# 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, [1] required measurement of the alveolar partial pressure PA.

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

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 physiological dead space volume 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.

## Derivation

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 VT = Vd + VA. The Bohr equation helps us find the amount of any expired gas, CO
2
, N2, O2, etc.

In this case we will focus on CO2.

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

VT x Fe = ( Vd x Fd ) + (VA x FA ).

This states that all of the CO2 expired comes from two regions, the dead space volume and the alveolar volume.
If we suppose that Fd = 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 Fe = Fraction expired CO2, and FA = 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 VT and by FA.

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

Where Ptot 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.

## References

1. Bohr C. Ueber die Lungenathmung. Skand Arch Physiol 1891; 2: 236 – 268.
2. Enghoff H. Volumen inefficax. Bemerkungen zur Frage des schädlichen Raumes. Upsala Läk.-Fören Förh, 1938;44:191-218. Article in German
3. Tipton, History of Exercise Physiology, p222
4. Respiratory Physiology: The Essentials, John B. West, 2005, 7th ed, Page 169
5. Davies, Andrew, and Carl Moores. The Respiratory System. Systems of the body. Edinburgh: Churchill Livingstone, 2003.