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A mortality kinetics approach to characterizing the fractionated wxposure-mortality response relationship of radon progeny
mecfims of again! and devalopmant
Mechanisms of Ageing and Development 83 (1995) 65-85
ELSEVIER
A mortality kinetics approach to characterizing fractionated exposure-mortality response relationship of radon progeny P.J. Neafsey*“, “School
qf Nursing,
bActuarial
Scienw,
Uniwrsity
Drparrmrnt
W.B. Lowrieb, F.T. Cross”
of Conne~~ticut. of’ Marhemurics.
Box
und Chemistry
U-59,
Uniwrsit~
CT 06269-3009. ‘Biology
the
Storrs.
06269-2059.
CT
q/ Connecticut.
Box
USA
U-9. Storrs.
USA
Deparrmenr,
Pac$c
Northnrst
Washington.
DC 99352,
Laboratory,
Richland.
USA
Received 23 January 1995; revision received 15 May 1995: accepted 28 May 1995
Abstract
The utility of mortality kinetics analysis in evaluating mortality data from fractionated exposure studies was demonstrated using radon-progeny induced extra mortality as an example. Gompertz (log-hazard) functions were used to characterize the mortality of male SPF Wistar rats exposed to radon progeny at 100 WL and 1000 WL for total exposures ranging from 20 to 10 240 WLM. There was an upward parallel displacement of the Gompertz functions following the period of radon exposure. The shape of the Gompertz functions for the exposed animals was consistent with a Gompertz model of toxicity resulting from short-term exposure, resulting in non-repaired injury that summates with natural (aging) injury. The parallel upward displacements (E,,) of the Gompertz functions showed an unexpected non-monotonic pattern for rats exposed at 1000 WL. The parallel upward displacements showed a sharp upward increase from 320 to 640 WLM, fell at 1280 WLM, and thereafter increased linearly to 10 240 WLM. These data suggest that the radon progeny exposure-mortality response is non-linear. In contrast, there was no significant parallel upward displacement of the Gompertz functions for rats exposed at 100 WL for total exposures of 20-1280 WLM. but a large displacement began at 2560 WLM total exposure. Ke~+~~ords: Dose-response;
* Corresponding 0047-6374/95/$09.50 SSDI
Gompertz;
Hazard
functions;
Mortality:
Radon-progeny
Author, Tel: + I 203 4863713; Fax: + I 203 4860512 0
1995
0047-6374(95)01606-Z
Elsevier Science Ireland Ltd. All rights reserved
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1. Introduction
The rationale for using survival models in toxicity studies is that no single endpoint (e.g. carcinogenesis) completely describes toxicant-induced damage. Since the mortality of a population is a measure of the combined effects of natural aging with environmental insults, techniques that can model time-dependent exposure patterns to toxicants may yield indices of net injury from combined neoplastic and non-neoplastic causes. Mortality data in toxicity studies are typically published as percentage survivorship or cumulative mortality plots, in terms of mean or median survival times, or as mean after survival (MAS). These variables do not characterize temporal mortality patterns such as the effect of fractionation of a given total exposure. Mortality kinetics analysis (also known as age-specific mortality rate analysis or hazard analysis) [I] integrates total causes of death related to dose. It can be used to characterize late-life toxic effects, relative biological effectiveness, and the temporal pattern of net injury from environmental exposure. The purpose of this paper is to demonstrate that Gompertz (log-hazard) functions are well suited to characterizing the effects of fractionation of a given total exposure. A mortality kinetics analysis of radon-induced mortality is used as an example. The use of Gompertz functions to characterize changes in mortality stemming from exposure to toxic agents evolved from work of Sacher and colleagues [2-lo]. The hazard function (instantaneous mortality rate, force of mortality, or instantaneous age-specific mortality rate), when multiplied by d-v, is the probability of death during the interval s to I + dx (where
h(x) =
- d[lnS(x)] = __.\1 dN. d.x
N, dx ’
(1)
where h(x) is the hazard function at age x, (also called the age-specific mortality rate), S(x) is the fraction of the population surviving at age x, and N., is the number of individuals surviving at age s. An age-specific mortality rate of 0.12 weeks-’ at age 50 weeks means that the probability of an individual dying in the 51st week is approximately 12%. Sacher and Brues [2,6-91 proposed that the hazard function is an exponential function of the mean intensity of physiological injury for homogeneous mammalian populations, kept free of preventable disease and housed in a uniform environment, h(x) = ke$“’
where h(x) is the hazard function at age x, k is a proportionality constant related to the inability of the population to withstand injury from the environmental conditions, and 4(x) is the mean intensity of injury of the population at age x. Sacher [5] defined injury in this context as any change which is detrimental to the
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status of vital system ‘states.’ He proposed that 4(s) is a weighted summation of tissue injury. resulting in mortality, from the natural aging process as well as from environmental insults. Taking Napierian logarithms, Eq. 2 yields the log-hazard function (hereafter called the Gompertz function) ln[h(.u)] = Ink
+
4(x-)
(3)
or G, = G,, +
$(x),
where G_, is termed the Gompertz transform(ation) or ‘Gompertzian’ [7] in honor of Benjamin Gompertz. Gompertz demonstrated that age-specific mortality rate increases exponentially for human populations between ages 35 and 85 [12]. In Eq. 4. G, is the Napierian logarithm of the hazard function at time s. The parameter G,, is the extrapolated intercept at time 0. Termed the vulnerability parameter, it is related to the vigor of the genotype in the environment. It measures the initial vulnerability of the population (before the onset of aging) to causes of disease and mortality occurring during senescence [ 10,131. A mathematical correspondence between the decline of physiological processes associated with aging and the Gompertz function was derived by Sacher and Trucco [ 141 and Strehler and Mildvan [15,16]. Sacher and Trucco [14] used equations for Brownian motion for a free particle to represent the physiological state of an organism as it moves through a configuration space with a limiting hypersurface which they termed the ‘lethal bound.’ A Gaussian distribution represented the location of the frequencies of mean physiological states of a homogeneous population within the configuration space. Their ‘stochastic model’ was used to predict that if the ‘mean physiological state’ of a group of individuals declined linearly, tlzen an exponential increase in the rate of mortality would follow. They were emphatic, however, in stating that their theory does not predict a linear decrease of physiological performance per se [14]. Strehler and Mildvan [ 15,161 used a Maxwell-Boltzman distribution to characterize the magnitude (‘harmfulness’) of environmental ‘stresses-challenges.’ They assumed a linearly decreasing capacity (‘vitality’) to do the work needed to circumvent the challenges. Their model assumes that mortality rate is proportional to the frequencies of stresses that exceed the ability of the individual to restore initial conditions. Their theory predicts the linear decline of physiological function that occurs with age and the corresponding exponential increase in mortality seen in humans between age 35 and 85 [15,17]. The thermodynamic interpretation of the Gompertz function was further developed by Lestienne [ 181 who integrated the concept of programmed longevity in to Strehler and Mildvan’s model. Lestienne characterized Gompertz distributions by the Gompertz slope and finite lifespan rather than the traditional Gompertz slope and Gompertz intercept.
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P.J. Neafsey e( al. / Mechanisms qf’ Agdng and Developmerlf 83 (1995) 65-85
There continues to be disagreement concerning the mathematical form of 4(x). Sacher [7] postulated that a linear Gompertz function would only be seen in homogeneous, inbred, mammalian populations housed in well-controlled environments, and that genetically heterogeneous populations, populations in heterogeneous environments, or populations exposed to environmental toxicants would likely exhibit a curvilinear Gompertz function. Numerous, more highly parameterized empirical models, that better characterize the mortality of heterogeneous populations, generate the linear Gompertz function as a special case [19%28]. Homogeneous, inbred rodent populations used in toxicology studies that are housed in uniform, well-controlled environments and fed nutritionally adequate diets exhibit the linear Gompertz function after weaning [1.29-331. Where appropriate, use of the linear Gompertz function helps to limit the number of parameters in potentially over-parameterized mathematical models of mortality 129-311. The linear Gompertz function is
G, =
G,
+
xx,
(5)
where c( is a first-order aging rate constant which specifies the rate at which the hazard function associated with initial vulnerability (G,) progresses with age (i.e. it characterizes senescent injury) [7]. Sacher [lo] was the first to propose that increments or decrements to mammalian injury resulting from exposure to exogenous agents are superimposable on senescent injury. The principle of superimposition of system responses widely used in pharmacokinetics [34] led Boxenbaum and colleagues [ 1,351 to develop modified Gompertz (log-hazard) functions which characterize the time course of induced injury from acute and chronic exposure to environmental toxicants. They compared Gompertz functions between control and treated groups of animals administered fixed dose rates of toxicant. The Gompertz functions were used to delineate the kinetics of injury production and dissipation for acute and chronic exposure. Boxenbaum et al. [l] described discrete classes of perturbations to the linear Gompertz relationship resulting from irreversible injury, reversible injury, and longevity hormesis (occurring independently of one another). They also illustrated how a heterogeneous population would alter the Gompertz function in some cases. The following three examples illustrate the shapes of the Gompertz functions for different exposure patterns: (1) single dose exposure; (2) chronic exposure; and (3) short-term exposure. The examples are meant to describe the shapes of the temporal mortality fluctuations. 1.1. Single-dose exposure
One of the perturbations to the Gompertz function investigated by both Sacher [7] and Boxenbaum et al. [l] is the case of short-term exposure to a single high dose of a toxicant which instantaneously results in non-repaired injury that summates with aging injury. The modified linear Gompertz function for acute toxicity is G,. = Go +
LXX +
Ed
(6)
P.J. Neafse?) et al.
