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Mortality-rate Crossovers and Maximum Lifespan in Advantaged and Disadvantaged Populations: Accelerated-mortality and Sudden-death Models
J. theor. Biol. (2000) 205, 171}180 doi:10.1006/jtbi.2000.2063, available online at http://www.idealibrary.com on
Mortality-rate Crossovers and Maximum Lifespan in Advantaged and Disadvantaged Populations: Accelerated-mortality and Sudden-death Models H. R. HIRSCH*-, X. LIU?
AND
T. M. WITTENA
* Department of Physiology, College of Medicine, ;niversity of Kentucky, ¸exington, K> 40536-0298, ;.S.A., ?Gulf =ar Health Center, =alter Reed Army Medical Center, =ashington, DC 20307, ;.S.A. and A
One population is advantaged relative to another by our de"nition if its survival function is greater at all ages. A population has a lifespan maximum if there is an age at which its survival function becomes exactly zero. Earlier work concerned conditions under which the mortalityrate functions of advantaged and disadvantaged populations displaying lifespan maxima always crossed. Here two survival models of populations having lifespan maxima are presented in which mortality-rate crossings between advantaged and disadvantaged subpopulations may fail to appear. One, the accelerated-mortality model, has a continuous survival function; in the other, the sudden-death model, the survival function is discontinuous. Both di!er from examples examined previously in that their mortality-rate functions become in"nite at their lifespan maxima. 2000 Academic Press
Introduction For many years considerable interest has been displayed in two applications of demography which bear upon important biological problems. First there is the question whether there is a maximum lifespan for each species which is set by a genetic program. Second, there are questions raised by the observation that age-speci"c mortality rates of certain populations cross in ways which are sometimes unanticipated (Vaupel & Yashin, 1985). Our recent work (Liu & Witten, 1995; Hirsch et al., 1996) suggests a biologically based connection between these two demo-
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graphic problems. In this paper, we further examine that relation. Two scenarios can be invoked to explain species lifespan. The "rst, here designated the programming scenario, is based on the assumption that the aging of an organism and its development are governed by a common genetic program. Individuals die at various ages from various causes, but the program determines a maximum age beyond which no member of the species survives. New treatments for diseases and improvements in the environment may rectangularize the survival function (Fries, 1980) but it remains bounded at its upper end. The concept of reliability underlies the second scenario. The genetic program which guides development ends at adulthood. The body, subject 2000 Academic Press
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to disease and environmental hazard, deteriorates because its components are not completely reliable (Witten, 1985). Repair takes place, but total repair, in terms of biological resources, is too expensive. Death ensues. The survival function approaches zero, but does not become identically zero at any speci"ed age. This reliability scenario is analogous to the way an automobile is constructed and eventually wears out (Vaupel, 1997). Demographic descriptions of the biology of lifespan are provided by Gavrilov & Gavrilova (1991) and by Wilmoth (1997). They indicate that, in principle, the programming scenario can be distinguished from the reliability scenario by examining the tail of the survival function for progressively larger populations. Unfortunately, this approach fails with respect to human and most other animal populations because of the sparsity of reliable mortality data on the very old. A less direct method which utilizes mortality-rate crossovers is investigated here. The use of a graph of intersecting mortalityrate functions as the sole illustration on the cover of the classic Gavrilov & Gavrilova monograph (1991) on the biology of lifespan suggests the importance of mortality-rate crossovers in longevity research. Reviews of studies of mortalityrate crossovers indicate that explanations of their occurrence fall into two schools of thought (Nam, 1994; Olshansky, 1995). Either the crossovers result from erroneous data and are not real (Coale & Kisker, 1986; Elo & Preston, 1994) or, after allowance is made for various reporting errors, a genuine phenomenon remains (Nam, 1994; Liu & Witten, 1995). We accept the biological reality of at least some of the reported mortality-rate crossovers for two reasons: (1) there are so many such reports that it is unlikely that all of them are in serious error and (2) most signi"cantly, crossovers have been observed in carefully controlled laboratory populations of non-human animals (Carey & Liedo, 1995; Heller et al., 1998). Biological explanations of crossovers are based on the assumption that the intersecting mortality-rate functions represent heterogeneous populations that di!er in frailty (Manton et al., 1979; Vaupel et al., 1979). A cohort of a genetically frail population experiences in its youth a mortality rate which is greater than that of a genetically
strong population. Selection eliminates less "t individuals from the frail population as its age increases. Its mortality rate increases less rapidly with age than it would otherwise because the surviving members of the population are the ones which are most "t. Fitness in this context can be interpreted in terms of age-related changes in physiological senescence parameters (Manton et al., 1994). Although the mortality rate of the strong population is initially lower than that of the frail population, its rate of increase may be greater, with the result that the mortality-rate functions cross. We have studied relationships between populations which obey the programming scenario with respect to maximum lifespan and which, in consequence, display crossovers resulting from heterogeneity with respect to frailty (Liu & Witten, 1995). Here, as in the earlier report, we consider two populations, one of which is advantaged with respect to the other in that its survival function is greater at all ages. The populations may be chosen in order to compare their genetic or other di!erences, e.g. whites vs. blacks (Manton et al., 1979) or females vs. males (Riggs, 1990). If the survival functions of the advantaged and disadvantaged populations are continuous and if the corresponding mortality-rate functions di!er initially and remain bounded at all ages, the mortality-rate functions must cross because the survival functions converge at the genetically determined maximum lifespan. The advantaged population initially has the lower mortality rate, but its mortality rate, sooner or later, must exceed that of the disadvantaged population in order to complete the required convergence of the survival functions. This is an example of mortality acceleration (Witten, 1989). Here we examine two survival models consistent with the programming scenario for which the mortality-rate functions of the advantaged and disadvantaged populations need not cross. One, the accelerated-mortality model, uses a continuous survival function. The other, the sudden-death model (Hirsch et al., 1996), employs a discontinuous survival function. Both are characterized by mortality-rate functions which become in"nite at the maximum lifespan. We begin with a brief review of survival and mortality-rate functions. Descriptions of the
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accelerated-mortality and sudden-death models are illustrated by examples in which the mortality-rate functions of advantaged and disadvantaged groups do not cross whether or not their survival functions are truncated by the imposition of a maximum lifespan. We discuss the distinctions between these examples and others (Liu & Witten, 1995) in which mortality-rate crossings occur. Survival and Mortality-rate Functions The survival function of a population, S (x), is de"ned as the proportion of a birth cohort surviving to age x. The instantaneous death rate at age x, also called the force of mortality or hazard rate, is represented by k(x): d [log S (x)] k(x)"! dx
(1a)
1 dS (x) . "! S (x) dx
(1b)
The use of eqns (1) relies upon the assumption that the death-rate and survival functions for a given population are continuous. However, it should be noted that a true survival function is a &&descending staircase'', discontinuous at the observed times of death and zero elsewhere. The use of a continuous function is satisfactory if the population size is su$ciently large. Since the survival function is monotonically decreasing, the force of mortality is nonnegative but is not necessarily less than or equal to one. Beyond 30 years of age, the mortality rate for human populations can be approximated by the classic Gompertz exponential function (Witten, 1989) k (x)"k ePV,
(2)
where the positive parameter k is the age independent hazard rate coe$cient, and the positive parameter r is the age-dependent mortality-rate coe$cient (Finch et al., 1990; Eakin et al., 1995). The corresponding Gompertz survival function can be obtained by integrating the
mortality-rate function, eqn (2), with the help of eqn (1a): S (x)"exp
k (1!ePV ) . r
(3)
The Gompertz law is used here for illustrative purposes and for reasons of simplicity. However, the examples examined below do not depend on its validity; any reasonable mortality rate function, e.g. the Weibull (1951), would lead to the same results. The use of a single exponential mortality-rate function is inconsistent with empirical data at &&oldest old'' ages (Economos, 1979; Vaupel et al., 1979; Carey et al., 1992; Curtsinger et al., 1992; Manton et al., 1994; Witten & Eakin, 1997). Current evidence indicates that human mortality rates at ages above 85 are less than that predicted by the Gompertz law. In extreme old age, mortality rates may level o! or even decline (Barrett, 1985; Riggs & Millecchia, 1992). In this age range, it is possible and sometimes convenient to represent survival by functions which are piecewise Gompertzian (Witten, 1988; Riggs & Millechhia, 1992; Fukui et al., 1993; Hirsch, 1994). Most organisms age, and for these the mortality-rate coe$cient r is positive (r'0). The more positive the value of r, the greater is the rate of aging. However, many unicellular organisms, e.g. bacteria, and some multicellular organisms do not age. The absence of ageing does not imply immortality. It simply means that the probability of death is independent of the age of the organism. For the non-ageing organism, r"0, and, from eqns (2) and (3), the mortality-rate and survival functions are
and
k(x)"k
(4)
S (x)"exp (!k x),
(5)
respectively. Equations (4) and (5) are sometimes called &&wild type'' because they can represent the survival of an organism in a wild environment in which survival is too brief for age-related factors to come into play. Negative values of r (r(0) are also possible. The life expectancy of a transistor actually increases with age. This is not true of any biological
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organism throughout its entire lifespan. However, the declines in old-age mortality rates cited above can be described by exponential functions for which r is negative (Hirsch, 1994). Equations (2)}(5) can be used to describe mortality rate and survival functions of advantaged and disadvantaged subpopulations. These are respectively denoted by the subscripts A and D. The survival functions are equal at the least and greatest values of age which are examined, but for all intermediate ages,
population is exactly zero. Otherwise stated, there is no force of mortality when there are no members of a population left to die. With the use of the Gompertz functions, eqns (2) and (3), and eqns (8),
S (x)'S (x). "
See Witten (1985) and Eakin and Witten (1995) for further discussion.
