Relation between PaO₂/F_IO₂ ratio and F_IO₂: a mathematical description

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 O₂ difference, the intrapulmonary shunt fraction, the oxygen index and the PaO₂/F_IO₂ ratio. Of these different indices the PaO₂/F_IO₂ 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 PaO₂/F_IO₂ measurement with PEEP set above 10 cmH₂O, but F_IO₂ was still not fixed [3]. Important for this discussion, the PaO₂/F_IO₂ ratio is influenced not only by ventilator settings and PEEP but also by F_IO₂. First, changes in F_IO₂ influence the intrapulmonary shunt fraction, which equals the true shunt plus ventilation–perfusion mismatching. At F_IO₂ 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_IO₂ increases if V/Q mismatch is a major component in inducing hypoxemia (e.g., chronic obstructive lung disease and asthma). Second, at an F_IO₂ of 1.0 absorption atelectasis may occur, increasing true shunt [4]. Thus, at high F_IO₂ 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 O₂ difference, the PaCO₂ level and the hemoglobin level, the relation between PaO₂/F_IO₂ ratio and F_IO₂ 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_IO₂ between 0.21 and 1.0. Five indices of O₂ exchange efficiency were calculated (PaO₂/F_IO₂, venous admixture, P(A-a)O₂, PaO₂/alveolar PO₂, and the respiratory index). They described a curvilinear shape of the curve for PaO₂/F_IO₂ ratio as a function of F_IO₂, but PaO₂/F_IO₂ ratio exhibited the most stability at F_IO₂ values ≥ 0.5 and PaO₂ values ≤ 100 mmHg, and the authors concluded that PaO₂/F_IO₂ 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 PaO₂/F_IO₂ and F_IO₂, however, underlines the limitations describing the intensity of hypoxemia using PaO₂/F_IO₂, and is thus of major importance for the clinician. The objective of this note is to describe the relation between PaO₂/F_IO₂ and F_IO₂ 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 O₂ content of an “ideal” capillary with the actual arterial O₂ content and taking into account what comes into the lung capillary, i.e., the mixed venous content. Cc′O₂ is the capillary O₂ content in the ideal capillary, CaO₂ is the arterial O₂ content, and CvO₂ is the mixed venous O₂ content,

$$ S=\frac{\dot{Q}s}{\dot{Q}t}=\frac{({\text{Cc}}^{\prime} {\text{O}}_2 - {\text{CaO}}_2)}{({\text{Cc}}^{\prime} {\text{O}}_2 - {\text{C}}\bar{\text{v}}\text{O}_2)} $$

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

$$ {\text{Cc}}^{\prime} {\text{O}}_2 - {\text{CaO}}_2 = \left(\frac{S}{1-S}\right)\times {\text{AVD}}. $$

Blood O₂ contents are calculated from PO₂ and hemoglobin concentrations as:

Equation of oxygen content (Equation 2)

$$ {\text{C}}{\text{O}}_2 = ({\text{Hb}}\times {\text{S}{\text{O}}_2\times 1.34)+(\text{P}{\text{O}}_2\times 0.0031)} $$

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 O₂ follows Henry's law – the amount of O₂ dissolved is proportional to its partial pressure. For each mmHg of PO₂ there is 0.003 ml O₂/dl dissolved in each 100 ml of blood. O₂ 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 O₂. Deriving blood O₂ content allows calculation of both Cc′O₂ and CaO₂ and allows Eq. 1 to be rewritten as follows:

$$ \begin{array}{ll} \left[({\text{Hb}}\times \text{S}{\text{c}}^{\prime} {\text{O}}_2\times 1.34)+(\text{P}{\text{c}}^{\prime} {\text{O}}_2\times 0.0031)\right]\\[1mm]-\big[({\text{Hb}}\times {\text{SaO}}_2\times 1.34)+({\text{PaO}}_2\times 0.0031)\big]\\=\displaystyle{\left(\frac{S}{1-S}\right)}\times {\text{AVD}}\end{array} $$

In the ideal capillary (c′), the saturation is 1.0 and the Pc′O₂ is derived from the alveolar gas equation:

$$ \text{P}{\text{c}}^{\prime} {\text{O}}_2={\text{PAO}}_2=(\text{P}_{\text{B}}-47)\times \text{F}_{\text{I}}{\text{O}}_2-\frac{{\text{PaCO}}_2}{R}. $$

