The large blood vessels typically run between the bones under the muscle. This is where they are best protected so that only deep cuts damages them. There are a few places where the blood vessels have to come right under the skin though. The elbow is one of these places. They can not go between the bones because the bones in the elbow move a lot in relation to each other creating pinch points. And there are no muscles or even fat in your elbow to protect the blood vessels. So they run just under the skin making it easy to insert a needle into them. You can even see your blood vessels through the skin.
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