Assessment of compensation: Boston and Copenhagen methods

This page acts as a footnote to the "Boston vs. Copenhagen" chapter from Acid-Base Physiology by Kerry Brandis. The aforementioned chapter in my opinion remains the definitive resource on the topic.

Acid-base compensation in the literature

Boston, Copenhagen and Stewart

Brandis' chapter explores the epistemology of acid-base interpretation systems by means of which we might be able to determine whether a patient has a single or mixed acid base disorder; i.e. whether there is a purely metabolic or a purely respiratory disturbance, or some mixture of the two. As it happens, there are two well-accepted systems for doing this, each with its own merits and demerits. These are the Boston and Copenhagen methods of acid-base interpretation.

There is also another not-so-well accepted system, the physicochemical method proposed by Peter Stewart - which possess a satisfying explanatory power as an instrument of academic physiology. Unfortunately, it is rather complicated, and difficult to apply at the bedside. Furthermore, there does not seem to be much of a difference in hard outcomes, regardless of which system one uses. Thus, this chapter will focus on the Boston and Copenhagen systems, which have equivalent validity as far as acid-base interpretation is concerned.

"Which is the system I need to rote-learn to pass my primaries?"

Such a question is expected from the fairweather intensivist, who will flee from the ICU as soon as a position opens in a more cushy training program. For the rest, one might remark that these analytical tools are all in common use, and any sufficiently advanced ICU trainee is expected to be intimately familiar with all of these systems.

However, the time-poor exam candidate may need to focus their attention on the area which would yield the greatest number of marks. The CICM Fellowship written paper inevitably features short answer questions requiring ABG analysis. These SAQs tend to present the exam candidate with only enough information to apply the Boston method- the Base Excess required for a Copenhagen interpretation is frequently not available. Thus, if one were to only learn one method of assessing mixed acid-base disorders, one could do worse than the Boston method. One would be in good company. Kerry Brandis also favours the Boston rules, and as a result of his influence several generations of anaesthetists have now hatched, all featuring an identically pro-Boston phenotype.

Even though the examiners seem to trend towards Boston, the college members continue to publish papers in support of Stewart. Certainly, Oh's Manual - the Grand Grimoire of Australian Intensive Care (and therefore the official CICM viewpoint) - explains acid-base balance in terms of Stewart's method (see Chapter 92 by T.J. Morgan).

Evolution of acid-base theory: a brief history

The rich history of this field and specifically the chronicles of the "Great Trans-Atlantic Acid-Base Debate" are elaborated upon at great lengths in a 1993 article by John W. Severinghaus, which is as free from bias as one can reasonably expect from an article written by somebody intimately involved in the fighting. If one is discouraged by the Scandinavian paywall, one can instead turn to the more recent article by a local (D.A. Story, from Melbourne; 2004).

Overall, the two methods arose as attempts to improve upon the early Van Slyke interpretation of acid-base disorders. In 1917 or thereabout, Donald D. Van Slyke published the first in a very popular series or articles, titled "Studies of Acidosis". Fortunately they are all available online as free full text articles.

After injecting some sulfuric acid into dogs (to model the effects of diabetic ketoacidosis) Van Slyke and Cullen observed that much of the buffering was done by haemoglobin and tissues, with serum bicarbonate accounting for about 30%. So the bicarbonate decreased, which suggested that there might be a sort of "bicarbonate deficit" which might be a therapeutic target if one wished to restore the acid-base balance to a normal pH.

Thus, the impression was formed that serum bicarbonate is a measure of metabolic acid-base disturbance. Sophistication was lacking. Is your bicarbonate low? That must be a metabolic acidosis; couldn't be anything else. This ancient approach totally neglected the fact that bicarbonate changes with changing CO2 concentration, or that chronic changes in one variable promote compensatory changes in the other.

In order to account for these compensatory changes, the empirical Boston school and the theoretical Copenhagen school take two different approaches. The Bostonites embrace the complexity of the whole live human organism, and reduce its acid-base behaviour to a selection of formulae and correction factors, which are derived from in vivo titration experiments. The Copenhagenoids take the opposite approach, and use mathematics to exclude the influence of respiratory disturbances, so that they may look upon the naked metabolic acid-base status.

