For all (most?) cancers, you need many mutations in one cell.
There is some probability that a mutation will trigger an internal check in the cell to detect bad mutations, and the cell will commit suicide.
There is also some probability that the cell will die accidentally before the second mutation.
There is some probability that the immune system will detect something fishy and kill the cell just in case.
If the time between each mutation is bigger, then you have more time to get lucky and remove the cell while it has a single mutation, before it can accumulate more mutations and become dangerous.
So a lower dose will increase the time between mutations and make the case of two accumulative bad mutations less common. Then at small doses, the effect is not linear and perhaps below some threshold the ability of the body to detect problems will fix all the cases before you notice.
The problem is that it's very difficult to measure the effect of very low doses for a long time. (You have to test the drug in some animal that lives for a long time like elephants instead of mice.) (And it's difficult to get funding and graduate students for a 20 year experiment.)
So the "linear non threshold model" is slightly pessimistic, it err on the side of caution, but it's probably good enough and not too alarmist.
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Technical note: There are actually no cells with zero mutations. All cells have some mutations. Some have less, some have more. Some mutations are dangerous, most are innocuous, a few may be advantageous. But to simplify the discussion, I just ignored this and imagine that usually the cells have no mutations.
It depends highly on the mechanism of carcinogenesis. Lead is a (possible) carcinogen, and accumulates even in relatively small quantities because it takes up residence in your bones and is slowly released back into your blood over years. Others may be eliminated rapidly and only cause damage in large or continuous doses. Frustratingly the answer is often "it depends". The very low end of the spectrum of exposures is often the most relevant to humans (due to our relatively low exposure to most things), but very hard to study, because the effects are often drowned out by other causes of morbidity/mortality.
In a side note, Organophosphates replaced Organochlorides for a very similar reason. They have a much faster elimination rate, reducing the time that they stay in the food chain (and hopefully reducing the exposure of non-target species). Organochlorines, on the other hand, were very stable, which made them easily stored, but also caused biomagnification, where animals further up the food chain started concentrating it in their bodies as they ate smaller animals that had been exposed. This caused serious environmental effects, leading to the shift to the pesticides we use today.
It's all about luck, you know? They'll all cause some sort of cellular damage. Cancer is when you get super unlucky and it's the right sort of cellular damage that leads to unchecked growth and disables apoptosis.
Well, your hypothesis is definitely scientific, since it can be disproven. Couple of counter-hypotheses:
- Consuming less food requires the body to use fat-reserves. Some of these might have been storing heavy metals. With a quick enough change in diet, these might pose a significant risk.
- Consuming less food might alter your gut microbiology. Disheveling this balance might create the risk of infection or even worse.
- Consuming less food (to be precise, less lipids), might alter, or even disable certain parts of the metabolism, often leading to deficiencies in supplies for the immune system, thus making us more vulnerable to 'poisons'.
I don't know which of the above hypotheses are correct, but I am sure they are just as (un)likely as yours is.
