) - 4.142) * 100

Db is the measured Hydrostatic average body density (gm/cc), generally corrected for Residual Volume of air remaining in the lungs.
W is the total body water as a fraction of body weight.

Katch added some knowledge to the field (F.I. Katch, 1967)

In the 1980s, various researchers recognized that those formulas, especially the popular 2-compartment models, regarding gender, age, race, health, and other issues, were sometimes quite incorrect. T.G. Lohman (1984, 1986, 1989) and D.H. Nielsen (1993) specifically publishing several papers regarding application to children.

During the 1990s, several 4-compartment models were presented. Two of the most respected are the 4-Comp models of body composition referred to as the Heymsfield 4-Comp and Baumgartner 4-Comp (S.B. Heymsfield, S. Lichtman, R.N. Baumgartner, J. Wang, Y. Kamen, A. Aliprantis, and R.N. Pierson Jr., Am. Jour. Clin. Nutr. 52: 52-58, 1990.).

The Heymsfield 4-Comp model formula is Bodyfat % = {[2.748 / D(P+F) - 2.051] [BW - (A + M)]/BW} * 100

The Baumgartner 4-Comp model formula is Bodyfat % = {[2.75 * (BV) - 0.714 * (A) + 1.148 * (M) - 2.05 * (BW)]/BW} * 100

D(P+F) is the density of protein-plus-fat compartment mixtures
BW is the body weight (kg)
A is the aqueous mass (kg)
M is the mineral mass (kg)
BV is the total body volume

It is generally assumed that neutron-activation analysis provides the most complete body composition analysis currently available. However, that process is very involved and expensive, and the Heymsfield 4-Comp model is commonly now considered the criterion to evaluate other models by.

Of the simpler formulas, only the 3-compartment Siri model is generally considered of reasonable accuracy.

From a pure Physics point of view, these various models seem all intrinsically flawed. They all involve assorted assumptions which clearly are not always true for all people. There also seems to be an implicit assumption regarding the actual scale of the bodyfat numbers generated by the formulas, which all seem to have been produced to agree with the basic Siri numbers from the 1950s. It is not clear that any of those formulas generate numbers that are actually strictly true "absolute percentages", but instead appear to be "relative percentages".

The primary basic research regarding actual composition seems to have all been done in the 1860s and 1870s in autopsy analysis. We suspect that the public would not tolerate newly dead human bodies being cut apart just to weigh the various parts, as being very disrespectful, which might explain why more recent data does not seem available.

We propose considering the body as being the sum total of each of the recognizable component parts, and therefore considering the (volume) percentages and densities of each of those components. The total of all the volume percentages would then necessarily equal 100%. Any adjustments to any individual component percentage (as for an adjustment for "big-boned" possibly being a slight increase in the volume percentage of that portion of the body which is bone), would then require an internal program correction of all the component percentages to still total 100%.

A simple questionnaire could provide any such adjustments of percentages or densities for a specific individual. Gender, race, elderly, youth, big-boned, delicate, athlete, sedentary, active, muscular, etc, can then easily each initiate adjustments to appropriate parameters of the formula, for a computer or web-site program to generate accurate Bodyfat Percentage results for that individual. The questionnaire can also ask about the success of expelling air from the lungs, permitting infirm or elderly people to get reasonably accurate results without exhaling at all (where the formula would use the Average lung capacity of 2.5 liters rather than the absolute minimum of 1.2 liters in its calculations).

The basic data for the formula then is used to multiply the volume percentages by the densities of each component part, to generate a prediction for a dry body weight for the specific individual. The person weighs him/herself to establish whether the predicted dry weight is correct, which is therefore an indication that the inputted questionnaire data was accurate.

Here are the starting data figures currently used for an Average Adult person (man):

ComponentMen Volume / Volume PercentageWomen Volume / Volume PercentageDensity gm/cc
Bone9% (13.5% by weight)9% (13.5% by weight)1.5
Blood7%7%1.060
Muscle36% (38% by weight)30% (32% by weight)1.0597
Air in LungsResidual Volume 1.2 liter / 1.4%Residual Volume 1.2 liter / 1.4%0.001
Gases in IntestineEstimated 0.2 liter / 0.2%Estimated 0.2 liter / 0.2%0.002
Brain1.5 liter / 1.6%1.3 liter / 1.6%1.3-1.4 kg / 0.93-1.0
Liver1/36 body weight / 2.8%1/36 body weight / 2.8%1.05
Stomach1/700 body weight / 0.15%1/700 body weight / 0.15%
Kidneys4.5-6 oz or 1/240 body weight or 0.4%4-5.5 oz or 1/240 body weight or 0.4%
Spleen1/320-1/400 body weight / 0.3%1/320-1/400 body weight / 0.3%
Heart1/169 body weight / 0.6%1/149 body weight / 0.7%
Lungs1/37 body weight / 2.7%1/43 body weight / 2.3%0.345 to 0.746
Skin4%4%
Intestines2%2%
Pancreas2-3.5 oz / 0.2%2-3.5 oz / 0.2%
Misc, cerebrospinal fluid, etc
Bodyfat26% (23.9% by weight)32% (29.4% by weight)0.918
TOTAL100%100%Average Body Density
These weight totals would be:
Men: 13.5% + 7.4% + 38.15% + 0 + 0 + 1.55% + 2.94% + 0.17% + 0.42% + 0.32% + 0.32% + 0.63% + 1.4% + 4.12% + 4.92% + 2.06% + 0.21% + 23.87% which gives here 101.98 or an average body density of 1.0198 gm/cc.
Women: 13.5% + 7.4% + 31.8% + 0 + 0 + 1.55% + 2.94% + 0.17% + 0.42% + 0.32% + 0.32% + 0.73% + 1.2% + 4.12% + 5.20% + 2.06% + 0.21% + 29.38% which gives here 101.32 or an average body density of 1.0132 gm/cc.

This would indicate that this woman would neutral float with a normal air load in her lungs (average density then being 0.9982 gm/cc) (in pool water that is density 0.9978 gm/cc) but would sink when she exhaled all her air. This man would sink in both cases, only being able to float with a significant in-breath of air.

The aqueous fraction of the fat-free body is slightly different for different people, being around 0.763 ±0.009 for young women, 0.723 ±0.014, for young men, 0.756 ±0.016 for elderly women, and 0.744 ±0.009 for elderly men. This suggests that slight adjustments should be made to the density figures for most of these components, due to gender and age.

This presentation was first placed on the Internet in December 2006.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago