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Моделирование смертности. Л.А. Гаврилов Center on Aging NORC and the University of Chicago Chicago, Illinois, USA. Questions of Actuarial Significance. How far could mortality decline go? (absolute zero seems implausible)

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Моделирование смертности

Л.А. Гаврилов

Center on Aging

NORC and the University of Chicago

Chicago, Illinois, USA


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Questions of Actuarial Significance

  • How far could mortality decline go?

    (absolute zero seems implausible)

  • Are there any ‘biological’ limits to human mortality decline, determined by ‘reliability’ of human body?

    (lower limits of mortality dependent on age, sex, and population genetics)

  • Were there any indications for ‘biological’ mortality limits in the past?

  • Are there any indications for mortality limits now?


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How can we improve the actuarial forecasts of mortality and longevity ?

By taking into account the mortality laws summarizing prior experience in mortality changes over age and time:

Gompertz-Makeham law of mortality

Compensation law of mortality

Late-life mortality deceleration


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The Gompertz-Makeham Law longevity ?

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

μ(x) = A + R e αx

A – Makeham term or background mortality

R e αx – age-dependent mortality; x - age

risk of death


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Gompertz Law of Mortality in Fruit Flies longevity ?

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

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


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Gompertz-Makeham Law of Mortality in Flour Beetles longevity ?

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

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


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Gompertz-Makeham Law of Mortality in Italian Women longevity ?

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

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


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How can the Gompertz-Makeham law be used? longevity ?

By studying the historical dynamics of the mortality components in this law:

μ(x) = A + R e αx

Makeham component

Gompertz component


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Historical Stability of the Gompertz longevity ?Mortality ComponentHistorical Changes in Mortality for 40-year-old Swedish Males

  • Total mortality, μ40

  • Background mortality (A)

  • Age-dependent mortality (Reα40)

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


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Predicting Mortality Crossover longevity ?Historical Changes in Mortality for 40-year-old Women in Norway and Denmark

  • Norway, total mortality

  • Denmark, total mortality

  • Norway, age-dependent mortality

  • Denmark, age-dependent mortality

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


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Predicting Mortality Divergence longevity ?Historical Changes in Mortality for 40-year-old Italian Women and Men

  • Women, total mortality

  • Men, total mortality

  • Women, age-dependent mortality

  • Men, age-dependent mortality

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


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Historical Changes in Mortality longevity ?Swedish Females

Data source: Human Mortality Database


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Extension of the Gompertz-Makeham Model Through the longevity ?Factor Analysis of Mortality Trends

Mortality force (age, time) =

= a0(age) + a1(age) x F1(time) + a2(age) x F2(time)


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Factor Analysis of Mortality Swedish Females longevity ?

Data source: Human Mortality Database


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Implications longevity ?

  • Mortality trends before the 1950s are useless or even misleading for current forecasts because all the “rules of the game” has been changed


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Preliminary Conclusions longevity ?

  • 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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Compensation Law of Mortality longevity ?(late-life mortality convergence)

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


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Compensation Law of Mortality longevity ?Convergence of Mortality Rates with Age

1 – India, 1941-1950, males

2 – Turkey, 1950-1951, males

3 – Kenya, 1969, males

4 - Northern Ireland, 1950-1952, males

5 - 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


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Compensation Law of Mortality (Parental Longevity Effects) longevity ? Mortality Kinetics for Progeny Born to Long-Lived (80+) vs Short-Lived Parents

Sons

Daughters


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Compensation Law of Mortality in Laboratory longevity ?Drosophila

1 – 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


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Implications longevity ?

  • 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


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The Late-Life Mortality Deceleration longevity ?(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.


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Mortality deceleration at advanced ages. longevity ?

  • 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.



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Mortality Leveling-Off in House Fly longevity ?Musca domestica

Based on life table of 4,650 male house flies published by Rockstein & Lieberman, 1959


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Non-Aging Mortality Kinetics in Later Life longevity ?

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


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Invertebrates: longevity ?

Nematodes, shrimps, bdelloid rotifers, degenerate medusae (Economos, 1979)

Drosophila melanogaster (Economos, 1979; Curtsinger et al., 1992)

Housefly, blowfly (Gavrilov, 1980)

Medfly (Carey et al., 1992)

Bruchid beetle (Tatar et al., 1993)

Fruit flies, parasitoid wasp (Vaupel et al., 1998)

Mammals:

Mice (Lindop, 1961; Sacher, 1966; Economos, 1979)

Rats (Sacher, 1966)

Horse, Sheep, Guinea pig (Economos, 1979; 1980)

However no mortality deceleration is reported for

Rodents (Austad, 2001)

Baboons (Bronikowski et al., 2002)

Mortality Deceleration in Animal Species


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Existing Explanations longevity ?of Mortality Deceleration

  • Population Heterogeneity (Beard, 1959; Sacher, 1966). “… sub-populations with the higher injury levels die out more rapidly, resulting in progressive selection for vigour in the surviving populations” (Sacher, 1966)

  • Exhaustion of organism’s redundancy (reserves) at extremely old ages so that every random hit results in death (Gavrilov, Gavrilova, 1991; 2001)

  • Lower risks of death for older people due to less risky behavior (Greenwood, Irwin, 1939)

  • Evolutionary explanations (Mueller, Rose, 1996; Charlesworth, 2001)


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Testing the “Limit-to-Lifespan” Hypothesis longevity ?

