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engorge (itself) on (someone or something)
To drink and fill itself with blood, as of an animal or insect. Unfortunately, it looks like the tick had already engorged itself on your blood before we found it.
engorge (itself) on someone or somethingand engorge (itself) with something
[for an animal] to drink its fill of blood. The vampire bat engorged itself on a number of creatures last night. Mosquitoes engorge on human victims. The mosquitoes engorged themselves with my blood all night long.