American Journal of EpidemiologyCopyright © 1998 by The Johns Hopkins University School of Hygiene and Public HealthAll rights reserved

Vol. 148, No. 8Printed in U.S.A.

Asymptomatic Incidence and Duration of Prostate Cancer

Ruth Etzioni,1 Raymond Cha,1 Eric J. Feuer,2 and Ori Davidov1

Prostate cancer is known as a disease with an extremely high prevalence relative to its clinical incidence inthe population. The combination of preciinicai incidence and duration that could yield this phenomenon is oftremendous interest to researchers trying to understand the natural history of the disease and to developefficient screening strategies. In this article, the authors present estimates of the age-specific asymptomaticincidence and average preciinicai duration of prostate cancer. The methodological approach is to first estimatethe age-specific incidence of new (stage AI) prostate cancers using preciinicai prevalence data from autopsystudies performed between 1941 and 1964 and clinical incidence data for the years 1960-1986 from theSurveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. Then, thepreciinicai prevalence estimates are divided by the derived preciinicai incidence estimates to yield estimatesof the average duration of asymptomatic disease. The estimated mean duration among white men is between11 and 12 years and appears to be approximately 1 year shorter for blacks than for whites. Comparison of thelifetime risks of preciinicai and clinical disease suggests that approximately 75% of prostate cancers will neverbecome diagnosed if clinical incidence remains at levels observed in 1984-1986, prior to the introduction ofprostate-specific antigen (PSA) screening in the population. Am J Epidemiol 1998; 148:775-85.

disease progression; natural history; prevalence; prostatic neoplasms; SEER program

The duration of the asymptomatic or latent period inprostate cancer has important public health implica-tions, particularly for the design and analysis ofscreening strategies. Indirect evidence, including highprevalence of the disease at autopsy relative to itsclinical incidence, suggests that the time from preciin-icai onset to diagnosis of prostate cancer can be rela-tively lengthy. For example, the proportion of menaged 65-69 years with histologic prostate cancer hasbeen estimated from autopsy studies to be at least10-20 percent (1, 2). In contrast, the annual clinicalincidence of prostate cancer in 1984-1988 was 0.37percent for black males aged 60-64 years and 0.73percent for those aged 65-69 years; the analogousfigures for white males were 0.23 percent and 0.46percent, respectively (3).

The goal of this paper is to estimate the length of theasymptomatic period in prostate cancer, that is, thetime from onset of the disease until the appearance ofsymptoms leading to its diagnosis. We also estimate

Received for publication July 30, 1997, and accepted for publi-cation March 20, 1998.

Abbreviations: PSA, prostate-specific antigen; SEER, Surveil-lance, Epidemiology, and End Results.

1 Fred Hutchinson Cancer Research Center, Seattle, WA.2 National Cancer Institute, Bethesda, MD.Reprint requests to Dr. Ruth Etzioni, Fred Hutchinson Cancer

Research Center, Program in Biostatistics, 1100 Fairview Avenue N,MPE-665, P. O. Box 19024, Seattle, WA 98109-1024.

the duration of the preciinicai period, which we defineas the time from onset of the disease until its clinicaldiagnosis, whether due to symptoms or not. In calen-dar periods in which no interventions exist to diagnoseasymptomatic prostate cancer, the preciinicai durationand the asymptomatic duration are the same. However,modalities that can be used to detect asymptomaticdisease have been developed more recently. In thisarticle, we use the more general term "preciinicaiperiod" to refer to the time from disease onset todiagnosis; when necessary, we distinguish betweentime to diagnosis and time to symptoms.

Previous estimates of the preciinicai duration ofprostate cancer have been presented by Whittemore etal. (4). These authors used autopsy data to estimate byage and race the mean volume of low-grade, latentcancer among men in the general population. Theythen hypothesized that the incidence of high-gradecancer among men at a given age would be propor-tional to the mean volume of low-grade cancer whenthe men were a few years younger. Based on a linearregression analysis, the average time to detection wasestimated to be approximately 7 years.

Work on longitudinal changes in prostate-specificantigen (PSA) in men with a subsequent diagnosis ofprostate cancer also provides information about thepreciinicai duration. Indeed, Morrell et al. (5) haveestimated that PSA curves in prostate cancer patients

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show a marked increase in slope beginning at roughly6.4 to 10.7 years prior to clinical diagnosis, dependingon the stage at diagnosis. Whittemore et al. (6), in amuch larger longitudinal PSA study, estimated that theincrease in slope occurs roughly 13 years prediagnosisfor men with subsequent stages B through D prostatecancer.

There are clearly some differences between previ-ously derived estimates of the preclinical duration inprostate cancer. In this article, we use a differentmethodological approach in an attempt to resolvethese differences and to provide more reliable infor-mation about the natural history of prostate cancer.

MATERIALS AND METHODS

Our methodological approach extends methods forestimating age-specific incidence from age-specificprevalence in the case of an irreversible disease that islifelong after diagnosis (7).

