Further Stewart Notes

First, why not try a Google search?

And now for a few other thoughts ...

What about whole blood?

If we look at whole blood - a mix of red cells and plasma - then things become more complex. If we take a sample of blood with everything at equilibrium then we can look at the red cell and plasma as two separate compartments, which interact. In each compartment, Stewart's principles must apply - if we know the independent variables, we can calculate the dependent ones!

It is thus possible to work out what is happening in the red cell compartment, just as we did for the plasma compartment. Things are more tricky in the red cell compartment, as there is an enormous amount of weak acid (in the form of haemoglobin), and the behaviour of this weak acid is, to put it mildly, very complex. I'm not aware of anyone who has sat down and accurately applied Stewart's approach to the red cell.

How do the two compartments interact? As Stewart pointed out, there are only two ways that the compartments can interact:

  1. by diffusion of CO2 across the red cell membrane (this is rapid, so the PCO2 in both compartments rapidly equilibrates);
  2. by movement of strong ions (notably the chloride ion).

The Traditional Approach

The traditional approach to acid-base concentrates on the Henderson-Hasselbalch (H-H) equation. This is simply a modification of one of the six equations we use in describing the relationship between the dependent variables we wish to calculate, and the independent variables that govern them. The H-H equation will of course always hold, but it cannot be used to explain the behaviour of dependent variables, which will be influenced by all of the independent variables in the acid-base system.

People who focus on the traditional approach are often vigorously critical of the physicochemical approach, but I have yet to read a valid mathematical criticism of Stewart's solution.

Note that there appear to be at least two "subdivisions" of the traditional approach - those who concentrate on plasma bicarbonate concentration as a measure of metabolic acid-base disturbance, and those who look at base excess as a measure of this disturbance. This is starkly illustrated by the lack of consensus in the Acid-base terminology document published in the Lancet way back when in 1965 (Lancet, 1965 2 1010-12).

Base Excess

How is base excess determined in a specimen of arterial blood? Formally, base excess is defined as "the amount of strong acid (or strong base) required to titrate whole blood to pH 7.40 at a standard PCO2 of 40mmHg". Note that no titration is done in real life - our 'blood gas machines' plug in measured values of Hb, pH and PCO2 and then use standardised algorithms to derive what base excess should be . It should be clear that base excess is the same as saying "How much do we have to change the SID in order to achieve a pH of 7.40?" - a 'traditionalist' will see this as "titrating strong acid/base"; a physicochemical fanatic will regard the strong ions administered with the "strong acid/base" as being the important component, as they reflect a change in SID!

What equation is used to calculate base excess? This equation has been termed the "Van Slyke equation" (See Scand J Cl Lab Invest Supp 77 37(146) 15-20, also Siggaard-Andersen, 1974). It tells us that:

deltacB'(B) = (1 - ctHb(B)*0.023) * (

deltactHCO 3 -(P) +

deltapH(P) * ( 2.30 * ctHb(B) + 7.7) ) .. Equation 1

This looks intimidating, but all we are doing is saying that:

The 'NCCLS recommendations' (Scand J Clin Lab Invest 1996 56 S 224 89-106) use a slightly different algorithm from Siggaard-Andersen's:

deltacB'(B) = (1 - ctHb(B) * 0.014 ) * (

deltactHCO3 -(P) +

deltapH(P)*(1.43 * ctHb(B) + 7.7) )

They take

deltapH as (pH - 7.40) and

deltactHCO 3 - as (cHCO 3 - - 24.8). Siggaard-Andersen variously uses 24.4 and 24.1 as values for a normal bicarbonate, and 0.023 or 0.0205 as the coefficient where they use 0.014.

It is even more instructive to get Siggaard-Andersen's book and read his derivation of Equation 1 (pages 44-51). The first thing you will note is that he makes a fair number of assumptions, perhaps the most telling being that plasma protein concentrations are normal !

