You might want to Google for thrips stuck in monitors, it's not that unusual, Here's
a link to Dell forums about someone with the same problem. If you also have a spare monitor around I suggest taking it apart for practice, decreasing the likeliness of screwing up when it matters.
As mentioned in forum (above link) The unit would have to be disassembled to clean it out. That is a job for a professional. Have you checked with a PC repair shop to see if they can do that (and guarantee their work in case they mess it up).
-I've already taken laptop screens apart, so I had a rough idea of what to do. ;-)
-If you want to attempt this, having taken apart electronics &c. is recommended. Please Note:
you may be able to get a replacement if your monitor/laptop is under warranty.