ABSTRACT - MEMBER | SOA · The purpose of this paper is to provide a survival analysis of persons in the various stages of HIV infection typically leading to AIDS and ultimately death.

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AIDS: SURVIVAL ANALYSIS OF PERSONS TESTING HIV +

HARRY H. PANJER

ABSTRACT

The purpose of this paper is to provide a survival analysis of persons in the various stages of HIV infection typically leading to AIDS and ultimately death. The model used is a continuous time Markov process with a constant intensity for each stage. It is shown that this model adequately describes the data which originated from a German longitudinal study. The data were . previously analyzed using less formal methods in the comprehensive paper of Cowell and Hoskins dealing with the effect of HIV infection on life insurance. This paper should be of special interest to health insurers since it deals with distribution of duration in each stage of progression of the disease.

INTRODUCTION

The progression of persons infected with the Human Immunodeficiency Virus (HIV) to Acquired Immune Deficiency Syndrome (AIDS) is being studied at the Centre for Internal Medicine of the University of Frankfurt [1]. The longitudinal study follows subjects in groups at high risk of AIDS through various stages from good health with an mv + status to death primarily caused by AIDS.

In the study 543 subjects were observed during the study period from 1982 through 1985; 377 were HIV + at the time they were initially observed; 307 were observed for at least three months of which 259 were HIV + upon initial observation.

The Walter Reed Staging Method [6] was used as a basis for classifying subjects:

la (At-Risk)

Ib (HIV +)

2a (LAS)

Healthy persons at risk for HIV infection, but testing negative;

Otherwise asymptomatic persons testing HIV + ; Persons with HIV infection and lymphadenopathy syndrome (LAS), together with moderate cellular immune deficiency;

2b (ARC) Patients with HIV infection and LAS, together with severe cellular immune deficiency (AIDS-Related Complex, or ARC);

3 (AIDS) Patients with AIDS.

The sixth stage was death.

This paper also appears in TSA 40, part I (1988): S 17-530.

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AIDS: SURVIVAL ANALYSIS

Table IA classifies the number of persons observed during the study by the length of time under observation and the initial stage; that is, the stage at the time the subject was first observed in the study. Table IB gives the corresponding number of persons whose condition worsened at least one stage during the study period. Table Ie gives the ratio of Table IA to Table 1B as a percentage. These tables are taken directly from the paper by Cowell and Hoskins [2, Part 2, p.19].

Range of Observation

Periods

Stage 1a

(At-Risk)

TABLE 1

Stage Ib

(HIV+)

Stage I 2a

(LAS)

Stage 2b

(ARC)

A) Number of Patients Observed bv Stage and ObservatIOn Period

3-6 months 6-12 months

12-24 months 24-36 months

All Periods

10 14 21 3

48

9 21 8 18 51 29 20 29 20

5 19 7

52 120 64

Stage 3

(AIDS)

6 9 7 I"

23

All Stages

54 121 97 35

307

B) Number of Pallents Observed Whose Health Worsened by at Least One Stagc, or Who Dlcd h 0 during t c bservation Period

3-6 months 1 1 3 0 4 9 6-12 months 6 10 20 3 6 45

12-24 months 9 15 14 10 5 53 24-36 months 2 4 14 4 O· 24

All Periods 18 30 51 17 15 131 C) Percentage of Pallents Obscrved Whose Health Worsened by at Least Onc Stage, or Who DIed d' hOb' P'd unng t c servatlon eno

3-6 months 10% 11% 14% 0% 67% 17% 6-12 months 43% 56% 39% 10% 67% 37%

12-24 months 43% 75% 48% 50% 71% 55% 24-36 months 67% 80% 74% 57% 0%" 69%

All Periods 38% 58% 42% 27% 65% 43%

"One patient with AIDS was still alive 28 months after diagnosis of Kaposi's sarcoma; all others with AIDS had died before the end of 24 months.

It should be noted that only information regarding the initial stage (that is, the stage at the time an individual entered the study) is available from these tables. An individual passing through several stages during the period of observation is indistinguishable from an otherwise identical person but moving only to the next stage.

