Age Dynamics of Body Mass and Human Lifespan

Abstract

Age dynamics of human body mass (0–90 years) is described as a function of periodic damped oscillations. Common regularities are found for the age-dependent changes of body mass, of entropic equivalent (a parameter equivalent to the thermodynamic entropy), and intensity of natural mortality. It is shown that the mass reflects the thermodynamic state of biological system and is measured oppositely directed to the entropy value. The second, third, and fourth extremes of the mass age dynamics correspond to the mean (70–75 years), the commonly accepted maximal (100–110 years), and the maximal known (140–150 years) human lifespan, while the mass oscillations cease at the age associated with the maximal known lifespan—about 145 years.

Full text

Age Dynamics of Body Mass and Human Lifespan
Gerasimov I. G., Ignatov D. Yu.
Age dynamics of human body mass (0–90 years) is described as a function of periodic
damped oscillations. Common regularities are found in the age changes of mass, of
entropic equivalent — a parameter equivalent to thermodynamic entropy, and of intensity
of natural mortality. It is shown that the mass reflects the biological system
thermodynamic state and is measured oppositely directed to the entropy value. The
second, third, and fourth extremes of the mass age dynamics correspond to the mean (70–
75 years), the commonly accepted maximal (100–110 years), and the maximal known
(140–150 years) human lifespan, while the mass oscillations cease at the age associated
with the maximal known lifespan — about 145 years.
INTRODUCTION
For the last few years the number of works, in which biological objects are described in terms of non-
equilibrium thermodynamics, has increased [1–5]. Many authors [1–3, 5–8] are right to suggest that increase of
entropy at aging leads to death of a biological system. The character of age changes of the thermodynamic state
seems to determine taxonomic and individual lifespan [9] that, according to the data of the work [10], is
inversely proportional to radical from metabolic entropy. However, practical checking of these concepts leads to
difficulties associated with evaluation of the biosystem entropy.
Thus, so far there is no commonly accepted opinion about change of entropy during the organism
development [2, 11, 12]. Some authors [13] consider without serious argumentation that the biosystem entropy
increases continuously. Such position indicates directly the organism aging at once after birth, as the higher
entropy determines unanimously the more aged state [14]. However, ―if the organism development had initially
begun from aging, not only phylogenetic evolution, but also the life itself would have been impossible‖ [6].
Indeed, assessment of entropy during both development, and aging of the biosystem has shown [1] that entropy
decreases during the biosystem development, while increases during aging. The cited work, to assess the
biosystem state on the basis of statements of the theory of thermodynamics of non-equilibrium processes [14],
used the entropic equivalent that is indefinite function of entropy and represents the sum of squares of elements
of the transformed correlation matrix. It is to be thought that the entropy dynamics is similar to EE changes.
Probably, dynamics of the system entropy could be evaluated by measuring macroscopic thermodynamical
parameters, such as temperature, pressure, volume. Since change of entropy occurs, in particular, due to transfer
of substance [15], and the complete change of entropy in ideal systems is calculated per mass unit [16], the mass
might possibly act as the system macroscopic parameter, whose dynamics is associated implicitly with the
entropy dynamics. In the case of biological systems the mass measurement has no principal difficulties, and age
changes of this parameter were studied in many works.
It has been shown that the human and animal body mass in all studied populations increases after birth up to
the maximal value, and then decreases [17–21]. The equations are derived, which describe adequately the age
dynamics of mass (ADM) in the process of its increase [21] and allow characterizing, in some cases, the mass
changes in a wider time ranges of the animal life [15]. Study of ADM is of doubtless interest for estimation of
lifespan, as the correlation is traced between the biomass increase and duration of the existence of cells both in
vivo and in vitro. Direct correlations were revealed be-tween the mean rate of the mass increase at the
embryonic period and the maximal lifespan of mammals [23]; between the mass that animals gain at puberty
and the main lifespan of the species [24, 25]; between an increase of cell mass in vitro and the lifespan of the
culture [23]. Besides, a correlation between the mass and lifespan of human has been shown [22]. Thus, the
change of mass and lifespan could correlate between each other and could be associated with change of entropy.
To check these suggestions, we have analyzed ADM in comparison with the age entropy dynamics and
intensity of human mortality.

