Living Forever is Hard, or, The Gompertz Curve

gwern

I recently recalled, apropos of the intermittent fasting/caloric restriction discussion, a very good blog post on mortality curves and models of aging:

For me, a 25-year-old American, the probability of dying during the next year is a fairly miniscule 0.03% — about 1 in 3,000. When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1 in 750, and so on. By the time I reach age 100 (and I do plan on it) the probability of living to 101 will only be about 50%. This is seriously fast growth — my mortality rate is increasing exponentially with age.

...This data fits the Gompertz law almost perfectly, with death rates doubling every 8 years. The graph on the right also agrees with the Gompertz law, and you can see the precipitous fall in survival rates starting at age 80 or so. That decline is no joke; the sharp fall in survival rates can be expressed mathematically as an exponential within an exponential:

$P(t) \approx e^{-0.003 e^{(t-25)/10}}$

Exponential decay is sharp, but an exponential within an exponential is so sharp that I can say with 99.999999% certainty that no human will ever live to the age of 130. (Ignoring, of course, the upward shift in the lifetime distribution that will result from future medical advances)

...There is one important lesson, however, to be learned from Benjamin Gompertz’s mysterious observation. By looking at theories of human mortality that are clearly wrong, we can deduce that our fast-rising mortality is not the result of a dangerous environment, but of a body that has a built-in expiration date.

gravityandlevity then discusses some simple models of aging and the statistical characters they have which do not match Gompertz's law:

'lightning' model: risk of mortality each period is constant; Poisson distribution:

What a crazy world! The average lifespan would be the same, but out of every 100 people 31 would die before age 30 and 2 of them would live to be more than 300 years old. Clearly we do not live in a world where mortality is governed by “lightning bolts”.
'accumulated lightning'; like in a video game, one has a healthbar which may take a hit each period; similar to above:

Shown above are the results from a simulated world where “lightning bolts” of misfortune hit people on average every 16 years, and death occurs at the fifth hit. This world also has an average lifespan of 80 years (16*5 = 80), and its distribution is a little less ridiculous than the previous case. Still, it’s no Gompertz Law: look at all those 160-year-olds! You can try playing around with different “lightning strike rates” and different number of hits required for death, but nothing will reproduce the Gompertz Law. No explanation based on careless gods, no matter how plentiful or how strong their blows are, will reproduce the strong upper limit to human lifespan that we actually observe.

What models do yield a Gompertz curve? gravityandlevity describes a simple 'cops and robbers' model (which I like to think of as 'antibodies and cancers'):

...in general, the cops are winning. They patrol randomly through your body, and when they happen to come across a criminal he is promptly removed. The cops can always defeat a criminal they come across, unless the criminal has been allowed to sit in the same spot for a long time. A criminal that remains in one place for long enough (say, one day) can build a “fortress” which is too strong to be assailed by the police. If this happens, you die.

Lucky for you, the cops are plentiful, and on average they pass by every spot 14 times a day. The likelihood of them missing a particular spot for an entire day is given (as you’ve learned by now) by the Poisson distribution: it is a mere $e^{-14} \approx 8 \times 10^{-7}$ .

But what happens if your internal police force starts to dwindle? Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day. Then the probability of them missing a criminal for an entire day decreases to $e^{-12} \approx 6 \times 10^{-6}$ . The difference between 14 and 12 doesn’t seem like a big deal, but the result was that your chance of dying during a given day jumped by more than 10 times. And if the strength of your police force drops linearly in time, your mortality rate will rise exponentially.

... The language of “cops and criminals” lends itself very easily to a discussion of the immune system fighting infection and random mutation. Particularly heartening is the fact that rates of cancer incidence also follow the Gompertz law, doubling every 8 years or so. Maybe something in the immune system is degrading over time, becoming worse at finding and destroying mutated and potentially dangerous cells.

...Who are the criminals and who are the cops that kill them? What is the “incubation time” for a criminal, and why does it give “him” enough strength to fight off the immune response? Why is the police force dwindling over time? For that matter, what kind of “clock” does your body have that measures time at all? There have been attempts to describe DNA degradation (through the shortening of your telomeres or through methylation) as an increase in “criminals” that slowly overwhelm the body’s DNA-repair mechanisms, but nothing has come of it so far.

This offers food for thought about various anti-aging strategies. For example, given the superexponential growth in mortality, if we had a magic medical treatment that could cut your mortality risk in half but didn't affect the growth of said risk, then that would buy you very little late in life, but might extend life by decades if administered at a very young age.

I recently recalled, apropos of the intermittent fasting/caloric restriction discussion, a very good blog post on mortality curves and models of aging:

For me, a 25-year-old American, the probability of dying during the next year is a fairly miniscule 0.03% — about 1 in 3,000. When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be about 1 in 750, and so on. By the time I reach age 100 (and I do plan on it) the probability of living to 101 will only be about 50%. This is seriously fast growth — my mortality rate is increasing exponentially with age.

