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There is usually no simple intervention that will close these holes and drainage tracts. When there is infection and drainage of pus, it is usually necessary to first get control of the infection and pus. This will typically require antibiotics and may require surgical opening of the tunnel.
A culture of the drainage can aide in selection of antibiotics.
It is usually only after the infection and purulent drainage is controlled that these holes and tunnels can heal.
The Gastroenterologist is correct that the care of these infected holes and tunnels would usually care by the surgeon.