What is Winter's formula?

Winter's formula predicts the PaCO2 a patient with a primary metabolic acidosis should reach through respiratory compensation: expected PaCO2 = (1.5 x HCO3) + 8, plus or minus 2 mm Hg. For a bicarbonate of 8 mEq/L, the expected PaCO2 is 18 to 22 mm Hg. A measured PaCO2 inside that window means appropriate compensation; a value above it means an added respiratory acidosis, and a value below it means an added respiratory alkalosis.

How do you use Winter's formula to find a mixed acid-base disorder?

First confirm a metabolic acidosis, usually acidemia with a low bicarbonate (in a mixed disorder the pH can be normal or high). Then calculate 1.5 x HCO3 + 8 and build a window of plus or minus 2 mm Hg. Compare the measured PaCO2: too high means the patient is not ventilating enough (a concomitant respiratory acidosis), and too low means something else is driving breathing, such as salicylates (a concomitant respiratory alkalosis). Finish with the anion gap and delta gap to look for a hidden metabolic process.

Can Winter's formula be used for metabolic alkalosis?

No. Winter's formula applies only to metabolic acidosis. In metabolic alkalosis, the lungs compensate by hypoventilating, and the Merck Manual expects PaCO2 to rise about 0.6 to 0.75 mm Hg (roughly 0.7) for every 1 mEq/L rise in bicarbonate, but not above about 55 mm Hg from compensation alone. A higher PaCO2 suggests an added respiratory acidosis and a lower one an added respiratory alkalosis. Respiratory acidosis and alkalosis have their own acute and chronic rules.

What is the last two digits of the pH rule?

In a well-compensated metabolic acidosis, the PaCO2 is roughly equal to the last two digits of the pH, so a pH of 7.22 suggests a PaCO2 near 22 mm Hg. The Merck Manual lists it as a rule of thumb. Checked against Winter's formula, it lands within about 2 to 3 mm Hg when bicarbonate is about 5 to 18 mEq/L and underestimates the expected PaCO2 above or below that range. It is a quick cross-check, not a replacement for the formula.

What is the expected compensation in respiratory acidosis and alkalosis?

The kidneys adjust bicarbonate. In acute respiratory acidosis, bicarbonate rises 1 to 2 mEq/L for every 10 mm Hg rise in PaCO2, and in chronic respiratory acidosis it rises 3 to 4 mEq/L. In acute respiratory alkalosis, bicarbonate falls 1 to 2 mEq/L for every 10 mm Hg fall in PaCO2, and in chronic respiratory alkalosis it falls 4 to 5 mEq/L. Renal compensation takes days, which separates acute from chronic.

Why is a normal PaCO2 abnormal in metabolic acidosis?

Because a patient with metabolic acidosis should be hyperventilating. With a bicarbonate of 12 mEq/L, Winter's formula predicts a PaCO2 of 24 to 28 mm Hg, so a measured PaCO2 of 40 means the lungs are not compensating, which is a concomitant respiratory acidosis. Causes include respiratory muscle fatigue, opioids or sedatives, brain stem lesions, neuromuscular weakness and COPD. In DKA, a PaCO2 rising toward normal can signal impending respiratory failure.

Winter's Formula: Expected PaCO2, Worked Examples & Traps

Winter's formula predicts the PaCO2 a patient with metabolic acidosis should reach by hyperventilating: expected PaCO2 = (1.5 × HCO3−) + 8, ± 2 mm Hg. A measured PaCO2 above that range means a respiratory acidosis is also present; a PaCO2 below it means a respiratory alkalosis is also present. Use it only after you have confirmed a primary metabolic acidosis (the other primary disorders have their own compensation rules), then pair it with the anion gap and delta gap to catch hidden metabolic disorders. Compensation never returns the pH fully to normal and never overshoots, so a normal or alkalemic pH with a low bicarbonate signals a mixed disorder.

Winter's formula in four steps

Confirm a metabolic acidosis: a low HCO3− (or a high anion gap), usually with a pH below 7.35; in a mixed disorder such as salicylate poisoning the pH can be normal or even high. (Normal: pH 7.35-7.45, PaCO2 about 40 mm Hg, HCO3− about 24 mEq/L; reference ranges vary by lab.)
Calculate the expected PaCO2 = 1.5 × HCO3− + 8, then build the ± 2 window.
Compare the measured PaCO2. Inside the window = appropriate compensation. Above = added respiratory acidosis. Below = added respiratory alkalosis.
Calculate the anion gap, correct it for albumin, and if it is high, check the delta gap.

