AP Aatrox. It sounds crazy, right?
Well, it is. There's really nothing to abuse here. If Blood Price proc'd Spellvamp, we'd be talking abuse, but this is just "WHAT ARE YOU DOING AATROX!?"

That said, I like playing AP Aatrox a lot. His AP and AD scalings on his E are nearly identical, and AP is a lot cheaper than AD. His ult has a 1.0 ratio and can hit a lot of people.

That said, here's a pro/con list:

Pros:

• A lot of early game damage
• Lots of poke that doesn't suck, so there's no need to go ballsdeep constantly
• Your ult has a 1.0 AP ratio and 400 base damage
• Zhonya's + passive 2gud

Cons:

• W scales off AD, so your sustain AND damage are gimped once you've blown your combo
• As a burst caster, you will die really quickly once it's out because of the aforementioned plus no tankiness
• Spamming your E for sieges is going to take a toll on your health, especially if you miss it
• Getting silenced will make you extremely sad because you can't escape or Zhonya's or burst people down and you will die just like that

Calculations/WIP theorycrafting/other in the following section.

- tested mid lane, poking with E and Q repositioning is useful, most midlaners can't outharass your sustain early on, go in at level 3-4, before they can get their base damages up (if it's just kass or akali push hard, hit level 6 first, and destroy them with E + W + ult + Ignite combo)

calculations section

Spirit Visage: not good, gives +7~25 bonus on W, which you won't be procing enough to justify this purchase

Lich Bane: not cost effective due to manaless champion, can't really spam abilities consistently for proc (given 12 seconds, you will lose at least 3 procs) not sure if I should keep

Hextech Gunblade (DOES NOT FACTOR IN MR):
you'll end up with 29% spellvamp/12% lifesteal, not sure if superfluous because your spells are AoE - but may be necessary for recovery on E and to mitigate W proc damage
calculations here:
E - 255 base damage at max rank + 36 from AP from gunblade - 291 damage. Multitarget ability - 1/3 * 29% = 9.67% - approx. 29 spellvamped. At max health at level 18, the cost of the ability will be 96. Spellvamp does not mitigate the cost.
How much AP do you need for the cost to be mitigated at 100% base health: need 96 spellvamp from 9.67%: Need approx 993 damage on E. Given 600 AP, you will have 255 + 360 = 615. You will need to hit two targets to mitigate an E from 100% health (given 600 AP).
For frame of reference:
30% HP E = 28.85 cost - mitigated as long as you hit something, even with just a hextech gunblade
50% HP E = 48.125 cost - mitigated from hitting two targets, or with 498 damage on E (405 AP)
75% HP E = 72.1875 cost - mitigated from hitting j targets with k AP
Format: j:k
4:baseline(Hextech Gunblade only)
3:baseline(Hextech Gunblade only)
2:197
1:impossible to achieve(?)

What if we factor the ult into the combo?
Max rank: 400 base damage + 1.0 AP. It has no cost!
A single target E from 100% health will cost you 67 health with just a hextech gunblade.
With just a hextech gunblade, ult will have 435 damage. Given the 9.67% spellvamp (dat multitarget), you will heal 42 health. Thus, the cost is mitigated down to 25 health.

What if we factor W into the combo (with/without ult):
With ult: Heals 40 more at max rank - you will gain health.
Without ult: Mitigates down to 27 health.
Extra from damage + lifesteal (hextech gunblade 40 AD factored in):
-1 autoattack: 18.24
-2 autoattacks: 36.48
-3 autoattacks: 54.72

A two-autoattack W + E from max health without ult will cause you to lose no health, and even gain health, given the enemy does not attack and has no resistances. Given resistances, I suspect a 3-autoattack-W will work, too.

Thus, my teamfight theory poses as follows:
Given a proper teamfight with your team there, if you Q in -> DFG assassination target -> ult -> Blood Price -> E -> Zhonya's, you will kill your target, debilitate the enemy team, and lose minimal health. Your team may then go in and scatter them while Zhonya's is still active, and you may go for E-spam sieges. If your passive is up, failing Zhonya's won't matter, and you may go for successive teamfights.

I'm not sure how to explain laning.

Reason for no extra health:
You don't want it if you want to maintain reasonable ability costs. You will have to rely on resistances. Once rito fixes resistances, it'll be easier, but for now, you'll have to deal with the League of Cleavers.

this theorycrafting is a work in progress and is subject to change at any moment

it's unformatted and not pretty because I'm just leaving this here so I don't have to clog up my Google Drive with spreadsheets