Can You Destroy This Tumor?

This is a semi-famous thought experiment:

"You are a doctor trying to destroy an inoperable stomach tumor in a patient. You can send a narrow beam of radiation at the tumor, but it destroys the tumor only at a high intensity, which would also destroy the surrounding healthy tissue. At a lower intensity it would spare the healthy tissue but also fail to kill the tumor."

Can you solve it? Karl Dunker was the first to propose the challenge, and it's meant to be tough--only about 10% of participants do so without resorting to their preferred search engine. I'll post the answer in the comments.