Eastern Roman Empire existed for almost a thousand years beyond the end of the original Western Empire.
About the bloodline: there was (a) Caesar and Imperator who could legitimately claim an unbroken line of succession to Augustus even at exactly the same time that Gutenberg was developing the printing press, and within 30 years of Columbus' discovery of the Americas.
It is no different from the Eastern Han and Western Han except Roman lasted 5 times longer to 1500 or so, almost to modern time.
