If the brain were to stop pumping the lungs, there would be something wrong with the brain that ought to be detected on an autopsy. I would think that the cause of death would be attributed to whatever went wrong with the brain.
The heart pumps the blood, the brain controls the breathing. If the heart stops for whatever reason, the brain no longer will function thus death from lack of breathing. A head injury can cause breathing to stop which will cause the heart to stop. An autopsy examines all aspects of the body including brain, heart, and lungs in fine detail.
an autopsy is usually performed when requested by the consultant to the patient, or the legal system if the cause of death is unknown and needs to be established. If the cause of death is known, there is no reason to perform an autopsy.
I think it's probably common in the US that autopsies are not performed on elderly or chronically ill individuals whose exact cause of death is not known but who obviously died of natural causes. The cause of death can be labeled as something like, "cardiopulmonary arrest" (which does not mean a heart attack; it simply means that the heart and lungs stopped). I believe this may be done in situations in which there does not seem to be any legal or medical need for an autopsy.
I agree with skydnsr and Ed it really isn't important how someone passes to my way of thinking. My mother died in middle of a big smile of either heart disease, stroke, kidney failure. My father died of what was told to us of lung cancer, but he had brain cancer, a heart attack, liver cancer, caused by prostate cancer. I'd simply center on keeping the family in comfort unless you think a crime was committed.
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