Clean medical department, bandaging artery first (3-7)
I believe the answer is:
pre-eminent
'first' is the definition.
(pre-eminent can mean first or leading)
'clean medical department bandaging artery' is the wordplay.
'clean' becomes 'preen' (preening is a kind of cleaning).
'medical department' becomes 'ENT' (abbreviation for Ear, Nose and Throat department).
'bandaging' indicates putting letters inside.
'artery' becomes 'MI' (resembles Mp motorway, a transport 'artery').
'preen'+'ent'='preenent'
'preenent' enclosing 'mi' is 'PRE-EMINENT'.
(Another definition for pre-eminent that I've seen is " Surpassing all others".)