1. Reliability Theory of Aging
Dr. Leonid A. Gavrilov, Ph.D.
Center on Aging
NORC and The University of Chicago
Chicago, Illinois, USA

2. What Is Reliability Theory?
Reliability theory is a general theory of systems failure.
Reliability theory is a body of ideas, mathematical models, and methods directed to predict, estimate, understand, and optimize the failure distribution of systems and their components.
Reliability theory allows researchers to predict the age-related failure kinetics for a system of given architecture (reliability structure) and given reliability of its components.
Reliability theory was historically developed to describe failure and aging of complex electronic (military) equipment, but the theory itself is a very general theory.

3. Some Representative Publications on Reliability Theory of Aging (Additional Reading)

5. The reliability theory of aging and longevity
The reliability theory of aging and longevity Journal of Theoretical Biology. 2001, 213,
The quest for a general theory of aging and longevity. Science SAGE KE (Science of Aging Knowledge Environment). 2003, 28, 1-10.
Reliability-engineering approach to the problem of biological aging. Annals of the New York Academy of Sciences. 2004, vol
Things fall apart: Engineering’s reliability theory explains human aging. IEEE Spectrum (in press).

6. Why Do We Need Reliability Theory?
Because reliability theory provides a common scientific language (general framework) for scientists working in different areas of aging research and anti-aging interventions.
Reliability theory helps to overcome disruptive specialization and it allows researchers to understand each other.

7. Most important, Reliability Theory goes to the heart of the Aging problem:
Aging, according to reliability theory, is what makes "old not as good as new", when the failure rates are increasing with age.
Non-aging objects are perfectly legitimate in reliability theory -- these are those objects, which do not deteriorate with age, when "old is as good as new", and when the failure rates are not increasing with age.
Thus, the attempts to stop aging are not against the laws of Nature -- this is not about stopping or reversing the physical time, but rather the efforts to keep "old as good as new" through proper maintenance, repair and parts replacement.
Thus, anti-aging interventions are perfectly legitimate according to Reliability Theory.

8. According to Reliability Theory: Aging is NOT just growing old Instead 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)

9. According to Reliability 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

10. Implications
Diseases are an integral part (outcomes) of the aging process
Aging without diseases is just as inconceivable as dying without death
Not every disease is related to aging, but every progression of disease with age has relevance to aging: Aging is a 'maturation' of diseases with age
Aging is the many-headed monster with many different types of failure (disease outcomes) Aging is, therefore, a summary term for many different processes.
Anti-aging interventions, therefore, should not be discouraged by their partial success limited to specific outcomes. There should be a complex of many different anti-aging interventions.

11. What are the Major Findings to be Explained?
Biogerontological studies found a remarkable similarity in survival dynamics between humans and laboratory animals:
Gompertz-Makeham law of mortality
Compensation law of mortality
Late-life mortality deceleration.

12. The Gompertz-Makeham Law
The Gompertz-Makeham law states that 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 + R0exp(α x)
A – Makeham term or background mortality
R0exp(α x) – age-dependent mortality

13. Exponential Increase of Death Rate with Age in Fruit Flies (Gompertz Law of Mortality)
Linear dependence of the logarithm of mortality force on the age of Drosophila.
Based on the life table for 2400 females of Drosophila melanogaster published by Hall (1969). Mortality force was calculated for 3-day age intervals.
Source: Gavrilov, Gavrilova,
“The Biology of Life Span” 1991

14. Age-Trajectory of Mortality in Flour Beetles (Gompertz-Makeham Law of Mortality)
Dependence of the logarithm of mortality force (1) and logarithm of increment of mortality force (2) on the age of flour beetles (Tribolium confusum Duval).
Based on the life table for 400 female flour beetles published by Pearl and Miner (1941). Mortality force was calculated for 30-day age intervals.
Source: Gavrilov, Gavrilova, “The Biology of Life Span” 1991

