It's entirely your choice. By all means title the doctors when they're introduced, and after this only have other characters use the title in dialogue, e.g.
DOCTOR MILES GRIFFIN, 30s, enters. Dashing good looks and a comforting bedside manner. The Nurses exchange approving glances.
GRIFFIN
Good morning!
NURSE #1
Good morning, Doctor Griffin.
Griffin checks the patient's notes.
GRIFFIN
Been feeling better, I see.
PATIENT
Yes, Doctor Griffin.
Griffin shines his flashlight in the patient's eyes.
GRIFFIN
Ah. Both working. Excellent.
DOCTOR ROGER HAMILTON, 50s, enters. A frosty curmugeon. He frowns at finding Griffin here.
NURSE #1
Er, good morning, Doctor Hamilton.
HAMILTON
There must be a mistake, Doctor
Griffin. I thought this was my
patient.
GRIFFIN
He was your patient, Doctor. Now
he's mine.
-Derek
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The practice of art isn't to make a living. It's to make your soul grow. ~The Complete Handbook of Novel Writing (Kurt Vonnegut)