Gompertz Slope Research Articles

Actuarial aging rates in human cohorts

Aging rate is an important characteristic of human aging. Attempts to measure aging rates through the Gompertz slope parameter lead to a conclusion that actuarial aging rates were stable during the most of the 20th century, but recently demonstrate an increase over time in the majority of studied populations. These findings were made using cross-sectional mortality data rather than by the analysis of mortality of real birth cohorts. In this study we analyzed historical changes of actuarial aging rates in human cohorts. The Gompertz parameters were estimated in the age interval 50-80 years using data on one-year cohort age-specific death rates from the Human Mortality Database (HMD). Totally, data for 2,294 cohorts of men and women from 76 populations were analyzed. Changes of the Gompertz slope parameter in the studied cohorts revealed two distinct patterns for actuarial aging rate. In higher mortality Eastern European countries actuarial aging rates showed continuous decline from the 1910 to 1940 birth cohort. In lower mortality Western European countries, Australia, Canada, Japan, New Zealand, and USA actuarial aging rates declined from the 1910th to approximately 1930th cohort and then increased. Overall, in 50 out of 76 populations (68%) actuarial aging rate demonstrated decreasing pattern of change over time. Compensation effect of mortality (CEM) was tested for the first time in human cohorts and the cohort species-specific lifespan was estimated. CEM was confirmed using cohort data and human cohort species-specific lifespan estimates were similar to the estimates obtained for the cross-sectional data published earlier.

Actuarial Aging Rates in Human Cohorts.

Aging rate is an important characteristic of human aging. Attempts to measure aging rates through the Gompertz slope parameter lead to a conclusion that actuarial aging rates were stable during the most of the 20th century, but recently demonstrate an increase over time in the majority of studied populations. These findings were made using cross-sectional mortality data rather than by the analysis of mortality of real birth cohorts. In this study we analyzed historical changes of actuarial aging rates in human cohorts. The Gompertz parameters were estimated in the age interval 50-80 years using data on one-year cohort age-specific death rates from the Human Mortality Database (HMD). Totally, data for 2,294 cohorts of men and women from 76 populations were analyzed. Changes of the Gompertz slope parameter in the studied cohorts revealed two distinct patterns for actuarial aging rate. In higher mortality Eastern European countries actuarial aging rates showed continuous decline from the 1910 to 1940 birth cohort. In lower mortality Western European countries, Australia, Canada, Japan, New Zealand, and USA actuarial aging rates declined from the 1910th to approximately 1930th cohort and then increased. Overall, in 50 out of 76 populations (68%) actuarial aging rate demonstrated decreasing pattern of change over time. Compensation effect of mortality (CEM) was tested for the first time in human cohorts and the cohort species-specific lifespan was estimated. CEM was confirmed using cohort data and human cohort species-specific lifespan estimates were similar to the estimates obtained for the cross-sectional data published earlier.

Trends in Human Species-Specific Lifespan and Actuarial Aging Rate.

The compensation effect of mortality (CEM) was tested and species-specific lifespan was estimated using data on one-year age-specific death rates from the Human Mortality Database (HMD). CEM was confirmed using this source of the data and human species-specific lifespan estimates were obtained, which were similar to the estimates published before. Three models (Gompertz-Makeham, Gompertz-Makeham with mean-centered age, and Gompertz) produced similar estimates of the species-specific lifespan. These estimates demonstrated some increase over time. Attempts to measure aging rates through the Gompertz slope parameter led to the conclusion that actuarial aging rates were stable during most of the 20th century, but recently demonstrated an increase over time in the majority (74%) of studied populations. This recent phenomenon is most likely caused by more rapid historical decline of mortality at the younger adult age groups compared to the older age groups, thus making the age gradient in mortality steeper over time. There is no biomedical reason to believe that human aging rates accelerated recently, so that the actuarial aging rate is probably not a good measure of true aging rate (rate of functional loss). Therefore, better measures of aging rate need to be developed.

