ISSN 20790570, Advances in Gerontology, 2012, Vol. 2, No. 3, pp. 196–202. © Pleiades Publishing, Ltd., 2012. Original Russian Text © S.V. Mylnikov, 2011, published in Uspekhi Gerontologii, 2011, Vol. 24, No. 4, pp. 563–569.
On Estimation of Parameters of Survival Curves and Classification of Geroprotectors S. V. Mylnikov St. Petersburg State University, St. Petersburg, Russia email:
[email protected] Received July 1, 2011
Abstract—To characterize survival curves, we suggest calculating the following three parameters: median life span, the slope of the survival curve, and expected maximum life span. Using these parameters, geroprotec tors can be classified as weak, moderate, and strong. Keywords: survival curve, geroprotector DOI: 10.1134/S2079057012030101
When analyzing hte mortality rate among groups of experimental animals, most studies estimate the mean life span and then compare the results using Student’s ttest. While it has long been shown that the distribu tion of individual longevity is nonGaussian [1], this approach has become popular in the literature. On the other hand, the Gompertz law of mortality [3] postulates an exponential dependence of death rate on age. Initially, the Gompertz law was deduced from analyzing tables of human mortality; it has also been proven to be applicable to experimental animals. Later, the Gompertz equation was supplemented by the Makeham term [5], which characterizes ageinde pendent ambient mortality. Also, it was considered obvious that, in a laboratory environment, the Make ham component can be neglected. However, in the lit erature, there are fairly frequent statements to the effect that the Gompertz law is only satisfied for part of the mortality curve; objective choice criteria are still not defined for that age interval. Moreover, it has been demonstrated in direct experiments that, among groups of large drosophila and nematodes, exponen tial growth appears to stop at some speciesspecific age [7]. The bibliography of discussions about which law (for example, Gompertz or Weibull) better describes mortality tables contains dozens of titles [1, 4, 6, 8]. At the same time, applied statistical software packages include analyses of survival curves based on Kaplan– Meier statistics, which use the logrank test for com parison. This method was developed for cases when a researcher does not have the time to wait for the death of the last of the longliving organisms. Moreover, it is extremely timeconsuming for groups that include more than 100 subjects. The analysis of the survival curve sets we obtained allowed us to propose a differ ent method for parameter determination and a new classification for geroprotectors.
MATERIALS AND METHODS The subjects of this study were Drosophila melano gaster lines of different origins, more specifically: wild type line Canton S and Lerik; lines selected for differ ences in reproductive function (HEM, LA–, LA+, HA–); lines with altered metabolism of second messengers (66, 155, 398); isogenic composite lines; and hybrid lines derived from the above. Mortality dynamic was charac terized using nonlinear regression analysis. Regression parameters were compared using the Ftest. Synthetic peptides were tested as geroprotectors by acting on the larvae of second and third instar. They were added into the culture medium, 0.04 mg per 1 g of body mass. Thus, the exposure time did not exceed two days. In control groups, physiological solution was added to the medium. RESULTS AND DISCUSSION Intensity of mortality was plotted against age, with the corresponding regression, for Canton S, 66, 155, 398, and their hybrid lines. Intensity of mortality in these cases can be satisfactorily described by Gomp ertz law. This can be confirmed by Table 1 data, which shows coefficients of determination in the regression model for different versions of the experiment. Since all of them exceed 70%, correlation between theoreti cal model and experimental data can be considered satisfactory. However, the comparison of regression models using the Ftest demonstrated that in some cases, the Makeham model is preferable. These conclusions are also true for series of breedings of HEM × Lerik, isogenic composite lines, and hybrid lines derived from Canton S and 66 lines (Table 1). Thus, the Gompertz–Makeham model better describes the observed correlation of the intensity of
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Table 1. Results of a comparative analysis of performance of Gompertz and Makeham models in mortality description of dif ferent drosophila lines Coefficients of determination, mean (minimum and maximum)
Line
The number of cases where the model is preferred, p < 0.01
Gompertz model
Makeham model
Gompertz
Makeham
0.90 0.72–0.97 0.89 0.69–0.99 0.87 0.57–0.99
0.92 0.75–0.99 0.90 0.71–0.99 0.88 0.57–0.99
17
3
6
2
12
4
Canton S, 66, 155, 398 and their hybrids Lerik, HEM and their hybrids Isogenic composite lines derived from Canton S and 66 lines
Table 2. Results of mortality modeling using Gompertz and Gompertz–Makeham models for isogenic composite lines derived from Lerik and HEM Coefficients of determination
Mean (minimum and maximum)
full details
truncated data
Gender
Females Males
Gompertz model
Makeham model
Gompertz model
Makeham model
0.40 0.08–0.70 0.43 0.08–0.95
0.43 0.08–0.71 0.43 0.08–0.95
0.84 0.40–0.99 0.91 0.71–0.99
0.86 0.53–0.99 0.93 0.71–0.99
stops growing exponentially. Regression analysis that takes into account all the experimental points (full details) gives low coefficients of determination (Table 2). Excluding points for older age resulted in good approximation by the Gompertz equation and magni fication of the coefficients of determination; however, such data truncation would be inevitably subjective. Thus, Gompertz model does not always adequately describe the mortality curves. This is most noticeable for a set of isogenic lines derived from lines HEM and