We begin by estimating the number of new cases ofasymptomatic disease in any given age interval. Figure1 illustrates the steps involved, which are described inmore detail in the rest of this section. The newly

No(Population)

I

Start of interval

(Symptom-freepopulation)

A.NooPo

(Prevalent)

Noo<1-Po>(Not prevalent)

(Lost cases)

I| N oo ( 1 - P o> - N |

N(new cases)

(Clinically diagnosed)(6)

NooPomo(Other-cause death)

(7)

(Nonsurvivingnew cases)

(8)

(Surviving new cases) I

1o1 o o oSymptom-free, prevalent)

(5) End of interval

FIGURE 1. Flowchart illustrating the main components of the cal-culation of N, the number of new cases in an interval. Numbers inparentheses refer to expressions in the text. We compute A/to and L,solve the final expression in the figure for n, and then obtain m. Thenumber of new cases N is given by the sum of m and n.

incident cases in a specific age interval consist of thosewho survive to the end of the interval and those whodo not. Formally, we denote the total number of newcases by N; N is equal to n + m, where n is the numberof surviving new cases and m is the number of newcases who do not survive. We derive an expression forn, which draws on mathematical development used inestimating the lifetime probability of a cancer diagno-sis (8). We then develop an approximation for m,which allows us to estimate the total number of newcases in the interval and the associated incidence.Finally, we divide the prevalence of asymptomaticdisease by the derived asymptomatic incidence toyield an estimate of the average sojourn time. Moreprecisely, we denote the age-specific preclinical inci-dence rate in age interval i by / and the prevalence byP{. An estimate of the average preclinical duration isthen given by the ratio of population prevalence toincidence (9), namely,

M = (1)

where age interval 1 is the lowest age interval at whichthe incidence and prevalence of disease are nonnegli-gible, k is the highest age interval in the population,and f, denotes the fraction of the population in agegroup i. In our calculations, the interval length is 1year and the lowest and highest age intervals are30-31 and 89-90 years, respectively.

To estimate n, the number of surviving new cases ina given age interval, we extend the methods of Leskeet al. (7), who estimated n as

n = NlPl - N0P0 NoPomo, (2)

where No and Po are the number who are alive and theprevalence of the condition at the beginning and end ofthe interval, Nl and P, are the number who are aliveand the prevalence of the condition at the end, and mo

is the probability of death due to other causes duringthe interval. Thus, the number of surviving new casesequals the number of prevalent cases at the end minusthe number of prevalent cases at the start plus thenumber of cases prevalent at the start who were lostduring the interval because of competing mortality.Cowen et al. (10) used a similar idea, but their calcu-lation neglected competing mortality. Note that we usethe term "prevalent" to refer to persons with asymp-tomatic disease.

Expression 2 is appropriate when the condition ofinterest is lifelong after diagnosis. In our case, thecondition of interest is preclinical prostate cancer,which is terminated when the disease is diagnosed.Termination of the condition of interest is a mecha-

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nism for removing prevalent cases from the interval inaddition to competing mortality. In this case, the num-ber of surviving new cases is given by

n = - NOPO + L, (3)

where L is the number of prevalent cases at the startwho are lost during the interval either through com-peting mortality or through clinical diagnosis of can-

cer, i.e.,

L = an N0P0m0 (4)

Here, ao is the number of clinical diagnoses in theinterval and mo is the probability of mortality due toother causes during the interval. Note that mo in thisexpression is not exactly the same as it is in expression2, since in this case clinical diagnosis is a competingrisk for other-cause mortality. Precise expressions forao and mo in expression 4 are given below.

This formulation assumes that all clinically diag-nosed cases are among the NOPO prevalent cases at thestart of the interval. We show in the Appendix that thisassumption does not lead to any loss of generality.

The prevalence estimates Po and Pl used in ourapplication were obtained from Carter et al. (1), whostudied 5,250 autopsies from the US literature (figure2). These estimates apply to a symptom-free popula-tion (men with a diagnosis of prostate cancer duringtheir lifetimes were excluded). Therefore, to estimatethe number of prevalent cases at the start and end ofeach age interval, we require the size of the symptom-free population (i.e., those without a prior prostatecancer diagnosis) at these times. We denote thesymptom-free population size at the start and end of agiven age interval by Noo and Nlo, respectively. Ex-pressions 3 and 4 still apply but with No and Nx

| Carter prevalence {

30 35 40 45 50 55 60 65 70 75 80 85

Age (years)

FIGURE 2. Age-specific prevalence of asymptomatic prostatecancer among symptom-free men at each age based on autopsystudies published between 1941 and 1966 (1).

replaced by Noo and Nlo and mo now equal to themortality probability in the symptom-free population.We assume that NO=NOO in the initial time interval.

Wun et al. (8) present expressions for a^ mo, andNlo, which take into account that death from othercauses and clinical incidence are competing risks act-ing on the symptom-free population (11). These au-thors assume exponential distributions of event timesin each age interval so that the probability of an eventin an interval of length i is 1 — exp(-n) , where r isthe rate of the event. Given Noo symptom-free indi-viduals at the start of an interval, we have

= Nooexp{- {qo + ro)}, (5)

where qo is the death rate from causes other thanprostate cancer and ro is the clinical incidence rate,respectively, among persons without symptoms at thestart of the interval. The rate qo is approximated by therate in the total population (8); qo=ma—mp, where ma

is the all-cause mortality rate and mp is the prostate-cancer mortality rate. Thus, deaths due to prostatecancer are assumed to occur among only those indi-viduals previously diagnosed with the disease and notamong persons without symptoms. This assumptionappears to be adequate, because the incidence of pre-viously undiagnosed cases with prostate cancer listedas a cause of death is extremely low (for example, theSurveillance, Epidemiology, and End Results (SEER)age-adjusted incidence of prostate cancer in 1984—1988 was 92.24 per 100,000, whereas the incidence ofdeath-certificate-only cases was 0.49 per 100,000 (3)).The rate ro of clinical diagnosis in the symptom-freepopulation is computed by multiplying r, the popula-tion clinical incidence rate, by (No/Noo), where No isthe population at the start of the interval. All clinicallydiagnosed cases in a given year are derived from thepool of asymptomatic, prevalent cases at the start; thatthis assumption leads to no loss of generality is shownin the Appendix.