Merck also lists rougher rules of thumb: expected PaCO2 ≈ HCO3− + 15, a fall of about 1.2 mm Hg in PaCO2 for every 1 mEq/L fall in HCO3−, or the last two digits of the pH. They only approximate Winter's formula (at an HCO3− of 8, "HCO3− + 15" gives 23, just outside the 18-22 window), so use Winter's when the answer hinges on a few mm Hg. The formula comes from Albert, Dell and Winters (Annals of Internal Medicine, 1967).

The two-digit shortcut: a cross-check only

In a well-compensated metabolic acidosis, the PaCO2 is roughly the last two digits of the pH (pH 7.22, PaCO2 about 22). Run through the Henderson-Hasselbalch equation, the rule lands within about 2-3 mm Hg of Winter's midpoint when HCO3− is roughly 5-18 mEq/L (a 2023 letter in the Southwest Journal of Pulmonary, Critical Care and Sleep reached a similar conclusion) and underestimates the expected PaCO2 when HCO3− is above 18 or below 5. Because 2-3 mm Hg can cross the edge of Winter's ± 2 window, let Winter's formula make the call.

The Q-bank vignette

A 27-year-old woman with type 1 diabetes ran out of insulin 3 days ago and now has abdominal pain, nausea and fast, deep breathing. She is drowsy and dry. Temperature 37.1°C, pulse 118/min, blood pressure 94/60 mm Hg, respirations 28/min.

Serum	Value	Arterial blood gas	Value
Sodium	132 mEq/L	pH	7.22
Chloride	96 mEq/L	PaCO2	20 mm Hg
Bicarbonate	8 mEq/L	HCO3−	8 mEq/L
Potassium	5.4 mEq/L
Glucose	560 mg/dL
Albumin	4.0 g/dL
Beta-hydroxybutyrate	Markedly elevated

Which of the following best describes her acid-base status?

A. High anion gap metabolic acidosis with appropriate respiratory compensation

B. High anion gap metabolic acidosis with a concomitant respiratory acidosis

C. High anion gap metabolic acidosis with a concomitant respiratory alkalosis

D. Normal anion gap metabolic acidosis from diarrhea

E. Primary respiratory alkalosis

F. High anion gap metabolic acidosis with a concomitant metabolic alkalosis

Working it through

pH 7.22 = acidemia; HCO3− 8 = primary metabolic acidosis.
Anion gap = 132 − (96 + 8) = 28 (albumin is normal, so no correction) = high.
Winter's formula: 1.5 × 8 + 8 = 20, window 18-22 mm Hg. Measured PaCO2 20 = appropriate.
Delta gap: 28 − 12 = 16; 16 + 8 = 24, a normal bicarbonate, so there is no hidden metabolic alkalosis or normal-gap acidosis.

Answer: A. This is diabetic ketoacidosis, and the deep, rapid Kussmaul respirations are the compensation, not a second disorder. Note the potassium: total-body potassium is depleted even when the serum level is normal or high, and insulin drives potassium into cells, so insulin is withheld until serum potassium is at least 3.3 mEq/L (Merck). More on the clinical side in DKA vs HHS.

Why every other choice is wrong

B. Concomitant respiratory acidosis: needs a PaCO2 above 22, say 30: she would not be ventilating enough, from tiring respiratory muscles, opioids, sedatives or alcohol, a brain stem stroke, neuromuscular weakness or COPD. Her PaCO2 of 20 sits in the window.
C. Concomitant respiratory alkalosis: needs a PaCO2 below 18, say 14: an extra drive to breathe beyond compensation, such as salicylate toxicity, hypoxemia, fever, pain or anxiety.
D. Diarrhea: causes a normal anion gap (hyperchloremic) acidosis: chloride rises as bicarbonate falls, and the gap stays near 12. Her gap is 28, and the ketones explain it. Sort normal-gap causes in anion gap vs non-anion gap acidosis.
E. Primary respiratory alkalosis: the pH would be above 7.45 (unless mixed), and renal compensation lowers HCO3− by only 4-5 mEq/L per 10 mm Hg fall in PaCO2 even when chronic. A 20 mm Hg drop could explain a bicarbonate of about 14-16 at most, never 8. Her low PaCO2 is the response, not the cause.
F. Concomitant metabolic alkalosis: would push the delta-gap sum above the normal bicarbonate range (vomiting is the usual culprit). Her sum is 24.

Worked examples: same formula, three verdicts

Patient	HCO3− (mEq/L)	Expected PaCO2 (mm Hg)	Measured PaCO2	pH	Verdict
DKA (the vignette)	8	18-22	20	7.22	Appropriate compensation
Lactic acidosis, given IV opioids	12	24-28	40	7.10	Added respiratory acidosis
Aspirin overdose	14	27-31	20	7.47	Added respiratory alkalosis

The "normal" PaCO2 trap. A PaCO2 of 40 looks normal, but with an HCO3− of 12 it should be 24-28. Failure to blow off CO2 is a respiratory acidosis; in DKA, a PaCO2 creeping up toward normal while the bicarbonate stays low warns that the patient is tiring.
The alkalemic acidosis. A low bicarbonate with a pH above 7.45 cannot be compensation, which never overshoots. Salicylates stimulate the medullary respiratory center (primary respiratory alkalosis) and separately cause an anion gap metabolic acidosis; per Merck, early blood gases show respiratory alkalosis, later a mixed picture, and finally a poorly compensated metabolic acidosis.

