Columbia University - Research Seminars in Epidemiology Reliability-Theory Approach to...

Columbia University - Research Seminars in Epidemiology

Reliability-Theory Approach

to Epidemiological Studies of Aging, Mortality and

LongevityDr. Leonid A. Gavrilov, Ph.D.

Dr. Natalia S. Gavrilova, Ph.D.

Center on Aging NORC and The University of Chicago

Chicago, Illinois, USA


What Is Reliability-Theory Approach?

Reliability-theory approach is based on ideas, methods, and models of a general theory of systems failure known as reliability theory.


Reliability theory was historically developed to describe failure and aging of complex electronic (military) equipment, but the theory itself is a very general theory based on probability theory and systems approach.


Why Do We Need Reliability-Theory Approach?

Because it provides a common scientific language (general paradigm) for scientists working in different areas of aging research, including epidemiological studies.

Reliability theory helps to overcome disruptive specialization and it allows researchers to understand each other.

Provides useful mathematical models allowing to explain and interpret the observed epidemiological data and findings.


Some Representative Publications on Reliability-

Theory Approach to Epidemiological Studies of

Aging, Mortality and Longevity


Gavrilov, L., Gavrilova, N. Reliability theory of aging and longevity. In: Handbook of the Biology of Aging. Academic Press, 6th edition, 2006, pp.3-42.


The Concept of System’s Failure

In reliability theory failure is defined as the event when a required function is terminated.


Failures are often classified into two groups:

degradation failures, where the system or component no longer functions properly

catastrophic or fatal failures - the end of system's or component's life


Definition of aging and non-aging systems in reliability

theory Aging: increasing risk of failure

with the passage of time (age).

No aging: 'old is as good as new' (risk of failure is not increasing with age)

Increase in the calendar age of a system is irrelevant.


Aging and non-aging systems

Perfect clocks having an ideal marker of their increasing age (time readings) are not aging

Progressively failing clocks are aging (although their 'biomarkers' of age at the clock face may stop at 'forever young' date)


Mortality in Aging and Non-aging Systems

Age

0 2 4 6 8 10 12

Ris

k o

f d

ea

th

1

2

3

Age0 2 4 6 8 10 12

Ris

k o

f D

eath

0

1

2

3

non-aging system aging system

Example: radioactive decay


According to Reliability Theory:

Aging is NOT just growing oldInstead

Aging is a degradation to failure: becoming sick, frail and dead

'Healthy aging' is an oxymoron like a healthy dying or a healthy disease

More accurate terms instead of 'healthy aging' would be a delayed aging, postponed aging, slow aging, or negligible aging (senescence)


Reliability-Theory Approach to Epidemiology of Aging (I) Focus on adverse health

outcomes, “health failures” (disability, disease, death) rather than any age-related changes

Focus on incidence of “health failures” rather than prevalence measures

Focus on age-specific incidence rates rather than age-aggregated (age-adjusted) measures


Reliability-Theory Approach to Epidemiology of Aging

(II) Very inclusive system approach

(not limited to humans). Extensive use of modeling:

Mathematical models Animal models Failure models for manufactured

items


Aging is a Very General Phenomenon!


Particular mechanisms of aging may be very different even across biological species (salmon vs humans)

BUT

General Principles of Systems Failure and Aging May Exist

(as we will show in this presentation)


Further plan of presentation

Empirical laws of failure and aging in epidemiology

Explanations by reliability theory

Links between reliability theory and epidemiologic studies


Empirical Laws of Systems Failure and

Aging


Stages of Life in Machines and Humans

The so-called bathtub curve for technical systems

Bathtub curve for human mortality as seen in the U.S. population in 1999 has the same shape as the curve for failure rates of many machines.


Failure (Mortality) Laws

1. Gompertz-Makeham law of mortality

2. Compensation law of mortality

3. Late-life mortality deceleration


The Gompertz-Makeham Law

μ(x) = A + R e αx

A – Makeham term or background mortalityR e αx – age-dependent mortality; x - age

Death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age.

risk of death Non-agingcomponent

Agingcomponent


Gompertz Law of Mortality in Fruit Flies

Based on the life table for 2400 females of Drosophila melanogaster published by Hall (1969).

Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991


Gompertz-Makeham Law of Mortality in Flour Beetles

Based on the life table for 400 female flour beetles (Tribolium confusum Duval). published by Pearl and Miner (1941).



Gompertz-Makeham Law of Mortality in Italian Women

Based on the official Italian period life table for 1964-1967.



Compensation Law of Mortality(late-life mortality

convergence)

Relative differences in death rates are decreasing with age, because the lower initial death rates are compensated by higher slope of mortality growth with age (actuarial aging rate)


Compensation Law of Mortality

Convergence of Mortality Rates with Age

1 – India, 1941-1950, males 2 – Turkey, 1950-1951, males3 – Kenya, 1969, males 4 - Northern Ireland, 1950-

1952, males5 - England and Wales, 1930-

1932, females 6 - Austria, 1959-1961, females

7 - Norway, 1956-1960, females

Source: Gavrilov, Gavrilova,“The Biology of Life Span” 1991


Compensation Law of Mortality (Parental Longevity Effects)

Mortality Kinetics for Progeny Born to Long-Lived (80+) vs Short-Lived Parents

Sons DaughtersAge

40 50 60 70 80 90 100

Lo

g(H

azar

d R

ate)

0.001

0.01

0.1

1

short-lived parentslong-lived parents

Linear Regression Line

Age

40 50 60 70 80 90 100

Lo

g(H

azar

d R

ate)

0.001

0.01

0.1

1

short-lived parentslong-lived parents

Linear Regression Line


Compensation Law of Mortality in Laboratory

Drosophila1 – drosophila of the Old

Falmouth, New Falmouth, Sepia and Eagle Point strains (1,000 virgin females)

2 – drosophila of the Canton-S strain (1,200 males)

3 – drosophila of the Canton-S strain (1,200 females)

4 - drosophila of the Canton-S strain (2,400 virgin females)

Mortality force was calculated for 6-day age intervals.

Source: Gavrilov, Gavrilova,“The Biology of Life Span” 1991


Epidemiological Implications

Be prepared to a paradox that higher actuarial aging rates may be associated with higher life expectancy in compared populations (e.g., males vs females)

Be prepared to violation of the proportionality assumption used in hazard models (Cox proportional hazard models)

Relative effects of risk factors are age-dependent and tend to decrease with age


The Late-Life Mortality Deceleration (Mortality Leveling-off,

Mortality Plateaus)

The late-life mortality deceleration law states that death rates stop to increase exponentially at advanced ages and level-off to the late-life mortality plateau.


Mortality deceleration at advanced ages.

After age 95, the observed risk of death [red line] deviates from the value predicted by an early model, the Gompertz law [black line].

Mortality of Swedish women for the period of 1990-2000 from the Kannisto-Thatcher Database on Old Age Mortality

Source: Gavrilov, Gavrilova, “Why we fall apart. Engineering’s reliability theory explains human aging”. IEEE Spectrum. 2004.


M. Greenwood, J. O. Irwin. BIOSTATISTICS OF SENILITY


Mortality Leveling-Off in House Fly

Musca domestica

Our analysis of the life table for 4,650 male house flies published by Rockstein & Lieberman, 1959.

Source: Gavrilov & Gavrilova.

Handbook of the Biology of Aging, Academic Press, 2006, pp.3-42.

Age, days

0 10 20 30 40

ha

zard

ra

te,

log

sc

ale

0.001

0.01

0.1


Non-Aging Mortality Kinetics in Later Life

If mortality is constant then log(survival) declines with age as a linear function

Source:

Economos, A. (1979). A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 2: 74-76.


Non-Aging Failure Kinetics of Industrial Materials in ‘Later Life’

(steel, relays, heat insulators)

Source:

Economos, A. (1979). A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 2: 74-76.