The later sections of this paper deal with a formal analysis of these data for the purpose of making inference about the distribution of lifetime for persons in the various stages.

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THE MODEL

As in Cowell and Hoskins [2, Part 2, pp. 4,5], it is assumed that individuals progress through the successive states in order and do not return to a previous state. In medical terms, this means that a person's condition (according to the Walter Reed Staging Method) can remain the same or deteriorate but can never improve.·

We model progression from stage to stage by assuming that a person in a given stage of HIV infection is subject to an intensity function (force of progression, hazard rate) that depends only upon the stage and not upon other factors such as age, sex and the length of time in the stage.

Let IJ-J,j = la, 1b, 2a, 2b, 3, denote the intensity function of progression from stage j to the next stage. Let 1j denote the time in stage j. Then, the probability that a person just entered stage j will remain in stage j for at least t years is

Pr{1j > t} = exp {- t IJ-Jdt} = e-II'-j;

the cumulative distribution function (cdf) of 1j is

FTit) = Pr {Tj :5 t} = 1 - e-I~j;

and the probability density function (pdf) of 1j is

f1/t) = IJ-j e -II'-j.

This is the exponential distribution with mean l/IJ-j and variance l/IJ-J.

(1)

(2)

(3)

A consequence of this model is that the times 1j, j = la, lb, 2a, 2b' 3, are stochastically independent. Furthermore, the memoryless property of the exponential distribution means that the length of time that a person has been in the current stage is irrelevant for our purposes and that the expected time of progression to the next stage is the same for all persons in the stage; that is, it is independent of the time already in the current stage. is, it is independent of the time already in the current stage. can be easily computed. For example, the random variable denoting the time from progression to AIDS (stage 3) from a positive HIV test (stage lb) is

(4)

'Seven of 307 subjects observed for more than three months improved at least one stage. This is attributed to the possibility of misdiagnosis of one of the stages. an event that can be expected to occur since some judgment is involved [1. p. 1178}.

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with mean

and variance

due to independence of the 1j's. The exact distribution function of Tlb + T2a + T 2b is easily obtained

by integration. We first obtain the pdf for a pair of exponential random variables.

~ ITI(t) + ~I ITit) (7) fJ.2 fJ.I fJ.) fJ.2

where * indicates the convolution operator. Consequently,

IT) + T2 • TJ(t) = IF) * 11'2 * IFJ(t)

(8)

In general, it can be shown [4, p. 79] that the distribution of T) + T2 + ... + T" is

(9)

Similarly, it can be shown that the probability that an individual in any stage (arbitrarily labeled 1) will pass through stages 2,3, ... , n -1 and be in stage n exactly t years later is

i~ C~: fJ.j} L~i fJ.j ~ fJ.Je ~i'. (10)

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This allows for exact evaluation of probabilities to answer questions such as "What is the probability that an HIV + individual will develop AIDS within three years?" or "Of 1,000 HIV + persons, how many can we expect to have AIDS or have died within five years?"

In the next two sections the parameters of the model are estimated and the model is tested for validity against the observed data given in Table 1.

ESTIMATION OF MODEL PARAMETERS

Consider a single stage with constant intensity Jl.. Since the times in each stage are independent, we consider the stages separately and drop the subscript j for notational convenience.

Notation:

Jl. "force of progression" to next stage, di number of persons progressing for those in observation period i,

i = 1, 2, 3, 4, ni

Pi =

qi

number of persons observed in observation period i, i = 1, 2, 3, 4, probability of not progressing if in observation period i, i = 1, 2, 3, 4, 1 - Pi = probability of progressing at least one stage.

The likelihood function is

L(Jl.) = i~1 (~:) (1 - Pi) dip/,i· di.

The loglikelihood function is

(11)

e(Jl.) = i~ {lOg (~:) + di 10g(1 - pJ + (ni - di)IOgPi}. (12)

Note that only Pi is a function of Jl. and will be specified later. The maximum likelihood estimator (MLE) tl of the parameter Jl. is ob

tained as the maximum of L(Jl.) or equivalently of e(Jl.).

