The story
Naturalized into medical English from neoclassical use of Latin in the professions. Exists in loanword form (retaining the Latin inflection, -am) and also in a further naturalized form with reanalysis in English morphology, per vagina = per + vagina, in parallel with other per + [English noun] constructions, such as per instructions or per statute.
Etymology from Wiktionary, CC BY-SA 4.0.