Gompertz Law Research Articles

The biodemography of human aging (Gompertz-Makeham law) applied to surgical mortality modelling: a retrospective national cohort study.

The Gompertz-Makeham law describes a characteristic pattern of mortality in human populations where death rate is near constant between age 18 and 30 years (Makeham's Law) and rises exponentially thereafter (Gompertz Law). This pattern has not been described in surgical populations, but if true, would have important implications for understanding surgical risk and design and interpretation of surgical risk models. The aim of this study was to determine if the Gompertz-Makeham law applies to perioperative mortality risk and the conditions under which it may apply. We examined the relationship between age and 1-month postoperative all-cause mortality risk in a 10-year New Zealand administrative dataset comprising of 5,615,100 surgical procedures from 2007 to 2016. The dataset includes patient and surgical factors including procedures, American Society of Anesthesiologist's physical status score (ASA-PS), diagnoses and other relevant details. Semi-logarithmic graphs of mortality risk and age were plotted. Linear regression models were fitted, with regression line slope and Pearson's correlation coefficient calculated. The primary outcome occurred in 114,782 (2.0%) of 5,615,100 included participants. The Gompertz-Makeham law seems to apply to the national surgical population as a whole (slope = 0.0241, R2 = 0.971). The law applies in all subgroups studied including sex, ASA-PS, surgical acuity, surgical severity category, cancer status and ethnicity (slopes 0.0066 to 0.0307, R2 0.771 to 0.990). Important interactions were found between age, mortality risk and three high risk groups (cancer diagnosis, ASA-PS 4-5 and high surgical severity). The Gompertz-Makeham law seems to apply in a national cohort of surgical patients. The inflection point for increased 1-month risk is apparent at age 30 years. A strict exponential rise in mortality risk occurs thereafter. This finding improves our understanding of surgical risk and suggests a concept-driven approach to improve modelling of age and important interactions in future surgical risk models.

Joint-Life Endowment Insurance Premium Calculation Based on Makeham Mortality Law

<p>This research determined life insurance premiums joint-life endowment insurance, that is the insurance that the benefit is paid if one of the insureds dies or all the insureds survive until the end of the contract. This premium calculation uses one of the famous mortality laws, that is Makeham. Makeham mortality law is an updated law of Gompertz's mortality law where the reason that makes Makeham different from Gompertz is the death calculation factor, because Gompertz only takes the cause of death from the age factor. Even though the death factor can also come from accidents, therefore, Makeham combined the two causes of death so that it is corresponded to the actual factors. The Makeham law will function to create the Makeham assumption mortality table with trials and errors method to construct Indonesia Mortality Table 2019 (TMI IV). This construction requires parameters that must be founded to determine the survival probability ( ) of the Makeham. After finding the , calculate the annual premium by dividing the net single premium by the life annuity. The joint-life endowment insurance premium calculation in this study uses 10-year protection with a 5.75% for interest rate and Rp1,000,000,000 for the benefit, and based on the research, the annual premium for this married couple is Rp78,016,785, where the premium calculated based on Makeham will get a higher premium than calculated based on TMI 2019, which gets an annual premium Rp77,410,954.</p>

Invariances in relations between aging, exposure to external hazards, and mortality reflected in life table aging rate (LAR) patterns examined through the lens of generalized Gompertz-Makeham law.

According to the Gompertz law, the age-dependent change in the logarithm of mortality (life-table aging rate, LAR) is equal to the population-averaged age-independent biological aging rate (γ), and LAR would be constant if aging were the only cause of mortality increase. However, LAR is influenced by population exposures to the external hazards. If they were constant, according to the Gompertz-Makeham law (GML), LAR would be below γ at lower ages and asymptotically and monotonically approach γ with increasing age. Actually, LAR trajectories derived from data on mortality in different countries and historical periods feature systematic undulations. In the present investigation, mortality-vs.-age trajectories were modeled based on a generalized GML (gGML). Unlike the canonical GML terms, which are population-specific constants, the respective terms of the gGML are represented with some population-specific functions of age. Invariant in gGML are the modes of translation of these functions into the dependency of mortality on age: linear for population exposure to the irresistible external hazards or exponential for population-averaged ability to withstand the resistible external and internal hazards. Modeling suggests that, at earlier ages, LAR undulations are attributable to changes in population exposures to the former hazards. However, only their unrealistically high levels can produce the transient increase in LAR at about 65 to 90 years. This pervasive undulation of LAR-vs.-age trajectory is rather caused by an increment in γ. Reasons to regard gGML as a genuine natural law, which defines relations between mortality, aging and environment, are discussed.