Compensation rules for every primary disorder

Winter's formula covers only one row of this table. Respiratory compensation for a metabolic disorder starts quickly; renal compensation for a respiratory disorder takes days, which is why respiratory disorders have separate acute and chronic rules.

Primary disorder	Compensation	Expected change (Merck unless noted)
Metabolic acidosis	Hyperventilation lowers PaCO2	Winter's formula: 1.5 × HCO3− + 8 ± 2 (about 1.2 mm Hg fall per 1 mEq/L fall in HCO3−)
Metabolic alkalosis	Hypoventilation raises PaCO2	0.6-0.75 mm Hg (about 0.7) rise per 1 mEq/L rise in HCO3−; PaCO2 should not climb above about 55 mm Hg from compensation alone
Acute respiratory acidosis	Buffering raises HCO3−	1-2 mEq/L per 10 mm Hg rise in PaCO2
Chronic respiratory acidosis	Kidneys retain HCO3−	3-4 mEq/L per 10 mm Hg rise
Acute respiratory alkalosis	Buffering lowers HCO3−	1-2 mEq/L per 10 mm Hg fall
Chronic respiratory alkalosis	Kidneys excrete HCO3−	4-5 mEq/L per 10 mm Hg fall

For practice with the other rows, see metabolic alkalosis and respiratory acidosis and alkalosis.

Anion gap and delta gap: finish the analysis

Anion gap = Na+ − (Cl− + HCO3−). Merck puts the typical gap at about 12 mEq/L; reference ranges vary by lab.
Correct for albumin. Lower the expected normal gap by 2.5 mEq/L for every 1 g/dL fall in albumin (Merck). Otherwise hypoalbuminemia hides a high-gap acidosis.
High-gap causes (Merck): ketoacidosis (diabetic, alcoholic, fasting), lactic acidosis (including D-lactate in short bowel syndrome), renal failure, rhabdomyolysis, and toxins such as methanol, ethylene glycol and salicylates. Many students memorize them as CAT MUDPILES.
Delta gap. Subtract the normal gap from the patient's gap and add the result to the measured HCO3−. A sum in the normal range means a pure high-gap acidosis. A sum above normal means a coexisting metabolic alkalosis (Merck); by the same logic, a sum well below normal points to a coexisting normal-gap (hyperchloremic) acidosis.

Example (DKA plus vomiting): Na 135, Cl 85, HCO3− 20. The gap is 30, so the delta gap is 18; 18 + 20 = 38, far above normal. The near-normal bicarbonate hides a severe ketoacidosis offset by a metabolic alkalosis from vomiting. Winter's formula still applies to the acidosis: expected PaCO2 = 1.5 × 20 + 8 = 38 ± 2. More layered cases are in mixed acid-base disorders.

How USMLE tests it: common traps

Using Winter's formula outside metabolic acidosis. Metabolic alkalosis and respiratory disorders have their own rules (table above).
Accepting a "normal" PaCO2. In metabolic acidosis, a PaCO2 of 40 is a respiratory acidosis.
Calling every low PaCO2 a respiratory alkalosis. Look at the pH and HCO3− first; a low PaCO2 with acidemia and a very low bicarbonate is usually compensation.
Forgetting that compensation never overshoots. A low bicarbonate with a normal or high pH means two primary processes; think salicylates.
Skipping the albumin correction. A "normal" gap in a hypoalbuminemic patient may really be high.
Leaning on the two-digit rule. It is only a rough check (within about 2-3 mm Hg) when HCO3− is about 5-18 mEq/L.

Key takeaways

Expected PaCO2 = 1.5 × HCO3− + 8 ± 2 in metabolic acidosis (Albert, Dell and Winters, 1967).
Measured above the window = added respiratory acidosis; below = added respiratory alkalosis.
Rougher shortcuts (PaCO2 ≈ HCO3− + 15; a 1.2 mm Hg fall per 1 mEq/L fall in HCO3−; the last two digits of the pH) are cross-checks only.
Compensation never fully corrects the pH and never overshoots.
Always add the anion gap (corrected for albumin) and the delta gap to find a third process.
DKA with a PaCO2 in the window = appropriate compensation (Kussmaul breathing); salicylates = mixed respiratory alkalosis plus anion gap metabolic acidosis.