Models for estimating empirical Gompertz mortality: With an application to evolution of the Gompertzian slope

AbstractUsing data from the human mortality database (HMD), and five different modeling approaches, we estimate Gompertz mortality parameters for 7,704 life tables. To gauge model fit, we predict life expectancy at age 40 from these parameters, and compare predicted to empirical values. Across a diversity of human populations, and both sexes, the overall best way to estimate Gompertz parameters is weighted least squares, although Poisson regression performs better in 996 cases for males and 1,027 cases for females, out of 3,852 populations per sex. We recommend against using unweighted least squares unless death counts (to use as weights or to allow Poisson estimation) are unavailable. We also recommend fitting to logged death rates. Over time in human populations, the Gompertz slope parameter has increased, indicating a more severe increase in mortality rates as age goes up. However, it is well‐known that the two parameters of the Gompertz model are very tightly (and negatively) correlated. When the slope goes up, the level goes down, and, overall, mortality rates are decreasing over time. An analysis of Gompertz parameters for all of the HMD countries shows a distinct pattern for males in the formerly socialist economies of Europe.

Reciprocal cross differences in Drosophila melanogaster longevity: an evidence for non-genomic effects in heterosis phenomenon?

Reciprocal cross effects (i.e., differences between reciprocal hybrids that are developed by reversing the strains from which the dam and the sire are taken) are commonly used as a measure of sex-linkage or maternal effects. However, the papers reporting parental effects on life span of experimental animals are scarce. In order to investigate the potential of parent-of-origin effects for the longevity of hybrids, we determined the life spans of the inbred lines of Drosophila melanogaster [Oregon-R (OR), Canton-S (CS) and Uman (Um)] that differ significantly in longevity, as well as the life span of the progeny from the reciprocal crosses among them. The hybridization caused the increase in both flies' mean and maximum life span mainly shifting the survival curves upward proportionally at all ages. This resulted in the reduction in the Gompertz intercept (frailty) whereas the Gompertz slope (the rate of aging) was predominantly unchanged. Better-parent heterosis was observed in hybrids between OR and Um inbred lines and the extent of heterosis was more pronounced in hybrids between CS and Um inbred lines if long-lived parent was used as the female parent, and short-lived parent was used as the male parent in the crossing scheme. Such discrepancy in life span between reciprocal crosses may indicate that non-chromosomal factors are significantly contributing to a heterotic response. Our data are in line with the previous reports suggesting the involvement of non-genomic factors, particularly epigenetic events attributed to hybridization, in the manifestation of heterosis.

Early cohort mortality predicts the rate of aging in the cohort: a historical analysis.

Early environmental influences on later-life health and mortality are well recognized in the doubling of life expectancy since 1800. To further define these relationships, we analyzed the associations between early-life mortality and both the estimated mortality level at age 40 and the exponential acceleration in mortality rates with age characterized by the Gompertz model. Using mortality data from 630 cohorts born throughout the 19th and early 20th century in nine European countries, we developed a multilevel model that accounts for cohort and period effects in later-life mortality. We show that early-life mortality, which is linked to exposure to infection and poor nutrition, predicts both the estimated cohort mortality level at age 40 and the subsequent Gompertz rate of mortality acceleration during aging. After controlling for effects of country and period, the model accounts for the majority of variance in the Gompertz parameters (about 90% of variation in the estimated level of mortality at age 40 and about 78% of variation in the Gompertz slope). The gains in cohort survival to older ages are entirely due to large declines in adult mortality level, because the rates of mortality acceleration at older ages became faster. These findings apply to cohorts born in both the 19th century and the early 20th century. This analysis defines new links in the developmental origins of adult health and disease in which effects of early-life circumstances, such as exposure to infections or poor nutrition, persist into mid-adulthood and remain evident in the cohort mortality rates from ages 40 to 90.

Variance in Death and Its Implications for Modeling and Forecasting Mortality.

The slope and curvature of the survivorship function reflect the considerable amount of variance in length of life found in any human population. Part is due to the well-known variation in life expectancy between groups: large differences according to race, sex, socioeconomic status, or other covariates. But within-group variance is large even in narrowly defined groups, and changes substantially and inversely with the group average length of life. We show that variance in length of life is inversely related to the Gompertz slope of log mortality through age, and we reveal its relationship to variance in a multiplicative frailty index. Our findings bear a variety of implications for modeling and forecasting mortality. In particular, we examine how the assumption of proportional hazards fails to account adequately for differences in subgroup variance, and we discuss how several common forecasting models treat the variance along the temporal dimension.

