Governor Jerry Brown recently signed a law requiring doctors to let a woman know if she has dense breast tissue—which can make it difficult to read a mammogram. Read more about this at the LA Times:
Doctors ... will have to explain that breast density is associated with a higher risk of breast cancer, that it makes mammograms harder to read and that “a range of screening options are available.” ... The knowledge could prompt those who have dense breasts to have more exams or opt for an MRI, which allows tumors to be seen clearly even when breast tissue is dense.
On the other hand, this adds more complication to the concern about over-testing, and the possibility of more "false positive" tests.
I'm looking forward to the day (and hope to see it in my lifetime) when the idea of a squeezing a woman's breasts as flat as possible to get a image with ionizing radiation seems as primitive as trepanning.