The Boston Method: Six bicarbonate-based bedside rules to assess compensation

These rules represent an attempt to predict what an appropriate "complete" compensation would be to a given acid-base disorder. The rules can be summarised as follows:

Acute Respiratory Acidosis

For every 10 mmHg increase in PaCO2, the HCO3- will rise by 1 mmol/L

In other words, expected HCO3 = 24 + ((PaCO2-40) / 10)

Chronic Respiratory Acidosis

For every 10 mmHg increase in PaCO2, the HCO3- will rise by 4 mmol/L

In other words, expected HCO3 = 24 + (4 × (PaCO2-40) / 10)

Acute Respiratory Alkalosis

For every 10 mmHg decrease in PaCO2, the HCO3- will fall by 2 mmol/L

In other words, expected HCO3 = 24 - (2 ×(40-PaCO2) / 10)

Chronic Respiratory Alkalosis

For every 10 mmHg decrease in PaCO2, the HCO3- will fall by 5 mmol/L

In other words, expected HCO3 = 24 - (5 ×(40-PaCO2) / 10)

Metabolic Acidosis

For complete compensation, expected PaCO2 = (1.5 × HCO3-) + 8

Metabolic Alkalosis

For complete compensation, expected PaCO2 = (0.7 × HCO3-) + 20

The bicarbonate used in these equations is the "actual bicarbonate", as calculated from the pCO2 and pH using the Henderson-Hasselbalch equation. The baseline bicarbonate value is assumed to be 24mmol/L, and the baseline "normal" CO2 is assumed to be 40mmHg.

These rules are based on whole-body acid-base titrations described by Schwartz and Relman in their 1963 NEJM article. However, not everybody seems to agree as to the exact formulae. For example, the formulae have been borrowed from Brandis, and are also found in the 2006 article by Lloyd and Freebairn. However, T.J Morgan's article contains a slightly different multiplier to describe metabolic alkalosis:

Metabolic alkalosis equation from Morgan, TJ (2003)

Expected PaCO2 = (0.9 × HCO3-) + 9

As well as this "0.9 and 9" rule, Oh's Manual also features a slightly different formula to represent chronic respiratory acidosis, apparently found in a 1980 article by Narins and Emmett:

Chronic respiratory acidosis equation from Oh's Manual (7th Ed)

expected HCO3 = 24 + (3.5 × (PaCO2-40) / 10)

Ok, so maybe the 3.5 multiplier is not for from Brandis' 4.0, but ultimately it's a source of inaccuracy.

With such diversity in formulae, one begins to wonder whether any of them are accurate. Who has the most accurate numbers, you start to wonder, and how did they determine at all those correction factors?

Why are all the equations different, and which one is correct?

It is difficult to say. The 1963 Schwartz and Relman article is unfortunately also paywalled, and one cannot make a comment about their methodology without a sizeable donation to the NEJM fund. One can, however, piece together certain facts about what they did. Schwartz and Relman argued the premise that to derive a "base excess" or "buffer base" value from titration of whole blood or plasma was a profoundly error-prone method, and that only in vivo bicarbonate titration could yield clinically accurate information regarding patients.

The in-vivo titration experiments were performed at Tufts university at Boston around 1962-1963. A series of healthy people were apparently exposed to recirculated CO2, and had their blood gases measured. Chronically alkalotic pregnant women, chronically acidotic smokers, and sick patients with metabolic acid-base disorders were all sampled, and the results of their ABGs were plotted. From these averaged results, the authors were able to determine the relationship between PaCO2 and HCO3- in each of the six cardinal disorders. In addition to these, the metabolic acidosis "1.5 plus 8" formula is also known as "Winters Formula", because it was also presented by Albert Dell and Winters in 1967. All of these formulae are based on the averaged data conglomerated into point plots, where lines of best fit are drawn to represent the relationship between HCO3- and CO2 under different circumstances. Being lines of best fit, these are subject to variation. This may explain the different correction factors featured by different authors.

