Dr. Matthew Rhoa is still haunted by one of his lowest moments as a physician. Several years ago, on the first leg of an international flight, he was just settling in for a nap when a flight attendant came on the public address system to ask, “Is there a doctor on the plane?”

Dr. Rhoa, who lives in San Francisco, didn’t push his call button. “As a gynecologist, I always waited for another doctor,” he said. “There’s never a need for a Pap smear at 30,000 feet.”

He fell asleep, only to be awakened an hour later by a second call for medical help. This time he answered, and at the back of the plane he found two anxious parents with their 18-month-old toddler, who had a cast on her broken leg and was crying inconsolably.

The girl’s toes were blue. Limbs can often swell in flight, and it was clear that the cast was much too tight. Dr. Rhoa slit the cast and pried it open. The girl stopped crying at once.