ISSN 20790570, Advances in Gerontology, 2015, Vol. 5, No. 3, pp. 141–146. © Pleiades Publishing, Ltd., 2015. Original Russian Text © A.A. Viktorov, V.D. Gladkikh, A.I. Ksenofontov, E.E. Morozova, 2014, published in Uspekhi Gerontologii, 2014, Vol. 27, No. 1, pp. 37–43.
Forecasting Environmental Health Risks Based on the Kinetic Theory of Aging of Living Systems A. A. Viktorova, V. D. Gladkikhb, A. I. Ksenofontovc, and E. E. Morozovad a
Applied Research Center for Medical and Ecological Safety, ul. Smol’naya 21/1, Moscow, 125493 Russia b Research and Development Center Pharmprotection, Russian Federal Medical and Biological Agency, Vashutinskoe sh. 11, Khimki, Moscow oblast, 141402 Russia c National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409 Russia d Central Children’s Clinical Hospital, Russian Federal Medical and Biological Agency, ul. Moskvorech’e 20, Moscow, 115309 Russia email: a
[email protected];
[email protected];
[email protected];
[email protected] Abstract—A method of iterative congruence for searching for the parameters of a kinetic mathematical model for aging of living systems using medical statistics data is developed. Its possibilities for describing risk functions of mortality and life expectancy for humans and animals depending on environmental factors are illustrated. The concept of forecasting environmental risks, i.e., risks to the population’s health from ecolog ical factors, is formulated. Keywords: aging, mathematical model, environmental health risk, iterative congruence
“It is difficult to avoid two things: dullness if you confine yourself to your specialty and inconsistency if you go beyond it.” Johann Wolfgang von Goethe DOI: 10.1134/S2079057015030133
One very important research and applied problem is a quantitative methodical approach to forecasting a population’s health depending on the effects of adverse environmental (ENV) factors [6, 10]. According to the opinion of Prof. B.A. Kurlyand skii [6], the development of domestic preventive toxi cology has been significantly “…influenced by state ideology, which denies the need to assess risks and forecast unavoidable losses. In the Soviet Union and, later, in Russia, the priority direction in the area of dangerous chemical compounds for many decades has been and still remains hygienic regulation, currently supplemented with ecological norms. The hygienic norms for chemical substances have undoubtedly played an important role in chemical safety in this country; however, this has led to onesidedness, like in any hypertrophied trend, and in several cases, to neg ative consequences.” Testing and adoption of an international method ology for assessing population health risks caused by chemical substances started in Russia in 1994 and is currently in progress [9]. In contrast, the authors of this work state that “… [the risks] should not be regarded as a realistic prediction of the morbidity or mortality rates in a particular exposed population. Taking this into account, the forecasted risk values in
the majority of cases should be used as a tool for com paring potential health damage under different condi tions of exposure to adverse environmental factors” [9]. Without a detailed discussion of the reasons why this methodology is unsuitable in practice for quanti tative forecasting of actually observed population health risks, let us note its three major flaws. (1) The accepted model of a biological system (BS) as a “dose–response” system is a black box–type system, implying that any processes within this box that main tain BS life activity are not considered. (2) This approach requires that dose–response toxicological information is obtained using laboratory animals and is then extrapolated to humans to simulate their bio logical response to external impacts. (3) The most important disadvantage is that the necessary toxico logical data for the overwhelming majority of sub stances in the ENV are absent [9]. All this interferes with consideration and use of the methodology as a model for quantitative estimation of population health risks, leaving it for risk factor assessment only. Meanhwile, statistical analysis, probability theory, and mathematical simulation of a population’s health, mortality, and pathology risks have been used in ger ontology, demography, and sociology since the early 19th century [1, 5, 7, 8, 15, 16]; as well, biophysical processes in the BS that determine aging have been
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studied. Unlike the results of toxicological studies [9], these data come from direct observations of the dynamics of aging, pathologies, and mortality of pop ulation cohorts living in particular localities and under particular social conditions, which reflect the actual factors of ENV impact on the biological response of an organism during its actual lifetime.
