This is the partial pressure of oxygen required to achieve 50% haemoglobin saturation. In the ABG machine, this value is extrapolated from the measured PaO_{2} and *s*O_{2}. It is represented as *p***50**, in contrast to the *p***50 _{(st)}** which is an idealised value calculated from the measured PaO

_{2}at a standard set of conditions (pH 7.40, pCO

_{2}40mmHg, and assuming the absence of dyshaemoglobins).

The oxygen-hemoglobin dissociation curve represents the affinity of hemoglobin for oxygen. The *p*50 value represents a mid-point in this curve, and gives us information regarding that affinity.

**In the adult, the normal p50 should be 24-28mmHg.**

The venerable Kerry Brandis gives 26.6 mmHg as the normal value for adult humans.

## Extrapolation of the *p*50 value using ABG machine algorithms

In brief, the process involves making a model of the oxyhaemoglobin dissociation curve on the basis of the measured variables, , and then calculating what the PaO_{2}would be at *s*O_{2} of 50%.

It is more difficult than it sounds. The reference manual for the local ABG analyser is a goldmine of information, and spells this out quite clearly. However, little physiological explanation is available there (it is after all an operations manual, not a physiology textbook). Thus, one turns to Ole Siggaard-Andersen's site which is the canonic resource for this topic, for many reasons but chiefly because he is the original author for many of the seminal papers and key equations described below. Therefore, to do this topic justice, the reader is directed to www.siggaard-andersen.dk.

## The tanh equation

This equation describes a hyperbolic tangent function, which acts as the mathematical model for the oxygen-haemoglobin dissociation curve. It was decribed in a seminal 1984 paper by both the Siggaard-Andersens as well as Wimberley and Gothgen.

In this model, the *x* and *y* coordinates of points along the curve are derived logarithmically from *s* and *p*, which are combined values. *s* is the combined saturation of oxygen and carbon monoxide, and *p* is the combined partial pressure of oxygen and carbon monoxide. This is done to account for the fact that haemoglobin binds carbon monoxide as well as (better than!) oxygen, and the presence of any carboxyhaemoglobin will alter the shape of the dissociation curve, making it more hyperbolic.

The equation has numerous components, and the components have subcomponents, and so on.

If one were a crazy person, one would represent the relationship in the following manner:

Obvious, this is not.

Those readers who do not intend to go on to a rich fulfilling career of designing ABG machines are unlikely to benefit from an indepth discussion of this mathematical quagmire. People who *do* wish to go around designing ABG machines would not be reading this site; likely they would be leafing absent-mindedly though Pure and Applied Chemistry. Therefore I can generate such ridiculous diagrams with the liberating expectation that nobody would ever try to genuinely* learn *anything from them.

In brief, the variables in the monstrous scrawl above are as follows:

- The
*x*and*y*coordinates are derived logarithmically from*s*(the combined saturation of oxygen and carbon monoxide), and*p*( the combined partial pressure of oxygen and carbon monoxide).*s*and*p*are derived from the measured variables - these are the solid foundations which validate the calculated curve.

- The
*x*° and*y°*coordinates represent the point of symmetry of the curve.*x*° is particularly interesting - it is the magnitude of the left and right shift, determined by the Bohr effects (i.e. the pH, pCO_{2}, dyshemoglobin levels and the concentration of 2,3-DPG) as well as the temperature.- Apart from a1 to a5,
*x*° is also determined by a6, an additional shift determined

*h°*and*k°*are constants.

## Calculating the shift of the oxyhemoglobin dissociation curve

First, one needs to select an oxyhaemoglobin dissociation curve to shift. The diagram below uses one which was generated using Severinghaus' classical data . Then, one needs to decise how far horisontally it has to be shifted, according to the combined effect of all the usual influences. Lastly, one calculates the position of a point, described by a set of coordinates (P_{0}, S_{0}), and forces the curve to pass through that point.

In the diagram above, the Radiometer variables (ac and a6) are used to describe the magnitude of the shift. (ac) can be though to represent "all causes", as in *"The calculation of the combined effect on the ODC position at 37 °C of all known causes for displacement"* . The (a6) is even more prosaic; its the 6th component of (ac), because ac = a1 + a2 + a3 + a4 + a5.

But wait, you say. The (ac) value is calculated with the use of cDPG! But, the ABG machine does not measure DPG. Where did this value come from?

This is unclear. Certainly the reference manual is silent on this matter. cDPG is estimated elsewhere to be 5mmol/L, which would yield an unsatisfying (a4= 0) result (i.e. why even include it).

