Mellemgaard K, Acta Physiol Scand 1966
In healthy seated adults breathing air, the alveolar–arterial oxygen difference rose with age, from a few mm Hg in young adults; the regression is often quoted as about 2.5 + 0.21 × age.
- Critical care
- Pulmonology
- Emergency medicine
A–a Gradient Calculator
Calculates the alveolar–arterial oxygen gradient from a blood gas, with the age-adjusted normal range.
Result
Enter FiO2, PaO2 and PaCO2.
Sources
4 studies
4
studies
243
citations
1,168
PubMed papers
Evidence
The measurements behind the normal range, the derivation of the alveolar gas equation, and the pneumocystis guidance that uses the gradient.
Known limitations
- Normal ranges apply on room air only.
- Assumes a respiratory quotient of 0.8 and body temperature of 37 °C.
- Needs an arterial sample and an accurately known FiO2.
- Shows that gas exchange is abnormal, not why.
Key papers
Curran-Everett D, Adv Physiol Educ 2006
An essay on the 1946 paper in which Fenn, Rahn and Otis set out the alveolar gas equation, and on how the equation explains pulmonary gas exchange and the limits of human performance, such as at altitude.
Adv Physiol Educ: Journal Metrics Score 2.3, Q1 · cited 32 times
National Institutes of Health-University of California Expert Panel for Corticosteroids as Adjunctive Therapy for Pneumocystis Pneumonia, N Engl J Med 1990
An expert panel recommended adjunctive corticosteroids for adults and adolescents with HIV and documented or suspected pneumocystis pneumonia with a room-air PaO2 below 70 mm Hg or an alveolar–arterial gradient of 35 mm Hg or more, started within 72 hours of anti-pneumocystis treatment.
N Engl J Med: Journal Metrics Score 29.8, Q1 · cited 385 times
Bozzette SA et al., N Engl J Med 1990
In 333 patients with AIDS and pneumocystis pneumonia randomized to early corticosteroids or none, those with confirmed or presumed pneumocystis pneumonia (251) had a lower 31-day risk of respiratory failure (14% vs 30%) and of death (11% vs 23%); intention-to-treat analysis of all 333 agreed. No benefit was shown in mild disease, a PaO2/FiO2 ratio above 350, equivalent to a room-air PaO2 above 75 mm Hg.
N Engl J Med: Journal Metrics Score 29.8, Q1 · cited 489 times
The normal values has been cited 243 times, and 1,168 PubMed articles mention the alveolar–arterial oxygen gradient in their title or abstract. Citation counts from OpenAlex and article counts from PubMed, retrieved November 14, 2026. Journal Metrics Scores and quartiles are from this site’s current release; follow the journal name for details.
After the result
Hypoxemia, raised gradient · Hypoxemia, normal gradient · Any severe hypoxemia
- Hypoxemia, raised gradient
- Hypoxemia, normal gradient
- Any severe hypoxemia
Look for a lung or circulation cause. Hypoventilation or low inspired oxygen. Treat first.
Next steps
Hypoxemia, raised gradient
- Look for ventilation–perfusion mismatch or shunt: chest X-ray, and CT pulmonary angiography if embolism is possible.
- If the PaO2 barely rises on high oxygen, suspect a shunt (ARDS, lobar collapse, an intracardiac shunt).
Hypoxemia, normal gradient
- Check the PaCO2. If it is high, look for sedative drugs, neuromuscular weakness, chest wall disease or obesity hypoventilation.
- Consider altitude or a low inspired oxygen; enter the barometric pressure if you are not at sea level.
Any severe hypoxemia
- Give oxygen and support breathing as needed; work out the gradient on a later gas if it cannot be done on room air.
Avoid common errors
5 common errors to avoid
- 1Supplemental oxygen
- 2Units must match
and 3 more
When the gradient misleads
1Supplemental oxygen
The normal gradient widens with FiO2, and nasal cannulas and masks deliver an uncertain FiO2. Room-air limits do not apply.
2Units must match
Enter PaO2 and PaCO2 in the same unit. Mixing kPa and mm Hg gives impossible results; the calculator converts both when you switch.
3Venous or air-exposed samples
A venous gas or a sample with air bubbles gives a misleading PaO2. The gradient needs an arterial sample.
4Respiratory quotient
The equation assumes a respiratory quotient of 0.8. Diet, feeding and fever change it (roughly 0.7 to 1.0), which shifts PAO2 by several mm Hg, more when the PaCO2 is high.
5Age matters
A gradient of 20 mm Hg is abnormal at 25 but within the normal range at 70 (limit 21.5 mm Hg). Enter the age for the adjusted limit.