Development of Hydroponic Application based on Web and Internet of Things for The Community to Monitor pH and Total Dissolved Solids

Construction of the Mortality Table with Gompertz's Law Using the 2019 TMI Reference

COMMISSIONERS METHOD FOR LAST SURVIVOR WHOLE LIFE INSURANCE RESERVES WITH GOMPERTZ LAW

Life insurance plays an important role in protecting one’s life in terms of financial risks. The right required reserve is one aspect that should be secured to obtain a good company’s financial health. A study reported that insufficient reserve is one of the primary causes of insolvency issues among bankrupted insurance companies. Therefore, this study aims to use the Commissioners method in calculating the reserve of the last-survivor whole life insurance policy The Gompertz mortality law is used to estimate Indonesian death probability of age . Gompertz law is employed due to its practical convenience in dealing with continuous case calculation. Linear Least Square Method will be utilized to estimate the Gompertz parameters and . This study uses secondary data from Indonesian Mortality Table IV published by OJK. The result of this study is that the reserve of multiple life status policies, such as last-survivor insurance, depends on the states of individuals of each pair. In this study, the reserve has the highest amount in cases where only the male person is alive. Average Relative Error values of the estimated Gompertz to the TMI IV show that the overall accuracy exhibits a deviation of up to 1.67%. MAPE values of Gompertz show that the estimated model best fits the TMI IV for the interval age of 0-60.

Five years later, with double the demographic data, naked mole-rat mortality rates continue to defy Gompertzian laws by not increasing with age

The naked mole-rat (Heterocephalus glaber) is a mouse-sized rodent species, notable for its eusociality and long lifespan. Previously, we reported that demographic aging, i.e., the exponential increase of mortality hazard that accompanies advancing age in mammals and other organisms, does not occur in naked mole-rats (Ruby et al., 2018), a finding that has potential implications for human healthy aging. The demographic data supporting that conclusion had taken over three decades to accumulate, starting with the original rearing of H. glaber in captivity. This finding was controversial since many of the animals in that study were relatively young. In the 5 years following that study, we have doubled our quantity of demographic data. Here, we re-evaluated our prior conclusions in light of these new data and found them to be not only supported but indeed strengthened. We additionally provided insight into the social dynamics of captive H. glaber with data and analyses of body weight and colony size versus mortality. Finally, we provide a phylogenetically proximal comparator in the form of lifespan data from our Damaraland mole-rat (Fukomys damarensis) colony and demographic meta-analysis of those data along with published data from Ansell’s mole-rat (Fukomys anselli). We found Fukomys mortality hazard to increase gradually with age, an observation with inferences on the evolution of exceptional lifespan among mole-rats and the ecological factors that may have accompanied that evolution.

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

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. 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. 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. 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.

Mortality of type 2 diabetes in Germany: additional insights from Gompertz models.

The Gompertz law of mortality proclaims that human mortality rates in middle to old ages grow log-linearly with age and this law has been confirmed at multiple instances. We investigated if diabetes mortality in Germany also obeys to the Gompertz law and how this information helps to communicate diabetes mortality more intuitively. We analyzed all statutory health-insured persons in Germany in 2013 that were aged 30years or older. Deaths in 2014 were recorded and given in 5-year age groups. We fitted weighted linear regression models (separately for females and males and for people with and without diabetes) and additionally computed the probability that a person with diabetes dies before a person of the same age and sex without diabetes, and the "diabetes age", that is, the additional years of mortality risk added to an individual's chronological age due to diabetes-related excess mortality. We included N = 47,365,120 individuals, 6,541,181 of them with diabetes. In 2014, 763,228 deaths were recorded, among them 288,515 with diabetes. Diabetes mortality followed nearly perfectly Gompertz distributions. The probability that a person with diabetes dies before a person without diabetes was 61.9% for females and 63.3% for males. Diabetes mortality for females and males aged 30years or older in Germany in 2014 followed the Gompertz law of mortality. The survival information of the population with diabetes during a large part of the lifespan can thus be reduced to the two parameters of the Gompertz distribution.

Estimation–Calibration of Continuous-Time Non-Homogeneous Markov Chains with Finite State Space

We propose a method for fitting transition intensities to a sufficiently large set of trajectories of a continuous-time non-homogeneous Markov chain with a finite state space. Starting with simulated data computed with Gompertz–Makeham transition intensities, we apply the proposed method to fit piecewise linear intensities and then compare the transition probabilities corresponding to both the Gompertz–Makeham transition intensities and the fitted piecewise linear intensities; the main comparison result is that the order of magnitude of the average fitting error per unit time—chosen as a year—is always less than 1%, thus validating the methodology proposed.

