Article

ISSN 0006-2979, Biochemistry (Moscow), 2024, Vol. 89, No. 2, pp. 341-355 © Pleiades Publishing, Ltd., 2024.

341

REVIEW

Exploring Patterns of Human Mortality and Aging:

A Reliability Theory Viewpoint

Leonid A. Gavrilov

1,2,a

* and Natalia S. Gavrilova

1,2

NORC at the University of Chicago, 60637 Chicago, IL, USA

Institute for Demographic Research, Federal Center of Theoretical and Applied Sociology,

Russian Academy of Sciences, 109028 Moscow, Russia

e-mail: lagavril@yahoo.com

Received December 14, 2023

Revised January 27, 2024

Accepted January 28, 2024

Abstract— The most important manifestation of aging is an increased risk of death with advancing age, a mortal-

ity pattern characterized by empirical regularities known as mortality laws. We highlight three significant ones:

the Gompertz law, compensation effect of mortality (CEM), and late-life mortality deceleration and describe new

developments in this area. It is predicted that CEM should result in declining relative variability of mortality

at older ages. The quiescent phase hypothesis of negligible actuarial aging at younger adult ages is tested and

refuted by analyzing mortality of the most recent birth cohorts. To comprehend the aging mechanisms, it is cru-

cial to explain the observed empirical mortality patterns. As an illustrative example of data-directed modeling

and the insights it provides, we briefly describe two different reliability models applied to human mortality pat-

terns. The explanation of aging using a reliability theory approach aligns with evolutionary theories of aging,

including idea of chronic phenoptosis. This alignment stems from their focus on elucidating the process of or-

ganismal deterioration itself, rather than addressing the reasons why organisms are not designed for perpetual

existence. This article is a part of a special issue of the journal that commemorates the legacy of the eminent

Russian scientist Vladimir Petrovich Skulachev (1935-2023) and his bold ideas about evolution of biological aging

and phenoptosis.

DOI: 10.1134/S0006297924020123

Keywords: aging, mortality, Gompertz model, compensation effect of mortality, mortality deceleration, reliability

theory of aging, evolutionary models of aging

* To whom correspondence should be addressed.

INTRODUCTION

The increasing interest in unraveling the intri-

cacies of aging underscores the necessity for a com-

prehensive theoretical framework. There has been a

significant surge in the volume of empirical data on

aging, reflecting a substantial expansion in our un-

derstanding of the aging processes. The exploration of

individual genes, pathways, and molecules in under-

standing the mechanisms that modulate aging has in-

deed seen significant progress. Researchers have made

strides in identifying the components involved in the

aging process. However, the challenge lies in compre-

hending how these various factors interact on a larger

scale to influence the aging processes. While we can

pinpoint specific genes, pathways, and molecules, the

integration of these elements into a comprehensive un-

derstanding of aging, including the emergence of func-

tional phenotypes like mortality laws, remains a com-

plex puzzle [1].

Evolutionary theories can provide a broader un-

derstanding of the aging phenomenon. Until recently,

there was a consensus among gerontologists that there

is no specific program of aging, because such program

could not appear when overwhelming majority of an-

imals in the wild and humans in the past did not sur-

vive to advanced ages [2-4]. However, this view was re-

vised in the last decade and there are more arguments

now in favor of aging as a program. Vladimir P. Sku-

lachev was among the researchers who promoted an

GAVRILOV, GAVRILOVA342

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

idea of programmed aging. He attempted to resurrect

Weismann’s theory of programmed death by proposing

the existence of a distinct self-destruction program for

entire organisms, termed “phenoptosis” [5-7]. This pro-

gram is believed to serve “crucial functions, cleansing

communities of organisms from undesirable individ-

uals”  [7]. As such, Weismann’s evolutionary theory of

programmed death continues to be a subject of scien-

tific debate and inquiry. Moreover, the ideas of harm-

ful role that older individuals play in population by

accumulating infections gain popularity among geron-

tologists  [6,  8]. Nevertheless, theories and hypotheses

about programmed aging primarily elucidate the rea-

sons organisms are not inherently designed to be flaw-

less and fall short in explaining the actual process of

age-related deterioration itself.

The primary limitation of evolutionary theories of

aging is that they rely on the concept of natural selec-

tion and the diminishing impact of natural selection

with age. However, aging is observed in non-replicat-

ing technical systems like automobiles, which lack the

capacity for procreation and are therefore unaffected

by evolution driven by natural selection.

Recognizing this broader context prompts the ex-

ploration of more universal explanations for aging that

transcend the constraints of evolutionary theories. Be-

yond overarching concepts, there is a crucial need to

explore empirical regularities in aging and mortality,

often referred to as mortality laws. This pursuit aims

to provide a comprehensive understanding of aging

phenomena that goes beyond the scope of traditional

evolutionary frameworks.

In this context, we examine the established mor-

tality laws and regularities alongside recent advance-

ments in the field of biodemography of aging. The ex-

isting patterns of mortality and aging are scrutinized

through the lens of reliability theory. Additionally, this

discussion encompasses the presentation of further

developments in reliability models of aging, contrib-

uting to an enhanced understanding of the underlying

mechanisms shaping the aging process.

MORTALITY LAWS IN THE BIOLOGY

OF AGING AND LIFESPAN

To gain a more precise understanding of the mech-

anisms behind an organism’s decline, we can explore

mortality patterns frequently referred to as mortali-

ty laws. Here we assume definition of aging applied

in reliability theory, which considers aging as a pro-

cess leading to increasing risk of death with age [9].

In ecology this process is also called an actuarial se-

nescence [10].

The Gompertz–Makeham law. In 1825, the Brit-

ish actuary Benjamin Gompertz unveiled a principle

of mortality, now recognized as the Gompertz law [2,

11-13]. According to the Gompertz law, human mortal-

ity rates double approximately every 8 years of adult

age:

μ(x) = Re

αx

, (1)

where x is age, R and α are parameters (Gompertz in-

tercept and Gompertz slope, since Gompertz law rep-

resents a straight line in semi-log coordinates).

An exponential (Gompertzian) increase of mortal-

ity rates with advancing age is observed across various

biological species, encompassing fruit flies [2, 14], flour

beetles like Tribolium confusum [2], mice  [14,  15], ba-

boons [16,  17], and others. Most importantly, this law

describes mortality of humans [2,  11-14,  18,  19].

In reality, organisms’ hazard rates can encompass

both non-aging and aging components, as seen in the

Gompertz–Makeham law of mortality [2,  11,  13,  19,  20]:

μ(x) = A + Re

αx

. (2)

Within this equation, the initial, age-independent

term (known as the Makeham parameter, “A”) signifies

the constant, “non-aging” facet of the hazard rate, pre-

sumably stemming from external causes of death, such

as accidents and acute infections. In biodemography

this constant term is called background mortality [2].

The subsequent, age-dependent term (the Gompertz

function, Re

αx

) represents the “aging” component, pre-

sumably arising from age-related degenerative diseases

like cancer and heart disease. This aging-related term

received the name of senescent mortality [21]. Carnes

and Olshansky proposed another way to classify total

mortality using causes of death. Mortality from exter-

nal causes of death and infections they called an extrin-

sic mortality while mortality from aging-related causes

they called an intrinsic mortality[22]. Thus, extrinsic

and intrinsic types of mortality may both depend on

age. It is important to note that the slope coefficient

α characterizes an “apparent aging rate,” indicating

the speed of age-related deterioration in mortality. If α

equals zero, there is no apparent aging, which means

that death rates remain constant with age.

Some authors have proposed a generalized form

of the Gompertz–Makeham law (GGM) [23]. In this

alternative perspective, the exponent, indicative of

stress resistance, is considered to be a function be-

yond a linear form. The Makeham term, traditionally

treated as a constant, is redefined to be associated

with mortality resulting from inherently irresistible

stresses and depends on age. This departure from the

standard Gompertz–Makeham law introduces a more

flexible approach to modeling mortality [23].

For technical systems one of the most popular

models for failure rate of aging systems is a Weibull

EXPLORING PATTERNS OF HUMAN MORTALITY AND AGING 343

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

Fig. 1. Mortality of Norwegian women in 2010 in semi-log scale and its linear fit according to the Gompertz law. Data source:

Human Mortality Database (www.mortality.org).

model, the power-function increase in failure rates

with age [2]:

μ(x) = αx

for x≥0, where α,β>0. (3)

The Weibull law, proposed by Swedish engineer

and mathematician W. Weibull in 1939 to describe the

strength of materials, is widely employed in charac-

terizing the aging and failure of technical devices [9].

Ithas also found occasional application in the study of

a limited number of biological species including nema-

todes C. elegans [24-26]. According to the Weibull law,

the logarithm of failure rates exhibits a linear increase

with the logarithm of age.

A comparative meta-analysis of 129 life tables for

fruit flies and 285 life tables for humans demonstrated

that the Gompertz law of mortality offers a more ac-

curate fit to the data for each of these two biological

species compared to the Weibull law [2].

While working with human data it was found that

mortality of women in semi-log scale demonstrated

slightly more concave trajectory than predicted by the

Gompertz model and mortality of men was somewhat

more convex [2]. Similar deviations from the Gompertz

law were observed by other researchers [27]. Further

analyses of local estimates of the Gompertz slope using

life table aging rate (LAR) found increase of LAR after

age 60 mostly among women [28,  29]. Recently accel-

eration of LAR after age 60 was found for both men

and women and for period and cohort mortality using

contemporary data [30]. Despite these small deviations

Gompertz law describes mortality remarkably well (see

Fig.1). There are indeed small fluctuations of mortal-

ity around the main Gompertz trajectory whereas the

general direction of mortality line remains stable. In

humans the Gompertz model fits both period and co-

hort data [31, 32].

In addition to the commonly used Gompertz and

standard two-parameter Weibull laws, another mor-

tality law known as the binomial law of mortality has

been proposed and theoretically justified through the

lens of system reliability theory [2, 9]. This particular

law is considered a special case of the three-parameter

Weibull function. Notably, it incorporates a negative

location parameter, offering a nuanced perspective on

the dynamics of mortality and aging within the frame-

work of reliability theory:

μ(x) = α (x

+ x)

. (4)

In this equation, the parameterx

is referred to as

the initial virtual age of the system [2, 9, 33]. This pa-

rameter, measured in units of time, represents the age

at which an initially ideal system would accumulate

damage equal to those of a real system at the starting

age (x=0).

When a system is in its undamaged state with an

initial virtual age of zero, the mortality rate adheres

to a power function of age, consistent with the charac-

teristics outlined in the Weibull law. However, as the

initial damage load accumulates, there is a deviation

from the Weibull law. This departure becomes more

GAVRILOV, GAVRILOVA344

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

pronounced with increasing levels of initial damage

load, eventually leading to a transition where the fail-

ure dynamics align with the quasi-Gompertz mortality

law [9].

Compensation law or compensation effect of

mortality. Another empirical finding, known as the

compensation law or compensation effect of mortality

(CEM), in its strong form pertains to mortality conver-

gence at older ages. This is when higher values for the

slope parameter α (in the Gompertz function) are offset

(compensated) by lower values of the intercept param-

eter R in different populations of a specific species:

ln(R) = ln(M) − Bα, (5)

where B andM represent species-specific constants.

This relationship is occasionally referred to as the

Strehler–Mildvan correlation [13,  34], although this

correlation was significantly affected by biases in pa-

rameter estimation arising from neglecting the age-in-

dependent mortality component, the Makeham term  A

[2,  20]. Parameter B is called the species-specific lifes-

pan, and parameter M is called the species-specific

mortality rate [2, 9,  35]. Recent estimates of parame-

ters B andM using data for contemporary human pop-

ulations showed that these estimates remain rather

stable over time [31,  32]. These parameters represent

the coordinates where all mortality trajectories con-

verge into a single point when extrapolated using the

Gompertz function [2]. This signifies that in disadvan-

taged populations within a specific species, high mor-

tality rates are compensated for by a slower apparent

“aging rate” (resulting in a longer mortality doubling

period). As a consequence of this compensation, rela-

tive differences in mortality rates have a tendency to

decrease with age within a given biological species.

The term “Compensation Effect of Mortality” was

introduced in 1978 when incorporating the Makeham

parameter yielded substantially different parameter

estimates for the Strehler–Mildvan correlation [35].

This effect is defined by the convergence of senescent

(age-dependent) mortality patterns at advanced ages

[2,  35]. CEM is observed not only for humans but for

some other species [2,  36,  37].

There were assertions that the Strehler–Mildvan

correlation arises as a statistical artifact of spurious cor-

relation between the estimates of the Gompertz param-

eters and does not exist in reality [38]. Nevertheless,

even when controlling for collinearity, the correla-

tion coefficients between the Gompertz parameters

experience only a modest reduction, yet the core cor-

relation remains intact [31]. Furthermore, it is worth

noting that the convergence of mortality trajectories

in a semi-logarithmic scale (CEM) at older ages can

be observed independently of the estimation of the

Gompertz parameters [31, 32]. Also, in human popula-

tions CEM is observed for both period and cohort data

on mortality [31, 32].

As mortality convergence is approached, a de-

crease in the relative variation of mortality is expected.

Consequently, we anticipate a decline in the relative

variation of mortality, assessed through metrics like

the coefficient of variation and the standard deviation

of the log of mortality, as the convergence point cor-

responding to the species-specific lifespan is reached.

This theoretical prediction will be subject to validation

in upcoming studies.

Late-life mortality deceleration. At more ad-

vanced ages (beyond age 70 in humans), sometimes a

phenomenon known as “old-age mortality decelera-

tion” occurs, wherein death rates increase with age at

a slower pace than expected based on the Gompertz

law [2,  21,  39-41]. Some biologists called this cessation

of mortality growth (non-aging state) “a revolution

for aging research” [42], although for humans it was

known by actuaries since the Gompertz times. This de-

celeration in mortality ultimately leads to “late-life

mortality leveling-off” and “late-life mortality plateaus”

at extremely old ages [2,  43,  44].

Actuaries, including Gompertz himself, were among

the first to observe this phenomenon. They proposed a

logistic formula to model the age-related increase in

mortality, aiming to address the decrease in mortality

growth at advanced ages.

Figure 2 illustrates mortality deceleration at older

ages using an example of mortality for cohort of U. S.

women born in 1886. Note that after age 90 years the

observed mortality deviates from the Gompertz law in

this particular case.

In humans mortality deceleration is almost always

observed for cross-sectional data. In the case of cohort

data, there is no consensus on the pattern of late-life

mortality trajectories. Earlier studies suggested that

mortality after age 80 displayed a slower increase

with age compared to the exponential Gompertz law.

This deceleration in mortality at advanced ages was ob-

served in both cohort and period data for various coun-

tries [21]. However, for U. S. cohort data, it was revealed

that mortality adheres to the Gompertz law within wide

age range of 80-106 years [45, 46]. Similar results were

recently obtained using French cohort data [47]. Other

scholars have found that the extent of mortality decel-

eration varies among different countries [48, 49]. The

primary issue with these studies was the attempt to

deduce the existence of only one conceivable mortal-

ity trajectory shape. Recently the existing controver-

sy about the shape of mortality at advanced ages was

resolved by studying long series of U. S. cohort data.

It turns out that mortality of earlier birth cohorts

(born before 1887) always demonstrates mortality de-

celeration. On the other hand, later birth cohorts fol-

low the Gompertz law up to ages 105-106 years [50].

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BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

Fig. 2. Mortality deceleration for cohort of U. S. women born in 1886. Data source: Human Mortality Database (www. mortality.org).

This trend called gompertzialization of mortality tra-

jectory is observed in other countries as well, although

with slower pace [51].

EXPLORING FURTHER ASPECTS OF AGING

As widely recognized, the aging phase constitutes

a substantial portion of the lifespan of many species,

necessitating that any mortality model offer an expla-

nation for this extended period of life. For modeling

the aging process, it is important to take into account

some additional age-related phenomena.

Similarity between mortality of biological and

technical systems. A notable parallel exists between

living organisms and technical devices in their age-re-

lated mortality patterns, often following what’s known

as the “bathtub curve”. [9]. This curve comprises three

distinct periods. Initially, mortality rates are high and

gradually decline with age, referred to as the “work-

ing-in” period or the phase of “burning-out” defective

components. A similar early period, known as “infant

mortality,” can be observed in most living organisms,

including humans. Following this, there is the “normal

working period,” characterized by relatively low and

stable failure rates. While this period exists in humans

as well, it tends to be relatively short, lasting approx-

imately 10-15 years before transitioning to the third

period known as “the aging period.” During this phase,

mortality rates increase inexorably with age, follow-

ing an explosive exponential trajectory, akin to the

Gompertz curve. For humans, this aging period typi-

cally spans from around 20 to 100 years. This similari-

ty in mortality patterns between technical and biolog-

ical systems is further emphasized by the presence of

a fourth common period at extremely advanced ages.

This phase is recognized in biology as “late-life mortal-

ity leveling-off” and is also observed in technical sys-

tems [52].

Keeping in mind this similarity in the phases of

age- specific mortality between biological and techni-

cal systems, Siler suggested an empirical equation that

described mortality of biological organisms during the

entire life period [53]. This equation considers three

mortality terms. The first term describes mortality de-

cline with age after birth and can be described by de-

clining exponential function. Two other terms are rep-

resented by already known background and senescent

mortality. The minimum of mortality around age 10

received lately an attention from researchers. It turns

out that the minimum of mortality measured in period

data looks very narrow while for cohort data mortality

may be flat for much longer period. This long period

of almost flat mortality was called a “quiescent phase”

[54] and was observed for cohorts born around the

1920s (Fig.3).

This phenomenon serves as a clear example of an

almost zero apparent (actuarial) aging rate, arising from

the counteracting influences of two opposing forces

that offset one another: the mitigating effect of mortal-

ity reduction (attributable to advancements in health-

care and improved living conditions) and the inexora-

ble force of aging [32]. Figure3 shows that mortality at

age 10 for 1920th cohort is almost the same as at ages

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BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

Fig. 3. Mortality for selected Swedish female birth cohorts. Data source: Human Mortality Database (www.mortality.org).

30-40 years, so that there is no clear minimum of mor-

tality around ages 10-40 years. This phase looks like

rather long “normal working period” in technical sys-

tems, but is rather a result of declining mortality due

to healthcare achievements and improvement of living

conditions [51].

The authors of the original study of quiescent

phase limited their analysis by extinct cohorts only[54].

However, mortality at younger ages can be studied

for more recent birth cohorts. Figure 3 shows that for

1950 and 1980 birth cohorts mortality is not flat and

slowly increases with age. It also looks like the posi-

tion of mortality minimum is shifting to younger ages

over time. When mortality from infectious diseases

was mostly eradicated and historical mortality decline

slowed down, the quiescent phase practically disap-

peared and mortality started to grow from very young

ages (8-12 years) without visible hump. It should be

noted that we use here data for women in order to

avoid substantial external mortality due to social fac-

tors, which is common for men at ages 18-25 years.

Figure 3 shows that mortality for more recent birth

cohorts can be described by two rather than three

phases with disappearing the “normal working peri-

od” or quiescent phase.

The concept of high initial damage load. In 1991

it was suggested that early developmental processes

in living organisms generate an exceptionally high

load of initial damage, comparable in magnitude to

the subsequent accumulation of age-related deterio-

ration throughout the entire adult lifespan [2]. While

this concept, known as the High Initial Damage Load

(HIDL) hypothesis [55], may appear counterintuitive,

it aligns with empirical observations of developmental

noise and substantial cell losses during early develop-

ment [56]. Recent advancements in molecular develop-

mental biology have recognized the stochastic nature

of development, often referred to as “developmental

noise.” This phenomenon has the potential to induce

phenotypic heterogeneity even in the absence of any

other alterations in genes or the environment [57].

Ina human body, approximately a hundred thousand

cells are generated every second through mitosis, and

a comparable number undergo a physiological suicide

process known as apoptosis. A significant portion of

cells produced during mammalian embryonic devel-

opment experiences physiological cell death before

the conclusion of the perinatal period [58]. Significant

cell losses during early development create conditions

for the uneven distribution of organisms based on the

number of remaining cells, which can be modeled us-

ing the binomial or even the Poisson distribution. [2].

The idea of high initial damage of biological sys-

tems recently received a further development. It was

found that the mortality and incidence of age-related

diseases exhibit a U-shaped curve, with the minimum

occurring before puberty. However, quantitative bio-

markers of aging, such as somatic mutations and DNA

methylation, do not follow this pattern and continue to

increase starting from birth or even before birth [59].

It was suggested that aging initiates early but is con-

cealed by early-life mortality decline. According to this

idea aging may be represented by the growth of the

sum of deleterious changes, the deleteriome [59, 60].

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BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

The concept of a high initial damage load also

suggests that events occurring early in life can impact

survival in later adulthood through the level of ini-

tial damage. This prediction has been substantiated

for early-life indicators like parental age at a person’s

conception and the month of a person’s birth [61-66].

There is an increasing body of evidence supporting the

notion of fetal origins of adult degenerative diseases

[67,  68] and the early-life programming of aging and

longevity [55, 69-71].

Loss of functional elements in an organism over

its lifespan. Aging is characterized by a progressive

loss of functional tissue influenced by a combination

of genetic and environmental factors, nutrition, and

lifestyle choices [72, 73].

The accumulation of damage in various cellular

structures, coupled with the loss of fully differentiated

and irreplaceable cells like neurons and cardiomyo-

cytes, should be considered as irreversible. If these ir-

replaceable components of an organism age and even-

tually perish, it follows that the organism as a whole

will experience aging as well. This emphasizes the crit-

ical role of cellular damage and the loss of vital com-

ponents in the overall aging of organisms [74].

Cell death mechanisms have traditionally been

categorized into two types: programmed cell death

(PCD) mechanisms, which require energy, and necrotic

cell death mechanisms, which do not [73,  75]. The in-

creased occurrence of PCD during aging is implicated

in the decline of the immune system, skeletal muscle

wasting (sarcopenia), loss of cells in the heart, and neu-

rodegenerative diseases. Throughout the aging pro-

cess, several tissues experience cell loss attributed to

either PCD or PCD-like processes. In mammals, there

is an aging-associated skeletal muscle atrophy known

as sarcopenia, characterized by reductions in muscle

fiber size and fiber loss [73]. Necroptosis, a regulated

form of cell death, plays a role in the genesis and pro-

gression of various life-threatening diseases, including

cancer, neurological disorders, cardiac myopathy, and

diabetes [73].

Research indicates a decline in the numbers of spe-

cialized cells with age, including a significant reduc-

tion in nephrons in healthy human kidneys with aging.

Comparing the youngest (18-29 years) and oldest (70-

75 years) age groups, there was a 48% decrease in the

number of nonsclerotic glomeruli, while cortical vol-

ume only decreased by 16% [76].

While there is substantial knowledge about spe-

cific molecular mechanisms of cell death [77], the age-

specific loss of cell numbers is less explored and is

presented in a limited number of publications [78-80].

Existing studies have uncovered the possibility that

some cells within the aging organism may exhibit

nonaging characteristics. Notably, in a diverse spectrum

of aging-related neurodegenerative conditions (18 di-

verse examples of inherited and acquired neurode-

generation including Parkinson’s disease), the rate of

neuronal death does not increase with age [79,  81,  82].

Neurons in different parts of brain of cognitively

healthy humans show constant rate of atrophy with

age [83]. These findings align with the observation that

“an impressive range of cell functions in most organs

remain unimpaired throughout the life span” [11], p.425.

Therefore, the current understanding of the kinetics of

cell loss with age indicates that an exponential distri-

bution (with constant mortality) is a plausible approx-

imation for the mechanism of loss in vital elements,

such as functional cells or telomeres.

EXPLAINING MORTALITY LAWS

THROUGH THE LENS OF RELIABILITY THEORY

There is a whole spectrum of models attempting

to explain the observed mortality phenomena. Hetero-

geneity models were among the first models attempt-

ing to explain human aging and mortality and became

popular after the pioneer publication by Beard in 1959

[84]. These models assume exponential increase for

baseline risk of death. Heterogeneity in these models

is introduced usually by postulating gamma distribu-

tion of individual risks [85-87]. Heterogeneity models

are famous for explaining late-life mortality decelera-

tion and mortality plateau as a result of mortality se-

lection. However, the exponential growth of mortality

risk with age is postulated in these models in advance.

Thus, it would be more interesting to consider a mod-

el, which can use organism’s structure and properties

to derive the observed empirical mortality regularities.

It is noteworthy that the phenomena of increasing

mortality with age, followed by a plateauing of mortal-

ity rates, are inherently anticipated features of reliabil-

ity models that conceptualize aging as the progressive

accumulation of damage or loss of vital elements with

age [2, 9, 33]. Reliability models of aging presented in

more detail earlier [2, 33] consider living organism

as a system with series-parallel reliability structure.

In a system of n independent components connected

in series, the entire system fails if any one of the com-

ponents fails. This means that the reliability of the sys-

tem is contingent on the reliability of each individual

component in series. It is clear that human organs like

heart, brain or liver can be considered as vital com-

ponents connected in series because failure of any of

these organs leads to organism’s death. On the other

hand, in a parallel system of n independent compo-

nents, the system fails only when all the components

simultaneously fail. In this configuration, the system

remains operational as long as at least one of the par-

allel components continues to function, providing a

redundancy that enhances overall system reliability.

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BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

It is also clear that functional cells in each vital organ

can be considered as components connected in paral-

lel, although in real organs the threshold for normal

functioning may be higher than one element (cell).

Thus, reliability models in the context of living organ-

isms consider the inherent functional structure and

dynamics of biological systems. These models acknowl-

edge and incorporate observed processes, such as the

loss of functional cells with age. This approach helps

in understanding the intricate relationships between

different elements within biological systems and their

impact on overall mortality.

Consider a parallel system comprising n non-ag-

ing elements, each characterized by a constant failure

rate denoted as µ and a reliability (survival) function

expressed as e

–µx

[9]. In such a scenario, the reliabil-

ity function for the entire parallel system can be de-

scribed as:

S(x) = 1 – (1 – p)

= 1 – (1 − e

−μx

)

. (6)

This equation applies to the most straightforward

scenario where the failures of individual elements are

statistically independent. As a result, the failure rate of

the entire system, denoted as µ

(x), can be expressed

in the following manner:

(x) =

−

dS(x)

S(x)dx

nμe

−μx

(1 − e

−μx

)

n−1

1 − (1 − e

−μx

)

, (7)

≈  nμ

n−1

when x << 1/µ (early-life period approxima-

tion, when 1 − e

–µx

 ≈ µx);

≈  μ when x >> 1/µ (late-life period approximation, when

1 − e

–µx

 ≈ 1).

Hence, the system’s failure rate initially exhibits

growth in accordance with an age-dependent power

function (following the Weibull law). According to this

model, systems with varying initial redundancy lev-

els (parameter n) will display distinct failure rates in

early stages, but these differences will diminish over

time as the rates converge toward the upper limit de-

termined by the rate of elements’ loss (parameter µ).

Consequently, the expected outcome of this model

aligns with the compensation law of mortality (in its

weak form).

The failure rate of a simple parallel system com-

posed of non-aging elements exhibits an increase with

age, contrary to the Gompertz law, with the initial fail-

ure kinetics following the Weibull law. This deviation

from the Gompertz law arises from the model’s as-

sumption that the system is constructed with initial-

ly ideal structures, where all elements are functional

from the outset. This limitation highlights the impor-

tance of considering real-world scenarios where com-

ponents may not be flawless initially and may experi-

ence variations in functionality over time, influencing

the overall failure kinetics of the system. In order to

obtain the quasi-Gompertz mortality growth we need

to consider models with distributed redundancy.

It is important to highlight that reliability models

align seamlessly with evolutionary models, including

the concept of programmed death. The accrual of dam-

age might follow a stochastic process, whereas the pa-

rameters governing this damage, such as the initial

redundancy level and the rate of damage, could be

preprogrammed. Evolutionary models elucidate why

organisms are constructed with distinct properties,

while reliability models specifically elucidate the pro-

cess of deterioration itself, which may be considered

as slow or chronic phenoptosis [71, 88].

Reliability model of initially homogeneous pop-

ulation. The model, which was published earlier [33]

examined a scenario where blocks (e.g., specific or-

gans) within each organism exhibit varying degrees of

redundancy, while the organisms themselves are ini-

tially considered initially identical to each other and

share an equal risk of death. This assumption can be

justified in the following cases [2]:

1.  Population homogeneity may occur when a rig-

id genetic program determines the initial degree of re-

dundancy for each block (organ) in the organism. This

situation may occur in the course of programmed cell

death during early development. In this case, the vari-

ability in block redundancy is not entirely random,

and the homogeneous models are applicable because

the genetically programmed distribution of blocks ac-

cording to their redundancy can be approximated by

the Poisson or binomial distribution.

2.  Organisms may have nearly identical initial dis-

tributions of the number of blocks with different re-

dundancy levels, even when the redundancy formation

mechanism is random. This occurs when the number

of irreplaceable (vital) blocks is very large. Consequent-

ly, the population is practically homogeneous in terms

of the risk of each organism dying, despite potential

heterogeneity in the risk of failure of individual blocks.

The simplest model within this family of reliabil-

ity models is the series-parallel structure with distrib-

uted redundancy within the organism. This model as

outlined in [33] considers the distribution of subsys-

tems based on initially functional elements, described

by the Poisson law due to a high initial damage load.

In such systems, the failure rate can be initially ap-

proximated by the exponential (Gompertz) law, with

subsequent mortality leveling-off [33]. In systems with

lower damage levels, where initially functional ele-

ments follow a binomial distribution, the failure rate

experiences initial growth in line with the binomial

law [2].

It is worth noting that there have been allegations

of errors in this straightforward model [89]. However,

it is important to point out that the authors failed to

acknowledge that this model was designed for a homo-

EXPLORING PATTERNS OF HUMAN MORTALITY AND AGING 349

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

geneous population and improperly recommended the

use of a formula tailored to a heterogeneous popula-

tion [89].

One interesting conclusion from the model of ini-

tially homogeneous population is related to the oppor-

tunity of estimating the rate of vital elements loss in

human organism or true aging rate. This rate is ap-

proximately equal to the inverse of the species-spe-

cific lifespan [1/B, Bcomes from equation(5)]. It was

found that the estimated species-specific lifespan is

stable over time and is equal to 95-97 years [2, 31].

Thus, the estimated rate of loss of vital elements is ap-

proximately equal to 1% per year. It is interesting that

this rate is consistent with the empirical estimates of

annual cell loss in several neural tissues– 0.6-1.6% [78].

Empirically estimated rate of telomere loss in periph-

eral blood mononuclear cells in humans is somewhat

lower – 0.5% base pairs per year [90], but still is of

similar magnitude. It was found that the rate of telo-

mere loss is a species-specific trait and is proportional

to the lifespan of animal species [90].

Reliability model of heterogeneous population.

Accounting for population heterogeneity leads us to an-

other model, which provides an explanation for all the

basic mortality laws, even in the simplest case where

the organism comprises a single vital block with n el-

ements.

The model considers the simplest case when the

organism consists of a single vital block with n ele-

ments connected in parallel with q being the proba-

bility that an element is initially functional. Then the

probability of encountering an organism with i initial-

ly functional elements out of a total number n of ele-

ments is given by the binomial distribution law.

The final formula for failure rate in heteroge-

neous population,µ

(x), is (see[2] for more detail):

(x) = −

F′(x)

1 − F(x)

nqμe

−μx

(1 − qe

−μx

)

n−1

1 − (1 − qe

−μx

)

, (8)

≈ Cnqμ

(1 − q + qμx)

n−1

for x << 1/µ;

≈ μ for x>>1/µ,

where C is a normalizing factor.

Thus, the hazard rate of a heterogeneous popula-

tion at first grows with age according to the binomial

law of mortality, then asymptotically approaches an

upper limit µ:

(x) ≈ Cn(qμ)

[

1 − q

qμ

+ x

]

n−1

 = Cn(qμ)

+ x)

n−1

, (9)

for x << 1 / μ

(x) ≈ μ for x >> 1 / μ

where x

= (1 − q) / qμ, a parameter which is called the

initial virtual age of the population. This parameter

has the dimension of time, and corresponds to the age

by which an initially homogeneous population would

have accumulated as much damage as a real popula-

tion actually possesses at the initial moment in time

(at x  =  0). In particular, when q = 1, i.e., when all the

elements in each organism are functional at the out-

set, the initial virtual age of the population is zero and

the hazard rate of population grows as a power func-

tion of age (the Weibull law). However, when the pop-

ulation is not initially homogeneous (q  <  1), we arrive

at the already mentioned binomial law of mortality.

Thus, the heterogeneous population model described

here can also provide a theoretical justification for the

binomial law of mortality.

Heterogeneity model also addresses the compen-

sation effect of mortality. The compensation effect of

mortality is evident when variations in mortality re-

sult from differences in the number of elements in the

organism(n) between populations, while other param-

eters, including the true aging rate(µ), remain nearly

identical for all populations of the same species [2, 9].

Figure  4 illustrates Gompertz, binomial and Weibull

models fitting mortality of Norwegian women born

in 1920. In the case of binomial model, it is assumed

that every block (e.g., vital organ) has on average 50

vital elements and the initial virtual age (indicator

of initial damage load) estimated using nonlinear re-

gression method is equal to 370 years. Note that bi-

nomial model fits mortality data almost as well as

the Gompertz model with similar values of Akaike

Information Criterion of goodness of fit (-466 and

-462 respectively) while Weibull model significantly

underestimates mortality at younger ages (see Fig.  4).

It is important to note that extrinsic mortality below

the age of 60 years is significant. However, adjusting

for extrinsic mortality using the Gompertz–Makeham

model becomes challenging due to the variation of

Makeham term over time in cohort data. Thus, even

this simplified heterogeneity model is able to fit aging-

related mortality of humans.

In summary, the model of a heterogeneous pop-

ulation provides an explanation for the regularities

of organism mortality: the initial quasi-exponential

growth in the hazard rate, followed by mortality decel-

eration, along with the compensation effect of mortali-

ty as follows from the model formulas [2].

The heterogeneous population model shares basic

conclusions with the initially homogeneous model of

series- connected blocks with varying degrees of redun-

dancy [33]. However, these are two distinct models.

Inthe first model, individual mortality risk is uniform

across all organisms and grows exponentially with age.

In the second model, there initially exist n subpopu-

lations of living organisms with differing death risks

that increase as a power function rather than expo-

nentially with age.

The fact that these two different models yield near-

ly identical interpretations of certain mortality phe-

nomena offers some reason for optimism. For instance,

GAVRILOV, GAVRILOVA350

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

Fig. 4. Fitting mortality of Norwegian women born in 1920 by binomial law and competing Gompertz(a) and Weibull(b) laws.

Data source: Human Mortality Database (www.mortality.org).

the compensation effect of mortality is a common fea-

ture of all these models, occurring only when the rate

of irreversible age changes (true aging rate) remains

constant within a species. This treatment of the com-

pensation effect of mortality is not unique to the mod-

els discussed here, as it is consistent with other models

as well [34,  35,  91,  92]. The existence of various com-

peting models does not hinder the reliable and mean-

ingful interpretation of many mortality phenomena,

as multiple models can reach agreement on several

aspects.

Further developments of reliability models. We

would like to mention here several studies developing

reliability models further. One of the first reliability

models of aging was suggested in 1978 and was based

on the phenomenon of linear decline in the function or

cells over time [91]. This simple model could explain

the Gompertz law, CEM and late-life mortality plateau.

This approach was developed further by Milne who

created “nested binomial” model also explaining exist-

ing mortality regularities [93]. These models explained

mortality in terms of probability of dying and did not

consider organism’s structure.

Laird and Sherrat extended the approach of ap-

plying reliability theory to aging of biological systems

described above by considering three alternative types

of element/genetic architecture [94]. In addition to the

“Parallel” model, they also presented a “Series” model

and a third type of model– a “Cascade” model, analo-

gous to the multi-stage model of disease progression in

which irreparable damage occurs in a strict sequence.

They showed that redundancy leads to actuarial se-

nescence in the Parallel and Cascade models but not

in the Series model. Finally, the authors attempted to

add evolutionary dynamics to their reliability model

and found that a population’s equilibrium redundancy

is sensitive to the environmental conditions that pre-

vailed during its evolution, such as the rate of extrin-

sic mortality [94]. For some reason, the authors did not

consider a series-parallel reliability model while only

this model is close to the real organism’s structure

where vital organs are connected in series and spe-

cialized cells in each organ are connected in parallel.

Theauthors do not attempt to fit their models to real

data, so it is difficult to check the model value.

Another model is able to explain all three mor-

tality regularities mentioned above. This is a simple

mathematical model combining the heterogeneity of

populations with an assumption that the mortality in

each subpopulation grows exponentially with age [95].

