The Fugue

So it’s been awhile since I estimated my residual lung volume (RV), and I figured it was time to do it again.

I’m a big guy, so my lungs hold a lot of air. When you’re blowing all your air into containers in the bathtub, and you’re ⅔ or more exhaled, is not the best time for pausing to move your straw to another container—or worse, refilling the container. This time I decided to use a balloon.

I took a deep breath, exhaled maxmially into the balloon, then blew the rest into the container through a straw. (Wait, did he just say “the rest” after “maximally exhaled”?) I heard that. Yes, you can’t exhale all of your air because the pressure in the balloon is higher than atmospheric pressure. In my case, I had another 400ml of air.

Then I emptied the air out of the balloon into the container. This is easy and leisurely once you figure out the trick, but it can seem next to impossible at first. Hint: don’t try to submerge the balloon. If you grab the lip of the balloon mouth only, and avoid pinching the neck, you can control the air flow very well. There, I had measured my vital lung capacity (VC).

It worked great, and compared with the last circus event when I measured VC it was much easier.

Now I had to figure out how to get from VC to RV. The clown who wrote http://hans.fugal.net/density kind of left this step vague. I’ve remedied that and added a page to my spreadsheet. For the curious, my RV is up from 2.0 liters to 2.2 liters, and a total lung capacity of 8.2 liters.

Remember that system I came up with for calculating body fat percentage using a gallon jug in a swimming pool? I always let the computer do the calculations for me—I have a little script that I run that updates my weight graph. But not everyone is as geeky as that, and formula is not that simple, and when you add units conversion in it gets downright hairy.

I finally figured out how to generate a nomogram. Now you have no excuses.

A couple days back I posted my idea for measuring body density and estimating
body
fat.
Dad, who has a set of skinfold calipers gave it a try and gave me comparative
results, and asked the question on everbody’s mind: just how accurate is it,
especially with that pretty blatant guess at residual lung volume?

So I took some time to learn how to account for uncertainty and take a stab at
pinning a confidence interval on the technique. First of all, I didn’t realize
how complicated uncertainty propogation is. Partial derivatives, squares and
square roots, etc. Luckily, I came across some lecture or presentation notes
detailing a sequential perturbation method (instead of an analytical method). I
could have talked Jacob into walking me through the partial derivatives, but
this method is easy to code and a find in and of itself. Read about it in this
PDF.

I coded up the formula and ran some test data through it. Here’s the equation
again for review: ρ = m / ((m + m_c)/ρ_w – (v_a +
v_c + v_r)) Here’s the values and uncertainty I attribute
to each variable:

• m = 121.29 ±0.02 kg
• ρ_w = 0.997 ±0.001 kg/l
• v_a = 1.13 ±ٍ0.01 l
• v_r = 1.87 ±0.5 l
• m_c = 0 ±0.02 kg
• v_c = 0 ±0.01 l

I didn’t actually use a counterbalance, but I included the uncertainty in
measuring its mass and volume as if I had, just for completeness. As suspected,
v_r has the largest uncertainty. I calculated the uncertainty if
v_r were magically accurate, and found that the uncertainty was
0.0014 kg/l. This translates to about 0.65% body fat with Siri’s equation
(ignoring the uncertainty inherent in that equation, which is a constant bias
accross measurements for one person on any given day).

Note that I give ρ_w this time, instead of whisking it away with a
magical 1 kg/l. I picked an average value between 72°F and 84°F (most pools are
in this range), with an uncertainty (due to water temperature) of about 0.001
kg/l. If you use 1 kg/l instead you are introducing a bias of about 0.9% body
fat. So I was wrong about that being insignificant.

Now, I found a better estimate
(why better? because it seems to come from a more reputable source than
Wikipedia) for residual lung volume: v_r = RV = 0.24 VC. So I may
have overestimated my RV last time by ½ liter.
(Update: I think that must be a typo on that page, they probably mean 24% or 28% of total capacity instead. This fits in much better with the rest of the literature that I have found, e.g. Quanjer and Paoletti.) That seems like a generous
uncertainty measure for RV, too. With that uncertainty factored in, we get an
uncertainty of about 2.1% body fat, or about 5% is you are on the slight side of average (the less you weigh, the more difference that 1/2 liter makes).

