Does sprinting through the game increase or decrease the amount of souls collected from enemies?

For new game plus, I was wondering if the character should just sprint through the entire game, without fighting many enemies. Sprinting though the whole game might create an experience entirely unique from the first playthough. Players can gain more souls by killing enemies, but at the same time, that makes it more likely that they will die and lose all their souls. So, does fighting or sprinting(with exception excellent soul farming/grinding spots) give the player more souls?

I'd say you're just as likely to die when you rush as you are if you stopped to fight every mob. At least with the mobs, if you're patient, you can get away with killing most of them without sustaining any damage. When you sprint, you're at the mercy of the area. For example, I'm very VERY familiar with Sen's Fortress, if I take my time, I can still make it through the level fairly quickly, but if I were to sprint through it all? There's a good chance I'd get hit by a swinging axe, or take corner too fast and end up running off a beam to my death.
Sens fortress is probably a bit TOO good of an example for what I'm trying to say, but you get the point.