A-a gradient calculator (alveolar–arterial oxygen)
Work out the alveolar–arterial oxygen gradient from a blood gas: enter FiO2, PaO2, PaCO2 and age to get the alveolar PO2 from the alveolar gas equation, the A–a gradient in mmHg and kPa, and whether it is above the age estimate of (age ÷ 4) + 4 mmHg.
Guides
- Normal A–a gradient by age, and what a raised gradient meansNormal A-a gradient by age from 20 to 80, where the (age/4) + 4 rule comes from, and how a normal or raised gradient sorts the causes of hypoxemia.
How this A–a gradient calculator works
This A–a gradient calculator measures how much oxygen pressure is lost between the alveoli and arterial blood. Arterial PO2 (PaO2) comes straight from a blood gas, but alveolar PO2 (PAO2) cannot be measured at the bedside, so the calculator estimates it with the alveolar gas equation, the relationship that Fenn, Rahn and Otis worked through in their 1946 study of alveolar air at altitude. PAO2 equals the fraction of inspired oxygen (FiO2) multiplied by barometric pressure minus water vapor pressure, minus arterial PCO2 divided by the respiratory quotient. The defaults are the values in Sarkar and colleagues' 2017 review of hypoxemia in Lung India: barometric pressure 760 mmHg at sea level, water vapor pressure 47 mmHg at 37 °C and a respiratory quotient of 0.8. With room air and a PaCO2 of 40 mmHg this gives a PAO2 of about 100 mmHg. The A–a gradient is that PAO2 minus the measured PaO2.
Alveolar PO2: PAO2 = FiO2 × (Patm − PH2O) − PaCO2 ÷ RQ
A–a gradient = PAO2 − PaO2
Age estimate (room air) = (age ÷ 4) + 4 mmHg. FiO2 above 1 is read as a percentage, so 21 and 0.21 give the same result. kPa are converted with 1 kPa = 7.50062 mmHg.
Worked example, as loaded above: a 60-year-old breathing room air has a PaO2 of 70 mmHg and a PaCO2 of 40 mmHg. PAO2 = 0.21 × (760 − 47) − 40 ÷ 0.8 = 149.7 − 50 = 99.7 mmHg, and the A–a gradient is 99.7 − 70 = 29.7 mmHg (3.96 kPa). The age estimate is 60 ÷ 4 + 4 = 19 mmHg, so this gradient is raised for age by 10.7 mmHg. In kPa the same patient's values are about 9.3 kPa for PaO2, 5.3 kPa for PaCO2, 101.3 kPa for barometric pressure and 6.3 kPa for water vapor; switching the unit converts the fields for you. On room air at sea level the equation simplifies to PAO2 ≈ 149.7 − 1.25 × PaCO2 in mmHg.
A–a gradient results and what they mean
An A–a gradient result is read against what is expected for the patient's age, because the gradient widens as people get older. Sarkar's review gives an A–a difference below 10 mmHg in a young person and attributes the rise with age to increasing ventilation–perfusion (V/Q) mismatch. This calculator compares the result with the bedside rule (age ÷ 4) + 4 mmHg, which Stein, Goldhaber and Henry used as one definition of a normal gradient in their 1995 analysis of the PIOPED study, and it shows Mellemgaard's 1966 regression line of 2.5 + 0.21 × age beside it. The rule is an estimate rather than a reference interval: we could not trace it to an original derivation. In a hypoxemic patient, a gradient within the estimate points to hypoventilation, which lowers PAO2 by raising PaCO2. A raised gradient points to the lung itself: V/Q mismatch, shunt or diffusion limitation, the three mechanisms Sarkar's review lists as widening the gradient.
| Result | Reading | What it suggests |
|---|---|---|
| At or below (age ÷ 4) + 4, room air | Within the age estimate | Gas exchange as expected for age. If PaO2 is low, look for hypoventilation: a high PaCO2 with an intact alveolar–capillary membrane. |
| Above (age ÷ 4) + 4, room air | Raised for age | V/Q mismatch (for example asthma, COPD, interstitial lung disease), shunt (pneumonia, pulmonary edema, ARDS) or diffusion limitation (emphysema, interstitial lung disease). |
| Any value on FiO2 above 0.21 | Age rule not valid | The normal gradient widens as FiO2 rises, so compare with earlier gradients on the same FiO2 rather than with the age rule. |
| Below 0 | Not physiological | Check the FiO2, the units of PaO2 and PaCO2, and that the sample was arterial. |
The example causes in the table are those listed in Sarkar's review for each mechanism. For a full table of age-expected values from 20 to 80 years and a step-by-step way to sort hypoxemia, see the guide to the normal A–a gradient by age.