(6)
k*(x)"
k (x) , 1!S (w)/S (x)
x)w,
(9a)
k ePV , 1!S(w)/S(x)
x)w,
(9b)
The Accelerated-mortality Model There is no maximum lifespan associated with the Gompertz model discussed above. The survival function approaches zero with increasing age but does not become exactly zero. There are many ways to truncate such a function to reduce it to zero at ages above a postulated maximum lifespan, w. One of the simplest is to subtract S (w) from the survival function and to renormalize it so that the new survival function has the value 1 at age zero. Thus,
S* (x)"
S (x)!S (w) , 1!S(w)
x)w,
0,
x'w,
(7)
where S* (x) is a truncated survival function displaying a maximum lifespan, and S (x) is the extended survival function from which it is derived. S* (x) is a continuous function, but its derivative is not. The truncated mortality-rate function is calculated in the same way as in eqns (1): d [log S* (x)] k* (x)"! dx 1 dS* (x) . "! S* (x) dx
(8a)
(8b)
If eqn (8b) were to be applied to the case x'w, a &&0/0'' indeterminate form would result. In fact, the force of mortality is unde"ned beyond the maximum lifespan because the size of the
MORTALITY IN HUMAN POPULATIONS
In large human populations, S (w) is quite small, and the truncated survival and mortalityrate functions di!er very little from their extended forms. In the world population of several billions, w is at least 122 years (Wilmoth, 1997), and S (w) is of the order of magnitude of 10\. The di!erence between S* (x) and S (x) is unobservable in practice. The truncation transformation, eqn (7), e!ectively &&stretches'' the ordinate of the survival function. If S(w) is small, the degree of stretch is small. Consider as an example the mortality data reported by Riggs (1990) for women in the United States during the period 1956}1987. Values of the Gompertz parameters, r"0.03727 and k "6.2748;10\, applicable to these data were obtained by Hirsch (1994). Substitution of the parameter values in eqns (2), (3), (7) and (9) allows calculation of the numerical values of the extended and trancated survival and mortalityrate functions. Suppose that the Gompertzian survival function based on Riggs (1990) data is truncated at a maximum lifespan of 85 yr. This value, although unrealistically low, provides an instructive illustration of the e!ects of truncation. Even though the survival fraction for women at age 85, 0.341 (Hirsch, 1994) is not small, the concavedownward shape of the survival function through age 85 is retained under the truncation transformation (Fig. 1). However, the transformation produces a qualitative change in the shape of the mortality-rate function (Fig. 2). The mortality
MAXIMUM LIFESPAN AND MORTALITY
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FIG. 1. Extended and truncated survival functions, SD (x) and S*D (x), respectively, based on Riggs' (1990) data for women. Accelerated-mortality model.
FIG. 3. Di!erence between male (disadvantaged) and female (advantaged) truncated mortality-rate functions, k*K (x) and k*D (x), respectively, in semi-logarithmic coordinates. Accelerated-mortality model. Based on Riggs' (1990) data as in Figs 1 and 2. The di!erence, k*K (x)!k*D (x), has the same sign, positive, at all ages, indicating that there are no crossovers between k*K and k*D.
FIG. 2. Extended and truncated logarithmic mortalityrate functions, kD(x) and k*D (x), respectively, corresponding to the survival functions for women shown in Fig. 1 (Riggs, 1990). Accelerated-mortality model. k*D (x) approaches a vertical asymptote, indicated by a dashed line, at age 85.
rate k (x) of the extended Gompertz function S (x) is a straight line in logarithmic coordinates, since it is a simple exponential; the mortality rate k* (x) of the truncated function S* (x) becomes in"nite due to the singularity introduced in the denominator of eqn (9b). The mortality rate in rectangular coordinates approaches a vertical asymptote hyperbolically at the maximum lifespan. The generality of this result is demonstrated in an appendix to the present report. We refer to survival and mortality-rate functions, truncated as in eqns (7) and (9), as the
accelerated-mortality model because of the everincreasing slope of the log mortality rate as age approaches the maximum lifespan. Equation (9b), in which the Gompertz law is truncated, is an example or special case of the acceleratedmortality model. It should be noted that accelerated mortality is related to but di!ers from the concept of acceleration of aging treated in an earlier paper (Witten, 1989). Riggs' (1990) data support the well-known observation that women are an advantaged population relative to men in developed countries in the 20th century, i.e. their survival at all ages is greater. In order to investigate crossovers in the extended and truncated mortality-rate functions, we compared his data for men in the United States during the period 1956}1987 with the data cited above for women. The reported death rates do not cross, and, as indicated in Fig. 3, no crossovers appear in the mortality-rate functions after they are truncated to reduce the maximum lifespan to the postulated value of 85 yr.
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MORTALITY IN NON-AGEING POPULATIONS
Similar results are obtained upon the application of the accelerated-mortality model to non-ageing populations. As indicated earlier, the mortality-rate and survival functions of nonaging organisms are given by eqns (4) and (5), respectively. As an example, consider a hypothetical non-aging organism for which the constant mortality rate k "0.02 yr\. This cor responds to a survival half-life of approximately 36.5 yr. If a maximum lifespan of 100 yr is imposed upon this organism with the use of the accelerated-mortality model, eqns (7) and (9), its mortality rate is no longer constant (Fig. 4); the organism now ages. The mortality rate accelerates upward, at "rst gradually, but, as the maximum lifespan approaches, quite rapidly. The population for which k "0.02 yr\ is advantaged relative to one for which k " 0.022 yr\. The half-life of the latter, disadvantaged, population is approximately 31.5 yr. Neither the exponential survival functions nor the constant mortality-rate functions of the advantaged and disadvantaged populations cross. More importantly, no crossovers occur in these functions when they are truncated with the use of the accelerated-mortality model. This is illustrated in Fig. 5, which shows that the di!erence between the mortality rates of the advantaged and disadvantaged populations, when truncated at 100 yr, has the same sign, negative, at all ages between 0 and 100.
FIG. 5. Di!erence in truncated mortality-rate functions between advantaged and disadvantaged populations for which the extended mortality-rate functions have constant values of 0.02 and 0.022 yr\, respectively. Acceleratedmortality model. The di!erence in the truncated functions has the same sign, negative, at all ages, indicating that the functions do not cross.