Formula
PAO2 = FiO2 × (Patm − 47) − PaCO2 ÷ 0.8; gradient = PAO2 − PaO2; normal on room air up to age ÷ 4 + 4.
FiO2 × 713 − PaCO2/0.8 − PaO2
≤ age/4 + 4raised
PAO2 = FiO2 × (Patm − 47) − PaCO2 ÷ 0.8; gradient = PAO2 − PaO2; normal on room air up to age ÷ 4 + 4.
Formulas and normal range
| Value | Formula |
|---|---|
| Alveolar PO2 (PAO2) | FiO2 × (barometric pressure − 47) − PaCO2 ÷ 0.8; at sea level, FiO2 × 713 − PaCO2 ÷ 0.8 |
| A–a gradient | PAO2 − PaO2 |
| Upper limit on room air | age ÷ 4 + 4 mm Hg (mean in healthy adults 2.5 + 0.21 × age, Mellemgaard 1966) |
| kPa | mm Hg ÷ 7.5; at sea level FiO2 × 95 − PaCO2 ÷ 0.8, all in kPa |
| Result | Gradient | Meaning |
|---|---|---|
| Normal gradient | up to age/4 + 4 mm Hg on room air | The gradient is within the normal range. If the patient is hypoxemic, a normal gradient points to hypoventilation (a high PaCO2) or low inspired oxygen, such as at altitude, rather than a lung problem. |
| Raised gradient | above age/4 + 4 mm Hg on room air | The gradient is wider than expected, so oxygen is not getting from the alveoli into the blood normally: ventilation–perfusion mismatch (pneumonia, pulmonary embolism, asthma, COPD), a shunt (atelectasis, ARDS, an intracardiac shunt) or, less often, a diffusion problem. |
Alveolar gas equation
Fenn, Rahn and Otis described the alveolar gas equation in 1946; the form used here is the simplified one, with a respiratory quotient of 0.8 and water vapour pressure of 47 mm Hg at body temperature.
Mellemgaard 1966
Measured the gradient in healthy people sitting and breathing air, and found that it rose with age, by about 0.21 mm Hg a year.
At altitude
Barometric pressure falls with height, to about 630 mm Hg at 1,500 m and about 530 mm Hg at 3,000 m; use the local measured value if you have it. Using 760 there overestimates PAO2 and the gradient.
Test cases
3 published cases, each also a test case
- 1.9
- 46.5
- 67.3
3 published cases; the calculator reproduces every gradient.
Worked examples
Each example is also a test case: the calculator must reproduce the gradient the source reports before any change to it can be published.
Hypercapnic failure, normal gradient: 1.9 mm Hg
1.9 mm HgA 31-year-old woman with Silver-Russell syndrome, a growth disorder, was admitted with breathlessness. She had scoliosis and very weak respiratory muscles, and her lung tissue was normal on CT apart from small lung volumes. On room air the PaO2 was 68.6 mm Hg and the PaCO2 63.4 mm Hg; the report gives an A–aDO2 of 1.88 mm Hg, within normal limits. Nocturnal noninvasive ventilation improved her respiratory failure.
Source and detailsHide details
“her alveolar-arterial oxygen gradient was within normal limits”
The calculator gives 1.88, shown as 1.9. Hypoxemia with a normal gradient is the pattern of pure hypoventilation.
Entered as: FiO2: 21 %; PaO2: 68.6 mm Hg; PaCO2: 63.4 mm Hg; Age (optional): 31 years.
Source: Hakamata M et al., Intern Med 2021 (PubMed · Free full text · DOI)
Pneumocystis pneumonia and pulmonary embolism: 46.5 mm Hg
46.5 mm HgA 23-year-old man came in with a week of diarrhea, fever and breathlessness; his oxygen saturation was 85% on room air and 95% on 5 L/min of oxygen by nasal cannula. The blood gas, for which the report does not give the FiO2, showed a PaO2 of 62 mm Hg and a PaCO2 of 33 mm Hg, and the report gives an A–a gradient of 46.5 mm Hg. CT showed bilateral ground-glass opacities and a pulmonary embolism. He was first treated as COVID-19, then found to have HIV and Pneumocystis pneumonia.
Source and detailsHide details
“alveolar-arterial (A-a) oxygen gradient of 46.5 mmHg”
The report does not state the FiO2 for this gas; room air reproduces the reported 46.5 exactly, so that is what the authors used. A PaO2 of 62 mm Hg would usually give a saturation above 90%, so the gas may have been drawn on oxygen; if so, the true gradient was larger still. The gradient is above 35 mm Hg, which, like a room-air PaO2 below 70 mm Hg, marks moderate to severe pneumocystis pneumonia in which adjunctive corticosteroids are recommended. Both the pneumonia and the embolism widen the gradient.
