Mass spectrometry has become quite powerful and I think it will be the tool to use for this soon(not breath per second, but it's possible for sure). It's completely possible right now to take a microscopic amount of blood and analyze it in 10 minutes for most available compounds in extreme resolution. The hiccup is actually the biomarkers now. Also probably money for a hospital to buy an instrument and hire/train a lab tech to use the instrument effectively. If you know the masses/identity of the blood biomarkers for certain diseases, you can do lc-ms(takes longer by a little bit) or even direct injection (quite fast) on more expensive instruments, to see them in the metabolic profile of the patient. To be honest they probably are using mass spectrometry to find the markers already.

Source: I do direct injection metabolomics of plant tissues for my masters.