Lifespan is the commonly accepted parameter of human health, detectable only after the death of an individual. However, knowing the parameters of kinetic equation (1), it is possible to forecast the lifespan of a BS, τmb, by solving the following equation: τ mb
MATERIALS AND METHODS
mr
– κD ( 1 – D )
m em
exp ⎛ – τ⎞ , ⎝ τ r⎠
(2)
0
The kinetic theory of aging of living systems was proposed by Viktorov et al. [2, 3]. This theory is based on two physical principles—adaptation of a BS to the ENV from birth to death and BS structuring–destruc turing as a way of its dynamic existence. The processes of BS structuring–destructuring, which determine the “gross kinetics” of BS aging, are described using the kinetics of autocatalytic chemical reactions [17], the rate of which depends on both the biological parame ters of the system and ENV parameters (physical, chemical, biological, and social), which in the general case change with time. The major factors that determine BS aging are genetic, ENV impact, and recombination of broken BS links. The effect of the ENV on a BS is taken into account by the parameter μ, reflecting the ENV impact in the BS destruction kinetics starting from birth: m ∂D μ(τ) = ( 1 – D ) d exp ∂τ 1 – ϑD
⎛ dD⎞
dτ = 1, ∫ ⎝ dτ ⎠
(1)
where D is the degree of BS destruction (risk), 0 ≤ D ≤ 1; K K κ = r = 0r, the ratio of the reaction rate constants K d K 0d for recombination of broken links and their destruc tion; μ, dimensionless parameters reflecting the ENV K impact; Kd,r = K0d,0rexp(–Eg); t and τ = ⎛ d⎞ t astro ⎝ n0 ⎠ nomical and biological (dimensionless) times; n0, ini tial concentration of functional links in the BS, i.e., the initial amount of physical links per BS unit vol ume, which determines the degree of BS aging; Eg = Ea , energy dimensionless parameter determining the kT level of innate resistance; Ea, link activation energy; k, Boltzmann’s constant; T, absolute temperature of the BS; md,r > 0, orders of reactions of link destruction and recombination; mem > 0, power coefficient of the term that takes into account depletion of BS adaptive reserves with an increase in the degree of aging, D. Without prejudice of generality, let us consider τr = ∞, since here we consider only cases of μ > μcr.
which allows an “astronomical” lifespan to be calcu Kd lated, tmb = τ mb . The value of the denominator in n0 the last equation can be found as the constant for a considered population according to the known aver age lifespan values in astronomical and biological (dimensionless) times. Kinetic equation (1) makes it possible to obtain the risk functions accepted in gerontology [1, 5], namely, τ dD D(τ) = mb ⎛ ⎞ dτ , mortality risk (or the cumulative 0 ⎝ dτ ⎠ distribution function for mortality); S(τ) = (1 – D(τ)), the probability of living for time period τ (or cumula dD , probability density tive survivorship function); dτ dD/dτ for mortality risk; and , mortality intensity 1 – D(τ) function (or risk function). Computation of these functions requires initial experimental statistical data in the maximum observation time range for at least of one of the mentioned functions. To search for parameters of kinetic model (1)—μ, κ, ϑ, md, mr, and mem—it is necessary to use numerical approaches, since analytic solutions in the general case are unfeasible. Taking into account the limited volume of initial experimental data, the search for model parameters is ambiguous. In order to select the most probable values, an iterative congruence method was elaboratedm which consists of three stages. The first stage is a preliminary assessment of model param eters based on the known experimental data and form of kinetic equations. The experimental data are most frequently represented as statistical distributions of mortality intensity, mortality risk, or survivorship functions. Analysis of Eq. (1) for the mortality inten sity in the adopted dimensionless coordinates allows us to write the following equations: μ = lnI0 at τ = 0; (3)
∫
μ ⎞ ϑ = ⎛ 1 – at τ = τ mb, m d = 1, m em > 0; (4) ⎝ ln I m⎠ μ ⎞ – ( 1 – D ) d exp ⎛ I ⎝ 1 – ϑD⎠ τ κ = , m m –1 D r ( 1 – D ) em m –1
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where I0, Iτ, and Im are the initial (τ = 0), current (τ), and maximum (τ = τmb) values of mortality intensity. The orders of biochemical reactions—md, mr, and mem—are selected within the range of their most likely changes from 0 to 3 [17]. The parameter values calcu lated using Eqs. (3)–(5) from the experimental data are their first approximation. The second stage is construction of nomograms for kinetic functions that vary the model parameters in the region of their most probable values (Fig. 1) with the help of a specially developed program. The third stage is identification of the experimental kinetic curves by the iterative congruence method, i.e., according to the best fit of the superposed experi mental and calculated functions and selection of parameters of the corresponding calculated variants as the final approximation. Once the model parameters are selected, the criti cal values μcr and Dcr are found according to Semenov [14] using the following conditions: dD = 0 and (6) dτ d ( dD/dτ) = 0. (7) dD Considering the most frequent case when md = 1 and mem ≥ 1, conditions (6) and (7) make it possible to obtain analytical equations allowing the critical values for parameters μcr and Dcr to be found: μ cr = ( 1 – ϑD cr ) ln ( κD cr ( 1 – D cr ) m
m em – 1
) and
(8)
m
ϑ ( 1 – D cr ) ln ( κD crr ( 1 – D cr ) em – 1 ) 1 – ϑD cr (9) 1–D – m r cr + m em D cr
= 1.
RESULTS AND DISCUSSION Figures 2–4 compare the computations and exper iments. Analysis shows that the proposed iterative congruence method successfully finds the parameters of the kinetic mathematical model of aging for humans and animals and forecasts the mortality risks and lifespan depending on the factors of ENV impact. The risk functions are the most sensitive to changes in parameter μ (see Fig. 1, curve 3). The parameter μ(τ) = μ ( τ ) , where μi(τ), in the general case i i reflects the effects of the ith influencing factors of dif ferent natures, intensities, and durations depending on time τ, to each of which the BS may display differ ent sensitivities [2]. In particular, the dependence of parameter μ of the kinetics of mortality risk of mice chronically exposed to radiation (see Fig. 3) on the radiation intensity is linear in the range of dose rates P
∑
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dD/dτ log (1 – D) 1.6 5 4 3 2
1.4
1
1.2 1.0 0.8 0.6 0.4 0.2 0 –0.2 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
09
1.0 τ
dD/dτ Fig. 1. Nomograms of mortality intensity for dif 1 – D(τ) ferent values of parameter μ in the kinetic equation of aging (the ordinate shows the common logarithm of mor μi ⎞ ∂D – 24D(1 – tality intensity), ⎛ ⎞ = (1 – D) exp ⎛ ⎝ ∂τ ⎠ i ⎝ 1 – 0.5D⎠ D)2. Changes in parameter μ: curve 1, μi = 1.55; curve 2, μi = 1.57; curve 3, μi = 1.60; curve 4, μi = 1.62; and curve 5, μi = 1.65.
of 12–56 cGy/day (μ = 5 + 8.2 × 10–3 P) with an accu racy of at least ±0.08% and μ does not depend on the age. However, in the case of acute radiation exposure (see Fig. 4), this parameter changes during an entire life: after the “load” is removed, the organism rehabil itates. The kinetic parameters in the model for popu lation aging are different for individual regions of the country (see Fig. 2). This suggests, and medical statis tics confirm, that each BS location and pattern of ENV impact create biological responses characteristic of this particular BS and that identification of the influencing factors can be a major task in searching for risk factors. The kinetic theory of aging of living systems intro duces a quantitative functional link between the influ encing ENV factors and the characteristics of a popu lation’s health as the equation for the gross kinetics of the aging process. The dependence of BS risk func tions on parameter μ allows the level of ENV impact μe to be determined; adherence to this level will ensure acceptable ecological risk for the population’s health. It is reasonable to specify the value of ENV impact with respect to μcr using the criticality factor κs = μe/μcr: the risk of ecological impact at κs = 1 will be
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dD/dτ 1–D 1
0.1
5 4
2
3
0.1
2 1
1
0.01
3
0.01
1 × 10–3
10–3
1 × 10–4 0 10–4 0
10
20
30
40
50
60
70 t, years
Fig. 2. Comparison of mortality intensity calculated according to the kinetic theory and actual data on mortal ity of the male population in Russia, Japan, and Sweden [15]. Dots show the actual values of mortality intensity for Russia and Japan. Curve 1 is an approximation of the sta tistical data for the mortality intensity of the Russian male population (data of 1994) by a piecewise smooth function; the kinetic equation with the parameters selected using the iterative congruence method is 3 2.6 ⎞ ∂D ⎛ = ( 1 – D ) exp – κD ( 1 – D ) ; the values of ⎝ 1 – ϑD⎠ ∂τ parameters κ and ϑ were changed with time in the range of 90.9 ≤ κ ≤ 121.4 and 0.06 ≤ ϑ ≤ 0.4. Curve 2 corresponds to the mortality intensity of the Japanese male population in 1991; the kinetic equation is 3 ∂D 1.4 ⎛ ⎞ = ( 1 – D ) exp – 31.4D ( 1 – D ) . Curve 3 is ⎝ 1 – 0.6D⎠ ∂τ the plot of mortality intensity calculated using the congru ence method based on the statistical medical data for the mortality of Sweden’s male population in 1900; the kinetic equation is 3 1.61 ∂D = ( 1 – D ) exp ⎛ ⎞ – 41.1D ( 1 – D ) . The ⎝ 1 – 0.49D⎠ ∂τ mortality intensity values shown on the ordinate axis are decreased a thousandfold.
minimal, and health risks increase with the criticality factor. The health risk values at a constant intensity of negative ENV impact during the life of a BS are not constant; the risk increases with the age of the BS. The possibility of numerically estimating ENV health risks to the population living in a certain area and exposed to an integrated effect of ENV factors of different nature with the help of integrated parameter μe results from the fact that the influencing factors are likely to induce similar biochemical reactions in the
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600
800
1000 1200 t, days
Fig. 3. Dependence of mortality intensity of laboratory mice on time of chronic exposure to radiation. Parameters of chronic exposure to radiation of mice selected by iterative congruence method Curve no.
Dose rate, cGy/day
μ
ϑ
κ
tmb, days
τmb
1 2 3 4 5
0.03 12 24 32 56
4.82 5.12 5.24 5.26 5.45
0.52 0.54 0.56 0.57 0.8
420 420 420 420 420
1200 800 800 600 400
6 × 10–3 4 × 10–3 4 × 10–3 3 × 10–3 2 × 10–3
Sensitivity of model to changes in parameters μ, ϑ, and κ, respectively, for dose rates P of 0.03, 12, and 24 cGy/day is shown by dashed, doted, and dotdash lines. The kinetic equation is dD/dτ = (1 – D)exp(μ/(1 – ϑD)) – κD(1 – D)2. The values of mortality intensity shown in the ordinate are decreased by a factor of 106. Changes in the parameters μ, ϑ, and κ Curve no. Dose rate, cGy/day 1 2 3
0.03 12 24
Change in parameter δμ = 1%, Δμ = ±0.051 δϑ = 5%, Δϑ = ±0.028 δκ = 10%, Δκ = ±42
BS and its systems (Fig. 5), and, as a consequence, similar integral gross kinetics of aging. In particular, this may explain why different kinds of chemical sub stances and their diversity, as a rule, induce, in the case of longterm exposure, similar nonspecific disease symptoms and syndromes, considerably more limited (by many orders of magnitude) in number [9]. Thus, it is likely that when computing the integrated μe, it is necessary to take into account its constituent μi with weight coefficients gi that reflect different biological ADVANCES IN GERONTOLOGY
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Mortality rate 0.7 3 2 1
0.6 0.5
1 × 10–3 0.4 0.3 1 × 10–4
0.2 0.1
1 × 10–5 0
200
400
600
800
1000 t, days
0
1 2 3
0 3 5
ϑ
κ
5.2 5.285 → 5.276 5.90 → 5.57
0.52 0.52 0.52
100 100 100
∑
The components of generalized environmental impact parameter μi, in a first approximation can be divided into components corresponding to the nature of influencing factors, namely, physical, chemical, biological, and social. The intensity of external impact that determines the first three factors can be physically measured; i.e., they can be expressed in quantitatively. Each of these factors reflects a biological response of the BS when it adapts to the ENV possessing the abovelisted influencing characteristics. Unlike the first three factors of ENV impact, the fourth factor of Vol. 5
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50
60
70
80 90 Age, years
social origin is the most intricate for quantitative description, whereas the consequences of its impact may be decisive for the health of both a person and the overall population.
