Shunt equation

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).

• Pulmonary shunt
• Ventilation/perfusion ratio
• Pulmonary contusion