The number of clinically diagnosed cases ao is givenby

a0 = Noo{\ - exp{( - qo + ro)}>qo + To

(6)

and the probability of noncancer death by

m0 = 1 - e x p { - (qo + ro)})- -• (7)

The total number of new cases N is approximatelyequal to n + m, where m is the number of new caseswho do not survive to the end of the interval. Weestimated m by using the approximation proposed by

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Leske et al. (7), namely, m = nuol{\ — uo)(l/2),where uo is the probability of dying from causesother than prostate cancer during the year, i.e.,uo = 1 — exp( — q0). A heuristic explanation forthe approximation is the following: If n is the number ofsurviving new cases, then «/(l — uo) represents the totalnew cases. Of these, nuol{\ — uo) are the "nonsurvivingnew cases," which includes those who did not surviveto their time of asymptomatic onset as well as thosewho survived their time of onset but died before theend of the interval. On average, half of the nonsurviv-ing new cases have a time of death that precedes theirtime of onset; the approximation is exact if time topreclinical onset and time to other-cause mortality areuniformly distributed within the interval. The numberof incident new cases is then approximately

N — n + m = n(1 + (1 - u0))

2(1 - u0) •(8)

N may be divided by an appropriate populationdenominator to yield the age-specific rate of asymp-tomatic onset in the interval. In our calculations, weused the population size at the start of the interval,although the population size at the midpoint of theinterval could also be used. The average preclinicalduration is then estimated by using expression 1.

In principle, our entire method can be applied inreverse if the goal is to generate the age-specificprevalence of preclinical disease given its incidence.Specifically, given lo the asymptomatic incidence ratein an interval, the number of new cases N is approx-imately No(l exp( — Q). The number of new caseswho survive to the end of the interval is n = 2N(l — uo)/(1 + (1 — uj) (compare with expression 8). Finally, thenumber of prevalent cases at the end of the intervalgiven the number prevalent at the start isNloPi = NOOPO - ao- NooPomo + n (compare withexpression 2).

In the next section, we illustrate application of boththe original and the reverse calculation. The originalcalculation enables us to estimate the incidence ofpreclinical disease by using data on preclinical prev-alence from a time when relatively few interventionswere available for early diagnosis of prostate cancer.However, it does not seem reasonable to use the sameprevalence data for later time periods. Indeed, recentchanges in medical practice patterns have had a sub-stantial impact on the frequency of a prostate cancerdiagnosis and, therefore, presumably also on the prev-alence of preclinical disease. In the application, weassume that preclinical incidence is relatively constantover time, since this is not likely to be affected bymedical interventions that lead to early diagnosis. Us-

ing our estimate of preclinical incidence from theearlier time period, we apply the reverse calculation toestimate preclinical prevalence and mean sojourn timein the later time periods.

RESULTS

Figure 3 presents estimates of the age-specific inci-dence of asymptomatic prostate cancer in white men.Table 1 shows the input data used in our calculations,and table 2 shows some intermediate results. We beginwith 5 million individuals at age 30 years, when No =Noo. The clinical incidence and mortality rates given in5-year intervals in table 1 are translated into 1-yearintervals by linearly interpolating between the mid-points of each interval, except for ages 30-32 yearsand ages 87-89 years, where the rates are assumed tobe constant because of a lack of precise information onmen younger than age 30 years and older than age 89years.

As an example of the sequence of calculations,consider the age interval 52-53 years. There are4,650,509 symptom-free individuals at the start of thisinterval (iVoo). The clinical incidence rate amongthese individuals, to five decimal places, is 0.00014(ro=rNo/Noo, where r = 0.000137 from table 1). Thenumber of symptom-free individuals at the end of theinterval is 4,618,287, since Nlo—Noo exp{ — (qo+ro)};qo is taken directly from table 1. The number ofclinically diagnosed cases, ao, is given by expression 6and is equal to 635; the number of deaths due to othercauses among cases prevalent at the start isNooPomo= 1,708, where mo is the probability of non-cancer death in the interval, given by expression 7.

0.06

0 . 0 5 • •

0.04 4-o

I 0.03 +o

~ 0.02 • •

0.01 • •

| Asymptomatic incidence]

I I I I I I I I I30 34 38 42 46 50 54 58 62 66 70 74 78 82 86

Age (years)

FIGURE 3. Annual age-specific incidence of asymptomatic pros-tate cancer in white men. The estimate is based on prevalence fromautopsy studies published between 1941 and 1966 (1), clinicalincidence from the Connecticut Surveillance, Epidemiology, andEnd Results (SEER) program (1960-1964) (12), mortality from 1986US life tables (13), and cancer mortality from SEER (1984-1988) (8).

