Additional Math Pages & Resources

Friday, January 14, 2011

How Fat Am I? Part V

This is the fifth blog on finding out how much body fat we have. The past 4 days we've looked at the easy methods of estimating body fat. Easy also means relatively inaccurate from a math perspective.

The definitive approach to determining body composition is called an autopsy. A person dies, someone cuts up the body, and weighs all the parts. Not a method anyone would choose! Although autopsies are often done to determine the cause of death, they are done very rarely for body composition research - perhaps only about 50 times in the past 50 years.

The best common approach to body composition measurement is called hydrostatic testing. After you breathe out, emptying your lungs of as much air as possible, operators in a fitness or research clinic will lower you into some water and measure the amount of water that is displaced. That will tell the testers precisely what your body volume is, and allow them to calculate how much of you is water.

(If this sounds familiar, it is the method I used to measure a copper ingot last summer. And it's how Archimedes discovered how to tell if a crown was made of silver or gold).

After you come up out of the water, the hydrostatic testers get to do some math. Lots and lots of math. How much air was still in your lungs, how much bone you have, etc.

Body Density = dry weight / [((dry weight - wet weight) / water density)- Residual Volume in Lungs - 0.1] 

I thought it would be simple, but after looking at a bunch of calculations, I decided I'm only a math publisher, not an exercise physiologist. I'm not going to put all the math in here. But here is an illustration and link to an extremely keen interactive way to learn about hydrostatic testing, from the Univ of Vermont.

While hunting around, I did find a business opportunity to operate a Mobile Hydrostatic Body Fat Testing Lab. It's essentially a big truck, like the one we use to deliver Excel Math. It contains a stainless tank, some scales, a heater, changing rooms, etc.

There is one more very accurate method - DEXA, or Dual Energy X-ray Absorbtiometry.

A subject reclines on a body scanner. X-rays come from below and pass upwards through the body to a detector overhead. Photons from two different kind of x-rays are measured. The ratios of each that reach the detector are compared to predict total body fat, fat-free mass, and total body bone mineral. This takes only 10 - 20 minutes and is useful for most people except those too large for the machine.
The main issue with DEXA testing is that the machines are large, they emit X-rays, and they take a whole room. And they cost a lot of money, say $50,000-150,000+.  Perhaps fine for labs or hospitals, but not for the home.

I suppose if you got the tank truck franchise you could put a DEXA machine in your truck too, and offer customers a choice of testing methods ... math would be useful to count all the money!