The ratio of the partial pressure of oxygen in arterial blood (PaO₂) to the inspired oxygen fraction (FiO₂) has been used to quantify the degree of abnormalities in pulmonary gas exchange. The ratio has been used in numerous experimental studies to quantify pulmonary gas exchange before and after therapeutic intervention (for example 123). The PaO₂/FiO₂ ratio has also been used in the clinical setting to classify patients' pulmonary gas exchange status, including the definitions of acute lung injury (ALI) (27 kPa ≤ PaO₂/FiO₂ < 40 kPa) and of adult respiratory distress syndrome (ARDS) (PaO₂/FiO₂ < 27 kPa) 45.

Despite its widespread use, the validity of the PaO₂/FiO₂ ratio as a tool for assessing pulmonary gas exchange has been questioned. Using mathematical models describing gas exchange, previous authors have simulated values of the PaO₂/FiO₂ ratio and have shown them to vary with the FiO₂ level 678. These theoretical analyses could lead us to believe that the PaO₂/FiO₂ ratio is a poor indicator of a patient's pulmonary gas exchange status in the clinic. This hypothesis is only true, however, if the simulations performed are indeed able to describe measured variations in the PaO₂/FiO₂ ratio, and if these variations happen within interesting ranges of FiO₂. The latter of these conditions is crucial in determining whether this ratio is a useful scientific and clinical parameter.

The ability of a particular simulation to accurately describe variation in the PaO₂/FiO₂ ratio depends upon the complexity of the mathematical models used. Gowda and Klocke 7 used the complex mathematical model included in the multiple inert gas elimination technique 9 to simulate changes in the PaO₂/FiO₂ ratio on varying FiO₂ levels. This complex model has the advantage of describing pulmonary gas exchange accurately; however, its complexity means that the model is not useful for describing an individual patient in the intensive care unit. Aboab and colleagues used a simple mathematical model where an 'effective' pulmonary shunt was used to describe all ventilation/perfusion (V/Q) abnormalities in the lung 6. This model has the advantage that values of 'effective shunt' can be estimated from clinical data. Values of 'effective shunt', however, are well known to vary with FiO₂, as shown previously 10. A single fixed value of 'effective shunt' may therefore not be able to simulate changes in the PaO₂/FiO₂ ratio accurately.

Mathematical models have been proposed recently that describe the gas exchange using two parameters: a shunt value, and a second parameter describing the V/Q ratio 1112. These parameter values can be estimated simply and noninvasively in the clinic 13, and have been shown to fit data from a range of mechanically ventilated patients and spontaneously breathing patients 141516. These models and techniques therefore provide tools that can both describe pulmonary gas exchange in the individual patient and potentially simulate changes in the PaO₂/FiO₂ ratio.

The purpose of the present article is to assess the relevance of variation in the PaO₂/FiO₂ ratio with the FiO₂ level. To do so, we determined whether changes in the PaO₂/FiO₂ ratio can be described accurately by either the 'effective shunt' model or a two-parameter model describing shunt and V/Q mismatch. Unlike previous studies that have examined changes in the PaO₂/FiO₂ ratio with FiO₂ theoretically through model simulation 678, the present analysis is performed both theoretically and experimentally by comparing model simulations with measured values of the PaO₂/FiO₂ ratio in a range of different patients. Simulations of the PaO₂/FiO₂ ratio performed with the two-parameter model are compared with those using the 'effective shunt' model to investigate whether the extra complexity of the two-parameter model is justified. The models are then used to simulate whether, and under which conditions, the PaO₂/FiO₂ ratio varies with FiO₂, to further investigate the discrepancies between the two models and whether such variation is clinically relevant.

Data were collected from 93 patients, most of these data being published previously 111415. Patients included postoperative surgical patients following gynaecological laparotomy 1114 and cardiac surgery 1415, those patients receiving intensive care therapy 14, normal subjects 14 and patients suffering from cardiac incompensation 14. Twenty-eight of these patients were mechanically ventilated and presented in the intensive care unit; the remaining 57 patients were breathing spontaneously. Some patients were studied on more than one occasion, giving a total of 120 patient cases. In addition, new data from a further eight mechanically ventilated intensive care patients studied at one or two positive end-expiratory pressure settings were included in the analysis, adding 14 additional patient cases – giving a total of 134 patient cases. All intensive care patients had disorders in pulmonary gas exchange either due to primary infectious involvement or due to a secondary pulmonary involvement as a consequence of severe sepsis or septic shock. Ethical approval was obtained from the relevant ethics committee for all studies, and informed written and oral consent was obtained for all patients.

