In 1980, Jack Brown was 38 and busy working in oilfield sales. When a lump appeared on his left groin, he ignored it until his wife, Bobbie, urged him to visit a doctor.

His New Iberia, Louisiana doctor removed the tumor, and a biopsy showed it was cancer. “It really rocked our world,” Bobbie says. “We thought cancer was the end. Jack started making final plans.”

Jack remembers asking the doctor what he suggested. “He suggested...