### 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]

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

## Derivation

Its derivation is based on the fact that only the ventilated gases involved in gas exchange ($V_a$) will produce CO2. 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 Vt = Vd + Va. 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 Fe as the fraction of expired CO2 and Fa as the fraction of expired alveolar CO2, and Fd as fraction of expired dead space volume CO2, we can say
Vt x Fe = ( Vd x Fd ) + (Va x Fa ). This merely means all the CO2 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]

$V_t \times F_e = V_a \times F_a$ Where Fe = Fraction expired CO2, and Fa = Alveolar fraction of CO2.
$V_t \times F_e = \left(V_t - V_d\right) \times F_a$ Substituted as above.
$V_t \times F_e = V_t \times F_a - V_d \times F_a$ Multiply out of the brackets.
$V_d \times F_a = V_t \times F_a - V_t \times F_e$ Rearrange.
$V_d \times F_a = V_t \times \left(F_a - F_e\right)$
$V_d/V_t = \frac\left\{F_a - F_e\right\}\left\{F_a\right\}$ Divide by Vt and by Fa.

The above equation makes sense because it describes the total CO2 being measured by the spirometer. The only source of the CO2 we are assuming to measure is from the alveolar space where CO2 and O2 exchange takes place. Thus alveolar's fractional component, Fa, will always be higher than the total CO2 content of the expired air, Fe, thus we will be always yielding a positive number.

Where Ptot is the total pressure, we obtain:

• $F_a \times Ptot = PaCO_2$ and
• $F_e \times Ptot = PeCO_2$

Therefore:

$V_d/V_t = \frac\left\{FaCO_2 - FeCO_2\right\}\left\{FaCO_2\right\} \times \frac\left\{Ptot\right\}\left\{Ptot\right\}$

This is simplified as:

$V_d/V_t = \frac\left\{PaCO_2 - PeCO_2\right\}\left\{PaCO_2\right\}$

## References

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