On each occasion patients were studied using four to eight different FiO₂ values, achieving arterial oxygen saturation (SaO₂) values in the range 85–100%. The FiO₂ values were selected on a patient-specific basis to cover this range, meaning that patients with more severe pulmonary disorders received higher FiO₂ levels. Steady state was achieved at each FiO₂ level either by waiting 5 minutes or by the presence of a stable end-tidal oxygen fraction over a 30-second period 13. At steady-state conditions, measurements were taken of ventilation (FiO₂, end-tidal oxygen fraction), of end-tidal carbon dioxide fraction, tidal volume, and respiratory frequency, and of arterial acid–base and oxygenation status (SaO₂, PaO₂, pH, partial pressure of carbon dioxide, haemoglobin, methaemoglobin, and carboxyhaemoglobin). In some patients it was necessary to administer subatmospheric oxygen fractions to achieve SaO₂ in the range 85–90%, which was achieved by mixing nitrogen with air in the inspiratory gas. In 18 experiments where all patients were breathing spontaneously, arterial blood gases were only measured at two levels of FiO₂. These patient cases were excluded from the current analysis, giving a total number of 116 patient cases for data analysis (51 mechanically ventilated patients, 65 spontaneously breathing patients). The PaO₂/FiO₂ ratio was calculated at each level of FiO₂.

Figures 2 and 3 illustrate the results of the theoretical analysis showing the effects of varying FiO₂ on model simulated values of SaO₂ and the PaO₂/FiO₂ ratio.

Figure 2a,b illustrates the effects of varying either the 'effective shunt' of the model of Aboab and colleagues 6 or the shunt value included in the two-parameter model, these being equivalent for ΔPO₂ = 0 kPa. Simulated increased shunt depresses the shoulder of the FiO₂ versus SaO₂ curve, and depresses and deforms the shape of the FiO₂ versus PaO₂/FiO₂ ratio curve. As a result, the relevant range of FiO₂ (thick solid part of lines) broadens with increases in shunt. The deformation in the PaO₂/FiO₂ ratio curve has a characteristic shape whereby the PaO₂/FiO₂ first falls and then gradually rises, explained as follows. On increasing the FiO₂ level, the partial pressure of oxygen in the lung capillary blood increases. As the lung capillary blood mixes with that shunted, the increase in the partial pressure of oxygen in the lung capillary blood helps to oxygenate the shunted blood, so that the PaO₂ value increases little and the PaO₂/FiO₂ ratio falls. On increasing the FiO₂ level further, the mixture of shunted and lung capillary blood reaches an SaO₂ value of about 98% where the arterial blood haemoglobin is almost saturated. Further increases in FiO₂ translate into increased PaO₂, and hence an increasing PaO₂/FiO₂ ratio. It should be noted that the range of FiO₂ giving 92–98% saturation may extend below atmospheric oxygen levels (FiO₂ = 0.21) in patients with only mild gas exchange abnormalities or in normal subjects. The simulations in Figure 2b show how the PaO₂/FiO₂ ratio changes with FiO₂ as found by Aboab and colleagues 6. For example, for a shunt value of 20% (see Figure 2b, points a and b) the PaO₂/FiO₂ ratio falls by 20.5 kPa, from 45.5 kPa to 25 kPa, over the relevant range of FiO₂.

Figure 3a,b illustrates the effects of varying the degree of V/Q mismatch in the two-parameter model. The effects of a V/Q mismatch on the SaO₂ or the PaO₂/FiO₂ ratio are quite different from the effects of shunt. The FiO₂ versus SaO₂ curves are shifted horizontally along the FiO₂ axis with increasing V/Q mismatch. The PaO₂/FiO₂ ratio is increased with increasing FiO₂ levels, as the absence of significant shunt means that arterial haemoglobin is saturated on small increases in FiO₂. The small dip in the PaO₂/FiO₂ ratio seen in these curves, particularly at the 0 kPa level, is due to the 5% shunt used in these plots. For the cases simulated in Figure 3, the PaO₂/FiO₂ ratio is quite sensitive to changes in FiO₂. Within the relevant range of FiO₂ (thick solid part of lines) for a ΔPO₂ value of 10 kPa (see Figure 3b, points a and b), the PaO₂/FiO₂ ratio increases by 8.3 kPa, from 32.9 kPa to 41.2 kPa.