The foregoing is a simple example of the scenario examined by Liu & Witten (1995) in which the mortality rates of the advantaged and disadvantaged populations are proportional. It illustrates the general result that crossovers do not occur if the mortality rates are in constant ratio. The Sudden-death Model Another simple way in which a population may display a maximum lifespan is by abrupt termination of its survival function at age w. Thus
S (x), x)w,
(10a)
x'w,
(10b)
S* (x)"
0
where S* (x) is the truncated survival function and is discontinuous. Equations (10) are illustrated in Fig. 6 with the use of the same data (Riggs, 1990) which were applied to the accelerated-mortality model in Fig. 1. The truncated mortality-rate function associated with eqns (10) is FIG. 4. Truncated mortality-rate function of a population for which the extended mortality-rate function has the constant value 0.02 yr\. Accelerated-mortality model. The mortality rate approaches a vertical asymptote, indicated by a dashed line, at age 100.
k* (x)"k (x)#S (w) u (x!w), x)w,
(11)
where u (x!w) is a unit impulse function at age x"w. An example of a truncated mortality-rate
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FIG. 6. Truncated survival function based on Riggs' (1990) data for women. Sudden-death model.
function based on the same data as Fig. 6 is shown in Fig. 7. In eqn (11), just as in eqns (8), the value of kH(x) is unde"ned if x'w. Once again this re#ects the concept that there is no force of mortality when there are no remaining members of the population. Since the survival function in eqn (10a) is assumed continuous except at age x"w, the mortality-rate function is de"ned for ages x(w. Owing to the use of the unit impulse function in eqn (11), the mortality rate is also de"ned at x"w. The unit impulse function, like the Dirac delta function, is in"nite where the value of its argument is zero and is zero elsewhere. It serves to represent the derivative of an otherwise continuous function at a point of discontinuity (Sokolniko! & Redhe!er, 1958). Thus w, the maximum lifespan, is the age at which sudden death strikes a population. Prior to age w, the existence of a lifespan maximum has no e!ect. Then, at age w, the mortality rate is very high for a very brief time, and the population is eliminated. In an earlier report (Hirsch et al., 1996) we called this a suicide model. As with the accelerated-mortality model, instantaneous truncation of the survival and mortality-rate functions would have very little e!ect if applied to a large population, such as the world population of
FIG. 7. Truncated mortality-rate function corresponding to the survival function for women show in Fig. 6 (Riggs, 1990). Sudden-death model. The vertical arrow at age 85 represents an impulse function. Prior to age 85, the extended and truncated mortality-rate functions are identical. At age 85, the mortality rate becomes in"nite, and the entire population dies.
humans, for which S(w) is very small. However, the sudden-death model could correctly describe populations for which aging is programmed. These might include may#ies, bamboo, salmon which die soon after spawning and populations of cells which die by apoptosis. The appearance of sudden death in a population can have no in#uence on the existence of mortality-rate crossovers between advantaged and disadvantaged populations, since the extended and truncated forms of the survival and mortality-rate functions are unchanged by the presence of a lifespan maximum for ages x(w. Crossovers will be present in the truncated mortality-rate functions obtained with the use of eqns (11) if and only if they are present in the original extended functions. Advantaged and disadvantaged populations having proportional mortality-rate parameters were discussed in one of our earlier reports (Liu & Witten, 1995). It is evident in the case of proportional constant mortality rates examined above and in the sudden-death model that the mortality-rate functions associated with such populations need not necessarily cross. Discussion The biological signi"cance of these results lies most importantly in their relevance to a
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previously explored linkage between the existence of lifespan maxima in populations and crossovers between their mortality rates. In an earlier paper (Liu & Witten, 1995), we examined a class of continuous survival functions for which the existence of a maximum lifespan required a crossover in the modality-rate functions of advantaged and disadvantaged subpopulations. Here we treat two classes of survival functions having maximum lifespans, one the continuous acceleratedmortality model, and the other, the discontinuous sudden-death model, for which mortalityrate crossings need not occur. We demonstrate that, despite the existence of a maximum lifespan for these classes, the mortality-rate functions of advantaged subpopulations need not cross those of disadvantaged subpopulations by providing examples in which such crossings do not occur. It should be observed that a crossover of the survival functions at a particular age imposes the condition that the mortality-rate functions must also cross at an earlier age (Hirsch, 1995, 1997). In Liu & Witten (1995), we discussed the consequences of maximum lifespan for advantaged and disadvantaged populations having continuous survival functions and mortality di!erences which are proportional over the lifespan. We concluded that the disadvantaged group would die out before the advantaged and that this result appeared to be inconsistent with what is known about human population dynamics. However, our results in the present work indicate that proportional mortality di!erences are consistent with survival functions having maximum lifespans if, as in the sudden-death model, the survival functions are discontinuous. Wilmoth (1997) states that the mortality rate must approach in"nity as age approaches the maximum lifespan. The approach to in"nity occurs in both the accelerated-mortality and sudden-death models but not in the scenarios examined by Liu & Witten (1995). In these, the mortality rates reached high but "nite values at the maximum age. The connection described between maximum lifespan and mortality-rate crossovers is associated with the failure of the mortality-rate functions to approach a vertical asymptote at the maximum lifespan. This is consistent with data cited earlier which demonstrate leveling of mortality rates in extreme old age.
In certain instances, e.g. those of programmed death in several nonhuman populations mentioned above, the sudden-death model appears to be an adequate choice. With respect to human populations, the applicability of a particular model is unclear. Wilmoth (1997) comments on the di$culty of observing and documenting the existence of a maximum lifespan with our present methods. The problem is that so few people live to ages approaching the maximum. Methods previously applied to small populations of laboratory animals (Hirsch & Peretz, 1984) as well as other methods currently under development may prove to be helpful in examining the demographic characteristics of this small group of survivors and, ultimately, in determining whether human populations have lifespan maxima. Conclusions 1. The existence of a maximum lifespan does not necessarily imply a crossing of the mortalityrate functions of advantaged and disadvantaged populations if the mortality-rate functions approach in"nity at the maximum lifespan. 2. Mortality-rate functions of advantaged and disadvantaged populations may be proportional throughout life and may therefore fail to cross if the survival functions of the populations are discontinuous. 3. If the survival function of a population is truncated as in the accelerated-mortality model, its mortality-rate function approaches a vertical asymptote at the maximum lifespan hyperbolically. 4. In most populations, such as the human, few members survive to an age at which a maximum, if it exists, might be observed. Distinctions among classes of survival functions, continuous or discontinuous, are di$cult to establish. Consequently, improved statistical techniques for examining such functions should be diligently pursued to achieve the goal of providing better data for populations studies used in gerontological research. REFERENCES BARRETT, J. C. (1985). The Mortality of Centenarians in England and Wales. Arch. Gerontol. Geriatrics 4, 211}218.