Fig. 1. Age dynamics of the body mass (1, 2), of body mass normalized to body height (3, 4) in men (1, 3) and women (2, 4). (I)
Approximation, (II) extrapolation from the equation (1); gaps between (I) and (II) curves indicate transfer to the right ordinate
scales: arrows indicate to which scale the curve belong. Abscissa: age (years), ordinates: the inner left and outer right — body
mass (kg), the outer left and inner right — body mass normalized to body height (kg/m).
MATERIALS AND METHODS
According to the data of work [26] on the human body mass and height in the range of 1–90 years, their mean
values for every decade, separately for men and women, as well as the corresponding values of the mass
standardized on body height (M/Н are calculated. Intensity of natural mortality (INM) was calculated by
subtracting its abiological component [27] from the total mortality intensity [13]. The data were processed using
software package Statistics for Windows. The confidence interval of the mean was calculated with the
confidence level of P = 0.95 (p < 0.05).
RESULTS AND DISCUSSION
The mean values of human mass and height de-pending on age are presented on Fig. 1. It is seen that values
of these parameters increase from the birth to 30 years, then decrease. Since the existence of the biological
system is coupled with a periodic change of physiological parameters relative to their equilibrium value [28] and
the amplitude of such oscillations decreases with age [13], we have described ADM as periodic function of
damping oscillations:
Y = A[e–tTcos(t + ) + 1],
(1)
where Y — mass (kg) or M/H (kg/m), A — the equilibrium value of Y at the age of t (years)  (kg for mass or
kg/m for M/H), T — increment of the damping (years–1), — frequency (rad/years), — phase (rad).
Coefficients of correlation between mass or M/H, on one hand, and t — on the other hand, which are calculated
from the equation (1), were in the range of r = 0.76–0.93 ( p < 0.05), and, hence, this equation describes ADM
adequately. The constants of the equation (1) calculated for mass and M/H separately for men and women as
well as their mean values are presented in Table 1. As seen from the Table, the values of each of the constants T,
, , as well as the oscillation period (, years) coincide within the limits of errors of the means. It is
wrong to find the mean value of the A constant due to its different dimensionality in the cases of mass and M/H.
Thus, oscillations of the mass and M/H differ exclusively by amplitude; therefore, only the dynamics of mass
will further be discussed.

Table 1. Constants of the equation (1)
Constant
Men
Women
The mean
body mass
M/H
body mass
M/H
A (kg for body mass or kg/m for M/H)
63.4
3.83
57.2
3.73
–
T (years–1)
0.0518
0.0884
0.0579
0.0949
0.073 ± 0.0343
 (rad/years)
0.0736
0.0912
0.0824
0.0865
0.083 ± 0.0120
 (years)
42.6
34.4
38.1
36.3
38 ± 5.6
 (rad)
3.18
2.76
3.10
2.76
3.0 ± 0.35
Note: M/H—the body mass normalized on body height.
As seen from Fig. 1, the mass increases after birth to the maximal value and then decreases, which is shown
in the age range of 0–70 years in different populations and is doubtless [17, 18, 20, 21, 29]. The decrease of this
parameter is determined both by a decrease of individual mass values at aging [30] and by elimination of
individuals with the higher mass from population [22]. We did not find in the available literature any data that
would allow estimating the mass changes after 70 years; however, it is also possible to determine the ADM
character at the elderly age and in long-livers, using the pro-posed model. Extrapolation of the mass value from
the equation (1) to the age of 90 years and more indicates an increment of mass at the age range of 70–105 years
(Fig. 1). With such change, the mean mass value will be higher in long-livers (older than 100 years) as
compared with elderly people, which is seen from the difference between the mass values in neighboring ADM
extremes (Table 2). The known facts confirm indirectly the made conclusion: the long-livers avoid the diets
promoting a decrease of mass [31] and have a higher fat content [32]. The tendency for a rise of mass and a
statistically significant increase of M/H at the age of 60 years, revealed in northern residents [29] also indicate a
possibility of a rise of mass at the older age. It is to be taken into account that the long-livers are a special elite
sample, whose representatives have physiological parameter values that are characteristic of the people younger
than 70 years [13, 33]. It is to be emphasized that such ―improvement‖ occurs not only due to death of people
with worse physiological parameters, but also due to a change of individual values of some of them, for
example, of arterial blood pressure [30]. In other words, the change with age both of individual values of mass
and of the means seems to have oscillatory pattern. Thus, the literature data indirectly confirm an increase of
mass after 70 years, which is calculated from equation (1).
Some authors think that dynamics of mass both at the postnatal and at the prenatal period can be described by
the same equation [21]. Also analysis of the fetal mass was not the task taken over in this work, it is to be noted
that the equation (1), unlike those proposed earlier, describes well the mass dynamics since the 6th month of
intrauterine development (Fig. 1).
To find the age, at which the mass is extreme (maximal or minimal), the first derivative of the equation (1)
was set equaled to zero:
Y ' = –A te–t T [cos(wt + f) + (w/T ) sin(wt + f)] = 0.
(2)
Since it is apparent that A te–tT ≠ 0, so
cos(wt + f) + (w/T) sin(wt + f) = 0,
(3)
hence,
tan(wt + f) = –(T / w)
(4)
or
t = (1/ w)[arctan(–T/w) – f] = n t,
(5)
where n = 0, 1, 2..., k—number of the extreme.