Figure 4 illustrates model simulations and measured data describing the changes in SaO₂ and the PaO₂/FiO₂ ratio on varying the FiO₂ level for six patients, selected from the 116 patient cases to represent typical cases. Measured values and the model simulations using the 'effective shunt' model and the two-parameter model are shown. The range of FiO₂ defined for each patient (giving SaO₂ = 92–98%) is shown, such that those patients with more severe lung diseases have a higher range than those with less severe disease. For each of the fits, values of model parameters are given along with the root mean square value describing the error in model fitting.

The average (± standard deviation) root mean square for fitting the two-parameter model to the data was 0.5 ± 0.4%, compared with 1.4 ± 1.0% (± SD) for the 'effective shunt' model. The results of the F test showed that the two-parameter model gave a statistically better fit to the data than the 'effective shunt' model (P < 0.005). In all cases the two-parameter model fitted the data either as well as or better than the 'effective shunt' model, as described by the root mean square. In cases where the 'effective shunt' model fitted the data well (for example, Figure 4a,d), the fits of the two models were almost identical. In other cases (for example, Figure 4b,c,e,f), the two-parameter model gave a much better fit to the data.

The plots of FiO₂ versus the PaO₂/FiO₂ ratio illustrated in Figure 4 also show that the two-parameter model is necessary to describe measured changes in the PaO₂/FiO₂ ratio, where a V/Q mismatch is present. For some patients (Figure 4b,c,e,f) the measured change in the PaO₂/FiO₂ ratio with FiO₂ had a very different form from that predicted by the 'effective shunt' model. The possibility of a patient being defined in different clinical groups dependent on the FiO₂ level can be seen, for example, in Figure 4d(ii). Here, according to the two-parameter model, an increase in the FiO₂ level from 0.21 to 0.43 decreased the PaO₂/FiO₂ ratio from 45 kPa to 34 kPa, resulting in a change in disease classification from mild hypoxemia to ALI.

Table 1 presents a confusion matrix showing the number of patient cases classified in the four disease groups and how this classification varied with changes in FiO₂ using the two models. The left-hand column presents the number of patient cases classified in each group at a low FiO₂ level. The table elements then describe the patient cases classified in each group at high FiO₂. Each element can therefore be interpreted to illustrate movement between groups; for example, of the 56 patient cases classified as normal at low FiO₂ level using the two-parameter model, 39 patients remain classified as normal at high FiO₂ levels.

Disease classification changed in 60 of 116 patient cases (~50%) according to the 'effective shunt' model, compared with 38 of 116 patient cases (~30%) according to the two-parameter model. With an increase in the FiO₂ level, but maintaining SaO₂ within the range 92–98%, according to the 'effective shunt' model the number of patient cases classified as ALI and ARDS changed from 14 to 40 (~186% increase) and from 18 to 38 (~111% increase), respectively. According to the two-parameter model, the number of patient cases classified as ALI and ARDS changed from 23 to 31 (~35% increase) and from 18 to 24 (~33% increase), respectively. According to the 'effective shunt' model, disease severity only increased with FiO₂ – whereas five patient cases changed classification to a less severe disease group according to the two-parameter model.

The present study has investigated the variation in the PaO₂/FiO₂ ratio with FiO₂, and the mathematical model complexity necessary to describe this variation. For the first time this analysis has been performed not only theoretically using mathematical model simulations, but also experimentally from measurements of the PaO₂/FiO₂ ratio taken at different FiO₂ levels.

