TL;DR the greatest risk to a long-lived organism is cancer, when cells start to reproduce incorrectly. To prevent this, not only you need a method to supply fresh stem cells to all organs but also a "check" (almost certainly an enzyme of some kind) to kill off runaway cells, to be replaced by core-produced stem cells. This mechanism does not protect the core itself: if it's overused, to the contrary, the core either withers away, or kills off the rest of the organism.
It would be a basic biologic protection system. The "core" would need to produce a sort of totipotent stem cells and, probably, keep the blood saturated in them. The stimulus to produce such cells would then be the depletion of such cells in the blood.
At the same time the organ has to "retire" the cells grown too old and weak, and implement some kind of regenerative check on all cell lines to prevent random mutations from devolving into cancers (see "integrity check", below). And/Or maybe the Hayflick limit for "normal" or fully grown cells is much lower, so they don't have the timetime to degenerate but rather go into apoptosis - they live fast, die early and leave a beautiful corpse.