Talk:Body fat excess: Difference between revisions

Revision as of 18:22, 17 January 2020

Work in progress

Body fat in the healthy reference population - a complementary route

In turn, M is the sum of the reference mass at a given height and excess body mass, M_E ≝ M-M°(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 . Thus Eq. 13 and 2 combined yield the definition for excess body mass, M_E ≝ M_FE + M_LE (Eq. 4).

Inserting Eq. 4 into Eq. 3,

Eq. 13:  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}

Eq. 14:  M_FE ≝ M_F - M°_F

Inserting Eq. 14 into Eq. 13 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. 15:  M = M° + M_F - M°_F + M_LE

Normalization of Eq. 15 for M° and considering that the body mass excess is BME=M/M°-1 (Eq. 5a), BFE = (M_F-M°_F)/M° (Eq. 5b), and BLE = M_LE/M° (Eq. 5c), yields Eq. 7 in the form of,

Eq. 16:  BME = BFE + BLE

By further normalization of Eq. 16 for BME, we obtain the summation of f_FE = BFE/BME (Eq. 9) and f_LE = BLE/BME (Eq. 10),

Eq. 17:  1 = f_FE + f_LE

where f_FE = 0.57 is the slope in Fig. 5b.

To derive the M_LE/M_FE ratio (Eq. 12), which is equal to BLE/BFE (Eq. 5b and 5c), Eq. 16 is divided by BFE and rearranged,

Eq. 18:  BLE/BFE = BME/BFE - 1

Eq. 18 is equivalent to Eq. 12, since BME/BFE = f_FE (Eq. 9).

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   ''Work in progress''
-=== Body fat in the healthy reference population - an alternative route ===
+=== Body fat in the healthy reference population - a complementary route ===
-:::: [[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>,
+:::: 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''°(Eq. 3). 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 . Thus Eq. 13 and 2 combined yield the definition for excess body mass, ''M''<sub>E</sub> <big>≝</big> ''M''<sub>FE</sub> + ''M''<sub>LE</sub></big> (Eq. 4).
- <big>'''Eq. 12''':  ''M'' <big>≝</big> ''M''<sub>L</sub> + ''M''<sub>F</sub></big>
+:::: Inserting Eq. 4 into Eq. 3,
-:::: 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''°(Eq. 2). 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 . Thus Eq. 12 and 2 combined yield the definition for excess body mass,
+  <big>'''Eq. 13''':  ''M'' = ''M''° + ''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. 12,
-  <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
-  <big>'''Eq. #6''':  ''M''<sub>FE</sub> <big>≝</big> ''M''<sub>F</sub> - ''M''°<sub>F</sub></big>
+  <big>'''Eq. 14''':  ''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,
- <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,
- <big>'''Eq. #8''':  BME = ''M''<sub>F</sub>/''M''° - ''M''°<sub>F</sub>/''M''° + ''M''<sub>LE</sub>/''M''°</big>
+:::: Inserting Eq. 14 into Eq. 13 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,
-:::: The excess lean mass normalized for ''M''° is a function of BME,
+ <big>'''Eq. 15''':  ''M'' = ''M''° + ''M''<sub>F</sub> - ''M''°<sub>F</sub> + ''M''<sub>LE</sub></big>
- <big>'''Eq. #9''':  ''M''<sub>LE</sub>/''M''° = ''f''(BME)</big>
+:::: Normalization of Eq. 15 for ''M''° and considering that the [[body mass excess]] is BME=''M''/''M''°-1 (Eq. 5a), BFE = (''M''<sub>F</sub>-''M''°<sub>F</sub>)/''M''° (Eq. 5b), and BLE = ''M''<sub>LE</sub></big>/''M''° (Eq. 5c), yields Eq. 7 in the form of,
-:::: Inserting Eq. #8 and #9 into Eq. #7.2 yields
+ <big>'''Eq. 16''':  BME = BFE + BLE</big>
- <big>'''Eq. #10''':  BME = ''M''<sub>F</sub>/''M''° - ''M''°<sub>F</sub>/''M''° + ''f''(BME)</big>
+:::: By further normalization of Eq. 16 for BME, we obtain the summation of ''f''<sub>FE</sub> = BFE/BME (Eq. 9) and ''f''<sub>LE</sub> = BLE/BME  (Eq. 10),
-:::: Solving for the measured variable ''M''<sub>F</sub> normalized for ''M''°,
+ <big>'''Eq. 17''':  1 = ''f''<sub>FE</sub> + ''f''<sub>LE</sub>
-  <big>'''Eq. #11''':  ''M''<sub>F</sub>/''M''° = BME - ''f''(BME) + ''M''°<sub>F</sub>/''M''°</big>
+:::: where ''f''<sub>FE</sub> = 0.57 is the slope in Fig. 5b.
-:::: which finally shows the equation derived to plot the normalized body fat mass as a function of BME,
+:::: To derive the ''M''<sub>LE</sub>/''M''<sub>FE</sub> ratio (Eq. 12), which is equal to BLE/BFE (Eq. 5b and 5c), Eq. 16 is divided by BFE and rearranged,
-  <big>'''Eq. #12''':  ''M''<sub>F</sub>/''M''° = (1-''f'')·BME + ''M''°<sub>F</sub>/''M''°</big>
+ <big>'''Eq. 18''':  BLE/BFE = BME/BFE - 1</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.
+:::: Eq. 18 is equivalent to Eq. 12, since BME/BFE = ''f''<sub>FE</sub> (Eq. 9).