Jerry Achan describes the day he collapsed from a stroke as “the luckiest day of my life.”

Without that stroke, doctors would never have arranged for him to have yearly CT scans as part of his follow-up regimen. First, the good news: The scans detected no signs of blocked arteries that could have led to another stroke. Now, the bad: One scan revealed multiple tumors scattered throughout his liver.

“I went in to because of...