A-a Gradient and Alveolar Gas Equation Calculator

The equation, term by term

PAO₂ = FiO₂ × (Patm − PH₂O) − PaCO₂ / R

Each term is doing something specific, and losing any one of them is a recognisable error.

• P_atm − P_H₂O. Inspired gas is fully saturated with water vapour by the time it reaches the alveolus, and that vapour exerts its own partial pressure which dilutes everything else. At body temperature it is 47 mmHg, and it depends only on temperature, so it does not fall with altitude. At sea level this leaves 760 − 47 = 713 mmHg for the dry gases.
• F_iO₂ × that. Oxygen’s share of the humidified inspired gas. Room air is 20.9 percent oxygen, so 0.21 × 713 = 149.73 mmHg. This is the inspired oxygen tension, P_iO₂, shown as its own readout above.
• − P_aCO₂ / R. Carbon dioxide arriving from the blood occupies space in the alveolus and displaces oxygen. Arterial carbon dioxide is used as a proxy for alveolar, which is legitimate because carbon dioxide diffuses so readily that the two are essentially equal.
• R, the respiratory quotient. Slightly less carbon dioxide is produced than oxygen consumed, so the gas volumes do not balance exactly. R is the ratio, 0.8 on a mixed diet, about 1.0 on pure carbohydrate and 0.7 on pure fat. Dividing 40 by 0.8 gives 50 rather than 40, so this correction is worth 10 mmHg and is not cosmetic.

Putting room air at sea level through it: 0.21 × (760 − 47) − 40 / 0.8 = 149.73 − 50 = 99.73 mmHg. That is where the textbook statement that alveolar oxygen is about 100 mmHg comes from.

The gradient, and what widens it

A-a gradient = PAO₂ − PaO₂. It is the oxygen that should have reached the blood but did not, so it measures how well the lung is transferring gas rather than how much oxygen is arriving.

Some gradient is normal, because of bronchial and coronary venous drainage into the systemic circulation and because ventilation and perfusion are never perfectly matched even in a healthy lung. It widens with age as elastic recoil is lost. Two estimates of the expected value are in circulation and examiners ask for either, so both are shown above:

• expected = age / 4 + 4
• expected = 2.5 + 0.21 × age

They agree closely in young adults, around 8 to 10 mmHg at 25, and give 19 to 24 mmHg at 80.

The clinical value of the gradient is that it splits the causes of a low arterial oxygen into two groups, and the split is genuinely useful:

• Wide gradient. The lung is the problem. Shunt, whether intrapulmonary as in consolidation or atelectasis, or intracardiac. Ventilation and perfusion mismatch, as in pulmonary embolism, asthma or chronic obstructive disease. Diffusion impairment, as in interstitial disease.
• Normal gradient with a low arterial oxygen. The lung is transferring gas properly and the problem lies outside it. Either hypoventilation, in which case the carbon dioxide will be high, from opioids, neuromuscular weakness or a central cause; or a low inspired oxygen tension, meaning altitude or a hypoxic gas mixture, in which case the carbon dioxide will be normal or low.

A worked contrast. Someone who has taken an opioid overdose might have an arterial oxygen of 55 mmHg and a carbon dioxide of 75. Alveolar oxygen is 0.21 × 713 − 75 / 0.8 = 149.73 − 93.75 = 56 mmHg, so the gradient is about 1 mmHg. The lung is fine; the patient is simply not breathing. No amount of examining the chest will explain the hypoxaemia, and the treatment is naloxone and ventilation rather than oxygen alone.

Why altitude is an input rather than a constant

Barometric pressure is a field above, not a hard coded 760, because getting this wrong invents disease. Pressure falls roughly exponentially with altitude:

• Sea level, about 760 mmHg.
• Shimla at around 2,200 m, about 580 to 590 mmHg.
• Leh at around 3,500 m, about 490 to 505 mmHg.

At 495 mmHg the lower pressure alone takes alveolar oxygen on room air down to 0.21 × (495 − 47) − 50 = 44.1 mmHg. Residents at altitude are also chronically hyperventilating, so their arterial carbon dioxide is low, around 30 rather than 40, and that partly offsets the fall: 0.21 × (495 − 47) − 30 / 0.8 = 56.6 mmHg. A healthy long term resident of Leh with an arterial oxygen of 50 therefore has a gradient of 6.6 mmHg, which is completely normal, and no lung disease at all. Computing their gradient against a sea level alveolar tension of 99.73 would report roughly 50 mmHg and send you looking for a shunt that does not exist.

When to use the P/F ratio instead

The A-a gradient has a real weakness: it is not comparable across different inspired oxygen concentrations. Raising the inspired fraction raises the alveolar tension, and a given shunt fraction then costs more absolute millimetres of mercury, so the gradient widens even though the lung has not changed. Comparing today’s gradient on 40 percent oxygen against yesterday’s on room air tells you very little.

The P/F ratio, arterial oxygen divided by the inspired fraction, is more stable across oxygen concentrations and is what the Berlin definition of acute respiratory distress syndrome uses: under 300 meets the oxygenation criterion, under 200 is moderate and under 100 is severe. On room air a normal arterial oxygen of 95 gives 95 / 0.21 = 452. Both numbers are shown above so the comparison is in front of you rather than requiring a second calculation.

Common mistakes

• Dropping the water vapour term. Using 760 instead of 713 raises alveolar oxygen by 10 mmHg and manufactures a gradient. The most frequent error with this equation.
• Entering the inspired oxygen as 21 rather than 0.21 in the bare formula. The field above takes a percentage and converts, but doing it by hand with 21 gives an absurd alveolar oxygen of 14,923 mmHg. The same error in a P/F ratio gives 4.5 and declares everyone to have severe respiratory failure.
• Using a pulse oximeter reading. The equation needs an arterial oxygen tension in mmHg from a blood gas, not a saturation in percent. A saturation of 95 percent corresponds to a tension of about 80 mmHg, and the relationship between them is the sigmoid oxyhaemoglobin dissociation curve rather than anything linear.
• Guessing the inspired fraction on nasal cannula. It varies with the patient’s own inspiratory flow, so the delivered fraction is genuinely uncertain. Any gradient computed from an assumed value on low flow oxygen carries that uncertainty, and a venturi mask or a stated ventilator setting is far more trustworthy.
• Reading a wide gradient as a diagnosis. It localises the problem to the lung and does not say which lung problem. Distinguishing shunt from mismatch needs the response to supplemental oxygen: a true shunt barely improves, whereas mismatch corrects readily.

Converting units first? Use the pressure conversion table.