A previously unrecognized peculiarity of late-life human mortality kinetics?

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A previously unrecognized peculiarity of late-life human mortality kinetics?

https://doi.org/10.21203/rs.3.rs-3316407/v1

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The late-life mortality kinetics in several species features the deceleration of mortality rate increase with age, which in humans starts at ca. 90 years. Being analyzed based on the Gompertz law µ(t)=µ₀*e^gt with its inbuilt linearity of the dependency of lnµ(t) on t, this feature is interpreted as reflecting the heterogeneity of a population where the frailer subjects die out earlier thus increasing the proportion of those whose dying out is slower and leading to a decrease in the demographic (actuarial) rate of aging. Using Human Mortality Database data related to France, Sweden and Japan in the periods 2020 and 1920 and to the cohorts born in 1920, it is shown in the present communication that the putative deceleration of the demographic (actuarial) rate of aging in humans is preceded by an acceleration, which starts at about 65 years, reaches a peak, which at about 85 years is ca. 30% higher than the pre-acceleration level, and decelerates thereafter until 90 years at least, when the pre-acceleration level is reached. This peculiarity cannot be explained with the predominant dying out of frailer subjects at earlier ages. Its plausible explanation may be the real acceleration of the (averaged) biological aging in humans at ages above 65-70 years, which conspicuously coincide with retirement. The biological implications of this finding are discussed in terms of the generalized Gomperts-Makeham law µ(t)=C(t)+µ₀e^f(t). In particular, the deceleration of mortality increase in the oldest-old subjects may be contributed to by the actual biological deceleration of aging.

actuarial aging rate

Gompertz law

biological aging

mortality plateau

retirement

Since humans aged above ca. 90 years have become numerous enough to matter at all and to make the analysis of mortality kinetics in such populations reasonably reliable, the apparent deceleration of the age-associated increase in death rate among the oldest old has been being attracting relentless attention, e.g. (Vaupel, Carey et al. 1998, Weitz and Fraser 2001, Bebbington, Green et al. 2012, Lai 2012, Barbi, Lagona et al. 2018, Newman 2018, Gavrilov and Gavrilova 2019, Dang, Camarda et al. 2023). The phenomenon, which is often associated with the so-called late-life mortality plateaus, may be explained assuming that either (i) the age-associated increases in the individual frailty decelerate at older ages and (ii) progressive age-associated changes in the composition of a population are such that those who are frailer initially or/and become frailer because of aging more rapidly die out earlier, which results in an increase in the proportion of those who die out at a slower rate (Vaupel, Carey et al. 1998, Avraam, Arnold et al. 2016, Németh and Missov 2018, Böhnstedt and Gampe 2019, Salinari and De Santis 2019). Logically, these two explanations, physiological and compositional (demographic), are not mutually exclusive. However, the latter one predominates currently. One reason for this may be that the age-associated increase in the mortality rate of adults is commonly believed to conform to the Gompertz law (GL) (Gompertz 1825, Gavrilov and Gavrilova 1991, Olshansky and Carnes 1997, Golubev 2009 and 2019, Bronikowski and Flatt 2010, Wrigley-Field 2014, Kirkwood 2015):

$${\mu }\left(t\right)={\mu }_{0}{\text{e}}^{\gamma \times t}$$

which may be written as

$$\text{ln}\left[\mu \left(t\right)\right]={\text{l}\text{n}\mu }_{0}+\gamma \times t$$

where γ is interpreted as the rate of aging and the associated increase in frailty, i.e. in the inability to withstand the causes of death. In a lnµ -vs-age plot, γ is the tangent of the slope of a straight line, which suggests that the age-associated increase in frailty occurs at a constant velocity. In studies where different species or genetic modifications of a species are compared, greater slopes of the linear approximations of lnµ(t)-vs-t datapoint series are interpreted as indicating higher rates of aging (Finch and Pike 1996, de Magalhaes, Cabral et al. 2005).

Because the linearity of the increase in frailty with increasing age is inbuilt in GL, there remain two options available to explain systematic deviations of empirical data on mortality, survivorship, and lifespan distribution from the patterns consistent with GL. One of the options is to introduce additional terms into GL. The simplest embodiment of such option implies that the single additive term does not depend on age. The result is known as the Gompertz-Makeham law (GML) (Makeham 1860, Gavrilov and Gavrilova 1991, Golubev 2004, 2019).

$${\mu }\left(t\right)=C+{\mu }_{0}{\text{e}}^{\gamma \times t}$$

The term C in GML may be interpreted as capturing how much the causes of death that produce the age-independent (accidental or extrinsic) mortality contribute to the total death rate.