...This data fits the Gompertz law almost perfectly, with death rates doubling every 8 years. The graph on the right also agrees with the Gompertz law, and you can see the precipitous fall in survival rates starting at age 80 or so. That decline is no joke; the sharp fall in survival rates can be expressed mathematically as an exponential within an exponential:

$P(t) \approx e^{-0.003 e^{(t-25)/10}}$

Exponential decay is sharp, but an exponential within an exponential is so sharp that I can say with 99.999999% certainty that no human will ever live to the age of 130. (Ignoring, of course, the upward shift in the lifetime distribution that will result from future medical advances)

...There is one important lesson, however, to be learned from Benjamin Gompertz’s mysterious observation. By looking at theories of human mortality that are clearly wrong, we can deduce that our fast-rising mortality is not the result of a dangerous environment, but of a body that has a built-in expiration date.

gravityandlevity then discusses some simple models of aging and the statistical characters they have which do not match Gompertz's law:

'lightning' model: risk of mortality each period is constant; Poisson distribution:

What a crazy world! The average lifespan would be the same, but out of every 100 people 31 would die before age 30 and 2 of them would live to be more than 300 years old. Clearly we do not live in a world where mortality is governed by “lightning bolts”.
'accumulated lightning'; like in a video game, one has a healthbar which may take a hit each period; similar to above:

Shown above are the results from a simulated world where “lightning bolts” of misfortune hit people on average every 16 years, and death occurs at the fifth hit. This world also has an average lifespan of 80 years (16*5 = 80), and its distribution is a little less ridiculous than the previous case. Still, it’s no Gompertz Law: look at all those 160-year-olds! You can try playing around with different “lightning strike rates” and different number of hits required for death, but nothing will reproduce the Gompertz Law. No explanation based on careless gods, no matter how plentiful or how strong their blows are, will reproduce the strong upper limit to human lifespan that we actually observe.

What models do yield a Gompertz curve? gravityandlevity describes a simple 'cops and robbers' model (which I like to think of as 'antibodies and cancers'):

...in general, the cops are winning. They patrol randomly through your body, and when they happen to come across a criminal he is promptly removed. The cops can always defeat a criminal they come across, unless the criminal has been allowed to sit in the same spot for a long time. A criminal that remains in one place for long enough (say, one day) can build a “fortress” which is too strong to be assailed by the police. If this happens, you die.

Lucky for you, the cops are plentiful, and on average they pass by every spot 14 times a day. The likelihood of them missing a particular spot for an entire day is given (as you’ve learned by now) by the Poisson distribution: it is a mere $e^{-14} \approx 8 \times 10^{-7}$ .

But what happens if your internal police force starts to dwindle? Suppose that as you age the police force suffers a slight reduction, so that they can only cover every spot 12 times a day. Then the probability of them missing a criminal for an entire day decreases to $e^{-12} \approx 6 \times 10^{-6}$ . The difference between 14 and 12 doesn’t seem like a big deal, but the result was that your chance of dying during a given day jumped by more than 10 times. And if the strength of your police force drops linearly in time, your mortality rate will rise exponentially.

... The language of “cops and criminals” lends itself very easily to a discussion of the immune system fighting infection and random mutation. Particularly heartening is the fact that rates of cancer incidence also follow the Gompertz law, doubling every 8 years or so. Maybe something in the immune system is degrading over time, becoming worse at finding and destroying mutated and potentially dangerous cells.

...Who are the criminals and who are the cops that kill them? What is the “incubation time” for a criminal, and why does it give “him” enough strength to fight off the immune response? Why is the police force dwindling over time? For that matter, what kind of “clock” does your body have that measures time at all? There have been attempts to describe DNA degradation (through the shortening of your telomeres or through methylation) as an increase in “criminals” that slowly overwhelm the body’s DNA-repair mechanisms, but nothing has come of it so far.

This offers food for thought about various anti-aging strategies. For example, given the superexponential growth in mortality, if we had a magic medical treatment that could cut your mortality risk in half but didn't affect the growth of said risk, then that would buy you very little late in life, but might extend life by decades if administered at a very young age.

This isn't an anti-aging strategy, but it is an anti-death strategy: low-dose aspirin. As explained in this New York Times article on December 6, 2010, "researchers examined the cancer death rates of 25,570 patients who had participated in eight different randomized controlled trials of aspirin that ended up to 20 years earlier".

Eight. Different. Randomized. Controlled. Trials. Twenty-five thousand people.