In addition he gives us equation 1 thus:

deltacB'(B) = (1 - ctHb(B)*0.023) * (

deltactHCO 3 -(P) + ( 2.30 * ctHb)B) + 7.7) *

deltapH(P) ) .. His equation (15)

Fixing the errant parenthesis, we note that he derives his Equation (15) from the following two equations:

deltacB'(B) =

deltactHCO 3 -(P) * (1 - ctHb(B)*0.0205) ) .. His equation (14)


dctHCO 3 -(P)/dpH(P) = -2.30 * ctHb(B) - 7.7 .. His Equation (11)

Even allowing for his sudden change from a coefficient of 0.0205 to 0.023, can you see how he does this? I unfortunately cannot!

Disregarding my mathematical ineptitude, we note that assumptions such as a normal plasma protein concentration are unlikely to hold in critically ill patients. The Van Slyke equation may hold in normals, but we should use it with caution in the critically ill!

Also note that there are many other variants of 'Base Excess' in the literature, some advocating use of Base Excess calculated to account for the whole extracellular fluid volume:

BE ecf = cHCO 3 - - 24.8 + 16.2*(pH - 7.4)

Wilkinson (Crit Care Med 1979 7(6) 280-1) uses a different formula, attributed to Severinghaus:

BE = 37 * e ((pH-7.4) + 0.345 * Y)/(0.55 - 0.09 * Y) - 1

where Y = ln (PCO 2 / 40)

The Nottingham Physiology Simulator (BJA 1998 81 327-32) uses a different variant of our first BE ecf equation:

BE ecf = cHCO 3 - - 24 + 11.6 *(pH - 7.4)

and so the confusion continues..

Some articles reviewed

Here are a few 'Stewart-related' papers we considered worth reading a few years ago. Some might now be a little dated!

Stewart - Ready for Prime Time?

There is little doubt in my mind that the Stewart approach makes sense, and provides a slightly better model of how acid-base works than does the conventional approach. I believe that Stewart provides a refinement of the conventional approach. Under many, perhaps most circumstances, the 'old-fashioned' approach works fine, but we should be aware of the exceptions (gross volume dilution with fluids which have a low SID; hypoalbuminaemia in association with metabolic acidosis) and invoke the physicochemical approach in these circumstances. This new approach also helps us explain how our therapeutic interventions work.

Much still needs to be done. We need a viable model based on physicochemical principles that can be consistently shown to be as good as or better than the older models. Ideally this model should also extend to assessment of whole blood acid-base status, and even allow us to predict whole-body pH changes in response to therapeutic interventions.

In addition, each clinician who makes therapeutic decisions should appreciate the limitations of the model they are using. He/she should also relate the model to the limitations in laboratory estimation of the numbers that go into the model. For example, in the hospital where I currently work, the standard deviation of the estimates of serum sodium concentration is 3 mmol/l. I don't believe I can trust a serum sodium of "170 mmol/l" as I have seen a repeat estimate on the same specimen come out as "177 mmol/l"! We have also known since 1977 that small variations in sampling technique may have profound effects on arterial blood gas analysis - Hansen and Simmons (ARRD 1977 115 1061-3) found substantial reductions in PCO 2 related to heparinisation of arterial blood gas samples. Be careful when you plug the numbers you obtain into any model, and then make dramatic alterations in clinical management based on small numbers, especially where there may be multiple sources of error! This point is well made by Swenson in an otherwise rather humdrum editorial that you can read online.

Use both Stewart and conventional approaches with caution!

Some references

  1. Stewart PA How to Understand Acid-Base. A Quantitative Acid-Base Primer for Biology and Medicine 1981 Edward Arnold. ISBN 0-7131-4390-8.

  2. Stewart PA Can J Physiol Pharmacol 1983 61 1444 Modern quantitative acid-base chemistry.

  3. Siggaard-Andersen O The Acid-Base Status of the Blood, 4ed. Munksgaard, 1974. ISBN 87 16 01567 3

  4. Jon Waters has also provided an excellent introduction to the Stewart approach.

  5. Here's a tribute to Peter Stewart!