Coughing up of blood is termed as

Coughing up of blood is termed Haemoptysis.
Haemoptysis is the medical term for coughing up blood from the lungs or bronchial tubes. It can range from small flecks of blood to a lot of blood.
It can be mild or indicate something serious. It is haemoptysis if the blood comes from the lungs or bronchial tubes (the tubes that bring air to the lungs).