Which of the following conditions leads to urine discharge from the umbilicus?

The presence of a urachal fistula is a condition that specifically results in urine discharge from the umbilicus. The urachus is a fibrous remnant of the allantois, which connects the fetal bladder to the umbilicus. Normally, this structure obliterates after birth and becomes a fibrous cord. However, if a urachal fistula remains patent, it creates an abnormal connection between the bladder and the umbilicus, allowing urine to exit through the navel.

In contrast, the patency of the urachus refers to a situation where the urachus does not fully close but does not have a direct connection suitable for urinary discharge as seen in a fistula. An umbilical hernia involves a protrusion of abdominal contents through the umbilical ring but does not involve the urinary tract or lead to urine discharge. An omphalocele is a condition involving herniation of abdominal organs through a defect at the umbilicus but like an umbilical hernia, it does not lead to urine discharge.

Thus, the unique ability of a urachal fistula to create a connection between the bladder and the umbilicus is what makes it the correct answer in this scenario.