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CAREY, J. R. & LIEDO, P. (1995). Sex-speci"c life table aging rates in large Med#y cohorts. Exp. Gerontol. 30, 315}325. CAREY, J. R., LIEDO, P., OROZCO, D. & VAUPEL, J. W. (1992). Slowing of mortality rates at older ages in large Med#y cohorts. Science 258, 457}461. COALE, A. J. & KISKER, E. E. (1986). Mortality crossovers: reality or bad data? Popul. Stud. 40, 389}401. CURTSINGER, J. W., FUKUI, H. H., TOWNSEND, D. R. & VAUPEL, J. W. (1992). Demography of genotypes: failure of the limited life span paradigm in Drosophila melanogaster. Science 258, 461}463. EAKIN, T. & WITTEN, M. (1995). A gerontological distance metric for analysis of survival dynamics. Mech. Ageing Dev. 78, 85}101. EAKIN, T., SHOUMAN, R., QI, Y., LIU, G. & WITTEN, M. (1995). Estimating parametric survival model parameters in gerontological aging studies: methodological problems and insights. J. Gerontol.: Biol. Sci. 50A, B166}B176. ECONOMOS, A. C. (1979). A non-gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE 2, 74}76. ELO, I. T. & PRESTON, S. H. (1994). Estimating African}American mortality from inaccurate data. Demography 31, 427}458. FINCH, C. E., PIKE, M. C. & WITTEN, M. (1990). Slow mortality rate accelerations during aging in some animals approximate that of humans. Science 249, 902}905. FRIES, J. F. (1980). Aging, natural death, and the compression of morbidity. New England J. Med. 303, 130}135. FUKUI, H. H., XIU, L. & CURTSINGER, J. W. (1993). Slowing of age-speci"c mortality rates in Drosophila melanogaster. Exp. Gerontol. 28, 585}599. GAVRILOV, L. A. & GAVRILOVA, N. S. (1991). ¹he Biology of ¸ifespan. Chur, Switzerland: Harwood. HELLER, D. A., AHERN, F. M., STOUT, J. T. & MCCLEARN, G. E. (1998). Mortality and biomarkers of aging in heterogeneous stocks (HS) mice. J. Gerontol.: Biol. Sci. 53A, B217}B230. HIRSCH, H. R. (1994). Can an improved environment cause maximum lifespan to decrease? Comments on lifespan criteria and longitudinal gompertzian analysis. Exp. Gerontol 29, 119}137. HIRSCH, H. R. (1995). Do intersections of mortality-rate and survival functions have signi"cance? Exp. Gerontol. 30, 147}167. HIRSCH, H. R. (1997). Intersections of mortality-rate and survival functions: model-independent considerations. Exp. Gerontol. 32, 287}296. HIRSCH, H. R., LIU, X. & WITTEN, M. (1996). Mortality-rate crossovers when suicide is genetically programmed. AGE 19, 172. HIRSCH, H. R. & PERETZ, B. (1984). Survival and aging of a small laboratory population of a marine mollusc, Aplysia californica. Mech. Ageing Dev. 27, 43}62. LIU, X. & WITTEN, M. (1995). A biologically based explanation for mortality crossover in human populations. Gerontologist 35, 609}615. MANTON, K. G., POSS, S. S. & WING, S. (1979). The black/white mortality crossover: investigation from the perspective of components of aging. Gerontologist 19, 291}300. MANTON, K. G., STALLARD, E., WOODBURY, M. A. & DOWD, J. E. (1994). Time-varying covariates in models of human mortality and aging: multidimensional general-
izations of the gompertz. J. Gerontol.: Biol. Sci. 49, B169}B190. NAM, C. B. (1994). Another look at mortality crossovers. Social Biol. 42, 133}142. OLSHANSKY, S. J. (1995). Mortality crossovers and selective survival in human and nonhuman populations. Gerontologist 35, 583}587. RIGGS, J. E. (1990). Longitudinal Gompertzian analysis of adult human mortality in the U.S., 1900}1986. Mech. Aging Dev. 54, 235}247. RIGGS, J. E. & MILLECCHIA, R. J. (1992). Mortality among the elderly in the U.S., 1956}1987: demonstration of the upper boundary to Gompertzian mortality. Mech. Aging Dev. 62, 191}199. SOKOLNIKOFF, I. S. & REDHEFFER, R. M. (1958). Mathematics of Physics and Modern Engineering. New York: McGraw-Hill. VAUPEL, J. W. (1997). Theories of mortality and advanced ages. In: Between Zeus and the Salmon (Wachter, K. W. & Finch, C. E., eds), pp. 17}37. Washington, DC: National Academy Press. VAUPEL, J. W., MANTON, K. G. & STALLARD, E. (1979). The impact of heterogeneity in invidual frailty on the dynamics of mortality. Demography 16, 439}454. VAUPEL, J. W. & YASHIN, A. I. (1985). Heterogeneity ruses: some surprising e!ects of selection of population dynamics. Am. Statist. 39, 176}185. WEIBULL, W. (1951). A statistical distribution of wide applicability. J. Appl. Mech. 18, 293}297. WILMOTH, J. R. (1997). In search of limit. In: Between Zeus and the Salmon (Wachter, K. W. & Finch, C. E., eds), pp. 38}64. Washington, DC: National Academy Press. WITTEN, M. (1985). A return to time, cells, systems and aging: III. Gompertzian models of biological aging and some possible roles for critical elements. Mech. Aging Dev. 32, 141}177. WITTEN, M. (1988). A return to time, cells, systems and aging: V. further thoughts on gompertzian survival dynamics*the geriatric years. Mech. Aging Dev. 46, 175}200. WITTEN, M. (1989). Quantifying the concepts of rate and acceleration/deceleration of aging. Growth Dev. Aging 53, 7}16. WITTEN, T. M. & EAKIN, T. (1985). Multiphasic models of survival: analysis of mortality rate change regions and the issue of "nite species lifespan. Exp. Gerontol. 32, 259}285.
APPENDIX Hyperbolic Approach of the Mortality-Rate Function to a Vertical Asymptote at the Maximum Lifespan in the Accelerated-Mortality Model Examination of mortality rates from truncated survival functions based on Riggs' (1990) data (Table A1) suggests that mortality-rate functions approximate hyperbolic curves as age approaches the maximum lifespan. The following simple derivation supports the validity of this observation in the accelerated-mortality model.
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TABLE A1 Comparison of the hyperbolic mortality function with truncated mortality-rate data x 75 80 83 84 84.5 85.8 84.9 84.95 84.98 84.99
k*D
k*K
eK
k*D
eD
0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 50.00 100.00
0.1033 0.2171 0.5273 1.0310 2.0329 5.0340 10.0344 20.0346 50.0347 100.0348
!3.22 !7.88 !5.19 !3.01 !1.62 !0.676 !0.343 !0.173 !0.0694 !0.0345
0.0847 0.1921 0.4983 1.0007 2.0020 5.0028 10.0030 20.0032 50.0032 100.0033
18.00 4.11 0.336 !0.072 !0.099 !0.055 !0.030 !0.016 !0.006 !0.003
x: age (yr). k*D (yr\)"1/(85!x): Hyperbolic mortality function, eqn (A.4). Approaches a vertical asymptote at age 85. k*K (yr\): Riggs' (1990) mortality rates for men truncated at age 85 with the use of eqn (7). eK"100(k*D/k*K!1): Percent error caused by replacement of the truncated mortality rates for men with the hyperbolic mortality function. k*D (yr\): Riggs' (1990) mortality rates for women truncated at age 85 with the use of eqn (7). eD"100 (k*D/k*D!1): Percent error caused by replacement of the truncated mortality rates for women with the hyperbolic mortality function.
An approximate form of the mortality-rate function, k*D (x), is de"ned by replacing the di!erentials dS* and dx in eqn (8b) with "nite di!erences: D
1
k* (x)"! S* (x)
DS* (x) Dx
,
(A.1)
where DS* (x)"S* (x)!S* (w),
(A.2)
and Dx"x!w.
(A.3)
With the use of eqn (7), S* (w)"0, and, from eqn (A.2), DS* (x)"S* (x). Simultaneous solution of eqns (A.1)}(A.3) then yields 1 k*D (x)"! . w!x
(A.4)
Thus, the approximate form of the mortality rate is hyperbolic. The corresponding survival
function, S*D, is linear: S*D (x)"1!x/w.
(A.5)
The relation between eqns (A.4) and (A.5) can be veri"ed by substitution of the derivative of eqn (A.5) into eqn (8b). It should be noted that the only approximation used to obtain eqn (A.4) is the substitution of "nite di!erences for di!erentials. The results in Table A1 provide two examples in which the hyperbolic approximation to the mortality-rate function corresponds closely to human data truncated with the use of the accelerated mortality model. The degree of agreement with the data for women is better than that for men, but, in both cases, the percent errors caused by the use of the hyperbolic functions are very small at ages which approach the maximum lifespan. The same conclusion is supported by a power-series expansion of the Gompertz function.
Mortality-rate Crossovers and Maximum Lifespan in Advantaged and Disadvantaged Populations: Accelerated-mortality and Sudden-death Models H. R. HIRSCH*-, X. LIU?
