After Tony Loth was diagnosed with tonsil cancer, a local specialist told him the best course of action was a radical neck dissection. He accepted this plan and went home. His doctor was the expert, after all.

Later that night, his wife, a dentist, drew out the procedure for him. Now he could see just how invasive it would be. “If that’s what it takes to beat cancer, I’ll do it,” Tony thought. But he wondered if there was an easier...