This is an especially important vein when the inferior vena cava (IVC) becomes obstructed, by providing a means of collateral venous return.
When a patient experiences portal hypertension, there can be congestion (backup) of blood that enters into the caval system via the thoracoepigastric vein.
When this occurs, there can be an externally visible dilation of the paraumbilical (and perhaps even the thoracoepigastric veins) which leads to the appearance of "Caput Medusae".
Caput medusae is a clinical sign that is recognized by the physician by the characteristic appearance of distended veins emanating from the umbilicus of the patient.
The shape of these veins and their arrangement around the umbilicus is said to resemble the snake-like hair of the mythological Greek Monster, Medusa.