Overall, one should remember that the Boston method is a qualitative approach, and the precise numbers generated by the equations do not matter as much as the broader pattern. For instance, after performing these mental gymnastics you might arrive at the conclusion that the patient's bicarbonate is lower than the expected bicarbonate, and thus there is a metabolic acidosis. But this conclusion will not lead you to a decision about whether this patient needs a dose of bicarbonate, nor will it help you decide how much bicarbonate may be required.

The Copenhagen Method: Four SBE-based bedside rules to assess compensation

The Copenhagen method rests on the use of Standard Base Excess to separate the respiratory and metabolic influences on in vivo acid base balance. Perhaps the most thorough exploration of this method was performed by a local - T.J Morgan (2003) -the same person who wrote the acid-base chapter from Oh's Manual.

This method also has a set of rules, which allow one to estimate exactly what sort of CO2 one should have, for any given SBE value. These rules were also derived from something resembling empiric in vivo measurements, albeit from retrospective meta-analysis of published data rather than prospectively collected data from asphyxiation experiments. Schlichtig Grogono and Severinghaus published this meta-analysis in 1998. These rules use PaCO2 measured in mmHg, and SBE measured in mEq/L.

Acute Respiratory Acidosis or Alkalosis

An acute change in PaCO2 will not change the Standard Base Excess.

So, expected SBE = 0... in other words, if there is any change in SBE then it cannot be due to acute respiratory acid-base disturbances - it must be of metabolic origin.

Working backwards, expected CO2 = 40 + (0 × SBE)

Chronic Respiratory Acidosis or Alkalosis

Expected change in SBE = 0.4 times the change in PaCO2

In other words, expected SBE = 0.4 × (PaCO- 40)

Working backwards, expected CO2 = 40 + (SBE / 0.4 )

Metabolic Acidosis

Compensatory change in PaCO2 will be proportional to the SBE.

In other words, expected CO2 = 40 + (1.0 × SBE)

Metabolic Alkalosis

Compensatory change in PaCO2 will be proportional to 0.6 times the SBE.

In other words, expected CO2 = 40 + (0.6 × SBE)

Thus, a patient with a chronic CO2 of around 50 would expect to have an SBE of around 4;  a patient with an SBE of around 10 would expect to have a CO2 of around 46; and so forth.

To this day, these rules of compensation are somewhat neglected in comparison to the Boston system. Indeed one can go though their professional life without ever having heard of them. This is curious, given that the majority of ABG analysers are customised in a manner which strongly favours the Copenhagen interpretation (Radiometer is a Danish company, their analysers are widely used and very popular, and most other ABG machines end up having to use the Copenhagen variables as a means of remaining competitive).

"Complete" and "typical" compensation

At by Alan Grogono (together with Severinghaus and Schlichtig, one of the authors of the 1998 paper mentioned above) one can see a  "the 5 to 3" ratio used to assess compensation. This ratio describes the rate of change in standard base excess.

Relationship of SBE  and PaCO2

A change of 6 mEq/L of SBE corresponds to a change of 10mmHg of PaCO2

This is the idealised relationship of SBE and CO2 which maintains a normal pH (7.4)

 According to this rule, a fully compensated metabolic alkalosis with an SBE of 10 should therefore result in a PaCO2 of around 57mmHg. This differs to the result which one would get using the empirical rules for compensation:  they would suggest a PaCO2 of around 46mmHg.

Why the difference? Well. The 5:3 ratio seems to be derived from the relationship of base excess and CO2 at a constant pH of 7.4, implying that to achieve complete compensation one must titrate the body fluids to complete neutrality.  This is chemically correct, but virtually unknown in the living organism. The "typical" compensation described by the empirically validated equations is a more accurate representation of what happens in reality.

In the case of metabolic alkalosis, one finds that alkalaemia is in fact a fairly weak depressant of ventilatory drive, which means that the ideal PaCO2 of 57mmHg would never be achieved in an alkalotic patient. One can imagine that the strategy of depressing the respiratory rate is going to eventually meet certain natural barriers. Consider a reductio ad absurdum: the patient with an SBE of 20 is not going to be happy with a respiratory compensation mechanism which results in a respiratory rate of one breath per minute. Hypoxia will intervene, for one. Similarly, in metabolic acidosis with an SBE of -20, the 5:3 rule would suggest a PaCO2 of around 6mmHg, which would probably represent a minute volume of around 25-30L/min, physiologically impossible in most circumstances.