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


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Implications longevity ?

  • 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.


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Latest Developments longevity ?

Was the mortality deceleration law overblown?

A Study of the Real Extinct Birth Cohorts in the United States


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Challenges in Hazard Rate Estimation At Extremely Old Ages longevity ?

  • Mortality deceleration may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect)

  • Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high

  • Ages of very old people may be highly exaggerated


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Challenges in Death Rate Estimation longevity ?at Extremely Old Ages

  • Mortality deceleration may be an artifact of mixing different birth cohorts with different mortality (heterogeneity effect)

  • Standard assumptions of hazard rate estimates may be invalid when risk of death is extremely high

  • Ages of very old people may be highly exaggerated


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U.S. Social Security Administration longevity ?Death Master File Helps to Relax the First Two Problems

  • Allows to study mortality in large, more homogeneous single-year or even single-month birth cohorts

  • Allows to study mortality in one-month age intervals narrowing the interval of hazard rates estimation


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What Is SSA DMF ? longevity ?

  • SSA DMF is a publicly available data resource (available at Rootsweb.com)

  • Covers 93-96 percent deaths of persons 65+ occurred in the United States in the period 1937-2003

  • Some birth cohorts covered by DMF could be studied by method of extinct generations

  • Considered superior in data quality compared to vital statistics records by some researchers


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Quality Control longevity ?

Study of mortality in states with better age reporting:

Records for persons applied to SSN in the Southern states, Hawaii and Puerto Rico were eliminated







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Crude Indicator longevity ?of Mortality Plateau (2)

Coefficient of variation for life expectancy is close to, or higher than 100%

CV = σ/μ

where σ is a standard deviation and μ is mean



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What are the explanations of mortality laws? of age

Mortality and aging theories


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Additional Empirical Observation: of ageMany 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).


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Like humans, nematode of ageC. elegans experience muscle loss

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.”

Body wall muscle sarcomeres

Left - age 4 days. Right - age 18 days


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What Should of agethe 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



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Stages of Life in Machines and Humans of age

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.

The so-called bathtub curve for technical systems


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Non-Aging Failure Kinetics of ageof Industrial Materials in ‘Later Life’(steel, relays, heat insulators)

Source:

A. Economos.

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


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Reliability Theory of age

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


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What Is Reliability Theory? of age

  • Reliability theory is a general theory of systems failure.


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The Concept of System’s Failure of age

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


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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.


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Aging and non-aging systems theory

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

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


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Mortality in Aging and Non-aging Systems theory

aging system

non-aging system

Example: radioactive decay


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According to Reliability Theory: theoryAging is NOT just growing oldInsteadAging 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)


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According to Reliability Theory: theory

  • Onset of disease or disability is a perfect example of organism's failure

  • When the risk of such failure outcomes increases with age -- this is an aging by definition


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The Concept of Reliability Structure across biological species (salmon vs humans)

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


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Two major types of system’s logical connectivity across biological species (salmon vs humans)

  • Components connected in series

  • Components connected in parallel

Fails when the first component fails

Ps = p1 p2 p3 … pn = pn

Fails when all components fail

Qs = q1 q2 q3 … qn = qn

  • Combination of two types – Series-parallel system


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Series-parallel Structure of Human Body across biological species (salmon vs humans)

  • Vital organs are connected in series

  • Cells in vital organs are connected in parallel


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Redundancy Creates Both Damage Tolerance and Damage Accumulation (Aging)

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

System with redundancy accumulates damage (aging)


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Reliability Model Accumulation (Aging)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-1early-life period approximation, when 1-e-kx kx

 klate-life period approximation, when 1-e-kx 1


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Failure Rate as a Function of Age Accumulation (Aging)in Systems with Different Redundancy Levels

Failure of elements is random


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Standard Reliability Models Explain Accumulation (Aging)

  • Mortality deceleration and leveling-off at advanced ages

  • Compensation law of mortality


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Standard Reliability Models Accumulation (Aging)Do Not Explain

  • The Gompertz law of mortality observed in biological systems

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


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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.


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Reliability structure of Mainframe Computer(a) technical devices and (b) biological systems

Low redundancy

Low damage load

High redundancy

High damage load

X - defect


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Models of systems with distributed redundancy Mainframe Computer

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


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

x0 = 0 - ideal system, Weibull law of mortality

x0 >> 0 - highlydamaged system,Gompertz law of mortality

Model of organism with initial damage load

Binomial law of mortality

- the initial virtual age of the system

where

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


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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.


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Statement of the HIDL hypothesis: parts than like ones built with pristine parts.(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.


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Spontaneous mutant frequencies with age in heart and small intestine

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


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Practical implications from intestinethe 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.


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Life Expectancy and Month of Birth intestine

Data source: Social Security Death Master File


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Acknowledgments intestine

This 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


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For More Information and Updates Please Visit Our intestineScientific and Educational Website on Human Longevity:

  • http://longevity-science.org


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Gavrilov, L., Gavrilova, N. intestineReliability theory of aging and longevity. In: Handbook of the Biology of Aging. Academic Press, 6th edition (published recently).


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