In the manga, when Natsuki comes round at X-Mas time, you are given the impression that Takumi was really young when she died, but again are given no more information about it.
If the movie menioned a cause then it was probably because some studio or director felt that it would dd depth, after all, lets be very honest here, after almsot 4 seasons of anime, several specials and 30+ manga (in Japan), just how much do we really know about the characters ? - not a lot.