I was in a very similar situation myself this application season. I was a biochemistry major in undergrad, but applied primarily to biomedical engineering programs. I have a lot of research experience, but nothing directly in BME (nor have I taken any programming classes, but I have taken calculus + diff eq). I have been accepted and/or interviewed at every program I applied to, and none of the POIs I spoke to seemed very concerned about my not being a BME major. In fact, I feel that coming from a different background--granted you have taken some quantitative classes--can actually be a plus as it sets you apart from other applicants. BME is very interdisciplinary, so I think there is definitely some demand for people who don't come from a traditional engineering background.