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So, let’s say Bill wants to gain immortality. Bill is a billionaire, and so invests a large portion of his estate to Brain Transplant research. After a bit, the doctors vivisect him, and encase his brain inside of a robotic shell. My question is, since his brain is still needing of nutrients, how would they go about feeding his brain to keep it alive?
Bill lives in the late 22nd century (2290s) when is wilder science is permitted
Searching for "How do they feed coma patients?" found this Quora post, which says that are two ways to feed a coma patient. One is a feeding tube to the stomach, but the other is
...they receive TPN (Total Parental [sic] Nutrition) which is carefully formulated by a pharmacist once blood work is performed each week or few days, etc. this is delivered through an implanted port, or a catheter tunneled under the skin into a vein. Essentially it is intravenous and the dressing is changed using sterile technique every 7 days.
So basically they inject food directly into the bloodstream. Presumably the robotic shell (in your story) still pumps blood through the brain to provide oxygen, so add nutrients to that blood. It's the same basic idea as with the coma patient.
Reality check:
Keeping a brain alive is a rather trivial exercise and Brythan's answer provides two common modes of feeding people who can't feed themselves. It's a "trivial exercise" because TPN and G-Tubes are placed in living bodies whose metabolic functions are working properly.
But Bill is looking to do something radically different indeed. He wants what appears to be an indeterminate existence of pure torture by sensory & communication deprivation. Tossing a brain into a soup of nutrient & oxygen rich fluids with a little pump attached to the circulatory system might do the trick of keeping his brain alive. And putting that into a robot, well, I don't know what the deal there is! The robot isn't going to be able to function as a surrogate body. All it could do is house Bill's brain and serve as a nutrient & waste storage receptacle.
This is similar in many respects to the experiments already done on whole body transplant. This is where Bill's old body would be surgically removed from his neck and a donor body would be attached to his neck. The main difference is that with a body transplant, Bill would, in theory, be able to see, hear, taste, feel (at least from the skin of his head and neck) and may be able to talk. A brain in a box will be able to do none of those things.
Why it's not going to happen: In the case of whole body transplant, the main reason why it's not been done is medical ethics. Technically, there are no surgical barriers to the procedure. The surgery is basically a complex of disarticulating cervical vertebrae & fusing them to the donated body; severing and reanastamosing blood vessels, trachea and esophagus. Severing and reanastamosing nerves (cervical & vagus nerves) is much trickier. So far so good. But the big hurdle to whole body transplant is what the hell can we do with the spinal cord itself? Bill's nervous system comprises a delicate and intimate network of interconnectivity: nerves and impulses pass from his brain stem down to every tiniest part of his body and back again. It's like looking inside a telephone closet:
Slice that with a sword and you'll be hard pressed to put it back together again! A telephone tech might be able to sort that rat's nest out, but the spinal cord doesn't come with convenient labels or easy to use splicers! Spinal regeneration is in its earliest infancy, and at best Bill will be a quadriplegic. He will also have to contend with a constant regimen of anti-rejection treatments. The first thing the transplanted body is going to do otherwise is set off its immune system to get rid of the foreign body, which in this case is Bill himself -- his head!
Your solution is a step in a different direction, but with even more hurdles to overcome. Bill doesn't want any part of his body to remain apart from his bare naked brain. The body transplant surgery would most likely lead to Bill's death. This surgery will lead to Bill's death.
Why it's even more not going to happen: Living brain excision is technically extremely challenging. As I'm sure you're aware, the brain is encased in a pretty solid mass of bone, called the skull. All of the blood vessels that serve the brain pass through the deepest, thickest part of it: the skull base.
The only way to get down there is to drill out all the bone:
Drilling away the bone will expose all those nasty little arteries and veins and nerves that Bill's surgeon will need to sever in order to free the brain from the skull. All of those structures pass through bone and even the tiniest nick can cause hemorrhage that can not be controlled and will spell brain death.
Each vessel will have to be immediately identified and anastamosed to a cardiopulmonary bypass machine. Once his brain is securely tethered to that device, then it's a matter of severing the brain from the nerves: the biggest one is of course the spinal cord itself. But the surgeons will also have to deal with the optic nerves, the otic nerves, and the many cranial nerves.
The brain is extremely delicate & squishy (see this video). Even very careful handling of it can cause major trauma to its tissues. The chances of getting the brain fully detached from the body, properly connected to its new mechanical receptacle and transfered to its new location all without the very slightest of mishap is just not possible even taking the best care.
