A gerontological distance metric for analysis of survival dynamics

Mechanisms of Ageing and Development 78 (1995) 85-101

ELSEVIER

A gerontological distance metric for analysis of survival dynamics T. Eakin,

M. Witten*

University of Texas System, Center For High Performance Computing, Department of Applications Research and Development, Pickle Research Campus. 1.154CMS. 10100 Burner Road, Austin, TX 78758-4497, USA

Received 29 July 1994;accepted 13 September 1994

Abstract A metric for quantifying a gerontological mapping ‘distance’ or displacement consistent with the historical concept of velocity of aging and with the more recent concept of acceleration of aging, is introduced using the paradigm of a simple linear dynamics system of elementary physics. This analysis is extended to recent analytical methods utilizing intrinsic or internal time scaling so that biological or gerontological similarity can be distinguished from chronological age similarity, not only among various intraspecies populations but also among interspecies populations which may not even have the same underlying mechanisms of senescence or survival distributions. Illustrative examples are provided and discussed. Also, applications involving the comparison of an individual from one population to an individual from another population, when both can be assessed with respect to their respective group properties, are considered. Keywords: Gerontology; Survival; Time scaling; Aging; Distance metric; Velocity of aging; Chronological age; Physiological age; Acceleration of aging; Biomarkers of aging; Intrinsic time; Extrinsic time; Dynamical systems

1. Introduction Which is older, a typical 20-year-old horse or a typical 20-day-old worm? Which pair are closer together in age, a typical 80-year-old man and his typical 54-year-old daughter or that typical 54-year-old daughter and her own typical 25-year-old son?

* Corresponding author, present address: Institute of Gerontology, 300 North Ingalls, University of Michigan, Ann Arbor, MI 48109, USA. Elsevier Science Ireland Ltd. SSDI 0047-6374(94)01508-J

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Which is older, a typical 6-month-old rat fed ad libitum or a 12-month-old rat fed on a restricted diet? Such questions are trivial when we think in terms of chronological age based on the standard solar calendar system. Yet we know that progressive frailty associated with aging and senescence is not a mere linear function of calendar time so that in a gerontological sense we must distinguish between chronological age and physiological or gerontological status. At first thought, it might appear meaningless to try comparing across species or between genetically nonhomogeneous populations. Nevertheless it is common for people to try estimating the human equivalent age of a pet by using some linear multiplicative factor based on relative maximum lifespans of species. A statement such as ‘my 8-year-old dog is 56 in human years’ would be an example. If aging mechanisms were identical other than being on a contracted or dilated extrinsic time scale then such estimations would be meaningful, but such an assumption is less valid the greater the evolutionary dissimilarity of the species being compared. However, if some generalized method of measuring the ‘distance’ in life traveled in terms of gerontological progression were available, then a basis for comparisons would be established. In this paper, we shall introduce a metric generated from dynamic variables of classical gerontology that addresses this issue. 2. Gerontology as a dynamical system In physics, the simplest dynamical system consists of a time unit, a time variable, a length or displacement unit, and a length variable along with its time derivatives. Analogously, we may consider the aging process as a dynamical system characterized by an underlying survival distribution from which homologous components can be constructed. In classical biological survival analysis, the typical procedure is to collect primary data as exact lifespans of all individuals in a given population. Transformations of this set of data then give related quantities which describe the system in terms of dynamic variables and characterizing parameters. The usual basic assumption is that, in the absence of accident or sacrifice, all deaths represent the culmination of aging and senescence processes. This may be debatable and a matter of semantics, but in theory, any particular identifiable internal cause of death could be designated as unrelated to senescence if desired and such cases classified along with accident and sacrifice as right censored data. Thus, for example, in a human population we might consider sudden death from a heart attack in an otherwise healthy individual as a right censored data point rather than as a natural death from the underlying force of mortality. After taking censored data into account, a survival distribution related to senescence can be constructed for any particular population. This can be either a nonparametric empirical distribution based explicitly on experimental data, or a theoretical parametric distribution based on modeling and optimization [5,6,12,26,29]. Historically the velocity of aging, also known as the mortality rate h, has been

