期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2021
卷号：118
期号：32
DOI：10.1073/pnas.2110693118
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：Bredberg and Bredberg (
1) suggest that some individuals age more slowly than others and that this accounts for the leveling off of death rates after age 100 y. They make this claim in a letter responding to Vaupel et al. (
2).
Bredberg and Bredberg (
1) vaguely describe their mathematical model without specifying formulas. Apparently, their model is based on
q
(
x
−
70
,
b
)
=
a
e
b
(
x
−
70
)
,
where
x
≥
70
is age and
q
(
x
−
70
,
b
)
is the annual probability of death at ages 70+ y for an individual with aging rate
b. Parameter
a
is a constant that the authors set at 0.021, and
b is normally distributed at age 70 y with mean of 1.107 and SD of 0.0091. The risk of death among survivors to age
x
is then given by
q
¯
(
x
−
70
)
=
∫
0
∞
q
(
x
−
70
,
b
)
⋅
π
(
x
−
70
,
b
)
d
b
,
where
π
(
x
−
70
,
b
)
is the probability distribution of
b
at age
x
≥
70
y. At age 70 y, this is the normal distribution specified by Bredberg and Bredberg (
1), but at higher ages the distribution is given by
π
(
x
−
70
,
b
)
=
π
(
0
,
b
)
⋅
s
(
x
−
70
,
b
)
∫
0
∞
π
(
0
,
b
)
⋅
s
(
x
−
70
,
b
)
d
b
,
where
s
(
x
−
70
,
b
)
is the chance of surviving from 70 y to age
x
>
70
y for individuals with rate of aging
b.
This model has serious deficiencies.
Normal distributions can take on negative values, but a negative rate of aging is implausible. Because the mean of the distribution Bredberg and Bredberg (
1) used is more than 11 SDs from zero, this fact is unlikely to have much impact, but it is a theoretical blemish. Perhaps the authors worked with a truncated normal distribution to only account for positive values.
In most cases, the so-called accelerated aging models—in which some individuals age more rapidly than others—lead to a decline in mortality at advanced ages, not a plateau (refs.
3–
6, among others). In particular, in the model described above which Bredberg and Bredberg (
1) may have used, the average annual risk of death reaches a maximum and then declines toward zero.
Furthermore, Bredberg and Bredberg (
1) do not cite research that suggests variation among individuals in rates of aging is low and perhaps close to zero (
7,
8). If individuals share the same rate of aging but differ in their initial mortality—parameter
a
in the model above—then death rates can approach a plateau (
9). Conversely, if a mortality plateau is approached at advanced ages, a plausible explanation is that individuals differ in their value of
a
but not
b (
10).
