I have heard that Botox can increase the circulation to the scalp and is effective for patterned balding

This treatment is being promoted in Germany and in Europe called AC-Therapy. It is supposed to decrease the muscle tension of the scalp which in turn decreases circulation to the scalp. Supposedly, with less circulation there is less oxygen and nutrition and this is therefore claimed as the cause for patterned hair loss in men.

We do know that the blood flow to the scalp is decreased with patterned balding, but this is because as the hair is lost or miniaturized, it needs less blood and therefore the body reduces the circulation to the in response to having less hair. There is no evidence that Botox will do anything other than paralyze the scalp and forehead muscles.