A-a gradient: the alveolar gas equation and the causes of hypoxaemia
A low arterial PO₂ does not say whether the lung is failing or the patient is under-breathing. The A-a gradient, calculated with the alveolar gas equation, does.
Calculate the alveolar–arterial O₂ gradient with the alveolar gas equation and use it to sort the causes of hypoxaemia.
The short version
- The A-a gradient is the alveolar PO₂ minus the arterial PO₂; the arterial value is measured and the alveolar value is calculated.
- On room air at sea level, alveolar PO₂ is about 150 − 1.25 × PaCO₂.
- A normal gradient points to hypoventilation or thin air (high altitude); a wide gradient points to V/Q mismatch, shunt or diffusion limitation.
- Compare the gradient with the expected value for age, and interpret it on room air.
- The age rule and the upper limit of normal vary between sources, so name the rule you use.
Hypoxaemia: five mechanisms, one number to sort them
A blood gas shows a low arterial PO₂ on room air. Is the lung failing at gas exchange, or is the patient simply not breathing enough? The oxygen number alone does not say which. The A-a gradient can.
There are five mechanisms of hypoxaemia:[1][4]
- low inspired oxygen
- hypoventilation
- diffusion limitation
- V/Q mismatch
- shunt
To sort them, compare the oxygen the alveolus should hold with what the blood actually carries. That comparison is the alveolar–arterial (A-a) gradient.[1]
The alveolar gas equation and the A-a gradient formula
The A-a gradient is the alveolar PO₂ (PAO₂) minus the arterial PO₂ (PaO₂). The arterial value is measured on the blood gas. The alveolar value has to be calculated, and PACO₂ is assumed to equal PaCO₂.[1][6]
PB = barometric pressure; PH₂O = water vapour pressure; R = respiratory quotient
PaO₂ is measured; PAO₂ is calculated
The convention used here is sea level: barometric pressure 760 mmHg and water vapour pressure 47 mmHg. On room air FiO₂ is 0.21, and R is 0.8.[1][2]
That reduces to one simple line. 760 − 47 = 713, and 713 × 0.21 is about 150. Dividing by 0.8 is the same as multiplying by 1.25.[1][6]
0.21 × 713 = 149.73, rounded to 150
What is a normal A-a gradient?
In a young adult the gradient is normally under about 10 mmHg. It rises with age.[1][6]
A rough upper limit is (age + 10) / 4. Some sources use a limit of under 20, and others use age / 4 + 4 or 2.5 + 0.21 × age, so name the rule you are using.[2][3][5][6]
Normal gradient: hypoventilation and high altitude
In hypoventilation PaCO₂ rises, and alveolar and arterial PO₂ fall together. The membrane is intact, so the gradient stays normal.[1][6]
- A 34-year-old man is breathing slowly after an opioid overdose. On room air: PaCO₂ 72, PaO₂ 50 mmHg.
- 1.25 × 72 = 90
- PAO₂ = 150 − 90 = 60
- Gradient = 60 − 50 = 10 mmHg
- Expected for age 34: (34 + 10) / 4 = 11 mmHg
- 10 is normal: low oxygen with a normal gradient means hypoventilation.
High altitude gives the same pattern. The air is thin, but the gradient is normal when PAO₂ is calculated with the local barometric pressure.[3][4][6]
Wide gradient: a problem in the lung itself
- A 55-year-old woman has pneumonia. On room air: PaCO₂ 32, PaO₂ 62 mmHg.
- She is overbreathing: 1.25 × 32 = 40
- PAO₂ = 150 − 40 = 110
- Gradient = 110 − 62 = 48 mmHg
- Expected for age 55: (55 + 10) / 4 ≈ 16 mmHg
- 48 against 16 is wide.
A wide gradient means V/Q mismatch, shunt or diffusion limitation: the problem is in the lung itself.[1][6] The last lesson showed how to split shunt from the rest with oxygen.
So the opening puzzle has an answer. One calculation told a lung problem from a breathing problem.
Using the A-a gradient to sort the causes of hypoxaemia
| A-a gradient | Mechanisms | Example above |
|---|---|---|
| Normal | Hypoventilation; low inspired oxygen (high altitude) | Opioid overdose |
| Wide | V/Q mismatch; shunt; diffusion limitation | Pneumonia |
- Get the numbersTake PaO₂ and PaCO₂ from a room-air blood gas.
- Calculate PAO₂On room air at sea level, 150 − 1.25 × PaCO₂.
- SubtractA-a gradient = PAO₂ − PaO₂.
- Compare with the expected valueUse (age + 10) / 4, or the rule you name, and read the result against the table above.
Practice question
Original · USMLE styleA 52-year-old man has 3 weeks of worsening breathlessness. At sea level, breathing room air (FiO₂ 0.21), his arterial blood gas shows pH 7.34, PaCO₂ 56 mmHg and PaO₂ 55 mmHg. Assume barometric pressure 760 mmHg, water vapour pressure 47 mmHg and a respiratory quotient of 0.8, and round 0.21 × 713 to 150.
Common questions
How do you calculate the A-a gradient?
Calculate the alveolar PO₂ with the alveolar gas equation, FiO₂ × (PB − PH₂O) − PaCO₂ / R, then subtract the measured arterial PO₂. On room air at sea level, alveolar PO₂ is about 150 − 1.25 × PaCO₂.
What is a normal A-a gradient?
In a young adult it is normally under about 10 mmHg, and it rises with age. A rough upper limit is (age + 10) / 4, though some sources use under 20, so name your rule.
What does a normal A-a gradient with low PaO₂ mean?
It points to hypoventilation or low inspired oxygen, such as high altitude. The membrane is intact, and the low PaO₂ reflects a low alveolar PO₂.
What does a wide A-a gradient mean?
It means V/Q mismatch, shunt or diffusion limitation: the problem is in the lung itself.
Why is the A-a gradient interpreted on room air?
The gradient widens as FiO₂ rises, and the expected-gradient formulas assume room air.
Transcript
Read the full narration, by chapter620 words
0:00A low PaO2: lung failure or not breathing enough?
A blood gas comes back with a low arterial PaO2 on room air. Is the lung failing at gas exchange, or is the patient simply not breathing enough? The oxygen number alone does not say which. One calculation can. Today: calculate the alveolar-arterial gradient and use it to sort the causes of hypoxaemia.
0:21The five mechanisms of hypoxemia
There are five mechanisms of hypoxaemia: low inspired oxygen, hypoventilation, diffusion limitation, V over Q mismatch and shunt. To sort them, compare the oxygen the alveolus should hold with what the blood carries.
0:37The A-a gradient and the alveolar gas equation
The alveolar-arterial gradient is the alveolar PaO2 minus the arterial PaO2. The alveolar gas equation: alveolar PaO2 equals FiO2, times the quantity barometric pressure minus water vapour pressure, then minus PaCO2 divided by R. Our convention is sea level: barometric pressure 760 millimetres of mercury, water vapour pressure 47. On room air FiO2 is 0.21, and R, the respiratory quotient, is 0.8. 760 minus 47 is 713. Times 0.21 is about 150. Dividing by 0.8 equals multiplying by 1.25. So on room air at sea level, alveolar PaO2 is about 150 minus 1.25 times the PaCO2.
1:37What is a normal gradient?
What gradient is normal? In a young adult, under about 10 millimetres of mercury. It rises with age; a rough upper limit is age plus 10, divided by four, though some sources use under 20, so name your rule. The gradient also widens as FiO2 rises, so we interpret it on room air.
1:59Example 1: opioid overdose (normal gradient)
Worked example one. A 34-year-old man is brought in after an opioid overdose, breathing slowly. On room air, PaCO2 72, arterial PaO2 50. 1.25 times 72 is 90. 150 minus 90 is 60. Gradient: 60 minus 50, 10. Expected for his age: 34 plus 10, 44, over four, 11. Ten is normal. Low oxygen, normal gradient: hypoventilation. High altitude is the same: thin air, but a normal gradient using local barometric pressure.
2:43Example 2: pneumonia (wide gradient)
Worked example two. A 55-year-old woman has pneumonia. On room air, PaCO2 32, arterial PaO2 62. She is overbreathing, so 1.25 times 32 is 40, and 150 minus 40 is 110. Gradient: 110 minus 62, 48. Expected: 55 plus 10, 65, over four, about 16. Forty-eight against 16 is wide. The problem is in the lung itself.