Differentiating the loglikelihood yields

U(Jl.) = ± di ( aPi ) + ni - di api aJl. i=t1-Pi aIL Pi aIL

= ± _ api {~ _ ni - di }.

i=1 aIL I-pi Pi (13)

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Solving

(14)

should yield the MLE ,1. We use a Newton-Raphson method to solve equation (14) for I-L. Let

(15)

(16)

Then

(17)

Beginning with an initial estimate of ,10 the successive estimates are obtained as

J.L" + I (18)

Now we make some assumptions regarding the exact time-on-study for the subjects. This is necessary since, for example, for a given individual observed for between 24 and 36 months, we require the exact time of observation. First, we assume that each subject was observed up to the midpoint of the interval, for example, 30 months for each subject in the 24-36 month interval. Then we repeat the exercise using a random censoring mechanism.

A. Assume exposure to the midpoint of the interval, then we obtain the probabilities

PI = f'" fJ£-1IJ dt = e-(318) II-

318

P2 = f'" p.e-IIJ dt = e-(3/4) II-

3/4

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Then

and


P3 = f'" jJ£-1Jl dt = e-(3/2) '" 3/2

P4 = f'" jJ£-1Jl dt = e·-(5/2) "'. 5/2

a2pi = t.· 2e- lJl;"

a p.2 '

where ti * is the midpoint of intelVal i.

(19)

(20)

(21)

B. Assume a uniform random censoring mechanism for each obselVation. Let t denote the exact time of obselVation. If the limits of the right-hand end of obselVation period i are ai and bi, we have ai<t<bi. Since we have no information about the relative likelihood of the possible censoring times t, we treat t as a random variable with a uniform (ai' bi )

distribution making all censoring times t equally likely. Then

fbi 1 f'" Pi = ai b

i _ a

i I jJ£-J.ld ds dt

fbi 1

= e- IU dt ai bi - ai

= ' e- IU dt 1 fb-

bi - ai ai

e-I'oai - e-I'bi qi = 1 - p.(b

i - a

i) ,

api aqi 1 - ~(a,e-I'oai - b,e-I'bi) - (e-I'oal - e-I'bi)

~2 -=-= ap' ap' bi - ai

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(22)

(23)


= {ae- I'<'i - be-/.lbi p} , , + ~ ,

f.L(bi - ai ) r-

1 iJ a,e- l4Ji - b,e-/.lbi

(b i - ai ) iJf.L f.L

= 1 -f.L(a/e- l4Ji - b/e-lI.hi)-(a,e 14Ji-bie lI.hi)

bi - ai f.L2

a/e-I'<'i - b/e-/.lbi _ ~ {iJiJ~:}. f.L(bi - a;) r- r-

The asymptotic variance of the MLE of f.L is obtained as 1

AsVar(p..) = ----

-E [::~J which is estimated by

AsVar(p..) = 1

evaluated at f.L = p...

(24)

(25)

(26)

(27)

The square root of the estimated asymptotic variance can be used as an estimate of the standard error of the estimate of f.L. It provides a measure of the reliability of the estimate p.. based on the observed data. Since under mild regularity conditions the MLE has an asymptotically normal distribution, approximate 95 percent confidence bounds can be calculated by adding and substracting 1.96 v' AsVar(p..) from the MLE p...

The asymptotic variance of any function g(p..) can be obtained as

AsVar (g(p..)) = AsVar (f.L){g'(p..)}2. (28)

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It is well known [5, p. 43] that in the case of an exponential distribution with data that are not censored or grouped, Jlll3 approaches normality more quickly (as the sampled size increases) than Jl. Because of this, we will base estimates of confidence bounds of various quantities on the confidence bounds of Jl1!3 using the assumption of normality of JlI!3. We obtain its asymptotic variance using (28) as

AsVar(JlI!3) = ASVar(Jl)~1-L -4!3. (29)

In the next section, confidence bounds on JlI!3 are transformed directly to obtain confidence bounds on related quantities. The reader interested in reviewing the properties of the maximum likelihood estimator should consult Cox and Hinkley [3, Ch. 9] or similar texts on statistics.