Entered as: FiO2: 21 %; PaO2: 62 mm Hg; PaCO2: 33 mm Hg; Age (optional): 23 years.
Source: Lim W et al., Case Rep Infect Dis 2021 (PubMed · Free full text · DOI)
Pneumonia in HIV on 24% oxygen: 67.3 mm Hg
67.3 mm HgA 65-year-old woman with HIV had pneumonia and severe colitis. On 24% oxygen the PaO2 was 73.3 mm Hg and the PaCO2 24.4 mm Hg; the report gives an alveolar–arterial gradient of 67.3 mm Hg. CT showed diffuse bilateral ground-glass opacities, and four pathogens were later found, including Pneumocystis jirovecii and Strongyloides. She died despite treatment.
Source and detailsHide details
“The alveolar-arterial gradient was 67.3 mmHg.”
The calculator gives 67.3, as reported. She was on a little extra oxygen, so the room-air limits do not strictly apply, but a gradient this wide is abnormal on 24%.
Entered as: FiO2: 24 %; PaO2: 73.3 mm Hg; PaCO2: 24.4 mm Hg; Age (optional): 65 years.
Source: Al-Shyoukh A et al., Cureus 2020 (PubMed · Free full text · DOI)
FAQ
6 questions and answers
- QHow do you calculate the A–a gradient?
- QWhat is a normal A–a gradient?
and 4 more answers
Questions
How do you calculate the A–a gradient?
+
First work out the alveolar oxygen tension: PAO2 = FiO2 × (760 − 47) − PaCO2 ÷ 0.8. Then subtract the arterial value: A–a gradient = PAO2 − PaO2. On room air with a PaCO2 of 40 mm Hg, PAO2 is 0.21 × 713 − 50 = about 100 mm Hg (99.7), so a PaO2 of 90 gives a gradient of about 10 mm Hg (9.7).
What is a normal A–a gradient?
+
On room air, roughly 5 to 10 mm Hg in young adults, rising with age. A common bedside upper limit is age ÷ 4 + 4 mm Hg, so about 14 mm Hg at 40 and 19 mm Hg at 60. In healthy seated adults Mellemgaard measured a mean of about 2.5 + 0.21 × age.
What does a normal gradient with low oxygen mean?
+
The lungs are transferring oxygen normally, so the cause is outside the lungs: hypoventilation (shown by a high PaCO2, for example from opioids, neuromuscular weakness or chest wall disease) or low inspired oxygen, such as at altitude.
What causes a raised A–a gradient?
+
Ventilation–perfusion mismatch (pneumonia, pulmonary embolism, asthma, COPD, interstitial lung disease), shunt (atelectasis, ARDS, pulmonary edema, an intracardiac or intrapulmonary shunt) and, less often, impaired diffusion.
Can I use the A–a gradient on oxygen?
+
With care. The normal gradient widens as FiO2 rises, and nasal cannulas and simple masks deliver an uncertain FiO2. On supplemental oxygen the PaO2/FiO2 ratio is the more usual measure of gas exchange.
Why is the A–a gradient used in pneumocystis pneumonia?
+
Treatment guidance for HIV-associated pneumocystis pneumonia defines moderate to severe disease, in which adjunctive corticosteroids are recommended, as a room-air PaO2 below 70 mm Hg or an A–a gradient of 35 mm Hg or more. This comes from the 1990 expert consensus and the trials behind it.
Further reading
7 references with PubMed and DOI links
7
references
3
free full text
From Mellemgaard K, Acta Physiol Scand 1966 to the newest case reports.
