Organs are made of cells, like pretty much the rest of your body.
A transplant procedure just hooks up the organ to the new body’s blood supply (veins and arteries), so that the cells inside the organ can have oxygen and food, and basically survive and do their jobs. That’s it.
There’s no “adapting”, the cells in the organ don’t change their DNA or makeup in any way, so basically they’ll live and divide and function based on the DNA that they have (which is, the donor’s DNA).
So, to answer your question, the organ “ages” according to the donor’s DNA and “age”.
But the biggest issue with transplanted organs is that the recipient’s immune system (which is in the blood), recognizes that the DNA of the organ is NOT from its own body, and attacks it as if it’s actually contamination. The immune system tries to kill all the cells in the organ, as if they were microbes.
To prevent this, every person receiving a transplant organ must go on medication to disable their own immune system from being so aggressive. Of course, if you disable a person’s immune system, they’ll get sick extremely easily, so you can’t disable it 100%. You can just reduce it a bit.
So the organ does eventually get attacked, because the immune system is not disabled, just reduced, so it does still attack the organ over time. 5 years, 10 years.
Basically, age isn’t a problem for the organ, as much as being “killed” by the new person’s immune system.
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