In reply to Norman Hadley:
Goodness me man, have you been living under a rock? Go and watch some educational documentaries like the Tomb Raider series, Cliffhanger (Stalone), The Descent (despite being about caving they carry some ice tools at one point)
Nobody but nobody runs with an ice axe, you run with a pair!! One in each hand holding the bottom of the shaft and flailing them as you run. On each of these runs you will also be required at least once to jump a gap slightly further than you can land, so the ice axes *just* catch the edge to save you, otherwise what's the point of having carried them?
Asking about running with an axe -- in the singular? It just beggars belief.