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''-''m''<sub>F</sub>,
+
:::: [[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> + ''m''<sub>F</sub></big>
+
  <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, ''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,
+
:::: 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> ''m''<sub>FE</sub> + ''m''<sub>LE</sub></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''° + ''m''<sub>FE</sub> + ''m''<sub>LE</sub></big>
+
  <big>'''Eq. 5''':  ''M'' = ''M''° + ''M''<sub>FE</sub> + ''M''<sub>LE</sub></big>
  
:::: 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
+
:::: 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''':  ''m''<sub>FE</sub> <big>≝</big> ''m''<sub>F</sub> - ''m''°<sub>F</sub></big>
+
  <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''-''m''°<sub>F</sub>), 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, ''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''° - ''m''°<sub>F</sub> + ''m''<sub>F</sub> + ''m''<sub>LE</sub></big>
+
  <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 = ''m''<sub>F</sub>/''M''° - ''m''°<sub>F</sub>/''M''° + ''m''<sub>LE</sub>/''M''°</big>
+
  <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''':  ''m''<sub>LE</sub>/''M''° = ''f''(BME)</big>
+
  <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 = ''m''<sub>F</sub>/''M''° - ''m''°<sub>F</sub>/''M''° + ''f''(BME)</big>
+
  <big>'''Eq. 10''':  BME = ''M''<sub>F</sub>/''M''° - ''M''°<sub>F</sub>/''M''° + ''f''(BME)</big>
  
:::: Solving for the measured variable ''m''<sub>F</sub> normalized for ''M''°,
+
:::: Solving for the measured variable ''M''<sub>F</sub> normalized for ''M''°,
  
   <big>'''Eq. 11''':  ''m''<sub>F</sub>/''M''° = BME - ''f''(BME) + ''m''°<sub>F</sub>/''M''°</big>
+
   <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''':  ''m''<sub>F</sub>/''M''° = (1-''f'')·BME + ''m''°<sub>F</sub>/''M''°</big>
+
   <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), ML [kg/x], is the fat-free body mass, and is thus defined as ML M-MF,
Eq. 2:  M  ML + MF
In turn, M is the sum of the reference mass at a given height and excess body mass, ME M-M°,
Eq. 3:  M  M° + ME
Excess body mass, ME, is due to accumulation of an excess fat mass, MFE, accompanied by a gain of excess lean mass, MLE, 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, MF, is defined as the sum of the reference fat mass and excess fat mass, MF M°F+MFE, 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 MF 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.