Organ a vital part of the arteries (3)
I believe the answer is:
ear
'organ' is the definition.
(the ear is an organ)
'a vital part of the arteries' is the wordplay.
'a vital part of' indicates the answer is hidden within the clue.
'EAR' can be found hidden inside 'the arteries'.
(Other definitions for ear that I've seen before include "Seed-bearing spike" , "'Give every man thine ..., but few thy voice' (Hamlet)" , "Van Gogh missing body part" , "Part of a cereal plant containing the seeds" , "Part of corn plant" .)