# Shunt equation

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 shunt fraction ${\displaystyle Q_{s}/Q_{t}}$, to the content of oxygen in venous, arterial, and pulmonary capillary blood.

${\displaystyle Q_{s}/Q_{t}=(Cc_{O_{2}}-Ca_{O_{2}})/(Cc_{O_{2}}-Cv_{O_{2}})}$ [1][2]
Where:
Qs = Pulmonary Physiologic Shunt (mL/min)
Qt = Cardiac Output (mL/min)
CCO2 = End-pulmonary-capillary Oxygen Content
CaO2 = Arterial oxygen content
CVO2 = Mixed Venous Oxygen Content

## Derivation

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

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

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

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

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

${\displaystyle Q_{t}\cdot Ca_{O_{2}}=Q_{s}\cdot Cv_{O_{2}}+(Q_{t}-Q_{s})\cdot Cc_{O_{2}}}$
Substitute Qc as above, CcO2 is the oxygen content of pulmonary (alveolar) capillary blood.

${\displaystyle Q_{t}\cdot Ca_{O_{2}}=Qs\cdot Cv_{O_{2}}+Q_{t}\cdot Cc_{O_{2}}-Qs\cdot Cc_{O_{2}}}$
Multiply out the brackets.
${\displaystyle Q_{s}\cdot Cc_{O_{2}}-Qs\cdot Cv_{O_{2}}=Q_{t}\cdot Cc_{O_{2}}-Qt\cdot Ca_{O_{2}}}$
Get the Qs terms and the Qt terms on the same side.
${\displaystyle Q_{s}\cdot (Cc_{O_{2}}-Cv_{O_{2}})=Q_{t}\cdot (Cc_{O_{2}}-Ca_{O_{2}})}$
Factor out the Q terms.

${\displaystyle {\dfrac {Q_{s}}{Q_{t}}}={\dfrac {Cc_{O_{2}}-Ca_{O_{2}}}{Cc_{O_{2}}-Cv_{O_{2}}}}}$
Divide by Qt and by (CcO2 - CvO2).