S(min) = 0
Given the apparent technological achievement of the culture that is able to make and use cyborgs, I think it would be quite fair to assume that the problem of biological organs has been solved. And in the real world, we're really quite close. Connecting a brain to a machine implies an extremely robust technological base, especially in the area of miniaturisation. As you mention in your query, artificial organs are a possibility and that's the possibility I'd advocate, but you complain of size...
Review of Organ Systems:
- Integument: skin and supporting tissues are not needed at all because the body is 100% mechanical
- Musculoskeletal: again, bones, muscles, tendons &c are not needed at all because their functions are assumed by mechanical means
- Digestive: the only organ system that needs to be fed is the brain and the neural net interface. More later.
- Kidney/Urinary/Liver: this system handles waste disposal. More later.
- Reproductive: unnecessary due to the body being mechanical, it can be dispensed with
- Cardiovascular/lymphatic: sole purpose is to pump blood around. More later.
- Respiratory: necessary for gas exchange & some acid/base equilibrium. More later.
Replacing the Necessaries:
We focus on the brain & neural network. We definitely need to feed the brain constantly; we need to keep it oxygenated; we need to remove wastes; we need to emulate some other functions like hormones. We can do away with the whole digestive system simply by supplying the cyborg body with a storage container that holds a concoction of appropriate nutrients (carbs, proteins, vitamins, minerals, etc). We can do away with the kidney & liver by use of what is essentially a dialysis machine: it removes impurities & filters wastes. We can do away with the CV system by means of a mechanical pump, artificial vessels and an artificial blood substitute. Likewise, we can do away with the respiratory system by means of an oxygen concentrator and an ECMO device.
Various synthetic hormones can similarly be stored in the cyborg's body and released to the brain through the artificial circulatory system as needed.
The essential breakthrough in the transformation from significantly biologically reliant cyborg prototypes to a minimally biological concept comes by means of the advances in miniaturisation. Being able to discern, categorise, separate, connect & programme all of the nerve connexions that exit (and enter) the brain has been a boon for all other areas of biocybernetics & engineering in general.
The Conundrum Solved:
The basic problem that "Borg" type cyborgs have is that they are primarily biological in nature. Therefore the various organ systems all need to be "full sized". This of course means that there is really no room inside the body for anything extra. This is why Borg have things sticking off their bodies. Even when we replace the musculoskeletal system with mechanical systems, the organs still need to be pretty close to full size just to maintain themselves! This, plus the fact that the mechanical systems need some kind of battery and some kind of fluid reservoirs, this would likely make the cyborg's body somewhat larger.
It is not until the microrevolution occurs that machines we already have, like the ECMO, like the dialysis machine, can be made small enough, literally crafted on a molecular level. This will allow for the manufacture for a relatively small & integrated unit to be housed within the cyborg body and still leave a whole lot of room for all the mechanicals, weapons systems, auxiliary processing systems, energy storage systems, etc.
The minimum size for the cyborg's vital organs is zero, because we've done away with the need for reliance on biological organs.
Technically, the brain itself is a vital organ and must be full size.