Working in dialysis?

I would like to work in dialysis.
I completed the course to become a nursing assistant with treatment care. This means I'm also authorized to administer subcutaneous injections. I have almost five years of professional experience. Now I'm working as a career changer at a doctor's office. If I get an injection license, which means I'm authorized to draw blood, can I also work in dialysis, or do I need to complete further training to become a medical assistant or a registered nurse?