Hi,
It's hard to predict how long the soreness will last after a core biopsy...Most probably the pain is due to nerve damage while having the procedure performed. It can take several weeks and sometimes even manymonths, but the discomfort should slowly improve along that time line.
You could apply some warm compresses, gently massage your breast, also wearing a good supportive bra can help in the healing process. You could take some OTC pain killers, like Acetaminophen or ibuprofen which usually address the pain related to this type of nerve injury.
I don’t think the titanium chip is causing the pain you are describing. This tiny chip is the size of a "Pinhead" It is not felt or supposed to cause any pain.
The something you mentioned starting with the letter “F” could be a Fibroadenoma, which is a benign solid tumor that can be removed if it enlarges too much or cause considerable pain. The “F” could also mean you have Fibrocystic breast changes. This condition is most often the cause of breast tenderness and burning sensation.
If the pain worsen, it would be a good idea to have your breast examined preferably by a Breast Specialist so that your condition can be evaluated. Breast Specialists are very experienced with all sorts of breast problems.
Best wishes and I hope you feel better soon.