The Exploding Chest is the last symptom of the Facehugger incubating inside a human body.
The Embryo will extract fluid and nutrients from the host to construct a placenta, but it's not like the human's. The placenta is like a weed, rooting itself into ever organ in the chest cavity.
Surgical extraction of the Embryo
Even if you could surgically remove the embryo, the placenta is like a cancer. The organs will begin shutting down one by one, and then the patient will die.