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

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

If F_iO₂ is small, or more specifically if ${\displaystyle F_{I}{\ce {O2}}(1-{\ce {RER}})\ll 1}$ then the equation can be simplified to:

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

where:

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

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

Doubling F_iO₂ will double p_iO₂.

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

$${\displaystyle {\begin{aligned}p_{A}{\ce {O2}}&=F_{I}{\ce {O2}}\left(PB-p{\ce {H2O}}\right)-p_{A}{\ce {CO2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\[4pt]&=p_{I}{\ce {O2}}-p_{A}{\ce {CO2}}\left(F_{I}{\ce {O2}}+{\frac {1-F_{I}{\ce {O2}}}{R}}\right)\\[4pt]&=p_{I}{\ce {O2}}-{\frac {V_{T}}{V_{T}-V_{D}}}\left(p_{I}{\ce {O2}}-p_{E}{\ce {O2}}\right)\\[4pt]&={\frac {p_{E}{\ce {O2}}-p_{I}{\ce {O2}}\left({\frac {V_{D}}{V_{T}}}\right)}{1-{\frac {V_{D}}{V_{T}}}}}\end{aligned}}}$$

Abbreviated alveolar air equation

$${\displaystyle p_{A}{\ce {O2}}={\frac {p_{E}{\ce {O2}}-p_{i}{\ce {O2}}{\frac {V_{D}}{V_{T}}}}{1-{\frac {V_{D}}{V_{T}}}}}}$$ 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)

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

Physiologic dead space over tidal volume (VD/VT)

V D V T = p A CO 2 − p E CO 2 p A CO 2 {\displaystyle {\frac {V_{D}}{V_{T}}}={\frac {p_{A}{\ce {CO2}}-p_{E}{\ce {CO2}}}{p_{A}{\ce {CO2}}}}} Intuitive Explanation

As 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.

Firstly, the partial pressure of inhaled oxygen is simply the fraction of inhaled oxygen multiplied by the atmospheric pressure ${\displaystyle F_{I}{\ce {O2}}*P_{{\ce {ATM}}}}$. Once oxygen enters the airways, we must account for the partial pressure of water vapor which is assumed to reach 100% saturation, hence ${\displaystyle F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})}$. Once the humidified atmospheric air reaches the alveoli, gas exchange takes place so we need to consider the amount of ${\displaystyle {\ce {O2}}}$ that enters the blood and ${\displaystyle {\ce {CO2}}}$ that leaves the blood. Conveniently, the arterial blood ${\displaystyle p_{a}{\ce {CO2}}}$ equals the alveolar blood ${\displaystyle p_{A}{\ce {CO2}}}$ and so this is a value we know. It would also be convenient if the same number of ${\displaystyle {\ce {CO2}}}$ and ${\displaystyle {\ce {O2}}}$ molecules were exchanged, in which case the alveolar gas equation would simply be ${\displaystyle p_{A}{\ce {O2}}\approx F_{I}{\ce {O2}}(P_{{\ce {ATM}}}-p{\ce {H2O}})-p_{a}{\ce {CO2}}}$. However in reality the number of ${\displaystyle {\ce {CO2}}}$ molecules exchanged differs slightly from the number of ${\displaystyle {\ce {O2}}}$ molecules, to correct for this the respiratory exchange ratio is used which is the ratio of ${\displaystyle {\ce {CO2}}}$ produced by the body to ${\displaystyle {\ce {O2}}}$ consumed by the body. Hence the alveolar gas equation becomes:

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

See also

• Pulmonary gas pressures