a .22 lr is a good training round, inexpensive to purchase and use and easy to learn because of the low impact of the round. It is the second worst cartridge available for self defense, with the most worthless one being the .22 short. The .22 short can barely be relied on to kill anything bigger than a rabbit without an absolute perfect hit in a vital region. A .22 lr isn't even great on twenty pound game without an immaculate hit.
You really can't use expanding bullets in the .22, it isn't going to penetrate and expand, one or the other, and penetration either way will be problematic.
In all seriousness, being shot by a .22 isn't much physically worse than being stabbed by a screwdriver. It's not going to cause a quick bleed out, that won't disturb the attacker. It isn't likely to tear big holes in organs such as heart, kidney, liver, again, only marginal wounds to them. Even a shot to the spine might fail after pushing through the entire body, it may deflect on the vertebra. A shot to the skull will be rather unlikely to kill immediately,either, it's simply too small, too light, too slow, again, it's going to be like being stabbed. There isn't enough energy being used to destroy tissues.
A .22 is better than nothing, and you have the chance of a series of hits, as opposed to a single or double tap. Remember, imo, it is better than nothing, so is a .177 pellet gun.
Myself, when I cant carry a centerfire, I carry a knife. A knife is a lot less likely to let you get a good first strike in, but a good blade will do more damage than a .22. You need a sharp, wide blade. It should have a curve to it, and it should be five to six inches long. A needle like point, a thin dagger blade, refer to the screwdriver. the curved blade, even a slight curve will allow the blade to sharpen better and slash better. A well built knife will do far more damage than a rimfire.