Calculating Aging: Analysis of Survival Curves in the Norm and Pathology, Fluctuations in Mortality Dynamics, Characteristics of Lifespan Distribution, and Indicators of Lifespan Variation.

The article describes the history of studies of survival data carried out at the Research Institute of Physico-Chemical Biology under the leadership of Academician V. P. Skulachev from 1970s until present, with special emphasis on the last decade. The use of accelerated failure time (AFT) model and analysis of coefficient of variation of lifespan (CVLS) in addition to the Gompertz methods of analysis, allows to assess survival curves for the presence of temporal scaling (i.e., manifestation of accelerated aging), without changing the shape of survival curve with the same coefficient of variation. A modification of the AFT model that uses temporal scaling as the null hypothesis made it possible to distinguish between the quantitative and qualitative differences in the dynamics of aging. It was also shown that it is possible to compare the data on the survival of species characterized by the survival curves of the original shape (i.e., "flat" curves without a pronounced increase in the probability of death with age typical of slowly aging species), when considering the distribution of lifespan as a statistical random variable and comparing parameters of such distribution. Thus, it was demonstrated that the higher impact of mortality caused by external factors (background mortality) in addition to the age-dependent mortality, the higher the disorder of mortality values and the greater its difference from the calculated value characteristic of developed countries (15-20%). For comparison, CVLS for the Paraguayan Ache Indians is 100% (57% if we exclude prepuberty individuals as suggested by Jones et al.). According to Skulachev, the next step is considering mortality fluctuations as a measure for the disorder of survival data. Visual evaluation of survival curves can already provide important data for subsequent analysis. Thus, Sokolov and Severin [1] found that mutations have different effects on the shape of survival curves. Type I survival curves generally retains their standard convex rectangular shape, while type II curves demonstrate a sharp increase in the mortality which makes them similar to a concave exponential curve with a stably high mortality rate. It is noteworthy that despite these differences, mutations in groups I and II are of a similar nature. They are associated (i) with "DNA metabolism" (DNA repair, transcription, and replication); (ii) protection against oxidative stress, associated with the activity of the transcription factor Nrf2, and (iii) regulation of proliferation, and (or these categories may overlap). However, these different mutations appear to produce the same result at the organismal level, namely, accelerated aging according to the Gompertz's law. This might be explained by the fact that all these mutations, each in its own unique way, either reduce the lifespan of cells or accelerate their transition to the senescent state, which supports the concept of Skulachev on the existence of multiple pathways of aging (chronic phenoptosis).

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.

Exploring Patterns of Human Mortality and Aging: A Reliability Theory Viewpoint.

The most important manifestation of aging is an increased risk of death with advancing age, a mortality pattern characterized by empirical regularities known as mortality laws. We highlight three significant ones: the Gompertz law, compensation effect of mortality (CEM), and late-life mortality deceleration and describe new developments in this area. It is predicted that CEM should result in declining relative variability of mortality at older ages. The quiescent phase hypothesis of negligible actuarial aging at younger adult ages is tested and refuted by analyzing mortality of the most recent birth cohorts. To comprehend the aging mechanisms, it is crucial to explain the observed empirical mortality patterns. As an illustrative example of data-directed modeling and the insights it provides, we briefly describe two different reliability models applied to human mortality patterns. The explanation of aging using a reliability theory approach aligns with evolutionary theories of aging, including idea of chronic phenoptosis. This alignment stems from their focus on elucidating the process of organismal deterioration itself, rather than addressing the reasons why organisms are not designed for perpetual existence. This article is a part of a special issue of the journal that commemorates the legacy of the eminent Russian scientist Vladimir Petrovich Skulachev (1935-2023) and his bold ideas about evolution of biological aging and phenoptosis.

The Gompertz Law emerges naturally from the inter-dependencies between sub-components in complex organisms

Understanding and facilitating healthy aging has become a major goal in medical research and it is becoming increasingly acknowledged that there is a need for understanding the aging phenotype as a whole rather than focusing on individual factors. Here, we provide a universal explanation for the emergence of Gompertzian mortality patterns using a systems approach to describe aging in complex organisms that consist of many inter-dependent subsystems. Our model relates to the Sufficient-Component Cause Model, widely used within the field of epidemiology, and we show that including inter-dependencies between subsystems and modeling the temporal evolution of subsystem failure results in Gompertizan mortality on the population level. Our model also provides temporal trajectories of mortality-risk for the individual. These results may give insight into understanding how biological age evolves stochastically within the individual, and how this in turn leads to a natural heterogeneity of biological age in a population.

