To unlock this trophy you must save all the Doctors that are held hostage and are scattered throughout Medical. This task is easy but some medical staff will require either combat or silent action to save them.
Dr. Chen
As you get closer to the doctor, you will get a little cutscene. After it, you will have to fight against several of Joker's henchman and once you finish, rescue the doctor.
Dr. Kellerman
When you arrive, you'll see two of Joker's henchman, knock them out. You'll want to grapple up and take out the vent cover. Go in and follow the path until you reach a platform. Go to the right of the platform and throw a Batarang so you can press the button and turn Fan 1 on. Jump across the gap, jump across the next gap, turn right and jump across that gap then look under the platform you were on before and hit the second button with a Batarang to turn on Fan 2. Finally, jump across to the next platform and throw a Batarang to cut the rope a Joker henchman is holding onto and he will break the roof of the house. Throw a Batarang at the button in the small room.
Dr. Young
As soon as you enter the X-ray room you will see a wall that can be blown up by explosive gel, place explosive gel on it but don't detonate it yet, crouch along the left sides and you'll reach another wall that can be exploded, place explosive gel and then detonate. This should take out the two groups of Joker henchman inside. Talk to Dr. Young to finish rescuing the doctor
Saving all doctors will earn you this trophy.