Compensation effect of mortality is a challenge to substantial lifespan extension of humans.

Despite frequent claims regarding radical extensions of human lifespan in the near future, many pragmatic scientists caution against excessive and baseless optimism on this front. In this study, we examine the compensation effect of mortality (CEM) as a potential challenge to substantial lifespan extension. The CEM is an empirical mortality regularity, often depicted as relative mortality convergence at advanced ages. Analysis of mortality data from 44 human populations, available in the Human Mortality Database, demonstrated that CEM can be represented as a continuous decline in relative mortality variation (assessed through the coefficient of variation and the standard deviation of the logarithm of mortality) with age, reaching a minimum corresponding to the species-specific lifespan. Through this method, the species-specific lifespan is determined to be 96-97 years, closely aligning with estimates derived from correlations between Gompertz parameters (95-98 years). Importantly, this representation of CEM can be achieved non-parametrically, eliminating the need for estimating Gompertz parameters. CEM is a challenge to lifespan extension, because it suggests that the true aging rate in humans (based on loss of vital elements, e.g., functional cells) remains stable at approximately 1% per year in the majority of human populations and is not affected by environmental or familial longevity factors. Given this rate of functional cell loss, one might anticipate that the total pool of functional cells could be entirely depleted by the age of 115-120 years creating physiological limit to human lifespan. Mortality pattern of supercentenarians (110 + years) aligns with this prediction.

Two Types of Survival Curves of Different Lines of Progeric Mice.

For most of their lifespan, the probability of death for many animal species increases with age. Gompertz law states that this increase is exponential. In this work, we have compared previously published data on the survival kinetics of different lines of progeric mice. Visual analysis showed that in six lines of these rapidly aging mutants, the probability of death did not strictly depend on age. In contrast, ten lines of progeric mice have survival curves similar to those of the control animals, that is, in agreement with Gompertz law, similar to the shape of an exponential curve upside down. Interestingly, these ten mutations cause completely different cell malfunctions. We speculate that what these mutations have in common is a reduction in the lifespan of cells and/or an acceleration of the transition to the state of cell senescence. Thus, our analysis, similar to the conclusions of many previously published works, indicates that the aging of an organism is a consequence of the aging of individual cells.

"Translating" All-Cause Mortality Rate Ratios or Hazard Ratios to Age-, Longevity-, and Probability-Based Measures.

Epidemiologists commonly use an adjusted hazard ratio or incidence density ratio, or a standardized mortality ratio, to measure a difference in all-cause mortality rates. They seldom translate it into an age-, time-, or probability-based measure that would be easier to communicate and to relate to. Several articles have shown how to translate from a standardized mortality ratio or hazard ratio to a longevity difference, a difference in actuarial ages, or a probability of being outlived. In this paper, we describe the settings where these translations are and are not appropriate and provide some of the heuristics behind the formulae. The tools that yield differences in "effective age" and in longevity are applicable when both 1) the mortality rate ratio (hazard ratio) is constant over age and 2) the rates themselves are log-linear in age. The "probability/odds of being outlived" metric is applicable whenever the first condition holds, and thus it provides no direct information on the magnitude of the effective age/longevity difference.

Recursive model for dose-time responses in pharmacological studies.

BACKGROUND: Clinical studies often track dose-response curves of subjects over time. One can easily model the dose-response curve at each time point with Hill equation, but such a model fails to capture the temporal evolution of the curves. On the other hand, one can use Gompertz equation to model the temporal behaviors at each dose without capturing the evolution of time curves across dosage. RESULTS: In this article, we propose a parametric model for dose-time responses that follows Gompertz law in time and Hill equation across dose approximately. We derive a recursion relation for dose-response curves over time capturing the temporal evolution and then specify a regression model connecting the parameters controlling the dose-time responses with individual level proteomic data. The resultant joint model allows us to predict the dose-response curves over time for new individuals. CONCLUSION: We have compared the efficacy of our proposed Recursive Hybrid model with individual dose-response predictive models at desired time points. We note that our proposed model exhibits a superior performance compared to the individual ones for both synthetic data and actual pharmacological data. For the desired dose-time varying genetic characterization and drug response values, we have used the HMS-LINCS database and demonstrated the effectiveness of our model for all available anticancer compounds.