References
- Mellemgaard K. The alveolar-arterial oxygen difference: its size and components in normal man. Acta Physiol Scand. 1966;67(1):10-20. PubMed · DOI
- 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. PubMed · DOI
- National Institutes of Health-University of California Expert Panel for Corticosteroids as Adjunctive Therapy for Pneumocystis Pneumonia. Consensus statement on the use of corticosteroids as adjunctive therapy for pneumocystis pneumonia in the acquired immunodeficiency syndrome. N Engl J Med. 1990;323(21):1500-1504. PubMed · DOI
- Bozzette SA, Sattler FR, Chiu J, Wu AW, Gluckstein D, Kemper C, et al. A controlled trial of early adjunctive treatment with corticosteroids for Pneumocystis carinii pneumonia in the acquired immunodeficiency syndrome. N Engl J Med. 1990;323(21):1451-1457. PubMed · DOI
- Hakamata M, Hokari S, Ohshima Y, Kagami M, Saito S, Motoike IN, et al. Chronic hypercapnic respiratory failure in an adult patient with Silver-Russell syndrome. Intern Med. 2021;60(12):1921-1926. PubMed · Free full text · DOI
- Lim W, Suhail M, Diaz K. Diagnostic challenges of Pneumocystis pneumonia during the COVID-19 pandemic: a case of a young patient with ground glass opacities and pulmonary embolism on chest CT. Case Rep Infect Dis. 2021;2021:5669543. PubMed · Free full text · DOI
- Al-Shyoukh A, Younis M, Warsame M, Gohar A. A rare case of multipathogenic pneumonia in a patient with human immunodeficiency virus. Cureus. 2020;12(7):e9307. PubMed · Free full text · DOI
Related pages
- Hunt and Hess scale
clinical grade of aneurysmal subarachnoid hemorrhage, I to V.
- Corrected calcium calculator
total calcium adjusted for albumin, in mg/dL or mmol/L.
- NIH Stroke Scale (NIHSS)
stroke severity from 15 bedside items, 0 to 42.
- Mean arterial pressure (MAP)
MAP from systolic and diastolic pressure, against the 65 mm Hg target.
- Pediatric Glasgow Coma Scale
eye, verbal and motor scores for preverbal children, 3 to 15.
- P/F ratio (PaO2/FiO2)
oxygenation ratio with the Berlin ARDS severity ranges.
- Sodium corrected for hyperglycemia
serum sodium corrected for high glucose, Katz or Hillier factor.
- Anion gap calculator
sodium − (chloride + bicarbonate), with the albumin-corrected gap.
- BUN/creatinine ratio
urea nitrogen to creatinine ratio, in US or SI units.
- Creatinine clearance (Cockcroft-Gault)
estimated creatinine clearance with actual, ideal or adjusted weight.
- Fick cardiac output
cardiac output and index from oxygen consumption and saturations.
- Absolute neutrophil count (ANC)
ANC from the white cell count, neutrophils and bands.
- HOMA-IR calculator
insulin resistance from fasting glucose and insulin, in US or SI units.
- Fractional excretion of sodium (FENa)
FENa from plasma and urine sodium and creatinine: prerenal, indeterminate or tubular pattern.
- Cincinnati Prehospital Stroke Severity Scale (CPSSS)
three-item stroke severity screen for large vessel occlusion, 0 to 4.
- Free water deficit
water deficit in hypernatremia from sodium, weight and body water.
- Winter’s formula
expected PaCO2 in metabolic acidosis, 1.5 × HCO3 + 8 ± 2.
- QTc
corrected QT interval by Bazett, Fridericia, Framingham or Hodges.
- PECARN head injury rule
head CT decision rule for children with minor head trauma.
- Maximum allowable blood loss
blood loss before hemoglobin or hematocrit reaches a chosen lowest value.
- Glasgow Coma Scale
eye, verbal and motor response, 3 to 15, with GCS-Pupils.
- Ideal and adjusted body weight
Devine ideal body weight and adjusted weight from height and sex.
- Pediatric endotracheal tube size
cuffed and uncuffed tube size and oral depth by age.
- Serum osmolality and osmolal gap
calculated osmolality, osmolal gap and ethanol correction.
- Maddrey discriminant function
severity of alcohol-associated hepatitis from prothrombin time and bilirubin.
- MELD score (MELD-Na, MELD 3.0)
MELD, MELD-Na and MELD 3.0 for chronic liver disease.
- ABCD² score for TIA
early stroke risk after a transient ischemic attack.
- NEXUS criteria (C-spine)
whether a patient with blunt trauma needs cervical spine imaging.
- Duke treadmill score
prognosis from a Bruce protocol exercise test.
- Canadian CT Head Rule
whether an adult with minor head injury needs CT.
- FIB-4 index
risk of advanced liver fibrosis from age, AST, ALT and platelets.
- New England Journal of Medicine
Which published the pneumocystis consensus and trial, with its current metrics.
MZ
About this page
Contributed by Dr. Meng Zhao (MD · Former Neurosurgeon · Medical AI Researcher), from the original publications listed here. Last updated November 14, 2026. LinkedIn
The calculator code is tested against every worked example on this page before each release.
This calculator supports, and does not replace, clinical judgment. It is meant for healthcare professionals and is not medical advice for patients. Report an error through the contact page.