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The Shunt equation quantifies the extent that venous blood bypasses oxygenation in the capillaries of the lung.

Shunt and dead space are terms used to describe conditions where either blood flow or ventilation does not meet the other in the lung as it should for gas exchange to take place. They can also be used to describe areas or effects where blood flow and ventilation are not properly matched though both may be present to varying extents. Some refer to shunt-effect or dead space-effect to designate the ventilation/perfusion mismatch states that are less extreme than absolute shunt or dead space.

The following equation relates the percentage of blood flow that is not exposed to inhaled gas, called the the shunt fraction $Q_{s} / Q_{t}$ , to the content of oxygen in venous, arterial, and pulmonary capillary blood.

Q_{s} / Q_{t} = (C c_{O_{2}} - C a_{O_{2}}) / (C c_{O_{2}} - C v_{O_{2}})

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Derivation

The blood entering the pulmonary system will have oxygen flux $Q_{t} \cdot C v_{O_{2}}$ , where $C v_{O_{2}}$ is oxygen content of the venous blood and $Q_{t}$ is the total cardiac output.

Similarly, the blood emerging from the pulmonary system will have oxygen flux $Q_{t} \cdot C a_{O_{2}}$ , where $C a_{O_{2}}$ is oxygen content of the arterial blood.

This will be made up of blood that bypassed the lungs ( $Q_{s}$ ) and that which went through the pulmonary capillaries ( $Q_{c}$ ). We can express this as
$Q_{t} = Q_{s} + Q_{c}$ .

We can solve for $Q_{c}$ :
$Q_{c} = Q_{t} - Q_{s}$ .

If we add the oxygen content of Qs to Qc we get the oxygen content of Qt:

$Q_{t} \cdot C a_{O_{2}} = Q_{s} \cdot C v_{O_{2}} + (Q_{t} - Q_{s}) \cdot C c_{O_{2}}$
Substitute Qc as above, CcO2 is content of capillary oxygen blood.

$Q_{t} \cdot C a_{O_{2}} = Q s \cdot C v_{O_{2}} + Q_{t} \cdot C c_{O_{2}} - Q s \cdot C c_{O_{2}}$
Multiply out the brackets.
$Q_{s} \cdot C c_{O_{2}} - Q s \cdot C v_{O_{2}} = Q_{t} \cdot C c_{O_{2}} - Q t \cdot C a_{O_{2}}$
Get the Qs terms and the Qt terms on the same side.
$Q_{s} \cdot (C c_{O_{2}} - C v_{O_{2}}) = Q_{t} \cdot (C c_{O_{2}} - C a_{O_{2}})$
Factor out the Q terms.

$\frac{Q_{s}}{Q_{t}} = \frac{C c_{O_{2}} - C a_{O_{2}}}{C c_{O_{2}} - C v_{O_{2}}}$
Divide by Qt and by (CcO2 - CvO2).

References

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Template:Respiratory physiology

↑ Respiratory Physiology: The Essentials, J. West, 2005, 7th ed, Page 169

[1] Respiratory Physiology: The Essentials, J. West, 2005, 7th ed, Page 169

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Portal:Algebra/Selected article/4

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Derivation

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