mortality and age for 21% out of 44 analyzed mortality curves. The 95% confidence interval for this frequency would be 10–35%. It can be stated that in at least 10% of laboratory experiments, the insects’ mortality curve would be described by the GompertzMakeham equa tion better than by the Gompertz equation. For example, Fig. 1 represents mortality graphs for females from one of eight isogenic composite lines derived from the HEM and Lerik lines. As you can see, the plot of intensity of mortality vs. age rather quickly Intensity of mortality 0.12
Full details
0.10 0.08 0.06 0.04 0.02 0 0
10
20
30
40
50
Intensity of mortality 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 60 70 10 0 5 Days
Truncated data
15
20
25
E Fig.1. Gompertz mortality plots for females from isogenic composite lines derived from Lerik and HEM. ADVANCES IN GERONTOLOGY
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30 Days
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% 100
5 6
7 8
50
50
0
0 25
0
50
75 Days
0
25
75 Days
50
Fig. 2. Results of mortality approximation using doseeffect model for males from isogenic composite lines derived from Canton S and 66 lines.
Lerik, in which we observed deviations from the pos tulated exponential growth of the intensity of mortal ity even in older groups of smaller size. The search for a different model led to the results presented below. In biology, the equation of a curve that starts at 100% and gradually declines to zero (it looks exactly like survival curve) was explicitly stated a long time ago. For our purposes, we modified the wellknown doseeffect relationship as follows: 100 Y = , [ ( MT 50 – X ) × HS ] 1 + 10 where MT50 is the mortality time of half of the group and HS (hill slope) is the survival curve slope (SCS). One can see that the parameter MT50 is a close analog of the mean life span; it is calculated by the least squares method. Thereafter, it is a standardized regres sion coefficient, and these coefficients can be com pared using the Ftest. The HS parameter statistically estimates the curve slope and, thus, indirectly assesses the maximum life span. If the function is considered to be satisfactory to describe the survival curves of the experimental ani mals, then we have the correct method for comparing the median life span (MeLS). It is especially important because the comparison of mean and maximum life spans of the control and experimental groups is essen
tial for characterizing the properties of potential gero protectors. Since the curve asymptotically approaches zero as the group’s age increases, we also suggest estimating the point on the X axis that corresponds to the value of a function equaling 1% of organisms left alive. We will call this age the “maximum life expectancy” (MLE). Point and interval estimates for that age can be obtained by solving the equation for X as follows: X = A – 1 log ⎛ 100 – 1⎞ . ⎠ B ⎝ Y Since the errors of the regression coefficients are known, calculating the error of X is not difficult. The method is included in most statistical software pack ages. Note also that the logarithm on the righthand side is approximately equal to 2 when Y = 1%, and approximately equal to 3 for Y = 0.1%. Some of our experimental results approximated by the doseeffect model are presented graphically in Fig. 2. The formulas above are easily obtained by rear ranging a standard logistic regression equation. There fore, this model can also be called a logistic model of mortality. The coefficients of determination for different models for the original lines, their hybrids, and isogenic compos ite lines are presented in Tables 3 and 4. The coefficients
Table 3. Results of comparative analysis of performance of Gompertz, Makeham, and doseeffect models in mortality description of different drosophila lines Coefficients of determination, mean (minimum and maximum) N
Line Canton S, 66, 155, 398, and their hybrids; Leric, HEM, and their hybrids Isogenic composite lines derived from Canton S and 66 lines
Gompertz model
Makeham model
Doseeffect model
0.90 0.69–0.99 0.87 0.57–0.99
0.90 0.71–0.99 0.88 0.57–0.99
0.99 0.96–0.99 0.98 0.97–0.99
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Table 4. Results of mortality modeling using Gompertz, Gompertz–Makeham and doseeffect models for isogenic com posite lines derived from Lerik and HEM Coefficients of determination Gender
Mean (minimum and maximum)
Full details
Truncated data
Full details;
N
Gompertz model Makeham model Gompertz model Makeham model Doseeffect model Females
0.40 0.08–0.70 0.43 0.08–0.95
Males
0.43 0.08–0.71 0.43 0.08–0.95
0.84 0.40–0.99 0.91 0.71–0.99
of determination for the doseeffect model are close to one, which suggests that the model adequately describes the process of the extinction of groups. A comparison of the cumulative relative frequen cies by the Kolmogorov–Smirnov criteria (Fig. 3) showed that the coefficients of determination from the Cumulative relative frequency (a) 1.0
8 8
In the Drosophila melanogaster survival curves we studied, MeLS ranged from 22 to 71 days and SCS Cumulative relative frequency (b) 1.0 λ = 1.5 p < 0.01
0.5
0.5
0.8 0.9 1.0 Coefficients of determination, R2 Cumulative relative frequency (c) 1.0 0.7
0 0.6
0.8 0.9 1.0 Coefficients of determination, R2 Cumulative relative frequency (d) 1.0
λ = 1.76 p < 0.01
0.7
λ = 2.7 p < 0.001
0.5
0 0.6
0.97 0.96–0.98 0.99 3.98–0.99
doseeffect model are statistically significantly higher than the one from the Gompertz model. Thus, the doseeffect model is preferred in the analysis of mor tality curves.