What to expect after: I don't think anyone knows what will actually happen to a person whose entire body has been cut away and removed. Assuming Bill survives the surgery and transferal procedures, what will his "life" be like? He has no eyes, so can not see; no ears, so can not hear; he has no spinal cord and no nervous system, so can not feel.
While he does however, have a spinal cord stump and various stumps of other nerves that may occasionally become stimulated, for the most part, he will be entirely and completely sundered from all outside existence. The effects of short and medium term sensory deprivation are pretty well known and relatively innocuous. Bill, assuming her survives the surgery, will be subject to an endless session of absolute sensory deprivation and with no hope at all of recovery in the near future.
I'd suggest that if the brain could ever be reconnected, say in three or four centuries to an android body, "Bill" as a person probably will be long gone from the brain. His personality, cognitive function, memory will probably all be shot.
What was the point: At this point in time, the question becomes one of what did it mean to "keep the brain alive" in light of Bill's desire to "gain immortality"? Is it enough that his non-functioning mind and no-longer-existing sense of self reside in a physically maintained shell? If "keeping the brain alive" satisfies, then there's your answer. But what about the immortality? Has that been granted as well?
I am reminded of the tale of Eos and Tithonos, she a goddess and he a mortal man. She begs of Zeus that he be made immortal, which Zeus grants, and therein lies the real answer to your question:
but when loathsome old age pressed full upon him, and he could not move nor lift his limbs, this seemed to her in her heart the best counsel: she laid him in a room and put to the shining doors. There he babbles endlessly, and no more has strength at all, such as once he had in his supple limbs.
Copy the brain to permanent storage.
From Neuromancer, by William Gibson.
He turned on the tensor beside the Hosaka. The crisp circle of light fell directly on the Flatline’s construct. He slotted some ice, connected the construct, and jacked in. It was exactly the sensation of someone reading over his shoulder.
He coughed. "Dix? McCoy? That you man?" His throat was tight.
"Hey, bro," said a directionless voice.
"It’s Case, man. Remember?"
"Miami, joeboy, quick study."
"What’s the last thing you remember before I spoke to you, Dix?"
"Nothin’."
"Hang on." He disconnected the construct. The presence was gone. He reconnected it. "Dix? Who am I?" "You got me hung, Jack. Who the fuck are you?"
"Ca — your buddy. Partner. What’s happening, man?"
"Good question."
"Remember being here, a second ago?"
No.’
"Know how a ROM personality matrix works?"
"Sure, bro, it’s a firmware construct."
"So I jack it into the bank I'm using, I can give it sequential, real time memory?"
"Guess so," said the construct.
"Okay, Dix. You are a ROM construct. Got me?"
"If you say so," said the construct. "Who are you?"
"Case."
"Miami," said the voice, "joeboy, quick study."
"Right. And for starts, Dix, you and me, we’re gonna sleaze over to London grid and access a little data. You game for that?"
"You gonna tell me I got a choice, boy?"
https://archive.org/stream/NeuromancerWilliamGibson/Neuromancer%20-%20William%20Gibson_djvu.txt
Dix was copied onto a ROM before his death. His consciousness, memories, instincts, sense of humor and cognitive abilities are all there. But as the clip shows, he is frozen at that spot. It can lay down memories while powered up but on being disconnected and reconnected it reverts to the saved version.
In the story the ROM of Dix is a character. There is only one, but there is no reason there could not be many. The ROM is hardware ("firmware")and does not require energy to persist. In the story the ROM of Dix was retrieved from a vault. A good way to survive the apocalypse.
Without some kind of neural interface technology that can graft to the nerve endings and propagate nerve signals in a way the brain can understand, there's not much point to this. The visual cortex, the motor cortex, the hind-brain all would need to be re-connected at some stage to something or he's just a prisoner in sensory deprevation.
If Bill thought there was a chance that technology was only a few years away or was even already in its infancy it might be worth trying? There would have to be a pretty incredible medical reason that this would be considered in the patient's best interest. Acute radiation poisoning perhaps? 10 Gy of radiation exposure is enough to be fatal in a few days, but less than 30 Gy of it doesn't cause neurological damage.
https://www.gizmodo.com.au/2012/07/giz-explains-what-nuclear-radiation-does-to-your-body/
As elemtilas points out there are pretty serious risks to this, from the procedure and afterwords. Keeping him in a medically induced coma might mitigate them. Coma patients who recover report that they sometimes experience dreams. There's also biological impairments - give up the body and now there's no bone marrow, no blood cell production, no immune system. No ability to even filter the nutrient solution. Given that NASA can't even keep bacteria out of their clean-rooms this is extremely risky - if bacteria enter the blood stream it is game over.
