Calculating Physiological Dead Space

[Bathrover]

The equation is:

Vd = Vt * (PaCO2 - PeCO2)/PaCO2

Vd * PaCO2 = Vt * (PaCO2 - PeCO2)

So reasoning, we are equating the number of moles of CO2:
PV = nRT, ie: P1V1 = P2V2
such that for example:

150 * PaCO2 = 500 * (PaCO2 - PeCO2)

150 * 40 = 500 (40 - PeCO2)

PeCO2 = 28 mm Hg

So this assumes non dead space airways exchange CO2 such that the retained pressure of CO2 post-expiratory is (PaCO2-PeCO2), so that if we multiply the retained CO2 pressure by tidal volume (total volume inspired/expired, assuming no air trapping) it equals the pre-exchange CO2 pressure (PaCO2) multiplied by dead space volume (Vd).

Also alveolar ventilation:

Alveolar Ventilation = (Vt - Vd) * RR

The rate (volume/time) of actual exchange occurring.

Is the above rationale sound or am I missing something?

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[CherryRedDracul]

It's not really "retained pressure of CO2" you're looking at. Your derivation heads generally in the right direction, but there are a few bits missing.

First of all, we know that tidal volume = dead space volume + alveolar volume, or in other words, air that participates in gas exchange and air that doesn't respectively. Vt = Vd + Va. We can rearrange that to

Va = Vt - Vd

Next, we turn transform that equation into the ideal gas law equation (a restatement of the law of conservation of mass) just like you did:

Va*PaCO2 = Vt*PtCO2 - VdPdCO2, where PtCO2 is the partial pressure of mixed expired air and PdCO2 is the partial pressure of dead space air AKA that of normal atmospheric air. The assumption is that the partial pressure of CO2 in the atmosphere is nearly 0 so we can drop part of the equation.

Va*PaCO2 = Vt*PtCO2 - Vd*0 --> Va*PaCO2 = Vt*PtCO2

Since we're interested in dead space volume and we can easily obtain tidal volume via spirometry, we can change Va to Vt - Vd

(Vt - Vd)*PaCO2 = Vt*PtCO2

And with a little bit of rearrangement and change the "t" to an "e", we your equation: Vd = Vt * (PaCO2 - PeCO2)/PaCO2

You're spot on with alveolar ventilation though. Hope I didn't make things more confusing!

 

[Bathrover]

Thanks this makes sense, the replacement of Vt-Vd with Va makes everything fit in place. I was having trouble conceptually understanding what Vt-Vd was referring to before reviewing my notes, thanks for confirming.