1Mechanisms of Age@ and Developmenr 83 (1995) 65-85
69
where eD is a term characterizing the additional irreversible injury at dose D. The addition of the permanent fixed injury term, sD, causes a steady-state parallel upward displacement of the Gompertz function (Fig. 1A). In the 1950s Upton et al. [38,39] plotted parallel Gompertz functions of mice exposed to an experimental thermonuclear detonation (high-energy y rays). Sacher [4] fitted a quadratic equation (apparently a phenomenological choice) to the upward displacement of the Gompertz intercepts from this data set. He stated that the form of the Gompertz functions, i.e. parallel lines, indicated that the injury was EXAhiW OF SINGLE DOSE EXFOSURE -KsEtlmm4Lwu(poBR 106 or, H UILE RATS
EXAMPLE OF CHRONIC EXPOSURE c.wcucMEcl_E~,35(4m,Y-R*TS
EXAMPLE OF SHORT-lERM EXT’OSLRE s!uxr-TEW @A- Am EXPOSR H-lws
Fig. 1. A: Example of single-dose exposure. Estimated Gompertzians for male SpragueeDawley rats subjected to a single 0.5 Gy whole-body exposure of fission neutrons over a 22-h period ( + ) and sham exposed controls (m). Raw mortality data were from Chmelevsky et al. [36] and LaFuma et al. [37]. Lines are the best fit of the data to Eq. 6. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Neafsey and Lowrie [32] with the permission of the copyright owner (Academic Press, Inc.). B: Example of chronic exposure. Estimated Gompertzians for female SpragueeDawley rats subjected to chronic methylene chloride (3500 ppm. 6 h/day, 5 days/week) begun at 8 weeks of age and continued for 2 years ( + ) and controls ( W). Raw mortality data were from Burek et al. [41]. Lines are the best fit of the data to generated from Eq. 10. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Fig. 8 of Neafsey et al. [29] with the permission of the copyright owner (Marcel Dekker, Inc.). C: Example of short-term exposure. Estimated Gompertzians for female B6CFI female mice subjected to 60 weekly whole body ;’ radiation exposure (4.5 Gy total dose) ( + ) and sham-exposed controls (m). Raw mortality data were from Thomson and Grahn [42]. Lines are the best fit of the data to Eq. 1I. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Fig. 6 of Neafsey and Lowrie [33] with the permission of the copyright owner (Radiation Research Society).
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not reparable and that the degree of upward displacement of the Gompertz functions indicated the degree of irreparable injury with dose. Neafsey and Lowrie [32] fit Eq. 6 to the data of Chmelevsky et al. [36] and LaFuma et al. [37] from rats subjected to single whole body exposures to fission neutrons. They used an a logarithmic-logistic function to relate .sDto dose: F = 'D
~2
0
!_ +D'
(7)
Q
By rearranging Eq. 7, they were able to use the model to estimate the dose of neutrons which produces a given displacement of the Gompertz function: (8) and then calculate the relative biological effectiveness (RBE) of neutrons radiation in terms of the degree of displacement: RBE+
to y
(9) N
where D,. is the dose (Gy) of y radiation for a given parallel upward displacement cD of the Gompertzian function and D, is the dose (Gy) of fission neutrons for the same parallel upward displacement of the Gompertz function. 1.2. Chronic exposure Chronic (constant zero-order) exposure to a toxic agent (e.g. lifespan toxicity studies), that results in irreparable injury, yields Gompertz functions with identical Gompertz intercepts but increased slopes, indicating a constant age-independent enhancement (y,) of the mean intensity of injury [7,29,30,40]: G, = G, + (a + yD).x
(10)
where yD is the cumulative toxicity with dose-rate D (Fig. 1B). Neafsey et al. [30] studied lifespan exposure to 500-3500 ppm methylene chloride (inhaled 6 h per day, 5 days per week) in female rats (raw mortality data from Burek et al. [41]). These mortality data generated a fan of Gompertz functions where yD was related to dose-rate by a logarithmic-logistic function. 1.3. Short-term exposure
When animals in toxicity studies are exposed for short periods of time and then allowed to live out their lifespans, Eq. 10 becomes G, = G,, + cIx + (j’D)b
(11)
P.J. Neafiey
Table 1 Exposure
Number
6
152 88 56 56 56 64 64
8
: Mechani.rms
uf Agring
and Derelnpnwnt
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71
regimens
Series
1
et a/.
118
64 32 32 32 96 544 480 384 192 96 96 192
of animals
Exposure
regimen,’
lOOO-WL radon progeny; I5 mg/m’ ore dust
Controls 100-WL
uranium
radon
progeny; 15 mg/m’ ore dust
uranium
Controls IOO-WL radon
progeny; 15 mg:m’ ore dust
Controls
uranium
Total exposure WLMh
Exposure time (days)
320 640 1280 2560 5120 10340 0 320 640 1280 2560 5120 0 20 40 80 160 320 640 0
2.24 4.48 8.96 17.92 35.84 71.68 0 22.4 44.8 89.6 179.2 358.4 0 1.4 2.8 5.6 11.2 22.4 44.8 0
‘In a 90-h week. “Working level (WL) is defined as any combination of the short-lived radon progeny in 1 I of air that will result in the ultimate emission of 1.3 x lo5 MeV of potential (2) energy (I WL = 2.08 x tom5 Jhm-‘). Working level month (WLM) is an exposure equivalent to 170 h at a I-WL concentration (1 WLM = 3.5 x IO ’ Jhm ‘).
where b is defined as the min (x, time of discontinuance) (Fig. 1C). Neafsey and Lowrie [33] fit Eq. 11 to Thomson and Grahn’s [42] raw mortality data of male and female mice subjected to 60 weekly whole body exposures to fission neutrons and ;’ rays. After the exposure ceased (420 days), a parallel upward displacement of the Gompertz functions was apparent for all doses. They termed the degree of displacement, sssD at ‘steady state’, i.e. the time after exposure ends. Note that sss,,/420 = 17~in Eq. 11. They substituted ksD for 6Din Eqs. 7 and 8 and calculated the RBE for neutrons compared to ;I radiation for short-term exposure. 1.4. Fractionated exposure When fractionated exposures are used in toxicity studies, groups of animals are exposed at different dose rates for varying periods of time, and effects from cumulative exposures achieved by different exposure patterns are of interest. The purpose of this paper is to demonstrate the utility of mortality kinetics analysis in evaluating mortality data from fractionated exposure studies. Mortality data were analyzed from male rats exposed to radon progeny at 100 WL and 1000 WL (in a 90-h week) (Table 1) for varying periods of time for total exposures ranging from
12
P.J. Neajkey
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20 to 10 240 WLM. It will be shown that Gompertz functions give rise to an informative visualization of the temporal nature of the changes in mortality when exposure is fractionated. 2. Methods The analyses in this paper are based on mortality data from several adult male SPF Wistar rat experiments conducted at the Pacific Northwest Laboratory. The details of the experiments may be found elsewhere [43,44]. Table 1 describes the historic series 6000, 7000, and 8000 experiments (hereafter referred to as series 6, 7, and 8, respectively), which were the basis for this paper. These experiments were designed to develop the relationships between response and exposure to radon progeny (at 100 WL in a 90-h week and 1000 WL in a 90-h week) and carnotite ore dust. The total exposure levels in WLM were chosen to mimic current and former conditions in uranium mines. Working level (WL) is defined as any combination of the short-lived radon progeny in 1 1 of air that will result in the ultimate emission of 1.3 x lo5 MeV of potential (CC)energy (1 WL = 2.08 x lo-’ Jhme3). Working level month (WLM) is an exposure equivalent to 170 h at a 1-WL concentration (1 WLM = 3.5 x lop3 Jhmw3). The exact time of death or sacrifice of each animal was known and the animals were allowed to live out their life spans. The mortality data were used to calculate estimated Gompertzians along with the associated estimated weight (w). The early time periods (O-300 days) were pooled to have estimated Gompertzians which had a reasonably large weight during the early ‘silent’ period when there were very few deaths [33]. If there are too few deaths in an interval, the weight, which is the reciprocal of the variance, is too low and the value (if any) of the estimated Gompertzian function is subject to too much statistical variation. Gompertzians during the silent period (O-300 days) were estimated using Eq. 12 (following). Gompertzians after the silent period were estimated at the midpoints of loo-day intervals (350, 450,...), where deaths represented at least 2% of the initial cohort numbers (but never less than two deaths) using the following equation [4,33,45]: lnSZ, = ln[ - (t)lnpj],
