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I think in privat clinics they do these tests and you can find the clinics in adds in the news pappers. I'm not sure about Hamad Medical Hospital, but you can ask since it is a general hospital.
You can test in all hospitals. Go to Al Rafa Polyclinic near Jaidah bridge.
GO FOR MEDICARE NEAR TO MALL ROUNDABOUT
Naseem Al Rabeeh Medical Centre,Opposite of Birla Public School 4652121