It seems that it may be possible to extrapolate a *c*DPG level from other findings. For one, it appears to be present in normal red cells at about 0.75:1 molar concentration ratio to haemoglobin. Additionally, published work by Samja et al (1981) has demonstrated that from empirical measurements a nomogram can be constructed, which allows one to calculate a cDPG level from known pO_{2}, pCO_{2} pH and *p*50 values. In short, methods to estimate this variable are available, but it is uncertain as to how this variable is derived in this specific situation, within the mysterious innards of the ABL800.

Siggaard-Andersen's site contains a brief entry on 2,3-DPG. There, an equation is offered to calculate the *c*DPG value using the *p*50_{(st)}. The reasoning is that by supplying all the left-shift parameters into the tanh equation, one is able to exclude everything *but * the *c*DPG and dyshaemoglobin from the equation, and because dyshaemoglobin is a measured parameter, one can be reasonably confident that the magnitude of the remaining left shift is due to *c*DPG alone.

Similarly, the *F*HbF (foetal haemoglobin) is a measured parameter (by absorption spectrophotometry, like the other haemoglobin species) but not all ABG machines report this value, and thus it is unclear what happens in that case (is it ignored?)

Anyway.

One has *some* variables to plug in, and some sort of (ac) is calculated, which is the shift of the curve at 37°C. Now, one must calculate the shift of the curve to fit to the position of a point, defined as (P_{0}, S_{0}).

These calculated coordinates represent the actual measured relationship between oxygen/carbon monoxide saturation and tension. (ac+a6), nominated as (a) by Siggaard-Andersen, is the total shift of the curve at a standard temperature of 37°C; thus the green curve in the diagram above represents the shape of a combined oxygen/carbon monoxide and haemoglobin dissociation curve at 37°C, accounting for all of Bohr's effects.

The curve is now shifted by another notch to correct for whatever the actual patient temperature is, using the equation

**b = 0.055 × ( T- T°) **

where **b** is the magnitude of the shift and ** T°** is 37°C. Essentially,

**b**changes by 0.055 for every degree difference.

So, now we have a shift due to temperature (b) and a shift due to everything else (a).

Now, one is finally able to plug these variables into the Tanh function. This will generate an oxygen/carbon monoxide dissociation curve which passes through a point which corresponds to directly measured variables from the blood sample.

Having arrived at a curve of some sort, one is now finally able to predict where the p50 will fall.

Voila.

## Sources of inaccuracy in the derived p50 parameter

There are several issues one notes if one explores these equations.

Firstly, in the flat part of the curve (i.e. beyond an *s*O_{2} of 97%) the accuracy tends to flounder, as large changes to all the *p*50-influencing variables tend to only cause very small changes to the *s*O_{2}.

Secondly, the lack of clarity regarding 2,3-DPG and foetal haemoglobin measurements makes it difficult to interpret the difference between *p*50 and *p*50_{st} (i.e. is it or isn't it due to 2,3-DPG levels and *F*HbF?)

Thirdly, the model assumes that the effects of all the Bohr factors (eg. pH, 2,3-DPG etc) are all linear and additive, whereas in fact they are not. For example, the influence of 2,3-DPG changes with pH and temperature.

Lastly, it seems sulfahemoglobin plays no role in any of the calculations, but it certainly plays a role in moving the dissociation curve around (it produces a right-shift).

Thus, the ABG-derived *p*50 value is closer to an empirical measurement, but is still an error-prone mathematical construct.

Briefly, below is a summary of situations which may give rise to a change in *p*50. The influences on the shape of the curve are discussed at great length elsewhere, and

## Increased *p*50: decreased Hb-O_{2} affinity; a right shift

In this situation, the decreased affinity of hemoglobin for oxygen improves the deposition of oxygen in the tissues, but impaires the extaction of oxygen from alveolar gas, and thus impaires the overall transport mechanism. In such circumstances improvements of alveolar oxygen content and V/Q matching may not result in any improvement in tissuen oxygen delivery.

**Causes of a right shift in the oxygen-hemoglobin dissociation curve**

- Acidosis
- Increased PaCO
_{2}(the Bohr Effect) - Increased temperature
- Increased 2,3-DPG (eg. in pregnancy)

## Decreased *p*50: increased Hb-O_{2} affinity; a left shift

In this situation, the increased affinity of hemoglobin improves the absorption of oxygen from the capillaries, but degrades the rate of its deposition in the tissues. Tissue oxygen extraction becomes impaired, and so tissue hypoxia may exist in spite of a well-oxygenated blood volume.

