A stochastic model for survival of early prostate cancer with adjustments for leadtime, length bias, and over-detection

Abstract

To compare the survival between screen-detected and clinically detected cancers, we applied a series of non-homogeneous stochastic processes to deal with leadtime, length bias, and over-detection by using full information on detection modes obtained from the Finnish randomized controlled trial for prostate cancer screening. The results show after 9-year follow-up the hazard ratio of prostate cancer death for screen-detected cases against clinically detected cases increased from 0.24 (95% CI: 0.16–0.35) without correction for these biases, to 0.76 after correction for leadtime and length biases, and finally to 1.03 (95% CI: 0.79–1.33) for a further adjustment for over-detection. Adjustment for leadtime and length bias but no over-detection led to a 24% reduction in prostate cancer death as a result of prostate-specific antigen test. The further calibration of over-detection indicates no gain in survival of screen-detected prostate cancers (excluding over-detected case as stayer considered in the mover–stayer model) as compared with the control group in the absence of screening that is considered as the mover. However, whether the model assumption on over-detection is robust should be validated with other data sets and longer follow-up.

1 Introduction

Leadtime bias, length bias, and over-detection of cancers are important issues in the evaluation of prostate cancer screening (Morrison, 1985; Walter and Stitt, 1987). They arise from the sojourn time, which is the duration of the pre-clinical detectable phase (PCDP), assuming that the temporal natural history of the disease follows a three-state model in which an individual's disease status is normal prior to the development of the disease, then passes through a PCDP and finally to the clinical phase (CP) when the disease becomes symptomatic (Day and Walter, 1984). Leadtime is the amount of time by which the detection of a cancer is advanced by screening. Length bias is inherent from the fact that tumors have different sojourn times, depending on their aggressiveness, and leads to the phenomenon that screen-detected cancers, particularly those detected at first screen, tend to have longer sojourn times than interval cancers (cancers diagnosed symptomatically between screens). Over-detected cases are defined as cancers with a sojourn time equal to infinity i.e., cases that would not have been diagnosed if there had been no screening.

Prostate cancer is the most common cancer in men in many industrialized countries (IARC, 2003) and has a slow natural course. Analyses comparing the survival of clinically detected cases with the survival of those screen-detected, therefore, need to be adjusted for the above biases. First, the early detection of cases may simply advance the date of diagnosis without prolonging life (see cases 1 and 2 in Fig. 1 where the earlier diagnosis of case 1 leads to 5 years of artificial leadtime), resulting in leadtime bias. The mean leadtime for prostate cancer has been estimated to be between 5 and 12 years (Auvinen et al., 2002; Draisma et al., 2003; Draisma and de Koning, 2003; Etzioni et al., 2002; Gann et al., 1995; Hugosson et al., 2000; Stenman et al., 1994; Törnblom et al., 2004). Even if early detection due to screening with the prostate-specific antigen (PSA) test does genuinely prolong life, when no adjustment is made for leadtime the associated survival benefit will be exaggerated (for example, case 3 as opposed to case 2 in Fig. 1 has additional 10 years of survival after correction for a 5-year leadtime instead of 15 years without correction). Second, empirical data on screen-detected and interval cases of prostate cancer ascertained within a population-based screening program provide valuable information as regards length bias. Screen-detected prostate cancers, particularly those detected at the first screen, tend to have a longer sojourn time (see case 4 in Fig. 1) than those arising clinically before the first screen (case 5 in Fig. 1). Similarly, prostate cancers arising after the first screen are more likely to be detected at subsequent screens if they have a longer sojourn time (case 6 compared with case 7 in Fig. 1). Interval cancers (cases diagnosed clinically in the interval between screens following a negative screen) are not affected by leadtime bias and have shorter sojourn times (case 7 in Fig. 1). These scenarios suggest that the distribution of sojourn time for screen-detected cases is different to that for interval cancers. Third, previous statistical models to adjust for these biases when comparing the survival of screen-detected and clinical cases of prostate cancer have not taken over-detection into account. This is a major weakness as some screen-detected prostate cancers progress so slowly that they would never produce symptoms (case 8 in Fig. 1), a major issue in prostate cancer screening with the PSA test (Etzioni et al., 2002).

Figure 1. Illustration of the natural course of disease (by age) adjusted for leadtime bias, length bias, and over-detection as a result of population-based screening for prostate cancer with PSA tests.

Unfortunately, leadtime, sojourn time, and over-detection cannot be directly observed because medical treatment interrupts the natural course of screen-detected cases, leaving the key details (the times at which the disease entered the PCDP and CP) unknown. Sophisticated statistical models are therefore required to estimate the unknown variables.

To make adjustment for these biases, which make the survival of screen-detected prostate cancer non-comparable to that of prostate cancer detected in the absence of screening, we propose a series of non-homogeneous Markov models to estimate leadtime, sojourn time, and the extent of over-detection, using empirical data from two rounds of a screening trial in Finland and also demonstrate that without adjusting for these biases the benefit of prostate screening can be seriously overstated.

The article is organized as follows. Section 2 summarizes the data. Section 3 begins by defining the notation and non-homogeneous stochastic models. We then present three bias-adjusted models with a step-by-step presentation of how leadtime, length bias, and over-detection are corrected, and the data that are required. The results are in Section 4. Section 5 gives a discussion of the novelty, strengths and limitations of the proposed model and the limitations of the proposed model, with respect to its application to prostate cancer screening.

