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As normal humans control their bodies on a gross level, demigods are aware and in control on a cellular level.

As an ordinary human, I have a modicum of awareness of my body. I can perceive an itchy place and scratch it. I am very aware of the state of and the alignment of my teeth, and position of my tongue. I receive consistent messages from bowels and bladder and I can intervene on their behalf when they signal their needs. Should I choose I can become aware of my breathing, a function which does not require my attention but over which I can take conscious control if I wish.

Your demigods have an even greater bodily awareness, down to the cellular level. Like my breathing, demigods can allow the cells to go about their business on autopilot and the cells generally get things done. However a demigod may choose to pay more attention to hormone levels, blood flow, sympathetic tone, and especially something like healing and regeneration. A demigod may choose to heal with a scar or not. Certainly an uncontrolled proliferation like cancer would be preempted by conscious volition. It may be that senescence of otherwise useful cells is also controlled consciously which give the demigods their long lives.

There are animals who are immune to cancer. According to this article, it is due to a specific protein called hyaluronan that causes skin elasticity and cancer resistance. The divine spark they have could cause, in addition to other abilities, an abundance of hyaluronan that provides cancer immunity(and elastic skin, I guess).

It is true that most cancer cells regenerate rapidly. However, that does not mean that rapidly regenerating cells lead to cancer. That's like saying "Horses eat a lot of food. My brother eats a lot of food. Therefore, my brother is a horse." The most rapidly regenerating cells in the body are not proportionally more likely to develop cancer. For example, the cells in hair follicles, nails, the mouth, digestive tract, and bone marrow all regenerate substantially faster than "typical cells". That's one of the reasons that people often look pale, get nauseous, and lose their hair when undergoing chemotherapy (most of which broadly targets rapidly growing cells).

As an example, the cells that line the stomach and intestines live less than a week, while, skin cells live a few weeks, and liver cells live from several months to over a year. If the likelihood of cancer was proportional to the cell turnover rate, you'd expect stomach and intestinal cancer to be the most common types, but they're not.

You might expect there to be some detectable increase in cancer rates as the cell replacement rate increases, because each division is a chance for a mutation that happens as a result of the division process itself. However, there are other, more common causes of cell mutation: radiation, environmental exposure to certain carcinogens, etc. It's even possible for rapidly dividing cells to be LESS susceptible to mutation, because the cells aren't exposed to the mutagen long enough, and when a cell dies, the mutagen gets removed from the body along with the debris from the dead cell.

As already mentioned by @RomainL, the immune system is responsible for keeping cancer in check. A stronger immune system (but not so strong as to cause autoimmune diseases) is the simplest way to keep cancer from developing. Also, the cells would probably need longer telomeres (which tell a cell when it's time to die for good). But there's SO much about our biology that is only barely understood, it's probably best not to get super specific about the mechanics of regeneration. Anything you come up with is likely to be proven incorrect, possibly very soon. Just keep it vague, and say the demigods bodies repair themselves much more rapidly than normal humans. Why no cancer? Answering that question would require teams of scientists to examine lots of demigods. If that's not happening in your world, then it's likely that no one knows the specifics, so don't try to explain it.

Cancers are complex diseases but one key aspect is that cancerous cells avoid detection by the host immune system, the simplest solution IMO would be to give your demi-god better immune system.

... I've been waiting so long to bust out my Naked Mole Rat knowledge unto this website. (The answer will be a bit simplified, since the Naked Mole Rat's mechanisms to prevent cancer/aging are very in-depth and complex)

Make em like Naked Mole Rats Baby!!

Naked Mole Rats are exceptionally long lived creatures (living up to the age of 30 with no real degradation to its health) when compared to similarly sized creatures, like the mouse (which only lives around 4.5 years at best).

Functions that make Naked Mole Rats very long lived

• Naked Mole Rat Ribosomes have a third piece in their ribosomes, which when compared to Mouse ribosomes make between 4 to 40 times less errors. This more precise translation of proteins reduces the chances of misfolded proteins occurring, which would otherwise reduce the lifespan of a Naked Mole Rat (since misfolded proteins are heavily theorized to contribute to the aging process)




• HMW-HA (High Molecular Weight Hyaluronic Acid) in Naked Mole rats is five times larger than in other animals, which leads to HMW-HA accumulating in the Naked Mole Rat's body (along with a combination of less active HA degrading enzymes). This accumulation of HMW-HA prevents Naked Mole Rat cells from growing close together. HMW-HA is also the chemical activates the anti-cancer response of gene p16, which stops Naked Mole Rat cells from proliferating when too many of them are close together.

