Bohr equation

The Bohr equation is used to quantify the ratio of physiological dead space to the total tidal volume, and gives an indication of the extent of wasted ventilation. The original formulation by Bohr,[1] required measurement of the alveolar partial pressure P_A.

${\displaystyle {\frac {\mathbf {V} _{\mathrm {d} }}{\mathbf {V} _{\mathrm {t} }}}={\frac {{\mathrm {P} _{\mathrm {A} }{\mathrm {CO} }_{\mathrm {2} }}-{\mathrm {P} _{\mathrm {e} }{\mathrm {CO} }_{\mathrm {2} }}}{\mathrm {P} _{\mathrm {A} }{\mathrm {CO} }_{\mathrm {2} }}}}$

The modification by Enghoff[2] replaced the mixed alveolar partial pressure of CO₂ with the arterial partial pressure of that gas.[3]

The Bohr equation, with Enghoff's modification, is commonly stated as follows:[4]

${\displaystyle {\frac {\mathbf {V} _{\mathrm {d} }}{\mathbf {V} _{\mathrm {t} }}}={\frac {{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}-{\mathrm {P} _{\mathrm {e} }{\mathrm {CO} }_{\mathrm {2} }}}{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}}}$

Here ${\displaystyle V_{d}}$ is the volume of the exhale that arises from the physiological dead space of the lung and ${\displaystyle V_{t}}$ is the tidal volume;

${\displaystyle P_{a\,{\ce {CO2}}}}$ is the partial pressure of carbon dioxide in the arterial blood, and

${\displaystyle P_{e\,{\ce {CO2}}}}$ is the partial pressure of carbon dioxide in the average expired (exhaled) air.

Its derivation is based on the fact that only the ventilated gases involved in gas exchange (${\displaystyle V_{A}}$) will produce CO₂. Because the total tidal volume (${\displaystyle V_{T}}$) is made up of ${\displaystyle V_{A}+V_{d}}$ (alveolar volume + dead space volume), we can substitute ${\displaystyle V_{A}}$ for ${\displaystyle V_{T}-V_{d}}$.

Initially, Bohr tells us V_T = V_d + V_A. The Bohr equation helps us find the amount of any expired gas, CO₂, N₂, O₂, etc.

In this case we will focus on CO₂.

Defining F_e as the fraction of CO₂ in the average expired breath, F_A as the fraction of CO₂ in the perfused alveolar volume, and F_d as the CO₂ makeup of the unperfused (and thus 'dead') region of the lung;

V_T x F_e = ( V_d x F_d ) + (V_A x F_A ).

This states that all of the CO₂ expired comes from two regions, the dead space volume and the alveolar volume.
If we suppose that F_d = 0 (since carbon dioxide's concentration in air is normally negligible), then we can say that:[5]

${\displaystyle V_{T}\times F_{e}=V_{A}\times F_{A}}$ Where F_e = Fraction expired CO₂, and F_A = Alveolar fraction of CO₂.

${\displaystyle V_{T}\times F_{e}=(V_{T}-V_{d})\times F_{A}}$ Substituted as above.

${\displaystyle V_{T}\times F_{e}=V_{T}\times F_{A}-V_{d}\times F_{A}}$ Multiply out the brackets.

${\displaystyle V_{d}\times F_{A}=V_{T}\times F_{A}-V_{T}\times F_{e}}$ Rearranging.

${\displaystyle V_{d}\times F_{A}=V_{T}\times (F_{A}-F_{e})}$

${\displaystyle V_{d}/V_{T}={\frac {F_{A}-F_{e}}{F_{A}}}}$ Divide by V_T and by F_A.

The only source of CO₂ is the alveolar space where gas exchange with blood takes place. Thus the alveolar fractional component of CO₂, F_A, will always be higher than the average CO₂ content of the expired air because of a non-zero dead space volume V_d, thus the above equation will always yield a positive number.

Where P_tot is the total pressure, we obtain:

• ${\displaystyle F_{A}\times P_{tot}=P_{A}{\ce {CO2}}}$ and
• ${\displaystyle F_{e}\times P_{tot}=P_{e}{\ce {CO2}}}$

Therefore:

${\displaystyle {\begin{aligned}V_{d}/V_{T}&={\frac {(F_{A}{\ce {CO2}}-F_{e}{\ce {CO2}})\times P_{tot}}{F_{A}{\ce {CO2}}\times P_{tot}}}\\&={\frac {P_{A}{\ce {CO2}}-P_{e}{\ce {CO2}}}{P_{A}{\ce {CO2}}}}\end{aligned}}}$

A common step is to then presume that the partial pressure of carbon dioxide in the end-tidal exhaled air is in equilibrium with that gas' tension in the blood that leaves the alveolar capillaries of the lung.