Intractable migraine is different from a usual migraine in that it lasts longer than a few hours and does not respond to ...

Though she didn't know it at the time, she was living with intractable migraine — a severe headache that can last for more than 72 hours and often doesn't respond to conventional treatments.

Refractory migraine, also called intractable migraine, refers to migraine that is resistant to multiple kinds of typical treatments. That’s what the word “refractory” means in this context.