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Ageing Dev. 78 (1995) 85-101

described as the negative density-normalized rate of change in survival, i.e if N(t) is the number of survivors, from an initial population N(O), at time I, then: x(t)

1 E

_-

dN(t) -

N(t)

1 =

dS(t 1 dt

-~

dt

__

s(t)

(2.1)

where the fraction survival S is merely:

N(t)

S(f) = ___

(2.2)

N(O) Notice, however, that: 1 dS(t) -----_---_ s(t) dt

d[ - In S(t)] (2.3)

dt

If we consider the transformed representation of the velocity of aging on the right hand side of (2.3) in the context of a simple dynamical system, we see that the argument of the time derivative, In [l/S(t)], corresponds to a distance or displacement variable. Therefore, let us define a gerontological distance from birth x(a) at age a by:

where we utilize the fact that S(0) is unity by definition so that x(0) = 0. Note that (2.4) is independent ofthefunctionalfirm of s(t), so that a comparative gerontological distance between an individual of age aI from a population with survival fraction distribution S, and an individual of age a2 from a population with survival fraction distribution S2 can be expressed as:

lAxI

= lx,(a,) - x1(+)1 =

In I

bk1-‘“[~1 I

(2.5)

Since the difference of two logarithms is equivalent to the logarithm of their quotient, we may express (2.5) as:

lAxI

= (ln[z]

1

(2.6)

We further note that successive differentiation of the gerontological distance x(a)

T. Eakin, M. Witten / Mech. Ageing Dev. 78 (1995) 85-101

88

with respect to the time variable age will yield back the mortality rate or velocity of aging: -

w4 dt

= X(u)

(2.7)

as would be expected from the generation of x(a) in (2.4), and: d’x(c) _ dG) _I dt dt2

= o(a)

(2.8)

where o(a) is the gerontological acceleration of aging as defined by Witten [28]. Thus, we have a simple gerontological dynamic system that is completely homologous in mathematical structure to an elementary linear dynamics system in physics, This also sets the stage for further analysis in terms of thermodynamical systems if we were to consider the statistical properties of gerontological variables in an ensemble of similar groups, for example in clonal populations. Witten [29] has introduced the concept of stochastic distributions of survival model parameters, which would also serve as part of the foundation for such developments. 3. Extension to dimensionless intrinsic time scaling The above formulation in time variables extrinsic to the population accommodates comparisons among typical individuals from groups with underlying survival values distributed over similar external time frames, e.g. the hypothetical 80-year-old human male and the hypothetical 54-year-old human female described in the introduction. However, if we wish to make a comparison involving individuals from populations whose survival distributions span substantially dissimilar ranges in extrinsic time [1,12,13,21,23], then a further refinement involving intrinsic time scaling will be convenient. A straightforward and mathematically compelling internal scaling procedure has been described previously [7,8] in which an intrinsic time variable r for a given population is generated by dividing the extrinsic time t by the life expectancy at birth, i.e: t ’ =

I;S(t)

(3.1)

dt

The intrinsic fraction survival 8 is then specified by: (3.2)

(B(r) f S(t)

Thus, the intrinsic dynamical system describing the aging process will have, in analogy to (2.4), an intrinsic gerontological distance given by: p(7) = In

1 __ [ @(r)

1

(3.3)

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and, corresponding to (2.6), a comparative intrinsic gerontological distance between a typical individual at intrinsic age a1 from an underlying intrinsic survival distribution 6t and a typical individual at intrinsic age a2 from an underlying intrinsic survival distribution a)2 given by:

lAh2’ = Iln [Gk]-In[dG](=Iln [2$]1

(3.4)

From definitions and identities given in (3.2) (3.3), and (3.4) it is easy to see that, as would be expected for a dimensionless variable, the magnitude of gerontological distance is independent of an extrinsic or intrinsic time basis, i.e: r(a) = x(a) and IArt

