KANDAHAR, Afghanistan—Under a dusty hospital tent where doctors yell over the roar of jet engines, Dr. John York studied an electronic image of a blood vessel in the neck of a soldier wounded by an improvised bomb. It looked like a balloon ready to pop. Too delicate to operate on directly. Dr. York would have to try a procedure that had rarely been attempted so close to a battlefield.

Using a sophisticated X-ray machine, he snaked a tube from an artery in the soldier's leg until it reached his neck. Dr. York threaded in a...