Importantly, C makes the lnµ-vs-age plots deviate from linearity so that the apparent rate of aging increases with increasing age, as shown in Fig. 1, which presents the results of numerical modeling, using Mathcad 13, at the values of the GMM parameters g = 0.1 and µ₀ = 0.00001 and variable C within ranges typical for humans.

Figure 1. Dependencies of mortality rate (upper right panel) and aging rate (lower left panel) on age at different levels of the Makeham parameter C, which is assumed to be either constant or decreasing linearly with increasing age. The dashed lines in the upper panel show the linear approximations of the plots constructed at C > 0.

It is easy to see in Fig. 1 that approximating the plots related to C > 0 with straight lines, i.e., assuming that the differences of C from zero are negligible, will make the slopes of the resulting lines decrease with increasing C making the impression that aging rates are different in the three cases where actually only C are different, that is the differences relate actually not to the properties of the organisms in the modeled populations but to the conditions they are in. Moreover, even a very small C, which produces seemingly negligible effect on the respective lnµ-vs-age plot, can distort the apparent dependence of aging rate on age quite markedly. When there is no C at work, that is C = 0, the apparent aging rate does not change with increasing age. The effect of C makes the apparent rate of aging significantly underestimated at lower ages, its estimates increasing only by middle ages up to the values deduced according to GMM.

Ignoring C in analyzing mortality or survival data is a source of much confusion (Golubev 2004) because it may lead to the erratic conclusion that the mean (median) lifespan has changed because of a change in the rate of aging whereas actually it has changed mainly because of changes in the age-independent (extrinsic, accidental) mortality, which is captured by C. Moreover, changes in C are associated with negative correlations between the estimates of g and µ₀ made assuming that C is zero. Thus, C is an important factor of the so-called Strehler-Mildvan correlation (Gavrilov and Gavrilova 1991). However, such correlations will result from whatever deviations of data from conformance to a model, including purely accidental or stochastic deviations, when data are treated using not only GM or GML (Tarkhov, Menshikov et al. 2017) but any nonlinear two- or three-parametric model to which the data do not conform exactly (Golubev, Panchenko et al. 2018).

These caveats taken into account, it may be shown that C is what mainly decreased over the last century in developed countries, whereas aging rate g and the initial mortality µ₀ remained relatively constant, and their negatively correlated deviations from constancy are attributable to data-treatment artifacts (Golubev 2012, Golubev 2019). For example, the difference between the plots showing lnµ vs age dependencies in French women in periods 1920 and 2020 (Fig. 1) is caused by a higher contribution of C to mortality in 1920. The contribution is more significant at younger ages because the age-dependent mortality becomes increasingly dominant at later ages. Importantly, in the plot related to the 1920 cohort, the downward slope at ages from 20 to 40 years may be attributable to the historical decrease in C.

The other option to deal with the deviations of empirical data from GL and GML, upon the assumption that g is constant, is to assume that the values of the parameter µ₀ or γ, or both are not the same for all subjects in a population and feature(s) some form of distribution. Clearly, any real population or cohort is heterogeneous, and its heterogeneity likely involves the properties that influence frailty and its age-associated changes. It is less clear how the resulting heterogeneity must be reflected in the parameters of mortality and survival as they depend on age. The most popular assumption is that µ₀ is gamma-distributed (Vaupel, Carey et al. 1998, Missov and Németh 2016, Németh and Missov 2018, Böhnstedt and Gampe 2019, Salinari and De Santis 2019). The resulting relationships are known as gamma-Gompertz and gamma-Gompertz-Makeham models. Both produce age-associated decreases in the slopes of lnµ-vs-age plots. The ages beyond which the decreases become noticeable depend on the numerical values of the parameters of a gamma-distribution.

