Thank you very much for the breakdown and the reference!
For some reason I assumed that this detector would be embedded into your circulatory system, so that it eventually scans all 5L of blood in your body rather than getting lucky with a 2mL draw. Now that I say it, it sounds pretty unrealistic, but I wonder if it's even remotely possible.
So if we want to get a little sci-fi but still within the realm of not-physically impossible...
If you wanted to monitor for a very specific piece of DNA or a small number thereof, you could have an inline piece of hardware that samples the blood continuously, trying to enrich for those. You could then use a hybridization-based (kinda like a microarray) test to check for the presence of that DNA. Right now, the relevant technology just isn't there. You'd need new materials (of the top of my head they'd need to be highly resistant to fouling, among many other properties), probably a completely new filtering/enrichment system (again fouling and clogging and specificity and sensitivity are huge problems here), and a decent, cheap, high sensitivity and reliability hybridization based detector. All of those things are quite hard, but I don't think they're completely impossible. I do think it's highly unlikely we'll be seeing this in the next 5-10 years though. But I could be wrong! :)
A similar device was proposed a while back-- a nanopore DNA sequencing array implanted into the heart that checks all circulating DNA. In context, the authors had proposed embedding some enzymes in every neuron to record action potentials, neurons would excrete the DNA into the bloodstream, and then you would recover the entire connectome and all the action potential history by DNA sequencing any circulating DNA that makes it to the heart.
For some reason I assumed that this detector would be embedded into your circulatory system, so that it eventually scans all 5L of blood in your body rather than getting lucky with a 2mL draw. Now that I say it, it sounds pretty unrealistic, but I wonder if it's even remotely possible.