3:19The rule in one card
A wide gradient means V over Q mismatch, shunt or diffusion limitation. The last lesson split shunt from the rest with oxygen. Back to that low PaO2: one calculation told a lung problem from a breathing problem.
3:35Practice question
Your turn. A 52-year-old man has three weeks of worsening breathlessness. At sea level on room air, his pH is 7.34, his arterial PaCO2 is 56 and his arterial PaO2 is 55. Take barometric pressure as 760, water vapour pressure as 47, FiO2 as 0.21, R as 0.8, and round the first term to 150. Which of the following is his alveolar-arterial gradient? A, 15. B, 25. C, 50. D, 80. E, 95. Pause here and decide.
4:24The options explained, and the answer
A is out: that is the expected gradient for his age, not his. C is out: it multiplies by R instead of dividing. D is out: 80 is the alveolar PaO2; the arterial value still has to be taken off. E is out: 95 forgets the carbon dioxide term. So the answer is B. 1.25 times 56 is 70. 150 minus 70 is 80. Eighty minus 55 is 25. That is wide against an expected gradient
5:01Recap and next lesson
of about 15. Three things to keep. One: gradient is alveolar minus arterial PaO2; room-air alveolar is about 150 minus 1.25 times PaCO2. Two: normal gradient means hypoventilation or thin air; wide means V over Q mismatch, shunt or diffusion limitation. Three: compare with the expected value for age, and interpret on room air. Next lesson: how to build a differential for acute shortness of breath.
Sources
Numbers in the text, e.g. [1], point here- Lung India. Mechanisms of hypoxemia (2017). pmc.ncbi.nlm.nih.gov/articles/PMC5234199/
- MSD Manual Professional. Equations for calculating alveolar oxygen pressure and A-a gradient. msdmanuals.com/professional/multimedia/table/equations-for-calculating-alveolar-oxygen-pressure-and-alveolar-to-arterial-oxygen-gradient
- MSD Manual. A-a gradient calculator. msdmanuals.com/iframe/manual/medical-calculators/aagradient.htm
- Silent hypoxemia in COVID-19 lung injury (five physiological causes of hypoxemia). pmc.ncbi.nlm.nih.gov/articles/PMC8328372/
- COVID-19 cohort using expected A-a gradient (age/4 + 4). pmc.ncbi.nlm.nih.gov/articles/PMC9078060/
- Wikipedia. Alveolar-arterial gradient (secondary). en.wikipedia.org/wiki/Alveolar%E2%80%93arterial_gradient
The practice question is original and its patient is synthetic. Every fact is checked against the sources above in independent accuracy reviews. Always follow your local guidelines and your own laboratory’s ranges.
Image credits. Lungs, airway and fluid plates: Servier Medical Art, CC BY 4.0 (adapted). Alveolus plate: NIAID NIH BioArt, public domain (bioart.niaid.nih.gov/bioart/15). The equation, number lines, slips, icons and all handwriting are drawn in code (original). Previous: EP 04 · V/Q mismatch: shunt vs dead space. Next: MS-CK-EMER-009 · Acute shortness of breath: the differential (clinical series).
Corrections. None so far. Spotted an error? Email info@medservatory.com with the lesson code MS-S1-RESP-005.
For exam study only, not medical advice. Read the medical disclaimer.
In this series
Playlist on YouTube- EP 01Lung volumes and capacities, explained5:26
- EP 02Obstructive vs restrictive lung disease: FEV1/FVC and loops5:23
- EP 03Lung compliance, Laplace and surfactant: why alveoli stay open5:23
- EP 04V/Q mismatch: shunt, dead space and the oxygen test5:18
- EP 05A-a gradient: the alveolar gas equation and the causes of hypoxaemia5:39
- EP 06Oxygen-hemoglobin dissociation curve: P50, shifts and content5:19
- EP 07CO₂ transport: bicarbonate, the chloride shift, Bohr vs Haldane5:29
- EP 08Central vs peripheral chemoreceptors and oxygen-induced hypercapnia5:27
- EP 09Hypoxic pulmonary vasoconstriction and cor pulmonale5:41
- EP 10CO, cyanide and methaemoglobin: why SpO₂ and PaO₂ fool you5:34