Alveolar gas equation

The alveolar gas equation is the method for calculating partial pressure of alveolar oxygen (P_AO₂). The equation is used in assessing if the lungs are properly transferring oxygen into the blood. The alveolar air equation is not widely used in clinical medicine, probably because of the complicated appearance of its classic forms. The partial pressure of oxygen (pO₂) in the pulmonary alveoli is required to calculate both the alveolar-arterial gradient of oxygen and the amount of right-to-left cardiac shunt, which are both clinically useful quantities. However, it is not practical to take a sample of gas from the alveoli in order to directly measure the partial pressure of oxygen. The alveolar gas equation allows the calculation of the alveolar partial pressure of oxygen from data that is practically measurable. It was first characterized in 1946.[1]

Assumptions

The equation relies on the following assumptions:

Inspired gas contains no carbon dioxide (CO₂)
Nitrogen (and any other gases except oxygen) in the inspired gas are in equilibrium with their dissolved states in the blood
Inspired and alveolar gases obey the ideal gas law
Carbon dioxide (CO₂) in the alveolar gas is in equilibrium with the arterial blood i.e. that the alveolar and arterial partial pressures are equal
The alveolar gas is saturated with water

Equation

p_{A}{\ce {O2}}=F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})-{\frac {p_{a}{\ce {CO2}}(1-F_{I}{\ce {O2}}(1-{\ce {RER}}))}{{\ce {RER}}}}

If $F_{I}{\ce {O2}}$ is small, or more specifically if $F_{I}{\ce {O2}}(1-{\ce {RER}})\ll 1$ then the equation can be simplified to:

p_{A}{\ce {O2}}\approx F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})-{\frac {p_{a}{\ce {CO2}}}{{\ce {RER}}}}

where:

Quantity	Description	Sample value
$p_{A}{\ce {O2}}$	The alveolar partial pressure of oxygen ( $p{\ce {O2}}$ )	107 mmHg (14.2 kPa)
$F_{I}{\ce {O2}}$	The fraction of inspired gas that is oxygen (expressed as a decimal).	0.21
P_ATM	The prevailing atmospheric pressure	760 mmHg (101 kPa)
$p{\ce {H2O}}$	The saturated vapour pressure of water at body temperature and the prevailing atmospheric pressure	47 mmHg (6.25 kPa)
$p_{a}{\ce {CO2}}$	The arterial partial pressure of carbon dioxide ( $p{\ce {CO2}}$ )	40 mmHg (5.33 kPa)
RER	The respiratory exchange ratio	0.8

Sample Values given for air at sea level at 37 °C.

Doubling $F_{i}{\ce {O2}}$ will double $P_{i}{\ce {O2}}$ .

Other possible equations exist to calculate the alveolar air.[2][3][4][5][6][7][8]

{\begin{aligned}P_{A}{\ce {O2}}&=F_{I}{\ce {O2}}\left(PB-P{\ce {H2O}}\right)-P_{A}C{\ce {O2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\&=P_{I}{\ce {O2}}-P_{A}C{\ce {O2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\&=P_{I}{\ce {O2}}-{\frac {V_{T}}{V_{T}-V_{D}}}\left(P_{I}{\ce {O2}}-P_{E}{\ce {O2}}\right)\\&={\frac {P_{E}{\ce {O2}}-P_{I}{\ce {O2}}\left({\frac {V_{D}}{V_{T}}}\right)}{1-{\frac {V_{D}}{V_{T}}}}}\end{aligned}}

Abbreviated alveolar air equation

P_{A}{\ce {O2}}={\frac {P_{E}{\ce {O2}}-P_{i}{\ce {O2}}{\frac {V_{D}}{V_{T}}}}{1-{\frac {V_{D}}{V_{T}}}}}

P_AO₂, P_EO₂, and P_iO₂ are the partial pressures of oxygen in alveolar, expired, and inspired gas, respectively, and VD/VT is the ratio of physiologic dead space over tidal volume.[9]

Respiratory quotient (R)

R={\frac {P_{E}{\ce {CO2}}(1-F_{I}{\ce {O2}})}{P_{i}{\ce {O2}}-P_{E}{\ce {O2}}-(P_{E}{\ce {CO2}}*F_{i}{\ce {O2}})}}

Physiologic dead space over tidal volume (VD/VT)

{\frac {V_{D}}{V_{T}}}={\frac {P_{a}{\ce {CO2}}-P_{E}{\ce {CO2}}}{P_{a}{\ce {CO2}}}}