Preliminaries: a generalized Gompertz-Makeham law

Strictly linear changes, including physiological decline during aging as reflected in the dependence of lnµ on age, are the least biologically plausible and thus are always only more or less acceptable, in practical terms, approximations to any real state of things. Based on this and more specific considerations derived from the possible constrains imposed by physics and chemistry on the primordial living objects upon their emergence from the physicochemical world, a generalized Gompertz-Makeham law (GGML) has been suggested (Golubev 2009, 2019):

$$\mu \left(t\right)=C\left(t\right)+{{\mu }}_{\text{o}}{\text{e}}^{\text{f}\left(t\right)}$$

From this standpoint, the canonical GL and GML are not laws but models, which may be derived from GGML upon the assumptions that f(t) = γt and C = 0 (GM) or f(t) = γt and C = const > 0 (GMM).

One implication of the GGML is that C is not just a constant added to GM to improve its fit to data, but is an essential term that captures indispensable aspects of the reality and thus has a substantial meaning and can never be actually equal to zero. Yet, a smaller C compared with the Gompertzian term of GMM will make the relationship between lnµ and f(t) closer to linearity (see Fig. 1) thus allowing more accurate inferences about the real age-dependent trajectory of functional decline, provided the heterogeneity of a population under examination may be ignored. Importantly, the relative contribution of C to the total mortality and, thus, any possible C-caused bias decreases with increasing age because of the increasing age-associated mortality. However, analyzing later ages is associated with greater noise because of smaller sample sizes. It is only in relatively recent times that the late-age human populations became sufficiently large and least affected by the age-independent mortality captured by C.

In the present attempt to derive, based on the GGML, the late-age patterns of the averaged individual physiological decline from the patterns of the late-age demographic (actuarial) mortality, Human Mortality Database (HMD) data related to France, Japan, and Sweden are used. This choice is based on that these populations are large and relatively homogenous, the estimates of C there are among the smallest in the world and, at the same time, the populations there markedly differ one from another in a number of other aspects. The period data relate to 2020, the last year before the COVID-19 pandemic, and to 1920, the starting year of the cohorts that reached the age of 100 years in 2020. The cohort data are used assuming that the counts of emigrants and immigrants and of the natives who are still alive are too small to care of licensing.

Procedures of HMD data treatment were initially such that they could avoid any data pretreatment biases and, therefore, were deliberately based on the raw data and were straightforward as much as possible, just primitive, not to say stupid. Instead of using the age-specific mortality rates precalculated in the HMD, the raw counts of those who died in a one-year age interval starting at an age A were divided by the counts of those at risk of death in the same age interval. The logarithm of the resulting ratio was subtracted from the logarithm obtained in the same way for the age interval starting at the age A + 1. Trends were derived by using the 5-years running average option available in MS Excel. The result obtained, for example, using raw HMD data on women in Japan in 2020 was not different from that obtained using µ data precalculated in HMD (Fig. 3). Alternatively, lnµ-vs-age plots were smoothed using the LOESS procedure implemented in the TableCurve2D (TC2D) software tool (fully functional trial version is available at https://systatsoftware.com/tablecurve2d/download-tablecurve2d-software/), and the smooth functions yielded in this way were differentiated using the same tool. The result of this procedure is also shown in Fig. 3. The trends obtained using all three procedures tested were unidirectional at an acceptable resolution. Therefore, only the last procedure was adopted for the subsequent data treatment.

In Fig. 3, the negative values of the rate of aging g in the period from infancy to teen ages correspond to the fact that development increases physiological capacities and thus the ability to resist the causes of death. The narrow peak of the apparent rate of aging before about 20 years of age corresponds to the rapid changes in the contribution of C to mortality in adolescents. After ca. 25 years, the apparent rate of aging increases until ca. 40 years, possibly because of the effect of C shown in Fig. 2. After ca. 65 years, the apparent rate of aging starts to accelerate despite that the contribution of C to mortality at those ages becomes least significant. The acceleration continues until ca. 90 years. Thereafter, a deceleration takes place.

Such patterns derived from data related to three different countries are shown in more detail in Fig. 4, where they are compared with those related to men and with the rates of aging (g) calculated according to GMM, i.e., upon the assumption that g does not depend on age once aging starts. To estimate g, µ vs. age data in the age interval 25 to 80 years obtained from HMD were loaded to TC2D and approximated with its inbuilt function y(x) = p + re^− x/s where -s^− 1 is analogous to g. It is more common to use the starting age of about 40 years or higher for estimating g according to GM or GMM. This choice is justified by that at later ages mortality patterns are more consistent with GM. However here the contribution of C to mortality is of special interest and thus younger starting ages are more adequate for assessing the contribution. Another consideration is that the apparent g is minimal at about 25 years, and this makes a reason to regard this age as the one when vulnerability to the age-dependent hazards captured by the Gompertz term of GMM starts to increase in humans.