AND
T. M. WITTENA
* Department of Physiology, College of Medicine, ;niversity of Kentucky, ¸exington, K> 40536-0298, ;.S.A., ?Gulf =ar Health Center, =alter Reed Army Medical Center, =ashington, DC 20307, ;.S.A. and A
One population is advantaged relative to another by our de"nition if its survival function is greater at all ages. A population has a lifespan maximum if there is an age at which its survival function becomes exactly zero. Earlier work concerned conditions under which the mortalityrate functions of advantaged and disadvantaged populations displaying lifespan maxima always crossed. Here two survival models of populations having lifespan maxima are presented in which mortality-rate crossings between advantaged and disadvantaged subpopulations may fail to appear. One, the accelerated-mortality model, has a continuous survival function; in the other, the sudden-death model, the survival function is discontinuous. Both di!er from examples examined previously in that their mortality-rate functions become in"nite at their lifespan maxima. 2000 Academic Press
Introduction For many years considerable interest has been displayed in two applications of demography which bear upon important biological problems. First there is the question whether there is a maximum lifespan for each species which is set by a genetic program. Second, there are questions raised by the observation that age-speci"c mortality rates of certain populations cross in ways which are sometimes unanticipated (Vaupel & Yashin, 1985). Our recent work (Liu & Witten, 1995; Hirsch et al., 1996) suggests a biologically based connection between these two demo-
-Author to whom correspondence should be addressed. 0022}5193/00/140171#10 $35.00/0
graphic problems. In this paper, we further examine that relation. Two scenarios can be invoked to explain species lifespan. The "rst, here designated the programming scenario, is based on the assumption that the aging of an organism and its development are governed by a common genetic program. Individuals die at various ages from various causes, but the program determines a maximum age beyond which no member of the species survives. New treatments for diseases and improvements in the environment may rectangularize the survival function (Fries, 1980) but it remains bounded at its upper end. The concept of reliability underlies the second scenario. The genetic program which guides development ends at adulthood. The body, subject 2000 Academic Press
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to disease and environmental hazard, deteriorates because its components are not completely reliable (Witten, 1985). Repair takes place, but total repair, in terms of biological resources, is too expensive. Death ensues. The survival function approaches zero, but does not become identically zero at any speci"ed age. This reliability scenario is analogous to the way an automobile is constructed and eventually wears out (Vaupel, 1997). Demographic descriptions of the biology of lifespan are provided by Gavrilov & Gavrilova (1991) and by Wilmoth (1997). They indicate that, in principle, the programming scenario can be distinguished from the reliability scenario by examining the tail of the survival function for progressively larger populations. Unfortunately, this approach fails with respect to human and most other animal populations because of the sparsity of reliable mortality data on the very old. A less direct method which utilizes mortality-rate crossovers is investigated here. The use of a graph of intersecting mortalityrate functions as the sole illustration on the cover of the classic Gavrilov & Gavrilova monograph (1991) on the biology of lifespan suggests the importance of mortality-rate crossovers in longevity research. Reviews of studies of mortalityrate crossovers indicate that explanations of their occurrence fall into two schools of thought (Nam, 1994; Olshansky, 1995). Either the crossovers result from erroneous data and are not real (Coale & Kisker, 1986; Elo & Preston, 1994) or, after allowance is made for various reporting errors, a genuine phenomenon remains (Nam, 1994; Liu & Witten, 1995). We accept the biological reality of at least some of the reported mortality-rate crossovers for two reasons: (1) there are so many such reports that it is unlikely that all of them are in serious error and (2) most signi"cantly, crossovers have been observed in carefully controlled laboratory populations of non-human animals (Carey & Liedo, 1995; Heller et al., 1998). Biological explanations of crossovers are based on the assumption that the intersecting mortality-rate functions represent heterogeneous populations that di!er in frailty (Manton et al., 1979; Vaupel et al., 1979). A cohort of a genetically frail population experiences in its youth a mortality rate which is greater than that of a genetically
strong population. Selection eliminates less "t individuals from the frail population as its age increases. Its mortality rate increases less rapidly with age than it would otherwise because the surviving members of the population are the ones which are most "t. Fitness in this context can be interpreted in terms of age-related changes in physiological senescence parameters (Manton et al., 1994). Although the mortality rate of the strong population is initially lower than that of the frail population, its rate of increase may be greater, with the result that the mortality-rate functions cross. We have studied relationships between populations which obey the programming scenario with respect to maximum lifespan and which, in consequence, display crossovers resulting from heterogeneity with respect to frailty (Liu & Witten, 1995). Here, as in the earlier report, we consider two populations, one of which is advantaged with respect to the other in that its survival function is greater at all ages. The populations may be chosen in order to compare their genetic or other di!erences, e.g. whites vs. blacks (Manton et al., 1979) or females vs. males (Riggs, 1990). If the survival functions of the advantaged and disadvantaged populations are continuous and if the corresponding mortality-rate functions di!er initially and remain bounded at all ages, the mortality-rate functions must cross because the survival functions converge at the genetically determined maximum lifespan. The advantaged population initially has the lower mortality rate, but its mortality rate, sooner or later, must exceed that of the disadvantaged population in order to complete the required convergence of the survival functions. This is an example of mortality acceleration (Witten, 1989). Here we examine two survival models consistent with the programming scenario for which the mortality-rate functions of the advantaged and disadvantaged populations need not cross. One, the accelerated-mortality model, uses a continuous survival function. The other, the sudden-death model (Hirsch et al., 1996), employs a discontinuous survival function. Both are characterized by mortality-rate functions which become in"nite at the maximum lifespan. We begin with a brief review of survival and mortality-rate functions. Descriptions of the
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accelerated-mortality and sudden-death models are illustrated by examples in which the mortality-rate functions of advantaged and disadvantaged groups do not cross whether or not their survival functions are truncated by the imposition of a maximum lifespan. We discuss the distinctions between these examples and others (Liu & Witten, 1995) in which mortality-rate crossings occur. Survival and Mortality-rate Functions The survival function of a population, S (x), is de"ned as the proportion of a birth cohort surviving to age x. The instantaneous death rate at age x, also called the force of mortality or hazard rate, is represented by k(x): d [log S (x)] k(x)"! dx
(1a)
1 dS (x) . "! S (x) dx
(1b)
The use of eqns (1) relies upon the assumption that the death-rate and survival functions for a given population are continuous. However, it should be noted that a true survival function is a &&descending staircase'', discontinuous at the observed times of death and zero elsewhere. The use of a continuous function is satisfactory if the population size is su$ciently large. Since the survival function is monotonically decreasing, the force of mortality is nonnegative but is not necessarily less than or equal to one. Beyond 30 years of age, the mortality rate for human populations can be approximated by the classic Gompertz exponential function (Witten, 1989) k (x)"k ePV,
(2)
where the positive parameter k is the age independent hazard rate coe$cient, and the positive parameter r is the age-dependent mortality-rate coe$cient (Finch et al., 1990; Eakin et al., 1995). The corresponding Gompertz survival function can be obtained by integrating the
mortality-rate function, eqn (2), with the help of eqn (1a): S (x)"exp
k (1!ePV ) . r
(3)
The Gompertz law is used here for illustrative purposes and for reasons of simplicity. However, the examples examined below do not depend on its validity; any reasonable mortality rate function, e.g. the Weibull (1951), would lead to the same results. The use of a single exponential mortality-rate function is inconsistent with empirical data at &&oldest old'' ages (Economos, 1979; Vaupel et al., 1979; Carey et al., 1992; Curtsinger et al., 1992; Manton et al., 1994; Witten & Eakin, 1997). Current evidence indicates that human mortality rates at ages above 85 are less than that predicted by the Gompertz law. In extreme old age, mortality rates may level o! or even decline (Barrett, 1985; Riggs & Millecchia, 1992). In this age range, it is possible and sometimes convenient to represent survival by functions which are piecewise Gompertzian (Witten, 1988; Riggs & Millechhia, 1992; Fukui et al., 1993; Hirsch, 1994). Most organisms age, and for these the mortality-rate coe$cient r is positive (r'0). The more positive the value of r, the greater is the rate of aging. However, many unicellular organisms, e.g. bacteria, and some multicellular organisms do not age. The absence of ageing does not imply immortality. It simply means that the probability of death is independent of the age of the organism. For the non-ageing organism, r"0, and, from eqns (2) and (3), the mortality-rate and survival functions are
and
k(x)"k
(4)
S (x)"exp (!k x),
(5)
respectively. Equations (4) and (5) are sometimes called &&wild type'' because they can represent the survival of an organism in a wild environment in which survival is too brief for age-related factors to come into play. Negative values of r (r(0) are also possible. The life expectancy of a transistor actually increases with age. This is not true of any biological
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H. R. HIRSCH E¹ A¸.
