Unfortunately you photo cropped out the tattoo. I only see a prominent vein. That said, dark skin can be very successfully treated with both nanosecond and picosecond technology. Initially dark ink is best broken up with nano and then when more faded, we switch to picosecond. We have great success with our Cutera Enlighten. A black tattoo treats the best but it will still take multiple treatments (4-6) as your body has to absorb the ink after the laser breaks up the ink molecules. After care is also important. The most important advice is to seek a medical based clinic who are experienced with all skin types and understand skin physiology.
You are best served by seeking several consultations, asking a lot of questions and before and after pictures. Perhaps the surgeon may have a patient you can talk with. If they are prominent, they can be treated with a combination sclerotherapy injection and interruption of the vein at the hairline. It will involve you compressing either side of the bridge of nose near the eye.
We do NOT recommend any laser removal for cosmetic eyebrow tattoos. There are metal oxides in the ink which turn anything from orange to green depending upon the ink.
Tattoo shading responds very well to laser removal as the ink molecules are small. We use the 1064 picosecond Cutera Enlighten with good results. It may still take several treatments spaced about 8 weeks apart plus good skin conditioning between.
For the best results in spider vein eliminations, the areas should be assessed with ultrasound and transillumination to find any "feeder" vein (reticulars) below the surface. If these are not treated, they can cause matting and recurrence of veins. Typically it can take 3-5 treatments for an area of vein. We recommend at least two about 4 weeks apart the reevaluate in 6+ weeks. We typically only use laser to "clean" up an area of fine veins after the complete treatments. Perhaps seek an opinion from an experienced medical office who specializes in this type of treatment. Matting can occur in any patients, often for unknown reasons. If there are no veins below the surface, IPL may help reduce the redness.