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Incidence of Prostate Cancer 779

TABLE 1. Input parameters used toand whites, by age in years

Age

30-3435-3940-4445-^950-5455-5960-6465-6970-7475-7980-8485-89

30-3435-3940-4445-4950-5455-5960-6465-6970-7475-7980-8485-89

Prevalence*

1,121.91,627.32,360.43,423.94,966.47,203.8

10,449.215,156.821,985.131,889.846,256.767,096.1

1,121.91,627.32,360.43,423.94,966.47,203.8

10,449.215,156.821,985.131,889.846,256.767,096.1

1960-1964#

0.20.20.76.0

13.740.0

113.9233.5409.8554.9773.2934.3

estimate the asymptomatic incidence of prostate cancer in blacks

Clinical incidence

1975-1979*

001.8

17.349.4

155.2313.8620.3874.3

1,166.51,310.61,439.3

00.21.34.6

23.473.0

176.5336.0550.6780.5

1,006.81,110.6

1984-1988$

Blacks

0.21.02.7

12.463.9

162.1370.2727.6

1,019.61,331.01,556.71,468.4

Whites

00.21.66.9

30.295.4

235.5462.5725.2.978.5

1,155.41,149.0

1990-1992**

004.2

15.6103.5323.5679.6

1,220.71,733.11,940.62,047.51,840.2

002.4

12.268.3

190.9479.4910.9

1,347.31,541.41,564.41,553.9

Prostate

csincormortality(1984-

1988)t,t

00.20.43.0

11.231.080.1

171.7329.9523.6839.3967.7

000.20.83.6

11.631.271.2

139.2246.0408.5616.8

rtM-CrallotJmortality(1986)t,§

448.0589.8797.0

1,034.41,524.62,075.43,075.64,074.15,797.08,038.1

11,854.715,488.1

180.6212.3295.6452.0746.3

1,221.51,939.82,908.74,602.16,988.1

10,825.718,576.1

PtePrib"?n1 i

0.1820.1380.1180.1060.1040.0970.0800.0690.0480.0320.0150.011

0.1630.1290.1070.1010.1060.1050.0900.0750.0550.0360.0200.013

• Computed at the midpoint of each age interval; based on autopsy studies published from 1941 to 1966 (1).f Annual figures.i Source: Miller etal. (3).§ Source: US life tables (13).H Source: US Census data (14).# Source: Connecticut Surveillance, Epidemiology, and End Results registry (12); not available for blacks.

** Source: Feuerand Wun (19).

Thus, the number of prevalent cases lost during theinterval because of clinical diagnosis or other cause ofdeath is L = 2,343 = 635 + 1,708. To obtain n, thenumber of surviving new • cases, we solveNxJ>i=NooPo+n - L to get n = 18,304; the totalnumber of incident cases is N = 18,372 given byexpression 8, and the incidence in the interval (notshown in the table) is / = N/No = 0.00395.

The estimates in figure 3 are based on autopsyprevalence data from Carter et al. (1), clinical inci-dence from the Connecticut SEER tumor registry for1960-1964 (12), all-cause mortality from 1986 USlife tables (13), and prostate cancer mortality from1984 to 1988 (3). We assumed a negligible incidenceof asymptomatic disease prior to age 30 years. Ourgoal was to estimate the average asymptomatic dura-tion in a population similar in terms of life expectancyto the US population in the 1980s. However, we usedhistorical clinical incidence data in an effort to elimi-

nate the impact of interventions leading to incidentaldiagnosis of cancer, such as transurethral resection ofthe prostate. Based on these input data, the averagesojourn time in a population with an age distributionsimilar to that of the US population in 1980 (14) is11.57 years.

Since the Connecticut SEER data from 1960 to 1964do not contain sufficient information on black cases,we obtained sojourn time estimates for blacks basedon clinical incidence in several other calendar timeperiods, namely 1975-1979, 1984-1988, and 1990-1992. This also allowed us to assess the impact ofdiffering medical practice patterns over time. We usedthe reverse calculation to estimate how the changingincidence of clinical disease would affect the age-specific prevalence of preclinical disease. In doing so,we assumed that the prevalence of preclinical diseaseat age 30 years would be the same as in Carter et al.(1), since medical practice patterns have changed little

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TABLE 2. Intermediate results of calculations used to estimate the age-specific incidence of asympto-matic prostate cancer in whites aged 30-34

Age(years)

3031323334

5051525354

No*

5,000,0004,990,9784,981,9734,972,9834,963,695

4,714,5204,684,9784,652,8824,618,2874,579,595

N- t

5,000,0004,990,9664,981,9484,972,9464,963,646

4,712,9294,683,0164,650,5094,615,4654,576,165

0.001810.001810.001810.001870.00193

0.006260.006840.007430.008360.00930

and 50-54 years

rJS

2.50E-062.50E-062.50E-062.46E-062.42E-06

0.000110.000120.000140.000190.00024

1212121212

499567635872

1,105

8794

101112125

1,2591,4731,7082,0562,440

L—

99106113124137

1,7582,0402,3432,9283,545

" f t

3,7384,0184,3204,6444,992

15,95417,09718,30419,73021,213

Ntt

3,7424,0234,3244,6484,996

16,00517,15718,37219,81421,312

* Population at the start of the interval.t Symptom-free population at the start of the interval.$ Mortality rate from other causes.§ Clinical incidence rate in the symptom-free population, from the Connecticut Surveillance, Epidemiology,

and End Results program (1960-1964) (13); rB= rNJN^.H Number of clinically diagnosed cases (text expression 6).# Number of noncancer deaths among prevalent cases at the start of the interval (text expression 7).