NUMERICAL RESULTS

Maximum likelihood estimates of the intensity function and the expected time to the next stage as well as upper and lower 95 percent confidence bounds were calculated using the methods described in the previous section. These calculations were carried out using both a midpoint departure assumption as well as a random censoring assumption. Table 2 indicates that the two assumptions produce virtually identical results. Consequently, we shall henceforth present results based on the midpoint method only.

TABLE 2

ESTIMATES AND 95% CONFIDENCE BOUNDS OF THE INTENSITY FUNCTION

AND THE EXPECTED TIME IN STAGE

A) Midpoint Method

Intensity Function 0.45 0.86 0.53 0.30 Lower Confidence Bound 0.27 0.57 0.40 0.18 Upper Confidence Bound 0.69 1.2 0.70 0.46 Expected Time in Stage 2.2 1.2 1.9 3.4 Lower Confidence Bound 1.4 0.81 1.4 2.2 Upper Confidence Bound 3.7 1.7 2.5 5.7

B) Random Censoring Method

Intensity Function 0.45 0.88 0.54 0.30 Lower Confidence Bound 0.27 0.58 0.40 0.18 Upper Confidence Bound 0.70 1.3 0.71 0.47 Expected Time in Stage 2.2 1.1 1.9 3.4 Lower Confidence Bound 1.4 0.79 1.4 2.1 Upper Confidence Bound 3.7 1.7 2.5 5.7

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Stage 3 (AIDS)

1.1 0.60 1.8 0.93 0.57 1.7

1.1 0.61 1.8 0.91 0.55 1.6


The expected time to the next stage is the average future length of time that any subject in a particular stage will wait before moving to the next stage. A consequence of the model is that this quantity does not depend upon the length of time a subject has been in the stage already; or stated equivalently, no aging of the subject occurs in any stage.

Table 3 gives the estimated cumulative distribution function and corresponding confidence limits for the time in any given stage. These numbers represent the proportion of persons who can be expected to have progressed to the next stage after 0.5, 1, 2, or 3 years in the stage. The estimates obtained by Cowell and Hoskins [2, Part 2, p. 25] are also given for comparative purposes. It should be noted that 11 out of 16 of their values fall within our confidence limits. Since we would expect about 15 out of 16 (that is, 95 percent) it would appear that our results are somewhat inconsistent with theirs. Furthermore, it should be noted that our estimates are all higher than those of Cowen and Hoskins.

TABLE 3

PROPORTION PROGRESSING TO NEXT STAGE IN SPECIFIED TIME

Sligo Time Proponiol'\ Lower Bound Upper Bound OJ'Wdl and Hoskins

1a 0.5 20% 13% 29% -(AI-Risk) 1 36 24 50 -

2 59 42 75 -3 74 55 87 -

Ib 0.5 35 25 46 10'7< (HIV+) 1 58 44 71 55

2 82 68 92 75 3 93 82 98 80

28 0.5 23 18 30 15 (LAS) 1 44 33 50 41

2 66 55 75 61 3 80 70 88 75

2b 0.5 14 8 21 5 (ARC) 1 26 16 37 10

2 45 30 60 51 3 59 41 75 58

3 0.5 42 26 58 26" (AIDS) I 66 45 83 45'

2 88 70 97 70' 3 96 84 99 80'

'Cowell and HoskinS used data from the U.S . Center for Disease Control In lieu of the Frankfurt Study data to obtain the stage 3 values [2, Part 2, pp. 3 and II].

Cowen and Hoskins [2, Part 2, p. 12] assumed the maximum length of observation periods throughout in order to offset the length of time from the

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l


onset of the stage to entry into the study. This will have the effect of decreasing the intensity function and the proportion progressing to the next stage. This should explain the downward bias in their results from ours. It should be noted that our model is memoryless and requires no assumption about the time-in-stage before entering the study.

