Final answer:
Endoscopic Retrograde Cholangiopancreatography (ERCP) is the procedure where a contrast medium is administered through an indwelling tube to visualize the biliary and pancreatic ducts.
Step-by-step explanation:
The cholangiographic procedure that utilizes an indwelling drainage tube for the administration of a contrast medium is known as Endoscopic Retrograde Cholangiopancreatography (ERCP). In an ERCP, the procedure involves placing a tube down the throat of a patient, into the stomach, and then guiding it into the small intestine. A dye is injected through this tube, which helps the doctor visualize the biliary tract's structure, including the common bile duct, other bile ducts, and the pancreatic duct, using X-ray imaging.