(12)
where In 0, is an estimate of the Gompertzian (i.e. G,, in Eq. 4) at the midpoint of the interval, and is used to approximate the Gompertz function. The fraction of the population surviving the age interval with length U,= t, + , - ti is denoted as pi. For censored data, p, = & where oi = Cr,/(N, - 1/2w,), and where d, and bvjare deaths and withdrawals during the interval, respectively. Elandt-Johnson and Johnson [45] state that oi is approximately unbiased and is also the maximum likelihood estimate for the probability of failure in the interval if the times at death are assumed to be distributed exponentially. Heuristically, the formula subtracts one-half of the interval for each withdrawal from the exposure. This assumes that
P.J. NeaJ.se?rt ul. )IMechanisms of’ Agruy
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83 (1995) 65-85
13
all withdrawals occur in the middle of the interval [45], or that the withdrawals occur uniformly throughout the interval. The heuristic argument for censoring can be extended if more is known about the times of withdrawal during the interval. Then the corresponding proportion of the period for each withdrawal is subtracted. This can be derived from the Balducci assumption [46], and is used widely in actuarial work [47]: (13)
The number of withdrawals in the interval (t;, ti+ ,) at assumed times t, I tj, < t,?... < t,, 2 t, $ , are \L’,,,u’,?,..., \Y,,,, respectively. In conventional (unweighted) least squares. it is tacitly assumed that variance estimates about each data point are equal. This is not the case in Gompertz analyses, in which cohorts are employed. Sacher [7,16] developed relationships to estimate the sampling variance. Statistical weights (IV) used for In Q, values were the reciprocals of the estimates of the variance ( V) [16,45]: (14) The rationale for using our modified Gompertz model falls under the more general category of the treatment of concomitant variables, which is detailed by Johnson and Elandt-Johnson [45] and is outlined and compared to the Cox model [48] in Neafsey and Lowrie [33]. It is reproduced in Appendix A for completeness. The linear Gompertz function was fitted to the data for the control groups. The Gompertz function slopes and intercepts of the three control groups were not statistically different (P = 0.05). Therefore, the control data were pooled and the pooled control Gompertz function was used in the remaining analyses. Inspection of plots of In Q, (estimated Gompertzians) versus time revealed that after exposure ended, the estimated Gompertz function of the exposed groups attained a steady-state parallel upward displacement from the Gompertz function of the pooled controls. The linear Gompertz function (G., = G,, + CIX)was fitted to the data for the control and exposed groups for the time period after the termination of exposure. The slopes of the Gompertz functions of the exposed groups were not significantly different (P = 0.05) from controls using the two-tailed t test for estimated parameters [49]. The linear Gompertz function was then refitted to the data for the exposed groups (post-exposure) with the constraint that the slope parameters, CI,have the same value as those of the pooled control group. The ‘steady-state’ time was attained a short time after exposure ceased. The degree of upward parallel displacement at steady state (E,,) was calculated as the difference between the estimated intercept (In Q,) of the fitted model for the pooled controls and the In R, value for each exposure group.
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The non-linear least-squares computer program PCNONLZN [50,51] was employed for curve fitting using initial estimates of parameters which were obtained graphically. The Nelder-Mead [52] algorithm was used to search the parameter space. Other least-squares optimization software such as SAS [53] and BMDP [54] have been used, and yield parameter estimates which are essentially the same as those found by the Nelder-Mead algorithm. The choice of PCNONLZN was based on convenience since the PC-based data files can be accessed directly. The NelderMead algorithm has provisions for preventing the optimization method from getting ‘stuck’ some distance from the global (local) minimum. The algorithm is very tolerant of poor initial estimates of the parameters at the expense of the relatively large number of iterations necessary for convergence. The Gauss-Newton method used in SAS (and also available in PCNONLZN) may diverge if the initial estimates are poor or the bounds on the parameters are too large. The Dunnett test for multiple comparisons [55] was used to compare estimated cSSparameters among exposure groups. Goodness-of-fit was verified in three ways. First, visual inspection of plots of weighted residuals versus x indicated relatively good randomness of scatter of data points about fitted curves [56]. Second, visual inspection of data points (In R, - x pairs) about the regression lines indicated good randomness-of-scatter [49]. Third, the computed x2 values were less than tabulated values (2 = 0.05) [ 1,111. 3. Results Figs. 2-4 illustrate the fitted Gompertz model for the series 6, 7, and 8 animals, respectively. Fig. 5 illustrates the steady-state parallel upward displacement (E,,) of the Gompertz functions with associated standard errors for the series 6 animals (1000 WL in a 90-h week). The parallel upward displacement of the Gompertz functions (E,,) rises sharply at 320 and 640 WLM total exposure, and is statistically significant compared to pooled controls (P < 0.05). At 1280 WLM, the displacement falls below the E,, at 320 WLM and from 1280 to 10 240 WLM, the displacement increases with total exposure in a linear fashion. Fig. 6 illustrates the steady-state parallel upward displacement of the Gompertz functions with associated standard errors for both the series 7 and 8 animals (100 WL in a 90-h week). There is no significant displacement from the control Gompertz function from 20 to 1280 WLM total exposure. The Gompertz functions for 2560 and 5120 WLM are significantly displaced above that of the pooled controls (P < 0.05). Fig. 7 simultaneously displays the cSSfor both the 1000 WL and the 100 WL exposed animals. At 1000 WL, total exposures below 1280 WLM resulted in greater cumulative injury resulting in mortality than did exposure at 100 WL.
6 1OOOV.t k320 WLMI
-6
-4
-3
;;
SERIES
Fig. 2. Estimated Gompertzians for series 6 animals exposed at 1000 WL (90 h, week) for total doses of 320 (1 ). 640 (*), and IO 740 (+) WLM. ( + ) are pooled controls. See Table 1 for exposure times in days. Lines are the best fit of the data
SERIES
,000
W
(1280 WMI
1280 (A), 2560 (I). to Eq. 11.
5
5120
( 2). 21
76
P.J. NeqFey et al. 1 Mechanisms of Ageing and Deoelopmenr 83 (1995) 65-85
Fig. 4. Estimated (m) WLM. ( +
1320 WLM,
SERIES
for series 8 animals exposed at 100 WL (90 h/week) for total doses of 20 (,_‘), 40 ( l ). 80 (*), See Table 1 for exposure times in days. Lines are the best fit of the data to Eq. I I.
) are pooled controls.
Gompertzians
SERIES .3 loow.
WU.4)
160 CA ). 320 CO), and 640
S lOOwL (640
P.J. Neqfiey et al. , Mechanisms oj’ Ageing und Developmerli 83 (1995) 65-85
SERIES 6 (1 OOOWL) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
in thousands DOSE M’LM) Estimated
degree of upward
parallel
displacement
of Gompertz
functions
(c,,) at steady state for
arnmats exposed at 1000 WL (90 h/week) for total doses from 320 WLM (,2.24 days) to 10 240 WLM (71.68 days). Points arc observed 8SSvalues and standard errors calculated from individual weighted least-squares analysis of Gompertzians from pooled control and exposed groups. The standard error of the Gompertz intercept (In Q,) is shown for pooled controls.
