First off I think this is a great hand to post. It looks to me like there are many close decisoins on all streets and I'd like to hear what everyone else thinks but here's my take, right or wrong.

Preflop--loose call that puts you in tough spots throughout the hand even though it is a bit shorthanded and premium hands are of slightly less concern plus you are getting good pot odds.

Flop--Check-call may be more prudent at this time. You have to respect the PFR and expect his raise to thin the field of middle pair type hands. You'd rather leave them in because you are drawing even though you have top pair. I can see betting the flop out as an option as well

Turn--If your aim is to push a player off AK then you should have either check-raised the flop or 3 bet when he raised it on the flop. That being said, there is no way of you knowing that you hand would improve so much on the turn. I don't hate the check-raise but I have a hard time seeing AK fold still, although it is not all that remote of a scenario.

River--Pot is huge but you are beat. For him to reraise the turn check-raise requires much more than AK. Looks more like AT than anything given his turn play but he could also easily have filled here. A fold is best but you'd better be awfully sure he's not running some kind of play on you--I for one believe him--you're beat and can save a bet.