= lAxI

(3.5)

With extrinsic variables thus recast as dimensionless intrinsic variables through intrinsic time scaling for each specific original population, we are now in a position to consider explicit survival models and to analyze the types of questions posed in the introduction. 4. The gerontological metric for classical parametric survival models When the survival fraction distributions present in (2.4), (2.5), (2.6), (3.3), and (3.4) come from parametric survival models, the gerontological distances can be expressed analytically in terms of relevant parameters. For example, a trivial model in which the extrinsic velocity of aging is merely a constant, k, will have an extrinsic fraction survival given by: S(t) = exp[-kt]

(4.1)

and, using (2.4) the extrinsic gerontological distance becomes: x(a) = ka

(4.2)

Since the expected life at birth is k-‘, the corresponding intrinsic fraction survival becomes: @(T)= exp[-Tl

(4.3)

with corresponding intrinsic gerontological distance:

da) = a

(4.4)

Thus, in this trivial exponential decay model the intrinsic distance is merely the intrinsic age.

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For the more complex but very commonly used Gompertz model [14-161, the fraction survival in extrinsic time is given by: S(t)=exp

[$

[1 -eYt]j

where h,-,is the so-called age independent mortality rate coefficient and y is the socalled age dependent mortality rate coeffkient. Substitution of (4.5) into (2.4) allows us to obtain the Gompertz-specific form for extrinsic gerontological distance:

x(u) =

-3L(eys -

1)

(4.6)

Y

For intrinsic time obtained from scaling extrinsic time by expected life at birth, the Gompertz survival fraction has been shown to be a one-parameter distribution [7,8] given by: B)(T)= exp (4 [1 - eU(‘~‘*9)rj)

(4.7)

where 6 is the ratio of extrinsic Gompertz parameters, i.e h&r, and U is the standard second confluent hypergeometric function [22]. Although the function U is not one commonly encountered, the special case here where the first two arguments are unity has an algebraic equivalent that is useful, i.e: U( 1, 1,d) = exr@M4)

(4.8)

The exponential integral function El is somewhat simpler than U, having only one argument. It is not analytically expressible in terms of elementary functions and its values must be obtained from tables or computationally based on its definition [24]:

Mb) =

Q, exp(-64

I

1

dx

X

It is available in the IMSL (now Visual Numerics) special function library as ‘El’, or alternatively the values for a given argument can be determined numerically using an algorithm such as that given by Amos [2]. In any event, the Gompertz-specific form for the intrinsic gerontologic distance can be obtained from (3.3) and (4.7) as: r(a) = 4]e U,t.+)a _ 1]

(4.10)

Another classic parametric survival model is the Weibull model [25] for which the survival fraction is given by:

S(t) = exp [-

(fi)]

(4.11)

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where c is the so-called age independent mortality rate coefficient and b is the socalled age dependent mortality rate coefficient. In theory based on mechanism, b takes on only integer values [30] but it is also often used as an empirical descriptor for curve fitting where it is optimized to a positive real value without the integer restriction. From (2.4) then, we can obtain the Weibull-specific form for extrinsic gerontological distances as: cab+l

x(a) = ~ b+l

(4.12)

The intrinsic time survival fraction distribution for the Weibull model, obtained from scaling extrinsic time by expected life at birth, has been shown to have a single parameter b, identical with the extrinsic parameter b [7,8]. It has the analytic form:

a)(T) = exp [-

k

(4.13)

(z)~]~+‘j

Survival Model Comparisons

I

1

I

I

I

I

1

I

I

Exponential Weibull ---Gompertz ---~ _ Phenomenological :,

,,,I ,,J’ ,,,’ I’ ,’ ,’

,..I ,.: .‘. ,.I

2.5

0

0

0.2

0.4

0.6

1.2 0.8 1 Intrinsic Chronological Age

1.4

1.6

,’

1.8

2

Fig. 1. Gerontological distance as a function of intrinsic chronological age for an exponential model, a Gompertz model with intrinsic parameter @I= 1, a Weibull model with intrinsic parameter b = 1, and a phenomenological model based on actual experimental data from a population of more than a million individual medflies [3].