λ = 0.16 p < 0.01
0 0.6
0.86 0.53–0.99 0.93 0.71–0.99
0.5
0.7
0.8 0.9 1.0 Coefficients of determination, R2
0 0.6
0.7
0.8 0.9 1.0 Coefficients of determination, R2
Gompertz model Doseeffect model Fig. 3. Cumulative relative frequency graphs of coefficients of determination for Gompertz and doseeffect models: (a) set of Can ton S, 66, 155, and 398 lines and their hybrids; (b) HEM, Lerik, and their hybrids; (c) isogenic lines 1–8; (d) isogenic lines A–H. ADVANCES IN GERONTOLOGY
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Table 5. Correlation coefficients of mortality parameters in different models Parameter
MeLS
SCS
R0
G
MeLS
1
0.0577
–0.389
0.193
SCS R0 G
– – –
1
–0.251 1 –
0.163 –0.657 1
– –
Note: Statistically significant correlation coefficients (p < 0.01) are in bold.
ranged from 73 to 133 days. Regression analysis showed no significant association (p > 0.05) between parameters MeLS and SCS in the doseeffect model. An analysis of correlations between parameters that characterize the extinction of drosophila groups (Table 5) revealed a weak MeLS–R0 correlation, in addition to the G–R0 correlation, which follows from the Strehler–Mildvan correlation. Thus, we can assume that two different models of mortality characterize the process of the extinction of groups from different points of view, as well as that these models are related through the parameter of the original rate of mortality. Thus, an analysis of the survival curve using the doseeffect model allows us to characterize the process of group extinction by two independent parameters, which in turn increases the power of analysis. We used the proposed method of analysis to assess the effect of synthetic peptide drugs on the survival of Drosophila melanogaster. The following results were obtained in the study of the drug’s effect on the sur vivalcurve parameters in the doseeffect model.
All studied synthetic peptides increased MeLS from 40 to 55% in females of the HA– line (Table 6). A reduction in SCS by 20% was only observed when Prostamax and Epithalamin were included to the diet. The inclusion of these drugs into the diet increased MeLS by 17%. The MeLS increase in males of the HA– line from 37 to 48% was observed under the influ ence of Vilon, Livagen, Prostamax, and Epitalon. The SCS for males of this line decreases under the influence of Cortagen and Prostamax and increases under the influence of Livagen. MLE increases under the influence Cortagen and Prostamax. The increase in MeLS in LA– line females was observed when they were subjected to Vilon, Livagen, Prostamax, Epitalon, and Epithalamin (Table 7). Three drugs, Vilon, Prostamax, and Epithalamin, increase SCS. Cortagen resulted in a decrease in MLE. In males of this line, the inthe crease in MeLS was observed under influence of Cortagen, Prostamax, and Epithalamin. The increase in SCS was observed under the influence of Cortagen, Livagen, Prostamax, and Epithalamin. Vilon resulted in a decrease in MLE. In LA+ line females, the exposure to peptide drugs resulted in a decrease in MeLS in five out of six exper imental versions (Table 8) and an increase in SCS in four out of six versions. A decrease in MLE was observed in four cases. In males from this line, MeLS increases when sub jected to Vilon and Cortagen. The decrease in SCS was only observed under the influence of Vilon. The increase in MLE was observed under the influence of Vilon and Livagen. In Table 9, we have divided the study drugs into one of three types of geroprotectors according to N.M. Emanuel’s classification [2]. According to this