When the A–a gradient is less reliable
The A–a gradient is only as good as the alveolar gas equation's assumptions, and several of them fail in sick patients. The equation assumes a steady state, a respiratory quotient close to 0.8 on a standard diet, and an alveolar PCO2 equal to the arterial PCO2, as Sarkar's review sets out. Raising FiO2 widens the normal gradient, because alveolar oxygen rises more than arterial oxygen, so a gradient measured on supplemental oxygen cannot be judged against a rule built for room air. Barometric pressure falls with altitude, so above sea level enter the local pressure instead of 760 mmHg. The age rule itself is an approximation: Mellemgaard studied 80 healthy seated subjects aged 15 to 75 years, as summarized in Murray and Nadel's textbook, and his regression line runs about 2 to 4.5 mmHg below (age ÷ 4) + 4 across that range. A normal gradient also does not exclude disease, as the PIOPED analysis of pulmonary embolism showed.
In that analysis Stein and colleagues studied 280 patients with pulmonary embolism proven by angiography and 499 in whom it was excluded. With normal defined as a gradient at or below age/4 + 4, 8% to 10% of patients with embolism had a normal A–a gradient, and comparable percentages of patients without embolism did too, so the authors concluded that a normal gradient did not exclude acute pulmonary embolism. With a cut-off of 20 mmHg the figure was 11% to 14%. Sarkar's review adds that prolonged hypoventilation can cause atelectasis and widen the gradient, so a raised gradient and a high PaCO2 can coexist.
Questions
What is a normal A–a gradient?
Below 10 mmHg in a young adult breathing room air, rising with age. The bedside estimate is (age ÷ 4) + 4 mmHg, for example 14 mmHg at 40 and 24 mmHg at 80. The guide to the normal A–a gradient by age has the full table.
What is the A–a gradient formula?
A–a gradient = PAO2 − PaO2, with PAO2 = FiO2 × (Patm − PH2O) − PaCO2 ÷ RQ. On room air at sea level with RQ 0.8 that becomes PAO2 ≈ 149.7 − 1.25 × PaCO2 mmHg.
Can I enter values in kPa?
Yes. Pick kPa and the pressure fields convert (1 kPa = 7.50062 mmHg); sea-level defaults become about 101.3 kPa and 6.3 kPa. The gradient is shown in both mmHg and kPa.
Does a normal A–a gradient rule out pulmonary embolism?
No. In the PIOPED analysis by Stein and colleagues, 8% to 10% of patients with angiographically proven embolism had a gradient at or below age/4 + 4.
Is my data sent anywhere?
No. The calculation runs in your browser, and the values you enter are not sent or stored.
Sources
- Fenn WO, Rahn H, Otis AB. A theoretical study of the composition of the alveolar air at altitude. Am J Physiol. 1946;146(5):637–653. doi:10.1152/ajplegacy.1946.146.5.637
- Curran-Everett D. A classic learning opportunity from Fenn, Rahn, and Otis (1946): the alveolar gas equation. Adv Physiol Educ. 2006;30(2):58–62. doi:10.1152/advan.00076.2005
- Sarkar M, Niranjan N, Banyal PK. Mechanisms of hypoxemia. Lung India. 2017;34(1):47–60. doi:10.4103/0970-2113.197116
- Mellemgaard K. The alveolar-arterial oxygen difference: its size and components in normal man. Acta Physiol Scand. 1966;67(1):10–20. doi:10.1111/j.1748-1716.1966.tb03281.x
- Gold WM, Koth LL. Pulmonary function testing. In: Murray and Nadel's Textbook of Respiratory Medicine. 6th ed. Elsevier; 2016:407–435.e18. doi:10.1016/B978-1-4557-3383-5.00025-7
- Stein PD, Goldhaber SZ, Henry JW. Alveolar-arterial oxygen gradient in the assessment of acute pulmonary embolism. Chest. 1995;107(1):139–143. doi:10.1378/chest.107.1.139