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Ageing Dev. 78 (1995) 85-101

where T is the standard complete gamma function [4]. Thus, the Weibull-specific form for the intrinsic gerontologic distance can be obtained from (3.3) and (4.13) as:

da) =

[r(~)qb+‘]

(4.14)

In general, parametric or phenomenological models will have low mortality rates during early years and high mortality in later years. Thus, compared to the exponential model, the gerontological distance will be less than the intrinsic chronological age early in life and will be greater than the intrinsic chronological age later in life, as is illustrated in Fig. 1. This same methodology can be readily applied to other distributions such as the logistic distribution [26]. 5. Illustrative examples using intraspecies populations First we consider applications involving comparisons among populations from the same species, thus with underlying survival distributions spanning extrinsic time ranges roughly within the same order of magnitude. We can attempt to answer the question in the introduction concerning intergenerational relatives by assuming that they are typical modern day Americans, for which recent U.S. Census Bureau life tables are available [ 111.The 1980 life table data for postpubertal ages (over extrinsic age 13) are well described by Gompertz distributions for both males and females [9,10]. Values for extrinsic expected remaining life at puberty and the intrinsic Gompertz parameter 4 are given in Table 1. Using these values, (3.1) and (4.7) we

Table 1 Survival distribution model parameters for various populations and their values Population 1980 post-puberty, U.S. male 1980 post-puberty, U.S. male 1980 post-puberty, U.S. female 1980 post-puberty, U.S. female Fischer 344 rat, Group A Fischer 344 rat, Group A Fischer 344 rat, Group R Fischer 344 rat, Group R Male flour beetle Male flour beetle Horse Horse Male C. eiegans DH26 Male C. eleguns DH26

Parameter

Value

i

58.8 years 0.0052 66.2 years 0.0014 689 days 0.0046 911 days 0.00008 2.016 0.0000732 day-3,0’6 0.0002 year-’ 0.173 year-’ 0.0115 day-’ 0.1308 day-’

Q

i & :

i 6 b C

ho i0

Y

All populations are tit to Gompertz models except for the male flour beetle which is tit to a non-integer Weibull model. Parameter values are taken from or derived from literature data [9,10,12,17,26].

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find that the 80-year-old man has an intrinsic postpubertal age of 1.139 and thus an intrinsic gerontological distance of 1.114. Similarly, his 54-year-old daughter has an intrinsic postpubertal age of 0.619 with a corresponding gerontological distance of 0.056; and the daughter’s 25-year-old son has an intrinsic postpubertal age of 0.204 with a corresponding gerontological distance of 0.008. These gerontological distances are tabulated, along with those from other illustrative examples to be discussed later, in Table 2. The gerontological distance between the old man and his daughter is thus 1.055 while that between the daughter and her son is 0.048. So even though the extrinsic chronological separation in age is smaller between the old man and his daughter, the intrinsic gerontological distance is greater. As another intraspecies example, let us try to determine the extrinsic chronological age of a typical individual from one population that is equivalent in gerontological distance to a typical individual of a given extrinsic chronological age coming from another population, genetically identical but subjected to a differing dietary regimen. The classic diet restriction data of Yu et al. (311, containing survival data for a population A of Fischer 344 rats fed ad libitum and for a population R of the same strain subjected to reduced food intake, can be used to formulate such a problem. These populations have been found to be satisfactorily described by a Gompertz distribution [9,10]. The expected life at birth and intrinsic Gompertz parameter (b for these populations, given in Table 1, can be derived from the original lifespan data deposited in the GAIA Project Multispecies Survival Database [ 181 using methodology described previously [7,8]. Suppose we want to know the extrinsic age of a typical diet restricted rat that corresponds gerontologically with a 600-day-old ad libitum fed rat. Using the extrinsic expected life from birth and intrinsic Gompertz parameter, we can determine from (3.1) and (4.10) that the 600-day-old rat from the A population has an intrinsic age of 0.87 with a corresponding intrinsic gerontological distance of 0.174. Thus we wish to find the extrinsic chronological age of a typical diet restricted rat which also corresponds to an intrinsic gerontological distance of 0.174. For this purpose, we first