New Trend in Old-Age Mortality: Gompertzialization of Mortality Trajectory.

There is great interest among gerontologists, demographers, and actuaries in the question concerning the limits to human longevity. Attempts at getting answers to this important question have stimulated many studies on late-life mortality trajectories, often with opposing conclusions. One group of researchers believes that mortality stops growing with age at extreme old ages, and that hence there is no fixed limit to the human life span. Other studies found that mortality continues to grow with age up to extreme old ages. Our study suggests a possible solution to this controversy. We found that mortality deceleration is best observed when older, less accurate life span data are analyzed, while in the case of more recent and reliable data there is a persistent mortality growth with age. We compared the performance (goodness of fit) of two competing mortality models - the Gompertz model and the Kannisto ("mortality deceleration") model - at ages of 80-105 years using data for 1880-1899 single-year birth cohorts of US men and women. The mortality modeling approach suggests a transition from mortality deceleration to the Gompertzian mortality pattern over time for both men and women. These results are consistent with the hypothesis about disappearing mortality deceleration over time due to improvement in the accuracy of age reporting. In the case of more recent data, mortality continues to grow with age even at very old ages. This observation may lead to more conservative estimates of future human longevity records.

Strehler-Mildvan correlation is a degenerate manifold of Gompertz fit.

Gompertz empirical law of mortality is often used in practical research to parametrize survival fraction as a function of age with the help of just two quantities: the Initial Mortality Rate (IMR) and the Gompertz exponent, inversely proportional to the Mortality Rate Doubling Time (MRDT). The IMR is often found to be inversely related to the Gompertz exponent, which is the dependence commonly referred to as Strehler-Mildvan (SM) correlation. In this paper, we address fundamental uncertainties of the Gompertz parameters inference from experimental Kaplan-Meier plots and show, that a least squares fit often leads to an ill-defined non-linear optimization problem, which is extremely sensitive to sampling errors and the smallest systematic demographic variations. Therefore, an analysis of consequent repeats of the same experiments in the same biological conditions yields the whole degenerate manifold of possible Gompertz parameters. We find that whenever the average lifespan of species greatly exceeds MRDT, small random variations in the survival records produce large deviations in the identified Gompertz parameters along the line, corresponding to the set of all possible IMR and MRDT values, roughly compatible with the properly determined value of average lifespan in experiment. The best fit parameters in this case turn out to be related by a form of SM correlation. Therefore, we have to conclude that the combined property, such as the average lifespan in the group, rather than IMR and MRDT values separately, may often only be reliably determined via experiments, even in a perfectly homogeneous animal cohort due to its finite size and/or low age-sampling frequency, typical for modern high-throughput settings. We support our findings with careful analysis of experimental survival records obtained in cohorts of C. elegans of different sizes, in control groups and under the influence of experimental therapies or environmental conditions. We argue that since, SM correlation may show up as a consequence of the fitting degeneracy, its appearance is not limited to homogeneous cohorts. In fact, the problem persists even beyond the simple Gompertz mortality law. We show that the same degeneracy occurs exactly in the same way, if a more advanced Gompertz-Makeham aging model is employed to improve the modeling. We explain how SM type of relation between the demographic parameters may still be observed even in extremely large cohorts with immense statistical power, such as in human census datasets, provided that systematic historical changes are weak in nature and lead to a gradual change in the mean lifespan.

Justifying the Gompertz curve of mortality via the generalized Polya process of shocks.

A new probabilistic model of aging that can be applied to organisms is suggested and analyzed. Organisms are subject to shocks that follow the generalized Polya process (GPP), which has been recently introduced and characterized in the literature. Distinct from the nonhomogeneous Poisson process that has been widely used in applications, the important feature of this process is the dependence of its future behavior on the number of previous events (shocks). The corresponding survival and the mortality rate functions are derived and analyzed. The general approach is used for justification of the Gompertz law of human mortality.

On some mortality rate processes and mortality deceleration with age.

