# 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. It is stated as follows:^{[1]}

## Derivation

Its derivation is based on the fact that only the ventilated gases involved in gas exchange () will produce CO_{2}. Because the Total tidal volume () is made up of (alveolar volume + dead space volume), we can substitute for .

Initially, Bohr tells us Vt = Vd + Va. 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 Fe as the fraction of expired CO_{2} and Fa as the fraction of expired alveolar CO_{2}, and Fd as fraction of expired dead space volume CO_{2}, we can say

Vt x fe = ( Vd x Fd ) (Va x Fa ). This merely means all the CO_{2} expired comes from two parts, the dead space volume and the alveolar volume.

If we suppose that Fd = 0 (since carbon dioxide concentration in air is normally negligible), then we can say that:^{[2]}

- Where F
_{e}= Fraction expired CO_{2}, and Fa = Alveolar fraction of CO_{2}.

- Substituted as above.

- Multiply out of the brackets.

- Rearrange.

- Divide by V
_{t}and by F_{a}.

The above equation makes sense because it describes the total CO_{2} being measured by the spirometer. The only source of the CO_{2} we are assuming to measure is from the alveolar space where CO_{2} and O_{2} exchange takes place. Thus alveolar's fractional component, F_{a}, will always be higher than the total CO_{2} content of the expired air, F_{e}, thus we will be always yielding a positive number.

Where Ptot is the total pressure, we obtain:

- and

Therefore:

This is simplified as:

## References

- ↑ Respiratory Physiology: The Essentials,
**John B. West**, 2005, 7th ed, Page 169 - ↑ Davies, Andrew, and Carl Moores. The Respiratory System. Systems of the body. Edinburgh: Churchill Livingstone, 2003.