Table 2 Gerontological distances computed for various typical individuals from populations illustrative examples

used in the

Population

Extrinsic age

Gerontological distance

1980 U.S. male 1980 U.S. female 1980 U.S. male Ad libitum rat (A) Diet restricted rat (R) Male flour beetle Medfly Horse Male C. elegans DH26

25 (12 post-puberty) years 54 (41 post-puberty) years 80 (67 post-puberty) years 600 days 752 days 30 days 7 days 20 years 20 days

0.008 0.056 1.111 0.174 0.174 0.0345 0.0343 0.0356 1.115

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T. Eakin. M. Witten /Mech. Ageing Dev. 78 (1995) M-101

need to invert (4.10) to give the intrinsic age in terms of intrinsic gerontological distance, i.e: 1 o = Ul,l,4)

ln

41

r(a) + 9 [

(5.1)

Again using extrinsic expected life from birth and intrinsic Gompertz parameter values for the R population as given in Table 1, it can be calculated that a value of 0.174 for a corresponds to an intrinsic age of 0.757 and thus an extrinsic chronological age of 752 days. We have now determined that a typical 600-day-old rat from the ad libitum population A is gerontologically equivalent to a typical 752day-old rat from the diet restricted population R.Similar correspondences in extrinsic age of typical individuals from these two rat populations matched by equal gerontological distances throughout the lifespan are presented in Table 3 and illustrated with a nomogram in Fig. 2. Fig. 3 illustrates a graph of the diet restricted rat extrinsic age in days vs. the ad libitum diet rat extrinsic age in days (open circles). The solid black diamonds correspond to the gerontologic distance for the given ad libitum or diet restricted rat extrinsic age in days. 6. IUustrative examples using interspecies populations Now we illustrate the use of the gerontological distance metric in making comparisons of gerontological status among typical individuals from various populaTable 3 Comparison of extrinsic chronological ages of typical individuals from an ad libitum rat population and from a diet restricted rat population [31] for equivalent gerontological distances Ad libitum age (days) 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900

Diet restricted age (days) 3.2 9.1 19.8 38.8 70.5 119.9 189.9 279.7 385.3 501.8 625.0 752. I 881.4 1011.8 1142.8 1274.0 1405.6 1537.2

Gerontological distance

0.0001 0.0002 0.0005 0.0010 0.0019 0.0037 0.0070 0.0134 0.0255 0.0485 0.0921 0.1749 0.3321 0.6305 1.1971 2.2726 4.3143 8.1903

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T. Eakin, M Witten / Mech. Ageing Dev. 78 (1995) M-IO1

Fischer 344 Rat

Fischer 344 Rat

Ad Iibitum

Diet Restricted

1500

__I.

,_....

1500

..: _.: __.’ _...

_..’ __.. ,:. __.’ __..

_,.’

1200

_: ._: _..’ _:.

___...

__..

-

1200

-

900

-

600

-

300

_..

_...’ __.’

_...

__... ,__...

__.. ,__...

900

600

.__..-___... ___...

___..._...___..___.-....

.. __.____..... .._____...-..____ _______..... ...___.___......___..___...... _____._...-~

.. .

300 .._ ...