A specific mortality rate process governed by the non-homogeneous Poisson process of point events is considered and its properties are studied. This process can describe the damage accumulation in organisms experiencing external shocks and define its survival characteristics. It is shown that, although the sample paths of the unconditional mortality rate process are monotonically increasing, the population mortality rate can decrease with age and, under certain assumptions, even tend to zero. The corresponding analysis is the main objective of this paper and it is performed using the derived conditional distributions of relevant random parameters. Several meaningful examples are presented and discussed.

A "universal" model of metastatic cancer, its parametric forms and their identification: what can be learned from site-specific volumes of metastases.

We develop a methodology for estimating unobservable characteristics of the individual natural history of metastatic cancer from the volume of the primary tumor and site-specific volumes of metastases measured before, or shortly after, the start of treatment. In particular, we address the question as to what information about natural history of cancer can and cannot be gained from this type of data. Estimation of the natural history of cancer is based on parameterization of a very general mathematical model of cancer progression accounting for primary tumor growth, shedding of metastases, their selection, latency and growth in a given secondary site. This parameterization assumes Gompertz (and, as a limiting case, exponential) growth of the primary tumor, exponential growth of metastases, and exponential distribution of metastasis latency times. We find identifiable parameters of this model and give a rigorous proof of their identifiability. As an illustration, we analyze a clinical case of renal cancer patient who developed 55 lung metastases whose volumes were measured through laborious reading of CT images. The model with maximum likelihood parameters provided an excellent fit to this data. We uncovered many aspects of this patient's cancer natural history and showed that, according to the model, onset of metastasis occurred long before primary tumor became clinically detectable.

[Discontinuity of the annuity curves. I. Departures from Gompertz law in Drosophila melanogaster Canton-S strain].

Analysis of male imago mortality in Drosophila melanogaster Canton-S strain demonstrated deviation from Gompertz law. Annuity curves were composed of five phases divided by sharp bends. All the phases seemed to be linear. We discuss the hypotheses of strict genetic determination of the phenomenon.

Mathematical modeling of the synergistic interplay of radiotherapy and immunotherapy in anti-cancer treatments.

Introduction: While radiotherapy has long been recognized for its ability to directly ablate cancer cells through necrosis or apoptosis, radiotherapy-induced abscopal effect suggests that its impact extends beyond local tumor destruction thanks to immune response. Cellular proliferation and necrosis have been extensively studied using mathematical models that simulate tumor growth, such as Gompertz law, and the radiation effects, such as the linear-quadratic model. However, the effectiveness of radiotherapy-induced immune responses may vary among patients due to individual differences in radiation sensitivity and other factors. Methods: We present a novel macroscopic approach designed to quantitatively analyze the intricate dynamics governing the interactions among the immune system, radiotherapy, and tumor progression. Building upon previous research demonstrating the synergistic effects of radiotherapy and immunotherapy in cancer treatment, we provide a comprehensive mathematical framework for understanding the underlying mechanisms driving these interactions. Results: Our method leverages macroscopic observations and mathematical modeling to capture the overarching dynamics of this interplay, offering valuable insights for optimizing cancer treatment strategies. One shows that Gompertz law can describe therapy effects with two effective parameters. This result permits quantitative data analyses, which give useful indications for the disease progression and clinical decisions. Discussion: Through validation against diverse data sets from the literature, we demonstrate the reliability and versatility of our approach in predicting the time evolution of the disease and assessing the potential efficacy of radiotherapy-immunotherapy combinations. This further supports the promising potential of the abscopal effect, suggesting that in select cases, depending on tumor size, it may confer full efficacy to radiotherapy.

[Age-specific regional characteristics of COVID-19 mortality in 2021]. / A COVID­19-halandóság életkor-specifikus regionális jellemzoi 2021-ben.