Table 2. Characteristic points of age dependences of the body mass, intensity of natural mortality (INM), entropic
equivalent (EE), and human lifespan
Age
The number and type
Change
at characteristic
of the characteristic point
of mass relative
point of age
for the parameter
Lifespan (years)
to the previous
dependence of the
body mass
EE
INM
extreme (kg)
body mass (years)
–2 ± 4.8
the 1st min
—
—
—
—
31
± 2.1
the 1st max
the 1st min
the 2nd min
—
—
69
± 8.3
the 2nd min
the 1st max
inflection
the mean, 70 [33]
–9.24 ± 1.0
107
± 13
the 2nd max
—
the 2nd max
commonly accepted maximal,
1.02 ± 0.12
100–110 [34, 35]
145
± 19
the 3rd min
—
—
maximal known, 40–160 [29, 36]
–0.11 ± 0.014
183
± 25
the 3rd max
—
—
—
0.015 ± 0.0072
Note: The age at characteristic points for INM and EE [1] was found out graphically. The mean lifespan in the USSR in 1970–
1972 is presented.
Fig. 2. Age dynamics of the body mass averaged for men and women (1 ) (black dots—extremes, p = 0.05), of entropy ((2 )—from
the work [1]), of the logarithm of intensity of natural mortality ((3 )—from the works [13, 27], and (3' )—from the work [38]).
Abscissa: age (years); ordinates: left—body mass (kg), right—the outer scale: the entropic equivalent (rel. units), the inner scale:
the logarithm of intensity of natural mortality (individuals/year).
On substituting values of T, , , and  (Table 1) into the equation (5), we obtained the age at the extremes of
mass separately for men and women. The age ranges and the corresponding characteristic points of age
dependences of mass (the means for men and women), EE, INM, and human lifespan are presented in Table 2;
these data indicate that the first minimum of mass in the calculation error range coincides both with the time of
oocyte fertilizations (–0.75 years), and with human birth time (0 years). The following three extremes of mass
correspond to the lifespan—the mean, the commonly accepted maximal, and the maximal known one (Table 2).
Thus, when approaching the age corresponding to one of the above-listed lifespans, the mass change rate tends
to zero and reaches it in the extreme point.
To reveal causes of the ADM oscillatory character, we compared the mass age dynamics with corresponding
changes of EE and INM. The mean mass (separately for men and women), EE [1], and INM depending on age
are presented in Fig. 2. It is seen that until 30 years (the approximate values of the age are given here and
further), there are an increase of mass and a decrease of EE. Later, from 30 to 70 years, EE increases, while the