Testing the “Limit-to-Lifespan” Hypothesis

Source: Gavrilov L.A., Gavrilova N.S. 1991. The Biology of Life Span


Epidemiological Implications

There is no fixed upper limit to human longevity - there is no special fixed number, which separates possible and impossible values of lifespan.

This conclusion is important, because it challenges the common belief in existence of a fixed maximal human life span.


Additional Empirical Observation:

Many age changes can be explained by cumulative effects of

cell loss over time Atherosclerotic inflammation -

exhaustion of progenitor cells responsible for arterial repair (Goldschmidt-Clermont, 2003; Libby, 2003; Rauscher et al., 2003).

Decline in cardiac function - failure of cardiac stem cells to replace dying myocytes (Capogrossi, 2004).

Incontinence - loss of striated muscle cells in rhabdosphincter (Strasser et al., 2000).


Like humans, nematode C. elegans experience muscle loss

Body wall muscle sarcomeres

Left - age 4 days. Right - age 18 days

Herndon et al. 2002. Stochastic and genetic factors influence tissue-specific decline in ageing C. elegans. Nature 419, 808 - 814.

“…many additional cell types (such as hypodermis and intestine) … exhibit age-related deterioration.”


What Should the Aging Theory Explain

Why do most biological species including humans deteriorate with age?

The Gompertz law of mortality

Mortality deceleration and leveling-off at advanced ages

Compensation law of mortality


The Concept of Reliability Structure

The arrangement of components that are important for system reliability is called reliability structure and is graphically represented by a schema of logical connectivity


Two major types of system’s logical connectivity

Components connected in series

Components connected in parallel

Fails when the first component fails

Fails when all

components fail

Combination of two types – Series-parallel system

Ps = p1 p2 p3 … pn = pn

Qs = q1 q2 q3 … qn = qn


Series-parallel Structure of Human Body

• Vital organs are connected in series

• Cells in vital organs are connected in parallel


Redundancy Creates Both Damage Tolerance and Damage Accumulation

(Aging)

System with redundancy accumulates damage (aging)

System without redundancy dies after the first random damage (no aging)


Reliability Model of a Simple Parallel

System

Failure rate of the system:

Elements fail randomly and independently with a constant failure rate, k

n – initial number of elements

nknxn-1 early-life period approximation, when 1-e-kx kx k late-life period approximation, when 1-e-kx 1

( )x =dS( )x

S( )x dx=

nk e kx( )1 e kx n 1

1 ( )1 e kx n



Failure Rate as a Function of Age in Systems with Different Redundancy

Levels

Failure of elements is randomSource: Gavrilov, Gavrilova, IEEE Spectrum. 2004.


Standard Reliability Models Explain

Mortality deceleration and leveling-off at advanced ages

Compensation law of mortality


Standard Reliability Models Do Not Explain

The Gompertz law of mortality observed in biological systems

Instead they produce Weibull (power) law of mortality growth with age:

μ(x) = a xb


An Insight Came To Us While Working With Dilapidated

Mainframe Computer The complex

unpredictable behavior of this computer could only be described by resorting to such 'human' concepts as character, personality, and change of mood.


Reliability structure of (a) technical devices and (b) biological

systems

Low redundancy

Low damage load

Fault avoidance

High redundancy

High damage load

Fault toleranceX - defect


Models of systems with distributed redundancy

Organism can be presented as a system constructed of m series-connected blocks with binomially distributed elements within block (Gavrilov, Gavrilova, 1991, 2001)


Model of organism with initial damage load

Failure rate of a system with binomially distributed redundancy (approximation for initial period of life):

x0 = 0 - ideal system, Weibull law of mortality

x0 >> 0 - highly damaged system, Gompertz law of mortality


( )x Cmn( )qk n 1 q

qkx +

n 1

= ( )x0 x + n 1

where - the initial virtual age of the systemx0 =1 q

qk

The initial virtual age of a system defines the law of system’s mortality:

Binomial law of mortality


People age more like machines built with lots of faulty parts than like ones built with

pristine parts.

