Relation between PaO2/FIO2 ratio and FIO2: a mathematical description

Jerome Aboab Bruno Louis Bjorn Jonson Laurent Brochard
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, theintrapulmonary shunt fraction, the oxygen index and the PaO2/FIO2 ratio. Of these different indices the PaO2/FIO2 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/FIO2 measurement with PEEP set above 10 cmH2O, but FIO2 was still not fixed [3]. Important for this discussion, the PaO2/FIO2 ratio is influenced not only by ventilator settings and PEEP but also by FIO2. First, changes in FIO2 influence the intrapulmonary shunt fraction, which equals the true shunt plus ventilation–perfusion mismatching. At FIO2 1.0, the effects of ventilation–perfusion mismatch are eliminated and true intrapulmonary shunt is measured. Thus, the estimated shunt fraction may decrease as FIO2 increases if V/Q mismatch is a major component in inducing hypoxemia (e.g., chronic obstructive lung disease and asthma). Second, at an FIO2 of 1.0 absorption atelectasis may occur, increasing true shunt [4]. Thus, at high FIO2 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/FIO2 ratio and FIO2 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 FIO2 between 0.21 and 1.0. Five indices of O2 exchange efficiency were calculated (PaO2/FIO2, venous admixture, P(A-a)O2, PaO2/alveolar PO2, and the respiratory index). They described a curvilinear shape of the curve for PaO2/FIO2 ratio as a function of FIO2, but PaO2/FIO2 ratio exhibited the most stability at FIO2 values = 0.5 and PaO2 values = 100 mmHg, and the authors concluded that PaO2/FIO2 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/FIO2 and FIO2, however, underlines the limitations describing the intensity of hypoxemia using PaO2/FIO2, and is thus of major importance for the clinician. The objective of this note is to describe the relation between PaO2/FIO2 and FIO2 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. CcO2 is the capillary O2 content in the ideal capillary, CaO2 is the arterial O2 content, and Cv¯O2 is the mixed venous O2 content,
S = Q.s Q.t = (CcO2 - CaO2) (CcO2 - C¯vO2)
This equation can be written incorporating the arteriovenous
difference (AVD) as:
CcO2 - CaO2 = S
1 - S× AVD.
Blood O2 contents are calculated from PO2 and hemoglobin concentrations as: Equation of oxygen content (Equation 2)
CO2 = (Hb × SO2 × 1.34) + (PO2 × 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 CcO2 and CaO2 and allows Eq. 1 to be rewritten as follows:
(Hb × ScO2 × 1.34) + (PcO2 × 0.0031)- (Hb × SaO2 × 1.34) + (PaO2 × 0.0031) = S
1 - S× AVD
In the ideal capillary (c), the saturation is 1.0 and the PcO2 is derived from the alveolar gas equation:
PcO2 = PAO2 = (PB - 47) × FIO2 - PaCO2 R .
This equation describes the alveolar partial pressure of O2 (PAO2) as a function, on the one hand, of barometric pressure (PB), from which is subtracted the water vapor pressure at full saturation of 47 mmHg, and FIO2, 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, ScO2 and SaO2 are bound with O2 partial pressure (PO2) PcO2 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 saturationand itsconformational 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)
SO2 = PO32 + 150PO2-1
× 23 400+ 1-1
This equation can be introduced in Eq. 1:
Hb × (PB - 47) × FIO2 - PaCO2 R 3 +150 (PB - 47) × FIO2 -
PaCO2 R -1 ×23 400+ 1-1 × 1.34+ (PB - 47) ×FIO2 - PaCO2 R × 0.0031
-Hb × PaO32
+ 150PaO2-1
× 23 400
+1-1
× 1.34+ (PaO2 × 0.0031)
= S
1 - S× AVD
Equation 1 modified gives a relation between FIO2 and PaO2 with six fixed parameters: Hb, PaCO2, the respiratory quotient R, the barometric pressure (PB), S and AVD.
The resolution of this equation was performed here with Mathcad® software, (Mathsoft Engineering & Education, Cambridge, MA, USA).
