The Two-Digit Rule and Winter’s Formula

To the Editor,

I recently wondered, “How does the “Two-digit rule” for determining the adequacy of respiratory compensation for metabolic acidosis relate to Winter’s formula?”  Many clinicians were taught this rule-of-thumb: the last two digits of the pH (those visible if you hold your thumb over the “7”) should provide an approximation of pCO₂ (in mm Hg) in the presence of normal respiratory compensation.  Although Winter’s formula is easy to remember and perform, the Two-digit rule is even easier.  Winter’s formula and the Two-digit rule have both been derived empirically from clinical data (1,2).  But the reason the Two-digit rule should work has never, to our knowledge, been adequately explained.  We wondered how the two rules were related.

We began with serum HCO₃ values ranging from 4 -18 mmol/L and used them to calculate the corresponding predicted pCO₂ values using Winter’s formula (3).  Next, we used the Henderson-Hasselbalch equation to calculate the corresponding pH for each [HCO₃^-]/pCO₂ pair and used the Two-digit rule to re-calculate the corresponding predicted pCO₂.  The predicted compensatory pCO2 calculated by the Two-digit rule and Winter’s formula were then compared. 

Table 1. Comparison of predicted compensatory respiratory response to metabolic acidosis by Winter’s formula and the Two-digit rule.  Results of the Two-digit rule in red font fall outside the 95% standard error range of Winter’s formula. (Click here to view Table 1 in a separate, enlarged window)

Estimates of pCO₂ by the Two-digit rule were then superimposed over Winter’s original graphical data, from which the Winter’s formula was derived (1).  See below. 

Figure 1.  Predicted pCO2 by the Two-digit rule (in red font) superimposed over the original graphic used to derive Winter’s formula (1). (Click here to view Figure 1 in a separate, enlarged window)

Note that the Two-digit rule provides pCO₂ estimates within two standard errors (+/- 2 mm Hg) of Winter’s formula, for most [HCO₃^-] values ranging from 5-18 mmol/L.  At a HCO₃ of 10 mmol/L, the Two-digit rule overestimates pCO₂ by 1mmHg compared to Winter’s formula.  At HCO₃ above 18 mmol/L and below 5 mmol/L, the Two-digit rule underestimates pCO₂ compared to Winter’s.

The apparently linear relationship between pH (the negative base-ten logarithm of [H+]) and related partial pressure of pCO2 in mm Hg is explained by the three relationships that link their association.

1) pH has a negative logarithmic association with [H⁺].  As most who have used the simplified form of the Henderson-Hasselbalch equation ([H+] = 24*pCO₂/HCO₃) will remember, this relationship is conveniently approximately linear over the narrow range of physiological pH values, such that [H+] values of 30, 40, 50 and 60 nmol/L approximately correspond to pHs of 7.50, 7.40, 7.30 and 7.20 respectively.  2) Central chemoreceptors in the ventral medulla (and other locations) respond to increasing brain interstitial [H+] by increasing ventilatory drive in a fashion that is also approximately linear in the physiological range of pH values (3).  See figure below.  3) Ventilatory drive is inversely linearly related to pCO₂.

(More precisely, Pa_CO2 = V_CO2*K/V_ALV , where V_CO2=the rate of CO₂ production, K, is a proportionality constant, and V_ALV=alveolar minute ventilation (total ventilation – dead space ventilation.)

Figure 2. The ventilatory response to changes in brain interstitial fluid pH as studied in conscious goats (3). (Click here to view Figure 2 in a separate, enlarged window)

Therefore, It's not surprising that the last two digits of the pH should have a positive, approximately linear correlation with pCO₂.  However, the convenience of the correlation (two digits of the pH equaling the pCO₂) is purely fortuitous. The Two-digit rule provides a good approximation of the expected compensatory pCO₂ as calculated by Winter’s formula for [HCO₃^-] ranging from 5-18 mmol/L.

Robert Raschke MD

Clinical Professor of Medicine, University of Arizona College of Medicine – Phoenix

Phoenix, AZ USA

References

1. Albert MS, Dell RB, Winters RW. Quantitative displacement of acid-base equilibrium in metabolic acidosis. Ann Intern Med. 1967 Feb;66(2):312-22. [CrossRef] [PubMed]
2. Fulop M. A guide for predicting arterial CO2 tension in metabolic acidosis. Am J Nephrol. 1997;17(5):421-4. [CrossRef] [PubMed] (This shows empirical derivation of two-digit rule.)
3. Fencl V, Miller TB, Pappenheimer JR. Studies on the respiratory response to disturbances of acid-base balance, with deductions concerning the ionic composition of cerebral interstitial fluid. Am J Physiol. 1966 Mar;210(3):459-72. [CrossRef][PubMed] (This shows relationship between resp drive and [H].)

Cite as: Raschke RA. Letter to the Editor: The Two-Digit Rule and Winter’s Formula. Southwest J Pulm Crit Care Sleep. 2023;27(1):11-13. doi: https://doi.org/10.13175/swjpccs031-23 PDF