** Number of prevalent cases lost because of clinical diagnosis or death.f t Number of surviving new cases (text expression 3).$$ Number of incident cases (text expression 8).

for this group in regard to prostate cancer care. Figure4 shows how the different clinical diagnosis ratesaffect the estimates of age-specific preclinical preva-lence in different calendar time periods.

Figure 4 and table 3 show that the asymptomaticprevalence and the average preclinical duration varywith age-specific clinical incidence, which haschanged considerably over time. In general, higherrates of clinical diagnosis lead to lower prevalenceestimates. For whites, the average sojourn time basedon the clinical incidence from 1960 to 1964 is higherthan the estimate based on later calendar periods. Theearlier estimate is probably closest to the true asymp-tomatic duration. The duration estimate based on theclinical incidence from 1975 to 1979 is about 6 monthslonger than that based on the clinical incidence from1984 to 1988. Both of these latter estimates are morereflective of time to detection than of a true asymp-tomatic duration. In fact, data from the National Hos-pital Discharge Survey (15) show an increasing fre-quency of transurethral resection of the prostate fromdischarge rates of roughly 800 per 100,000 in 1973and 900 per 100,000 in 1977 to a peak of approxi-mately 2,300 per 100,000 in 1986. The sojourn timeestimates reflect these changing practices; since a sideeffect of transurethral resection of the prostate is thedetection of asymptomatic cancer, the interval frompreclinical onset to diagnosis is shorter in these latercalendar periods. Finally, the duration estimates basedon the clinical incidence in 1990-1992 are substan-tially lower than those based on the clinical incidence

from 1984 to 1988 for both blacks and whites. Thelater interval reflects a period during which the PSAscreening test for prostate cancer was being used in thepopulation. Consequently, incidence rates during thisperiod were higher than previously observed, and theaverage time from onset of disease to detection wascorrespondingly reduced.

A comparison of the estimates for black and whitemales based on SEER clinical incidence from 1975 to1979, 1984 to 1988, and 1990 to 1992 shows that theaverage sojourn time for black males is almost 1 yearshorter, supporting the belief that prostate cancer, onaverage, may be more aggressive and have highergrowth rates in blacks than in whites.

DISCUSSION

This paper presents a novel approach to estimatingthe asymptomatic incidence and duration of prostatecancer. The incidence calculation uses data on theage-specific prevalence of preclinical disease fromautopsy studies and on the age-specific incidence ofclinical disease from SEER. Our results suggest thatthe average asymptomatic duration for whites is 11-12years, which is between the published estimates (5, 6)of the time from initial acceleration of the PSA growthcurve to clinical diagnosis of prostate cancer. Thus,these results, taken together, support the notion thatthe initial acceleration of PSA growth probably occursat roughly the time of disease onset.

The methods presented here extend the work of

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u.uie •

0.014-

0.012 •

o 0.01 •

•g 0.008 •

u- 0.006 •

0.004 •

0.002 •

0<

-• -1960-1964

- • -1975 -1979

- • -1984 -1988

-*—1990-1992

*

/ / / /f ///30 35 40 45 50 55 60 65 70 75 80 85 89

Age (years)

oocm

Inci

d

0.02.

0.015-

0.01 •

0.005 •

0 -

-•-1975-1979-•-1984-1988-*-1990-1992

/

/ .

/ /J/1 1 h--i—i—

30 35 40 45 50 55 60 65 70 75 80 85 89

Age (years)

u.o -

0.7-

0 6 -o r> c

•— r\ A

2 0.3-

0.1 -

0 J

—Carter— 1975-1979— 1984-1988— 1990-1992

/ /

/ /1/

* r-Til—I—I—I—I—h—I—30 35 40 45 50 55 60 65 70 75 80 85 89

Age (years)

1975-1979

1984-1988

1990- 1992

Carter

30 35 40 45 50 55 60 65 70 75 80 85 89

Age (years)

FIGURE 4. Age-specific Surveillance, Epidemiology, and End Results (SEER) program incidence and corresponding asymptomaticprevalence by race and calendar period. Top left: incidence, whites; bottom left: prevalence, whites; top right: incidence, blacks; and bottomright: prevalence, blacks. Asymptomatic incidence is as shown in figure 3; asymptomatic prevalence at age 30 years is from Carter et al. (1);all-cause mortality is from 1986 US life tables (13); and cancer mortality is from SEER (1984-1988) (8).

Leske et al (7), who estimated incidence from preva-lence when the disease of interest is lifelong afterdiagnosis. In our case, the "disease of interest" is theasymptomatic phase of prostate cancer; since theasymptomatic period is terminated by clinical diagno-sis, it is not necessarily a lifelong condition. Ourmethods directly address this problem by consideringclinical diagnosis and other-cause mortality as com-peting risks for removal of asymptomatic cases fromthe prevalence pool. An alternative approach would beto use methods developed by Podgor and Leske (16)for estimating disease incidence when mortality ratesdiffer for prevalent and nonprevalent cases. In thiscase, other-cause mortality and clinical diagnosiswould be considered collectively as "mortality" in theprevalent group. This approach requires solution of anonlinear equation and so may be computationallymore complex than our approach, which may be im-

TABLE 3. Estimates* of the average sojourn time, by periodof clinical incidence, among populations of blacks and whiteswith age distributions similar to those in the US population in1980

Population Mean preclinical duration (years)

Whites1960-19641975-19791984-19881990-1992

Blacks1975-19791984-19881990-1992

11.5711.4510.989.09

10.409.987.63

* Estimates are based on prevalence from autopsy studies pub-lished between 1941 and 1966 (1), Surveillance Epidemiology, andEnd Results clinical incidence data from several calendar periods(3, 12), all-cause mortality from 1986 US life tables (13), andprostate cancer mortality from 1984 to 1988 (3).

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782 Etzioni et al.

plemented by using a spreadsheet program such asEXCEL (Microsoft Corporation, Redmond, Washing-ton). Figure 5 shows that both approaches yield similarresults in the present problem.

The mathematical relation, average duration equalsprevalence divided by incidence, has a long history inthe epidemiologic literature (9). An implicit assump-tion is that the condition of interest is progressive inthe sense that it will terminate unless prevented, forexample, by competing mortality (17). Therefore, thisapproach is not valid for nonprogressive diseases orfor diseases that can regress. Given the possibility thatprostate cancer cases may exist in whom the tumormight remain indolent no matter how long they lived(infinitely indolent), this is a limitation of our ap-proach. Based on this scenario, the quantity of interestis presumably the mean asymptomatic duration amongprogressive cases. Our mean duration estimates mayoverestimate this quantity, but the extent of overesti-mation is impossible to gauge; little is known aboutthe fraction of prevalent, asymptomatic cases that areinfinitely indolent. Certainly, given an estimate of theage-specific prevalence of infinitely indolent disease,the methods presented in our paper may be appliedusing appropriately adjusted prevalence estimates toyield an estimate of the mean asymptomatic durationamong progressive cases.

A second assumption in using the prevalence/inci-dence approach is that incidence and mean durationare constant over time, so that the current number of

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o 0.05-uc.§ 0.04 •u- 0.03 •

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0.01 •

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- • - Leske

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/y30 35 40 45 50 55 60 65 70 75 80 85 89

Age (years)

FIGURE 5. Comparison of age-specific asymptomatic incidencewith that estimated using the method of Podgor and Leske (16).Estimates are based on prevalence from Carter et al. (1); clinicalincidence from the Connecticut Surveillance, Epidemiology, andEnd Results (SEER) program (1960-1964) (12); mortality from 1986US life tables (13); and cancer mortality from SEER (1984-1988) (8).For comparison purposes, the denominator of our age-specificasymptomatic incidence is the population size in the middle of theage interval, approximated by the average of the population sizes atthe start and end of the interval.

prevalent cases, which derive from prior incidence,may be compared with the current number of incidentcases to give a valid result. In other words, the prev-alent cases in the numerator must have arisen as aresult of an incidence process that is the same as thatin the denominator. The need for this second assump-tion invalidates age-specific sojourn time estimatesthat derive from the ratio of age-specific prevalence toincidence, unless age-specific incidence and durationare constant. The assumption is implicit in the calcu-lations leading to table 3, although the duration of timeto detection of cancer may be not be constant in thelater time periods. Therefore, strictly speaking, theseestimates are hypothetical, representing a steady-statesituation in which asymptomatic incidence, clinicalincidence, and mean time to detection are not chang-ing over time.

The identity relating prevalence, incidence, and du-ration specifies that the prevalence of the condition ofinterest should be divided by the incidence of newcases of that condition. This yields an estimate of theaverage duration of the condition. Thus, to estimatethe asymptomatic period for any cancer, the preva-lence of asymptomatic disease should be divided bythe incidence of asymptomatic disease. If the averageasymptomatic duration is fairly short, for example, inlung cancer, most preclinical cases will progress tobecome clinically diagnosed within their lifetimes. Inthis case, observed or expected clinical incidence willbe a reasonable surrogate for preclinical incidence.However, in cancers such as prostate cancer that mayprogress slowly in a nontrivial fraction of cases, manyasymptomatic cases will not become diagnosed duringtheir lifetimes; therefore, it will not be adequate toapproximate the asymptomatic incidence by the ob-served clinical incidence. Indeed, the use of clinicalincidence will lead to a denominator that is too lowrelative to the numerator, and the mean sojourn timewill be overestimated. In recognition of this issue,much of the effort reported on in this paper has fo-cused on obtaining an unbiased estimate of asymptom-atic incidence, which we assume is more or less un-changing over time.

The use of autopsy data to estimate prevalence leadsto an estimate of average sojourn time that can beinterpreted as the time from disease onset to symp-toms, under the assumption that an autopsy will iden-tify disease if it is present. This is in contrast to the useof prevalence rates as estimated from screening data,which leads to an estimate that can be interpreted asthe time from screen-detectable disease onset to symp-toms. Therefore, it would be expected that studiesbased on screening data would yield slightly shorterduration estimates than those based on autopsy studies.

