I've started wearing a tungsten carbide wedding band because of some of the metalwork I do...?

I kept losing my other wedding rings when I took them off or 'destroying' them beyond use when I didn't. The tungsten ring is pratically indestructible. Which brings to mind a question. What happens if I injure my hand and it swells. Can they cut the darn thing off, or will they have to remove my finger? Has anybody had any experience with this?