**Causes of a left shift in the oxygen-hemoglobin dissociation curve**

- Alkalosis
- decreased PaCO
_{2} - Decreased temperature
- Decreased 2,3-DPG (eg. in stored blood)

Unusual haemoglobin species can also alter this value if the "total" unfractionated hemoglobin is being examined. Foetal hemoglobin (FHb) methaemoglobin (MetHb) and carboxyhemoglobin (COHb) all increase the affinity of haemoglobin for oxygen, and will decrease the apparent *p*50 even if the "natural" hemoglobin *p*50 will remain the same. The arterial blood gas analyser will dutifully disregard these subtleties, and present you with an empirical *p*50 value for the blood sample you supplied, whatever mixture of freakish alien hemoglobin it contains. This was a trap set by the CICM Fellowship examiners in Question 6.2 from the second paper of 2010.

## The relevance of p50 to the critically ill population

"Who gives a flying fuck", would yell the pragmatic intensivist, enraged by the general pfuffery of this increasingly academic discussion. "Who looks at that number anyway?" Indeed. Rarely if ever does an ICU doctor receive a phone call from emergency, ED staff specialist on the line wringing their hands with concern over an abnormal *p*50 value. However, this topic has attracted some attention from intensive care celebrities such as Myburgh and Worthley. Their investigation into this issue has revealed that critically ill patients on average have a higher oxygen affinity than the normals (*p*50 of 24.5 vs 26.6).

The implications of this are on oxygen delivery to tissues, which is ultimately the objective of all your resuscitation efforts. With a uselessly clingy haemoglobin, a heroic effort to restore normal oxygen saturation will not yield a satisfying improvement in the surrogate markers of tissue perfusion. This has the greatest influence in the context of massive transfusion, where one's organism is suddenly inundated with stored red cells near expiry, which have been chilled to 4°C, at a pH of 6.0 or so, and totally depleted of 2,3-DPG. In this situation, the *p*50 suddenly looks important. Sure, you might feel confident looking at the haemoglobin result ("look, its 80, they aren't bleeding any more - well done everybody") but in actual fact all of that haemoglobin is almost totally useless, and will remain so while the storage-damaged red cells gradually recover their function.

### References

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"

Malmberg, P. O., M. P. Hlastala, and R. D. Woodson. "Effect of increased blood-oxygen affinity on oxygen transport in hemorrhagic shock." *Journal of Applied Physiology* 47.4 (1979): 889-895.

Woodson, Robert D. "Physiological significance of oxygen dissociation curve shifts." *Critical care medicine* 7.9 (1979): 368-373.

MACDONALD, ROSEMARY. "Red cell 2, 3‐diphosphoglycerate and oxygen affinity." *Anaesthesia* 32.6 (1977): 544-553.

Huber, Fabienne L., et al. "Does venous blood gas analysis provide accurate estimates of hemoglobin oxygen affinity?." *Annals of hematology* 92.4 (2013): 517-521.

Kokholm G. "Simultaneous measurements of blood pH, pCO2, pO2 and concentrations of hemoglobin and its derivates--a multicenter study." Scand J Clin Lab Invest Suppl. 1990;203:75-86.

Samaja, Michele, et al. "Equations and nomogram for the relationship of human blood p50 to 2, 3-diphosphoglycerate, CO2, and H+." *Clinical chemistry* 27.11 (1981): 1856-1861.

Ostrander, L. E., et al. "A computer algorithm to calculate P50 from a single blood sample." *Biomedical Engineering, IEEE Transactions on* 4 (1983): 250-254.

Ekeloef, N. P., J. Eriksen, and C. B. Kancir. "Evaluation of two methods to calculate p50 from a single blood sample." *Acta anaesthesiologica scandinavica* 45.5 (2001): 550-552.

SIGGAARD‐ANDERSEN, M. A. D. S., and Ole SIGGAARD‐ANDERSEN. "Oxygen status algorithm, version 3, with some applications." *Acta Anaesthesiologica Scandinavica* 39.s107 (1995): 13-20.

Siggaard-Andersen, O., and M. Siggaard-Andersen. "The oxygen status algorithm: a computer program for calculating and displaying pH and blood gas data." *Scandinavian Journal of Clinical & Laboratory Investigation* 50.S203 (1990): 29-45.

Siggaard-Andersen, Ole, et al. "A mathematical model of the hemoglobin-oxygen dissociation curve of human blood and of the oxygen partial pressure as a function of temperature." *C**linical chemistry* 30.10 (1984): 1646-1651.

Siggaard-Andersen, Ole, Mads Siggaard-Andersehf, and Niels Fogh-Andersen. "The TANH-equation modified for the hemoglobin, oxygen, and carbon monoxide equilibrium." *Scandinavian Journal of Clinical and Laboratory Investigation* 53.s214 (1993): 113-119.

McLellan, Stuart A., and Timothy S. Walsh. "Red blood cell 2, 3-diphosphoglycerate concentration and in vivo P50 during early critical illness*."*Critical care medicine* 33.10 (2005): 2247-2252.