It was stated by Rudolph that Astarte had only two more weeks to live, because of the strain Rhododactylos familiar was putting on her body/soul/mana pool/whatever.
After defeating Rudolph, Kojou bites her
and says that he would put the familiar under his control, so her lifespan would increase. The bite mark is shown in the picture:
But in later story arcs this seems to be completely ignored. Astarte acts and uses Rhododactylos on her own.