Biomedical engineering is the application of traditional engineering to medicine not mechanical or electrical engineering to medicine exclusively. You can study CS and be a biomedical engineer. There are BME programs that focus on computational biology or bioinformatics and have nothing to do with MechE. BMEs don't only design hip joints and artificial organs.
You can study Chemical engineering and be a BME. There are BME programs that deal with artificial tissues and biomaterials.
BME is not some form of ME or EE or anything -- yes some schools do make it an extension of those majors (ME/EE + Medicine) but it doesn't have to be.
Check out:
http://www.bme.jhu.edu/academics/ugrad/ugrad_home.htm
that's one approach to BME. (Hopkins has 4 areas BMEs can focus in. 2 of them have to do with EE or ME.)