The use of a two-parameter model of gas exchange to describe variation in the PaO₂/FiO₂ ratio has been investigated. This model has been shown, using an F test, to provide a statistically better fit to oxygenation data than an 'effective shunt' model, even when taking into account the degrees of freedom lost due to the presence of an extra parameter. This improvement in fit can be seen in the plots shown in Figure 4, which were selected to illustrate a variety of patient cases. In four of these six cases (Figure 4b,c,e,f), simulations using the 'effective shunt' model do not describe the measured variation in the PaO₂/FiO₂ ratio with varying FiO₂ level. Interpretation of the PaO₂/FiO₂ ratio changes in these four examples using the 'effective shunt' model would result in an overestimation of the changes in the PaO₂/FiO₂ ratio when varying the FiO₂ level. In the remaining two cases (Figure 4a,d) the 'effective shunt' model and the two-parameter model provide an equivalent description of the data. The case presented in Figure 4a represents a normal subject with no V/Q mismatch problem and 5% shunt, whilst the case shown in Figure 4d represents a patient with little V/Q mismatch such that all oxygenation problems can be explained by shunt.

In general, use of the 'effective shunt' model to simulate changes in the PaO₂/FiO₂ ratio results in an overestimate of the number of patient cases changing disease classification upon increasing FiO₂, as illustrated in Table 1. Approximately 50% of the patient cases change classification using the 'effective shunt' model, in comparison with 30% using the two-parameter model.

For five patient cases, the change in disease classification simulated by the 'effective shunt' model was in the opposite direction to that shown by the measured PaO₂/FiO₂ ratio – the 'effective shunt' model simulating an incorrect degree of disease severity. In these patient cases the V/Q mismatch was the major cause of hypoxemia according to the two-parameter model, and this model was necessary to simulate these changes in the PaO₂/FiO₂ ratio. The difference in the direction of disease classification provided by these two models can be understood by looking at Figure 4e. For this case, the 'effective shunt' model simulation would result in a reduction in the PaO₂/FiO₂ ratio on increasing the FiO₂ level in comparison with both the raw data and the two-parameter model simulation.

The necessary criteria for diagnosing ALI and ARDS include acute onset of respiratory failure, bilateral infiltrates seen on a frontal chest radiograph and no clinical evidence of left atrial hypertension in addition to the PaO₂/FiO₂ ratio limits 4. In the present study, patient cases were classified only from the PaO₂/FiO₂ ratio. The sole difference between the criteria for ALI and ARDS, however, is the level of hypoxemia quantified by the PaO₂/FiO₂ ratio.

The present article has shown that the PaO₂/FiO₂ ratio depends on both the FiO₂ level and the SaO₂ level, and that, for changes in FiO₂ corresponding to an SaO₂ range of 92–98%, 30% of patients change disease classification due to variation in the PaO₂/FiO₂ ratio. The clinical and scientific utility of the PaO₂/FiO₂ ratio therefore seems doubtful, and at the very least the FiO₂ level at which the PaO₂/FiO₂ ratio is measured should be specified when quantifying the effects of therapeutic interventions or when specifying diagnostic criteria for ALI and ARDS. Perhaps more appropriate would be to replace the single-parameter PaO₂/FiO₂ ratio description with two parameters, a parameter to describe the oxygenation problem due to V/Q mismatch and one to describe oxygenation problems due to shunt. Indeed, Riley and Cournand 20 recognized in the 1950s that oxygenation problems should be divided in this way. With the ability to identify two-parameter models rapidly using pulse oximetry data 14 and simple clinical methods 13, their clinical application seems timely.

• The variation in the PaO₂/FiO₂ ratio with the FiO₂ level is scientifically and clinically relevant.

• The variation in the PaO₂/FiO₂ ratio with the FiO₂ level cannot be explained with an 'effective shunt' model, and requires a more complex, two-parameter, model.

ALI = acute lung injury; ARDS = acute respiratory distress syndrome; ΔPO₂ = drop in oxygen pressure from the ventilated alveoli to the mixed blood leaving the lung capillaries oxygen; fA2 = fraction of ventilation to a compartment receiving 90% of nonshunted perfusion; FiO₂ = inspired oxygen fraction; PaO₂ = partial pressure of oxygen in arterial blood; SaO₂ = arterial oxygen saturation; V/Q = ventilation/perfusion.

SK, SA and SER are all shareholders of Mermaid Care APS, a company involved in the development of equipment for the measurement of pulmonary gas exchange. SA is a board member of Mermaid Care APS. All other authors declare that they have no competing interests.

All authors contributed to the conception and design of the study. SK, KE and CA contributed to the data collection and clinical interpretation of the results. DSK, BWS, SA and SER contributed to the mathematical modelling, data analysis and technical interpretation of the results, including statistical analysis. DSK and SER drafted the manuscript, with all other authors being involved in its revision and approval.