In summary, in the living organism "complete compensation" does not occur: pH is never normalised to 7.4 because from an evolutionary standpoint, the development of compensatory mechanisms never viewed this as a worthwhile goal. The aim of compensatory mechanisms is to restore some sort of physiological normality, so that various enzyme systems may function well enough for the organism to survive.

A real example of what happens when you get confused.

Behold, a horrific ABG.


Let us attempt to reason through the compensation in this example.

The expected bicarbonate for this scenario, according to the "1 for 10" Boston rule, should be somewhere around 35.5 mmol/L. This "compensation" would be the direct consequence of carbonic anhydrase processing CO2 into HCO3- and H+.

However, the reported bicarbonate is 22.5mmol/L. Though "normal", this is still much lower than the expected value for this level of hypercapnea. Therefore, a metabolic acidosis must also be present, right? We know that the relationship of CO2 and HCO3- has a certain predictability to it. Thus, 22.5mmol/L must be the level to which the HCO3- has risen after the hypecapnea developed. In fact the HCO3- must have risen by around 11 mmol/L if we follow the "1 for 10" rule, and must therefore have been around 11.5mmol/L back when the CO2 was normal.

This is interesting. The base excess tells us there is no metabolic acid-base disturbance, but the rules of compensation suggest that there is. Why do these methods contradict each other?

Well. The answer lies in the little (st) symbol in the corner. The HCO3- (st) presented on the ABG print-out is the Copenhagen "standard bicarbonate", rather than the "actual bicarbonate". The actual bicarbonate is calculated from the measured PaCO2 and pH values using the Henderson-Hasselbalch equation, and is therefore the "gospel" bicarbonate (i.e. there can be no bicarbonate value other than this, for these measured values). The standard bicarbonate is a derived variable, using the actual base excess and various correction factors to estimate what the bicarbonate would be if the the PaCO2 was 40mmHg, and the PaO2 was 100mmHg. By rejecting the respiratory influences on buffer systems, those grim Norsemen invalidated their bicarbonate value, making it essentially useless for the purposes of Bostonian bedside assessment.

This is a bit of a problem. Does this mean we should never use the standard bicarbonate to assess acid-base compensation?

Yes. Yes it does.

The discrepancy between standard bicarbonate and actual bicarbonate is proportional to the derangement of PaCO2. The greater the derangement, the more inaccurate the compensation estimates based on the combination of Boston rules and Copenhagen measurements.

Thus, one should not make any attempt to combine these methods.

Instead, one needs to commit to one school or another.

For example, we may commit to the Copenhagen method to estimate compensation if we are unwilling to calculate the actual bicarbonate. On the other hand, if we enjoy mathematics for their own sake, we can calculate the actual bicarbonate ourselves, and use the Boston rules. In practice, both methods will yield the same answer. Observe the following interpretations:

Assessment of compensation with the Copenhagen rules

Using the Copenhagen rules, we begin with the assumption that the acute change in base excess in response to a respiratory disturbance is zero. Using this school of thought, one must conclude that ABE and CO2 are independent of one another, and that the former represents the metabolic component and the latter the respiratory.

In this fashion, we arrive at the conclusion that this patient has a primarily respiratory acidosis, with a negligible metabolic component.

Assessment of compensation with the Boston rules

In order to use the Boston rules, we must calculate the actual bicarbonate from the pH and PaCO2. Using this handy online calculator (which features a rearranged Henderson-Hasselbalch equation) we can calculate that at a pH of 6.956 and PaCO2 of 155mmHg, the HCO3- is 33.4mmol/L. The expected bicarbonate for this scenario, acording to the "1 for 10" Boston rule, should be somewhere around 35.5 mmol/L. The actual bicarbonate value is only slightly lower; therefore, we again conclude that this patient has a primarily respiratory acidosis, coexisting with a minor metabolic acidosis.