4. Discussion
The parallel upward displacements of the Gompertz functions at steady state for the series 6 animals exposed at 1000 WL (Figs. 2 and 5) illustrate the imperative in dose-response studies for carefully selected exposures in the low dose range. Had the 320 and 640 WLM total exposure studies not been conducted, the striking linearity of the upward displacement of the Gompertz functions for the 1280; 2560; 5120; and 10 240 WLM groups might have led to a serious underestimation of the response below 1280 WLM at 1000 WL and a conclusion that at 1000 WL total exposure-mortality response (upward parallel displacement of the Gompertz loghazard functions) is linear. A simultaneous fit of Eq. 11 for pooled controls and the 1280, 2560, 5120, and 10 240 WLM groups produced estimated parameters and standard errors of: In Q,= - 10.34 & 0.19; a=6.85 x lop3 & 3.14 x 10-j days ‘; and ~~,oOOw,_ = 2.44 x lop2 f 3.08 x lo- 3 days I. The model standard error was 0.37 (26 d.f.) and the model weighted sums of squares (WSS) was 3.64. The x2 value for the model was calculated by summing the calculated x2 value for each exposure group and the pooled control group and comparing the summed value to the tabulated value for (k - Y - 1) degrees for freedom at the x = 0.05 level of significance, where k is the sum of the number of cells for all doses and Y is the total number of model parameters (i.e. r = 3 for Eq. 11). For a discussion of
P.J. NrufSey
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the use of the x z goodness-of-fit test as it is applied to mortality models see Lee [ 1l] and Boxenbaum et al. [I]. The computed x2 value was 68.88 (65 d.f.). The tabulated x2 value at a = 0.05 is 84.80 for 65 d.f. The relatively low coefficients of variation, small mode1 standard error, low WSS and x’ value indicated consistency with a linear mode1 for E,, with WLM when the lowest total exposure groups were omitted. When all of the exposure groups at 1000 WL and pooled controls were fitted simultaneously to Eq. 11, the model standard error increased to 0.44 (40 d.f.) and the WSS increased to 8.82. The standardized residuals for the 320 and 640 WLM groups were large and all positive. The calculated x2 value was 173.48 (109 d.f.) compared to the tabulated value of 133 at x = 0.05 for 109 d.f. The hypothesis that the displacement of the Gompertz functions at steady state with total exposure is linear must be rejected when all exposed groups at 1000 WL are included in the model. The life-shortening effects of exposure to 640 WLM at 1000 WL compared to 1280 WLM, are not easily explained. The rats exposed for 4.48 days at 1000 WL (640 WLM) had significantly more injury resulting in death than the rats exposed for 8.96 days (1280 WLM). Whether this is (1) a statistical anomaly; (2) due to some type of induced repair process; or (3) due to killing of some cells that incur fatal injury, is unknown. These mortality data do suggest that the radon progeny exposure-mortality response at 1000 WL is not only non-linear but not even a monotonic function. Future studies using larger sample sizes and including a total SERIES
7,8 ( lOOWL)
1.25
P g & u P (I>
1.oo
0.75
0.50
B 5
0.25
1
0.00
-0.25 0
1000
2000
3000
4000
5000
6000
DOSE (WLM) Fig. 6. Estimated degree of upward parallel displacement of Gompertz functions (c,,) at steady state for animals exposed at 100 WL (90 h/week) for total doses from 20 WLM (1.4 days) to 5120 WLM (358.4 days). Points are observed c,, values and standard errors calculated from individual weighted leastsquares analysis of Gompertzians from pooled control and exposed groups. The standard error of the Gompertz intercept (In iW,,) is shown for pooled controls.
80
P.J. Neafsey et al. / Mechanisms of Ageing and Development 83 (1995) 65-85
2.00 1.75 E e f? E v)
k
1.50 1.25 1.00
z
0.75
9 3
0.50
&
0.25
0.0
1.5
3.0
4.5
6.0 7.5 in thousands DOSE (WLM)
9.0
10.5
12.0
Fig. 7. Estimated degree of upward parallel displacement of Gompertz functions (a,) for exposures at 1000 WL (0) and 100 WL (m).
exposure group at 960 WLM (6.72 days) are necessary to establish the shape of the exposure-response relationship at 1000 WL in the O-1500 WLM region. The total WLM exposure levels analyzed here were chosen to mimic former and current conditions in uranium mines. The level of exposures was relatively high and moderately short term (Table 1). We caution against using these data to extrapolate risk for the very low, highly protracted exposures of residential radon progeny. Lifetime radon exposures in a typical house produce cumulative radon progeny exposures of about 20 WLM. At the EPA 4-pCi/l radon action level, they are about 80 WLM. In the data analyzed here, there appeared to be no excess mortality in rats subjected to acute exposure to radon progeny for cumulative exposures typical in residences (Fig. 6), although excess rat lung cancers were produced. In this study, protracting exposure decreased the life shortening effect of radon progeny below total exposures of 1,280 WLM. Moolgavkar et al. [44] reported a similar finding for total exposures of 320- 10 000 WLM at 100 and 1000 WL. In one recent French experiment, adult rats given 25 WLM at high exposure rates showed excess lung cancers but no excess was found when the 25 WLM was protracted over the remaining rat lifespan [57,58]. While it is tempting to speculate that the parallel upward displacements of the Gompertz functions at steady state for the series 7 and 8 animals exposed at 100 WL (Figs. 3, 4 and 6) suggest a threshold-type exposure-mortality response, several caveats are offered. First, the assumption that protraction of very low cumulative exposure (such as that received in residences) will also decrease the life-shortening effect of radon progeny may not hold. The life-shortening effect of neutron
P.J. Neafsey
r? itl.
: Mechanisms
o/ Ageinp and Development
83 (1995) 65-85
81
exposure has been reported to be enhanced by protraction of exposure for relatively high total exposures of 1.6-2.4 Gy [59-611, although no augmentation of the life-shortening effect was seen with protraction of lower total exposures (0.1-0.5 Gy). Second, the majority of radon-progeny induced rat lung tumors are considered to be non-fatal and, unlike human lung cancers, are not the cause of early death. There are a plethora of concomitant variables which might affect human mortality response to radon progeny [62]. Among these variables are: frequency and duration of cigarette smoking, passive exposure to cigarette smoke, diet (e.g. consumption of antioxidants such as beta carotene, vitamin E and ascorbic acid), exposure to dust (e.g. arsenic), genetic resistance to cancer, ventilation in buildings, oral and nasal breathing patterns, age at exposure, and concentration and activitysize distributions of radon progeny. Moolgavkar et al. 1441contend that exposureresponse with respect to radon progeny, will depend on the particular exposure regimen (rate of exposure, total exposure, and timing of exposure) and particular response (e.g. probability of tumor, time to tumor, MAS, or age-specific mortality rate) chosen. The data reported here reinforce this contention. Mortality kinetics analysis must therefore be considered an adjunct to radon-progeny exposure-response analysis. It is not a substitute to tumor based models of risk. Acknowledgements
This work was supported in part by the U.S. Department of Energy under Contract DE-AC06-76RL01830 and by the University of Connecticut Research Foundation. Appendices Appendix
A:
Conwomitant
uariubles
A concomitant variable can be continuous (e.g. age, dose, etc.) or discrete (e.g. gender), and is part of the data. Parameters are determined from the data. Let y’ = (~3,.J~~....._v,,) be a row vector of concomitant variables (y is a column vector) and the hazard rate be i(t;y) 2 0. The hazard rate may depend on unknown parameters, so we write E.(t: JJ) = i,(t; _v; B),
(15)
where B’ = (II,, B, ,.... B,,) is a row vector of parameters. Theoretically, a different model could be used for each treatment group in an experiment. If the models are unrelated, they are difficult to fit and explain. A typical paradigm for an experiment uses one model with several parameters and/or concomitant variables. The choice of parameters allows a different sub-model for the control group and each treatment group. Suppose there are y groups (e.g. treatments). Then the hazard rate for the jth group is
82
P.J. Neafsey
ij(t; V) =
ij(t;
_V;
et al. / Mechanisms
of Ageing and Development
83 (1995) 65-85
(16)
Bj),
where j= 1, 2,..., q. Let Y’~= (JJ,~,yzir..., vPi) be the row vector of observed values of the concomitant variables for the ith individual. Once observed from the data, these values are considered fixed within the experiment at hand. There are two commonly used special cases of the general model: the additive model and the multiplicative model. The additiue model is assumed to be
where R is called the underlying hazard rate. It is assumed that the functions h,(t) are functions of time only, and the functions q,,(_v,,)= q,,(yP, B,>)p = 1, 2,...,s do not depend on t. Johnson and Elandt-Johnson [45] present, among others, a model using the Gompertz hazard rate as the underlying hazard rate. The general wzultiplicatiuemodel is assumed to be /i(t; y,)
= n(t).q(?,,;
B)
(18)
or In /Z(t; y,) = In R(t) + In
q(y,;
B),
(19)
where q has a known form and A is unspecified or of a known form (e.g. Gompertz) with parameters. This model is called the proportional hazard rate model. It is commonly assumed that the function ~1is not a function of t. If we assume that qCyi; B)
=
ecB’Y)
(20)
or
In q&i; B) = WY)
(21)
where B’y is the inner product of the vectors y and B, we have the Cox model [48]. Our models (Eqs. 4, 5, 10, 11) are similar to a Cox model with f = .x, In A(t) = Go + at
(22)
To have a Cox model in Eq. 6, Q, would have to be a linear function which does not fit the data as well as does Eq. 7. More complicated versions of Eq. 5 may be used to characterize other mortality effects of exposure to toxic agents, e.g. early paradoxical effects (longevity hormesis) [29,30,33] or food restriction [31] where (I is a function of time (age) as well as the concomitant variables.
P.J. Neajiey
et al.
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of Ageing und De~elopmenr
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83 (1995) 65-85
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U.S. Department of Energy. Washington, DC. 1988. S.H. Moolgavkar, F.T. Cross. G. Luebeck and G.E. Dagle, A two-mutation model for radon-induced lung tumors in rats. Radiat. Ref., 1Zi (1990) 28-37. R.C. Elandt-Johnson and N. Johnson, Survival Models and Data Analysis. Wiley, New York. 1980. D. London, Survival Models, Actex Co., Winsted, CT, 1986. N.L. Bowers, Jr., H.U. Gerber, J.C. Hickman, D.A. Jones and C.J. Nesbitt. Actuarial Mathematics. Society of Actuaries, Chicago, 1986. D. Cox and R.D. Oakes, Ana!vsis of Survival Data. Chapman and Hall. London, 1984.