0

Extrinsic

-

0

Chronological Age (days)

Fig. 2. Nomogram comparing equivalent gerontological points for an ad Iibitum fed rat population and a diet restricted rat population [31] on an extrinsic chronological age scale.

tions of different species where extrinsic time lifespan ranges are not necessarily similar nor even of the same order of magnitude. An example that allows us to consider some underlying survival distributions other than the Gompertz is a comparison of the gerontological distance of a typical month-old male flour beetle with that of a typical week-old medfly. Using data from Pearl et al. [20], Wilson [26] has found that male flour beetles have an underlying survival distribution that closely tits a non-integer Weibull distribution with the parameter b in (4.11) having magnitude 2.016 and the parameter c having magnitude 0.0000732 when r is express-

96

T. Eakin. M. W&en /Meeh.

0

200

400

Ageing Dev. 78 (1995) M-IO1

600

600

1000

Ad Lib&urn Rats (age in days)

Fig. 3. In this graph, the open circles illustrate the extrinsic age equivalence of ad libitum rats to diet restricted rats [31]. For any given extrinsic age on the ad libitum axis, draw a vertical line to the open circles and read the corresponding value of extrinsic age for the diet restricted rats from the left-hand vertical axis. The solid black diamonds represent the gerontologic distance for the given age in days. The right-hand vertical axis expresses the gerontologic distance on a log scale. For any given extrinsic age on the ad libitum axis, draw a vertical line to the solid diamonds and read the corresponding value of gerontologic distance from the right-hand vertical axis.

ed in days. Thus, by (4.12) the gerontological distance for t = 30 days is 0.0345. The underlying survival distribution for medflies has been measured experimentally for an initial population size in excess of a million individuals [3]. This initial population size is so massive that its own phenomenological survival fraction distribution can be considered as the true underlying distribution, particularly since simple models such as Gompertz or Weibull do not seem to be adequate for describing mortality over the entire lifespan range [26,27]. Therefore, it is simply a matter of noting that after 7 days there were 1 163 026 survivors from an initial population of 1 203 646. Using (2.2) and (2.4) the gerontological distance can be computed for the typical week-old medfly as 0.0342. We see then that although the typical month-old male flour beetle has a marginally larger gerontological distance, the difference is a mere 0.002. Since this difference is less than 1% we can say that these two typical individuals have essentially the same gerontological status even though there is a factor greater than four in their relative chronological ages. As a last example, let us return to the very first question asked at the beginning of the introduction. Comparing a typical horse with a typical worm will show how the gerontological distance metric can be applied to underlying populations having

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78 (1995) 85-101

1.2

80 year old U. S. male human 1.0

5 month old male flour beetle 0.8

0.6

700 day old

a&iY&tn

rat

0.4

2 week old medfly 0.2

0.0

20 year old horse

Gerontological Distance Fig. 4. Gerontological status of various typical individuals from various species displayed on the same gerontological distance line.

not only an extreme difference in expected life at birth but also having extreme differences in typical size dimensions, physiology, and taxonomic classification. Extrinsic Gompertz parameters can be obtained for a horse population using initial mortality rate (IMR) and time for mortality rate to double (MRD) given by Finch et al. [12], and are given in Table 1. Then from (4.6), it can be determined that a typical 20-year-old horse has a gerontological distance of 0.0356. For a worm population, let us choose males of the DH26 strain of the nematode Cuenorhabditis eleguns for which Johnson [ 171 has likewise reported the IMR and MRD. The cor-

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Ageing Dev. 78 (1995) 85-101

Man

Horse 100

-

100

80

-

80

60

-

60

40

-

40

20

-

20

0

-

0

Extrinsic Chronological Age

(years)

Fig. 5. Nomogram comparing equivalent gerontological points for horse and man (1980 U.S. male) on an extrinsic chronological age scale.

responding extrinsic Gompertz parameters are also given in Table 1, and using (4.6) the gerontological distance for a typical 20-day-old DH26 male is found to be 1.115. We thus find in this comparison that the typical 20-year-old horse has a much smaller gerontological distance than does the typical 20-day-old worm, even though the horse has a chronological age that is 365 times as great. 7. Comparing specific individuals In the development of this gerontological distance metric and in the presentation of examples, we have been careful to emphasize that the application refers to ‘typical