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The accuracy of autopsies and the autopsy preva-lence of prostate cancer are topics of some debate. Forexample, while Carter et al. (1) found a prevalence ofapproximately 2-4 percent among men aged 40-49years, a recent article (18) reported small foci ofhistologic cancer in 34 percent of patients in this agegroup. In addition, Carter et al. did not find a dissim-ilar prevalence in black men and white men, which isin contrast to the investigation of Guileyardo et al. (2)but is consistent with the findings of Sakr et al. (18).Our methods are applicable to any autopsy dataset thatpresents age-specific prevalence within sufficientlyfine age intervals, and the paper by Carter et al. was,to our knowledge, the best source from this point ofview. However, we truncated the asymptomatic inci-dence at age 89 years, since we were not comfortableextrapolating the Carter et al. prevalence curve beyondthis bound. Given the years in which the autopsies inthe Carter et al. study were performed, we assume thatthe prevalences represent the prevalence of prostatecancer in the (living) population in the absence ofdiagnostic interventions such as transurethral resectionof the prostate or screening. In general, application ofautopsy prevalence to the living population assumesthat death due to causes other than prostate cancerdoes not happen with a greater or lesser frequency inprevalent cases than in cancer-free individuals. Inci-dentally, this assumption is also needed to be able tointerpret the ratio of prevalence to incidence as anestimate of the mean of the duration that would havebeen observed if not for competing mortality.

Given the uncertainty of the prevalence figures fromCarter et al. (1), we conducted a sensitivity analysis inwhich we evaluated the impact of inflating the prev-alence on our results by using the data for whites andthe clinical incidence from Connecticut SEER (1960-1964). Figure 6 indicates that the asymptomatic inci-dence results are highly sensitive to the prevalencevalues. When prevalence was 20 percent higher thanour original input, asymptomatic incidence increasedby 15-20 percent; when prevalence was 50 percenthigher, asymptomatic incidence increased by 40-50percent. However, as might be expected, the estimatesof average duration are rather less affected. We ob-tained 11.64 years instead of 11.57 years for an in-crease in prevalence of 20 percent and 11.72 years foran increase of 50 percent. We also evaluated thesensitivity of our results to the assumption that theother-cause mortality rate in the symptom-free popu-lation could be approximated by the rate in the popu-lation. Specifically, we set other-cause mortality in thesymptom-free population to a value 20 percent higherand then 20 percent lower than that in the populationas a whole. Average duration estimates corresponding

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Age (years)

FIGURE 6. Sensitivity analysis: age-specific incidence of asymp-tomatic disease, inflating the age-specific prevalence (Po) by 20%and 50%.

to clinical incidence levels among whites from theConnecticut registry in 1960-1964 were 11.42 and11.72, respectively, which show relatively little sensi-tivity to this assumption. Finally, we considered thesensitivity of the results to the population age distri-bution weights / j . In our example, the weights werechosen to reflect the approximate age distribution inthe United States in 1980 leading, in a sense, to anestimate of sojourn time standardized to the 1980 USpopulation. It turns out that the sojourn time estimateis relatively insensitive to the exact choice of theseweights, because the ratio of prevalence to incidencewithin each age stratum is fairly constant across agesin our example. In fact, it can be shown analyticallythat if Pi=cli for all i, then the ratio in expression 1will be independent of the weights/j.

The asymptomatic incidence estimates are useful inunderstanding the relative frequency of diagnosed andundiagnosed prostate cancers. For instance, the meth-ods of Feuer et al. (19) can be applied to our preclin-ical incidence estimates to compute the lifetime prob-ability of developing preclinical prostate cancer. Thelifetime probability for whites (up to age 90 years) isapproximately 36 percent. The same methods mayalso be used to compute the lifetime probability of acancer diagnosis; for example, based on the incidenceof prostate cancer in 1984-1986, the lifetime proba-bility (up to age 90 years) of a prostate cancer diag-nosis was estimated to be approximately 9.3 percent.This suggests that roughly 75 percent of asymptomaticprostate cancers will never be diagnosed if diagnosticcapabilities remain at levels similar to those in 1984-1986, just prior to the introduction of PSA screening inthe population. While this may seem staggeringlyhigh, we note that based on the observed incidence in1960-1964 (3), the lifetime probability of a prostate

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784 Etzioni et al.

cancer diagnosis was only 6 percent. This suggeststhat 83 percent of prostate cancers would never bediagnosed if few or no diagnostic interventions ex-isted. It is interesting to note that the lifetime proba-bility of a cancer diagnosis based on the observedincidence in 1990-1992 is approximately 18 percent;thus, with diagnosis occurring at a rate similar to thatobserved in recent years, the fraction of asymptomaticprostate cancers that will never be diagnosed is re-duced to 50 percent. However, the rates of diagnosisobserved in these later years are highly unstable be-cause of the introduction of prostate cancer screeningin the population during this time.

The asymptomatic incidence and sojourn time esti-mates are biologically plausible and are consistentwith the literature on PSA growth in prostate cancercases. They confirm what has already been suspectedfor some time, namely, that prostate cancer is a rela-tively slow-growing neoplasm, and they suggest thatamong whites, 50-75 percent of new cases are un-likely to surface clinically. The estimates should beuseful to researchers studying the natural history of thedisease and designing effective and cost-effectivescreening programs.

ACKNOWLEDGMENTS

Research was supported by National Institutes of Healthgrant R29 CM0221 (R. Etzioni), by contract NCI N01CN-05230 from the National Cancer Institute (R. Etzioni and R.Cha), and by National Research Service Award 5 F32 CA71133-02(0. Davidov).

The authors thank Benjamin F. Hankey from the Surveil-lance Program, Division of Cancer Prevention and Control,National Cancer Institute, for suggesting the calculation oflifetime risk of asymptomatic prostate cancer that will neverbe detected.

REFERENCES

1. Carter HB, Piantadosi S, Isaacs JT. Clinical evidence for andimplications of the multistep development of prostate cancer.J Urol 1990;143:742-6.