So, Dad, let’s bump your score up by about 1% for the density of water and then
tack an uncertainty of 2% onto it, you have a body fat of 26.3% ±2%. I’m no
expert on using calipers, but one paper’s
abstract indicates that the
skinfold method uncertainty is about 3%. I’ve seen 10% tossed around casually
too, but have no reliable source to back that up. That puts the two methods
within the appropriate reach of eachother, which is heartening. It’s
interesting to note that BMI is overestimating Dad’s fat, because he’s more
lean than the average couch potato. Imagine the difference if the subject were
someone completely nuts, like a young triathlete, who has body fat of about
15%. Even better, if you are such a nut you could do the experiment and post
your results (and BMI) here as a comment for us to see.

When you try to lose weight, what you are really trying to do is lose fat.
Weighing yourself is a first approximation of your progress, but a better
indicator is your body fat percentage (%BF). Unfortunately, measuring %BF can
be expensive and/or difficult. It doesn’t need to be so. All you need is a body
of water (e.g. a swimming pool), a gallon jug, and your bathroom scale.

One of the most accurate ways to measure body fat is hydrostatic weighing. You are weighed underwater and on land, and your body’s density is determined. Then body fat is estimated from the measured density. This is the same basic technique that we will use, but we don’t require an underwater scale or special tank.

First the how, then I’ll give you the physics. Get in the pool and exhale all
the air you can, and allow yourself to sink. You will sink unless you’re
particularly obese. Take note of the sensation of sinking. Then do the same
thing but with your lungs full. Take note of the sensation of floating. Now, we
want to reach the point of neutral buoyancy when your lungs are empty, where
you are neither sinking nor floating. You will be weightless under the water.
Take the gallon jug and hold it under the water, then exhale completely. If the
jug is full of air, you will probably float (unless you are quite lean, in
which case you’ll need two jugs). Keep adding water and repeating until you
reach neutral buoyancy. If you sink, add air (pour out some water). If you
float, add water. Once you’ve found the magic amount of water, use this
equation to calculate your body density (ρ):

where m is your mass (what the scale tells
you), v is the volume of you and your buoy combined, and v_air is the
volume of air in your buoy. If you have ¾ gallon of water in your gallon jug,
then v_air is ¼ gallon. ρ_w=1 kg/liter for all the
precision we need.

Once you have density, you may like to estimate your body fat. The equation for that is Siri’s equation, which says

This equation assumes your lungs are completely empty, which they can’t be, so we need to introduce a term for the residual volume of your lungs. This is about ¼ of your total lung capacity, or ⅓ of your vital lung capacity. You can measure your vital lung capacity with a balloon or a by blowing air through a straw into an inverted container filled with water. The average residual lung capacity for an american male is 1.2 liters; mine is about 1.9 liters. So we can adjust the formula for density as follows:

If you do this experiment you will probably find that your estimated %BF is not too far from your BMI, which is a statistical tool for estimating %BF. It can be wildly inaccurate for statistical outliers (e.g. people who are actually in shape), but it’s easy to calculate and a decent sanity check in this case.

Here’s what’s going on. We’re using Archimedes’ principle: the buoyant force on a submerged object is equal to the weight of the fluid displaced. When the buoyant force balances the force of gravity, we have neutral buoyancy. The buoyant force is expressed as . Substitute weight for the buoyant force and solve for the volume of the body (v = v_body + v_air), then substitute that into the definition of density (m/v), and you get the formula I gave you above (if you consider the mass of air and the gallon jug as negligible). I glossed over that—if you’d like me to go into more detail say so in the comments.

If you’re particularly obese and don’t sink when you exhale completely, then all is not lost. You just need some counterbalance. The modified equation is: