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Biological implications of the Weibull and Gompertz models of aging
The Journals of Gerontology; Washington; Feb 2002; Robert E Ricklefs; Alex Scheuerlein;

Part Name:  Biological sciences and medical sciences
Part Number:  Series A,
Volume:  57A
Issue:  2
Start Page:  B69-B76
ISSN:  10795006
Subject Terms:  Models
Aging
Biology
Abstract:
Gompertz and Weibull functions imply contrasting biological causes of demographic aging. The terms describing increasing mortality with age are multiplicative and additive, respectively, which could result from an increase in the vulnerability of individuals to extrinsic causes in the Gompertz model and the predominance of intrinsic causes at older ages in the Weibull model.

Full Text:
Copyright Gerontological Society of America, Incorporated Feb 2002
[Headnote]
Gompertz and Weibull functions imply contrasting biological causes of demographic aging. The terms describing increasing mortality with age are multiplicative and additive, respectively, which could result from an increase in the vulnerability of individuals to extrinsic causes in the Gompertz model and the predominance of intrinsic causes at older ages in the Weibull model. Experiments that manipulate extrinsic mortality can distinguish these biological models. To facilitate analyses of experimental data, we defined a single index for the rate of aging (omega) for the Weibull and Gompertz functions. Each function described the increase in aging-related mortality in simulated ages at death reasonably well. However, in contrast to the Weibull omega^sub W^, the Gompertz omega^sub G^ was sensitive to variation in the initial mortality rate independently of aging-related mortality. Comparisons between wild and captive populations appear to support the intrinsiccauses model for birds, but give mixed support for both models in mammals.

SENESCENCE (or aging) is a decline of physiological function with age. This decline is manifested in populations as an increase in mortality rate at older ages, which is often referred to as actuarial senescence (AS). In the absence of detailed studies on organism function, the increase in mortality rate with age has been used to compare the rate of aging in different populations and species of animals (1,2). AS also directly influences population growth potential and measures the strength of natural selection to postpone aging and its demographic consequences. Thus, increase in mortality rate with age has figured prominently in evolutionary studies of aging (3-6). When many populations are compared, it is most useful to describe the rate of aging by a single index for each population (7). This is usually accomplished by fitting a mathematical function to the relationship between rate of mortality and age or, alternatively, to the relationship between the proportion of individuals surviving and age. The coefficients of an aging model fitted to the data are used to describe the course of AS. Ideally, the rate of aging should be represented by a single index having units of 1/time (i.e., time^sup -1^). Many mathematical functions have been used to describe actuarial senescence (8,9). The most prominent of these are the Gompertz and Weibull equations.

The Gompertz and Weibull models differ in the way that early adult mortality and age-dependent mortality are related. Gerontologists ought to prefer the function that represents the underlying causes of increasing mortality with age most accurately (9,10). However, because both models are commonly used, it is also important to understand the relationship between the coefficients of the two functions (10,11). In this contribution, we distinguish essential properties of the two models, show how their coefficients are related, use simulated data sets to show the basic interchangeability of the two models and the circumstances under which they differ, and discuss some biological arguments for preferring one or the other function. Most of these points have been discussed in the literature; however, distinctions are often based on fine points of model fitting rather than the biological processes represented by the models. We argue that biological considerations should be paramount in distinguishing between models of aging, as they are likely to identify issues for future research.

Characteristics of Gompertz and Weibull Models of Aging

The Gompertz and Weibull models of AS differ primarily in the way in which age-independent and age-dependent components of mortality are related to each other. Both models ignore the typical decline in mortality that accompanies growth and development prior to maturity, although this component of mortality can be added to either model, as shown, for example, by Witten (12). Both models also incorporate a minimum mortality rate suffered by young adults prior to the onset of their physiological decline. This is usually referred to as the initial mortality rate (m^sub 0^). The models differ in the way in which mortality increases with age. In the Gompertz model, aging-related mortality increases exponentially as a multiple of the initial mortality m^sub 0^. In the Weibull model, the aging-related component of mortality is a power function of age that is added to the initial mortality rate. Thus, the initial mortality rate may be zero in the Weibull model, but it must be a positive number in the Gompertz model. Biologically, the Gompertz model implies that the increase in mortality rate with age represents increasing vulnerability to causes of mortality suffered by young adults. The exponential term of the Gompertz equation describes how rapidly this vulnerability increases with age. The vulnerability model assumes that the probability of death of each individual rises as its physiological function declines with age. In contrast, the Weibull model implies that causes of death of young adults and old individuals are different, independent, and additive. In addition, the Weibull model incorporates death that is due to catastrophic intrinsic causes whose probability increases with age. Thus, the Weibull function has been used in conjunction with failure-time models in which failure depends on the occurrence of one or more rare events, such as genetic mutations or cell deaths (10,13,14).

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Mathematical characterization.-Instantaneous, or exponential, mortality rate (m) can range between 0 and infinity. In both the Gompertz and Weibull models, m increases continuously without limit. Some models of aging-related mortality, such as the logistic function (8,9), have upper mortality plateaus and thus better describe the leveling of mortality rate at old age observed in large cohorts of flies and humans (15-19). However, the deceleration of the mortality rate among the oldest old likely reflects, at least in part, heterogeneity in aging processes among individuals (19-22) rather than a deceleration in the probability of death of a single individual. Regardless, we shall restrict this discussion to the nonasymptotic Gompertz and Weibull functions because of the practical consideration that small cohorts of individuals normally do not survive long enough to show marked deceleration of mortality rate. In addition, as we indicate below, in the Weibull model the rate of increase in the mortality rate slows with increasing age and thus can describe most survival data adequately.

DISCUSSION

How Well Does Each Model Retrieve the Input Parameters?

One objective of our simulations was to determine how well the Gompertz and Weibull models fit the same data sets. We found that both equations could be fitted reasonably well to simulated data, regardless of which model was used to generate the data. Thus, the two models of actuarial aging are roughly equivalent in their ability to characterize aging-related mortality. We also found that variation in the Gompertz estimates for a given simulation was generally lower than that for Weibull estimates, regardless of whether the curves were generated from Gompertz or Weibull models. Gompertz models yielded better retrieval of input m^sub 0^, especially when the input values were low. The Gompertz equation also yielded less variable estimates of omega when initial mortality m^sub 0^ was high, but not otherwise. Thus, it would appear that the Gompertz equation provides somewhat more consistent parameter estimates for a particular sample of ages at death, although both equations appear to produce unbiased estimates of parameter values under a variety of parameter values. Additional simulations (not shown) indicate that parameter estimates vary less as cohort (sample) size increases, to the point that differences in the quality of the fits for each equation largely disappear for samples of 1000 or more.

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Table 2.

The Weibull function tends to give more consistent estimates of the rate of aging for data in which the rate of aging is low. Because the rate of aging decreases more slowly than initial mortality in natural populations of birds, a larger proportion of individuals die of intrinsic causes in species with low rates of aging (2). Thus, the better performance of the Weibull model under these conditions argues in favor of its use for studying actuarial senescence in long-lived organisms.

Differences Between Gompertz and Weibull Models of Aging

The most important difference between the Gompertz and Weibull models is that the increase in mortality that resuits from senescence is a multiple of the initial mortality rate in the first case and is independent of the initial mortality rate in the second case. This difference has a parallel in the biological basis for actuarial senescence. The initial mortality (m^sub 0^) rate applies to individuals prior to the onset of physiological senescence for which causes of death are largely extrinsic to the organism: accidents of life that strike individuals independently of their age. Such causes of mortality include predation, physical trauma from accidents, starvation resulting from failed food supplies, extreme weather conditions, and infectious diseases. Aging may cause an increase in mortality rate above the initial level in two ways. First, general physiological decline at advancing age may increase the individual's vulnerability to the same extrinsic causes of mortality that affect young adults. Second, physiological aging may result in disease states that kill the individual independently of extrinsic mortality factors. Deaths resulting from cancers, stroke, heart disease, severe autoimmune disease, and other intrinsic causes fall into this category. Although such intrinsic aging processes may increase the vulnerability of the individual to extrinsic mortality factors, death is inevitable regardless of extrinsic agents, whose intensity has only minor direct influence on the individual's age at death.

Whether actuarial senescence in animals expresses an increase in death from extrinsic or intrinsic causes can be determined, in principle, by manipulating the strength of extrinsic causes of death. If actuarial senescence resulted from increasing vulnerability to extrinsic mortality factors, the mortality rate at a particular age would vary in direct proportion to the mortality of presenescent individuals (m^sub 0^) in the population. If actuarial senescence resulted from disease processes that cause death irrespective of external conditions, then the increase in mortality with age would be independent of m^sub 0^.

These two possibilities have mathematical parallels in the Gompertz and Weibull functions. Suppose that the aging parameters gamma (Gompertz) and alpha and beta (Weibull) represent intrinsic physiological changes in the organism that presumably are independent of most extrinsic causes of mortality in the environment. This is not to say that many environmental factors, such as radiation, diet, toxins, and stress, do not influence the rate of physiological aging. However, to the extent that aging-related mortality increases independently of the intensity of external mortality, measured values of aging parameters should be independent of variation in the value of mo in a particular population.

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Table 4.

If the exponential parameter (gamma) of the Gompertz equation represented the rate of increase in vulnerability of individuals to primarily extrinsic mortality factors that affect young adults, then the rate of mortality would be the product of this exponential term and the intensity of initial causes of mortality (m^sub 0^). Accordingly, the mortality rate at a particular age (m^sub x^) would vary in direct proportion to the extrinsic mortality rate (m^sub 0^). If aging-related causes of death were primarily intrinsic, then deaths over and above the initial mortality (m^sub x^ - m^sub 0^) would be largely independent of the environment. As a consequence, variation in environmental conditions causing a change in mo would require a compensating change in the fitted Gompertz aging parameter gamma.