INTRODUCTION: In most countries, COVID-19 mortality increases exponentially with age, but the growth rate varies considerably between countries. The different progression of mortality may reflect differences in population health, the quality of health care or coding practices. OBJECTIVE: In this study, we investigated differences in age-specific county characteristics of COVID-19 mortality in the second year of the pandemic. METHOD: Age-specific patterns of COVID-19 adult mortality were estimated according to county level and sex using a Gompertz function with multilevel models. RESULTS: The Gompertz function is suitable for describing age patterns of COVID-19 adult mortality at county level. We did not find significant differences in the age progression of mortality between counties, but there were significant spatial differences in the level of mortality. The mortality level showed a relationship with socioeconomic and health care indicators with the expected sign, but with different strengths. DISCUSSION AND CONCLUSION: The COVID-19 pandemic in 2021 resulted in a decline in life expectancy in Hungary not seen since World War II. The study highlights the importance of healthcare in addition to social vulnerability. It also points out that understanding age patterns will help to mitigate the consequences of the epidemic. Orv Hetil. 2023; 164(17): 643-650.

Protective Effects of Familial Longevity Decrease With Age and Become Negligible for Centenarians.

It is known that biological relatives of long-lived individuals demonstrate lower mortality and longer life span compared to relatives of shorter-lived individuals, and at least part of this advantage is likely to be genetic. Less information, however, is available about effects of familial longevity on age-specific mortality trajectories. We compared mortality patterns after age 50 years for 10 045 siblings of US centenarians and 12 308 siblings of shorter-lived individuals (died at age 65 years). Similar comparisons were made for sons and daughters of longer-lived parents (both parents lived 80 years and more) and shorter-lived parents (both parents lived less than 80 years) within each group of siblings. Although relatives of longer-lived individuals have lower mortality at younger ages compared to relatives of shorter-lived individuals, this mortality advantage practically disappears by age 100 years. To validate this observation further, we analyzed the survival of 3 408 US centenarians born in 1890-1897 with known information on maternal and paternal life span. We found using the Cox proportional hazards model that both maternal and paternal longevity (life span 80+ years) is not significantly associated with survival after age 100 years. The results are compatible with the predictions of reliability theory of aging suggesting higher initial levels of system redundancy (reserves) in individuals with protective familial/genetic background and hence lower initial mortality. Heterogeneity hypothesis is another possible explanation for the observed phenomena.

Deep longitudinal phenotyping of wearable sensor data reveals independent markers of longevity, stress, and resilience.

Biological age acceleration (BAA) models based on blood tests or DNA methylation emerge as a de facto standard for quantitative characterizations of the aging process. We demonstrate that deep neural networks trained to predict morbidity risk from wearable sensor data can provide a high-quality and cheap alternative for BAA determination. The GeroSense BAA model was trained and validated using steps per minute recordings from 103,830 one-week long and 2,599 of up to 2 years-long longitudinal samples and exhibited a superior association with life-expectancy over the average number of steps per day in, e.g., groups stratified by professional occupations. The association between the BAA and effects of lifestyles, the prevalence of future incidence of diseases was comparable to that of BAA from models based on blood test results. Wearable sensors let sampling of BAA fluctuations at time scales corresponding to days and weeks and revealed the divergence of organism state recovery time (resilience) as a function of chronological age. The number of individuals suffering from the lack of resilience increased exponentially with age at a rate compatible with Gompertz mortality law. We speculate that due to the stochastic character of BAA fluctuations, its mean and auto-correlation properties together comprise the minimum set of biomarkers of aging in humans.

Nutrient supply, cell spatial correlation and Gompertzian tumor growth.

Gompertzian tumor growth can be reproduced by mitosis, related to nutrient supply, with local spatial cell correlations. The global energy constraint alone does not reproduce in vivo data by the observed values of the nutrient expenditure for the cell activities. The depletion of the exponential growth, described by the Gompertz law, is obtained by mean field spatial correlations or by a small word network among cells. The well-known interdependence between the two parameters of the Gompertz growth naturally emerges and depends on the cell volume and on the tumor density.

Deciphering death: a commentary on Gompertz (1825) 'On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies'.