2 Data

The Finnish population-based prostate cancer screening trial, which is part of the European Randomized Study of Screening for Prostate Cancer (ERSPC), offers an opportunity to assess how the three types of bias might distort the evaluation of screening benefit. Beginning in 1996, 32 000 men were randomly allocated to the screening arm and another 48 458 men were randomly assigned to the control. Subjects in the screen arm were invited to 4-yearly PSA screening and men with a serum PSA concentration of 3.0 ng/mL or higher were referred for secondary screening test or diagnostic examinations. The study design of the trial has been detailed elsewhere (Finne et al., 2003).

Follow-up data were obtained from 20 796 subjects recruited at their first screen, and followed until prostate cancer diagnosis, death, emigration from the study area, or end of the second screening round. Cause of death was obtained from death certificates. All cancers detected in the screening arm were followed up from the date of diagnosis until death or to the end of 2005. In addition, follow-up data were obtained on 757 cases of prostate cancer diagnosed in the control arm during the first four years since randomization. Cases diagnosed later than this were excluded to avoid contamination due to opportunistic screening, which was much more common in the later phase of the trial. The classification of study participants by detection mode is shown in Fig. 2.

Figure 2. Classification of study participants by detection modes and the corresponding likelihood function (the details of notations, see below) for the length-bias-adjusted model including over-detected cases.

Screen-detected cases of prostate cancer were cases detected in the first (prevalence) or second (incidence) screening rounds. Clinically detected cases were cancers from the control arm. A proportion of these would have arisen because of opportunistic screening (contamination) so as to avoid overestimating the survival of clinical cases focal cancers (those stage T1c according to the TNM staging system) were excluded. Such cases are, by definition, neither palpable nor visible on imaging, but found by needle biopsy because of elevated PSA. Not all T1c tumors are detected due to opportunistic PSA screening though, because PSA testing is also performed for the differential diagnosis of benign prostate disease. Therefore, because the data did not distinguish between these two detection modes, the frequency of T1c tumors in the control arm was compared with that of the general population in the same period (using data from the Finnish Cancer Registry) and the “excess” T1c cancers in the control arm considered as focal cancers and randomly removed.

3 Methodology

3.1 Notations and model specification

Let be a discrete random variable representing a stochastic process depicting the evolution of prostate cancer over time t with state space , where

• (i)
  0=normal,

• (ii)
  1=pre-clinical detectable phase (PCDP),

• (iii)
  2=clinical phase (CP),

• (iv)
  3=prostate cancer death (PCa death), and

• (v)
  4=other causes of death (OCD).

Suppose we have k=1, 2,…, m screening rounds, where a_k denotes an individual's age at the k-th screen, or age at diagnosis if a cancer. Further, let a_kc denote age at diagnosis of interval cancer (prostate cancer diagnosed between screens due to clinical symptoms or signs), or one of cancers in the CP (state 2); a_kc is a (uncensored) time and lies between a_k and a_k+1. Finally, we use a_t to denote age at the end of follow-up.

The Markov model is shown in Fig. 3A and B, and is completed through specification of intensity matrices (Cox and Miller, 1965). We use two intensity matrices: (i) Q(·) for those with potential to progress to the CP (so-called mover); and (ii) Q^S(·) for those without potential to progress to the CP (so-called stayer). The reason for two matrices is to capture the over-detection problem: it is assumed that one group of individuals can never progress to state 2 (stayers) and that these are different from the group who do not progress to state 2 but could have done (movers)

• 

and

• 

where PCa refers to prostate cancer; λ₀(·), λ₁(·), and λ₂(·) represent the incidence rates of pre-clinical prostate cancer (state 0→state 1), the transition rate from the PCDP to the CP (state 1→state 2), which determines the distribution of sojourn time, and the hazard rate of prostate cancer death among prostate cancers in the CP (state 2→state 3); and u₀(·), u₁(·), and u₂(·) are three hazard rates of death from other causes (state 4) for respectively subjects in state 0, state 1, and state 2. Some transitions are not allowed, such as progression from the PCDP to the CP (state 1→state 2), see case 8 in Figs. 1 and 3B. Note that the reason of modeling all the transition rates (λ(·) and u(·)) as a function of transition time is to allow for a non-homogeneous stochastic model. The transition probability matrix, which is a function of the above intensity matrices and the time of transition history, can be derived from the backward Kolmogorov equation (Cox and Miller, 1965), and has been used for breast cancer screening (Chen et al., 2000; Wu et al., 2004).

Figure 3. Diagrams of Markov models of natural history and prognosis of prostate cancer adjusted for leadtime, length bias, and over-detection. (A) A Markov model combining the natural history and prognosis of prostate cancer. The model is similar to that for progressive prostate cancer cases, also known as movers. (B) A Markov model for non-progressive prostate cancer cases, also known as stayers. White arrows denote direct observations from the follow-up of screen-detected cases. Dotted arrows denote unobserved transitions, which divided the observed survival time into leadtime and post-leadtime survival time. OST: observed survival time; LT: leadtime; PST: post-leadtime survival time.

