I don't think I've heard "veins" used that way.
Area of research - 86 million hits on Google, with plural "areas" around 2 million
Field of research - 86 million, but only 400,000 for the plural "Fields"
Lines of research - 35.6 million Google hits
However, a respectable number of hits for vein and veins: 738,000 and 387,000, respectively.
Student or Learner