mass de-creases, after which the mass increase corresponds to the EE decrease. The extreme points for EE are
within the limits of error for calculated extremes of mass (Fig. 2; Table 2). Thus, at the age interval between 10
and 80 years, the changes of mass and EE are antibathical (directed oppositely). Since both parameters (mass
and EE) characterize the thermodynamic state of biosystem [2], the common regularity in their changes is not
random. Whereas by definition, EE changes symbathically (unidirectionally) towards the biosystem entropy, the
mass dynamics is antibathical to the latter. We are to emphasize that these regularities are obtained for healthy
individuals and cannot be used at analysis of the pathological processes leading to a rise of mass, for example, at
obesity.
As to ADM and INM dynamics, INM decreases at the age intervals of 0–10 and 20–30 years on the
background of a rise of mass (see Fig. 2). Be-tween 30 and 70 years, an increase of INM corresponds to the
mass decrease, whereas at the interval of 70–105 years, a deceleration of the INM in-crease and its following
decrease are observed on the background of the mass increase. The characteristic points for INM, in particular
the inflection at the age about 70 years (see, for example, the plot in work [36]), are present within the limits of
mass extreme calculation error (see Fig. 2; Table 2). Hence, in the interval of 0–105 years an increase of INM
corresponds to a decrease of mass, while a decrease or growth deceleration of INM — to its in-crease. An
exception is the period of puberty (10–20 years) that seems to be accompanied by the processes leading to a rise
of INM. Apparently, INM depends on a relative number of people, whose entropy value is close to critical (not
compatible with life), and, hence, the common regularities of age changes of mass and INM are regular. Thus,
an increase of mass and, probably, a decrease of the biosystem entropy are accompanied by a decrease of INM
or deceleration of its rise, and, on the contrary, an increase of entropy leads to an increase of INM.
The second minimum of mass and, probably, the first global critical thermodynamic state of the biosystem
correspond to the age of the mean lifespan (see Fig. 2). The subsequent increase of mass (a decrease of entropy)
that lasts until the age of maximal lifespan seems to be due to intensive death of individuals with worse
physiological characteristics [35] and high entropy values. A decrease of mass after reaching maximal lifespan,
under condition that the correlation between changes of mass and INM would be preserved in the long-livers’
sample, would lead to further rise of INM, which can be observed on the mortality curve after 105 years (see
Fig. 2), i.e., to even more intensive death of long-livers. Such rapid decrease of the non-numerous long-livers’
sample allows explaining why this age is associated with the maximal lifespan.
By extrapolating the mass to the age over 90 years, we have found that amplitude of its oscillations be-comes
equal to zero within the limits of the calculation error (± 0.05 kg) approximately at 145 years (Table 2). It is to
be emphasizes that this age corresponds to the minimum of mass and, apparently, to the second global critical
thermodynamic state. Since the existence of any biological system is impossible without oscillatory processes,
the lack of them seems to mean cessation of functioning of the system as the biological one. In this case the age
of 140–150 years (the maximal known lifespan), indeed, is limiting for the human species.
Since the age changes both of mass [17–21] and of intensity of mortality [34] are similar in human and
various mammalian species, the revealed regularities: the oscillatory character of ADM, correspondence of the
mass extremes to the mean and maximal lifespans, as well as the correlation be-tween changes of mass and INM
— also seem to be characteristic of animals.
Thus, ADM is described by periodic function of damped oscillations. It is shown that the mass reflects the
thermodynamic state of biological sys-tem and changes oppositely directed to its entropy. Extremes of human
ADM correspond to the mean (70 years), the commonly accepted (100–110 years) and the maximal known
(140–160 years) lifespan, while oscillations of mass cease at the age associated with the maximal lifespan —
about 145 years.