Table 4 gives the expected time to death of persons in any stage and 95 percent confidence bounds on these expected times. For computational reasons, the method used to calculate confidence bounds here is based on the assumption of asymptotic normality of these estimators rather than asymptotic normality of jll!3 as in the previous calculations. Consequently, these confidence bounds will be slightly inconsistent with those developed previously but should still give the reader some measure of the degree of reliability of the life expectancies.

TABLE 4

LIFE EXPECTANCY OF A PERSON IN ANY STAGE

Stage Life Expectancy Lower Ilound Upper Ilound Cowell and Hoskins·

1a (At-Risk) 9.6 7.5 12 -Ib (HIV+) 7.3 5.5 9.2 11.1 2a (LAS) 6.2 4.4 8.0 8.8 2b (ARC) 4.3 2.6 • 6.0 6.7 3 (AIDS) 0.93 0.44 1.4 2.1

.. 'Obtamed by addlllon of components found m Cowell and Hoskins 12, ParI 2, p. 121.

As mentioned above, Cowell and Hoskins' methodology results In an upward bias in the life expectancies as well.

To this point we have not tested the validity of our constant intensity Markov process model. We do this by fitting or predicting the number of persons progressing to the next stage of the exposure base in Table lA and comparing the results statistically with those of Table lB. The results of these calculations are given in Table 5.

TABLE 5

PREDICTED (ACTUAL) PROGRESSIONS BASED ON THE MODEL

Observation Stage la Stage Ih Stage 2. Stage 2h Stage .1 Period (At-Risk) (HIV+) (LAS) (ARC) (AIDS)

3·6 months 1.5

i! 2.5 1) 3.8 3) 0.8(0) 2.0(4i 6·12 months 4.0 8.6 10~ 16.9 20l 5.8t 5.0(6

12·24 months 10.3 14.5 15 16.0 14 7.1 10) 5.6~5 24·36 months 2.0 4.4 4) 14.0 14 3.74) 0.90)

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The fit appears to be remarkably good in the sense that it predicts the number of progressions accurately. A chi-squared statistic of 10.1 in comparison with a Xt4 variable (14 degrees of freedom, since 20 cells and 5 parameters, 14 = 20 - 5 - 1) at any reasonable significance level indicates that the model adequately describes the data.

This conclusion does not mean that the model used here precisely describes the physical phenomena underlying the progression of subjects through the various stages. The addition of further data in the near future could well require further refinements to the model. However, such refinements cannot be justified yet on the basis of the data available in this paper.

Table 6 presents a model multistage "life table" for persons who are in stage 1b (HIV +). At each duration, it gives the distribution by stage of a cohort of persons who were HIV + initially. The differences between Cowell and Hoskins' methodology and ours are reflected in this table as well (see Cowell and Hoskins [2, Part 2, p. 26]).

TABLE 6

PERCENT DISTR IBUTION BY STAGe: AND YEA RS SINCE HIV INFECT/O!'

YClIrl'i since Slage Ih Slage 2. SI.~e 2h Stage 3 HIV Infeelin" (HIV +) (LAS), ,ARC) (AIDS) Dead

0 100.0 0.0 0.0 0.0 0.0 0.5 64.9 30.5 4.4 0.2 0.0 I 42.2 43 .1 13.2 1.2 0.4 1.5 27 .4 45 .9 22 .6 2.8 1.4 2 17.8 43.4 30.7 4.7 3.4 2.5 11.5 38.7 36.7 6.7 6.5 3 7.5 33.1 40.5 8.3 10.5 3.5 4.9 27.6 42.5 9.6 15.4 4 3.2 22.6 42.8 10.5 20.8 4.5 2.1 18.3 42.0 11.0 26.7 5 1.3 14.6 40.3 11.1 32.6 6 0.6 9.1 35.3 10.6 44 .4 7 0.2 5.6 29.6 9.4 55.2 8 0.1 3.4 24.0 8.0 64.6 9 0.0 2.0 19.1 6.5 72.4

