The alveolar gas equation

A patient is in respiratory distress, SaO2 deteriorates and you wonder how to turn round the situation. Apart from other diagnostics, you get an arterial blood gas and analyse pH, pCO2 and elytes. Yet the core value is pO2 and this is the hardest to analyse. There are no normal values for pO2, unless you know the FiO2, the temperature and the age of the patient. The one most important medical equation for intensive care is the alveolar gas equation (AGE), which is actually quite easy to derive:

1. O2 streams into the alveolus with the breathing gas, which consists of ambient air (possibly enriched with O2) and water vapor (saturated by the nose, trachea and bronchi) – so it comes with a partial pressure of

  •  (atmospheric pressure – water pressure) x inspiratory O2 quotient

Comments:

  • Now we are used to getting atmospheric pressure as hPa and not in Torr, but our BGA values tend to be in Torr (aka mmHg), so you have to convert the units. In Augsburg, atmospheric pressure is about 1020 hPa, which amounts to 768 mmHg. Water pressure is 47 mmHg.
  • The inspiratory O2 quotient is easy, if no O2 is supplied (21%), but gets problematic if nasal prongs or a mask without reservoir are used, because they don’t provide a predictable FiO2. Nasal prongs deliver from 30% to at most 40% (the latter only if you turn up the flow to ridiculous levels). Normal face masks deliver between 40% und 60%, again depending on the flow. Non-rebreather (reservoir) masks can deliver 100% only, if you (can) close both inlet valves, otherwise you end up with (about) 80%.

2. Now O2 not only streams into the alveolus through breathing, but also is taken up by the blood, but the amount is hard to measure (we don’t know much about our own work, so we can’t  know how much the patient works either, so we cannot compute how much oxygen he needs). But we get that information from the CO2 in the alveolus, because every CO2 has to have been metabolized from O2 through glycolysis. Depending on what we burn with O2, we get between 1,2 and 0,7 CO2 for every O2 (this is the respiratory quotient), but it is usually taken as 0,8 in intensive care. So we spend O2 in an amount proportional to

  • alveolar CO2 / respiratory quotient

The funny thing is that CO2 diffuses exceptionally well through the various alveolar layers, so we may assume that alveolar CO2 is roughly equal to arterial CO2, so that O2 spent is proportional to

  • arterial CO2 / respiratory quotient

Since dividing by 0,8 is about as good as multiplying by 1,25 or maybe 1,2, the latter expression reads

  • 1,2 * PaCO2

[Here comes the mathematician: we talked about income and spent O2 being proportional to these expressions, but not about the real values. That you can still compute with pressures rather than volumes is not completely obvious – but follow the complete math here.]

3.The whole AGE reads

  • P_alveolar O2 = FiO2 * (P_atm – P_H2o) – 1,2 * PaCO2 = FiO2 * 721 – 1,2 * PaCO2    [values for Augsburg, 11th floor, stroke unit]

This, of course, holds only under steady state conditions (O2 and CO2 have time to diffuse, be metabolized and so on; no change in diet) and proper ICU diets.

Comments:

  • I find it not too hard to do the math in my head, but ICU docs usually just ignore the CO2-part, using FiO2 * 600 as a rough estimate for expected alveolar PO2. In fact, the Horowitz index is a very crude simplification of the AGE. If you know that high CO2(as in COPD) and low atmospheric pressures (such as on Mount Everest)  obscur things, you may actually use these simplifications in practice.
  • How do you use the AGE? You compare your arterial PO2 to the expected, i.e., alveolar PO2, subtract the higher from the lower (forming the alveolar-arterial oxygen difference PAaO2) and use this as a marker for the severity of your lung problem. Note that real hypoventilation (as in being sedated) should not really increase your PAaO2.
  • Nowadays, we know that advanced lung disease may interfere with a lot of the assumptions of the AGE, so beware with ARDS and COPD patients.
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