In the historical period 2020, the apparent rates of aging in all cases increase in the age periods from ca. 25 to 50 years, most probably because of the C-effect shown in Fig. 1, no matter how small the age-independent (external, accidental) mortality C may be in 2020. In the period 1920, the C-effect is more protracted because C is higher. In the cohorts 1920, the increases in the apparent rate of aging before the age of about 40 are steeper because of the combined effects of the relative decrease in the contribution of C to mortality with increasing age and the actual decrease in C in the respective historical period. The small deviations of the plots from the general trends are likely caused by data noisiness amplified by differentiation included in data treatment procedures. With that, the seemingly regular undulations of the plots related to women only in Sweden and, at a smaller scale, to men only in Japan are not readily interpretable.

Figure 4 is supplemented below with a plot (Fig. 5) derived from HMD data on mortality in England and Wales in 2012 pretreated in (Tang, Dodd et al. 2021) using adaptive P-spline procedure, which allowed the authors to protract mortality trend up to 120 years. The resulting plots presented in (Tang, Dodd et al. 2021) were digitized and the datapoint series obtained in this way were treated as described before, except for the Eigen decomposition was used for smoothing. With all differences concerning data treatment, an increase in the rate of aging starting at about 65 years of age is similar to that seen in Fig. 4. The decrease that occurs after a peak at about 85 years discontinues after 100 at a level of aging rate similar to its pre-peak value.

Taken together, the observations reported here and interpreted in terms of GGML suggest, with account of the contribution of the C-effect, that the rate of aging in humans is almost constant in the age period from about 40 (and possibly even 25) years until ca. 65–70 years, then it increases, reaches a peak at ca. 85 years, and starts to decrease thereafter to achieve the pre-peak values at about 90 years. Data noisiness above this age precludes making inferences about what may occur further on.

The alternative interpretations are that the composition of a population may depend on age or/and that C may start to change dramatically at ca. 65 years of age.

In the former case, it is understandable in principle how the preferential dying out of frailer subjects may result in an increase in the proportion of those who are less frail and thus die out less rapidly making demographic aging slower. However, the increase in the proportion of the subjects whose aging rate is higher within the age period about 65–90 years is hard to interpret in this way.

In the latter case, the assumption that C is not necessarily constant is generally consistent with GGML. In a recent paper (Lenart, Psalmon et al. 2023), it was suggested that learning may improve survival, and this effect was captured by an age-dependent term added to the GMM, that is the term confers age-dependency to the part of mortality not accounted for by the Gompertz term and thus makes C dependent on age in the final account. Assuming that C was relatively small in France, Sweden and Japan in 2020, especially with account of its vanishing relative contribution to the late-age mortality, it is hard to envision, upon the assumption that aging rate is constant, how such embodiment of GGML can account for the late-age peculiarities reported here, even if C does continue to decrease after 70 due to, say, continuing acquisition of wisdom.

Noteworthy, in all cases shown in Figs. 4 and 5 the rates of aging are generally higher in females than in males. This is still a highly controversial issue (Lenart, Kuruczova et al. 2019). There are grounds to suggest that females live longer than males not because their aging rate estimated according to GM is slower but because their initial mortality is lower in humans (Lenart, Kuruczova et al. 2017, Golubev 2019) and mice (Kunstyr and Leuenberger 1975, Golubev 2019), and the two parameters are negatively correlated according to the compensation effect of mortality (Golubev 2019). It is also worth mentioning that aging rate seems to start decreasing in women after ca. 50, the age of the menopause. However, the same, although less pronounced, may be seen in the plots related to men, except for the period 2020 in Japan, where the difference between men and women is clear cut.

The admission that the peculiarities of demographic aging reported here do reflect an actual increase in the averaged individual physiological aging rate in humans aged above ca. 65 years raises the question why may it be so? Conspicuous in this regard is the coincidence of this age with retirement. Can changes in life associated with retirement affect humans as negatively as to increase their rate of aging? The issue of relationships between retirement and changes in human mortality is being avidly discussed for obvious socioeconomic reasons. Some studies do suggest a negative impact of retirement on human mental and physical health (Fitzpatrick and Moore 2018, Das 2023). However, meta-analyses of relevant literature have led to the conclusion that the impact of retirement on aging is highly context-dependent and controversial (Schaap, de Wind et al. 2018, Sewdas, Wind et al. 2020), whereas the findings presented here are quite consistent across several counties and two widely separated time periods.