References

Curran-Everett D (June 2006). "A classic learning opportunity from Fenn, Rahn, and Otis (1946): the alveolar gas equation". Adv Physiol Educ. 30 (2): 58–62. doi:10.1152/advan.00076.2005. PMID 16709734. S2CID 42010762.
Raymond L, Dolan W, Dutton R, et al: Pulmonary function and gas exchange during altitude hypoxia (abstract). Clin Res 19:147, 1971
Spaur WH, Raymond LW, Knott MM, et al: Dyspnea in divers at 49.5 ATA: Mechanical not chemical in origin. Undersea Biomed Res 4:183-198, 1977
Rossier P-H, Blickenstorfer E: Espace mort et hyperventilation. Helv Med Acta 13:328-332, 1946
Riley RL, Lilienthal JL Jr, Proemmel DD, et al: On the determination of the physiologically effective pressures of oxygen and carbon dioxide in alveolar air. Am J Physiol 147:191-198, 1946
McNicol MW, Campbell EJM: Severity of respiratory failure: arterial blood gases in untreated patients. Lancet 1:336-338, 1965
Begin R, Renzetti AD Jr: Alveolar-arterial oxygen pressure gradient: I. Comparison between an assumed and actual respiratory quotient in stable chronic pulmonary disease; Relationship to aging and closing volume in normal subjects. Respir Care 22:491-500, 1977
Suwa K, Geffin B, Pontoppidan H, et al: A nomogram for deadspace requirement during prolonged artificial ventilation. Anesthesiology 29:1206-1210, 1968
Fenn WO, Rahn H, Otis AB: A theoretical study of the composition of alveolar air at altitude. Am J Physiol 146:637-653, 1946

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[pmid16709734-1] Curran-Everett D (June 2006). "A classic learning opportunity from Fenn, Rahn, and Otis (1946): the alveolar gas equation". Adv Physiol Educ. 30 (2): 58–62. doi:10.1152/advan.00076.2005. PMID 16709734. S2CID 42010762.

[2] Raymond L, Dolan W, Dutton R, et al: Pulmonary function and gas exchange during altitude hypoxia (abstract). Clin Res 19:147, 1971

[3] Spaur WH, Raymond LW, Knott MM, et al: Dyspnea in divers at 49.5 ATA: Mechanical not chemical in origin. Undersea Biomed Res 4:183-198, 1977

[4] Rossier P-H, Blickenstorfer E: Espace mort et hyperventilation. Helv Med Acta 13:328-332, 1946

[5] Riley RL, Lilienthal JL Jr, Proemmel DD, et al: On the determination of the physiologically effective pressures of oxygen and carbon dioxide in alveolar air. Am J Physiol 147:191-198, 1946

[6] McNicol MW, Campbell EJM: Severity of respiratory failure: arterial blood gases in untreated patients. Lancet 1:336-338, 1965

[7] Begin R, Renzetti AD Jr: Alveolar-arterial oxygen pressure gradient: I. Comparison between an assumed and actual respiratory quotient in stable chronic pulmonary disease; Relationship to aging and closing volume in normal subjects. Respir Care 22:491-500, 1977

[8] Suwa K, Geffin B, Pontoppidan H, et al: A nomogram for deadspace requirement during prolonged artificial ventilation. Anesthesiology 29:1206-1210, 1968

[9] Fenn WO, Rahn H, Otis AB: A theoretical study of the composition of alveolar air at altitude. Am J Physiol 146:637-653, 1946

Respiratory physiology
Respiration	breath inhalation exhalation obligate nasal breathing respiratory rate respirometer pulmonary surfactant compliance elastic recoil hysteresivity airway resistance bronchial hyperresponsiveness constriction dilatation mechanical ventilation
Control	pons pneumotaxic center apneustic center medulla dorsal respiratory group ventral respiratory group chemoreceptors central peripheral pulmonary stretch receptors Hering–Breuer reflex
Lung volumes	VC FRC Vt dead space CC PEF calculations respiratory minute volume FEV1/FVC ratio Lung function tests spirometry body plethysmography peak flow meter nitrogen washout
Circulation	pulmonary circulation hypoxic pulmonary vasoconstriction pulmonary shunt
Interactions	ventilation (V) Perfusion (Q) Ventilation/perfusion ratio V/Q scan zones of the lung gas exchange pulmonary gas pressures alveolar gas equation alveolar–arterial gradient hemoglobin oxygen–hemoglobin dissociation curve (Oxygen saturation 2,3-BPG Bohr effect Haldane effect) carbonic anhydrase (chloride shift) oxyhemoglobin respiratory quotient arterial blood gas diffusion capacity (DLCO)
Insufficiency	high altitude death zone oxygen toxicity hypoxia