Plethysmography. That word! Quite possibly the hardest word to pronounce in the English language. Right up there with “thistle” and “chrysanthemum.” One of those dreadful words with legitimate “th’s” in them that thound like mithtakes and remind me that I therioulthy thuffered in thixth grade from a lithp.

Thuper. Theven years of thpeech therapy dithmantled in a thingle paragraph.

But beyond the manner in which it forces me to emotionally decompensate into a bullied preteen, plethysmography is like physics magic! And I’ve discovered that even those who use it daily as part of their PFTs often don’t really understand how mathematically awesome it is underneath.

From here on in I will refer to plethysmography as simply ‘pleth’ because it is just as annoying to type as it is to pronounce.

Pleth measures changes in volume. In respiratory therapy, we use it to calculate the FRC, Functional Residual Capacity, of the lungs. But it can also be used to calculate blood volume changes in the limbs, cerebral blood flow, and a more recent study, an exercise in “duh”, used penile pleth to measure changes in blood volume in the penis showing that, duh, the more homophobic men are, the more aroused they are by gay porn.

But that’s another story. Never mind. Anyway…. <–Sondheim Quote. Felt appropriate here.

To obtain lung volumes, we put the patient in a sealed box of a known volume that has a pressure transducer in it to measure changes in pressure. The patient is connected to an outside air source and another pressure transducer is placed at the opening of the mouth.

And if you believe in the physics – AND YOU SHOULD! – knowing the starting volume and the pressures of the box and the mouth opening gives you enough information to calculate the lung volume at end tidal expiration (not forced expiration), aka the FRC.

Our story begins with the Ideal Gas Law. It’s good to have ideals. And laws. Gas… not so much.

This law is expressed by the equation:  PV = nrT ,  where P,V and T are Pressure, Volume and Temperature respectively.

n = number of moles of gas. Neither a rodent nor a spy, a mole in chemistry is 6.02×10^23 p/mol.
r = Boltzmann constant = 1.38066×10^23 J/K

Both of these constants have rich, interesting science and history behind them but all you really need to know is that, in a closed system, they are both constant. And hey, if they are always constant why not just combine them into one and call it “k” for “constant” because “constant” starts with…. k? It must be a foreign language thing. I blame the Germans. But just roll with it. K? (It actually has to do with the standard conventions of proportionality mathematics.)

We’re left with: PV = kT

Well to make things even simpler, in our little body box closed system, T also is a constant. Or it becomes constant after a minute or so when the body heat of the person and the ambient temp in the box reaches an equilibrium. Until that happens you just make the patient sit there, trapped, on the edge of claustrophobic panic.

So with T constant, we can roll it right into k and reduce the equation to:

PV = k.

Pressure and Volume are inversely proportional. Meaning that as the pressure rises, the volume falls. And vice versa, if the volume rises, the pressure falls. It’s a simple, linear relationship. This was observed by scientist Robert Boyle’s in 1662 when he published his findings and labeled it ‘Boyle’s law’, stating that the volume of an ideal gas is inversely proportional to its absolute pressure at a constant temperature. They call these factoids of physics “laws” but they are more like “absolute truths”. You can break a law. You can’t break *this*.

So in a system where there is a change in volume

P1*V1 = P2*V2 <– This is everything. Know it. Live it. Love it. Become one with it.

Now back to our box:

The patient is asked to breathe normally from the mouthpiece for a while (because being trapped in a box breathing through a tube makes that so easy) until the end tidal baseline becomes steady and consistent. Just when they get comfortable, on exhale, you occlude the mouth piece. It’s probably best to warn them that this is going to happen to avoid unnecessary flailing and eye bulging.

When the patient tries to breathe against the occluded mouthpiece their thoracic cage expands and contracts and the pressure being measured at the oral opening rises and falls in proportion.

Also, the volume of the box falls and rises and its pressure also varies accordingly.

Here’s where it get’s fun! Or even *more* fun. Cuz, I’m already there.

Starting with the pressure and volume of the box during the occlusion maneuver: We know the initial Pressure – P1. We know the initial Volume – V1. And we measure the pressure after the thoracic volume change on an attempted inhale – P2.

And since P1*V1 = P2*V2, this allows us to solve for V2.

V2 – V1 = the change in volume of the box. Which is – TA DA!!! – the change in volume of the thoracic cage on attempted inhale.

Now to the patient! Like the box, we know the pressure before inhale – P1, and the pressure after inhale – P2. And we now know the change in the Volume of the lung – dV (d is for Delta. Normally a triangle but I don’t know how to make that character on here).

So to solve for the volume of the lung at baseline tidal exhale, or FRC, (V1), we are left with this simple equation.

P1 * V1 = P2 * (V1 + dV)

Now just plug in the known values and, voila, you have the V1, or the FRC of the lungs. With the FRC, you can now extrapolate total lung volume, a critical measure to help diagnose pulmonary diseases!

Wasn’t that fun? I thought it was fun. Fun, fun, fun. Plethysmography is fun!

Now I have to call my speech therapist.



Author: elijoi

Humanist, Rationalist, Writer, Web Developer, Table Tennis Junky, Composer, Lyricist, Actor, Singer, and very recently with a mid-life career change, a Respiratory Therapist

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