Winters' formula for metabolic acidosis

Clinical background

In metabolic acidosis the body responds with hyperventilation, which lowers the PaCO₂ and thereby counteracts the fall in pH. Whether this respiratory compensation is adequate — that is, whether it lies at the level physiology predicts — is central to distinguishing a pure metabolic acidosis from a mixed acid–base disturbance. Without a quantitative reference it is tempting to read a low PaCO₂ as a sign that the patient is compensating well, when in fact it may mask a concurrent respiratory alkalosis. Conversely, a PaCO₂ that appears "not that high" may still be too high in relation to the degree of acidosis and thereby indicate a concurrent respiratory acidosis.

Winter's formula provides that frame of reference. It calculates the expected PaCO₂ from the bicarbonate concentration and gives a tolerance range. If the measured PaCO₂ falls outside the range, a respiratory disturbance is present in addition to the metabolic acidosis already known.

Calculating Winter's formula

The formula relates the expected PaCO₂ to the serum bicarbonate concentration:

$$\text{PaCO}_2 = 1{.}5 \times [\text{HCO}_3^-] + 8 \pm 2 \text{ mmHg}$$

where $[\text{HCO}_3^-]$ is the serum or plasma bicarbonate in mEq/L. The calculated value is the midpoint, and ±2 mmHg gives the conventional tolerance range within which compensation is regarded as adequate.

The formula was derived by Albert, Dell and Winters and published in 1967 [1]. The study examined the quantitative relationship between acid–base equilibrium and respiratory compensation in patients with metabolic acidosis. The authors established a linear relationship between bicarbonate and PaCO₂ that has since become the dominant clinical reference for assessing compensation in metabolic acidosis. The cohort consisted of patients with varying degrees of metabolic acidosis, and the relationship was expressed as a regression line whose slope and intercept gave the coefficients 1.5 and 8 respectively.

Interpretation in practice

The assessment consists of comparing the measured PaCO₂ with the range the formula gives. Three scenarios cover the possibilities:

A quick bedside aid is Fulop's rule: with adequate compensation the PaCO₂ (in mmHg) should correspond to the two digits after the decimal point in the pH. If the pH is 7.28, a PaCO₂ of around 28 mmHg is expected. This rule is a rough approximation and works best in moderate acidosis; it does not replace the formula in borderline cases.

Validation and performance

Winter's formula has not undergone large prospective external validation studies in the modern sense. Its spread rests rather on decades of clinical use and on the fact that it reproduces the physiology satisfactorily for patients with acute metabolic acidosis of moderate to severe degree.

The most systematic challenge comes from Marano, who has evaluated several compensation formulae in chronic haemodialysis patients. In a study of 180 blood gas samples from haemodialysis patients with mild metabolic acidosis (HCO₃⁻ ≥ 14 mEq/L), Marano found that Winter's formula had a larger prediction error than simpler alternatives, particularly the formula PaCO₂ = [HCO₃⁻] + 15 [2]. An earlier study of 291 blood gas samples from the same population confirmed the pattern: Winter's formula had larger deviations than both the "1.2 rule", which describes the relationship between the fall in bicarbonate and the fall in PaCO₂, and the simple formula [HCO₃⁻] + 15 [3]. The author noted, however, that once the bicarbonate falls below approximately 12 mEq/L the formulae converge and in practice give the same predicted PaCO₂, meaning that Winter's formula performs better in severe acidosis, precisely the situation for which it was originally derived.

In summary: the formula is well established for acute metabolic acidosis with a low bicarbonate, but performs less well in mild acidosis and in populations with chronic metabolic acidosis, above all haemodialysis patients.

Limitations

Winter's formula applies to metabolic acidosis and must not be used for metabolic alkalosis, where a different relationship between bicarbonate and PaCO₂ holds. Nor is it applicable to respiratory acid–base disorders as the primary diagnosis.

The formula was derived from patients with relatively severe acidosis. In mild metabolic acidosis (HCO₃⁻ near 20 mEq/L or higher) the prediction error is greater, and alternative formulae may be more accurate in this range [2,3]. This is particularly relevant for chronic haemodialysis patients, who often have a stable, mild metabolic acidosis in which respiratory compensation follows a somewhat different relationship.

The tolerance range of ±2 mmHg is a convention, not a statistically derived confidence limit from a modern validation cohort. This means that patients whose PaCO₂ lies just at the boundary may be difficult to classify with confidence.

The commonest misuse is to interpret a PaCO₂ within the expected range as proof that the patient is "stable", when the formula says only that the respiratory compensation is proportionate. The severity and cause of the underlying metabolic acidosis must be assessed separately.

Another pitfall is to apply the formula before establishing that a metabolic acidosis is in fact present. If the primary disturbance is a respiratory acidosis with metabolic compensation, Winter's formula gives a misleading answer. The primary disturbance is identified by assessing the pH, PaCO₂ and bicarbonate together, not by the formula alone.

References

1. Albert MS, Dell RB, Winters RW. Quantitative displacement of acid-base equilibrium in metabolic acidosis. Ann Intern Med. 1967;66(2):312–22. PMID: 6016545
2. Marano M. Evaluation of the expected ventilatory response to metabolic acidosis in chronic hemodialysis patients. Hemodial Int. 2018;22(2):180–183. PMID: 28834137
3. Marano M, D'Amato A, Marano S. A very simple formula to compute pCO2 in hemodialysis patients. Int Urol Nephrol. 2015;47(4):691–694. PMID: 25613433