Accumulating Insoluble Protein and Rate of Aging

It has been proposed that during aging there is a progressive accumulation of insoluble cross-linked protein, which impedes cellular function by excluding macromolecular colloids (e.g., chromosomes and transport vesicles) from part of the water volume. The exponential increase with age in mortality rate, according to the Gompertz law, is correlated with the logarithmic decrease in water volume available to a colloid in a polymer network, according to the Ogston-Laurent equation. The Gompertz slope is proposed to be a measure of the rate of insoluble protein accumulation. According to this model, aging is retarded and life span prolonged by impeding the accumulation of the insoluble protein.

Gompertz law and aging as exclusion effects

The exponential increase with age in mortality rate, the Gompertz law, indicates that the decrease in vitality and viability linked to aging depends on phenomena with exponential or logarithmic dynamics. Gompertz slope (alpha) is assumed to be a measure of aging rate, provided the studied cohort is homogeneous and in a supporting environment. The law provides no clue about the cause of aging, but may be formally correlated with various physical or mathematical functions. A possible correlation between the Ogston-Laurent exclusion equation and human aging is examined. An increase with age of an inert cross-linked insoluble protein network is assumed to result in a logarithmic decrease in water volume available to colloidal macromolecules. In this model, alpha is assumed to be a measure of the rate of accumulation of the polypeptide network.

Gompertz law and aging as exclusion effects

The exponential increase with age in mortality rate, the Gompertz law, indicates that the decrease in vitality and viability linked to aging depends on phenomena with exponential or logarithmic dynamics. Gompertz slope (alpha) is assumed to be a measure of aging rate, provided the studied cohort is homogeneous and in a supporting environment. The law provides no clue about the cause of aging, but may be formally correlated with various physical or mathematical functions. A possible correlation between the Ogston-Laurent exclusion equation and human aging is examined. An increase with age of an inert cross-linked insoluble protein network is assumed to result in a logarithmic decrease in water volume available to colloidal macromolecules. In this model, alpha is assumed to be a measure of the rate of accumulation of the polypeptide network.

Critique of the demographic evidence for 'late-life non-senescence'.

Although the Gompertz formula accurately describes observed mortality distributions over most of their extent, their 'tail' is much longer than that of a Gompertz curve fitted to the whole data set. A simple candidate explanation is that the longest-lived subset of any population will necessarily be enriched in individuals that age more slowly than the average of that population. However, some investigators have suggested that, instead, individuals actually cease to senesce after a certain age. Here, using a new approach to determining the best-fit degree of heterogeneity in the Gompertz slope parameter, it is shown that observed distributions can in fact be fit quite accurately by purely 'heterogeneous Gompertz' curves. Either explanation may therefore be correct.

Penna ageing model and improvement of medical care in 20th century

In Penna's model for biological ageing we reduce after equilibration the influence of acquired diseases and get, similar to reality, the observed increase of the Gompertz slope for the human mortality curves.

Changes in neuroendocrine elements in bronchial mucosa in chronic lung disease in adults.

It is not clear whether there is any association between metaplasia of the bronchial epithelium and changes in the distribution of neuroendocrine cells. This study examined, by immunohistological techniques, the distribution of neuroendocrine cells and juxtamucoscal nerve fibres in bronchial biopsies showing metaplastic changes. Bronchial biopsies from 12 subjects with epithelial metaplasia associated with bronchiectasis and diffuse pulmonary fibrosis were examined by conventional light microscopy and immunohistological techniques for protein gene product 9.5 (PGP), chromogranin A and B (CAB), serotonin, vasoactive intestinal peptide (VIP), substance P (SP), calcitonin gene-related peptide (CGRP), calcitonin (CT), and gastrin releasing peptide (GRP). Regions of non-metaplastic epithelium contained numerous PGP and serotonin immunoreactive cells. Sub-populations of these cells displayed CAB, CGRP, CT, and GRP immunoreactivity. Metaplastic epithelium contained only a few weakly stained PGP, serotonin, CAB, GRP, CT and CGRP immunoreactive cells in six cases. Metaplastic epithelium was characterised by a high number of CAB-containing cells in six cases and in these biopsies prominent PGP-containing nerve bundles were seen in the subepithelial layer beneath the metaplastic epithelium. The distribution patterns of neuroendocrine cells and neuronal elements vary between areas of normal and metaplastic epithelium and within areas of metaplastic epithelium. Neuronal hyperplasia was associated with an increase in the number of CAB-containing cells within the metaplastic epithelium.

