The quantity a, the ratio of total mineral to protein, is discussed below under “Density—Total Body Water Method.” The standard deviations o, and o,, include the uncertainty in the exact composition of the reference body but more particularly reflect dispersion in body composition for the population. They are, in effect, measures of the deviation of individuals from a fixed reference. The standard deviation in d,, the reference body density, is derived from o, and o.,, (See Appendix 2). The value of o,, is estimated from the combined data of Keys und Brozgek et al., (1953), and Siri (1956). (1958), Behnke (1954); Behnke, Densitometric Method A correlation between corporeal density and fatness was suspected as early as 1901 by Stern (1901), but lacking an accurate technique for measuring body density, he could not establish a well-defined relationship.' By improving the underwater weighing method for determining density of the body by Archimedes’ principle, and compensating for lung volume, Behnke, et al. (1942) were able to demonstate a high correlation between overweight and density. Using this method Rathbun and Pacé (1945) formulated a quantitative relationship between body density and depot fat in guinea pigs by comparison with direct chemical analysis. The semi-empirical expression derived by these investigators has the form f ==(a/d)— b, in which d is body density and a and b are empirical constants, The constants derived for humans on the basis of the guinea pig studies, which were related to body specific gravity rather than density, were a == §.648 and b == 6.044. These values are still widely used although they contain a systematic error because they are based on an incorrect value of fat density. Keys and Brozek (1953) and Behnke (1954) later proposed somewhat different values based on more extensive though indirect human data and the correct fat density. The formula for estimating fat from density alone is derived from the general formulations in the Section on General Principles. It requires that all adult humans be identical in composition except for individual differences in their proportions of adipose tissue. Thus the individual is necessarily regarded as a reference body of standard composition to which adipose tissue of some prescribed composition has been appended or from which it has been removed. The formulas in the Section on General Principles are greatly \ editor’ Comment ¢J.4)5 In the hlatory of the densitometric snalyale of indy composition one should net overlogk the costribuulon of Wo Kohtrauach (Methudik sur quantitativen Hestimmuang der Kirperstoffe in vivo, Arhertaphymoal., 2, 28-46 (1930) > Zur Kenntnis dew Treiningszustandes, Arbeitspayaso 2, 46-60. 229 simplified for the densitometric method if expressed in terms of the density of the reference body d, and that of the generalized adipose tissue, d,. An individual who differs from the reference body by a proportion of adipose tissue A is characterized by a mean body density d, related to A by (5) 1 - a = A d, + 1—A ee ee Rearranging terms, the estimating equation for adipose tissue difference becomes 1 dd, | dy (6) Ama (4 "acd aa The difference that is pure fat is then Af == Af,, whereus the total proportion of fat in the individual is f = Af, + (1 — A)f,, or more explicitly, (7) 4 peng ( d,—d, _. d,d, fi — f. . 4 ay fi ofo_ Eqs. (6) and (7) are entirely general but still retain the form f =(a/d)— 6 that was proposed originally, The examples of numerical working forms of these equations may now be evaluated first on the basis of the Minnesota standard man, and then on the basis of the fat-free reference body. For the first of these, d, = 1.063 gm/ce, f, — 0.14, and f, — 0.62; hence, (8) A= Bibs —- 8.245 (9) = A208 3.817 These are essentially the equations proposed by Keys and Brozek (1953) except for small differences in the constants because fewer decimal places are used in d, and dy. If, on the other hand, the fat-free body is the correct reference, then d,, = 1.1 gm/ce, d, = 0.90 gm/ce, f, = 0.0, and f, = 1.0, and the fat estimating equation becomes (10) fA 49° 4.500 It is of interest, before examining the uncertainty in the method, to compurethe values for fat derived from these and similar numerical formulas that have been proposed. For a man of density 1.050 gm/cc, the original Rathbun-Pace formula yields 23.9%, Keys and Brozek’s version, which is the same as Eq. (9) above, gives 18.9%, whereas Eq. (10) above gives 21.5%. For a density of 1.000, the total fat estimated by these two formulas differs by 6% body weight. 230 JISSVID WSITOUV LAW NOLLRLLIN fooOT WAGOLIO WAWALdAS Q ON 6 TON NOLLRELLAIN Pet V.V.U. LIDTMNANY