They found (read the article) that low-dose aspirin dramatically decreased the risk of death from solid tumor cancers. Again, this ("risk of death") is the gold standard - many studies measure outcomes indirectly (e.g. tumor size, cholesterol level, etc.) which leads to unpleasant surprises (X shrinks tumors but doesn't keep people alive, Y lowers cholesterol levels but doesn't keep people alive, etc.). Best of all is this behavior: "the participants in the longest lasting trials had the most drastic reductions in cancer death years later."

Not mentioned in the article is the fact that aspirin is an ancient drug, in use for over a century with side effects that, while they certainly exist, are very well understood. This isn't like the people taking "life-extension regimens" or "nootropic stacks", who are, as far as I'm concerned, finding innovative ways to poison themselves.

Yet the article went on to say this:

But even as some experts hailed the new study as a breakthrough, others urged caution, warning people not to start a regimen of aspirin without first consulting a doctor about the potential risks, including gastrointestinal bleeding and bleeding in the brain (hemorrhagic strokes).

“Many people may wonder if they should start taking daily aspirin, but it would be premature to recommend people starting taking aspirin specifically to prevent cancer,” said Eric J. Jacobs, an epidemiologist with the American Cancer Society.

I'm a programmer, not a doctor - but after looking around, I concluded that the risks of GI bleeding were not guaranteed fatal, and the risks of hemorrhagic strokes were low in absolute terms. Also, aspirin is famously effective against ischemic strokes. According to Wikipedia: "Although aspirin also raises the risk of hemorrhagic stroke and other major bleeds by about twofold, these events are rare, and the balance of aspirin's effects is positive. Thus, in secondary prevention trials, aspirin reduced the overall mortality by about a tenth."

So unless aspirin's risks are far more grave than I've currently been led to believe, as far as I'm concerned, people saying "hey, even if you're not subject to aspirin's well-known contraindications, you shouldn't start low-dose aspirin just yet" are literally statistically killing people. Cancer is pretty lethal and we're not really good at fixing it yet, so when we find something that can really reduce the risk (and there aren't many - the only other ones I can think of are the magical substances known as not-smoking and avoiding-massive-doses-of-ionizing radiation), we should be all over that like cats on yarn.

I make damn sure to take my low-dose aspirin every day. I started it before reading this article on the advice of my doctor who thought my cholesterol was a little high - I'm almost 28, so it'll have many years in which to work its currently poorly understood magic.

That said, this reduces the risk of one common cause of death (two or three if you throw in heart attacks and ischemic strokes). There are lots of others out there. Even if you could avoid all of them (including the scariest one, Alzheimer's - it's insanely common, we have no fucking clue what causes it or how to stop it, and it annihilates your very self - even if cryonics is ultimately successful, advanced Alzheimer's is probably the true death), humans pretty clearly wear out with an upper bound of 120 years. Maybe caloric restriction can adjust that somewhat. But I think I'll sign up for cryonics sooner rather than later - I'm in favor of upgrading probability from "definitely boned" to "probably boned but maybe not".

And I think I have my answer:

Last week, researchers in London reported that they had analyzed nine randomized studies of aspirin use in the United States, Europe and Japan that included more than 100,000 participants. The study subjects had never had a heart attack or stroke; all regularly took aspirin or a placebo to determine whether aspirin benefits people who have no established heart disease.
In the combined analysis, the researchers found that regular aspirin users were 10 percent less likely than the others to have any type of heart event, and 20 p

... (read more)

3gwern15y

I wonder how the aspirin trials would look in regular people (I haven't checked, are any of your randomized trials in normal people?). Mike Darwin on aspirin; [...]

31Scott Alexander15y

The meta-analysis you cite is moderately convincing, but only moderately. They had enough different analyses such that some would come out significant by pure chance. Aspirin was found to have an effect on 15-year-mortality significant only at the .05 level, and aspirin was found not to have a significant effect 20-year-mortality, so take it with a grain of salt. There was also some discussion in the literature about how it's meta-analyzing studies performed on people with cardiac risk factors but not bleed risk factors, and so the subjects may have been better candidates for aspirin than the general population. The Wikipedia quote you give is referring to secondary prevention, which means "prevention of a disease happening again in someone who's already had the disease". Everyone agrees aspirin is useful for secondary prevention, but there are a lot of cases where something useful for secondary prevention isn't as good for primary. In primary prevention, aspirin doesn't get anywhere near a tenth reduction in mortality (although it does seem to have a lesser effect). I would say right now there's enough evidence that people who enjoy self-experimentation are justified in trying low-dose aspirin and probably won't actively hurt themselves (assuming they check whether they're at special risk of bleeds first), but not enough evidence that doctors should be demonized for not telling everyone to do it.

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Living Forever is Hard, or, The Gompertz Curve

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Living Forever is Hard, or, The Gompertz Curve

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