X-Message-Number: 1450
Date: 15 Dec 92 22:38:08 EST
From: "Steven B. Harris" <>
Subject: CRYONICS Gompertz and Fruit Flies
Harris Comments about D. Lubkins's Note:
This Gompertz hubbub is an interesting sidelight in geronto-
logy, which regularly results from the fact that we talk about
mortality both in terms of the time dependence of mortality risk
functions, otherwise known as fractional mortality rates 1/N(t)-
*(dN/dt), but ALSO about mortality in terms of simple numbers of
a population left alive N(t)-- population survival curves. One
is the slope of the other, and to get at one from the other, you
need a bit of math, and the two functions should be clearly kept
separate when talking about aging, so as not to confuse.
Benjamin Gompertz was an insurance actuary who in 1825
noticed that your risk of dying in any given year increased
exponentially with time. If other non-time dependent stuff is
left out (getting bombed in a war, or starving, or dying of
plague, or getting eaten by wild animals), this equation should
(when integrated and graphed) produce a pretty good look at what
aging in the absence of external causes looks like, in terms of
population survival. It's sort of a square curve, and is what we
find in developing countries. Start with a population of N
people or animals, and follow over time:
N
100|............................
.
% | .
.
| .
.
| .
|
| .
| .
.
| .
| .
| .
| .
.
|
.
|
.
-------------------------------------------------------
TIME
Theoretical Curve, with Gompertz mortality, Makeham term zero
But if you REALLY do the experiment with animals in good
conditions, or REALLY look at population curves in developed
countries, the curve is non-Gompertzian at the end, and looks
like this:
N
100|............................
.
% | .
.
| .
.
| .
|
| .
| .
.
| .
| .
| .
| .
.
|
.
| .
. .
-----------------------------------------------------
TIME
Actual curve
The fact that the last 10-12% left alive of any naturally
occurring population depart from the Gompertzian exponentially
increasing mortality risk, and instead tend toward a time-
independent (albeit high) CONSTANT mortality risk function, is
something that has been known about for a long, long time (if my
reference books were here instead of the office, I could tell you
how long). There is a discussion of it, for instance, in Wal-
ford's book Maximum Lifespan, which was published a decade ago.
There Walford notes that for the very oldest old, the survival
function (mortality curve) changes, and begins to looks like the
survival function for drinking glasses being washed in a re-
staurant-- it's a simple inverse exponential population decline
with no time dependence at all on risk for individuals (in other
words in the case of glasses it doesn't matter how old a glass
is-- it has the same change of being broken with each washing).
Although Walford doesn't go into it, anyone with rudimentary
calculus will recognize that exponentially decreasing numbers in
a cohort imply constant risk for individual cohort members with
time (think, for instance, of radioactive decay). That little
"exponential tail" on the end of survival curves in lab animals
is something you will see in every single paper where lab animals
have been cared for in good conditions for the last 30 years, at
least. In the last paper I wrote on the subject (1990-- Journal
of Gerontology), I had to discard my oldest 10% of survivors to
get good Gompertz plots. Everybody does this.
The odd thing about fruit flies, which has just been dis-
covered by looking at a million of them, is that the non-Gom-
pertzian exponential tail of the last 10% of flies goes WAAAY out
(at least) to times *twice* as long as it took to get from 0% to
90% mortality. That's amazing, for it shows that for at least
that long, there really isn't any of what we would call "aging"
(time dependent increases in mortality risk) after a fruit fly
has made it to the privileged cohort of the flies that outlive
90% of their brothers. Something of this sort does seem to
happen to people, but it's obviously not quite so pronounced,
since the 90% mortality figure applies to us in our late 80's,
and there aren't any people (proven by good records) who've made
it to anything near 160, even though we've kept statistics on a
much larger number of people than we have fruit flies.
Still, there are a few pieces of comfort in all this: the
time-dependent Gompertzian exponential *increase* in mortality
risk does eventually stop at high ages, so perhaps some of what
we call "aging" is not really due to processes as relentless and
implacable as we had thought. The bad news, of course, is that
the constant mortality risk we're left with after the Gompertz
equation is through with us at about age 90 is something like 30%
per year, and even if your yearly mortality risk never gets any
higher than that for the rest of your life, it's easy to see that
30 or 40% chance of dying in the next year, year after year, will
get you in the end, before too long. Worse still, for humans
(but not fruit flies), there's obviously something ELSE that adds
on to this to make sure that something will get us even before we
hit 120 or so. Still, in the interim, since constance of risk
suggests mortality due more to external than internal causes, it
may be that we'll learn to avoid the stressors that push 30% of
very, very old people over the edge each year. Then, by the time
the ultimate Grim Reaper gets to us finally at 120, we may have
learned something from the fruit flies. And if not, then there's
always cryonics.
Steve
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