At age 27, Nadine Beech was a picture of health and notably, a non-smoker. So when she was diagnosed with non-small cell lung cancer in 1997, she was in complete disbelief.

Nadine’s doctor also didn’t initially detect the disease. “I started spitting up blood after a water skiing accident, and my doctor thought I had a bruised lung,” Nadine says. She saw several doctors in Kansas City, Missouri, before one discovered the 8-centimeter...