When my twin brother, James, was diagnosed with acute myeloid leukemia in June 2013, everyone was shocked. We were only 19 at the time. Cancer does not run in our family. And before that, he’d been completely healthy.

My brother and I do everything together. We have the same friends, enjoy the same hobbies and hang out all the time. We aren’t just siblings; we’re best friends.

But the leukemia was already so advanced by...