15. Age-Trajectory of Mortality in Italian Women (Gompertz-Makeham Law of Mortality)
Dependence of the logarithm of mortality force (1) and logarithm of increment of mortality force (2) on the age of Italian women.
Based on the official Italian period life table for Mortality force was calculated for 1-year age intervals.
Source: Gavrilov, Gavrilova,
“The Biology of Life Span” 1991

16. The Compensation Law of Mortality
The Compensation law of mortality (late-life mortality convergence) states that the relative differences in death rates between different populations of the same biological species are decreasing with age, because the higher initial death rates are compensated by lower pace of their increase with age

17. Compensation Law of Mortality Convergence of Mortality Rates with Age
1 – India, , males
2 – Turkey, , males
3 – Kenya, 1969, males
4 - Northern Ireland, , males
5 - England and Wales, , females
6 - Austria, , females
7 - Norway, , females
Source: Gavrilov, Gavrilova,
“The Biology of Life Span” 1991

18. Compensation Law of Mortality in Laboratory 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

19. 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.
An immediate consequence from this observation is that 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 (‘biological limit’ to human longevity).

20. Mortality at Advanced Ages
Source: Gavrilov L.A., Gavrilova N.S. The Biology of Life Span:
A Quantitative Approach, NY: Harwood Academic Publisher, 1991

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

23. Mortality deceleration at advanced ages.
Risk of death (in log scale) is plotted as a function of age. Note that after age 95, the observed risk of death (black line) deviates from the Gompertz law (grey line) leveling-off at extreme ages.
Source: Gavrilov, Gavrilova, “Things fall apart: Engineering’s reliability theory explains human aging”. IEEE Spectrum (in press).

24. Mortality Leveling-Off in Drosophila
Non-Gompertzian mortality kinetics of Drosophila melanogaster
Source: Curtsinger et al., Science, 1992.

25. Non-Gompertzian Mortality Kinetics of Four Invertebrate Species
Non-Gompertzian mortality kinetics of four invertebrate species: nematodes, Campanularia flexuosa, rotifers and shrimp.
Source: A. Economos A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 1979, 2:

26. Non-Gompertzian Mortality Kinetics of Three Rodent Species
Non-Gompertzian mortality kinetics of three rodent species: guinea pigs, rats and mice.
Source: A. Economos A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 1979, 2:

27. Non-Gompertzian Mortality Kinetics of Three Industrial Materials
Non-Gompertzian mortality kinetics of three industrial materials: steel, industrial relays and motor heat insulators.
Source: A. Economos A non-Gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE, 1979, 2:

28. Aging is a Very General Phenomenon!

29. What Should the Aging Theory Explain:
Why do most biological species deteriorate with age?
Specifically, why do mortality rates increase exponentially with age in many adult species (Gompertz law)?
Why does the age-related increase in mortality rates vanish at older ages (mortality deceleration)?
How do we explain the so-called compensation law of mortality (Gavrilov & Gavrilova, 1991)?

30. Redundancy Creates Both Damage Tolerance and Damage Accumulation (Aging)

31. Explanations of Aging Phenomena Using Reliability Theory:
Consider a system built of non-aging elements with a constant failure rate k. If these n elements are mutually substitutable, so that the failure of a system occurs only when all the elements fail (parallel construction in the reliability theory context), the cumulative distribution function for system failure, F(n,k,x), depends on age x in the following way:
Therefore, the reliability function of a system, S(n,k,x), can be represented as:
Consequently, the failure rate of a system (n,k,x), can be written as follows:
 nknxn-1
when x << 1/k (early-life period approximation, when 1-e-kx  kx);
 k
when x >> 1/k (late-life period approximation, when 1-e-kx  1)

32. Source: Gavrilov, Gavrilova, Science SAGE KE. 2003, 28, 1-10.

33. Why Organisms May Be Different From Machines?

34. Differences in reliability structure between (a) technical devices and (b) biological systems
Each block diagram represents a system with m serially connected blocks (each being critical for system survival, 5 blocks in these particular illustrative examples) built of n elements connected in parallel (each being sufficient for block being operational). Initially defective non-functional elements are indicated by crossing (x).
The reliability structure of technical devices (a) is characterized by relatively low redundancy in elements (because of cost and space limitations), each being initially operational because of strict quality control. Biological species, on the other hand, have a reliability structure (b) with huge redundancy in small, often non-functional elements (cells).

