What tissue / organ is the most vulnerable to an attack by nanites?
Assume the nanites are very primitive -- something would could conceptualize today.
Assume the nanites can communicate via the selected answer from How do Nanites receive, store, and execute instructions?
Nanites could have 'weapons'. From physical hooks and spikes to rend and pierce to chemical /enzymes that can alter, dissolve or mark for destruction. Things that could be difficult to achieve and differ per body chemistry. They have a weapon that is universal and only requires motive power though. Swarming.
All organs will die from a lack of blood supply. The brain and heart are simply the most vulnerable and vital on the short term, with the brain being exceptionally sensitive. The blood goes through thinner and thinner vains, which is where your nanites will attack. They simply bunch up together, clogging a hair vain. This can even be done without any method to hold together. With enough together they can simply block by getting stuck on each other and the blood vessel wall.
Blocking the flow to key areas like parts of the brain stem or several places of the heart and you're done. This could also lead to targeted attacks on the brain, killing specific areas to modulate behaviour.
This would give the opportunity for the nanites to be both relatively simple and large. The size allows more easy blocking as well as simpler/more technology to fit.
The lungs.
Nanites would function just like another virus. The body has made a lot of passive protections to at least slow them down, a virus won’t instantly pass through your skin for example, that takes time. It is also not inconceivable that bacteria and the white bloodcells manage to absorb nanites and damage them enough to be inoperable. Nanites are usually less metal robots but just some molecules forming a virus like structure to move and act.
But the most common and easiest way into a vital bodypart is just the air ways. You get nanites into the lungs through the air and you can do all sorts of things to them. Simple and effective without the need to go through the bloodstream or something similar.
The Short answer is the same Tissue/Organs that are most vulnerable to other forms of attack
Let's take the Heart for example - Penetrating wounds, Trauma to the chest, Electrical shock etc. etc. The Heart, despite being buried in the second-most protected area of our Body, is vulnerable to all sorts of things that can stop/interfere with it. And given it's importance - that makes it vulnerable.
A Nanite, even a primitive one could probably use the electrical signal from a Heartbeat or just the regular muscle contractions (they are pretty distinctive) to home in on the heart.
From there - the Nanites could drill a hole in the heart, they could clog up the blood vessels that feed the heart itself, they could seal the valves, they could severe the aorta etc. etc.
Same deal with the Brain - it's arguably the most important organ (although me writing this is using my brain to say my brain is the most important organ...) and the most well protected organ - yet concussion, TBI, Strokes, Brain Bleeds, penetrating injuries etc. etc.
Once Nanites get into the Brain they could do all sorts of mischief... Even without intending to damage the tissue, the very fact the Nanites are in the Brain area, traversing your neurons could be enough to cause fatal brain damage