This equation describes the alveolar partial pressure of O₂ (PAO₂) 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_IO₂, to get the inspired O₂ fraction reaching the alveoli, and on the other hand of PaCO₂ and the respiratory quotient (R) indicating the alveolar partial pressure of PCO₂. Saturation, Sc′O₂ and SaO₂ are bound with O₂ partial pressure (PO₂) Pc′O₂ and PaO₂, by the oxy-hemoglobin dissociation curve, respectively. The oxy-hemoglobin dissociation curve describes the relationship of the percentage of hemoglobin saturation to the blood PO₂. This relationship is sigmoid in shape and relates to the nonlinear relation between hemoglobin saturation and its conformational changes with PO₂. A simple, accurate equation for human blood O₂ dissociation computations was proposed by Severinghaus et al. [9]:

Blood O₂ dissociation curve equation (Equation 4)

$$ \text{S}{\text{O}}_2=\Bigg(\left(\bigg(\text{P}{\text{O}}_2^3+150\text{P}{\text{O}}_2\bigg)^{-1}\times 23\,400\right)+1\Bigg)^{-1} $$

This equation can be introduced in Eq. 1:

$$ \begin{array}{ll}\Bigg[\Bigg(\text{Hb}\times\Bigg(\bigg(\bigg(\bigg((\text{P}_{\text{B}}-47)\times \text{F}_{\text{I}}\text{O}_2 - \displaystyle{\frac{{\text{PaCO}}_2}{R}}\bigg)^3 \\+\,150\,\bigg((\text{P}_{\text{B}}-47)\times \text{F}_{\text{I}}{\text{O}}_2-\displaystyle{\frac{{\text{PaCO}}_2}{R}}\bigg)\bigg)^{-1}\\\times\,23\,400\bigg)+1\bigg)^{-1}\times 1.34\Bigg)+\Bigg(\bigg((\text{P}_{\text{B}}-47)\\\times\,\text{F}_{\text{I}}{\text{O}}_2-\displaystyle{\frac{{\text{PaCO}}_2}{R}}\bigg)\times 0.0031\Bigg)\Bigg]\\-\Bigg[\Bigg({\text{Hb}}\times\bigg(\bigg(\bigg({\text{PaO}}_2^3+150{\text{PaO}}_2\bigg)^{-1}\times 23\,400\bigg)\\+\,1\bigg)^{-1}\times 1.34\bigg)+({\text{PaO}}_2\times 0.0031)\Bigg)\Bigg]\\=\left(\displaystyle{\frac{S}{1-S}}\right)\times {\text{AVD}}\end{array} $$

Equation 1 modified gives a relation between F_IO₂ and PaO₂ with six fixed parameters: Hb, PaCO₂, 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_IO₂ and PaO₂/F_IO₂ ratio. As previously mentioned, numerous factors, notably nonpulmonary factors, influence this curve: intrapulmonary shunt, AVD, PaCO₂, respiratory quotient and hemoglobin. The relationship between PaO₂/F_IO₂ and F_IO₂ 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 PaO₂/F_IO₂ ratio varies considerably with changes in F_IO₂. At both extremes of F_IO₂, the PaO₂/F_IO₂ is substantially greater than at intermediate F_IO₂. In contrast, at extremely high shunt (≅ 60%) PaO₂/F_IO₂ ratio is greater at low F_IO₂ and decreases at intermediate F_IO₂, but does not exhibit any further increase as inspired F_IO₂ 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 PaO₂/F_IO₂ ratio for a given F_IO₂. AVD can vary substantially with cardiac output or with oxygen consumption.

Fig. 1

Relation between PaO₂/FIO₂ and FIO₂ for a constant arterio-venous difference (AVD) and different shunt levels (S)

Fig. 2

Relation between PaO₂/FIO₂ and FIO₂ for a constant shunt (S) level and different values of arterio-venous differences (AVD)

These computations therefore illustrate substantial variation in the PaO₂/F_IO₂ index as F_IO₂ 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_IO₂ results in denitrogenation atelectasis. Despite this limitation, large nonlinear variation and important morphologic differences of PaO₂/F_IO₂ ratio curves vary markedly with intrapulmonary shunt fraction and AVD variation. Thus, not taking into account the variable relation between F_IO₂ and the PaO₂/F_IO₂ 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_IO₂ 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_IO₂ to 1.0. In 17 patients (41%), the hypoxemia criterion for ARDS persisted after this change (PaO₂/F_IO₂ < 200 mmHg), while in the other 24 patients (58.5%) the PaO₂/F_IO₂ had become greater than 200 mmHg after changing the F_IO₂, 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_IO₂ will alter the PaO₂/F_IO₂ 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 PO₂/F_IO₂ with F_IO₂ occurs only after F_IO₂ increase to > 0.6 (depending on the AVD value). Thus, the use of the PO₂/F_IO₂ ratio as a dynamic variable should be used with caution if F_IO₂, as well as other ventilatory settings, varies greatly.