organism throughout its entire lifespan. However, the declines in old-age mortality rates cited above can be described by exponential functions for which r is negative (Hirsch, 1994). Equations (2)}(5) can be used to describe mortality rate and survival functions of advantaged and disadvantaged subpopulations. These are respectively denoted by the subscripts A and D. The survival functions are equal at the least and greatest values of age which are examined, but for all intermediate ages,
population is exactly zero. Otherwise stated, there is no force of mortality when there are no members of a population left to die. With the use of the Gompertz functions, eqns (2) and (3), and eqns (8),
S (x)'S (x). "
See Witten (1985) and Eakin and Witten (1995) for further discussion.
(6)
k*(x)"
k (x) , 1!S (w)/S (x)
x)w,
(9a)
k ePV , 1!S(w)/S(x)
x)w,
(9b)
The Accelerated-mortality Model There is no maximum lifespan associated with the Gompertz model discussed above. The survival function approaches zero with increasing age but does not become exactly zero. There are many ways to truncate such a function to reduce it to zero at ages above a postulated maximum lifespan, w. One of the simplest is to subtract S (w) from the survival function and to renormalize it so that the new survival function has the value 1 at age zero. Thus,
S* (x)"
S (x)!S (w) , 1!S(w)
x)w,
0,
x'w,
(7)
where S* (x) is a truncated survival function displaying a maximum lifespan, and S (x) is the extended survival function from which it is derived. S* (x) is a continuous function, but its derivative is not. The truncated mortality-rate function is calculated in the same way as in eqns (1): d [log S* (x)] k* (x)"! dx 1 dS* (x) . "! S* (x) dx
(8a)
(8b)
If eqn (8b) were to be applied to the case x'w, a &&0/0'' indeterminate form would result. In fact, the force of mortality is unde"ned beyond the maximum lifespan because the size of the
MORTALITY IN HUMAN POPULATIONS
In large human populations, S (w) is quite small, and the truncated survival and mortalityrate functions di!er very little from their extended forms. In the world population of several billions, w is at least 122 years (Wilmoth, 1997), and S (w) is of the order of magnitude of 10\. The di!erence between S* (x) and S (x) is unobservable in practice. The truncation transformation, eqn (7), e!ectively &&stretches'' the ordinate of the survival function. If S(w) is small, the degree of stretch is small. Consider as an example the mortality data reported by Riggs (1990) for women in the United States during the period 1956}1987. Values of the Gompertz parameters, r"0.03727 and k "6.2748;10\, applicable to these data were obtained by Hirsch (1994). Substitution of the parameter values in eqns (2), (3), (7) and (9) allows calculation of the numerical values of the extended and trancated survival and mortalityrate functions. Suppose that the Gompertzian survival function based on Riggs (1990) data is truncated at a maximum lifespan of 85 yr. This value, although unrealistically low, provides an instructive illustration of the e!ects of truncation. Even though the survival fraction for women at age 85, 0.341 (Hirsch, 1994) is not small, the concavedownward shape of the survival function through age 85 is retained under the truncation transformation (Fig. 1). However, the transformation produces a qualitative change in the shape of the mortality-rate function (Fig. 2). The mortality
MAXIMUM LIFESPAN AND MORTALITY
175
FIG. 1. Extended and truncated survival functions, SD (x) and S*D (x), respectively, based on Riggs' (1990) data for women. Accelerated-mortality model.
FIG. 3. Di!erence between male (disadvantaged) and female (advantaged) truncated mortality-rate functions, k*K (x) and k*D (x), respectively, in semi-logarithmic coordinates. Accelerated-mortality model. Based on Riggs' (1990) data as in Figs 1 and 2. The di!erence, k*K (x)!k*D (x), has the same sign, positive, at all ages, indicating that there are no crossovers between k*K and k*D.
FIG. 2. Extended and truncated logarithmic mortalityrate functions, kD(x) and k*D (x), respectively, corresponding to the survival functions for women shown in Fig. 1 (Riggs, 1990). Accelerated-mortality model. k*D (x) approaches a vertical asymptote, indicated by a dashed line, at age 85.
rate k (x) of the extended Gompertz function S (x) is a straight line in logarithmic coordinates, since it is a simple exponential; the mortality rate k* (x) of the truncated function S* (x) becomes in"nite due to the singularity introduced in the denominator of eqn (9b). The mortality rate in rectangular coordinates approaches a vertical asymptote hyperbolically at the maximum lifespan. The generality of this result is demonstrated in an appendix to the present report. We refer to survival and mortality-rate functions, truncated as in eqns (7) and (9), as the
accelerated-mortality model because of the everincreasing slope of the log mortality rate as age approaches the maximum lifespan. Equation (9b), in which the Gompertz law is truncated, is an example or special case of the acceleratedmortality model. It should be noted that accelerated mortality is related to but di!ers from the concept of acceleration of aging treated in an earlier paper (Witten, 1989). Riggs' (1990) data support the well-known observation that women are an advantaged population relative to men in developed countries in the 20th century, i.e. their survival at all ages is greater. In order to investigate crossovers in the extended and truncated mortality-rate functions, we compared his data for men in the United States during the period 1956}1987 with the data cited above for women. The reported death rates do not cross, and, as indicated in Fig. 3, no crossovers appear in the mortality-rate functions after they are truncated to reduce the maximum lifespan to the postulated value of 85 yr.
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H. R. HIRSCH E¹ A¸.
MORTALITY IN NON-AGEING POPULATIONS
Similar results are obtained upon the application of the accelerated-mortality model to non-ageing populations. As indicated earlier, the mortality-rate and survival functions of nonaging organisms are given by eqns (4) and (5), respectively. As an example, consider a hypothetical non-aging organism for which the constant mortality rate k "0.02 yr\. This cor responds to a survival half-life of approximately 36.5 yr. If a maximum lifespan of 100 yr is imposed upon this organism with the use of the accelerated-mortality model, eqns (7) and (9), its mortality rate is no longer constant (Fig. 4); the organism now ages. The mortality rate accelerates upward, at "rst gradually, but, as the maximum lifespan approaches, quite rapidly. The population for which k "0.02 yr\ is advantaged relative to one for which k " 0.022 yr\. The half-life of the latter, disadvantaged, population is approximately 31.5 yr. Neither the exponential survival functions nor the constant mortality-rate functions of the advantaged and disadvantaged populations cross. More importantly, no crossovers occur in these functions when they are truncated with the use of the accelerated-mortality model. This is illustrated in Fig. 5, which shows that the di!erence between the mortality rates of the advantaged and disadvantaged populations, when truncated at 100 yr, has the same sign, negative, at all ages between 0 and 100.
FIG. 5. Di!erence in truncated mortality-rate functions between advantaged and disadvantaged populations for which the extended mortality-rate functions have constant values of 0.02 and 0.022 yr\, respectively. Acceleratedmortality model. The di!erence in the truncated functions has the same sign, negative, at all ages, indicating that the functions do not cross.