Table 6. Parameters of survival curves of HA– lines in different variants of experiment Females Variant of exposure Control group Vilon Cortagen Livagen Prostamax Epitalon Epithalamin
MeLS, days 26.3 ± 0.63 39.0 ± 0.781*) 38.6 ± 0.691*) 36.8 ± 0.621*) 40.8 ± 0.721*) 38.5 ± 0.481*) 39.1 ± 0.511*)
Males
X/Y SCS MLE, days couple (days–1 × 102) –0.044 ± 0.0026 –0.037 ± 0.0021 –0.037 ± 0.0019 –0.040 ± 0.0020 –0.035 ± 0.00172*) –0.039 ± 0.0015 –0.035 ± 0.00132*)
94.7 ± 2.43 99.7 ± 2.77 104.9 ± 3.23 96.6 ± 2.66 111.3 ± 3.091*)
26
100.8 ± 2.30 110.0 ± 2.841*)
24
24 22 25 24
26
MeLS, days 20.8 ± 0.75 30.9 ± 0.641*) 20.4 ± 1.13 23.6 ± 0.351*) 28.4 ± 0.621*) 30.8 ± 0.701*) 22.9 ± 0.59
SCS (days–1 × 102)
MLE, days
–0.041 ± 0.0029 –0.039 ± 0.0020 –0.027 ± 0.002011*)
90.9 ± 2.60 92.5 ± 2.74 112.9 ± 5.4311*)
–0.063 ± 0.00301*) –0.033 ± 0.00152*)
79.80 ± 2.85 105.1 ± 2.6822*)
–0.040 ± 0.0023 –0.041 ± 0.0022
96.8 ± 2.92 90.7 ± 2.89
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Table 7. Parameters of survival curves of LA– lines in different versions of experiment Females Variant of exposure Control group
MeLS, days
X/Y SCS SCS MLE, days MeLS, days couple (days–1 × 102) (days–1 × 102)
19.3 ± 0.46 21.1 ± 0.452*)
–0.054 ± 0.0029
19.7 ± 0.44 24.9 ± 0.711*)
–0.070 ± 0.00452*)
Prostamax
26.8 ± 0.661*)
–0.086 ± 0.00971*)
Epitalon
23.2 ± 0.721*)
–0.069 ± 0.0071
Epithalamin
27.2 ± 0.451*)
–0.070 ± 0.00451*)
Vilon Cortagen Livagen
Males
–0.065 ± 0.0040
–0.066 ± 0.0063
68.8 ± 2.19 56.5 ± 1.57 51.7 ± 2.812*) 59.1 ± 1.96 65.0 ± 2.65 56.9 ± 2.31 67.7 ± 2.08
21 19 15 21 21 19 21
20.5 ± 1.68 23.3 + 1.26 28.0 ± 0.451*) 25.0 ± 0.63 27.2 ± 0.612*) 25.9 ± 1.05 27.2 ± 0.491*)
–0.044 ± 0.0089 –0.071 ± 0.0135 –0.187 ± 0.03031*) –0.110 ± 0.01571*) –0.166 ± 0.03361*) –0.097 ± 0.0207 –0.194 ± 0 .03651*)
MLE, days 63.1 ± 11.9 44.6 ± 4.301*) 51.7 ± 2.83 50.4 ± 2.88 56.1 ± 5.68 52.5 ± 5.39 50.8 ± 4.57
X/Y couple 13 15 19 15 15 15 15
Table 8. Parameters of survival curves of LA+ lines in different versions of experiment Females Variant of exposure Control group
MeLS, days
X/Y SCS MLE, days couple (days–1 × 102)
28.2 ± 0.68 28.6 ± 0.71 26.0 ± 0.402*)
–0.038 ± 0.0022
Livagen
25.8 ± 0.502*)
–0.064 ± 0.00401*)
Prostamax
21.1 ± 0321*) 21.6 ± 0.591*)
–0.081 ± 0.00421*) –0.058 ± 0.0041
16.9 ± 0361*)
–0.076 ± 0.00451*)
Vilon Cortagen
Epitalon Epithalamin
Males
–0.044 ± 0.0027
–0.073 ± 0.00421*)
103.0 ± 3.94 102.1 ± 3.99 80.7 ± 2.101*) 88.1 ± 4.60 65.0 ± 2.731*) 78.4 ± 2.292*) 76.9 ± 2.941*)
24
MeLS, days
SCS MLE, days (days–1 × 102)
17.7 + 1.12 25.0 ± 1.321*)
–0.042 ± 0.0048
22
24.2 ± 1.331*)
–0.032 ± 0.0031
21
21.7 ± 1.62 19.3 ± 0.96 17.2 ± 0.92 17.1 ± 0.63
–0.028 ± 0.0032
24
18 21 22
–0.025 ± 0.00212*)
–0.042 ± 0.0039 –0.048 ± 0.0049 –0.049 ± 0.0035
108.1 ± 5.31 145.8 ± 7.501*) 115.1 ± 6.17 131.3 ± 8.482*) 100.8 ± 4.00 93.1 ± 5.42 81.9 ± 1.931*)
X/Y couple 27 26 26 25 26 24 21
Note: In Tables 6–8, 1*) p < 0.001; 2*) p < 0.01.