Longitudinal Gompertzian analysis of stroke mortality in the U.S., 1951–1986: Declining stroke mortality is the natural consequence of competitive deterministic mortality dynamics

Age-adjusted mortality rates for stroke in the United States from 1951 to 1986 were subjected to longitudinal Gompertzian analysis. Age-adjusted stroke mortality rate distributions were determined by a variable environmental factor and a constant Gompertz slope. Compared to 1951 values, the environmental factor in 1986 had declined (improved) 49.8% for men and 59.1% for women. This was associated with a 51.9% and 31.4% decrease in the annual crude mortality rate from stroke for men and women respectively. However, the Gompertz slope remained remarkably constant from 1951 to 1986; 0.050152 for men and 0.048341 for women. The constant Gompertz slope for age-adjusted mortality rate distributions for stroke is in sharp contrast to the increasing Gompertz slope which occurs with an improving environment in ‘degenerative’ diseases and aging in general. These findings suggest that the recent dramatic decline in overall stroke mortality is the natural consequence of competitive deterministic mortality dynamics. As the overall environment becomes more conducive to human survival, Gompertzian diseases with converging mortality rate distributions must increase as causes of human mortality at the expense of diseases with constant Gompertz slopes.

Relationship between γ-irradiation, life span, metabolic rate and accumulation of fluorescent age pigment in the adult male housefly, Musca domestica

The patterns of fluorescent age pigment (FAP) accumulation in response to γ-irradiation were examined in adult male houseflies under conditions of relatively high and low levels of physical activity- Flies were exposed to 0, 20, 40, and 66 kR and their level of physical activity was altered by permitting or restricting flight activity. Under conditions of high activity the mean life span of flies exposed to 20 kR and 40 kR was greater than in the controls. The mean life span of high activity flies exposed to 66 kR and all groups of low activity irradiated flies was less than the control life span. The force of mortality as indicated by Gompertz slope was greater in high activity controls than in unirradiated flies maintained under low activity conditions. Gompertz slope was similar in corresponding high and low activity groups exposed to radiation. Radiation decreased the rate of FAP accumulation in high activity groups, but increased the rate of FAP accumulation under low activity conditions. The low activity control and low activity flies exposed to 20 kR had a lower FAP accumulation than the corresponding high activity groups at 14 days of age. At this age, low activity groups of flies exposed to 40 kR and 66 kR had concentrations of FAP that were similar to the corresponding high activity groups. These results indicate that radiation exposure decreased longevity by causing damage rather than by increasing the rate of aging.

General theory of mortality and aging.

A theory of the kinetics of death is presented which is based upon the experimentally determined Gompertz function and the postulates that the distribution of stress magnitudes is a Maxwell-Boltzmann distribution and an organism dies when stress magnitude exceeds the organism's maximum ability to compensate therefor. The theory predicts a zero-order loss of function versus age. This is borne out in human males by independent observation. The theory permits several independent calculations of the value of B, the percent of loss per year of physiologic function. The calculated values range from 0.9 to 1.4 percent per year and agree closely with the observed rates in human males. The theory predicts an inverse linear relationship between Gompertz slope and ln R/ sub 0/ (intercept), which is closely confirmed by observation. The theory predicts that the mean ratio of maximum reserve capacity to average demand lies between 7 and 11. Independent physiologic measurement data are in reasonable agreement with these values. The theory predicts the observed effects of prolonged or instantaneous exposure of experimental animals to ionizing radiation. The relative deleteriousness (D) of various national environments can be calculated. They have been found to differ by approximately 50 percent. We have beenmore » unable to make an independent test of these relative values. Despite the fact that it is derived for a homogeneous population the theory is shown to be not inconsistent with individual variability within a population. (auth)« less