I fly missions in the United States for Angel Flight Northeast (
www.angelflightne.org). Besides the list of scheduled flights from which we select flights, we periodically receive emails looking for pilots to sign up for organ transplant flights.
These missions are spontaneous, time-sensitive, patient-oriented missions whereby the patient is notified that a (typically) matching kidney or lung has become available (because the donor just died). The patient then starts at the top of a list of volunteer pilots and calls a pilot to fly the patient and a family member to the destination city within a time frame of normally 3 to 6 hours for the transplant.
When flying these missions, we use the "Lifeguard" callsign, which in the US is a callsign that indicates medical priority on-board. Given an airliner with 400 people or a Lifeguard flight with one patient aboard, both competing for a vector on the ILS, my understanding is that the airliner will be moved out of the way to allow the Lifeguard flight the priority.
Additionally, on the local frequencies I have heard medical helicopters, most likely transporting accident victims to local hospitals, use the "Lifeguard" callsign.