Contrary to the other answer posted, this paper shows via microscope images that the tattoo ink is in fact absorbed into cells, and forms small intracellular round granules.
Electron microscopy of untreated tattoos revealed membrane-bound pigment granules, predominantly within fibroblasts and macrophages, and occasionally in mast cells. These granules contained pigment particles ranging from 2-in diameter.
Images a, c and e are images of toluidine blue stained cells, while images b, d and f are images of unstained cells showing the tattoo pigments.
Images a and b are prior to the laser treatment, c and d directly after the treatment, and e and f 90 days post treatment with the laser.
The reason why the tattoo marks persist is not because the pigments are deposited extracellularly, but that they are deposited intracellularly.
The pigments form intracellular granules that are not broken down, and therefore in the absence of external forces such as a laser, the pigments will remain there for long periods of time.