Talk:Body fat excess

Work in progress

Body fat in the healthy reference population

The healthy reference population, HRP, is a population without excess body fat (no overweight) nor underweight (no ‘negative excess’ body fat) over the entire allometric range of height, H, and corresponding reference body mass, M°. The HRP has a body mass excess of zero, BME=0. By extrapolation of the relation between normalized body fat mass, M_F/M°, and BME to the intercept at BME=0 (Fig. 5a), therefore, we obtain the fraction of body fat in the HRP, M°_F/M°, which is higher in women (0.27) compared to men (0.15). Three resilient features are most remarkable in the linear (or very nearly linear) plots of M_F/M° as a function of BME (Fig. 5a),

 Eq. 1:  MF/M° = slope·BME + intercept

1. Whereas the allometric function of H/M^A in the HRP (M=M°) is practically identical in females and males, the intercept M°_F/M° is nearly twice in women than men.
2. The slopes are practically identical in women and men. This provides a key rationale of the concept of body fat excess, BFE. Body mass excess, BME, constitutes a linearly proportional measure of BFE for women and men, over large ranges of height and a remarkable diversity of evolutionary backgrounds. However, no generalization to all ethnic populations is possible.
3. The linear slope can be rationalized by a minimum model of BME related to accumulation of body fat. Note that the range of underweight with negative BME requires an entirely different analysis. For interpretation of Eq. 1, consider the total body mass as the sum of the reference body mass, M°, and excess body mass, M_E,

Eq. 2:  M ≝ M° + ME

Excess body mass results from excess fat mass, M_FE, and excess non-fat (lean) mass, M_LE,

Eq. 3:  ME = MFE + MLE

It is useful to summarize some simple definitions of excess mass normalized for M°,

Eq. 4a	Body mass excess:	BME = ME/M° 	=  M/M°-M°/M°
Eq. 4b	Body fat mass excess:	BFE = MFE/M° 	=  MF/M°-M°F/M°
Eq. 4c	Body lean mass excess:	BLE = MLE/M° 	=  ML/M°-M°L/M°

Dividing Eq. 2 by M° and inserting definitions of Eq. 3 and Eq. 4,

Eq. 5:  M/M° = 1 + BFE + BLE

Solving for BFE and inserting Eq. 4a,

Eq. 6:  BFE = BME - BLE

M°_F/M° is the intercept of Eq. 1, which can be expressed by inserting BFE from Eq. 4b in the simplified form (Fig. 5b),

Eq. 7:  BFE = slope∙BME

Therefore, the slope is the proportionality constant between BFE and BME. In other words, the slope is the fraction, f_FE, of body fat excess contributing to total body mass excess,

Eq. 8:  fFE = BFE/BME

Since the slope is less than one, f_FE<1, non-fat (lean) mass contributes to the total body mass excess. BLE (Eq. 4c) is due to the ‘weight-lifting’ effect: long-term adjustments to excess weight are known to induce heavier and larger (but not longer) bones, an increase in low-aerobic capacity muscle mass, and possibly other non-fat contributors to excess lean body mass, M_LE. The linear slope (Eq. 6; Fig. 5) requires that BLE is a linear function of BME (compare Eq. 8),

Eq. 9:  fLE = BLE/BME

Substituting BLE = f_LE∙BME in Eq. 6,

Eq. 10:  BFE = (1-fLE)∙BME

From Eq. 7 the slope is f_FE=0.57 (Fig. 5b), and f_FE = 1- f_LE (Eq. 10); thus f_LE=0.43. A slope f_FE of 0.5 would suggest that BFE and BLE contribute equally to BME. f_FE>0.5 indicates that BFE accounts for a higher contribution than BLE to BME.

An alternative route

Lean body mass of an individual (object), M_L [kg/x], is the fat-free body mass, and is thus defined as M_L ≝ M-M_F,

Eq. #2:  M ≝ ML + MF

In turn, M is the sum of the reference mass at a given height and excess body mass, M_E ≝ M-M°(Eq. #2). Excess body mass, M_E, is due to accumulation of an excess fat mass, M_FE, accompanied by a gain of excess lean mass, M_LE, which includes increased bone mineral density, added bone mass and muscle mass due to the mechanical 'weight-lifting effect' (Iwaniec 2016 J Endocrinol). Thus Eq. #2 and 2 combined yield the definition for excess body mass,

Eq. #4:  ME ≝ MFE + MLE

Inserting Eq. #4 into Eq. #2,

Eq. #5:  M = M° + MFE + MLE

The fat mass, M_F, is defined as the sum of the reference fat mass and excess fat mass, M_F ≝ M°_F+M_{FE, hence}

Eq. #6:  MFE ≝ MF - M°F

Inserting Eq. #6 into Eq. #5 yields body mass as the sum of the reference mass minus reference fat mass (which is the reference lean mass, M°_L = M-M°_F), plus the total body fat mass and the excess lean mass,

Eq. #7:  M = M° - M°F + MF + MLE

Normalization for M° and considering that the body mass excess is BME=M/M°-1,

Eq. #8:  BME = MF/M° - M°F/M° + MLE/M°

The excess lean mass normalized for M° is a function of BME,

Eq. #9:  MLE/M° = f(BME)

Inserting Eq. #8 and #9 into Eq. #7.2 yields

Eq. #10:  BME = MF/M° - M°F/M° + f(BME)

Solving for the measured variable M_F normalized for M°,

 Eq. #11:  MF/M° = BME - f(BME) + M°F/M°

which finally shows the equation derived to plot the normalized body fat mass as a function of BME,

 Eq. #12:  MF/M° = (1-f)·BME + M°F/M°

In this plot (Fig. 1), the slope equals (1-f), and the intercept is the fat mass normalized for the reference mass at a given height in the HRP.