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.