2. Guileyardo JM, Johnson WD, Welsh RA, et al. Prevalence oflatent prostatic carcinoma in two US populations. J NatlCancer Inst 1980;65:311-16.

3. SEER cancer statistics review: 1973-1990. Bethesda, MD: USDepartment of Health and Human Services, Public HealthService, National Institutes of Health, National Cancer Insti-tute, 1993. (DHHS publication no. (NIH)93-2789).

4. Whittemore AS, Lele C, Friedman GD, et al. Prostate-specificantigen as predictor of prostate cancer in black men and whitemen. J Natl Cancer Inst 1995;87:354-60.

5. Morrell CH, Pearson JD, Carter HB, et al. Estimating un-known transition times using a piecewise nonlinear mixed-effects model in men with prostate cancer. J Am Stat Assoc1995;90:45-53.

6. Whittemore AS, Keller JB, Betensky R. Low-grade, latentprostate cancer volume: predictor of clinical cancer incidence?J Natl Cancer Inst 1991,83:1231-5.

7. Leske MC, Ederer F, Podgor M. Estimating incidence from

age-specific prevalence in glaucoma. Am J Epidemiol 1981;113:606-13.

8. Wun LM, Merrill RM, Feuer EJ. Estimating lifetime andage-conditional risks of developing cancer. Lifetime DataAnal 1998;4:169-86.

9. Freeman J, Hutchison GB. Prevalence, incidence and duration.Am J Epidemiol 1980; 112:707-23.

10. Cowen ME, Chartrand M, Weitzel WF. A Markov model of thenatural history of prostate cancer. J Clin Epidemiol 1994;47:3-21.

11. Gail M. A review and critique of some models used in com-peting risk analysis. Biometrics 1975;31:209-22.

12. Forty-five years of cancer incidence in Connecticut: 1935-79.Natl Cancer Inst Monogr 1986;70:1-706.

13. National Center for Health Statistics: vital statistics of theUnited States, 1986. Vol II: Mortality, Part A. Washington,DC: US GPO, 1988. (DHHS publication no. (PHS)88-1122).

14. Yax LK. US Census Bureau, national population estimates(online), 1998. Available from Homepage of US CensusBureau: http://www.census.gov. Accessed 1998, April 21.

15. Potosky AL, Kessler L, Gridley G, et al. Rise in prostaticcancer incidence associated with increased use of transurethralresection. J Natl Cancer Inst 1990;82:1624-8.

16. Podgor MJ, Leske MC. Estimating incidence from age-specific prevalence for irreversible diseases with differentialmortality. Stat Med 1986;5:573-8.

17. Albert A, Gertman PM, Louis TA. Screening for the earlydetection of cancer. 1. The temporal natural history of aprogressive disease state. Math Biosci 1978;40:l-59.

18. Sakr WA, Haas GP, Cassin BF, et al. The frequency ofcarcinoma and intraepithelial neoplasia of the prostate inyoung male patients. J Urol 1993,150:379-85.

19. Feuer EJ, Wun LM. DEVCAN: Probability of DEVelopingCANcer software, version 3.11. Bethesda, MD: US Depart-ment of Health and Human Services, Public Health Service,National Institutes of Health, National Cancer Institute, 1994.

APPENDIX

Consider first the situation in which it is assumedthat new cases may not become clinically diagnosed.Let uo be the probability of competing mortality in theinterval. By assumption, uo is the same for new casesas for the population in general. To simplify mathe-matical development, we redefine our notation as fol-lows: Let n( 1 — uo) be the number of new cases whosurvive to the end of the interval (previously ri). As-sume that new cases cannot develop and become clin-ical in the same interval. Then, as in Leske et al. (7),the number of new cases who survive to their time ofpreclinical onset but die before the end of the intervalis approximately

m = n—.

Therefore, the total number of new cases who surviveto their time of preclinical onset is approximately

N = - u0) + m = n(l - uo/2).(Al)

Now suppose that new cases may become clinicallydiagnosed. Let co be the fraction of new cases whobecome clinically diagnosed by the end of the interval,

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and let n(l — MO)(1 - co) be the number of new caseswho survive to the end of the interval without dying orbeing clinically diagnosed. The number of new caseswho survive to their time of preclinical onset but thendie of other causes without being clinically diagnosedis approximately n{\ - co)uj2, and the number ofnew cases who survive to their time of preclinicalonset but then become clinically diagnosed beforepossibly dying of other causes is approximately «(1 —uol2)co. Therefore, the total number of new cases whosurvive to their time of preclinical onset is

N = n(l - «o)(l - c0) + nil - co)uol2

+ n{\ - uol2)co = n{\ - uo/2), (A2)

which is the same as expression 1. Therefore, regard-less of the probability that a new case may develop and

become clinically diagnosed in the same interval, thenumber of new cases in the numerator of the asymp-tomatic incidence remains the same. In the notation ofexpression 8, n and m may change, but their sumremains the same. Intuitively, this result may bethought of as follows: The cases making up the asymp-tomatic incidence in a given interval consist of allthose newly developing cases who survive to theirtime of preclinical onset. The only event that willprevent a new case from reaching his or her time ofpreclinical onset is competing mortality. However, thisis independent of the probability co that a new casewill become clinically diagnosed in the same intervalas his or her preclinical onset. Consequently, the num-ber of new cases who survive to their time of preclin-ical onset is effectively independent of co.

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