Usage patterns
〈物〉が気管に入るfor something to go down the windpipe (choking)
The food went down the wrong pipe and I started coughing.
気管が詰まるfor the windpipe to become blocked
A foreign object got stuck in his windpipe and he had trouble breathing.
気管を切開するto perform a tracheotomy
The doctor performed a tracheotomy to secure the patient's breathing.