How the transition rates contained in the two matrices can identify the three biases of interest is delineated as follows. As the total survival time of prostate cancer in the PCDP includes the leadtime (LT) from the PCDP (state 1) to the CP (state 2) and post-leadtime survival time (PST) from the CP (state 2) to prostate cancer death (state 3), both survival times cannot be separated using empirical data only on the observed survival times (OST) without using a stochastic model (see Fig. 3A). The model parameters λ₁(·) and λ₂(·) correspond to the leadtime and the post-leadtime. For leadtime adjustment, we need to examine only the screen-detected prostate cancers.

For length-bias adjustment, because the leadtime distribution is derived from the sojourn time distribution, λ₁(·) can be used to obtain an unbiased estimate of the sojourn time distribution; this can then be used to adjust for length bias, provided the stochastic model is able to accurately capture the natural history of prostate cancer. This requires data on screen-detected prostate cancers (with long sojourn times) (e.g., cases 4 and 6 in Fig. 1) and interval cancers (short sojourn time) (case 7 in Fig. 1).

It should be noted that our definition of over-detection differs from the traditional clinical definition in which deaths from other causes occurring before the clinical diagnosis of prostate cancer are considered over-detection because it assumes that pre-clinical prostate cancers cannot be detected before death. The drawback of this assumption is that other causes of death in subjects free of symptomatic cancer include three possibilities: (i) non-susceptible to prostate cancer (not allowed to enter into the PCDP); (ii) non-progressive prostate cancer (not allowed to progress from the PCDP to the CP); and (iii) progressive prostate cancer but not surfacing to the CP before dying from other causes. Prostate cancers in the PCDP are occult and the three possibilities cannot be distinguished using the conventional definition. By using data from a population-based randomized controlled trial, our model is able to separate these three possibilities by using the parameters of λ₀(·), λ₁(·), u₀(·), u₁(·), and the proportion of Q and Q^S matrices that captures over-detection (see the over-detection adjustment model described below). The two parameters, λ₀(·) and u₀(·), in each matrix are tailored for identifying susceptibility to prostate cancer. The parameter of λ₁(·) in Q but not in Q^S together with u₁(·) are intended to separate progressive prostate cancers (e.g. from the PCDP to the CP) from non-progressive ones. The empirical data on the control group are further required for this part of the analysis.

3.2 Bias-adjusted model

In this section we present the step-by-step explanations of how much information can be gleaned from population-based cancer screening. We start with a leadtime-adjusted model based on screen-detected cases only. We then extend the model to consider length bias using data from screen-detected and interval cancers. The final model took over-detection into account by further exploiting data from the control group. The details of each model are described below.

3.2.1 Leadtime bias adjustment model

In this leadtime bias adjustment model, the total survival time for screen-detected cases is decomposed into leadtime and post-leadtime. We use a Markov model, based on the natural history part of Fig. 3A, to estimate the expected leadtime using screen-detected cases and normal subjects. Recall that λ₀(·) and λ₁(·) represent the pre-clinical incidence rate (from state 0 to state 1) and the instantaneous transition rate from the PCDP (state 1) to the CP (state 2), respectively. Additionally, since only screen-detected cases have leadtime gains, the data we use here to estimate the leadtime comprise no clinically detected cases, i.e. interval cancers. Using the method proposed by Chen et al. (2000) and Wu et al. (2004), λ₀(·) and λ₁(·) can be estimated without relying on interval cancers. After obtaining the estimate of the annual transition rate from the PCDP to the CP, λ₂(·), the prognosis part was incorporated into the Markov model (the gray part in Fig. 3A) to estimate post-leadtime survival based on screen-detected cases only. A naïve approach is to approximate post-leadtime survival (PST, the dotted arrow in Fig. 3A) by subtracting the estimated leadtime (LT) from observed survival time (OST, the white arrow in Fig. 3A), which starts from the PCDP until the end of follow-up, including death from prostate cancer, other causes, and censored status, whichever came first.

To accommodate the possibility of a non-constant hazard for the risk of prostate cancer death, a Weibull distribution was used to model post-leadtime survival. The transition rate, which is dependent on age after correction for leadtime, can be expressed as

• 

where λ₂₀ is the scale parameter, and γ₂ is the shape parameter of the Weibull distribution. To estimate the parameters of the Markov model for the natural history and prognosis parts shown in Fig. 3A, data on the follow-up of the screen-detected cases are required. When the distributions of sojourn time and the CP are specified by the exponential and Weibull distributions, respectively, at the end of follow-up at age a_t from a subject detected by screen at age a_k the probability of being alive is

• (1)

the density function of being dead from prostate cancer is

• (2)

and the density function of being dead from other causes is

• (3)

where u₁(·) is the hazard rate from the PCDP to other causes of death, P₁₁(a_k, a_t) is the probability of staying in state 1 (PCDP) from a_k to a_t, P₁₂(a_k, a_t) is the probability of transition from state 1 (PCDP) at a_k to state 2 (CP) and staying in state 2 up to a_t, and f₁(·) and f₂(·) are the probability density functions for death from prostate cancer and death from other causes at time a_t, respectively.

3.2.2 Length-bias adjustment model

Different types of prostate cancer may have different sojourn times. As the distribution of leadtime depends on the distribution of sojourn time, the estimate of λ₁(·) in the above leadtime adjustment model is biased (because screen-detected cancers tend to have longer sojourn times). In order to get an unbiased estimate of λ₁(·) as well as an unbiased estimate of the hazard rate of λ₂(·), we incorporate interval cancers, which tend to have shorter sojourn times.

A five-state stochastic process (Fig. 3A) is proposed to estimate transition parameters related to the leadtime, making allowance for different estimates for the sojourn time by combining screen-detected cases (long sojourn times) and interval cancers (short sojourn times). The post-leadtime estimate can be obtained from prognosis part of the model based on screen-detected cases only.

Without assuming any particular distribution for the transition rates, λ's, in the second (k=2) screen at a₂ the probability of remaining in the normal state since the first screen until the second screen is denoted by

• (4)

and the probability of being diagnosed as a screen-detected case at a₂ is

• (5)

where P₀₀ and P₁₁ are, respectively, the probabilities of staying in state 0 (normal) and state 1 (PCDP).

The density function that incorporates the probability of interval cases (short sojourn time) diagnosed at age a_kc can be expressed as

• (6)

where f_c (a_kc) is the derivative of the probability of being negative at a_k (age at the k-th screen) and being diagnosed clinically due to symptoms and signs at a_kc.

For the first screen, the probability of being screened as normal or asymptomatic prostate cancer has to be conditional on time until the disease is clinically recognized being longer than the time to screen. Hence the conditional probability of being screened as normal or as an asymptomatic case at the first screen can be expressed as

• (7)

• (8)

respectively, recalling that a₁ is the age at the first screen.

We assume that the rates of entering the PCDP, of the transition from the PCDP to the CP, and of the transition from the CP until prostate cancer death follow three Weibull distributions (Weibull (λ₀₀, γ₀), Weibull (λ₁₀, γ₁), and Weibull (λ₂₀, γ₂) as functions of age, and that three rates from normal, PCDP, and CP to other causes of death follow three exponential distributions. Weibull distribution, which allows for the transition rate changing with time, can accommodate non-homogeneous Markov model. Then Eqs. (4)–(6) can be rewritten as

• (9)

• (10)

• (11)

The probabilities of having three outcomes for the follow-up of screen-detected cases, include

• (12)

for the subject still alive

• (13)

for prostate cancer death and

• (14)

for other causes of death.

Note that if γ equals 1, Eqs. (9)–(14) are reduced to exponential distributions. Here, we also assume that u₀(·), u₁(·), and u₂(·) follow exponential distribution. Although the proposed non-homogeneous model can be parameterized by the application of a Weibull distribution, estimation of parameters is time consuming and the estimates might have a large variance. For these reasons, we also adopted a piecewise method. Because the derivation of piecewise model is straightforward but laborious, the formulae used for the piecewise method are given in Appendix A.

In our first analysis only the last phase followed a Weibull distribution whereas the times from normal to entering the PCDP, and from the PCDP to the CP, were modeled as exponential distributions. Following expressions (1)–(8) for the model with two exponential distributions, Eqs. (9)–(14) for the Weibull model, or equations for the piecewise method (see Appendix A), the likelihood function for the model in Fig. 3A is

• 

where and are the ages at the first and k-th screen for the i-th subject, , and are the ages at diagnosis of interval cancer, and the age at the end of follow-up for the i-th subject, respectively. and are indicator variables if the i-th subject was normal or screen-detected at the first screen. The counterparts for the k-th screen are and . The indicator of denotes interval cancer. , , and are indicator variables for screen-detected cases who are, respectively, alive, dead from prostate cancer, and dead from other causes, after the k-th screen for the i-th subject.

Based on the estimated transition parameters of λ₂₀ and λ₂, the prostate cancer survival function after correcting for leadtime and length bias, simultaneously making allowance for other causes of death, can be calculated as

• (15)

where a_k is age at the k-th screen, and a_t is age at the end of follow-up.

3.2.3 Length-bias model with over-detection

3.2.3.1 Model Formulation

Although the length-bias model considers different progression rates of natural history, it cannot capture the heterogeneity stemming from the over-detection of cases that would not have come to light in the absence of screening (Fig. 3B). The corresponding sojourn times are therefore defined as infinity. The estimate of post-leadtime would be biased if these cases are not singled out.

To account for these over-detected cases, the length-bias model was extended to a mover–stayer model by using two intensity matrices. Recall the “mover” who has the potential to progress from the PCDP to the CP (Fig. 3A) has a Q(·) matrix, whereas the “stayers” who are not capable of moving to the CP and would never die from prostate cancer (Fig. 3B) has a Q^S(·) matrix. We therefore constructed two Markov models, one for the movers (Fig. 3A) and the other for the stayers (Fig. 3B).

Equations (4)–(11) are still used to define the likelihoods for the movers with long or short sojourn times. The conditional probabilities of stayers being screened as normal or as asymptomatic cases at the first screen at a₁ shown in Eqs. (7) and (8) can be expressed as

• (16)

• (17)

respectively, where

• (18)

in which P₀₀ is the probability of staying in the normal stage for both movers and stayers, and is the probability of staying in the PCDP specifically for stayers (and different from that for movers).