As the number of bad components, the initial damage load, increases [bottom to top], machine failure rates begin to mimic human death rates.

Source: Gavrilov, Gavrilova, IEEE Spectrum. 2004


Statement of the HIDL hypothesis:

(Idea of High Initial Damage Load )

"Adult organisms already have an exceptionally high load of initial damage, which is comparable with the amount of subsequent aging-related deterioration, accumulated during the rest of the entire adult life."

Source: Gavrilov, L.A. & Gavrilova, N.S. 1991. The Biology of Life Span: A Quantitative Approach. Harwood Academic Publisher, New York.


Why should we expect high initial damage load in biological systems?

General argument:-- biological systems are formed by self-assembly without helpful external quality control.

Specific arguments:

1. Most cell divisions responsible for DNA copy-errors occur in early development leading to clonal expansion of mutations

2. Loss of telomeres is also particularly high in early-life

3. Cell cycle checkpoints are disabled in early development


Birth Process is a Potential Source of High Initial

Damage Severe hypoxia and asphyxia just

before the birth.

oxidative stress just after the birth because of acute reoxygenation while starting to breathe.

The same mechanisms that produce ischemia-reperfusion injury and the related phenomenon, asphyxia-reventilation injury known in cardiology.


Mutation Frequencies are Already High Early in Life

Spontaneous mutant frequencies with age in heart and small intestine of mouse

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35Age (months)

Mu

tan

t fr

eq

uen

cy (

x10

-5)

Small IntestineHeart

Source: Presentation by Jan Vijg at the IABG Congress, Cambridge, 2003


Practical implications from the HIDL hypothesis:

"Even a small progress in optimizing the early-developmental processes can potentially result in a remarkable prevention of many diseases in later life, postponement of aging-related morbidity and mortality, and significant extension of healthy lifespan."

Source: Gavrilov, L.A. & Gavrilova, N.S. 1991. The Biology of Life Span: A Quantitative Approach. Harwood Academic Publisher, New York.


Implications for Epidemiological Studies

If the initial damage load is really important, then we may expect significant effects of early-life conditions (like season-of-birth, birth order, or paternal age at conception) on late-life morbidity and mortality


Month of Birth

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

life

exp

ecta

ncy

at

age

80, y

ears

7.6

7.7

7.8

7.9

1885 Birth Cohort1891 Birth Cohort

Life Expectancy and Month of BirthData source: Social Security Death Master File

Published in:

Gavrilova, N.S., Gavrilov, L.A. Search for Predictors of Exceptional Human Longevity. In: “Living to 100 and Beyond” Monograph. The Society of Actuaries, Schaumburg, Illinois, USA, 2005, pp. 1-49.


Birth Order and Chances to Become a Centenarian

Cases - centenarians born in the United States between 1890 and 1899

Controls – their siblings born in the same time window

Model:

Log(longevity odds ratio)= ax + bx2 + cz + d

where x – birth order; z – family size; a, b, c, d – parameters of polynomial regression model


Birth Order and Survival to 100

Source:

Gavrilova, N.S., Gavrilov, L.A. Search for Predictors of Exceptional Human Longevity. In: “Living to 100 and Beyond” Monograph. The Society of Actuaries, Schaumburg, Illinois, USA, 2005, pp. 1-49.

Birth order

1 2 3 4 5 6 7 8 9 10

Od

ds

to b

eco

me

a ce

nte

nar

ian

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FemalesMales


Genetic Justification for Paternal Age Effects

Advanced paternal age at child conception is the main source of new mutations in human populations.

James F. Crow, geneticistPNAS USA, 1997, 94(16): 8380-6

Professor Crow (University of Wisconsin-Madison) is recognized as a leader and statesman of science. He is a member of the National Academy of Sciences, the National Academy of Medicine, The American Philosophical Society, the American Academy of Arts and Sciences, the World Academy of Art and Science.


