Living Forever is Hard, or, The Gompertz Curve


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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:

  1. '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”.

  2. '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.