Well, they didn't really keep to any story, as it wasn't based on an established Batman arc. The Nolan movies actually rebooted a lot of canon. But they drew a lot thematically from the comics is what I'm saying.

To be honest, my one gripe with the Batman movies is they don't have enough humor, so they don't really keep the spirit of comic books. Comics are campy by nature, that's just part of their charm. Batman tends to be one of the darker ones, just because the title character broods so often, but it has its lighter moments, too. I think the comic book movie that really nailed what it was trying to accomplish was Iron Man.