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individuals. Survival distributions, which form the basis for constructing the metric, are manifestations of aggregate properties of large populations and thus reference to specific individuals can only be within the context of a group average. However, the more homogeneous the underlying population in terms of genetics, gender, environment, exposure to external risk factors, etc., the more likely it is that any particular individual will closely resemble a group composite. The gerontological distance we assign to a particular individual based exclusively on its extrinsic chronological age is therefore not necessarily the most meaningful representation of its own specific gerontological status. For this reason, various investigators are trying to establish a panel of biomarkers, e.g. IGF and glucocorticoid levels, which will characterize an individual’s physiological status with respect to significant manifestations of aging and senescence. Some attempts at addressing this problem by examining varying time-covariates and their effects on Gompertz parameters may be found in Manton et al. [19]. If it eventually becomes possible to correlate composite internal physiological status with aggregate mortality risk, then a specified individual may be assigned an extrinsic ‘physiological age’ from which a more meaningful gerontological distance can be derived. 8. Concluding remarks

We have seen that with the addition of the concept of gerontological distance as defined in (2.4), we may now view survival processes in the frame of a simple dynamical system and thereby provide a mechanism for evaluating the relative gerontological status of typical individuals from populations of varying underlying survival distributions. For example, various typical members of diverse populations can be displayed along a gerontological distance line as shown in Fig. 4, or various stages of gerontological status between chronological age markers of two dissimilar populations can be illustrated in the form of nomograms as is shown in Fig. 5. However, it should be remembered that such analyses are based exclusively on longevity and do not directly address issues of functional capacities or physiological condition. Acknowledgements

We would like to thank R. Shouman and F. Bookstein for observations, suggestions, and comments concerning this material. The University of Texas System CHPC has provided the computer resources utilized in calculations and manuscript preparation. The work has been supported by research grant PHS lRO1 AGllO79 from the National Institute of Aging. References [ll R.E. Albert, S.A. Benjamin and R. Shukla, Life span and cancer mortality in the beagle dog and humans. Mech. Ageing Dev., 74 (1994) 149-159. (2) D.E. Amos, Computation of exponential integrals. ACM Trans. Math. Software, 6 (1980) 365-377. [31 J.R. Carey, P. Liedo, D. Orozco and J.W. Vaupel, Slowing of mortality rates at older ages in large medfly cohorts. Science. 258 (1992) 457-461.

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141 P.J. Davis, Gamma function and related functions, In M. Abramowitz and IA. Stegun, (eds.). Handbook ofMathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover

Publications, Inc., New York, 1965, pp. 253-293. 151 T. Eakin, R. Shouman, Y. Qi, G. Liu and M. Witten, Issues of Parametric Estimation in Multiparametric Gerontologic Survival Models. UT-CHPC Technical Report Series, XCHPC TR-19920002, 1992. [6] T. Eakin, R. Shouman, Y. Qi, G. Liu and M. Witten, Estimating parametric survival model parameters in gerontological aging studies: methodological problems and insights. J. Gerontol., (1994) in press. [7) T. Eakin, An Intrinsic Time Scaling for Survival Analyses of Biological Populations. UT-CHPC Technical Report Series, UCHPC TR-1992-0009, 1992. [8] T. Eakin, Intrinsic time scaling in survival analysis: applications to biological populations. Bull. Math. Biol., (1994) in press. [9] T. Eakin and M. Witten, Intrinsic Prolate Rectangularity in Transition Regions of Survival Curves: How Square is the Survival Curve of a Given Species? UT-CHPC Technical Report Series, #CHPC TR-1993-0006, 1993. [IO] T. Eakin and M. Witten, How square is the survival curve of a given species? Exp. Gerontol., (1994) in press. [ 1I] J.F. Faber and A.H. Wade, Life Tables for the United States: 1900-2050 Actuarial Study No. 89, U.S. Department of Health and Human Services, SSA Pub]. No. 1l-1 1536, 1983. [12] C.E. Finch, M.C. Pike and M. Witten, Slow mortality rate accelerations during aging in some animals approximate that of humans. Science, 249 (1990) 902-905. [I31 R.J.M. Fry, Extrapolation from experimental systems to man: a review of the problems and possibilities, In: To Address a Proposed Federal Radiation Research Agenda. NIH, DHEW, Washington, DC, 1980, pp. 1l-136. [14] B. Gompertz, A sketch on the analysis and the notation applicable to the value of life contingencies. Phil. Trans. R. Sot., 110 (1820) 214-294.