In 1825, the actuary Benjamin Gompertz read a paper, 'On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies', to the Royal Society in which he showed that over much of the adult human lifespan, age-specific mortality rates increased in an exponential manner. Gompertz's work played an important role in shaping the emerging statistical science that underpins the pricing of life insurance and annuities. Latterly, as the subject of ageing itself became the focus of scientific study, the Gompertz model provided a powerful stimulus to examine the patterns of death across the life course not only in humans but also in a wide range of other organisms. The idea that the Gompertz model might constitute a fundamental 'law of mortality' has given way to the recognition that other patterns exist, not only across the species range but also in advanced old age. Nevertheless, Gompertz's way of representing the function expressive of the pattern of much of adult mortality retains considerable relevance for studying the factors that influence the intrinsic biology of ageing. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

Biodemography of old-age mortality in humans and rodents.

The growing number of persons living beyond age 80 underscores the need for accurate measurement of mortality at advanced ages and understanding the old-age mortality trajectories. It is believed that exponential growth of mortality with age (Gompertz law) is followed by a period of deceleration, with slower rates of mortality increase at older ages. This pattern of mortality deceleration is traditionally described by the logistic (Kannisto) model, which is considered as an alternative to the Gompertz model. Mortality deceleration was observed for many invertebrate species, but the evidence for mammals is controversial. We compared the performance (goodness-of-fit) of two competing models-the Gompertz model and the logistic (Kannisto) model using data for three mammalian species: 22 birth cohorts of U.S. men and women, eight cohorts of laboratory mice, and 10 cohorts of laboratory rats. For all three mammalian species, the Gompertz model fits mortality data significantly better than the "mortality deceleration" Kannisto model (according to the Akaike's information criterion as the goodness-of-fit measure). These results suggest that mortality deceleration at advanced ages is not a universal phenomenon, and survival of mammalian species follows the Gompertz law up to very old ages.

Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population.

The widely-known Gompertz law of mortality states the exponential increase of mortality with age in human populations. Such an exponential increase is observed at the adulthood span, roughly after the reproductive period, while mortality data at young and extremely old ages deviate from it. The heterogeneity of human populations, i.e. the existence of subpopulations with different mortality dynamics, is a useful consideration that can explain age-dependent mortality patterns across the whole life-course. A simple mathematical model combining the heterogeneity of populations with an assumption that the mortality in each subpopulation grows exponentially with age has been proven to be capable of reproducing the entire mortality pattern in a human population including the observed peculiarities at early- and late-life intervals. In this work we fit this model to actual (Swedish) mortality data for consecutive periods and consequently describe the evolution of mortality dynamics in terms of the evolution of the model parameters over time. We have found that the evolution of the model parameters validates the applicability of the compensation law of mortality to each subpopulation separately. Furthermore, our study has indicated that the population structure changes so that the population tends to become more homogeneous over time. Finally, our analysis of the decrease of the overall mortality in a population over time has shown that this decrease is mainly due to a change in the population structure and to a lesser extent to a reduction of mortality in each of the subpopulations, the latter being represented by an alteration of the parameters that outline the exponential dynamics.

OPTIMALITY THEORY, GOMPERTZ' LAW, AND THE DISPOSABLE SOMA THEORY OF SENESCENCE.

The "disposable soma" theory for the evolution of senescence suggests that senescence arises from an optimal balancing of resources between reproduction and somatic repair. Dynamic programming models are constructed and analyzed to determine the optimal relationship between reproduction, diversion of resources from repair, and added senescent mortality. Of particular interest is the relationship between the repair-reproduction trade-off and the form of the mortality-rate-versus-age curve predicted. The models analyzed in the greatest detail assume that the relationship between reproduction and added senescent mortality does not change with age. These suggest that mortality should increase at an increasing rate with age, but may approach a linear rate as mortality becomes very high. General results are derived for the shape of the mortality curves early and late in the senescing part of the life span, and mortality curves for specific trade-off functions are illustrated. An exponential increase in death rate with age (Gompertz' Law) corresponds to only one of many possible relationships between reproduction and aging. The "Law" is unlikely to hold generally if the disposable soma theory accounts for a large fraction of the observed senescent increase in mortality with age. However, support for the generality of Gompertz' Law is weak, and other theories have not produced an evolutionary explanation for the law. The disposable soma theory is consistent with some of the exceptions to Gompertz' Law that have been observed.