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A mammogram is used to examine the internal structure of the breast ("mammary gland"). A specialised x-ray is obtained of the breast, and the x-ray films are interpreted by the radiologist.
The basic theory behind mammography is that different types of tissue stop x-rays by different amounts. In particular, breast cancers tend to stop x-rays much more than normal breast tissue. Also, some forms of breast cancer contain tiny flecks of calcium, which stop x-rays very well. These abnormal densities and certain sorts of calcification are what the radiologist searches for.
You should also be aware that sometimes the breast tissue of some women can also stop x-rays very efficiently. This means that occasionally abnormalities are hidden, or masked inside normal tissue. In particular, the breasts of younger women often appear very dense on mammograms.
You should also know that the mammogram is very good at detecting small and subtle abnormalities that develop over time. It is therefore, very important to bring previous mammograms with you. The radiologist is looking for small changes, which by themselves may be unremarkable, but may be very significant if they were not there one or two years ago.