Through the Looking Glass (A-a, a/A, P:F)
Imagine you’re window shopping. The store isn’t open, but you can see the item that you’re interested in through a few different windows. The more angles you can view the item from, the more information you gather about it. It might look good from straight on, but another angle shows you something you don't like.
This is how we can think of the A-a gradient, a/A ratio, and P:F ratio - different views of the same item. These formulas, however, are not the complete view of oxygenation. Oxygenation is a huge topic, and there are more common values that we use:
How well is the patient saturating?
Is there dyspnea?
How do the lungs sound (auscultation) or look (POCUS)?
Where is the patient on the OHCD?
What does the ABG show?
As we move forward, it's important to keep all measures of oxygenation in perspective. None of them are perfect, and none of them show the whole picture of oxygenation when considered in isolation. We form a general impression of the oxygenation status of a patient based on many findings, some of which are objective, while others might be rather subjective. In discussing formulas like the alveolar gas equation, gradients, and ratios, there is value in understanding their purpose, not just in using them. By this, I mean that increasing our understanding of oxygenation about difficult things like what we'll discuss in this blog helps us think more deeply about any patient, regardless of what lab values we have to work with. You don't have to actually use these values to gain value from learning them.
One time, we put an A-a gradient problem up on a whiteboard at a station and waited to see if anyone could figure it out. Perhaps we were channeling some Good Will Hunting vibes.
Shout out to Steve McGaughey, who ended up being our Will Hunting (paramedic - not janitor).
The point of posting the problem was that it was difficult because the A-a gradient is difficult. Luckily for us, there are plenty of online calculators now.
You can type in values and get answers within seconds, but without understanding what the formula contains, the results probably won't mean very much to you. So, what's in the A-a gradient formula? It starts off with deceiving simplicity:
A-a Gradient = PAO2 - PaO2.
PaO2 is familiar to us because that's the number we get from arterial blood gases. PaO2 represents about 1.5% of the oxygen in the blood - the oxygen not attached to the hemoglobin (the remaining 98.5% is saturated on hemoglobin which is measured with SPO2).
What about the PAO2? 'P-BIG A-O2' is the pressure of the oxygen in the alveoli, which we'll need the alveolar gas equation to figure out. (Skip down to the next subheading and check that out before going any further if your brain prefers to work more linearly.)
Now we can simply say that the A-a gradient is the difference (in mmHg) between the oxygen in the alveoli and the arterial blood. Oxygen is always moving down its pressure gradient, so there is always going to be some difference between the PAO2 and the PaO2. Which should be higher? The PAO2 should always be higher since oxygen is moving down its pressure gradient into the body. The question is, how much difference should there be? Before we get to specific numbers, let's consider the principle behind this. Consider a couple of examples:
The PAO2 is high, and the PaO2 is low.
The PAO2 is low, and the PaO2 is low.
Here we have two hypoxemic patients (both have low PaO2 values). However, the PAO2 is high in one case and low in the other.
For the patient with the high PAO2 (patient 1), we would have to ask ourselves why oxygen is present in the lungs but is not getting into the arteries. Asking: "what is causing the big difference between the two?" Is the same as asking: "what is causing the increase in the A-a Gradient?"
For the patient with the low PAO2 (patient 2), we would have to ask ourselves why the PAO2 is so low. A low PAO2 value will cause a low PaO2 value (lack of oxygen in the lungs will cause a lack of oxygen in the blood). This patient might have a normal A-a gradient (there isn't much of a difference between oxygen in the lungs and arteries), but they're still hypoxemic. In this case, we might ask: "why is this patient not getting enough oxygen?"
Pictured above are two hypoxemic patients. Notice how the patient with the elevated A-a gradient (top) seems to have trouble getting oxygen through the A/C membrane (sounds like a V/Q mismatch). Meanwhile, the patient at the bottom just doesn't have enough oxygen coming into the alveoli to diffuse a substantial amount into the blood (sounds like hypoventilation or high altitude).
Here is a quick infographic that tries to summarize what might cause a hypoxemic patient to have an elevated or normal A-a gradient, and what a normal value might be. This was compiled from information found at LITFLand Deranged Physiology, which both had great reference pages for these formulas.
Alveolar Gas Equation for PAO2
To figure out how much pressure oxygen is exerting in the alveoli, we have to account for any opposing pressures, as well as the oxygen that has already left. This sounds like a tall order, but if you checked out the calculators above for A-a gradient or a/A ratio, you can see it's done with only a couple of numbers that you would have on hand anyway (and some things are assumed).
PAO2 = PO2 entering - PO2 leaving
Ah yes, another definition that needs defining. How much PO2 is entering? there are three factors that influence this.
FiO2 (% of oxygen being delivered)
The point in bringing this PO2 entering up is to see what's inside of the formula, not how to do it. However, if you want to see the formula for PO2 entering, it would be:
PAO2 Entering = 0.21 x (760 - 47)
This would be for a patient on room air (0.21), at sea level (760), with normal water pressure in their lungs. FiO2 and barometric pressure make sense, but what's with the water pressure? We have to account for the isothermic saturation boundary that warms and humidifies our inhaled air. What about PO2 leaving?