It should be noted stayers cannot be distinguished from movers who are still alive at the end of study period (censored cases). To tackle this problem, we propose a latent-variable approach, where the proportion of stayers is . In other words, we classify the underlying population into two parts, those with potential for progression (movers), and those without potential for progression (stayers), depending on the k-th screen. The proportion of stayers changes with each screening round because pre-clinical cancers that were detected have been removed from the screened population at the previous screen. Therefore, the likelihood for being screened as normal or as asymptomatic cases for both movers ( and , referring to Eqs. (7) and (8)) and stayers ( and , referring to Eqs. (16) and (17)) at the first screen can be expressed as

• (19)

• (20)

where π₁ is the proportion of stayers among the invited asymptomatic population at first screen.

Consequently, the probability for being screened as normal or as asymptomatic case at the later screens can be expressed as

• (21)

• (22)

Furthermore, because stayers would not progress to the CP, the conditional probability for stayers progressing to the CP equals zero. Hence, the probability for interval cases with reference to Eq. (16) can be expressed as

• 

Regarding the likelihood of follow-up of prostate cancer cases detected at the first screen, the probability and the density functions can be derived by weighting Eqs. (12)–(14) with the proportion of stayers among screen-detected cases at the k-th screen, denoted by π_Ck, as

• 

for being alive at, where , which is the probability of staying in state 1 (PCDP) of stayers

• 

for being death from prostate cancer, and

• 

for being death from other causes of death, where

• 

Taking competing risks into account, the prostate cancer survival rate for movers is

• (15)

It should be noted that Eq. (23) is tailored for the comparison of the survival of screen-detected cases after correcting length bias and over-detection with an unscreened control group. The likelihood of each detection mode is in the footnote of Fig. 2.

Survival for all screen-detected cases of prostate cancer, taking account of competing risks is

• (24)

where π_Ck represents the proportion of stayers among the screen-detected cases at the k-th round of screening. The formulae for π_k and π_Ck are detailed in Appendix B. The cumulative risk of prostate-cancer death since diagnosis was calculated for screen-detected and clinically detected cases.

Using Eqs. (15), (23), and (24), we computed the expected number of deaths from prostate cancer after correcting for leadtime, length and over-detection bias.

3.2.3.2 E-M algorithm procedure

To reduce the identifiability problem that result from having correlated parameters, we adopted an expectation-maximum (E-M) likelihood algorithm iteration technique (Dempster et al., 1977) to estimate π_C1, λ₀, λ₁, λ₂, and λ₃. The iterative procedures are detailed in Appendix C.

3.3 Other methods used

Uncorrected survival analysis between screen-detected and clinically detected prostate cancers was first implemented. Prostate cancer survival analysis was performed using the life-table method, without adjustment for leadtime, length bias, and over-detection. The hazard ratios of death from prostate cancer for screen-detected cases compared with clinically detected cases and their 95% confidence intervals (CI) were calculated. Survival after correcting leadtime, length bias, and over-detection were also calculated to compare with the uncorrected survival.

4 Results

There were 1122 screen-detected and 757 clinically detected cases with a mean follow-up of 5.3 years after diagnosis (SD 2.2, median 5.4, Table 1), after 87 T1c cancers were excluded for reasons described in Section 2. The 9-year prostate cancer survival was 92.1% for screen-detected and 79.3% for clinically detected cases (Fig. 4). A lower case-fatality rate was observed in screen-detected cases compared with clinically detected cancers after 9-year follow-up, with a hazard ratio of 0.24 (95% CI: 0.16–0.35).

Figure 4. Prostate cancer survival curves of screen-detected and clinically detected prostate cancer cases adjusted for leadtime bias, length bias, and over-detection in the Finnish population-based prostate cancer screening randomized controlled trial, 1996–2005.

Table 1. Number of prostate cancer cases by detection mode and year of diagnosis identified between 1996 and 2003. Follow-up occurred upon death from prostate cancer or until the end of 2005. Subjects were enrolled in the Finnish population-based prostate cancer screening randomized trial.

Year of diagnosis	Screen detected			Clinically detected (control group)
	Population	PCaa)	PCa deathsb)	Population	PCa	PCa deaths
a) ^a) PCa: Prostate cancer. b) ^b) PCa deaths: The number of prostate cancer cases diagnosed in the specified year and who died during the study period.
1996	5051	130	12	12 252	28	10
1997	5256	155	3	22 980	69	15
1998	5378	132	8	34 944	124	19
1999	5111	170	8	46 606	179	23
2000	3833	89	2	34 352	162	12
2001	4144	148	2	23 629	120	8
2002	4240	175	2	11 659	75	4
2003	4028	123	1	–	–	–
Total	37 041	1122	38	186 420	757	91

4.1 Leadtime and length-bias adjustment model

Based on the Markov model in Fig. 3A assuming the two exponential distributions for λ₀ and λ₁, the annual progression rate from the PCDP to the CP was estimated as 0.19 (95% CI: 0.17–0.21), and the mean leadtime was 5.24 (95% CI: 4.82–5.74) years after adjustment for length bias (Table 2). The scale and shape parameters for the rate of prostate cancer death following the Weibull distribution were estimated as 6.14×10⁻⁶ (95% CI: 4.96×10⁻⁶–7.32×10⁻⁶) and 2.65 (95% CI: 2.57–2.73). The shape parameter implies that the hazard of prostate cancer death increased with age.