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Reassessment? National
Mechanisms of Ageing and Development 83 (1995) 65-85
ELSEVIER
A mortality kinetics approach to characterizing fractionated exposure-mortality response relationship of radon progeny P.J. Neafsey*“, “School
qf Nursing,
bActuarial
Scienw,
Uniwrsity
Drparrmrnt
W.B. Lowrieb, F.T. Cross”
of Conne~~ticut. of’ Marhemurics.
Box
und Chemistry
U-59,
Uniwrsit~
CT 06269-3009. ‘Biology
the
Storrs.
06269-2059.
CT
q/ Connecticut.
Box
USA
U-9. Storrs.
USA
Deparrmenr,
Pac$c
Northnrst
Washington.
DC 99352,
Laboratory,
Richland.
USA
Received 23 January 1995; revision received 15 May 1995: accepted 28 May 1995
Abstract
The utility of mortality kinetics analysis in evaluating mortality data from fractionated exposure studies was demonstrated using radon-progeny induced extra mortality as an example. Gompertz (log-hazard) functions were used to characterize the mortality of male SPF Wistar rats exposed to radon progeny at 100 WL and 1000 WL for total exposures ranging from 20 to 10 240 WLM. There was an upward parallel displacement of the Gompertz functions following the period of radon exposure. The shape of the Gompertz functions for the exposed animals was consistent with a Gompertz model of toxicity resulting from short-term exposure, resulting in non-repaired injury that summates with natural (aging) injury. The parallel upward displacements (E,,) of the Gompertz functions showed an unexpected non-monotonic pattern for rats exposed at 1000 WL. The parallel upward displacements showed a sharp upward increase from 320 to 640 WLM, fell at 1280 WLM, and thereafter increased linearly to 10 240 WLM. These data suggest that the radon progeny exposure-mortality response is non-linear. In contrast, there was no significant parallel upward displacement of the Gompertz functions for rats exposed at 100 WL for total exposures of 20-1280 WLM. but a large displacement began at 2560 WLM total exposure. Ke~+~~ords: Dose-response;
* Corresponding 0047-6374/95/$09.50 SSDI
Gompertz;
Hazard
functions;
Mortality:
Radon-progeny
Author, Tel: + I 203 4863713; Fax: + I 203 4860512 0
1995
0047-6374(95)01606-Z
Elsevier Science Ireland Ltd. All rights reserved
66
P.J. Neqfbe)
et cd. /)/Mechanisms
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und Droelopmrnr
83 (1995)
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1. Introduction
The rationale for using survival models in toxicity studies is that no single endpoint (e.g. carcinogenesis) completely describes toxicant-induced damage. Since the mortality of a population is a measure of the combined effects of natural aging with environmental insults, techniques that can model time-dependent exposure patterns to toxicants may yield indices of net injury from combined neoplastic and non-neoplastic causes. Mortality data in toxicity studies are typically published as percentage survivorship or cumulative mortality plots, in terms of mean or median survival times, or as mean after survival (MAS). These variables do not characterize temporal mortality patterns such as the effect of fractionation of a given total exposure. Mortality kinetics analysis (also known as age-specific mortality rate analysis or hazard analysis) [I] integrates total causes of death related to dose. It can be used to characterize late-life toxic effects, relative biological effectiveness, and the temporal pattern of net injury from environmental exposure. The purpose of this paper is to demonstrate that Gompertz (log-hazard) functions are well suited to characterizing the effects of fractionation of a given total exposure. A mortality kinetics analysis of radon-induced mortality is used as an example. The use of Gompertz functions to characterize changes in mortality stemming from exposure to toxic agents evolved from work of Sacher and colleagues [2-lo]. The hazard function (instantaneous mortality rate, force of mortality, or instantaneous age-specific mortality rate), when multiplied by d-v, is the probability of death during the interval s to I + dx (where
h(x) =
- d[lnS(x)] = __.\1 dN. d.x
N, dx ’
(1)
where h(x) is the hazard function at age x, (also called the age-specific mortality rate), S(x) is the fraction of the population surviving at age x, and N., is the number of individuals surviving at age s. An age-specific mortality rate of 0.12 weeks-’ at age 50 weeks means that the probability of an individual dying in the 51st week is approximately 12%. Sacher and Brues [2,6-91 proposed that the hazard function is an exponential function of the mean intensity of physiological injury for homogeneous mammalian populations, kept free of preventable disease and housed in a uniform environment, h(x) = ke$“’
where h(x) is the hazard function at age x, k is a proportionality constant related to the inability of the population to withstand injury from the environmental conditions, and 4(x) is the mean intensity of injury of the population at age x. Sacher [5] defined injury in this context as any change which is detrimental to the
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et al.
: Mechmisms
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and Drvrlopmtv~r
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67
status of vital system ‘states.’ He proposed that 4(s) is a weighted summation of tissue injury. resulting in mortality, from the natural aging process as well as from environmental insults. Taking Napierian logarithms, Eq. 2 yields the log-hazard function (hereafter called the Gompertz function) ln[h(.u)] = Ink
+
4(x-)
(3)
or G, = G,, +
$(x),
where G_, is termed the Gompertz transform(ation) or ‘Gompertzian’ [7] in honor of Benjamin Gompertz. Gompertz demonstrated that age-specific mortality rate increases exponentially for human populations between ages 35 and 85 [12]. In Eq. 4. G, is the Napierian logarithm of the hazard function at time s. The parameter G,, is the extrapolated intercept at time 0. Termed the vulnerability parameter, it is related to the vigor of the genotype in the environment. It measures the initial vulnerability of the population (before the onset of aging) to causes of disease and mortality occurring during senescence [ 10,131. A mathematical correspondence between the decline of physiological processes associated with aging and the Gompertz function was derived by Sacher and Trucco [ 141 and Strehler and Mildvan [15,16]. Sacher and Trucco [14] used equations for Brownian motion for a free particle to represent the physiological state of an organism as it moves through a configuration space with a limiting hypersurface which they termed the ‘lethal bound.’ A Gaussian distribution represented the location of the frequencies of mean physiological states of a homogeneous population within the configuration space. Their ‘stochastic model’ was used to predict that if the ‘mean physiological state’ of a group of individuals declined linearly, tlzen an exponential increase in the rate of mortality would follow. They were emphatic, however, in stating that their theory does not predict a linear decrease of physiological performance per se [14]. Strehler and Mildvan [ 15,161 used a Maxwell-Boltzman distribution to characterize the magnitude (‘harmfulness’) of environmental ‘stresses-challenges.’ They assumed a linearly decreasing capacity (‘vitality’) to do the work needed to circumvent the challenges. Their model assumes that mortality rate is proportional to the frequencies of stresses that exceed the ability of the individual to restore initial conditions. Their theory predicts the linear decline of physiological function that occurs with age and the corresponding exponential increase in mortality seen in humans between age 35 and 85 [15,17]. The thermodynamic interpretation of the Gompertz function was further developed by Lestienne [ 181 who integrated the concept of programmed longevity in to Strehler and Mildvan’s model. Lestienne characterized Gompertz distributions by the Gompertz slope and finite lifespan rather than the traditional Gompertz slope and Gompertz intercept.
68
P.J. Neafsey e( al. / Mechanisms qf’ Agdng and Developmerlf 83 (1995) 65-85
There continues to be disagreement concerning the mathematical form of 4(x). Sacher [7] postulated that a linear Gompertz function would only be seen in homogeneous, inbred, mammalian populations housed in well-controlled environments, and that genetically heterogeneous populations, populations in heterogeneous environments, or populations exposed to environmental toxicants would likely exhibit a curvilinear Gompertz function. Numerous, more highly parameterized empirical models, that better characterize the mortality of heterogeneous populations, generate the linear Gompertz function as a special case [19%28]. Homogeneous, inbred rodent populations used in toxicology studies that are housed in uniform, well-controlled environments and fed nutritionally adequate diets exhibit the linear Gompertz function after weaning [1.29-331. Where appropriate, use of the linear Gompertz function helps to limit the number of parameters in potentially over-parameterized mathematical models of mortality 129-311. The linear Gompertz function is
G, =
G,
+
xx,
(5)
where c( is a first-order aging rate constant which specifies the rate at which the hazard function associated with initial vulnerability (G,) progresses with age (i.e. it characterizes senescent injury) [7]. Sacher [lo] was the first to propose that increments or decrements to mammalian injury resulting from exposure to exogenous agents are superimposable on senescent injury. The principle of superimposition of system responses widely used in pharmacokinetics [34] led Boxenbaum and colleagues [ 1,351 to develop modified Gompertz (log-hazard) functions which characterize the time course of induced injury from acute and chronic exposure to environmental toxicants. They compared Gompertz functions between control and treated groups of animals administered fixed dose rates of toxicant. The Gompertz functions were used to delineate the kinetics of injury production and dissipation for acute and chronic exposure. Boxenbaum et al. [l] described discrete classes of perturbations to the linear Gompertz relationship resulting from irreversible injury, reversible injury, and longevity hormesis (occurring independently of one another). They also illustrated how a heterogeneous population would alter the Gompertz function in some cases. The following three examples illustrate the shapes of the Gompertz functions for different exposure patterns: (1) single dose exposure; (2) chronic exposure; and (3) short-term exposure. The examples are meant to describe the shapes of the temporal mortality fluctuations. 1.1. Single-dose exposure
One of the perturbations to the Gompertz function investigated by both Sacher [7] and Boxenbaum et al. [l] is the case of short-term exposure to a single high dose of a toxicant which instantaneously results in non-repaired injury that summates with aging injury. The modified linear Gompertz function for acute toxicity is G,. = Go +
LXX +
Ed
(6)
P.J. Neafse?) et al.