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) Resolution of the equation The equation results in a nonlinear relation between FIO2 and PaO2/FIO2 ratio. As previously mentioned, numerous factors, notably nonpulmonary factors, influence this curve: intrapulmonary shunt, AVD, PaCO2, respiratory quotient and hemoglobin. The relationship between PaO2/FIO2 and FIO2 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/FIO2 ratio varies considerably with changes in FIO2. At both extremes of FIO2, the PaO2/FIO2 is substantially greater than at intermediate FIO2. In contrast, at extremely high shunt (~=60%) PaO2/FIO2 ratio is greater at low FIO2 and decreases at intermediate FIO2, but does not exhibit any further increase as inspired FIO2 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/FIO2 ratio for a given FIO2. AVD can vary substantially with cardiac output or with oxygen consumption. These computations therefore illustrate substantial variation in the PaO2/FIO2 index as FIO2 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 FIO2 results in denitrogenation atelectasis. Despite this limitation, large nonlinear variation and important morphologic differences of PaO2/FIO2 ratio curves vary markedly with intrapulmonary shunt fraction and AVD variation. Thus, not taking into account the variable relation between FIO2 and the PaO2/FIO2 ratio couldintroduce 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 FIO2 may to some extent participate in these discrepancies. A studyby 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 ventilatorsettingschiefly consisted of increasing FIO2 to 1.0. In 17 patients (41%), the hypoxemia criterion for ARDS persisted after this change (PaO2/FIO2 < of =" 20%."> 0.6 (depending on the AVD value). Thus, the use of the PO2/FIO2 ratio as a dynamic variable should be used with caution if FIO2, as well as other ventilatory settings, varies greatly. References
1. Murray J,Matthay MA, Luce J, FlickM (1988) An expanded definition of the adult respiratory distress syndrome. Am Rev Respir Dis 138:720–723
2. Bernard GR, Artigas A, Brigham KL, Carlet J, Falke K, Hudson L, Lamy M,
Legall JR, Morris A, Spragg R (1994) The American–European Consensus Conference on ARDS. Definitions, mechanisms, relevant outcomes, and clinical trial coordination. Am J Respir Crit Care Med 149:818–824
3. Ferguson ND, Davis AM, Slutsky AS, Stewart TE (2005) Development of a clinical definition for acute respiratory distress syndrome using the Delphi technique. J Crit Care 20:147–154
4. Santos C, Ferrer M, Roca J, Torres A, Hernandez C, Rodriguez-Roisin R (2000) Pulmonary gas exchange response to oxygen breathing in acute lung injury. Am J Respir Crit Care Med 161:26–31
5. Gowda MS, Klocke RA (1997) Variability of indices of hypoxemia in adult respiratory distress syndrome. Crit Care Med 25:41–45
6. Whiteley JP, Gavaghan DJ, Hahn CE (2002) Variation of venous admixture, SF6 shunt, PaO2, and the PaO2/FIO2 ratio with FIO2. Br J Anaesth 88:771–778
7. Whiteley JP, Gavaghan DJ, Hahn CE
(2002) Mathematical modelling of oxygen transport to tissue. J Math Biol 44:503–522
8. Berggren S (1942) The oxygen deficit of arterial blood caused by nonventilating
parts of the lung. Acta Physiol
Scand 11:1–92
9. Severinghaus J (1979) Simple, accurate equations for human blood O2 dissociation computations. J Appl Physiol
46:599–602
10. Esteban A, Fernandez-Segoviano P, Frutos-Vivar F, Aramburu JA, Najera L,
Ferguson ND, Alia I, Gordo F, Rios F (2004) Comparison of clinical criteria for the acute respiratory distress syndrome with autopsy findings. Ann Intern Med 141:440–445
11. Ferguson ND, Kacmarek RM, Chiche JD, Singh JM, Hallett DC, Mehta S, Stewart TE (2004) Screening of ARDS patients using standardized ventilator settings: influence on enrollment in a clinical trial. Intensive Care
Med 30:1111–1116