Table 2. Estimated progression rates (per year) of prostate cancer cases, adjusted for both leadtime and length bias. The extended model with over-detection adjustment is also shown. Subjects were enrolled in the Finnish population-based prostate cancer screening randomized trial, 1996–2005.

Parameters	Estimate	95% CIa)
a) ^a) CI, confidence interval. b) ^b) The proportion of over-detection, 40.45%, was estimated using a mover–stayer model.
(1) Leadtime- and length-bias-adjustment model I
Pre-clinical incidence rate (λ0) (Exponential)	0.0075	0.0070–0.0081
Annual progression rate (λ1) (Exponential)	0.1908	0.1741–0.2074
Rate of prostate cancer death (λ2) (Weibull)
Scale (λ20)	6.14×10^−6	4.96×10^−6–7.32×10^−6
Shape (γ2)	2.6496	2.5673–2.7329
Rate of other causes of death (u1) from PCDP	0.0230	0.0184–0.0276
(2) Leadtime- and length-bias-adjustment model II
Pre-clinical incidence rate (λ0) (Piecewise)
55–58 y/o (λ01)	0.0079	0.0062–0.0096
59–62 y/o (λ02)	0.0070	0.0060–0.0079
63–66 y/o (λ03)	0.0094	0.0083–0.0106
67+ y/o (λ04)	0.0082	0.0072–0.0092
Annual progression rate (λ1) (Piecewise)
55–62 y/o (λ11, λ12)	0.2821	0.2378–0.3264
63+ y/o (λ13, λ14)	0.1218	0.1025–0.1410
Rate of prostate cancer death (λ2) (Weibull)
Scale (λ20)	5.59×10^−6	1.94×10^−6–9.23×10^−6
Shape (γ2)	2.7243	2.5665–2.8821
Rate of other causes of death (u1) from PCDP	0.0195	0.0156–0.0235
(3) Extended model with over-detection adjustmentb)
The proportion of stayer	40.45%	31.95–48.95%
Pre-clinical incidence rate (λ0) (Piecewise)
55–58 y/o (λ01)	0.0009	0.0006–0.0013
59–62 y/o (λ02)	0.0047	0.0040–0.0054
63–66 y/o (λ03)	0.0069	0.0059–0.0078
67+ y/o (λ04)	0.0119	0.0108–0.0130
Annual progression rate (λ1) (Piecewise)
55–62 y/o (λ11, λ12)	0.1376	0.0964–0.1787
63+ y/o (λ13, λ14)	0.1340	0.1110–0.1570
Rate of prostate cancer death (λ2) (Weibull)
Scale (λ20)	2.01×10^−5	2.19×10^−8–4.02×10^–5
Shape (γ2)	2.5086	2.2855–2.7318
Rate of other causes of death
From normal (u0)	0.0094	0.0087–0.0101
From PCDP (u1)	0.0159	0.0127–0.0191
From CP (u2)	0.0213	0.0164–0.0261

By relaxing the assumption of two exponential distributions for λ₀ and λ₁ with two Weibull distributions, the estimates were as follows: λ₀, scale (λ₀₀)=2.54×10⁻⁴ (95% CI: 0–6.72×10⁻⁴), shape (γ₀)=1.82 (95% CI: 1.42–2.23); λ₁, scale (λ₁₀)=30.86 (95% CI: 0–81.81), shape (γ₁)=0.11 (95% CI: 0–0.23); and λ₂, scale (λ₂₀)=1.33×10⁻⁵ (95% CI: 2.95×10⁻⁶–2.36×10⁻⁵), shape (γ₂)=2.76 (95% CI: 2.55–2.97).

It can be seen that the confidence interval was wide for scale of λ₁. This might suggest unstable estimation. The alternative considered in Section 3 was to apply a piecewise method to allow for non-constant hazards. Table 2 lists the estimates from the piecewise method. The results were very similar to using the three Weibull distributions, but the estimates were more tightly estimated.

The leadtime and length-bias corrected analysis yielded a hazard ratio of 0.61 (95% CI: 0.45–0.81) for prostate cancer death of screen-detected versus clinically detected prostate cancer (Table 3) assuming two exponential distributions. After the relaxation of these exponential assumptions using piecewise method, the hazard ratio was elevated to 0.76 (95% CI: 0.58–1.00), which was close to 0.79 (95% CI: 0.60–1.04) using three Weibull distributions (data not shown).

Table 3. Hazard ratios for prostate cancer death and the corresponding 95% CI in screen-detected prostate cancer cases compared with clinically detected cases. The models adjust for both leadtime and length bias using constant and non-constant hazard models. The extended model with over-detection adjustment is also shown. Subjects were enrolled in the Finnish population-based prostate cancer screening randomized trial, 1996–2005.

