Shalom and thank you for your question!
The short answer is that it is probably permissible, but it is important to understand why.
The process of making tattoos may be a similar process, whereby marks of writing are made on the skin and then the skin is pierced (or cut in some way,) for the dye to pentrate right through to the flesh.
According to Jewish law, Halacha, it makes no difference whether the dye or the incision comes first. The practice of making tattoos has a historical connection with idol worship. (Beliefs that are contradictory to Torah values can be in the category of idol worship.)
There is discussion among the earlier legal codifiers as to whether or not a tattoo which lasts a long time is forbidden by the law of the written Torah, or by Rabbinic decree, and whether or not the prohibition refers to a permanent tattoo.
In a recent Halachic publication called ‘Techumin,’ (Volunes 10 and 18,) there are two articles dealing with the topics of permanent makeup and ‘tattoos’ in the eyebrow area. These articles, which discuss the processes involved and their Halachic ramifications, present various reasons for permitting these things.
There are differring opinions in Halacha concerning the issue of tattoos. Is the prohibition only against tattoos involving written letters? In this case a tattoo that leaves a different kind of mark on the skin, other than writing, is not forbidden. It is also possible that if one’s intention is not to make a tattoo at all, no prohibition is involved. This is especially so if the mark is not permanent. Also, since the prohibition is based on idol worship, and there is definitely no such intention, it could be that there is no prohibition, and if there was no intention of making a tattoo at all, there may be no prohibition.
In principle, performing an act of writing in the flesh is definitely prohibited, but there are opinions that if it is not permanent it is permitted.
So, to summarize, if the intention is not to make a tattoo, and no writing is performed, it is permissible, especially when it is not permanent.
It is worthwhile making sure that the person who performs the treatment is reliable and that the treatment is safe. We also have a mitzvah to guard our health!