This is generally a fixed expression in French, meaning that something or someone is reaching some "end". For example, if an appliance "arrive en fin de course", it means that it's just about shot (on its last leg).
And well, with medicine, the idea could be that the drug has almost run its course
If a woman's lipstick "arrive en fin de course", it means that she's just about out of it (hardly any left coming out of the tube).
I'm not saying that everyone else is wrong, but it seems funny to me that they that would use "arrivé en fin de course" to mean "administered at the end of treatment".
I just googled another example where they're talking about replacing a worn-out syringe. Part of the sentence reads: relayer une seringue arrivée en fin de course. .... hospitalisation en réanimation, nous avons noté pour chaque médicament administré au PSE : le ...
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V...