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Variation in the PaO_{2}/FiO_{2} ratio with FiO_{2}: mathematical and experimental description, and clinical relevance
Critical Care volume 11, Article number: R118 (2007)
Abstract
Introduction
Previous studies have shown through theoretical analyses that the ratio of the partial pressure of oxygen in arterial blood (PaO_{2}) to the inspired oxygen fraction (FiO_{2}) varies with the FiO_{2} level. The aim of the present study was to evaluate the relevance of this variation both theoretically and experimentally using mathematical model simulations, comparing these ratio simulations with PaO_{2}/FiO_{2} ratios measured in a range of different patients.
Methods
The study was designed as a retrospective study using data from 36 mechanically ventilated patients and 57 spontaneously breathing patients studied on one or more occasions. Patients were classified into four disease groups (normal, mild hypoxemia, acute lung injury and acute respiratory distress syndrome) according to their PaO_{2}/FiO_{2} ratio. On each occasion the patients were studied using four to eight different FiO_{2} values, achieving arterial oxygen saturations in the range 85–100%. At each FiO_{2} level, measurements were taken of ventilation, of arterial acid–base and of oxygenation status. Two mathematical models were fitted to the data: a oneparameter 'effective shunt' model, and a twoparameter shunt and ventilation/perfusion model. These models and patient data were used to investigate the variation in the PaO_{2}/FiO_{2} ratio with FiO_{2}, and to quantify how many patients changed disease classification due to variation in the PaO_{2}/FiO_{2} ratio. An F test was used to assess the statistical difference between the two models' fit to the data. A confusion matrix was used to quantify the number of patients changing disease classification.
Results
The twoparameter model gave a statistically better fit to patient data (P < 0.005). When using this model to simulate variation in the PaO_{2}/FiO_{2} ratio, disease classification changed in 30% of the patients when changing the FiO_{2} level.
Conclusion
The PaO_{2}/FiO_{2} ratio depends on both the FiO_{2} level and the arterial oxygen saturation level. As a minimum, the FiO_{2} level at which the PaO_{2}/FiO_{2} ratio is measured should be defined when quantifying the effects of therapeutic interventions or when specifying diagnostic criteria for acute lung injury and acute respiratory distress syndrome. Alternatively, oxygenation problems could be described using parameters describing shunt and ventilation/perfusion mismatch.
Introduction
The ratio of the partial pressure of oxygen in arterial blood (PaO_{2}) to the inspired oxygen fraction (FiO_{2}) 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 [1–3]). The PaO_{2}/FiO_{2} 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 ≤ PaO_{2}/FiO_{2} < 40 kPa) and of adult respiratory distress syndrome (ARDS) (PaO_{2}/FiO_{2} < 27 kPa) [4, 5].
Despite its widespread use, the validity of the PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio and have shown them to vary with the FiO_{2} level [6–8]. These theoretical analyses could lead us to believe that the PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio, and if these variations happen within interesting ranges of FiO_{2}. 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 PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio on varying FiO_{2} 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 FiO_{2}, as shown previously [10]. A single fixed value of 'effective shunt' may therefore not be able to simulate changes in the PaO_{2}/FiO_{2} 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 [11, 12]. 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 [14–16]. These models and techniques therefore provide tools that can both describe pulmonary gas exchange in the individual patient and potentially simulate changes in the PaO_{2}/FiO_{2} ratio.
The purpose of the present article is to assess the relevance of variation in the PaO_{2}/FiO_{2} ratio with the FiO_{2} level. To do so, we determined whether changes in the PaO_{2}/FiO_{2} ratio can be described accurately by either the 'effective shunt' model or a twoparameter model describing shunt and V/Q mismatch. Unlike previous studies that have examined changes in the PaO_{2}/FiO_{2} ratio with FiO_{2} theoretically through model simulation [6–8], the present analysis is performed both theoretically and experimentally by comparing model simulations with measured values of the PaO_{2}/FiO_{2} ratio in a range of different patients. Simulations of the PaO_{2}/FiO_{2} ratio performed with the twoparameter model are compared with those using the 'effective shunt' model to investigate whether the extra complexity of the twoparameter model is justified. The models are then used to simulate whether, and under which conditions, the PaO_{2}/FiO_{2} ratio varies with FiO_{2}, to further investigate the discrepancies between the two models and whether such variation is clinically relevant.
