Possible Historical Traces in the Doctrina Addai (Analecta Gorgiana)

The Teaching of Addai is a Syriac document convincingly dated by some scholars in the fourth or fifth century AD. I agree with this dating, but I think that there may be some points containing possible historical traces that go back even to the first century AD, such as the letters exchanged by king Abgar and Tiberius. Some elements in them point to the real historical context of the reign of Abgar 'the Black' in the first century. The author of the Doctrina might have known the tradition of some historical letters written by Abgar and Tiberius.