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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 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^[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]

{\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 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 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]

V_{T}\times F_{e}=V_{A}\times F_{A}

Where

F e

= Fraction expired CO₂, and

F A

= Alveolar fraction of CO₂.

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 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:

$F_{A}\times P_{tot}=P_{A}{\ce {CO2}}$ 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.

References

^ Bohr C. Ueber die Lungenathmung. Skand Arch Physiol 1891; 2: 236 – 268.
^ Enghoff H. Volumen inefficax. Bemerkungen zur Frage des schädlichen Raumes. Upsala Läk.-Fören Förh, 1938;44:191-218. Article in German
^ Tipton, History of Exercise Physiology, p222
^ 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.

[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.

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