HOUSTON—On the wall of the medical office of Jeffrey S. Brown is a photograph of Carl Lewis, the nine-time Olympic gold medalist. Lewis is one of several former or current patients of Brown's who have climbed the Olympic podium, including Galen Rupp, who won a silver medal in the 10,000 meters at the London Olympics.

"The patients I've treated have won 15 Olympic gold medals," said Brown.

Among...