efficiencies of these influencing factors, similar to computating the total biological effect of radiation of different nature and energy [4]: μe = g μ . How i i i ever, this assumption requires further comprehensive studies and reliable confirmation.
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Fig. 5. Mortality rate of male population of Tula in 1996 for different nosological entities normalized to the total mor tality for each nosological entity depending on age (statis tical data for mortality rates according to [7]).
Model parameters for acute radiation exposure of mice μ
20
Total for all nosological entities Cardiovascular diseases Respiratory diseases Neoplasms
Fig. 4. Dependence of mortality intensity of laboratory mice on times after acute radiation exposure. The kinetic equation with aging parameters selected by the iterative congruence method for the control mouse group (curve 1) 2 5.2 ∂D is ⎛ ⎞ = ( 1 – D ) exp ⎛ ⎞ – 100D ( 1 – D ) . For ⎝ 1 – 0.52D⎠ ⎝ ∂τ ⎠ cases of acute radiation exposure, μ was approximated with a piecewise smooth function; for a dose of 3 Gy (curve 2), the external influencing parameter μ was decreased from 5.285 to 5.276 every 100 days and for a dose of 5 Gy (curve 3), from 5.90 to 5.57 every 50 days. The values of mortality intensity in the ordinate are decreased by a factor of 107.
Curve no. Dose, Gy
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In the terminology of [11–13, 15], we refer to the parameter of the biological response of a BS to the social state as psychoemotional stress. Study of the human psychoemotional state falls into the scope of social psychology. Psychoemotional stress as the degree of depth of such a state has deep social roots and no less important consequences for human health and mortality risk. This is especially illustrative by the drastic deterioration of the demographic situation in Russia, increase in population mortality risk, and reduction in lifespan after the collapse of the Soviet Union [15]. At that time, in the 1990s, the processes associated with the unjust and, in many respects, criminal privatization of national wealth commenced in this country, as well as the yetunheardof differen tiation of society according to income, impoverish ment of the majority of the population, inequality in the actual legal position of different social cohorts in society, and cultural, industrial, national, and political crises. For the overwhelming majority of the impover ished population, this has led to high psychoemotional stress, presenting as constant anxiety, absolute eco nomic insecurity, meaninglessness of wasted life, deep
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disappointment, apathy, and depression, causing stress and aggression [11–13, 15]. This particular factor led to the demographic crisis of that time and drastically reduced lifespan, since other factors at least did not increase because of the industrial crisis. Such a demographic state of the Rus sian population is first and foremost determined by personal income, i.e., by impoverishment of the majority of the population. Long ago, Karl Marx wrote in this regard that the last cause of all real crises is always poverty and limited expenditures of the masses. A quantitative assessment of the psychoemotional stress parameter and the range of potential changes in its numerical values is a future task solvable only with the involvement of psychological research data.
8. 9.
10.
11.
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Translated by G. Chirikova
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