May it be that the age period when retirement usually occurs is not as arbitrary as it may seem and could be adopted based on some routine experience, which is however poorly structured in the scientific terms? Indeed, there are data suggesting that some physiological indices show an accelerated decrease after ca. 65–70 years of age. In a Chinese cohort aged above 60 years at the onset of the study (Han, Hu et al. 2022), the decrease in the basal metabolic rate markedly accelerated after 6 years of follow up. Age-specific world records in different track-and-field and weight-lifting disciplines decrease more rapidly at ages above 70 than before (Berthelot, Bar-Hen et al. 2019). It may be hypothesized that the age-dependent physiological decline may be compensated for by mobilizing body reserves, which may become exhausted by about 70 years of age. Importantly, master athletes are those who are most able to mobilize their reserves.

Especially demonstrative in this regard are the results of a longitudinal study of swimming performance in humans, which clearly show a marked acceleration of the worsening of the performance after 70 years of age (Donato, Tench et al. 2003). Swimming as any other physical activity depends not only on the muscles, but also on the capacities of the lungs, heart, circulation, nerves and brain, and on the conditions of the mitochondria and the stiffness of the extracellular matrix. The consequences of the compromised conditions of all this involve the immune system and thus the ability to resist cancer, infections and autoimmune hazards (Duggal, Niemiro et al. 2019).

If an increase in the rate of the demographic aging does reflect an increase in the rate of the physiological aging, why is it not that the subsequent decrease in the rate of the demographic aging in the oldest old reflects the averaged deceleration of the physiological rate of aging among them? One of the causes of such deceleration may be a decrease in the remaining nonmodified substrate where the covalent nonenzymatic modifications of slowly turning over macromolecules can accumulate to result in aging (Golubev 2009, 2021; Golubev, Hanson et al. 2017). For example, the trajectories of the increases in skin and eye-lens autofluorescence caused by advanced glycation end product accumulation with age is more consistent with a logarithmic or logistic dose-response fashion (not shown) rather than with a linear fashion presented in the original plots published in (Cahn, Burd et al. 2014, Isami, West et al. 2018).

Of course, the above reasoning makes sense only upon the assumption that GM and GMM are consistent with the biological reality, which is captured by GGML, rather than are just handy mathematical tools invented by actuaries. The latter standpoint is implicit in the attempts to use other core functions for treating mortality and survival data, e.g. (Avraam, Arnold et al. 2016, Canudas-Romo, Mazzuco et al. 2018, Mazzuco, Scarpa et al. 2018). As it is claimed for example (Stroustrup 2018), “… despite its popularity the Gompertz model is frequently out-performed by several other two-parameter distributions. These include the Weibull model that assumes a polynomial increase in the risk of death over time, as well as the inverse-Gaussian model, which notably has a compelling theoretic grounding in the statistical physics of random walks. Other parametric alternatives include Gompertz-Makeham, log-logistic, and lognormal models”. For example, the Weibull distribution is claimed to be preferable as a model of C. elegans lifespan variability (Oswal, Martin et al. 2022). It is easy to see, however, that f(t) = x + ylnt in the GGML makes the lnµ-vs-lnt plot linear, that is consistent with the Weibull distribution. Indeed, the time trajectories of C. elegans mortality shown in (Oswal, Martin et al. 2022) may be fitted with a logarithmic function pretty well.

Finally, is there any evidence that physiological aging rates undergo significant changes over lifespans in other species? May these changes account for the different species-specific trajectories of mortality described in (Jones, Scheuerlein et al. 2014)? May these trajectories be interpreted in terms of GGML? In this regard, it is important that H. sapiens and C. elegans afford uniquely huge populations and cohorts for examination thus providing for reducing the noisiness of data, which is amplified by any data treatment procedures that involve differentiation required to estimate the rates of changes in whatever it may be. Still, the present findings suggest that the beliefs that aging rate is constant and the age-independent mortality is negligible, which underly by default many models and procedures currently used for treating mortality and survival data, need being checked up in every specific case.

This research received no specific funding. The author declares no conflict of interest.

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A previously unrecognized peculiarity of late-life human mortality kinetics?

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Introduction

Preliminaries: a generalized Gompertz-Makeham law

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