# Failure Laws In Survival Studies

Attempts to develop a fundamental quantitative theory of aging, mortality, and lifespan have deep historical roots. In 1825, the British actuary Benjamin Gompertz discovered a law of mortality (Gompertz, 1825), known today as the Gompertz law (Finch, 1990; Gavrilov and Gavrilova, 1991; Olshansky and Carnes, 1997; Strehler, 1978). Specifically, he found that the force of mortality increases in geometrical progression with the age of adult humans. According to the Gompertz law, human mortality rates double about every 8 years of adult age.

Gompertz also proposed the first mathematical model to explain the exponential increase in mortality rate with age (Gompertz, 1825). In reality, system failure rates may contain both nonaging and aging terms as, for example, in the case of the Gompertz-Makeham law of mortality (Finch, 1990; Gavrilov and Gavrilova, 1991; Makeham, 1860; Strehler, 1978):

In this formula the first, age-independent term (Makeham parameter, A) designates the constant, ''nonaging'' component of the failure rate (presumably due to external causes of death, such as accidents and acute infections), while the second, age-dependent term (the Gompertz function, Reax) designates the ''aging'' component, presumably due to deaths from age-related degenerative diseases like cancer and heart disease.

The validity of the Gompertz-Makeham law of mortality can be illustrated graphically, when the logarithms of death rates without the Makeham parameter (^x — A) are increasing with age in a linear fashion. The log-linear increase in death rates (adjusted for the Makeham term) with age is indeed a very common phenomenon for many human populations at ages 35-70 years (Gavrilov and Gavrilova, 1991).

Note that the slope coefficient a characterizes an ''apparent aging rate'' (how rapid is the age-deterioration in mortality)—if a is equal to zero, there is no apparent aging (death rates do not increase with age).

At advanced ages (after age 70), the ''old-age mortality deceleration'' takes place—death rates are increasing with age at a slower pace than expected from the Gompertz-Makeham law. This mortality deceleration eventually produces ''late-life mortality leveling-off'' and ''late-life mortality plateaus'' at extreme old ages (Curtsinger et al., 1992; Economos, 1979; 1983; Gavrilov and Gavrilova, 1991; Greenwood and Irwin, 1939; Vaupel et al., 1998). Actuaries (including Gompertz himself) first noted this phenomenon and proposed a logistic formula for mortality growth with age in order to account for mortality fall-off at advanced ages (Perks, 1932; Beard, 1959; 1971). Greenwood and Irwin (1939) provided a detailed description of this phenomenon in humans and even made the first estimates for the asymptotic value of human mortality (see also review by Olshansky, 1998). According to their estimates, the mortality kinetics of long-lived individuals is close to the law of radioactive decay with half-time approximately equal to 1 year.

The same phenomenon of ''almost nonaging'' survival dynamics at extreme old ages is detected in many other biological species. In some species the mortality plateau can occupy a sizable part of their life (see Figure 5.5).

Biologists have been well aware of mortality leveling-off since the 1960s. For example, Lindop (1961) and Sacher (1966) discussed mortality deceleration in mice. Strehler and Mildvan (1960) considered mortality deceleration at advanced ages as a prerequisite for all mathematical models of aging to explain. Later A. Economos published a series of articles claiming a priority in the discovery of a ''non-Gompertzian paradigm of mortality'' (Economos, 1979; 1980; 1983; 1985). He found that mortality leveling-off is observed in rodents (guinea pigs, rats, mice) and invertebrates (nematodes, shrimps, bdelloid rotifers, fruit flies, degenerate medusae Campanularia Flexuosa). In the 1990s the phenomenon of mortality deceleration and leveling-off became widely known after some publications demonstrated mortality leveling-off in large samples of Drosophila melanogaster (Curtsinger et al., 1992) and medflies Ceratitis capitata (Carey et al., 1992), including isogenic strains of Drosophila (Curtsinger et al., 1992; Fukui et al., 1993; 1996). Mortality plateaus at advanced ages are observed for some other insects: house fly Musca vicina and blowfly Calliphora erythrocephala (Gavrilov, 1980), bruchid

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