You use a tool called an elastrator.
Sorry for the hijack, but this made me incredibly sad. Poor kitty.
Dracoi, poor kitty indeed, and sadly not uncommon. 
To follow up on brossa’s post, lying in one position for a long time on a hard surface can cause widespread rhabdomyolysis (skeletal muscle breakdown). This doesn’t just cause muscle damage and cramping: the lysed cells release their contents, including potassium (which is normally present at much higher concentrations within cells than outside cells) and myoglobin. If it gets severe enough, the elevated potassium level can cause heart arrhythmias. Also, myoglobin can damage the kidneys if there is enough floating around, and the dehydration often present in these scenarios exacerbates the problems of impaired renal function and electrolyte abnormalities. This can happen if, say, an elderly person falls, breaks a hip, and lies on the floor, unable to move themselves to a phone, until they are found.
Don’t be so fast to abandon that mechanism . . . That being said, although it’s theoretically possible to lose a limb by cutting off its blood supply (say, by passing out drunk in a position where the blood flow to the limb is compromised), it’s much likely that one loses a (large) patch of skin.
As an example, imagine that someone has a stroke, becomes paralyzed, and thus cannot move. Let’s say they’re on their back. Those points of the body (skin) in contact with the floor (often in the form of ‘bony prominences’) are being subjected to continuous and uninterrupted pressure. Very often, that pressure is higher than the blood pressure. That means that blood won’t be able to flow into the ‘squashed’ tissues. If it persists long enough, the squashed tissue (e.g. skin) will die (so-called pressure necrosis).
Get that thing away from me.
At least one guy on Dirty Jobs argued that this was more inhumane than just cutting it off, judging by recovery time of the sheep they used it on.
The lead singer of megadeth, after years of risky behavior and drug abuse, once fell asleep with his arm hanging over the back of a hard bench (which put a lot of pressure on I believe his radial nerve). After waking up and it taking a long time to get use of his hand again he was told he’d never play guitar again. However after a year or so of rehab he was back. So that was ironic that the drugs didn’t hurt him but sleeping on a bench did.
Personally I get this all the time and it worries me. But carpal tunnel syndrome involves compression too and as far as I know people don’t get amputations or lose a limb due to that.