Construction of the Mortality Table with Gompertz's Law Using the 2019 TMI Reference

The Indonesian Mortality Table, which was specifically based on the real conditions of Indonesia, was successfully created for the first time in 1993. The mortality table was called the Indonesian Mortality Table (IMT) I of 1993. In its preparation from IMT II to IMT III, the time span was long enough that the Asosiasi Asuransi Jiwa Indonesia (AAJI) and Persatuan Aktuaris Indonesia (PAI) formed a team to prepare the Indonesian Mortality Table IV (IMT IV) which studied data during the study period between 2013-2017. Finally, the IV Indonesian Mortality Table was compiled in 2019. In this study, the mortality table used was the 2019 Indonesian Mortality Table issued by Persatuan Aktuaris Indonesia (PAI), as well as a modification of the 2019 IMT with Gompertz Law. In Gompertz's Law, one method that can be used is the least power square method. This study will show the suitability of Gompertz's Law against TMI 2019 for men and TMI 2019 for women, as well as to determine the suitability of Gompertz's Law using the Least Square Method for IMT 2019 for men and IMT 2019 for women.

Demographic indicators, models, and testing

The use of simple demographic indicators to describe mortality dynamics can obscure important features of the survival curve, particularly during periods of rapid change, such as those caused by internal or external factors, and especially at the oldest or youngest ages. Therefore, instead of the generally accepted Gompertz method, other methods based on demographic indicators are often used. In human populations, chronic phenoptosis, in contrast to age-independent acute phenoptosis, is characterized by rectangularization of the survival curve and an accompanying increase in average life expectancy at birth, which can be attributed to advances in society and technology. Despite the simple geometric interpretation of the phenomenon of rectangularization of the survival curve, it is difficult to notice one, detecting changes in the optimal coefficients in the Gompertz-Makeham law due to high computational complexity and increased calculation errors. This is avoided by calculating demographic indicators such as the Keyfitz entropy, the Gini coefficient, and the coefficient of variation in lifespan. Our analysis of both theoretical models and real demographic data shows that with the same value of the Gini coefficient in the compared cohorts, a larger value of the Keyfitz entropy indicates a greater proportion of centenarians relative to average life expectancy. On the contrary, at the same value of the Keyfitz entropy, a larger value of the Gini coefficient corresponds to a relatively large mortality at a young age. We hypothesize that decreases in the Keyfitz entropy may be attributable to declines in background mortality, reflected in the Makeham term, or to reductions in mortality at lower ages, corresponding to modifications in another coefficient of the Gompertz law. By incorporating dynamic shifts in age into survival analyses, we can deepen our comprehension of mortality patterns and aging mechanisms, ultimately contributing to the development of more reliable methods for evaluating the efficacy of anti-aging and geroprotective interventions used in gerontology.

Retracted: Classical and Bayesian Inference of Marshall-Olkin Extended Gompertz Makeham Model with Modeling of Physics Data

Retracted: Classical and Bayesian Inference of Marshall-Olkin Extended Gompertz Makeham Model with Modeling of Physics Data

Evaluation of four tumour growth models to describe the natural history of meningiomas.

The incidence of newly diagnosed meningiomas, particularly those diagnosed incidentally, is continually increasing. The indication for treatment is empirical because, despite numerous studies, the natural history of these tumours remains difficult to describe and predict. This retrospective single-centre study included 294 consecutive patients with 333 meningiomas who underwent three or more brain imaging scans. Linear, exponential, power, and Gompertz models were constructed to derive volume-time curves, by using a mixed-effect approach. The most accurate model was used to analyse tumour growth and predictors of rapid growth. The Gompertz model provided the best results. Hierarchical clustering at the time of diagnosis and at the end of follow-up revealed at least three distinct groups, which can be described as pseudoexponential, linear, and slowing growth with respect to their parameters. Younger patients and smaller tumours were more frequent in the pseudo-exponential clusters. We found that the more "aggressive" the cluster, the higher the proportion of patients with grade II meningiomas and who have had a cranial radiotherapy. Over a mean observation period of 56.5 months, 21% of the tumours moved to a cluster with a lower growth rate, consistent with the Gompertz's law. Meningiomas exhibit multiple growth phases, as described by the Gompertz model. The management of meningiomas should be discussed according to the growth phase, comorbidities, tumour location, size, and growth rate. Further research is needed to evaluate the associations between radiomics features and the growth phases of meningiomas. No funding.