[15] B. Gompertz, On the nature of the function expressive of the law of human mortality and on a new mode of determining the value of life contingencies. Phi/. Trans. R. Sot.. lZ5 (1825) 513-585. 1161 B. Gompertz, A supplement to two papers published in the transactions of the Royal Society, on the science connected with human mortality; the one published in 1820, and the other in 1825. Phil. Trans. R. Sot., 52 (1862) 511-559. [I71 T.E. Johnson, Increased life-span of age-l mutants in Caenorhabditis elegans and lower Gompertz

rate of aging. Science, 249 (1990) 908-912. [18] G. Liu, R. Shouman, Y. Qi, T. Eakin and M. Witten, GAIA project. Access and Submission to the Multispecies Database. UT-CHPC Technical Report Series, #CHPC TR-1992-0012, 1992. [I91 K.G. Manton, E. Stallard, M.A. Woodbury and J.E. Dowd, Time-varying covariates in models of human mortality and aging: multidimensional generalizations of the Gompertz. J. Gerontol., 49 (1994) B169-Bl90. [20] R. Pearl, T. Park and J.R. Miner, Experimental studies on the duration of life. IX. Life tables for the flour beetle Tribolium confusum Duval. Am. Nat., 75 (1941) 5-19. 121) J.M. Robine and K. Ritchie, Explaining fruit fly longevity. Science, 260 (1993) 1665. [22] L.J. Slater, Confluent hypergeometric functions. In M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Pubhcations, Inc., New York, 1965, pp. 503-535. [23] J.B. Storer, T.J. Mitchell and R.J.M. Fry, Extrapolation of the relative risk of radiogenic neoplasms across mouse strains to man. Rodiat. Res., 114 (1988) 331-353. [24] C.G. van der Laan and N.M. Temme, Calculations of Special Functions: The Gamma Function. The Exponential Integrals and Error-like Functions, Mathematisch Centrum, Amsterdam. 1984. [25] W. Weibull, A statistical distribution of wide applicability. J. Appl. Mech., I8 (1951) 293-297. [26] D.L. Wilson, The analysis of survival (mortality) data: fitting Gompertz, Weibull and logistic functions. Mech. Ageing Dev., 74 (1994) 15-33. [27] M. Witten, A return to time, cells, systems and aging: V. Further thoughts on Gompertzian survival dynamics - The geriatric years. Mech. Ageing Dev., 46 (1988) 175-200.

T. Eakin. hf. Wiiten/Mech.

Ageing Dev. 78 (1995) 85-101

10

[28] M. Witten, Quantifying the concepts of rate and acceleration/deceleration of aging. Growrh Dev. Aging, 53 (1989) 7-16. 1291 M. Witten, Might stochasiticity and sampling variation be a possible explanation for variation in clonal population survival curves. Mech. Ageing Dev., 73 (1994) 223-248. [30] M. Witten, Interpreting the Gompertz and Weibull Models of Aging. C/T-CHPC Technical Report Series, #UT-CHPC TR-1993-0003, 1993.

[31] B.P. Yu, E.J. Masoro, I. Murata, H.A. Bertrand and F.T. Lynd, Life span study of SPF Fischer 344 male rats fed ad libitum or restricted diets: longevity, growth, lean body mass and disease. .I. Geronroi., 37 (1982) 130-141.