Side Effects:

Giving your Demigods the ability to produce HMW-HA like the Naked Mole Rat's would cause physical changes to the appearance of a Demigod. Their skin would become very elastic, much like the Naked Mole Rat's (and they ain't too pretty.)

Sources

Last Image obtained from this site

Website with data on the Naked Mole Rat's Unique Ribosomes

More in depth look at Naked Mole Rat's anti-cancer Mechanisms

To simplify it (a lot) you will get cancer if:

• there is a faulty gene already present that practically ensures a person
  gets cancer at some point


• the cell is exposed to something that triggers the wrong mutation


• the combination of the above: there is a preexisting genetic risk, and then the right (wrong) trigger happens.

Side note: not all mutations are "bad". If some mutations weren't beneficial, then evolution won't happen. In fact, it's pretty much a random roulette: any mutation that won't kill an organism before it menages to procreate will get passed down to its offspring. The mutations that increase the chance of procreation happening are more likely to get passed down the line, but that's not essential. The mutation doesn't have to be beneficial. The only condition is, I think, that the given mutation must be at least not deadly until the organism manages to reproduce. The living organisms (humans included) end up with a lot of random stuff this way.

So, aside from cancer, the demigods may end up with different mutations, some of those they may even want to keep. For example someone loses an eye, then after regenerating it they end up with a vividly yellow eye, or the one that sees an unusual light spectrum (eg. ultraviolet).

What would stop them from performing the process a few times, until they end up with good/acceptable result?

The point to consider:

Usually cells can only divide a limited number of times, until their telomeres are too short and they can divide no longer. In a way, this can protect against cancer (which is a cell that divides uncontrollably) but I've read somewhere that sometimes cancerous cells protect their telomeres which makes them sort of immortal...?

If the demigods have a very long/unlimited lifespan and are not ageing or ageing at a different rate that can also mean that their telomeres are:

1. not shortening at all


2. shortening at a slower rate than in normal humans

Option 1) will make a scenario of "regenerate until I'm pleased with the result" possible, and will make your demigods virtually immortal.

Option 2) I think is more interesting. It means regeneration costs something. The more you regenerate, the faster you'll age and the sooner you'll die. If you regenerate your eyes too many times you may end up blind etc. It also makes a scenario of "regenerate until I'm pleased with the result" possible, but there will be limitations, for example you only try until you end up with something not deadly. This means there will be people with bad/random results: disfigured, scarred, with patchy skin, horns, six fingers etc. Because if they want to fix it, they have to pay with their lifespan, and many won't be willing to do that.

An idea:

To limit the possibility of a mutation happening, the exposure to mutagens may be deliberately limited. For example there may exist special chambers you're suppose to go in to regenerate. There's no light in them, and the air is filtered. The forced isolation will be psychologically traumatic, but still a lot safer than not using the chamber. You'll regenerate outside the regeneration chamber only if you have no other choice. You don't go in sick if possible, as that will increase the risk of mutations (for example if you have a flu and your injury is limited enough that you can wait until the virus is out of your system, you'll wait). You go in completely naked. Before you go in, you're disinfected. You eat only a nasty nutritious paste through your entire stay in the regeneration chamber etc. Regeneration tanks may be another option instead of the chamber in this case but it's been done to death. Or maybe the tanks are only used when the damage is very significant.

An issue worth considering:

What if a person regenerates their reproductive organs and a mutation happens that affects the reproductive cells? They may even not be aware of it until a child with a mutation is born...

(I also like the @Willk's idea of demigods being aware/in control of the regeneration proces, at least partially, for example they can trigger it or temporarily stop it, and at a higher level of skill even control it entirely)

The demigods are immortal individuals that protect our reality from eldritch abominations, such as Cthulhu, Niggurath, etc, on a plane that exists between our world and theirs. These battles against these eldritch forces and their minions happen on a daily basis to prevent them from breaching our world.

This plane that exists between our world and their is immaterial (or nonmaterial). In this plane the only things that exist are minds and concepts.

When the demigods step into this plane their mortal bodies evaporate into nothingness. They live and fight in this plane as a conceptualisation of themselves. Without the limits of physical matter they are able to quickly regenerate from wounds or injuries, these wounds are simply the concepts of wounds so can be suppressed by a sufficiently bright concept of non-wound.

When they return to the mortal world they create a new physical vessel for themselves. If their old body had cancer it would not matter to their new one.

The only thing to watch out for is a disease of the mind, if one of your heroes lives with cancer for a long period the idea that they are dieing might creep into their image or conceptualisation of themselves, placing them in real danger.