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## Introduction

The acute respiratory distress syndrome (ARDS) is characterized by severe hypoxemia, a cornerstone element in its definition. Numerous indices have been used to describe this hypoxemia, such as the arterial to alveolar O_{2} difference, the intrapulmonary shunt fraction, the oxygen index and the PaO_{2}/F_{I}O_{2} ratio. Of these different indices the PaO_{2}/F_{I}O_{2} ratio has been adopted for routine use because of its simplicity. This ratio is included in most ARDS definitions, such as the Lung Injury Score [1] and in the American–European Consensus Conference Definition [2]. Ferguson et al. recently proposed a new definition including static respiratory system compliance and PaO_{2}/F_{I}O_{2} measurement with PEEP set above 10 cmH_{2}O, but F_{I}O_{2} was still not fixed [3]. Important for this discussion, the PaO_{2}/F_{I}O_{2} ratio is influenced not only by ventilator settings and PEEP but also by F_{I}O_{2}. First, changes in F_{I}O_{2} influence the intrapulmonary shunt fraction, which equals the true shunt plus ventilation–perfusion mismatching. At F_{I}O_{2} 1.0, the effects of ventilation–perfusion mismatch are eliminated and true intrapulmonary shunt is measured. Thus, the estimated shunt fraction may decrease as F_{I}O_{2} increases if V/Q mismatch is a major component in inducing hypoxemia (e.g., chronic obstructive lung disease and asthma). Second, at an F_{I}O_{2} of 1.0 absorption atelectasis may occur, increasing true shunt [4]. Thus, at high F_{I}O_{2} levels (> 0.6) true shunt may progressively increase but be reversible by recruitment maneuvers. Third, because of the complex mathematical relationship between the oxy-hemoglobin dissociation curve, the arterio-venous O_{2} difference, the PaCO_{2} level and the hemoglobin level, the relation between PaO_{2}/F_{I}O_{2} ratio and F_{I}O_{2} is neither constant nor linear, even when shunt remains constant.

Gowda et al. [5] tried to determine the usefulness of indices of hypoxemia in ARDS patients. Using the 50-compartment model of ventilation–perfusion inhomogeneity plus true shunt and dead space, they varied the F_{I}O_{2} between 0.21 and 1.0. Five indices of O_{2} exchange efficiency were calculated (PaO_{2}/F_{I}O_{2}, venous admixture, P(A-a)O_{2}, PaO_{2}/alveolar PO_{2}, and the respiratory index). They described a curvilinear shape of the curve for PaO_{2}/F_{I}O_{2} ratio as a function of F_{I}O_{2}, but PaO_{2}/F_{I}O_{2} ratio exhibited the most stability at F_{I}O_{2} values ≥ 0.5 and PaO_{2} values ≤ 100 mmHg, and the authors concluded that PaO_{2}/F_{I}O_{2} ratio was probably a useful estimation of the degree of gas exchange abnormality under usual clinical conditions. Whiteley et al. also described identical relation with other mathematical models [6, 7].

This nonlinear relation between PaO_{2}/F_{I}O_{2} and F_{I}O_{2}, however, underlines the limitations describing the intensity of hypoxemia using PaO_{2}/F_{I}O_{2}, and is thus of major importance for the clinician. The objective of this note is to describe the relation between PaO_{2}/F_{I}O_{2} and F_{I}O_{2} with a simple model, using the classic Berggren shunt equation and related calculation, and briefly illustrate the clinical consequences.

## Berggren shunt equation (Equation 1)

The Berggren equation [8] is used to calculate the magnitude of intrapulmonary shunt (*S*), “comparing” the theoretical O_{2} content of an “ideal” capillary with the actual arterial O_{2} content and taking into account what comes into the lung capillary, i.e., the mixed venous content. Cc′O_{2} is the capillary O_{2} content in the ideal capillary, CaO_{2} is the arterial O_{2} content, and CvO_{2} is the mixed venous O_{2} content,

This equation can be written incorporating the arterio-venous difference (AVD) as:

Blood O_{2} contents are calculated from PO_{2} and hemoglobin concentrations as:

## Equation of oxygen content (Equation 2)

The formula takes into account the two forms of oxygen carried in the blood, both that dissolved in the plasma and that bound to hemoglobin. Dissolved O_{2} follows Henry's law – the amount of O_{2} dissolved is proportional to its partial pressure. For each mmHg of PO_{2} there is 0.003 ml O_{2}/dl dissolved in each 100 ml of blood. O_{2} binding to hemoglobin is a function of the hemoglobin-carrying capacity that can vary with hemoglobinopathies and with fetal hemoglobin. In normal adults, however, each gram of hemoglobin can carry 1.34 ml of O_{2}. Deriving blood O_{2} content allows calculation of both Cc′O_{2} and CaO_{2} and allows Eq. 1 to be rewritten as follows:

In the ideal capillary (c′), the saturation is 1.0 and the Pc′O_{2} is derived from the alveolar gas equation:

This equation describes the alveolar partial pressure of O_{2} (PAO_{2}) as a function, on the one hand, of barometric pressure (P_{B}), from which is subtracted the water vapor pressure at full saturation of 47 mmHg, and F_{I}O_{2}, to get the inspired O_{2} fraction reaching the alveoli, and on the other hand of PaCO_{2} and the respiratory quotient (*R*) indicating the alveolar partial pressure of PCO_{2}. Saturation, Sc′O_{2} and SaO_{2} are bound with O_{2} partial pressure (PO_{2}) Pc′O_{2} and PaO_{2}, by the oxy-hemoglobin dissociation curve, respectively. The oxy-hemoglobin dissociation curve describes the relationship of the percentage of hemoglobin saturation to the blood PO_{2}. This relationship is sigmoid in shape and relates to the nonlinear relation between hemoglobin saturation and its conformational changes with PO_{2}. A simple, accurate equation for human blood O_{2} dissociation computations was proposed by Severinghaus et al. [9]:

## Blood O_{2} dissociation curve equation (Equation 4)

This equation can be introduced in Eq. 1:

Equation 1 modified gives a relation between F_{I}O_{2} and PaO_{2} with six fixed parameters: Hb, PaCO_{2}, the respiratory quotient *R*, the barometric pressure (P_{B}), *S* and AVD. The resolution of this equation was performed here with Mathcad^{®} software, (Mathsoft Engineering & Education, Cambridge, MA, USA).

## Resolution of the equation

The equation results in a nonlinear relation between F_{I}O_{2} and PaO_{2}/F_{I}O_{2} ratio. As previously mentioned, numerous factors, notably nonpulmonary factors, influence this curve: intrapulmonary shunt, AVD, PaCO_{2}, respiratory quotient and hemoglobin. The relationship between PaO_{2}/F_{I}O_{2} and F_{I}O_{2} is illustrated in two situations. Figure 1 shows this relationship for different shunt fractions and a fixed AVD. For instance, in patients with 20% shunt (a frequent value observed in ARDS), the PaO_{2}/F_{I}O_{2} ratio varies considerably with changes in F_{I}O_{2}. At both extremes of F_{I}O_{2}, the PaO_{2}/F_{I}O_{2} is substantially greater than at intermediate F_{I}O_{2}. In contrast, at extremely high shunt (≅ 60%) PaO_{2}/F_{I}O_{2} ratio is greater at low F_{I}O_{2} and decreases at intermediate F_{I}O_{2}, but does not exhibit any further increase as inspired F_{I}O_{2} continue to increase, for instance above 0.7. Figure 2 shows the same relation but with various AVDs at a fixed shunt fraction. The larger is AVD, the lower is the PaO_{2}/F_{I}O_{2} ratio for a given F_{I}O_{2}. AVD can vary substantially with cardiac output or with oxygen consumption.

These computations therefore illustrate substantial variation in the PaO_{2}/F_{I}O_{2} index as F_{I}O_{2} is modified under conditions of constant metabolism and ventilation–perfusion abnormality.

## Consequences

This discussion and mathematical development is based on a mono-compartmental lung model and does not take into account dynamic phenomena, particularly when high F_{I}O_{2} results in denitrogenation atelectasis. Despite this limitation, large nonlinear variation and important morphologic differences of PaO_{2}/F_{I}O_{2} ratio curves vary markedly with intrapulmonary shunt fraction and AVD variation. Thus, not taking into account the variable relation between F_{I}O_{2} and the PaO_{2}/F_{I}O_{2} ratio could introduce serious errors in the diagnosis or monitoring of patients with hypoxemia on mechanical ventilation.

Recently, the accuracy of the American–European consensus ARDS definition was found to be only moderate when compared with the autopsy findings of diffuse alveolar damage in a series of 382 patients [10]. The problem discussed here with F_{I}O_{2} may to some extent participate in these discrepancies. A study by Ferguson et al. [11] illustrated the clinical relevance of this discussion. They sampled arterial blood gases immediately after initiation of mechanical ventilation and 30 min after resetting the ventilator in 41 patients who had early ARDS based on the most standard definition [2]. The changes in ventilator settings chiefly consisted of increasing F_{I}O_{2} to 1.0. In 17 patients (41%), the hypoxemia criterion for ARDS persisted after this change (PaO_{2}/F_{I}O_{2} < 200 mmHg), while in the other 24 patients (58.5%) the PaO_{2}/F_{I}O_{2} had become greater than 200 mmHg after changing the F_{I}O_{2}, essentially “curing” them of their ARDS in a few minutes. Of note, outcome varied greatly between the “persistent” and “transient” ARDS groups. There was a large difference in mortality, and duration of ventilation, favoring the “transient” ARDS group. Thus, varying F_{I}O_{2} will alter the PaO_{2}/F_{I}O_{2} ratio in patients with true and relative intrapulmonary shunt of ≥ 20%. In clinical practice, when dealing with patients with such shunt levels, one should know that the increasing PO_{2}/F_{I}O_{2} with F_{I}O_{2} occurs only after F_{I}O_{2} increase to > 0.6 (depending on the AVD value). Thus, the use of the PO_{2}/F_{I}O_{2} ratio as a dynamic variable should be used with caution if F_{I}O_{2}, as well as other ventilatory settings, varies greatly.

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Aboab, J., Louis, B., Jonson, B. *et al.* Relation between PaO_{2}/F_{I}O_{2} ratio and F_{I}O_{2}: a mathematical description.
*Intensive Care Med* **32**, 1494–1497 (2006). https://doi.org/10.1007/s00134-006-0337-9

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DOI: https://doi.org/10.1007/s00134-006-0337-9