Materials and methods
Data were collected from 93 patients, most of these data being published previously [11, 14, 15]. Patients included postoperative surgical patients following gynaecological laparotomy [11, 14] and cardiac surgery [14, 15], those patients receiving intensive care therapy [14], normal subjects [14] and patients suffering from cardiac incompensation [14]. Twentyeight 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 endexpiratory 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 FiO_{2} values, achieving arterial oxygen saturation (SaO_{2}) values in the range 85–100%. The FiO_{2} values were selected on a patientspecific basis to cover this range, meaning that patients with more severe pulmonary disorders received higher FiO_{2} levels. Steady state was achieved at each FiO_{2} level either by waiting 5 minutes or by the presence of a stable endtidal oxygen fraction over a 30second period [13]. At steadystate conditions, measurements were taken of ventilation (FiO_{2}, endtidal oxygen fraction), of endtidal carbon dioxide fraction, tidal volume, and respiratory frequency, and of arterial acid–base and oxygenation status (SaO_{2}, PaO_{2}, pH, partial pressure of carbon dioxide, haemoglobin, methaemoglobin, and carboxyhaemoglobin). In some patients it was necessary to administer subatmospheric oxygen fractions to achieve SaO_{2} 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 FiO_{2}. 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 PaO_{2}/FiO_{2} ratio was calculated at each level of FiO_{2}.
Mathematical models
The data were analysed using two mathematical models of gas exchange: the 'effective shunt' model, used by Aboab and colleagues [6]; and the twoparameter model [11, 13, 14], the equations of which have been published previously ([14] electronic supplement). Figure 1 illustrates how these models differ in their representation of pulmonary gas exchange. The 'effective shunt' model includes one ideally ventilated and perfused alveolar compartment plus a compartment representing pulmonary shunt. The twoparameter model includes two alveolar compartments incorporating V/Q inequality with the addition of a shunt compartment.
In the 'effective shunt' model, oxygenation problems are described by a single parameter ('effective shunt') quantifying the blood flowing through the lungs without being oxygenated. In the twoparameter model, a shunt parameter is included along with the parameter fA2 describing the fraction of ventilation to a compartment receiving 90% of nonshunted perfusion. An fA2 value of 0.9 gives ideal V/Q matching, while lower fA2 values indicate V/Q mismatching. An fA2 value can be transformed into a ΔPO_{2} value, which describes the drop in oxygen pressure from the ventilated alveoli to the mixed blood leaving the lung capillaries; that is, the value in blood prior to the mixing of shunt. As such, ΔPO_{2} describes the extra oxygen pressure required at the mouth to remove an oxygenation problem due to V/Q mismatch; that is, ΔPO_{2} = 20 kPa means air plus 20% inspired oxygen (FiO_{2} = 0.41) is required.
Mathematical model simulations and statistical analysis
The 'effective shunt' model and the twoparameter model were used in three ways.
A theoretical comparison was performed between model simulations of changes in SaO_{2} and the PaO_{2}/FiO_{2} ratio with variation in FiO_{2} using the two mathematical models. To do so, simulations were performed for different values of model parameters.
The models were fitted to the data from each patient in turn using the leastsquares method, and the root mean square of the residuals was calculated for each of the fits. Model fits were illustrated by plotting simulated and measured values of SaO_{2} and the PaO_{2}/FiO_{2} ratio versus FiO_{2}. A statistical comparison of the 'goodness' of fit of the two models to the data was performed using an F test [17].