So what the hell are we supposed to do?

Driven to eye-clawing madness by the nightmarish complexity of these systems, an ICU trainee might ask, which system is best?

Well, either is reasonable. The Blood Gas Database contains several examples of real ABGs which are interpreted using both methods simultaneously; these examples serve to illustrate the interchangeability of the two methods, and indeed they tend to arrive at the same conclusions.

All this bicarbonate and base excess gibberish may of course be an artifact of a bygone era, when ABG machines were the size of townhouses and were powered by human sacrifice. The age of the Boston and Copenhagen debate was a primitive time, which saw ancient physiology professors stampeding around intensive care units like dinosaurs, roaring and raking the air with their rudimentary forelimbs. In this enlightened iAge we have achieved a significant (and massively underrated) refinement of our understanding of acid-base physiology, which is represented by the Stewart Quantitative Model. Contrary to the normal principles of science, this elegant approach increases the complexity of the subject matter, and burdens the user with a massively increased intensity of computation, which - thus far - has not been assumed by ABG machine microchips. Some say that in the coming decades we may witness a shift away from primitive mid-20th century methods, burying the Transatlantic debate along with its crappy equations, and embrace the Stewart model as the standard. Others point to the fact that over three decades have already passed and we are still submerged up to our knees in base excesses and bedside rules, without any significant deterioration in the quality of patient care. Time will tell whether abstract academics will be victorious over blunt clinical pragmatism.


Device-specific information in all these ABG pages refers to the ABG machine used in my home unit.

Other machines may have different reference ranges and different symbols.

For my ABG analyser, one can examine this handy operations manual.

There is also an even more handy reference manual, but one needs to be an owner of this equipment before one can get hold of it. Its called the "989-963I ABL800 Reference Manual"

Ole Siggaard-Andersen's website was also an excellent source of information for this topic.

Park, M., et al. "Clinical utility of standard base excess in the diagnosis and interpretation of metabolic acidosis in critically ill patients." Brazilian Journal of Medical and Biological Research 41.3 (2008): 241-249.

Schlichtig, Robert, Alan W. Grogono, and John W. Severinghaus. "Human PaCO2 and standard base excess compensation for acid-base imbalance."Critical care medicine 26.7 (1998): 1173-1179.

Schwartz, William B., and Arnold S. Relman. "A Critique of the Parameters Used in the Evaluation of Acid-Base Disorders: Whole-Blood Buffer Base and Standard Bicarbonate Compared with Blood pH and Plasma Bicarbonate Concentration." New England Journal of Medicine 268.25 (1963): 1382-1388.

Severinghaus, John W. "Siggaard-Andersen and the “great trans-Atlantic acid-base debate”." Scandinavian Journal of Clinical and Laboratory Investigation53.s214 (1993): 99-104.

Story, David A. "Bench-to-bedside review: A brief history of clinical acid–base."Critical Care 8.4 (2004): 253.

Brackett Jr, Newton C., Jordan J. Cohen, and William B. Schwartz. "Carbon dioxide titration curve of normal man: Effect of increasing degrees of acute hypercapnia on acid-base equilibrium." New England Journal of Medicine 272.1 (1965): 6-12.

NARINS, ROBERT G., and MICHAEL EMMETT. "Simple and mixed acid-base disorders: a practical approach." Medicine 59.3 (1980): 161-182.

Van Slyke, Donald D., and Glenn E. Cullen. "Studies of acidosis I. The bicarbonate concentration of the blood plasma; its significance, and its determination as a measure of acidosis.Journal of Biological Chemistry 30.2 (1917): 289-346.

Freebairn, R., and P. Lloyd. "Using quantitative acid-base analysis in the ICU." Critical Care and Resuscitation 2006; 8: 19-30

Kishen, R., et al. "Facing acid–base disorders in the third millennium–the Stewart approach revisited." International Journal of Nephrology and Renovascular Disease 7 (2014): 209-217.

ALBERT, MORRIS S., RALPH B. DELL, and ROBERT W. WINTERS. "Quantitative displacement of acid-base equilibrium in metabolic acidosis."Annals of internal medicine 66.2 (1967): 312-322.