Old doctor who’s after your blood? (5)

I believe the answer is:

leech

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'old doctor' is the definition.
(I've seen this before)

I can't explain the remainder of the clue.

(Other definitions for leech that I've seen before include "Medicinal bloodsucker" , "Blood-sucking parasite" , "Small worm once used for blood-letting, or a parasitical person" , "Blood sucker" , "Archaic word for physician" .)

I've seen this clue in The Times.

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