35. 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 The Biology of Life Span:
A Quantitative Approach. Harwood Academic Publisher, New York.

36. Why should we expect high initial damage load ?
General argument: --  In contrast to technical devices, which are built from pre-tested high-quality components, biological systems are formed by self-assembly without helpful external quality control.
Specific arguments:
Cell cycle checkpoints are disabled in early development     (Handyside, Delhanty,1997. Trends Genet. 13, )
extensive copy-errors in DNA, because most cell divisions   responsible for  DNA copy-errors occur in early-life   (loss of telomeres is also particularly high in early-life)
ischemia-reperfusion injury and asphyxia-reventilation injury   during traumatic process of 'normal' birth

37. Birth Process is a Potential Source of High Initial Damage
During birth, the future child is deprived of oxygen by compression of the umbilical cord and suffers severe hypoxia and asphyxia.
Then, just after birth, a newborn child is exposed to oxidative stress because of acute reoxygenation while starting to breathe. It is known that acute reoxygenation after hypoxia may produce extensive oxidative damage through the same mechanisms that produce ischemia-reperfusion injury and the related phenomenon, asphyxia-reventilation injury.
Asphyxia is a common occurrence in the perinatal period, and asphyxial brain injury is the most common neurologic abnormality in the neonatal period that may manifest in neurologic disorders in later life.

38. Spontaneous mutant frequencies with age in heart and small intestine
Source: Presentation of Jan Vijg at the IABG Congress, Cambridge, 2003

39. 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."
"Thus, the idea of early-life programming of aging and longevity may have important practical implications for developing early-life interventions promoting health and longevity."
Source: Gavrilov, L.A. & Gavrilova, N.S The Biology of Life Span:
A Quantitative Approach. Harwood Academic Publisher, New York.

40. Season of Birth and Female Lifespan 8,284 females from European aristocratic families born in Seasonal Differences in Adult Lifespan at Age 30
Life expectancy of adult women (30+) as a function of month of birth (expressed as a difference from the reference level for those born in February).
The data are point estimates (with standard errors) of the differential intercept coefficients adjusted for other explanatory variables using multivariate regression with categorized nominal variables.

41. Mortality Kinetics in Highly Redundant Systems Saturated with Defects
Failure rate of a system is described by the formula:
where n is a number of mutually substitutable elements (connected in parallel) organized in m blocks connected in series;
k - constant failure rate of the elements;
i - is a number of initially functional elements in a block;
λ - is a Poisson constant (mean number of initially functional elements in a block).
Source: Gavrilov L.A., Gavrilova N.S. The reliability theory of aging and longevity. Journal of Theoretical Biology, 2001, 213(4):

42. Dependence of the logarithm of mortality force (failure rate) on age for binomial law of mortality
Source: Gavrilov, Gavrilova, Journal of Theoretical Biology. 2001, 213,

43. Failure Kinetics in Mixtures of Systems with Different Redundancy Levels Initial Period
The dependence of logarithm of mortality force (failure rate) as a function of age in mixtures of parallel redundant systems having Poisson distribution by initial numbers of functional elements (mean number of elements,  = 1, 5, 10, 15, and 20.

44. Failure Kinetics in Mixtures of Systems with Different Redundancy Levels “Big Picture”
The dependence of logarithm of mortality force (failure rate) as a function of age in mixtures of parallel redundant systems having Poisson distribution by initial numbers of functional elements (mean number of elements,  = 1, 5, 10, 15, and 20.

45. Strategies of Life Extension Based on the Reliability Theory
Increasing durability of components
Increasing redundancy
Maintenance and repair
Replacement and repair

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

47. 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.

48. Acknowledgments 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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