# Bohr equation

The Bohr equation, named after Danish physician Christian Bohr (1855–1911), describes the amount of physiological dead space in a person's lungs. This is given as a ratio of dead space to tidal volume. It differs from anatomical dead space as measured by Fowler's method as it includes alveolar dead space.

## Description

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, required measurement of the alveolar partial pressure P$_A$.
    ${\frac {\mathbf {V} _{\mathrm {d} }}{\mathbf {V} _{\mathrm {t} }}}={\frac {{\mathrm {P} _{\mathrm {A} }{\mathrm {CO} }_{\mathrm {2} }}-{\mathrm {P} _{\mathrm {e} }{\mathrm {CO} }_{\mathrm {2} }}}{\mathrm {P} _{\mathrm {A} }{\mathrm {CO} }_{\mathrm {2} }}}$
The modification by Enghoff replaced the mixed alveolar partial pressure of CO$_2$ with the arterial partial pressure of that gas.
The Bohr equation, with Enghoff's modification, is commonly stated as follows:
    ${\frac {\mathbf {V} _{\mathrm {d} }}{\mathbf {V} _{\mathrm {t} }}}={\frac {{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}-{\mathrm {P} _{\mathrm {e} }{\mathrm {CO} }_{\mathrm {2} }}}{\mathrm {P} _{\mathrm {a} }{\mathrm {CO} }_{\mathrm {2} }}}$
Here $V_{d}$ is the volume of the exhale that arises from the physiological dead space of the lung and $V_{t}$ is the tidal volume;
       $P_{a\,{\ce {CO2}}}$ is the partial pressure of carbon dioxide in the arterial blood, and
       $P_{e\,{\ce {CO2}}}$ is the partial pressure of carbon dioxide in the average expired (exhaled) air.

## Derivation

Its derivation is based on the fact that only the ventilated gases involved in gas exchange ($V_{A}$) will produce CO$_2$. Because the total tidal volume ($V_{T}$) is made up of $V_{A}+V_{d}$ (alveolar volume + dead space volume), we can substitute $V_{A}$ for $V_{T}-V_{d}$.
Initially, Bohr tells us V$_T$ = V$_d$ + V$_A$. The Bohr equation helps us find the amount of any expired gas, CO$_2$, N$_2$, O$_2$, etc.
In this case we will focus on CO$_2$.
Defining F$_e$ as the fraction of CO$_2$ in the average expired breath, F$_A$ as the fraction of CO$_2$ in the perfused alveolar volume, and F$_d$ as the CO$_2$ 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 CO$_2$ 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:
    $V_{T}\times F_{e}=V_{A}\times F_{A}$ Where F$_e$ = Fraction expired CO$_2$, and F$_A$ = Alveolar fraction of CO$_2$.
    $V_{T}\times F_{e}=(V_{T}-V_{d})\times F_{A}$ Substituted as above.
    $V_{T}\times F_{e}=V_{T}\times F_{A}-V_{d}\times F_{A}$ Multiply out the brackets.
    $V_{d}\times F_{A}=V_{T}\times F_{A}-V_{T}\times F_{e}$ Rearranging.
    $V_{d}\times F_{A}=V_{T}\times (F_{A}-F_{e})$
    $V_{d}/V_{T}={\frac {F_{A}-F_{e}}{F_{A}}}$ Divide by V$_T$ and by F$_A$.
The only source of CO$_2$ is the alveolar space where gas exchange with blood takes place. Thus the alveolar fractional component of CO$_2$, F$_A$, will always be higher than the average CO$_2$ 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:
* $F_{A}\times P_{tot}=P_{A}{\ce {CO2}}$ and
* $F_{e}\times P_{tot}=P_{e}{\ce {CO2}}$
Therefore:
    ${\begin{aligned}V_{d}/V_{T}&={\frac {(F_{A}{\ce {CO2}}-F_{e}{\ce {CO2}})\times P_{tot}}{F_{A}{\ce {CO2}}\times P_{tot}}}\\&={\frac {P_{A}{\ce {CO2}}-P_{e}{\ce {CO2}}}{P_{A}{\ce {CO2}}}}\end{aligned}}$
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.