Forecasting Individual Patients' Best Time for Surgery in Colon-Rectal Cancer by Tumor Regression during and after Neoadjuvant Radiochemotherapy.

The standard treatment of locally advanced rectal cancer is neoadjuvant chemoradiotherapy before surgery. For those patients experiencing a complete clinical response after the treatment, a watch-and-wait strategy with close monitoring may be practicable. In this respect, the identification of biomarkers of the response to therapy is extremely important. Many mathematical models have been developed or used to describe tumor growth, such as Gompertz's Law and the Logistic Law. Here we show that the parameters of those macroscopic growth laws, obtained by fitting the tumor evolution during and immediately after therapy, are a useful tool for evaluating the best time for surgery in this type of cancer. A limited number of experimental observations of the tumor volume regression, during and after the neoadjuvant doses, permits a reliable evaluation of a specific patient response (partial or complete recovery) for a later time, and one can evaluate a modification of the scheduled treatment, following a watch-and-wait approach or an early or late surgery. Neoadjuvant chemoradiotherapy effects can be quantitatively described by applying Gompertz's Law and the Logistic Law to estimate tumor growth by monitoring patients at regular intervals. We show a quantitative difference in macroscopic parameters between partial and complete response patients, reliable for estimating the treatment effects and best time for surgery.

'Skeletal Age' for mapping the impact of fracture on mortality.

Fragility fracture is associated with an increased risk of mortality, but mortality is not part of doctor-patient communication. Here, we introduce a new concept called 'Skeletal Age' as the age of an individual's skeleton resulting from a fragility fracture to convey the combined risk of fracture and fracture-associated mortality for an individual. We used the Danish National Hospital Discharge Register which includes the whole-country data of 1,667,339 adults in Denmark born on or before January 1, 1950, who were followed up to December 31, 2016 for incident low-trauma fracture and mortality. Skeletal age is defined as the sum of chronological age and the number of years of life lost (YLL) associated with a fracture. Cox's proportional hazards model was employed to determine the hazard of mortality associated with a specific fracture for a given risk profile, and the hazard was then transformed into YLL using the Gompertz law of mortality. During the median follow-up period of 16 years, there had been 307,870 fractures and 122,744 post-fracture deaths. A fracture was associated with between 1 and 7 years of life lost, with the loss being greater in men than women. Hip fractures incurred the greatest loss of life years. For instance, a 60-year-old individual with a hip fracture is estimated to have a skeletal age of 66 for men and 65 for women. Skeletal Age was estimated for each age and fracture site stratified by gender. We propose 'Skeletal Age' as a new metric to assess the impact of a fragility fracture on an individual's life expectancy. This approach will enhance doctor-patient risk communication about the risks associated with osteoporosis. National Health and Medical Research Council in Australia and Amgen Competitive Grant Program 2019.

Hazard-selfsimilarity of diffusions’ first passage times

A recent study introduced a novel approach to the exploration of diffusions’ first-passage times (FPTs): selfsimilarity. Specifically, consider a general diffusion process that runs over the non-negative half-line; initiating the diffusion at fixed positive levels, further consider the diffusion’s FPTs to the origin. Selfsimilarity means that the FPTs are spanned by an intrinsic scaling of their initial levels. The recent study addressed two types of selfsimilarity: stochastic, scaling the FPTs in ‘real space’; and Laplace, scaling the FPTs in ‘Laplace space’. The Laplace selfsimilarity manifests an underlying sum-like structure. Shifting from the sum-like structure to a max-like structure—a-la the shift from the Central Limit Theorem to Extreme Value Theory—this study addresses a third type of selfsimilarity: hazard, scaling the FPTs in ‘hazard space’. A comprehensive analysis of hazard-selfsimilarity is established here, including: the universal distribution of the FPTs; the dramatically different statistical behaviors that the universal distribution exhibits, and the statistical phase transition between the different behaviors; the characterization of the generative diffusion dynamics, and their universal Langevin representation; and the universal Poissonian statistics that emerge when the initial levels are scattered according to the statistical steady-state of the generative diffusion dynamics. The analysis unveils the following linkages: of the universal distribution to the Gumbel, Gompertz, and Frechet laws; of the universal Langevin representation to diffusion in quadratic and logarithmic potentials; and of the universal Poissonian statistics to non-normalizable densities, to the maxima of the exponential law, and to the harmonic Poisson process.