1Mechanisms of Age@ and Developmenr 83 (1995) 65-85
69
where eD is a term characterizing the additional irreversible injury at dose D. The addition of the permanent fixed injury term, sD, causes a steady-state parallel upward displacement of the Gompertz function (Fig. 1A). In the 1950s Upton et al. [38,39] plotted parallel Gompertz functions of mice exposed to an experimental thermonuclear detonation (high-energy y rays). Sacher [4] fitted a quadratic equation (apparently a phenomenological choice) to the upward displacement of the Gompertz intercepts from this data set. He stated that the form of the Gompertz functions, i.e. parallel lines, indicated that the injury was EXAhiW OF SINGLE DOSE EXFOSURE -KsEtlmm4Lwu(poBR 106 or, H UILE RATS
EXAMPLE OF CHRONIC EXPOSURE c.wcucMEcl_E~,35(4m,Y-R*TS
EXAMPLE OF SHORT-lERM EXT’OSLRE s!uxr-TEW @A- Am EXPOSR H-lws
Fig. 1. A: Example of single-dose exposure. Estimated Gompertzians for male SpragueeDawley rats subjected to a single 0.5 Gy whole-body exposure of fission neutrons over a 22-h period ( + ) and sham exposed controls (m). Raw mortality data were from Chmelevsky et al. [36] and LaFuma et al. [37]. Lines are the best fit of the data to Eq. 6. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Neafsey and Lowrie [32] with the permission of the copyright owner (Academic Press, Inc.). B: Example of chronic exposure. Estimated Gompertzians for female SpragueeDawley rats subjected to chronic methylene chloride (3500 ppm. 6 h/day, 5 days/week) begun at 8 weeks of age and continued for 2 years ( + ) and controls ( W). Raw mortality data were from Burek et al. [41]. Lines are the best fit of the data to generated from Eq. 10. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Fig. 8 of Neafsey et al. [29] with the permission of the copyright owner (Marcel Dekker, Inc.). C: Example of short-term exposure. Estimated Gompertzians for female B6CFI female mice subjected to 60 weekly whole body ;’ radiation exposure (4.5 Gy total dose) ( + ) and sham-exposed controls (m). Raw mortality data were from Thomson and Grahn [42]. Lines are the best fit of the data to Eq. 1I. Time on the abscissa refers to that period following the initiation of the experiment. Reproduced from Fig. 6 of Neafsey and Lowrie [33] with the permission of the copyright owner (Radiation Research Society).
70
P.J. Neafsey et al. / Mechanisms of’ Ageing and Development 83 (1995) 65-85
not reparable and that the degree of upward displacement of the Gompertz functions indicated the degree of irreparable injury with dose. Neafsey and Lowrie [32] fit Eq. 6 to the data of Chmelevsky et al. [36] and LaFuma et al. [37] from rats subjected to single whole body exposures to fission neutrons. They used an a logarithmic-logistic function to relate .sDto dose: F = 'D
~2
0
!_ +D'
(7)
Q
By rearranging Eq. 7, they were able to use the model to estimate the dose of neutrons which produces a given displacement of the Gompertz function: (8) and then calculate the relative biological effectiveness (RBE) of neutrons radiation in terms of the degree of displacement: RBE+
to y
(9) N
where D,. is the dose (Gy) of y radiation for a given parallel upward displacement cD of the Gompertzian function and D, is the dose (Gy) of fission neutrons for the same parallel upward displacement of the Gompertz function. 1.2. Chronic exposure Chronic (constant zero-order) exposure to a toxic agent (e.g. lifespan toxicity studies), that results in irreparable injury, yields Gompertz functions with identical Gompertz intercepts but increased slopes, indicating a constant age-independent enhancement (y,) of the mean intensity of injury [7,29,30,40]: G, = G, + (a + yD).x
(10)
where yD is the cumulative toxicity with dose-rate D (Fig. 1B). Neafsey et al. [30] studied lifespan exposure to 500-3500 ppm methylene chloride (inhaled 6 h per day, 5 days per week) in female rats (raw mortality data from Burek et al. [41]). These mortality data generated a fan of Gompertz functions where yD was related to dose-rate by a logarithmic-logistic function. 1.3. Short-term exposure
When animals in toxicity studies are exposed for short periods of time and then allowed to live out their lifespans, Eq. 10 becomes G, = G,, + cIx + (j’D)b
(11)
P.J. Neafiey
Table 1 Exposure
Number
6
152 88 56 56 56 64 64
8
: Mechani.rms
uf Agring
and Derelnpnwnt
83 (1995)
65-85
71
regimens
Series
1
et a/.
118
64 32 32 32 96 544 480 384 192 96 96 192
of animals
Exposure
regimen,’
lOOO-WL radon progeny; I5 mg/m’ ore dust
Controls 100-WL
uranium
radon
progeny; 15 mg/m’ ore dust
uranium
Controls IOO-WL radon
progeny; 15 mg:m’ ore dust
Controls
uranium
Total exposure WLMh
Exposure time (days)
320 640 1280 2560 5120 10340 0 320 640 1280 2560 5120 0 20 40 80 160 320 640 0
2.24 4.48 8.96 17.92 35.84 71.68 0 22.4 44.8 89.6 179.2 358.4 0 1.4 2.8 5.6 11.2 22.4 44.8 0
‘In a 90-h week. “Working level (WL) is defined as any combination of the short-lived radon progeny in 1 I of air that will result in the ultimate emission of 1.3 x lo5 MeV of potential (2) energy (I WL = 2.08 x tom5 Jhm-‘). Working level month (WLM) is an exposure equivalent to 170 h at a I-WL concentration (1 WLM = 3.5 x IO ’ Jhm ‘).
