Well if the strain is severe enough there will be enough local swelling that it could put some pressure on the gland, I assume you mean the mammary gland? But the muscle tissue and the entire mammary complex is totally seperate.
Forgive me if I am way off, but once I had my nipples hurt for several days after training chest (the same thing you describe but more topical) and I figured out I was dragging the bar of my nipples when I became fatigued while bench pressing, now I was 15 but I do remember doing it.