Paternal Age and Risk of Schizophrenia

Estimated cumulative incidence and percentage of offspring estimated to have an onset of schizophrenia by age 34 years, for categories of paternal age. The numbers above the bars show the proportion of offspring who were estimated to have an onset of schizophrenia by 34 years of age.

Source: Malaspina et al., Arch Gen Psychiatry.2001.


Daughters' Lifespan (30+) as a Functionof Paternal Age at Daughter's Birth

6,032 daughters from European aristocratic families born in 1800-1880

Life expectancy of adult women (30+) as a function of father's age when these women were born (expressed as a difference from the reference level for those born to fathers of 40-44 years).

The data are point estimates (with standard errors) of the differential intercept coefficients adjusted for other explanatory variables using multiple regression with nominal variables.

Daughters of parents who survived to 50 years.

Paternal Age at Reproduction

15-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59

Lif

es

pa

n D

iffe

ren

ce

(y

r)

-4

-3

-2

-1

1

0

p = 0.04


Contour plot for daughters’ lifespan (deviation from cohort mean) as a function of paternal lifespan (X axis) and paternal age at daughters’ birth (Y axis)

7984 cases

1800-1880 birth cohorts

European aristocratic families

Distance weighted least squares smooth

40 50 60 70 80 90

Paternal Lifespan, years

20

25

30

35

40

45

50

55

60

65

Pat

erna

l Age

at

Per

son'

s B

irth

, yea

rs

3 2 1 0 -1 -2 -3


Daughters’ Lifespan as a Function of Paternal Age at Daughters’ Birth

Data are adjusted for other predictor variables

Daughters of shorter-lived fathers (<80), 6727 cases

Daughters of longer-lived fathers (80+), 1349 cases

Paternal Age at Person's Birth

15-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59

Lif

esp

an D

iffe

ren

ce (

yr)

-4

-3

-2

-1

1

0

Paternal Age at Person's Birth

15-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59

Lif

esp

an D

iffe

ren

ce (

yr)

-4

-2

2

4

0


Preliminary Conclusion

Being conceived to old father is a risk factor, but it is moderated by paternal longevity

It is OK to be conceived to old father if he lives more than 80 years

Epidemiological implications: Paternal lifespan should be taken into account in the studies of paternal-age effects


Conclusions (I) Redundancy is a key notion for

understanding aging and the systemic nature of aging in particular. Systems, which are redundant in numbers of irreplaceable elements, do deteriorate (i.e., age) over time, even if they are built of non-aging elements.

An apparent aging rate or expression of aging (measured as age differences in failure rates, including death rates) is higher for systems with higher redundancy levels.


Conclusions (II) Redundancy exhaustion over the life course explains

the observed ‘compensation law of mortality’ (mortality convergence at later life) as well as the observed late-life mortality deceleration, leveling-off, and mortality plateaus.

Living organisms seem to be formed with a high load of initial damage, and therefore their lifespans and aging patterns may be sensitive to early-life conditions that determine this initial damage load during early development. The idea of early-life programming of aging and longevity may have important practical implications for developing early-life interventions promoting health and longevity.


AcknowledgmentsThis study was made possible thanks to:

generous support from the National Institute on Aging, and

stimulating working environment at the Center on

Aging, NORC/University of Chicago


For More Information and Updates Please Visit Our Scientific and Educational

Website on Human Longevity:

http://longevity-science.org


Possible Change of Paradigm in Epidemiology

Mortality of Swedish females

Before the 1960s After the 1960s

Age

0 20 40 60 80

Lo

g (

Haz

ard

Rat

e)

10-4

10-3

10-2

10-1

190019201930194019501960

Age

0 20 40 60 80

Lo

g (

Ha

zard

Ra

te)

10-4

10-3

10-2

10-1

19601970198019902000


Mortality Before the 1960s Can be Modeled Using the Gompertz-

Makeham lawBy studying the historical dynamics of the mortality components in this law:

μ(x) = A + R e αx

Makeham component Gompertz component


Historical Stability of the Gompertz

Mortality Component Before the 1980s

Historical Changes in Mortality for 40-year-old Swedish Males1. Total mortality,

μ40 2. Background

mortality (A)3. Age-dependent

mortality (Reα40)

Source: Gavrilov, Gavrilova, “The

Biology of Life Span” 1991


Predicting Mortality Crossover

Historical Changes in Mortality for 40-year-old Women in Norway and Denmark

1. Norway, total mortality

2. Denmark, total mortality

3. Norway, age-dependent mortality

4. Denmark, age-dependent mortality



Predicting Mortality Divergence

Historical Changes in Mortality for 40-year-old Italian Women and Men

1. Women, total mortality

2. Men, total mortality3. Women, age-

dependent mortality4. Men, age-dependent

mortality



Hypothesis of Death Quota for Total Mortality

The Idea of Non-Specific Vulnerability Intermediate State

Normal state of organism

State of nonspecific vulnerability

Death

Extremesituationsproducingbackgroundmortality

Various diseases and “causes” of death

Aging(limiting stage)


Mortality Decline After the 1960s May Be a Result of Improvement in Early-Life

Conditions

Birth cohorts of Swedish women (Source of data: HMD)

Age

40 50 60 70 80 90

Mor

talit

y R

ate,

log

scal

e

0.001

0.010

0.100

1880 Birth Cohort1890 Birth Cohort1900 Birth Cohort1910 Birth Cohort1920 Birth Cohort


Extension of the Gompertz-Makeham Model Through the

Factor Analysis of Mortality Trends

Mortality force (age, time) = = a0(age) + a1(age) x F1(time) + a2(age) x

F2(time)

Where:

• ai(age) – a set of numbers; each number is fixed for specific age group

• Fj(time) – “factors,” a set of standardized numbers; each number is fixed for specific moment of time (mean = 0; st. dev. = 1)


Factor Analysis of Mortality Trends

Swedish Females

Calendar Year

1900 1920 1940 1960 1980 2000

Mo

rta

lity

fa

cto

r s

co

re

-2

0

2

4 Makeham-like factor 1("young ages")Gompertz-like factor 2("old ages")

“Factor analysis of the time series of mortality confirms the preferential reduction in the mortality of old-aged and senile people [in recent years]…” Gavrilov, Gavrilova, The Biology of Life Span, 1991.

Data source for the current slide: Human Mortality Database


Testing hypothesis of statistical independence between causes of death

Based on 179 values of male mortality at age 55-64 from 26 countries (WHO)

Cause of death

VIIIth revision of ICD

Correlation coefficient with total mortality Coefficient

of mortality amplificatio

n (2)/(1)

Expected value*

(1)

Observed value

(2)

Ischaemic heart disease (A83) +0.770 +0.606 0.79Cerebrovascular disease +0.246 +0.345 1.40Lung cancer (A51) +0.215 +0.601 2.80Cirrhosis of liver (A102) +0.139 +0.027 0.19Bronchitis, emphysema, asthma (A93) +0.121 +0.603 4.98

Stomach cancer (A47) +0.114 +0.148 1.30Diseases of arteries, arterioles and capillaries (A86)

+0.036 +0.680 18.89 * Expected value = std(cause of death)/std(all causes) Preston, Nelson, 1974


A Broader View on the Historical Changes in

Mortality

Swedish Females

Data source: Human Mortality Database

Age

0 20 40 60 80

Lo

g (

Ha

zard

Ra

te)

10-4

10-3

10-2

10-1

1925196019802000


Preliminary Conclusions

There was some evidence for ‘ biological’ mortality limits in the past, but these ‘limits’ proved to be responsive to the recent technological and medical progress.

Thus, there is no convincing evidence for absolute ‘biological’ mortality limits now.

Analogy for illustration and clarification: There was a limit to the speed of airplane flight in the past (‘sound’ barrier), but it was overcome by further technological progress. Similar observations seems to be applicable to current human mortality decline.

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