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# Lean body mass calculation: walk-through

This is a worked example of calculating (estimating) the lean body mass for a hypothetical patient:

• ht: 125 cm
• wt: 23kg
• gender: 'f'
• age: 7yrs

The formula for lean body mass(girls) is:

#### Equation 1

$ln(LBM) = -3.8345 + 0.954 \times \ln(height) + 0.6515 \times \ln(weight) - 0.0102 \times (BMI z-score)^2$

Looks pretty straight forward— except for getting a z-score for the body mass index (BMI). Lets do that first.

$BMI = \frac{wt}{ht^2}$ $BMI = \frac{23}{1.25^2}$ $BMI = \frac{23}{1.5625}$ $BMI = 14.72$

Next, we need the equation for the BMI z-score and the appropriate data from the CDC's published tables:

#### Equation 2

$Z = \frac{((\frac{BMI}{M})^{L}) - 1}{L\cdot S}$

and the age-and-gender-specific L, M, S data:

$age (months) = 12 \times 7$ $age (months) = 84$ $L = -2.93, M= 15.45, S = 0.11$

So the full BMI z-score equation is now:

$Z = \frac{((\frac{14.72}{15.45})^{-2.93}) - 1}{-2.93\cdot 0.11}$
$Z = \frac{(0.9528^{-2.93}) - 1}{-0.3223}$
$Z = \frac{(1.1522) - 1}{-0.3223}$
$Z = \frac{0.1522}{-0.3223}$
$Z = \frac{0.1522}{-0.3223}$
$Z = -0.4722$

Back to equation 1:

$ln(LBM) = -3.8345 + 0.954 \times \ln(height) + 0.6515 \times \ln(weight) - 0.0102 \times (BMI z-score)^2$

and now substituting in our values:

$ln(LBM) = -3.8345 + 0.954 \times \ln(125) + 0.6515 \times \ln(23) - 0.0102 \times (-0.4722)^2$
$ln(LBM) = -3.8345 + 0.954 \times 4.8283 + 0.6515 \times 3.1355 - 0.0102 \times (-0.4722)^2$
$ln(LBM) = -3.8345 + 4.6062 + 2.0428 - 0.0102 \times 0.2230$
$ln(LBM) = -3.8345 + 4.6062 + 2.0428 - 0.0023$
$ln(LBM) = 0.7717 + 2.0428 - 0.0023$
$ln(LBM) = 2.8122$
$LBM = exp(2.8122)$
$LBM = 16.6465$

### References

Limitations of Expressing Left Ventricular Mass Relative to Height and to Body Surface Area in Children.
Foster BJ, Gao T, Mackie AS, Zemel BS, Ali H, Platt RW, Colan SD.
J Am Soc Echocardiogr. 2012 Dec 22. pii: S0894-7317(12)00938-8. [Epub ahead of print]

# disclaimer

page updated: 14 Jan 2013