Both models were then used to analyse the variation in the PaO_{2}/FiO_{2} ratio over a range of FiO_{2} levels. This analysis had two aims: first, to evaluate the significance of any difference between the two models when fitted to the data; and second, to investigate whether the simulated variation in the PaO_{2}/FiO_{2} ratio was relevant. The relevant range was defined on an individual patient basis as the FiO_{2} range that resulted in a simulated value of SaO_{2} within the range 92–98%. The variation in the PaO_{2}/FiO_{2} ratio was then used to quantify the number of patients changing disease classification as a result of varying FiO_{2} levels according to the two models across the defined FiO_{2} range, these results being presented in a confusion matrix [18]. Patients were classified into disease groups at the lowest and highest FiO_{2} level in the range, according to the following criteria: ARDS (PaO_{2}/FiO_{2} < 27 kPa) [4, 5], ALI (27 kPa ≤ PaO_{2}/FiO_{2} < 40 kPa) [4, 5], and normal (PaO_{2}/FiO_{2} > 47 kPa) [19]. Those patients falling outside these categories are defined here as having mild hypoxemia (40 kPa ≤ PaO_{2}/FiO_{2} < 47 kPa).
Results
Figures 2 and 3 illustrate the results of the theoretical analysis showing the effects of varying FiO_{2} on model simulated values of SaO_{2} and the PaO_{2}/FiO_{2} 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 twoparameter model, these being equivalent for ΔPO_{2} = 0 kPa. Simulated increased shunt depresses the shoulder of the FiO_{2} versus SaO_{2} curve, and depresses and deforms the shape of the FiO_{2} versus PaO_{2}/FiO_{2} ratio curve. As a result, the relevant range of FiO_{2} (thick solid part of lines) broadens with increases in shunt. The deformation in the PaO_{2}/FiO_{2} ratio curve has a characteristic shape whereby the PaO_{2}/FiO_{2} first falls and then gradually rises, explained as follows. On increasing the FiO_{2} 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 PaO_{2} value increases little and the PaO_{2}/FiO_{2} ratio falls. On increasing the FiO_{2} level further, the mixture of shunted and lung capillary blood reaches an SaO_{2} value of about 98% where the arterial blood haemoglobin is almost saturated. Further increases in FiO_{2} translate into increased PaO_{2}, and hence an increasing PaO_{2}/FiO_{2} ratio. It should be noted that the range of FiO_{2} giving 92–98% saturation may extend below atmospheric oxygen levels (FiO_{2} = 0.21) in patients with only mild gas exchange abnormalities or in normal subjects. The simulations in Figure 2b show how the PaO_{2}/FiO_{2} ratio changes with FiO_{2} as found by Aboab and colleagues [6]. For example, for a shunt value of 20% (see Figure 2b, points a and b) the PaO_{2}/FiO_{2} ratio falls by 20.5 kPa, from 45.5 kPa to 25 kPa, over the relevant range of FiO_{2}.
Figure 3a,b illustrates the effects of varying the degree of V/Q mismatch in the twoparameter model. The effects of a V/Q mismatch on the SaO_{2} or the PaO_{2}/FiO_{2} ratio are quite different from the effects of shunt. The FiO_{2} versus SaO_{2} curves are shifted horizontally along the FiO_{2} axis with increasing V/Q mismatch. The PaO_{2}/FiO_{2} ratio is increased with increasing FiO_{2} levels, as the absence of significant shunt means that arterial haemoglobin is saturated on small increases in FiO_{2}. The small dip in the PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio is quite sensitive to changes in FiO_{2}. Within the relevant range of FiO_{2} (thick solid part of lines) for a ΔPO_{2} value of 10 kPa (see Figure 3b, points a and b), the PaO_{2}/FiO_{2} 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 SaO_{2} and the PaO_{2}/FiO_{2} ratio on varying the FiO_{2} 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 twoparameter model are shown. The range of FiO_{2} defined for each patient (giving SaO_{2} = 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 twoparameter 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 twoparameter model gave a statistically better fit to the data than the 'effective shunt' model (P < 0.005). In all cases the twoparameter 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 twoparameter model gave a much better fit to the data.
The plots of FiO_{2} versus the PaO_{2}/FiO_{2} ratio illustrated in Figure 4 also show that the twoparameter model is necessary to describe measured changes in the PaO_{2}/FiO_{2} ratio, where a V/Q mismatch is present. For some patients (Figure 4b,c,e,f) the measured change in the PaO_{2}/FiO_{2} ratio with FiO_{2} 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 FiO_{2} level can be seen, for example, in Figure 4d(ii). Here, according to the twoparameter model, an increase in the FiO_{2} level from 0.21 to 0.43 decreased the PaO_{2}/FiO_{2} 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 FiO_{2} using the two models. The lefthand column presents the number of patient cases classified in each group at a low FiO_{2} level. The table elements then describe the patient cases classified in each group at high FiO_{2}. Each element can therefore be interpreted to illustrate movement between groups; for example, of the 56 patient cases classified as normal at low FiO_{2} level using the twoparameter model, 39 patients remain classified as normal at high FiO_{2} 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 twoparameter model. With an increase in the FiO_{2} level, but maintaining SaO_{2} 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 twoparameter 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 FiO_{2} – whereas five patient cases changed classification to a less severe disease group according to the twoparameter model.
