Memory leak is the likely culprit like was already posted. One program that can help you do many things, one of which is to tell you how much memory is being used by a program is https://technet.microsoft.com/en-us/sysinternals/bb896653.aspx
It will tell you how much cpu, and memory different processes are using, allowing you to monitor if DAI is building up memory use when it shouldn't.
The above process explorer will also tell you if something else is starting to run at that time and is taking cpu time away from the game.
Another issue that MAY be happening(not near as likely) is heat buildup. You may have a slow build of heat in gpu/cpu that can be shed once the game is shut down, and doesn't build up enough to cause issue until you play long term again. Again this isn't likely, because most likely it would happen quicker the second time due to all the related components being hotter already. But a program like speedfan is good to have around anyway, and is handy for this situation.
If you have dual monitor screens, then having both of these programs up and running while playing would make monitoring both for the issue easy.