Talk:Body fat excess

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Body fat in the healthy reference population

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. 3:  M ≝ M° + ME

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 3 combined yield the definition for excess body mass,

Eq. 4:  ME ≝ MFE + MLE

Inserting Eq. 4 into Eq. 3,

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.