What is the cause to have a swollen vein in my left breast?

Probably normal. The breasts normally have veins but sometimes they can become very prominent. They tend to enlarge during menstrual cycles as well as from sun exposure or in fair skinned women. Usually nothing is done for these veins. Occasionally breast veins in pregnant women can develop inflammation. If breast veins are bothersome, they could be injected with sclerotherapy.
Asymmetry need check. Even though plenty of perfectly benign reasons for that swelling may exist, any asymmetrical growth, swelling or tenderness of the breast must be checked by a physician. Physical examination and breast ultrasound are likely to be obtained.

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