where b is defined as the min (x, time of discontinuance) (Fig. 1C). Neafsey and Lowrie [33] fit Eq. 11 to Thomson and Grahn’s [42] raw mortality data of male and female mice subjected to 60 weekly whole body exposures to fission neutrons and ;’ rays. After the exposure ceased (420 days), a parallel upward displacement of the Gompertz functions was apparent for all doses. They termed the degree of displacement, sssD at ‘steady state’, i.e. the time after exposure ends. Note that sss,,/420 = 17~in Eq. 11. They substituted ksD for 6Din Eqs. 7 and 8 and calculated the RBE for neutrons compared to ;I radiation for short-term exposure. 1.4. Fractionated exposure When fractionated exposures are used in toxicity studies, groups of animals are exposed at different dose rates for varying periods of time, and effects from cumulative exposures achieved by different exposure patterns are of interest. The purpose of this paper is to demonstrate the utility of mortality kinetics analysis in evaluating mortality data from fractionated exposure studies. Mortality data were analyzed from male rats exposed to radon progeny at 100 WL and 1000 WL (in a 90-h week) (Table 1) for varying periods of time for total exposures ranging from
12
P.J. Neajkey
et al. 1 Mechanisms
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and Development
83 (1995)
65-85
20 to 10 240 WLM. It will be shown that Gompertz functions give rise to an informative visualization of the temporal nature of the changes in mortality when exposure is fractionated. 2. Methods The analyses in this paper are based on mortality data from several adult male SPF Wistar rat experiments conducted at the Pacific Northwest Laboratory. The details of the experiments may be found elsewhere [43,44]. Table 1 describes the historic series 6000, 7000, and 8000 experiments (hereafter referred to as series 6, 7, and 8, respectively), which were the basis for this paper. These experiments were designed to develop the relationships between response and exposure to radon progeny (at 100 WL in a 90-h week and 1000 WL in a 90-h week) and carnotite ore dust. The total exposure levels in WLM were chosen to mimic current and former conditions in uranium mines. Working level (WL) is defined as any combination of the short-lived radon progeny in 1 1 of air that will result in the ultimate emission of 1.3 x lo5 MeV of potential (CC)energy (1 WL = 2.08 x lo-’ Jhme3). Working level month (WLM) is an exposure equivalent to 170 h at a 1-WL concentration (1 WLM = 3.5 x lop3 Jhmw3). The exact time of death or sacrifice of each animal was known and the animals were allowed to live out their life spans. The mortality data were used to calculate estimated Gompertzians along with the associated estimated weight (w). The early time periods (O-300 days) were pooled to have estimated Gompertzians which had a reasonably large weight during the early ‘silent’ period when there were very few deaths [33]. If there are too few deaths in an interval, the weight, which is the reciprocal of the variance, is too low and the value (if any) of the estimated Gompertzian function is subject to too much statistical variation. Gompertzians during the silent period (O-300 days) were estimated using Eq. 12 (following). Gompertzians after the silent period were estimated at the midpoints of loo-day intervals (350, 450,...), where deaths represented at least 2% of the initial cohort numbers (but never less than two deaths) using the following equation [4,33,45]: lnSZ, = ln[ - (t)lnpj],
(12)
where In 0, is an estimate of the Gompertzian (i.e. G,, in Eq. 4) at the midpoint of the interval, and is used to approximate the Gompertz function. The fraction of the population surviving the age interval with length U,= t, + , - ti is denoted as pi. For censored data, p, = & where oi = Cr,/(N, - 1/2w,), and where d, and bvjare deaths and withdrawals during the interval, respectively. Elandt-Johnson and Johnson [45] state that oi is approximately unbiased and is also the maximum likelihood estimate for the probability of failure in the interval if the times at death are assumed to be distributed exponentially. Heuristically, the formula subtracts one-half of the interval for each withdrawal from the exposure. This assumes that
P.J. NeaJ.se?rt ul. )IMechanisms of’ Agruy
md Dedopmmt
83 (1995) 65-85
13
all withdrawals occur in the middle of the interval [45], or that the withdrawals occur uniformly throughout the interval. The heuristic argument for censoring can be extended if more is known about the times of withdrawal during the interval. Then the corresponding proportion of the period for each withdrawal is subtracted. This can be derived from the Balducci assumption [46], and is used widely in actuarial work [47]: (13)
The number of withdrawals in the interval (t;, ti+ ,) at assumed times t, I tj, < t,?... < t,, 2 t, $ , are \L’,,,u’,?,..., \Y,,,, respectively. In conventional (unweighted) least squares. it is tacitly assumed that variance estimates about each data point are equal. This is not the case in Gompertz analyses, in which cohorts are employed. Sacher [7,16] developed relationships to estimate the sampling variance. Statistical weights (IV) used for In Q, values were the reciprocals of the estimates of the variance ( V) [16,45]: (14) The rationale for using our modified Gompertz model falls under the more general category of the treatment of concomitant variables, which is detailed by Johnson and Elandt-Johnson [45] and is outlined and compared to the Cox model [48] in Neafsey and Lowrie [33]. It is reproduced in Appendix A for completeness. The linear Gompertz function was fitted to the data for the control groups. The Gompertz function slopes and intercepts of the three control groups were not statistically different (P = 0.05). Therefore, the control data were pooled and the pooled control Gompertz function was used in the remaining analyses. Inspection of plots of In Q, (estimated Gompertzians) versus time revealed that after exposure ended, the estimated Gompertz function of the exposed groups attained a steady-state parallel upward displacement from the Gompertz function of the pooled controls. The linear Gompertz function (G., = G,, + CIX)was fitted to the data for the control and exposed groups for the time period after the termination of exposure. The slopes of the Gompertz functions of the exposed groups were not significantly different (P = 0.05) from controls using the two-tailed t test for estimated parameters [49]. The linear Gompertz function was then refitted to the data for the exposed groups (post-exposure) with the constraint that the slope parameters, CI,have the same value as those of the pooled control group. The ‘steady-state’ time was attained a short time after exposure ceased. The degree of upward parallel displacement at steady state (E,,) was calculated as the difference between the estimated intercept (In Q,) of the fitted model for the pooled controls and the In R, value for each exposure group.
74
P.J. Neafsey et al. / Mechanisms of Age@
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The non-linear least-squares computer program PCNONLZN [50,51] was employed for curve fitting using initial estimates of parameters which were obtained graphically. The Nelder-Mead [52] algorithm was used to search the parameter space. Other least-squares optimization software such as SAS [53] and BMDP [54] have been used, and yield parameter estimates which are essentially the same as those found by the Nelder-Mead algorithm. The choice of PCNONLZN was based on convenience since the PC-based data files can be accessed directly. The NelderMead algorithm has provisions for preventing the optimization method from getting ‘stuck’ some distance from the global (local) minimum. The algorithm is very tolerant of poor initial estimates of the parameters at the expense of the relatively large number of iterations necessary for convergence. The Gauss-Newton method used in SAS (and also available in PCNONLZN) may diverge if the initial estimates are poor or the bounds on the parameters are too large. The Dunnett test for multiple comparisons [55] was used to compare estimated cSSparameters among exposure groups. Goodness-of-fit was verified in three ways. First, visual inspection of plots of weighted residuals versus x indicated relatively good randomness of scatter of data points about fitted curves [56]. Second, visual inspection of data points (In R, - x pairs) about the regression lines indicated good randomness-of-scatter [49]. Third, the computed x2 values were less than tabulated values (2 = 0.05) [ 1,111. 3. Results Figs. 2-4 illustrate the fitted Gompertz model for the series 6, 7, and 8 animals, respectively. Fig. 5 illustrates the steady-state parallel upward displacement (E,,) of the Gompertz functions with associated standard errors for the series 6 animals (1000 WL in a 90-h week). The parallel upward displacement of the Gompertz functions (E,,) rises sharply at 320 and 640 WLM total exposure, and is statistically significant compared to pooled controls (P < 0.05). At 1280 WLM, the displacement falls below the E,, at 320 WLM and from 1280 to 10 240 WLM, the displacement increases with total exposure in a linear fashion. Fig. 6 illustrates the steady-state parallel upward displacement of the Gompertz functions with associated standard errors for both the series 7 and 8 animals (100 WL in a 90-h week). There is no significant displacement from the control Gompertz function from 20 to 1280 WLM total exposure. The Gompertz functions for 2560 and 5120 WLM are significantly displaced above that of the pooled controls (P < 0.05). Fig. 7 simultaneously displays the cSSfor both the 1000 WL and the 100 WL exposed animals. At 1000 WL, total exposures below 1280 WLM resulted in greater cumulative injury resulting in mortality than did exposure at 100 WL.
6 1OOOV.t k320 WLMI
-6
-4
-3
;;
SERIES
Fig. 2. Estimated Gompertzians for series 6 animals exposed at 1000 WL (90 h, week) for total doses of 320 (1 ). 640 (*), and IO 740 (+) WLM. ( + ) are pooled controls. See Table 1 for exposure times in days. Lines are the best fit of the data
SERIES
,000
W
(1280 WMI
1280 (A), 2560 (I). to Eq. 11.
5
5120
( 2). 21
76
P.J. NeqFey et al. 1 Mechanisms of Ageing and Deoelopmenr 83 (1995) 65-85
Fig. 4. Estimated (m) WLM. ( +
1320 WLM,
SERIES
for series 8 animals exposed at 100 WL (90 h/week) for total doses of 20 (,_‘), 40 ( l ). 80 (*), See Table 1 for exposure times in days. Lines are the best fit of the data to Eq. I I.
) are pooled controls.
Gompertzians
SERIES .3 loow.
WU.4)
160 CA ). 320 CO), and 640
S lOOwL (640
P.J. Neqfiey et al. , Mechanisms oj’ Ageing und Developmerli 83 (1995) 65-85
SERIES 6 (1 OOOWL) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
in thousands DOSE M’LM) Estimated
degree of upward
parallel
displacement
of Gompertz
functions
(c,,) at steady state for
arnmats exposed at 1000 WL (90 h/week) for total doses from 320 WLM (,2.24 days) to 10 240 WLM (71.68 days). Points arc observed 8SSvalues and standard errors calculated from individual weighted least-squares analysis of Gompertzians from pooled control and exposed groups. The standard error of the Gompertz intercept (In Q,) is shown for pooled controls.
