Talk:Body fat excess

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Body fat in the healthy reference population

Lean body mass, M_L [kg], is the fat-free body mass, and is thus defined as M_L ≝ M-m_F,

M ≝ ML + mF                          (Eq. 2)

In turn, M is the sum of the reference mass at a given height and excess body mass, M_E ≝ M-M°,

M ≝ M° + ME                                 (Eq. 3)

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,

ME ≝ mFE + mLE                         (Eq. 4)

Inserting Eq. 4 into Eq. 3,

M = M° + mFE + mLE                           (Eq. 5)

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}

mFE ≝ mF - m°F                           (Eq. 6)

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,

M = M° - m°F + mF + mLE                 (Eq. 7)

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

BME = mF/M° - m°F/M° + mLE/M°             (Eq. 8)

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

mLE/M° = f(BME)                                             (Eq. 9)

Inserting Eq. 8 and 9 into Eq. 7.2 yields

BME = mF/M° - m°F/M° + f(BME)                   (Eq. 10) 

Solving for the measured variable m_F normalized for M°,

 mF/M° = BME - f(BME) + m°F/M°             (Eq. 11)

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

 mF/M° = (1-f)·BME + m°F/M°                 (Eq. 12)

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.