# Difference between revisions of "Talk:Body fat excess"

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=== Body fat in the healthy reference population === | === Body fat in the healthy reference population === | ||

− | :::: [[Lean body mass]] of an individual (object), ''M''<sub>L</sub> [kg/x], is the fat-free body mass, and is thus defined as ''M''<sub>L</sub> <big>≝</big> ''M''-'' | + | :::: [[Lean body mass]] of an individual (object), ''M''<sub>L</sub> [kg/x], is the fat-free body mass, and is thus defined as ''M''<sub>L</sub> <big>≝</big> ''M''-''M''<sub>F</sub>, |

− | <big>'''Eq. 2''': ''M'' <big>≝</big> ''M''<sub>L</sub> + '' | + | <big>'''Eq. 2''': ''M'' <big>≝</big> ''M''<sub>L</sub> + ''M''<sub>F</sub></big> |

:::: In turn, ''M'' is the sum of the reference mass at a given height and excess body mass, ''M''<sub>E</sub> <big>≝</big> ''M''-''M''°, | :::: In turn, ''M'' is the sum of the reference mass at a given height and excess body mass, ''M''<sub>E</sub> <big>≝</big> ''M''-''M''°, | ||

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<big>'''Eq. 3''': ''M'' <big>≝</big> ''M''° + ''M''<sub>E</sub></big> | <big>'''Eq. 3''': ''M'' <big>≝</big> ''M''° + ''M''<sub>E</sub></big> | ||

− | :::: Excess body mass, ''M''<sub>E</sub>, is due to accumulation of an excess fat mass, '' | + | :::: Excess body mass, ''M''<sub>E</sub>, is due to accumulation of an excess fat mass, ''M''<sub>FE</sub>, accompanied by a gain of excess lean mass, ''M''<sub>LE</sub>, 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, |

− | <big>'''Eq. 4''': ''M''<sub>E</sub> <big>≝</big> '' | + | <big>'''Eq. 4''': ''M''<sub>E</sub> <big>≝</big> ''M''<sub>FE</sub> + ''M''<sub>LE</sub></big> |

:::: Inserting Eq. 4 into Eq. 3, | :::: Inserting Eq. 4 into Eq. 3, | ||

− | <big>'''Eq. 5''': ''M'' = ''M''° + '' | + | <big>'''Eq. 5''': ''M'' = ''M''° + ''M''<sub>FE</sub> + ''M''<sub>LE</sub></big> |

− | :::: The fat mass, '' | + | :::: The fat mass, ''M''<sub>F</sub>, is defined as the sum of the reference fat mass and excess fat mass, ''M''<sub>F</sub> <big>≝</big> ''M''°<sub>F</sub>+''M''<sub>FE, hence |

− | <big>'''Eq. 6''': '' | + | <big>'''Eq. 6''': ''M''<sub>FE</sub> <big>≝</big> ''M''<sub>F</sub> - ''M''°<sub>F</sub></big> |

− | :::: 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''°<sub>L</sub> = ''M''-'' | + | :::: 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''°<sub>L</sub> = ''M''-''M''°<sub>F</sub>), plus the total body fat mass and the excess lean mass, |

− | <big>'''Eq. 7''': ''M'' = ''M''° - '' | + | <big>'''Eq. 7''': ''M'' = ''M''° - ''M''°<sub>F</sub> + ''M''<sub>F</sub> + ''M''<sub>LE</sub></big> |

:::: Normalization for ''M''° and considering that the [[body mass excess]] is BME=''M''/''M''°-1, | :::: Normalization for ''M''° and considering that the [[body mass excess]] is BME=''M''/''M''°-1, | ||

− | <big>'''Eq. 8''': BME = '' | + | <big>'''Eq. 8''': BME = ''M''<sub>F</sub>/''M''° - ''M''°<sub>F</sub>/''M''° + ''M''<sub>LE</sub>/''M''°</big> |

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

− | <big>'''Eq. 9''': '' | + | <big>'''Eq. 9''': ''M''<sub>LE</sub>/''M''° = ''f''(BME)</big> |

:::: Inserting Eq. 8 and 9 into Eq. 7.2 yields | :::: Inserting Eq. 8 and 9 into Eq. 7.2 yields | ||

− | <big>'''Eq. 10''': BME = '' | + | <big>'''Eq. 10''': BME = ''M''<sub>F</sub>/''M''° - ''M''°<sub>F</sub>/''M''° + ''f''(BME)</big> |

− | :::: Solving for the measured variable '' | + | :::: Solving for the measured variable ''M''<sub>F</sub> normalized for ''M''°, |

− | <big>'''Eq. 11''': '' | + | <big>'''Eq. 11''': ''M''<sub>F</sub>/''M''° = BME - ''f''(BME) + ''M''°<sub>F</sub>/''M''°</big> |

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

− | <big>'''Eq. 12''': '' | + | <big>'''Eq. 12''': ''M''<sub>F</sub>/''M''° = (1-''f'')·BME + ''M''°<sub>F</sub>/''M''°</big> |

:::: 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. | :::: 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. |

## Revision as of 20:38, 15 January 2020

Work in progress

### 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},

- Lean body mass of an individual (object),

Eq. 2:M≝M_{L}+M_{F}

- In turn,
*M*is the sum of the reference mass at a given height and excess body mass,*M*_{E}≝*M*-*M*°,

- In turn,

Eq. 3:M≝M° +M_{E}

- 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,

- Excess body mass,

Eq. 4:M_{E}≝M_{FE}+M_{LE}

- Inserting Eq. 4 into Eq. 3,

Eq. 5:M=M° +M_{FE}+M_{LE}

- 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}

- The fat mass,

Eq. 6:M_{FE}≝M_{F}-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,

- 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,

Eq. 7:M=M° -M°_{F}+M_{F}+M_{LE}

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

- Normalization for

Eq. 8: BME =M_{F}/M° -M°_{F}/M° +M_{LE}/M°

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

- The excess lean mass normalized for

Eq. 9:M_{LE}/M° =f(BME)

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

Eq. 10: BME =M_{F}/M° -M°_{F}/M° +f(BME)

- Solving for the measured variable
*M*_{F}normalized for*M*°,

- Solving for the measured variable

Eq. 11:M_{F}/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:M_{F}/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.

- In this plot (Fig. 1), the slope equals (1-