PO2 leaving is estimated based on our PaCO2 (from our ABG). We assume that CO2 leaving the alveoli into the lung to be exhaled is equivalent to the O2 going into the blood - the PO2 leaving. Also, there is a constant of 0.8 that we divide by, which is something called our respiratory quotient (RQ). The RQ is estimated at 0.8 (sometimes 0.82) based on the patient's diet. The RQ moves closer to 1.0 the more carbohydrates you're body is consuming for energy. If your body is running mostly on fatty acid metabolism, your RQ will be closer to 0.7.
Assuming a normal PaCO2 of 40, and RQ of 0.8, the PO2 leaving would be calculated as:
PO2 Leaving = 40 / 0.8
Now we have all the pieces of the puzzle together in order to calculate the A-a gradient. We have our PAO2, which consisted of the difference between the PO2 entering and leaving. We also have the PaO2 from our ABG. Here are a couple of examples that start off with the alveolar gas equation and then finish with the A-a Gradient at the bottom. The final value that is left over is the A-a Gradient.
Again, the value in learning these formulas is not to do them by hand, but to understand what variables are inside of them. This helps us understand oxygenation in general. For example, consider what would happen if we drastically changed the PaCO2 levels in the alveolar gas equation.
We seemed to have caused a depressed A-a gradient due to the PaCO2 being very high. If we actually did the math by hand, this new PaCO2 would have given us a PAO2 of 49.73, and we see in the calculator above that the PaO2 was only 45. This gives us a new dimension to consider hypoventilation in the context of oxygenation, beyond just that of hemoglobin saturation and desaturation on the OHDC. We may also consider the elevation, age, and FiO2 in the A-a gradient as well. What about the a/A ratio?
The a/A Ratio
Now that we have our alveolar gas equation, we can simply flip the numbers around and do something totally different. The a/A ratio is the arterial oxygen (PaO2) divided by the alveolar oxygen (PAO2). This gives us the percent of the oxygen that made it through the A/C membrane into the blood. This should generally be greater than 75-80%, according to most literature.
To quickly clarify these values so far:
The A-a gradient was the difference in mmHg between the PAO2 and the PaO2.
The a/A ratio is the percent of the oxygen that is making it from alveoli to capillary.
Here in this picture, we've flipped the image to make this a little easier to comprehend (left to right). The PaO2 (80) divided by our PAO2 (100) gave us 80% diffusion.
Here is an interesting case where we can look at both A-a gradient and a/A ratio:
If we only evaluated the A-a gradient, it would seem elevated at 100 mmHg (an elevated value for anyone). However, when we look at the a/A ratio, we can see that 86% of the oxygen is making it through the membrane, which is a great number. This is an example of false elevation on the A-a Gradient due to using very large numbers. The A-a gradient is generally going to be elevated in patients with high PAO2 values (patients who are on high FiO2). This is why you may hear someone say that high amounts of FiO2 cause an increased A-a Gradient, which initially sounds counterintuitive, but is true when considering the absolute numbers. The a/A ratio tells the whole story, by evaluating if the percent that is making it from alveoli to capillary is appropriate.
This picture tells a completely different story:
Here we have an A-a gradient of 350 (very elevated), and the a/A ratio is only 50%. Even though this patient is not hypoxemic by any standard, seeing this value would still cause concern because the patient seems to have a substantial amount of V/Q mismatch. Perhaps they are developing ARDS or pneumonia, or maybe they have a PFO (any of the items listed above as increasing A-a gradient).
A low ratio of oxygen making it through the A/C membrane is very similar to the A-a gradient being elevated. Both mean that we're having some type of communication problem between air and blood (V/Q mismatch).
The P:F Ratio
For our last value, the P:F ratio, we isolate the FiO2 that we're administering. Whatever our FiO2 is, there is a predictable result that we should be getting from the PaO2. Generally, a healthy person will raise their PaO2 by 4-5x the FiO2 that they're receiving.
From the above example:
21% gave us about 80-100
50% gave us about 200-250
100% gave us about 400-500
Just knowing the appropriate ratio is actually pretty helpful when evaluating a ventilated patient. If you have the patient on a set FiO2, you can check the PaO2 on the ABG to see if they're getting at least 4-5x the FiO2 you have set. If we see a much lower PaO2, we would likely conclude that the patient has some type of V/Q mismatch, preventing some of the oxygen from diffusing into the blood.
Now that we understand the general idea of the P:F ratio, the formula won't be much of a surprise.
PaO2 / FiO2 = P:F Ratio.
Example: 100 / 0.21 = 476
The lower the P:F ratio goes, the worse it is. Also, if you have a patient that is on 1.0 FiO2, the P:F ratio will always be equal to whatever the PaO2 is (because you're dividing by 1.0).
No math is needed. 😉
The number you get from the formula above is