Bohr equation

frac{VD}{VT}  =  frac{PaCO2-PECO2}{PaCO2}

This describes the amount of physiological dead space in the lungs, it’s given as a ratio and a typical value is 0.2 – 0.35.

bohr.jpg

All expired carbon dioxide (FE) must, by definition, come from alveolar gas (VA x FA) not dead space gas (VD).

Therefore to derive the Bohr equation (explanatory notes in brackets):

VT  x  FE  =  VA  x  FA

VT  x  FE  =  (VT-VD)  x  FA   (As VA=VT-VD)

VT  x  FE  =  VT  x  FA  –  VD  x  FA  (Multiply out brackets)

VD  x  FA  =  VT  x  FA  –  VT  x  FE  (Rearrange)

VD  x  FA  =  VT  x  (FA-FE)  (Simplify)

frac{VD}{VT}  =  frac{(FA-FE)}{FA}  (Divide by VT and FA)

The fraction of CO2 in the alveoli (FA) can’t realistically be measured, so the partial pressure of CO2 in the blood is used as it is virtually identical. The fraction of expired CO2 can easily be measured as a partial pressure with a capnometer.

frac{VD}{VT}  =  frac{PaCO2-PECO2}{PaCO2}

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Dead space

The volume within the respiratory system that does not participate in gas exchange.

Anatomical dead space
The volume of the conducting airways (~150mls)

Physiological dead space
Volume of alveoli that do not eliminate carbon dioxide, plus the anatomical dead space volume

Dead-Space

In health the anatomical dead space volume is the same as the physiological dead space volume, however in lung disease and other clinical situations the physiological dead space may get larger.

Measuring alveolar ventilation

Because no gas exchange occurs in the anatomical dead space all carbon dioxide in expired gases must come from alveolar gas.

VCO2 = VA  x  FCO2

Therefore VA  =   frac{VCO2}{FCO2}

The partial pressure of carbon dioxide in the alveoli is proportional to the fraction of carbon dioxide (FCO2) multipled by K, a constant.

VA  =  frac{VCO2}{PCO2}  x  K

This is the alveolar ventilation equation. The alveolar ventilation is inversely proportional to the partial pressure of carbon dioxide.

The Bohr equation is used to calculate the fraction of dead space in the lungs.

Fick’s law of diffusion

The rate of transfer of a gas through a sheet of tissue is proportional to the tissue area and the difference in gas partial pressures between the two sides, and inversely proportional to tissue thickness.

Vgas  alpha  frac{A}{T}  x  D  x  (P1-P2)

Where D is the diffusion constant (Graham’s law), stating that diffusion is proportional to solubility but inversely proportional to the square root of the molecular weight (or density).

D  alpha  frac{Solubility}{sqrt{MW}}

Carbon dioxide diffuses around twenty times more rapidly than oxygen does due to increased solubility despite a higher molecular weight.

Summary

The rate of diffusion through a tissue is proportional to the cross sectional area but inversely proportional to the thickness.

The rate of diffusion is proportional to the pressure differential between either side of the tissue.

The more soluble a substance is the more rapidly it will diffuse, but the larger it’s molecular weight the slower it will diffuse.