classification, the code for the first type of geroprotec tor should have the form +/0/+ (increase in the mean and maximum life span at unchanged SCS). This code was observed once when the LA+ line males were sub jected to Cortagen. The geroprotector code of the sec ond type should have the form +/–/+ (increase in mean and maximum life span and decrease in SCS). This code was observed four times. Geroprotectors of the second type were Prostamax, which acts on females and males of the HA– line; Epithalamin, ADVANCES IN GERONTOLOGY
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which acts on females of the HA– line; and Vilon, which acts on males of the LA+ line. The third type of geroprotector code should have the form +/+/0 (increase in mean life span at unchanged maximum life span and increase in SCS). The third type of gero protectors were Livagen, which acts on males of the HA– line; Cortagen, which acts on males of the LA– line; Prostamax, which acts on females and males of the LA– line; and Epithalamin, which acts on females and males of the LA– line.
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Table 9. Geroprotective action of studied drugs according to MeLS/SCS/MLE Lines HA–
Drug
Vilon Cortagen Livagen Prostamax Epitalon Epithalamin
LA–
LA+
Females
Males
Females
Males
Females
Males
+/0/0 +/0/0 +/0/0 +/–/+ +/0/0 +/–/+
+/0/0 +/0/0 +/+/0 +/–/+ +/0/0 0/0/0
+/0/0 0/+/– +/0/0 +/+/0 +/0/0 +/+/0
0/0/– +/+/0 0/+/0 +/+/0 0/0/0 +/+/0
0/0/0 –/+/– –/+/0 –/+/– –/0/– –/+/–
+/–/+ +/0/+ 0/0/+ 0/0/0 0/0/0 0/0/–
Note: (+) indicates a significant increase compared to control groups, (–) indicates a significant decrease compared to control groups, and (0) indicates no effect.
CONCLUSIONS A number of cases remain outside of this classifica tion, e.g., the +/0/0 code only increases the life expectancy. There are ten such cases in this table. We suggest calling these geroprotectors “weak.” We pro pose calling geroprotectors “moderate” if they increase the median life span and decrease the slope of the survival curve, but do not increase the expected maximum life span. We propose to call geroprotectors strong if they increase the median life span, decrease the slope of the survival curve, and increase the expected maximum life span. REFERENCES 1. Gavrilov, L.A. and Gavrilova, N.S., Biologiya prod olzhitel’nosti zhizni (Biology of the Life Duration), Moscow: Nauka, 1991. 2. Emanuel, N.M. and Obukhova, L.K., Types of Experi mental Delay in Aging Pattern, Exp. Geront., 1978, vol. 13, nos. 1–2, pp. 25–29.
3. Gompertz, B., On the Nature of the Function Expres sive of the Law of Human Mortality and on a New Model of Determining Life Contingencies, Philos. Trans. R. Soc. Lond. A, 1825, vol. 115, pp. 513–585. 4. Juckett, D.A. and Rosenberg, B., Comparison of the Gompertz and Weibul Functions as Descriptors for Human Mortality Distributions and Their Intersections, Mech. Aging Dev., 1993, vol. 69, nos. 1–2, pp. 1–31. 5. Makeham, W.M., On the Law of Mortality and the Construction of Annuity Tables, J. Inst. Actuaries, 1860, vol. 8, pp. 301–310. 6. Riggs, J.E., The Gompertz Function: Distinguishing Mathematical from Biological Limitations, Mech. Aging. Dev., 1993, vol. 69, nos. 1–2, pp. 33–36. 7. Vaupel, J.W., Johnson, T.E., and Lithgow, G.J., Rates of Mortality in Populations of Caenorhabditis elegans, Science, 1994, vol. 263, no. 5147, pp. 668–671. 8. Wilson, D.L., A Comparison of Methods for Estimat ing Mortality Parameters from Survival Data, Mech. Aging Dev., 1993, vol. 66, no. 3, pp. 269–281.
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