In contrast, in the Weibull model, aging-related mortality is independent of the intensity of extrinsic mortality. If aging-related mortality were intrinsically caused and if extrinsic mortality were reduced experimentally even to nil (m^sub 0^ = 0), mortality rate would still increase with age as a result of disease processes that eventually resulted in death, and the estimates of alpha, beta, and omega^sub W^ would not vary. However, if aging-related mortality reflected increased vulnerability to extrinsic causes, then omega^sub W^ would vary in relation to m^sub 0^.

Evaluating Gompertz and Weibull Models by Using Biological Rationales

In the human population, the causes of deaths of young adults and old individuals differ, Excluding infant mortality, these causes are mostly extrinsic in the case of the young and intrinsic in the case of the old (27). Captive populations of rhesus macaques show a similar pattern (28). This suggests that the Weibull model may have a stronger biological rationale than the Gompertz model, but in the absence of a suitable experiment we cannot determine how the mortality rate at a particular age would change in response to a change in m^sub 0^. A relevant experiment is performed when animals are brought into captivity in laboratories or zoos, where extrinsic causes of mortality are minimized. Accordingly, the Gompertz model predicts that the increase in mortality rate as a function of age should diminish in proportion to the decrease in m^sub 0^. The intrinsic-mortality model predicts that the age-dependent increase in mortality should remain the same in captivity as in nature. Thus, for the Weibull function, the aging parameters alpha and beta should be independent of variation in m^sub 0^; that is, they should be the same in captive and natural populations. For the Gompertz function, gamma should increase to compensate for the decrease in m^sub 0^ and maintain a constant aging-dependent component of mortality.

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Figure 3.

One comparison of Weibull parameters between wild and captive populations of birds showed that although m^sub 0^ decreased markedly in captivity, omega^sub W^ remained unchanged (26). Unfortunately, the sample size in this comparison was small and few of the species in the wild and captivity were closely matched. Ricklefs and Scheuerlein (25) compared Weibull aging parameters of 12 conspecific or congeneric pairs of mammals in the wild and in captivity. In this case, 9 of the pairs exhibited lower values of omega^sub W^ in captivity than in the wild. This was true for all the species in the sample that inhabit open savannalike environments in nature where a decrease in physiological function with age is likely to reduce an individual's ability to hunt prey or escape predators. Thus, for many species of mammals the increase in mortality rate with age may reflect increasing vulnerability to extrinsic mortality factors. Nonetheless, the rate of aging remained relatively high in captivity in the absence of extrinsic mortality factors experienced in the wild, and so some component of aging-related mortality may also be intrinsic. One of the difficulties with studies of captive populations is that initial mortality rates (m^sub 0^) are only partly reduced in captivity. Thus, captivity may impose novel mortality factors, perhaps related to stress and contagious disease, which confound analyses of aging processes and may affect the course of aging.

Conclusions

The Gompertz and Weibull functions make clear distinctions between the manner in which mortality rate increases with age within a population. From an empirical standpoint, each function appears to fit age-at-death data equally well, particularly when sample sizes are large. However, the Weibull function appears to lend itself better to a single parameter (omega) describing the rate of aging in comparative studies when aging-related mortality has intrinsic causes rather than simply reflecting vulnerability to extrinsic causes. Comparisons of the rate of aging between wild and captive populations should allow one to distinguish between the Gompertz and Weibull functions on biological grounds, but results are equivocal because of (a) difficulties in finding suitable phylogenetically matched comparisons, (b) novel sources of mortality in captivity, and (c) mixed results from available comparisons. Our understanding of the causes of aging-related mortality can be guided by considering the biological implications of the mathematical functions we use to describe aging data, The distinction between intrinsic and extrinsic causes of death is difficult but also has meaning for the way mortality relates to the processes of normal aging in organisms. If aging-related mortality primarily reflected intrinsic causes that kill regardless of extrinsic factors, then each individual would maintain a high level of personal fitness until his or her relatively sudden death. If aging-related mortality reflected increasing vulnerability to extrinsic causes of death, then normal aging would be accompanied by continual deterioration of function. These Weibull-like and Gompertz-like scenarios have very different implications for how we view normal aging and the prospects for human life span and the health of the elderly population.

ACKNOWLEDGMENTS

This study was supported by National Institutes of Health Grant AG16895-01 (R. E. Ricklefs). We are grateful to David Wilson, James Curtsinger, and two anonymous reviewers for instructive comments on the manuscript.

[Reference]
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[Reference]
Received March 23, 2001
Accepted October 8, 2001
Decision Editor: John A. Faulkner, PhD

[Author note]
Robert E. Ricklefs and Alex Scheuerlein
Department of Biology, University of Missouri, St. Louis.

[Author note]
Address correspondence to Robert E. Ricklefs, Department of Biology, University of Missouri-St. Louis, 8001 Natural Bridge Road. St. Louis, MO 63121-4499. E-mail: ricklefs@umsl.edu



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