The foregoing is a simple example of the scenario examined by Liu & Witten (1995) in which the mortality rates of the advantaged and disadvantaged populations are proportional. It illustrates the general result that crossovers do not occur if the mortality rates are in constant ratio. The Sudden-death Model Another simple way in which a population may display a maximum lifespan is by abrupt termination of its survival function at age w. Thus
S (x), x)w,
(10a)
x'w,
(10b)
S* (x)"
0
where S* (x) is the truncated survival function and is discontinuous. Equations (10) are illustrated in Fig. 6 with the use of the same data (Riggs, 1990) which were applied to the accelerated-mortality model in Fig. 1. The truncated mortality-rate function associated with eqns (10) is FIG. 4. Truncated mortality-rate function of a population for which the extended mortality-rate function has the constant value 0.02 yr\. Accelerated-mortality model. The mortality rate approaches a vertical asymptote, indicated by a dashed line, at age 100.
k* (x)"k (x)#S (w) u (x!w), x)w,
(11)
where u (x!w) is a unit impulse function at age x"w. An example of a truncated mortality-rate
177
MAXIMUM LIFESPAN AND MORTALITY
FIG. 6. Truncated survival function based on Riggs' (1990) data for women. Sudden-death model.
function based on the same data as Fig. 6 is shown in Fig. 7. In eqn (11), just as in eqns (8), the value of kH(x) is unde"ned if x'w. Once again this re#ects the concept that there is no force of mortality when there are no remaining members of the population. Since the survival function in eqn (10a) is assumed continuous except at age x"w, the mortality-rate function is de"ned for ages x(w. Owing to the use of the unit impulse function in eqn (11), the mortality rate is also de"ned at x"w. The unit impulse function, like the Dirac delta function, is in"nite where the value of its argument is zero and is zero elsewhere. It serves to represent the derivative of an otherwise continuous function at a point of discontinuity (Sokolniko! & Redhe!er, 1958). Thus w, the maximum lifespan, is the age at which sudden death strikes a population. Prior to age w, the existence of a lifespan maximum has no e!ect. Then, at age w, the mortality rate is very high for a very brief time, and the population is eliminated. In an earlier report (Hirsch et al., 1996) we called this a suicide model. As with the accelerated-mortality model, instantaneous truncation of the survival and mortality-rate functions would have very little e!ect if applied to a large population, such as the world population of
FIG. 7. Truncated mortality-rate function corresponding to the survival function for women show in Fig. 6 (Riggs, 1990). Sudden-death model. The vertical arrow at age 85 represents an impulse function. Prior to age 85, the extended and truncated mortality-rate functions are identical. At age 85, the mortality rate becomes in"nite, and the entire population dies.
humans, for which S(w) is very small. However, the sudden-death model could correctly describe populations for which aging is programmed. These might include may#ies, bamboo, salmon which die soon after spawning and populations of cells which die by apoptosis. The appearance of sudden death in a population can have no in#uence on the existence of mortality-rate crossovers between advantaged and disadvantaged populations, since the extended and truncated forms of the survival and mortality-rate functions are unchanged by the presence of a lifespan maximum for ages x(w. Crossovers will be present in the truncated mortality-rate functions obtained with the use of eqns (11) if and only if they are present in the original extended functions. Advantaged and disadvantaged populations having proportional mortality-rate parameters were discussed in one of our earlier reports (Liu & Witten, 1995). It is evident in the case of proportional constant mortality rates examined above and in the sudden-death model that the mortality-rate functions associated with such populations need not necessarily cross. Discussion The biological signi"cance of these results lies most importantly in their relevance to a
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H. R. HIRSCH E¹ A¸.
previously explored linkage between the existence of lifespan maxima in populations and crossovers between their mortality rates. In an earlier paper (Liu & Witten, 1995), we examined a class of continuous survival functions for which the existence of a maximum lifespan required a crossover in the modality-rate functions of advantaged and disadvantaged subpopulations. Here we treat two classes of survival functions having maximum lifespans, one the continuous acceleratedmortality model, and the other, the discontinuous sudden-death model, for which mortalityrate crossings need not occur. We demonstrate that, despite the existence of a maximum lifespan for these classes, the mortality-rate functions of advantaged subpopulations need not cross those of disadvantaged subpopulations by providing examples in which such crossings do not occur. It should be observed that a crossover of the survival functions at a particular age imposes the condition that the mortality-rate functions must also cross at an earlier age (Hirsch, 1995, 1997). In Liu & Witten (1995), we discussed the consequences of maximum lifespan for advantaged and disadvantaged populations having continuous survival functions and mortality di!erences which are proportional over the lifespan. We concluded that the disadvantaged group would die out before the advantaged and that this result appeared to be inconsistent with what is known about human population dynamics. However, our results in the present work indicate that proportional mortality di!erences are consistent with survival functions having maximum lifespans if, as in the sudden-death model, the survival functions are discontinuous. Wilmoth (1997) states that the mortality rate must approach in"nity as age approaches the maximum lifespan. The approach to in"nity occurs in both the accelerated-mortality and sudden-death models but not in the scenarios examined by Liu & Witten (1995). In these, the mortality rates reached high but "nite values at the maximum age. The connection described between maximum lifespan and mortality-rate crossovers is associated with the failure of the mortality-rate functions to approach a vertical asymptote at the maximum lifespan. This is consistent with data cited earlier which demonstrate leveling of mortality rates in extreme old age.
In certain instances, e.g. those of programmed death in several nonhuman populations mentioned above, the sudden-death model appears to be an adequate choice. With respect to human populations, the applicability of a particular model is unclear. Wilmoth (1997) comments on the di$culty of observing and documenting the existence of a maximum lifespan with our present methods. The problem is that so few people live to ages approaching the maximum. Methods previously applied to small populations of laboratory animals (Hirsch & Peretz, 1984) as well as other methods currently under development may prove to be helpful in examining the demographic characteristics of this small group of survivors and, ultimately, in determining whether human populations have lifespan maxima. Conclusions 1. The existence of a maximum lifespan does not necessarily imply a crossing of the mortalityrate functions of advantaged and disadvantaged populations if the mortality-rate functions approach in"nity at the maximum lifespan. 2. Mortality-rate functions of advantaged and disadvantaged populations may be proportional throughout life and may therefore fail to cross if the survival functions of the populations are discontinuous. 3. If the survival function of a population is truncated as in the accelerated-mortality model, its mortality-rate function approaches a vertical asymptote at the maximum lifespan hyperbolically. 4. In most populations, such as the human, few members survive to an age at which a maximum, if it exists, might be observed. Distinctions among classes of survival functions, continuous or discontinuous, are di$cult to establish. Consequently, improved statistical techniques for examining such functions should be diligently pursued to achieve the goal of providing better data for populations studies used in gerontological research. REFERENCES BARRETT, J. C. (1985). The Mortality of Centenarians in England and Wales. Arch. Gerontol. Geriatrics 4, 211}218.
