In competitions against the clock, some athletes display an ability to seize control. Think of the Clark-Kent-to-Superman routines that John Elway and Michael Jordan often pulled in the final seconds.

But Iram Leon stands on the sidelines of his own race against time. Lodged in his brain is an untreatable and inoperable cancerous tumor that statistics suggest will kill him before he is 40, eight years from now. Medical science is advancing at a rate that doesn't preclude the development of a treatment, but it's not clear if...