It has been proven that this model is capable of repro-

ducing the entire mortality pattern in a human popula-

tion including the observed peculiarities at early- and

late-life intervals. The authors found that the evolution

of the model parameters validates the applicability of

the compensation law of mortality to each subpopula-

tion separately. This study has indicated that the pop-

ulation structure changes so that the population tends

to become more homogeneous over time[95].

In a separate study, the authors systematically

examined the practical utility of redundancy models

for investigating the mechanisms of aging in a quanti-

EXPLORING PATTERNS OF HUMAN MORTALITY AND AGING 351

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

tative manner [96]. The authors analyzed predictions

of the reliability model of homogeneous population de-

scribed in the previous section [33]. They showed that

redundancy models fit the data well and argue that this

is a strength of redundancy models over non-mecha-

nistic models because (i) when contrasting aging pat-

terns can be understood within the framework of a

single mechanistic model this indicates that the mod-

el may capture the essence of the aging process, and

(ii) redundancy parameter inference may teach us

something about the underlying mechanisms and can

as such be used to develop new hypotheses[96].

In summarizing it can be concluded that reliabil-

ity models of aging continue to be developed further

and are able to explain existing mortality regularities

outlined earlier.

CONCLUDING REMARKS

In this article, we examined various empirical

phenomena associated with the aging process and in-

troduced new insights into describing patterns of mor-

tality using reliability theory concept.

A substantial body of research on aging has yield-

ed a multitude of important and varied discoveries,

prompting the need for a unified theoretical frame-

work to consolidate this wealth of knowledge. Evolu-

tion-based theories of aging, grounded in the concept

of diminishing natural selection with advancing age,

demonstrate the practical applicability of broad theo-

retical principles in the field of aging research [97-99].

V. P. Skulachev has played a significant role in advanc-

ing evolutionary concepts. His contributions have been

instrumental in shaping contemporary perspectives

on aging, suggesting the presence of a specific pro-

gram or mechanism that influences the aging process.

Skulachev’s work has added depth to the exploration

of evolutionary theories, challenging traditional views

and paving the way for further investigations into the

molecular and genetic aspects of aging. He posited that

if aging is indeed programmed, it could be delayed, pre-

vented, or potentially reversed through interventions

that disrupt the execution of this program, similar to

our ability to intervene in cell death programs [5, 100].

His perspectives have generated numerous hypotheses

characterizing aging as a programmed process.

In this article, our aim was to offer a more com-

prehensive explanation of aging as a process of de-

terioration, extending beyond reproductive species,

through the application of the general systems fail-

ure theory known as reliability theory. This approach

aligns seamlessly with evolutionary theories, including

those related to programmed aging, when consider-

ing aging as a chronic phenoptosis. It became evident

that redundancy plays a central role in understanding

aging, particularly within a systemic framework. Sys-

tems that incorporate redundancy in essential compo-

nents inevitably undergo degradation (i.e., aging) over

time, even if constructed from elements that do not

age. Redundancy has a dual impact, enhancing dam-

age tolerance to reduce mortality and extend lifespan,

while also allowing the accumulation of damage, thus

giving rise to the aging phenomenon.

Systems with higher redundancy levels exhibit a

higher apparent aging rate or expression of aging, all

else being equal. This insight offers valuable perspec-

tive on the observation of negligible senescence in cer-

tain environments, revealing that some cases of negli-

gible senescence may be attributed to a lack of system

redundancies. For example, birds exhibit long lifespans

compared to their weight and low actuarial aging rate

(negligible senescence). However, they have relatively

high mortality at younger ages suggesting low level of

redundancy (cell reserves) [9]. On the other hand, com-

plex, redundant systems designed for greater durabili-

ty may exhibit more pronounced expressions of aging.

Throughout their lifespans, organisms deplete their

cells and reserve capacity, providing a potential ex-

planation for phenomena such as the CEM, mortality

convergence in older ages, late-life mortality deceler-

ation, leveling-off, and mortality plateaus. Organisms

appear to start their lives with an initial load of dam-

age [55, 59], and their lifespan and aging patterns can

be sensitive to early-life conditions that determine this

initial damage load during development. This concept

of early-life programming has potential implications

for interventions aimed at promoting health and lon-

gevity.

Aging is an intricate phenomenon, and adopting a

holistic approach, incorporating reliability theory, may

assist in analyzing, comprehending, and potentially

managing it. Today, gerontologists realize that a top-

down systemic approach is necessary to fully under-

stand and explain the phenomenon of aging [1].

Acknowledgments. We express our deepest grati-

tude to the late Professor Vladimir Petrovich Skulachev

(1935-2023), a distinguished Russian scientist who

served as our invaluable scientific mentor and advisor

from the 1970s onward. This article is part of a spe-

cial journal Issue dedicated to honoring his memory.

Professor Skulachev played a pivotal role in inspiring

the creation of our book, “The Biology of Life Span,”

referenced in this article, for which he served as the

scientific editor [2]. Furthermore, his encouragement

led to a significant collaborative study with him on the

variability of human life history traits [101].

Contributions. Leonid A. Gavrilov designed the

study, analyzed and interpreted results, and edited the

manuscript. Natalia S. Gavrilova conducted statistical

analyses and prepared the manuscript.

GAVRILOV, GAVRILOVA352

BIOCHEMISTRY (Moscow) Vol. 89 No. 2 2024

Funding. This work was partially supported by

the National Institutes of Health (project no. NIH

R21AG054849).

Ethics declarations. This work does not contain

any studies involving human and animal subjects.

The authors of this work declare that they have no

conflicts of interest.

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