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CAREY, J. R. & LIEDO, P. (1995). Sex-speci"c life table aging rates in large Med#y cohorts. Exp. Gerontol. 30, 315}325. CAREY, J. R., LIEDO, P., OROZCO, D. & VAUPEL, J. W. (1992). Slowing of mortality rates at older ages in large Med#y cohorts. Science 258, 457}461. COALE, A. J. & KISKER, E. E. (1986). Mortality crossovers: reality or bad data? Popul. Stud. 40, 389}401. CURTSINGER, J. W., FUKUI, H. H., TOWNSEND, D. R. & VAUPEL, J. W. (1992). Demography of genotypes: failure of the limited life span paradigm in Drosophila melanogaster. Science 258, 461}463. EAKIN, T. & WITTEN, M. (1995). A gerontological distance metric for analysis of survival dynamics. Mech. Ageing Dev. 78, 85}101. EAKIN, T., SHOUMAN, R., QI, Y., LIU, G. & WITTEN, M. (1995). Estimating parametric survival model parameters in gerontological aging studies: methodological problems and insights. J. Gerontol.: Biol. Sci. 50A, B166}B176. ECONOMOS, A. C. (1979). A non-gompertzian paradigm for mortality kinetics of metazoan animals and failure kinetics of manufactured products. AGE 2, 74}76. ELO, I. T. & PRESTON, S. H. (1994). Estimating African}American mortality from inaccurate data. Demography 31, 427}458. FINCH, C. E., PIKE, M. C. & WITTEN, M. (1990). Slow mortality rate accelerations during aging in some animals approximate that of humans. Science 249, 902}905. FRIES, J. F. (1980). Aging, natural death, and the compression of morbidity. New England J. Med. 303, 130}135. FUKUI, H. H., XIU, L. & CURTSINGER, J. W. (1993). Slowing of age-speci"c mortality rates in Drosophila melanogaster. Exp. Gerontol. 28, 585}599. GAVRILOV, L. A. & GAVRILOVA, N. S. (1991). ¹he Biology of ¸ifespan. Chur, Switzerland: Harwood. HELLER, D. A., AHERN, F. M., STOUT, J. T. & MCCLEARN, G. E. (1998). Mortality and biomarkers of aging in heterogeneous stocks (HS) mice. J. Gerontol.: Biol. Sci. 53A, B217}B230. HIRSCH, H. R. (1994). Can an improved environment cause maximum lifespan to decrease? Comments on lifespan criteria and longitudinal gompertzian analysis. Exp. Gerontol 29, 119}137. HIRSCH, H. R. (1995). Do intersections of mortality-rate and survival functions have signi"cance? Exp. Gerontol. 30, 147}167. HIRSCH, H. R. (1997). Intersections of mortality-rate and survival functions: model-independent considerations. Exp. Gerontol. 32, 287}296. HIRSCH, H. R., LIU, X. & WITTEN, M. (1996). Mortality-rate crossovers when suicide is genetically programmed. AGE 19, 172. HIRSCH, H. R. & PERETZ, B. (1984). Survival and aging of a small laboratory population of a marine mollusc, Aplysia californica. Mech. Ageing Dev. 27, 43}62. LIU, X. & WITTEN, M. (1995). A biologically based explanation for mortality crossover in human populations. Gerontologist 35, 609}615. MANTON, K. G., POSS, S. S. & WING, S. (1979). The black/white mortality crossover: investigation from the perspective of components of aging. Gerontologist 19, 291}300. MANTON, K. G., STALLARD, E., WOODBURY, M. A. & DOWD, J. E. (1994). Time-varying covariates in models of human mortality and aging: multidimensional general-
izations of the gompertz. J. Gerontol.: Biol. Sci. 49, B169}B190. NAM, C. B. (1994). Another look at mortality crossovers. Social Biol. 42, 133}142. OLSHANSKY, S. J. (1995). Mortality crossovers and selective survival in human and nonhuman populations. Gerontologist 35, 583}587. RIGGS, J. E. (1990). Longitudinal Gompertzian analysis of adult human mortality in the U.S., 1900}1986. Mech. Aging Dev. 54, 235}247. RIGGS, J. E. & MILLECCHIA, R. J. (1992). Mortality among the elderly in the U.S., 1956}1987: demonstration of the upper boundary to Gompertzian mortality. Mech. Aging Dev. 62, 191}199. SOKOLNIKOFF, I. S. & REDHEFFER, R. M. (1958). Mathematics of Physics and Modern Engineering. New York: McGraw-Hill. VAUPEL, J. W. (1997). Theories of mortality and advanced ages. In: Between Zeus and the Salmon (Wachter, K. W. & Finch, C. E., eds), pp. 17}37. Washington, DC: National Academy Press. VAUPEL, J. W., MANTON, K. G. & STALLARD, E. (1979). The impact of heterogeneity in invidual frailty on the dynamics of mortality. Demography 16, 439}454. VAUPEL, J. W. & YASHIN, A. I. (1985). Heterogeneity ruses: some surprising e!ects of selection of population dynamics. Am. Statist. 39, 176}185. WEIBULL, W. (1951). A statistical distribution of wide applicability. J. Appl. Mech. 18, 293}297. WILMOTH, J. R. (1997). In search of limit. In: Between Zeus and the Salmon (Wachter, K. W. & Finch, C. E., eds), pp. 38}64. Washington, DC: National Academy Press. WITTEN, M. (1985). A return to time, cells, systems and aging: III. Gompertzian models of biological aging and some possible roles for critical elements. Mech. Aging Dev. 32, 141}177. WITTEN, M. (1988). A return to time, cells, systems and aging: V. further thoughts on gompertzian survival dynamics*the geriatric years. Mech. Aging Dev. 46, 175}200. WITTEN, M. (1989). Quantifying the concepts of rate and acceleration/deceleration of aging. Growth Dev. Aging 53, 7}16. WITTEN, T. M. & EAKIN, T. (1985). Multiphasic models of survival: analysis of mortality rate change regions and the issue of "nite species lifespan. Exp. Gerontol. 32, 259}285.
APPENDIX Hyperbolic Approach of the Mortality-Rate Function to a Vertical Asymptote at the Maximum Lifespan in the Accelerated-Mortality Model Examination of mortality rates from truncated survival functions based on Riggs' (1990) data (Table A1) suggests that mortality-rate functions approximate hyperbolic curves as age approaches the maximum lifespan. The following simple derivation supports the validity of this observation in the accelerated-mortality model.
180
H. R. HIRSCH E¹ A¸.
TABLE A1 Comparison of the hyperbolic mortality function with truncated mortality-rate data x 75 80 83 84 84.5 85.8 84.9 84.95 84.98 84.99
k*D
k*K
eK
k*D
eD
0.10 0.20 0.50 1.00 2.00 5.00 10.00 20.00 50.00 100.00
0.1033 0.2171 0.5273 1.0310 2.0329 5.0340 10.0344 20.0346 50.0347 100.0348
!3.22 !7.88 !5.19 !3.01 !1.62 !0.676 !0.343 !0.173 !0.0694 !0.0345
0.0847 0.1921 0.4983 1.0007 2.0020 5.0028 10.0030 20.0032 50.0032 100.0033
18.00 4.11 0.336 !0.072 !0.099 !0.055 !0.030 !0.016 !0.006 !0.003
x: age (yr). k*D (yr\)"1/(85!x): Hyperbolic mortality function, eqn (A.4). Approaches a vertical asymptote at age 85. k*K (yr\): Riggs' (1990) mortality rates for men truncated at age 85 with the use of eqn (7). eK"100(k*D/k*K!1): Percent error caused by replacement of the truncated mortality rates for men with the hyperbolic mortality function. k*D (yr\): Riggs' (1990) mortality rates for women truncated at age 85 with the use of eqn (7). eD"100 (k*D/k*D!1): Percent error caused by replacement of the truncated mortality rates for women with the hyperbolic mortality function.
An approximate form of the mortality-rate function, k*D (x), is de"ned by replacing the di!erentials dS* and dx in eqn (8b) with "nite di!erences: D
1
k* (x)"! S* (x)
DS* (x) Dx
,
(A.1)
where DS* (x)"S* (x)!S* (w),
(A.2)
and Dx"x!w.
(A.3)
With the use of eqn (7), S* (w)"0, and, from eqn (A.2), DS* (x)"S* (x). Simultaneous solution of eqns (A.1)}(A.3) then yields 1 k*D (x)"! . w!x
(A.4)
Thus, the approximate form of the mortality rate is hyperbolic. The corresponding survival
function, S*D, is linear: S*D (x)"1!x/w.
(A.5)
The relation between eqns (A.4) and (A.5) can be veri"ed by substitution of the derivative of eqn (A.5) into eqn (8b). It should be noted that the only approximation used to obtain eqn (A.4) is the substitution of "nite di!erences for di!erentials. The results in Table A1 provide two examples in which the hyperbolic approximation to the mortality-rate function corresponds closely to human data truncated with the use of the accelerated mortality model. The degree of agreement with the data for women is better than that for men, but, in both cases, the percent errors caused by the use of the hyperbolic functions are very small at ages which approach the maximum lifespan. The same conclusion is supported by a power-series expansion of the Gompertz function.