The alveolar gas equation

This chapter is most relevant to Section F9(iv) from the 2017 CICM Primary Syllabus, which expects the exam candidates to be able to "understand the common respiratory equations". Though there are no specific CICM primary or Fellowship exam questions which ask about this equation directly, virtually every ABG question requires the candidate to consider the A-a gradient, making the alveolar gas equation essential.

In short, this equation describes the concentration of gases in the alveolus, and thus allows us to make educated guesses as to the effectiveness of gas exchange. One can use this to calculate the tension-based indices of oxygenation, such as A-a gradient or the a/A ratio (which is expressed as a percentage). The ABG machine frequently does this work for you, provided you have entered the FiO₂ and have specified that your sample is "arterial". The result is usually reported as pO₂(a/A).

Alveolar gas equation

An excellent article exploring the history of this equation discusses the original 1946 paper by Fehn, Rahn, and Otis. The modern form of the equation is as follows:

Thus, on room air and at sea level, we can assume certain constants.

PAO₂ = (0.21 x (760 - 47)) - (PaCO₂ x 1.25)

Thus:

PAO₂ = (149 - (PaCO₂ x 1.25)

Thus, the patient with a relatively normal PaCO2 (say, 40) :

PAO₂ = (149 - 50)

So, a normal person should have a PAO2 of around 99 mmHg.

Or, for a patient with normal PaCO₂ and an increased FiO₂:

PAO₂ = (FiO2 x 713) - 50

Of course, it is possible to have a strange respiratory quotient, but for this we would need to measure the total body VO₂ and VCO₂, which can only be accomplished by means of indirect calorimetry.

So, what should your PAO₂ be at any given FiO₂? In mmHg, the values are as follows:

FiO2 21%	100
FiO2 30%	164
FiO2 40%	235
FiO2 50%	307
FiO2 60%	378
FiO2 70%	449
FiO2 80%	520
FiO2 90%	592
FiO2 100%	662

In a nutshell, one can say that for every 10% increase in FiO₂, the PAO₂ will rise by about 71-72 mmHg.

Atmospheric gas mixture

Of course it would be amiss for us not to ask: why does the Earth titrate its FiO₂ to 21% (or, more precisely, 20.9%)?

Thankfully, this would not be the first ridiculous digression in physiology. For instance,  John F. Nunn has written a chapter (Chapter 1 of Nunn's Respiratory Physiology) about the atmosphere. In it,  he is grateful that greenhouse gases have allowed the existence of surface water for the last 4000 million years, and he laments that the sun "proceeds remorselessly towards becoming a red giant, which will ultimately envelop the inner planets". Unfortunately the rest of the textbook proceeds soberly along a straight and predictable path.

A better introduction into the subject would probably be afforded by The Chemical Evolution of the Atmosphere and Oceans, by Heinrich D. Holland.  The author confesses on page 2 that "the range of topics considered in the book is uncomfortably large", and that due to the mass of information "chaos was a continuous threat" during the assembly of the manuscript. In any case, for a monograph written between the years 1968 and 1981, this is a fine work. It is deserving of attention from anybody who has finished with their final CICM exams and still has some enthusiasm for the written word.