Years of follow-up	Without adjustment		Both leadtime and length bias adjustment, Model I (Constant hazards)		Both leadtime and length bias adjustment, Model II (Non-constant hazards)		Leadtime, length bias, and over-detection adjustment (over-detection: 40.45%)a) (Non-constant hazards)
	HRb)	95% CI	HR	95% CI	HR	95% CI	HR	95% CI
a) ^a) The proportion of over-detection, 40.45%, was estimated using a mover–stayer model. b) ^b) HR: hazard ratio.
1	0.21	0.08–0.58	0.59	0.28–1.24	0.84	0.43–1.67	1.19	0.62–2.25
2	0.10	0.04–0.24	0.52	0.32–0.84	0.69	0.44–1.08	0.96	0.63–1.46
3	0.15	0.08–0.28	0.59	0.40–0.89	0.76	0.52–1.11	1.06	0.74–1.51
4	0.16	0.10–0.28	0.57	0.40–0.82	0.75	0.54–1.05	1.02	0.74–1.40
5	0.18	0.11–0.29	0.56	0.41–0.78	0.74	0.54–1.00	0.99	0.74–1.32
6	0.21	0.14–0.31	0.57	0.42–0.78	0.73	0.55–0.97	0.98	0.75–1.29
7	0.22	0.15–0.33	0.59	0.44–0.80	0.74	0.56–0.99	1.01	0.77–1.32
8	0.24	0.16–0.35	0.61	0.45–0.81	0.76	0.57–1.00	1.03	0.79–1.34
9	0.24	0.16–0.35	0.61	0.45–0.81	0.76	0.58–1.00	1.03	0.79–1.33

4.2 Adjustment for over-detection

During the study period, 128 interval cancers (n_IC) out of 20 209 men with negative finding at first screen were ascertained and 168 prostate cancers (n_RE) were diagnosed among 11 073 refusers. Additionally, 587 subjects were identified with prostate cancers in the first rounds of screen (n_SD) from 20 796 attendees to the first screen. After nine iterations, the proportion of over-detection among first screen-detected cases, π_C1, converged to 40.45% (95% CI: 31.95–48.95%). For proportional-hazards regression analysis, if over-detected cases were included in the light of Eq. (24), the hazard ratio was 0.59 (95% CI: 0.44–0.79). After allowing for over-detection in conjunction with leadtime and length-bias adjustment pursuant to Eq. (23), the hazard ratio of prostate cancer death for all screen-detected versus clinically detected prostate cancer was inflated to 1.03 (95% CI: 0.79–1.33) (Table 3).

4.3 Comparison of adjusted and unadjusted survival curves

The predicted survival curves corresponding to step-by-step adjustments for leadtime, length bias, and over-detection in turn are shown in Fig. 4. The survival curve after correcting for leadtime and length bias was lower than the observed survival curve from the screen-detected cases with or without relaxing the assumption of exponential distributions. Including over-detected cases (stayer), the corrected survival was close to the leadtime and length-bias-adjusted curve. However, the survival curve for progressive prostate cancer (mover) only with adjustment for leadtime and length bias was close to that of clinically detected cases from the control group.

5 Discussion

We proposed a series of non-homogeneous stochastic models to estimate the mixture proportion of over-detection (non-progressive prostate cancers) and transition parameters governing the temporal natural history of progressive prostate cancer. By relating these estimates to the prognosis part of prostate cancer death making allowance for other causes of death, the benefit of screening was assessed by comparing the survival of screen-detected cancers with that of clinically detected cancers after correcting for leadtime, length and over-detection bias simultaneously. In contrast to previous statistical methods, our model not only provides a substantial advance in the statistical modeling used in this area, but also provides the novelty of estimating three biases simultaneously by making use of data from a population-based screening program.

Our model was applied to data from the population-based Finnish prostate cancer screening trial. The uncorrected 9-year survival among the screen-detected cases was substantially higher than the clinically detected cases. However, the impression of a substantial screening benefit was distorted by leadtime and length bias, and over-detection: the results reporting the impact of screening frequently published in the literature should be interpreted with caution (Brenner and Arndt, 2005; Efstathiou et al., 2006; Jang et al., 2006; Paquette et al., 2002; Quaglia et al., 2003; Tarone et al., 2000; Ung et al., 2002). Correction of survival rates for leadtime, length bias or over-detection using our proposed model is important to help assess the true benefit of screening on the survival.

From a methodological viewpoint the proposed model has several advantages in comparison to the model proposed by Walter and Stitt (1987). First, our model is more flexible because it relaxes the assumption of exponential distributions with a constant hazard, not only for the hazard rate of death, but also for age-dependent pre-clinical incidence rates, and the leadtime distribution by using the Weibull distribution or a piecewise-constant hazard instead. Other distributions with non-constant hazards such as gamma distribution could be considered in the future. This point is important for tumors such as prostate cancer having age-dependent pre-clinical incidence rate, non-constant sojourn times, and non-constant hazard rate of death over time. Second, we also incorporated competing risks. As mentioned above, modeling other causes of death together with our stochastic model for natural history of prostate cancer can separate subjects with non-susceptible to prostate cancer from non-progressive prostate cancer. The proportion of stayers was increased to 62% if other causes of death from normal subjects were not taken into account. Third, our model can deal with leadtime, length and over-detection bias simultaneously, which has not been possible with previous methods. More importantly, length bias has been corrected by considering different detection modes including the incorporation of interval cancers (representing tumors with short sojourn times), and with an adjustment for over-sampling of tumors with long sojourn times at the first screen. An unbiased estimate of sojourn time was then applied to correcting the survival among the screen-detected cases.