REFERENCES
1. Gerasimov, I.G., Use of Entropic Characteristics for Assessment of the Organism Biological Age and
Functional State, Probl. Staren. Dolgolet., 1996, vol. 6, nos. 1–2, pp. 32–35.
2. Zotin, A.I., Termodinamicheskaya osnova reaktsii organizma na vneshnie i vnutrennie faktory
(Thermodinamical Basis of Organism Responses to External and Internal Factors), Moscow, 1988.
3. Gerasimov, I.G., Entropy of Biological Systems, Probl. Staren. Dolgolet., 1998, vol. 8, no. 2, pp. 39– 44.
4. Kalmykova, Z.I., Buldakov, L.A., and Tokarskaya, Z.V., Changes of Entropy of Dog Heart Mass during
Inhalation of Transuranium Radionucleotides, Med. Radiol., 1992, vol. 36, no. 5, pp. 27–30.
5. Riggs, J.Å., Aging and Mortality: Manifestations of Increasing Informational Entropy of the Genome?
Mech. Ageing Dev., 1993, vol. 66, pp. 249–256.
6. Arshavskii, I.A., Mechanism of Onto- and Gerontogenesis in Representatives of the Class of Mammals,
Usp. Fiziol. Nauk, 1998, vol. 29, no. 4, pp. 93–107.
7. Kheiflik, L., Mortality and Immortality on the Cellular Level, Biokhimiya, 1997, vol. 62, pp. 1380– 1393.
8. Riggs, J.Å., Carcinogenesis, Genetic Instability and Genomic Entropy: Insight Derived from Malignant
Brain Tumor Age Specific Mortality Rate Dynamics, J. Theor. Biol., 1994, vol. 170, pp. 331–338.
9. Hershey, D. and Lee, W., Excess Entropy Production as a Marker in Longevity Analysis, Age, 1988, no.
11, pp. 173–176.
10. Sacher, G.A., Maturation and Longevity in Relation to the Cranial Capacity in Hominid Evolution, An-
tecedent of Man and after. Primates: Functional Morphology and Evolution, 1975, pp. 417–441.
11. Nikolis, G. and Prigozhin, I., Samoorganizatsiya v neravnovesnykh sistemakh (Self-Organization in Non-
Equilibrium Systems), Moscow, 1979.
12. Presnov, E.V., Formalism of Non-Equilibrium Phenomenological Thermodynamics, Termodinamika
biologicheskikh protsessov (Thermodynamics of Biological Processes), Moscow, 1976, pp. 25–43.
13. Frolkis, V.V. and Muradyan, Kh.K., Starenie, evoly-utsiya i prodlenie zhizni (Aging, Evolution and
Prolongation of Life), Kiev, 1992.
14. Prigozhin, I., Ot sushchestvuyushchego k voznikayushchemu: Vremya i slozhnost’ v fizicheskikh nauka-kh
(From Existing to Originating: Time and Complexity in Physical Sciences), Moscow, 1985.
15. Zotin, A.I., Termodinamicheskii podkhod k problemam razvitiya i stareniya (Thermodinamical Approach
to the Problems of Development and Aging), Moscow, 1974.
16. de Gross, S. and Masur, P., Neravnovesnaya termodinamika (Non-Equilibrium Thermodynamics),
Moscow, 1964.
17. Morfologiya cheloveka (Human Morphology), Nikityuk, V.A. and Chtetsov, V.P., Eds., Moscow, 1990.
18. Snider, V.S., Cook, V.J., and Naseet, E.S., Chelovek. Mediko-biologicheskie dannye (Human. The Medico-
Biological Data (Publication no. 23 of the International Commission on Radiological Protection)),
Moscow, 1977.
19. Spencer, R.R., Organ/Body Weight Loss with Aging: Evidence for Coordinated Involution, Med. Hypoth.,
1996, vol. 46, no. 2, pp. 59–62.
20. Lamb, M., Biologiya stareniya (Biology of Aging), Moscow, 1980.
21. Zotina, R.S. and Zotin, A.I., Differential Equations of Developmental Biology, Termodinamika
biologicheskikh protsessov (Thermodynamics of Biological Processes), Moscow, 1976, pp. 83–92.
22. Samars, Ò.Ò. and Storms, L.H., Impact of Height and Weight on Life Span, Bull. World Health Organ.,
1992, vol. 70, pp. 259–267.
23. Prokhorov, L.Yu., Estimation of Relations of Some Parameters Depending on Lifespan of ―Permanently
Aging‖ Cultures and Maximal Lifespan of Mammals, Tsitologiya, 1999, vol. 41, no. 10, pp. 900– 913.
24. Colder, W.A., Body Size, Mortality, and Longevity, J. Theor. Biol., 1989, vol. 102, no. 3, pp. 135–144.
25. Penzes, L. and Csernai, L., On the Interrelationship between Lifespan, Constitution and Information, ZFA,
1980, vol. 35, pp. 285–296.
26. Vlasov, Yu.A. and Okuneva, G.N., Krovoobrashchenie i gazoobmen cheloveka: spravochnoe rukovodstvo
(Human Blood Circulation and Gas Exchange: A Reference Book), Novosibirsk, 1992.
27. Voitenko, V.P., Elementary Events of Aging and Their Systemic Generalization, Nadezhnost’ i
elementarnost’ sobytiya stareniya biologicheskikh ob”ektov (Reliability and Elementarity of Events in
Aging Processes of Biological Objects), Kiev, 1986, pp. 76–89.
28. Kassil’, G.N., Vnutrennyaya sreda organizma (Internal Medium of an Organism), Moscow, 1983.

29. Solonin, Yu.G. and Chermnykh, N.A., Physiologic Parameters of Northern Men at Aging, Klinich.
Gerontol., 1999, no. 4, pp. 59–63.
30. Frolkis, V.V., Starenie i biologicheskie vozmozhnosti organizma (Aging and Biological Capabilities of an
Organism), Moscow, 1975.
31. Johnson, Ì.A., Brown, M.À., and Poon, L.W., Nu-tritional Patterns of Centenarians, Int. Jagring Hum.
Dev., 1992, vol. 34, no. l, pp. 57–76.
32. Paolisso, G., Gambardella, A., and Balbi, V., Body Composition, Body Fat Distribution, and Resting
Metabolic Rate in Healthy Centenarians, Am. J. Clin. Nutr., 1995, vol. 62, pp. 746–750.
33. Kakhachidze, T.N., The State of Structure of Loose Connective Tissue in the process of Aging of Human
Organism (in Aged, Old Ages, and Long-Living Age), Cand. Sci. Dissertation, Tbilisi, 1970.
34. Gavrilov, L.A. and Gavrilova, N.S., Biologiya prodolzhitel’nosti zhizni (Biology of Lifespan), Moscow,
1991.
35. Lamb, M., Biologiya stareniya (Biology of Aging), Moscow, 1980.
36. Assessment of Contribution of the Main Classes of Death Causes to the Total Mortality by Kinetic Method,
Obshchie problemy biologii. Populatsionnaya gerontologiya (General Problems of Biology. Populational
Gerontology), Moscow, 1987, vol. 6, pp. 130–135.