The Line? A Canadian TV series?

Probably the long line of people, which probably seemed to never get shorter, waiting to be seen and treated --since you seem to prefer Brit English, you may think of "the line" as "the queue". Americans call it a line, and JAMA is an American medical journal.

Several other possibilities, tho less likely ones: an assembly line or aa production line in a factory is where products get put together, passing from one work station to another often by a conveyor belt or something similar. The parts move past right at the edge of human ability to assemble them. That might have been what it felt like to the doctors, trying to treat 400 patients
a day.

Also in wars up to World War II, there was a "front line" or "the line", the place where soldiers were strung out--usually in a line or several lines--which was how they met the enemy. The place where workers come in contact with customers is also somethimes called "the line" by analogy. So it's conceivable they were talking about themselves as the front-line workers in the emergency rooms or operating rooms meeting the vicitms of the disaster at closest quarters.

Thank you MJ.