Discussion
The present study has investigated the variation in the PaO_{2}/FiO_{2} ratio with FiO_{2}, 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 PaO_{2}/FiO_{2} ratio taken at different FiO_{2} levels.
The use of a twoparameter model of gas exchange to describe variation in the PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio with varying FiO_{2} level. Interpretation of the PaO_{2}/FiO_{2} ratio changes in these four examples using the 'effective shunt' model would result in an overestimation of the changes in the PaO_{2}/FiO_{2} ratio when varying the FiO_{2} level. In the remaining two cases (Figure 4a,d) the 'effective shunt' model and the twoparameter 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 PaO_{2}/FiO_{2} ratio results in an overestimate of the number of patient cases changing disease classification upon increasing FiO_{2}, as illustrated in Table 1. Approximately 50% of the patient cases change classification using the 'effective shunt' model, in comparison with 30% using the twoparameter 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 PaO_{2}/FiO_{2} 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 twoparameter model, and this model was necessary to simulate these changes in the PaO_{2}/FiO_{2} 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 PaO_{2}/FiO_{2} ratio on increasing the FiO_{2} level in comparison with both the raw data and the twoparameter 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 PaO_{2}/FiO_{2} ratio limits [4]. In the present study, patient cases were classified only from the PaO_{2}/FiO_{2} ratio. The sole difference between the criteria for ALI and ARDS, however, is the level of hypoxemia quantified by the PaO_{2}/FiO_{2} ratio.
Conclusion
The present article has shown that the PaO_{2}/FiO_{2} ratio depends on both the FiO_{2} level and the SaO_{2} level, and that, for changes in FiO_{2} corresponding to an SaO_{2} range of 92–98%, 30% of patients change disease classification due to variation in the PaO_{2}/FiO_{2} ratio. The clinical and scientific utility of the PaO_{2}/FiO_{2} ratio therefore seems doubtful, and at the very least the FiO_{2} level at which the PaO_{2}/FiO_{2} 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 singleparameter PaO_{2}/FiO_{2} 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 twoparameter models rapidly using pulse oximetry data [14] and simple clinical methods [13], their clinical application seems timely.
Key messages

The variation in the PaO_{2}/FiO_{2} ratio with the FiO_{2} level is scientifically and clinically relevant.

The variation in the PaO_{2}/FiO_{2} ratio with the FiO_{2} level cannot be explained with an 'effective shunt' model, and requires a more complex, twoparameter, model.
Abbreviations
 ALI:

acute lung injury
 ARDS:

acute respiratory distress syndrome
 DOPCCO:

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
 IOF:

inspired oxygen fraction
 PPOAB:

partial pressure of oxygen in arterial blood
 AOS:

arterial oxygen saturation
 V/Q:

ventilation/perfusion.
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Acknowledgements
This work was partially supported by the Programme Commission on Nanoscience, Biotechnology and IT under the Danish Council for Strategic Research.
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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.
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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.
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Karbing, D.S., Kjærgaard, S., Smith, B.W. et al. Variation in the PaO_{2}/FiO_{2} ratio with FiO_{2}: mathematical and experimental description, and clinical relevance. Crit Care 11, R118 (2007). https://doi.org/10.1186/cc6174
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DOI: https://doi.org/10.1186/cc6174
Keywords
 Acute Lung Injury
 Patient Case
 Adult Respiratory Distress Syndrome
 Arterial Oxygen Saturation
 Alveolar Compartment