My champion with life steal gear is currently stuck in an endless battle vs apothecary. They are both just standing healing themselves up.

The main issue here is the problem with how much speed is a factor in arena. Whoever has the highest speed will have the upper hand.

Apothecary with his speed boost will continue to be higher in speed than my champion with speed stats on his gear. Doesn't matter how much damage he deals as long as apothecary is faster to heal up.

You need to either nerf life steal gear or apothecary. Remove speed as a factor in arena and replace with coin toss system for turns so that you don't end up in endless battles. 

The only solution now is for me to leave the battle and take a drop in arena score which is unfair.