10 0.0 1.2 14.9 5.2 78.7 11 0.0 0.7 11.5 4.1 83.7 12 0.0 0.4 8.8 3.2 87.6 13 0.0 0.2 6.7 2.5 90.6 14 0.0 0.1 5.1 1.9 92.9 15 0.0 0.1 3.8 1.4 94 .7 16 0.0 0.1 2.8 1.1 96.0 17 0.0 0.0 2.2 0.8 97 .0 18 0.0 0.0 1.6 0.6 97.8 19 0.0 0.0 1.2 0.5 98.3 20 0.0 0.0 0.9 0.3 98 .8 21 0.0 0.0 0.7 0.3 99.1 22 0.0 0.0 0.5 0.2 99.3 23 0.0 0.0 0.4 0.1 99.5 24 0.0 0.0 0.3 0.1 99.6 25 0.0 0.0 0.2 0.1 99 .7

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OBSERVATIONS AND CONCLUSIONS

The purpose of this paper was twofold. First, the numerical results should be of interest to various parties, including insurers, dealing with the issue of AIDS. Second, the usefulness of simple parametric survival models was demonstrated. Their validity can be tested, standard errors and confidence bounds on parameters and related quantities can be calculated easily and they have intuitive appeal because of their inherent smoothness.

In the case discussed in this paper, five parameters were estimated using twenty independent pieces of data. In methods such as those used by Cowell and Hoskins, in effect, twenty quantities are estimated by twenty independent data points. This makes the results highly sensitive to the data. Parmetric models with an inherent smoothing function are more robust under small changes to the data. Furthermore, for this data set the parametric model implicitly projects beyond the longest observation period (three years).

The constant intensity model selected in this study is very simple. Although it is justified on the basis of the data provided, it is probably reasonable to expect that intensity functions would increase by duration-instage, that is, that subjects age or deteriorate making progression to the next stage more probable with duration in stage. This would require the introduction of more parameters. Furthermore, the issue of left-hand censoring becomes important and difficult to handle. How long a person has been in a particular stage prior to diagnosis has a direct effect on the likelihood function since it affects the level of the intensity function to be used at the time of diagnosis (entry into the study). Adding complexity to the model requires the use of more parameters in the model. As more data become available, it will be necessary to present the data in a format that will allow the user to extract information about the parameters.

Finally, a few comments regarding the format of the data in this study would be appropriate. By providing only information on the subjects' initial stage, no contribution is made to inferences about subsequent stages for persons who may have passed through several stages of the data. If this is done the standard errors of the estimates of the intensities of stages lb, 2a, 2b and 3 can be discussed and the reliability of the various estimates increased. I hope that this will be done in subsequent reports.

REFERENCES

1. BRODT, H.R., HELM, E.B., WENER, A., JOETTEN, A., BERGMANN, L., KLUVER, A., AND STILLE W. "Spontanverlauf der LAVn-ITLV-III-Infektion; Verlaufsbeobachtungen

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bei Personen aus AIDS-Risikogruppen," Deutsche Medizinische Wochenschrift, Stutlgart, m (1986):1175-80_

2. COWELL, M.1. AND HOSKINS, W.H. "AIDS, HIV Mortality and Life Insurance ." In The Impact of AIDS on Ufe and Health Insurance Companies: A Guide for Practicing Actuaries, Report of the Society of Actuaries Task Force on AIDS. Itasca, HI.: Society of Actuaries, 1988.

3. Cox, D.R. AND HINKLEY, D.V. Theoretical Statistics. London: Chapman and Hall, 1974.

4. EVERllT, B. AND HAND, D. Finite Mixture Distributions. London: Chapman and Hall, 1981.

5. KALBFLEISCH, J.G. Probability and Statistical Inference, Vol. 2, 2d ed. New York: Springer-Verlag, 1985.

6. REDFIELD, R., WRIGHT, D.C. AND TRAMONT, E.C. "The Walter Reed staging classification for IITL V-HIlLA V Infection," New England Journal of Medicine 3) 4, No.2 (1986):131-32.

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ABSTRACT - MEMBER | SOA · The purpose of this paper is to provide a survival analysis of persons in the various stages of HIV infection typically leading to AIDS and ultimately death.

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