Actually BME is a field where you can start working or study with any science background, i.e. physics, chemistry, biology, medical science, mathematics and engineering. But the best way is to get a BME degree anyway. If you are from a non-engineering background, the course will be designed for you based on that. You probably will have to take more engineering/mathematics courses. Similarly, if you are from engineering background like me, biology or medical science courses will be more emphasized in your coursework. So nothing to worry. This is broadly interdisciplinary field and really exciting and potential one. So you are welcome!