4. Discussion
The parallel upward displacements of the Gompertz functions at steady state for the series 6 animals exposed at 1000 WL (Figs. 2 and 5) illustrate the imperative in dose-response studies for carefully selected exposures in the low dose range. Had the 320 and 640 WLM total exposure studies not been conducted, the striking linearity of the upward displacement of the Gompertz functions for the 1280; 2560; 5120; and 10 240 WLM groups might have led to a serious underestimation of the response below 1280 WLM at 1000 WL and a conclusion that at 1000 WL total exposure-mortality response (upward parallel displacement of the Gompertz loghazard functions) is linear. A simultaneous fit of Eq. 11 for pooled controls and the 1280, 2560, 5120, and 10 240 WLM groups produced estimated parameters and standard errors of: In Q,= - 10.34 & 0.19; a=6.85 x lop3 & 3.14 x 10-j days ‘; and ~~,oOOw,_ = 2.44 x lop2 f 3.08 x lo- 3 days I. The model standard error was 0.37 (26 d.f.) and the model weighted sums of squares (WSS) was 3.64. The x2 value for the model was calculated by summing the calculated x2 value for each exposure group and the pooled control group and comparing the summed value to the tabulated value for (k - Y - 1) degrees for freedom at the x = 0.05 level of significance, where k is the sum of the number of cells for all doses and Y is the total number of model parameters (i.e. r = 3 for Eq. 11). For a discussion of
P.J. NrufSey
et ul. I( Meclranims
of’ Aping
und
Drorlopnwnt
83 (1995)
19
65-85
the use of the x z goodness-of-fit test as it is applied to mortality models see Lee [ 1l] and Boxenbaum et al. [I]. The computed x2 value was 68.88 (65 d.f.). The tabulated x2 value at a = 0.05 is 84.80 for 65 d.f. The relatively low coefficients of variation, small mode1 standard error, low WSS and x’ value indicated consistency with a linear mode1 for E,, with WLM when the lowest total exposure groups were omitted. When all of the exposure groups at 1000 WL and pooled controls were fitted simultaneously to Eq. 11, the model standard error increased to 0.44 (40 d.f.) and the WSS increased to 8.82. The standardized residuals for the 320 and 640 WLM groups were large and all positive. The calculated x2 value was 173.48 (109 d.f.) compared to the tabulated value of 133 at x = 0.05 for 109 d.f. The hypothesis that the displacement of the Gompertz functions at steady state with total exposure is linear must be rejected when all exposed groups at 1000 WL are included in the model. The life-shortening effects of exposure to 640 WLM at 1000 WL compared to 1280 WLM, are not easily explained. The rats exposed for 4.48 days at 1000 WL (640 WLM) had significantly more injury resulting in death than the rats exposed for 8.96 days (1280 WLM). Whether this is (1) a statistical anomaly; (2) due to some type of induced repair process; or (3) due to killing of some cells that incur fatal injury, is unknown. These mortality data do suggest that the radon progeny exposure-mortality response at 1000 WL is not only non-linear but not even a monotonic function. Future studies using larger sample sizes and including a total SERIES
7,8 ( lOOWL)
1.25
P g & u P (I>
1.oo
0.75
0.50
B 5
0.25
1
0.00
-0.25 0
1000
2000
3000
4000
5000
6000
DOSE (WLM) Fig. 6. Estimated degree of upward parallel displacement of Gompertz functions (c,,) at steady state for animals exposed at 100 WL (90 h/week) for total doses from 20 WLM (1.4 days) to 5120 WLM (358.4 days). Points are observed c,, values and standard errors calculated from individual weighted leastsquares analysis of Gompertzians from pooled control and exposed groups. The standard error of the Gompertz intercept (In iW,,) is shown for pooled controls.
80
P.J. Neafsey et al. / Mechanisms of Ageing and Development 83 (1995) 65-85
2.00 1.75 E e f? E v)
k
1.50 1.25 1.00
z
0.75
9 3
0.50
&
0.25
0.0
1.5
3.0
4.5
6.0 7.5 in thousands DOSE (WLM)
9.0
10.5
12.0
Fig. 7. Estimated degree of upward parallel displacement of Gompertz functions (a,) for exposures at 1000 WL (0) and 100 WL (m).
exposure group at 960 WLM (6.72 days) are necessary to establish the shape of the exposure-response relationship at 1000 WL in the O-1500 WLM region. The total WLM exposure levels analyzed here were chosen to mimic former and current conditions in uranium mines. The level of exposures was relatively high and moderately short term (Table 1). We caution against using these data to extrapolate risk for the very low, highly protracted exposures of residential radon progeny. Lifetime radon exposures in a typical house produce cumulative radon progeny exposures of about 20 WLM. At the EPA 4-pCi/l radon action level, they are about 80 WLM. In the data analyzed here, there appeared to be no excess mortality in rats subjected to acute exposure to radon progeny for cumulative exposures typical in residences (Fig. 6), although excess rat lung cancers were produced. In this study, protracting exposure decreased the life shortening effect of radon progeny below total exposures of 1,280 WLM. Moolgavkar et al. [44] reported a similar finding for total exposures of 320- 10 000 WLM at 100 and 1000 WL. In one recent French experiment, adult rats given 25 WLM at high exposure rates showed excess lung cancers but no excess was found when the 25 WLM was protracted over the remaining rat lifespan [57,58]. While it is tempting to speculate that the parallel upward displacements of the Gompertz functions at steady state for the series 7 and 8 animals exposed at 100 WL (Figs. 3, 4 and 6) suggest a threshold-type exposure-mortality response, several caveats are offered. First, the assumption that protraction of very low cumulative exposure (such as that received in residences) will also decrease the life-shortening effect of radon progeny may not hold. The life-shortening effect of neutron
P.J. Neafsey
r? itl.
: Mechanisms
o/ Ageinp and Development
83 (1995) 65-85
81
exposure has been reported to be enhanced by protraction of exposure for relatively high total exposures of 1.6-2.4 Gy [59-611, although no augmentation of the life-shortening effect was seen with protraction of lower total exposures (0.1-0.5 Gy). Second, the majority of radon-progeny induced rat lung tumors are considered to be non-fatal and, unlike human lung cancers, are not the cause of early death. There are a plethora of concomitant variables which might affect human mortality response to radon progeny [62]. Among these variables are: frequency and duration of cigarette smoking, passive exposure to cigarette smoke, diet (e.g. consumption of antioxidants such as beta carotene, vitamin E and ascorbic acid), exposure to dust (e.g. arsenic), genetic resistance to cancer, ventilation in buildings, oral and nasal breathing patterns, age at exposure, and concentration and activitysize distributions of radon progeny. Moolgavkar et al. 1441contend that exposureresponse with respect to radon progeny, will depend on the particular exposure regimen (rate of exposure, total exposure, and timing of exposure) and particular response (e.g. probability of tumor, time to tumor, MAS, or age-specific mortality rate) chosen. The data reported here reinforce this contention. Mortality kinetics analysis must therefore be considered an adjunct to radon-progeny exposure-response analysis. It is not a substitute to tumor based models of risk. Acknowledgements
This work was supported in part by the U.S. Department of Energy under Contract DE-AC06-76RL01830 and by the University of Connecticut Research Foundation. Appendices Appendix
A:
Conwomitant
uariubles
A concomitant variable can be continuous (e.g. age, dose, etc.) or discrete (e.g. gender), and is part of the data. Parameters are determined from the data. Let y’ = (~3,.J~~....._v,,) be a row vector of concomitant variables (y is a column vector) and the hazard rate be i(t;y) 2 0. The hazard rate may depend on unknown parameters, so we write E.(t: JJ) = i,(t; _v; B),
(15)
where B’ = (II,, B, ,.... B,,) is a row vector of parameters. Theoretically, a different model could be used for each treatment group in an experiment. If the models are unrelated, they are difficult to fit and explain. A typical paradigm for an experiment uses one model with several parameters and/or concomitant variables. The choice of parameters allows a different sub-model for the control group and each treatment group. Suppose there are y groups (e.g. treatments). Then the hazard rate for the jth group is
82
P.J. Neafsey
ij(t; V) =
ij(t;
_V;
et al. / Mechanisms
of Ageing and Development
83 (1995) 65-85
(16)
Bj),
where j= 1, 2,..., q. Let Y’~= (JJ,~,yzir..., vPi) be the row vector of observed values of the concomitant variables for the ith individual. Once observed from the data, these values are considered fixed within the experiment at hand. There are two commonly used special cases of the general model: the additive model and the multiplicative model. The additiue model is assumed to be
where R is called the underlying hazard rate. It is assumed that the functions h,(t) are functions of time only, and the functions q,,(_v,,)= q,,(yP, B,>)p = 1, 2,...,s do not depend on t. Johnson and Elandt-Johnson [45] present, among others, a model using the Gompertz hazard rate as the underlying hazard rate. The general wzultiplicatiuemodel is assumed to be /i(t; y,)
= n(t).q(?,,;
B)
(18)
or In /Z(t; y,) = In R(t) + In
q(y,;
B),
(19)
where q has a known form and A is unspecified or of a known form (e.g. Gompertz) with parameters. This model is called the proportional hazard rate model. It is commonly assumed that the function ~1is not a function of t. If we assume that qCyi; B)
=
ecB’Y)
(20)
or
In q&i; B) = WY)
(21)
where B’y is the inner product of the vectors y and B, we have the Cox model [48]. Our models (Eqs. 4, 5, 10, 11) are similar to a Cox model with f = .x, In A(t) = Go + at
(22)
To have a Cox model in Eq. 6, Q, would have to be a linear function which does not fit the data as well as does Eq. 7. More complicated versions of Eq. 5 may be used to characterize other mortality effects of exposure to toxic agents, e.g. early paradoxical effects (longevity hormesis) [29,30,33] or food restriction [31] where (I is a function of time (age) as well as the concomitant variables.
P.J. Neajiey
et al.
;
Mechanisms
of Ageing und De~elopmenr
83
83 (1995) 65-85
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