Call the airline directly...
Tell them your situation...
They will ask for "proof", they will want a name and phone # of the doctor treating the patient.
You won't fly for free, but you will get an immediate flight at a MUCH reduced rate.
I know from experience.
Back in August of '95, my parents were on vacation in Florida with some friends.
While they were at Disney World, my Father suffered a massive heart attack. I was at a local dragstrip all day and when I got home around midnight, the phone was ringing...
It was my Mom, she said Dad was in the hospital and they were "working" on him. She said not to panic, and that she'd call back when she knew how he was...
In the meantime, I called several different airlines until I found one that could fly myself and three younger brothers to Orlando if needed. I don't remember which airline it was anymore, but they said "no problem, we just need the doctors name and phone #"
After waiting 1 or 2 hours, the phone finally rang. It was my Dads buddy, and he said "Art, you and your brothers ought to get down here."
The doctors were still "working" on him and weren't sure if he'd make it yet...
I called the airline back and secured a flight from Philly to Orlando.
My brothers and I were in Orlando that morning...
The airlnie even arranged to have my one brother flown in from Norfolk VA where he was living at the time.
Luckily, my Dad survived and is now in better shape than before. But only after a Quadruple by-pass.
Bottom line is...
Shop around different airlines, some will help and some won't...