Our approach can be compared with an intention-to-treat analysis. This is tailored for evaluation of effectiveness of population-based cancer screening which requires the denominator. In our study, we did not have this objective. However, we calculated the prostate cancer mortality rate by dividing the number of prostate cancer deaths with person–year at risk of all study subjects instead of only including the prostate cancer cases, and the rate ratio of the screen arm was 1.05 (95% CI: 0.81–1.35) compared with the control arm. This indicates no screening benefit. To do intention-to-treat analysis, the date of randomization was used as starting point of survival time which would not lead to leadtime bias, length bias, and over-detection to calculate the unbiased estimate for the effectiveness of screening. The results of intention-to-treat are close to our final model estimate, 1.03, based on prostate cancer cases. This corroborates our proposed model for leadtime, length bias, and over-detection. However, whether the similar results would be seen in other population-based service screening programs should be verified.

An advantage of using a parametric method to model the survival is that the predicted long-term survival after correcting leadtime and length bias can be easily derived from the parameter estimates. The predicted 10-year, 15-year, and 20-year survival adjusted for leadtime bias, length bias, and over-detection were calculated as 76.9, 67.5, and 59.4%. These estimates are consistent with those reported in the literature for early, moderately differentiated prostate cancer (Albertsen et al., 2005; Johansson et al., 1997, 2004), which further supports the validity of our results. However, as our follow-up period is not long enough these predicted results should be validated in the future. Moreover, our mover–stayer model that accounts for over-detection can also estimate the survival of the mover after correcting leadtime and length bias. This information is very useful for dealing with over-detection, which is a major challenge in evaluation of prostate cancer screening.

There are several concerns that should be addressed. First, the analysis is subject to the model being right, in particular that over-detection can be modeled using a mover–stayer model, and the assumptions made following the proposed model. The assumed distributions on pre-clinical incidence rate, sojourn time, and rate of prostate cancer death play a crucial role in leadtime and length-bias adjustment. The survival benefit was reduced from 39 to 24% (see Table 3) when the constant hazards on three distributions were relaxed by considering non-constant hazards with the piecewise method or the Weibull distribution. This means the results may be different if various distributions are applied. Since the results based on the piecewise method were close to those using three Weibull distributions that are able to accommodate different shapes of hazard function we believe the survival benefit calibrated with leadtime and length bias using the piecewise method may slightly different from those using other distributions. Another key assumption in our mixture mover–stayer model is that the stayer is modeled to represent over-detected cases that are characterized by non-progressive prostate cancer with infinite sojourn time. The proportion of stayers was captured by applying this mixture model to both data from the screening arm (including stayer and mover) and the control arm (only mover). More importantly, if the mixture model assumption is not robust, then the estimated benefit of survival might be changed. As was seen in Table 3, the survival benefit as a result of PSA test after correcting for leadtime, length bias, and over-detection lies between 0 and 24% depending on whether and the extent of over-detection is considered. In Fig. 4, we also see the survival benefit would be overestimated and was close to that with adjustment for leadtime and length using exponential distribution without considering over-detection when the stayers were included. In addition to different distributions applied to leadtime and length bias, the interpretation of this finding should be taken with great caution. Note that over-detection resulting from non-progressive cancers is not equivalent to the cured progressive prostate cancer due to early detection through screening and then early treatment as seen in 24% reduction in prostate cancer death after correction for leadtime and length bias using the piecewise method. Over-detection of non-progressive cancers would not have been detected in the absence of screening like the control arm in the randomized controlled trial. Therefore, this provides a justification for excluding stayers when the comparison of survival was made between the screen-detected cancers and clinically detected cancers from the control group that only consists of the mover according to our definition of mover–stayer model. Second, since the mean sojourn time is generally long for the temporal natural history of prostate cancer, 9-year follow-up data may not be sufficient to reflect true benefit of PSA screening after correcting for these three biases. In particular, insufficient follow-up time may render the distinction impossible between the stayer and the mover with sojourn time longer than follow-up time (censored cases of natural history) that are possible to be cured if they can be detected by screening and undergo early treatment. This would further obscure the benefit of screening. Longer follow-up data are required to validate the efficacy after the correction of these biases. Third, the true time entering the CP is not known. It might be affected by the awareness of subjects for interval cancer, or cancers from non-participants to render the observed time to the CP earlier or later as compared with true time to enter the CP (delayed or never diagnosed). As both these are likely, the expected time to enter the CP would not be substantially affected. Moreover, since the control group, clinically detected cancer used for the comparator of survival, has similar problem, the bias caused by this concern might be balanced.

In conclusion, the proposed stochastic model is a useful method to correct for leadtime, length and over-detection biases when comparing survival between screen-detected and clinically detected cases. Uncorrected estimates of survival overstate the benefit of screening. We demonstrated that the observed gain in survival resulting from screening after nine years of follow-up of screen-detected prostate cancers from the Finnish PSA-based trial possibly only reflects leadtime, length bias, and over-detection. An approximate 24% reduction in Pca death due to PSA test was achieved if there is lacking of over-detection. The further calibration of over-detection indicates no gain in survival of screen-detected prostate cancers in comparison with the control group in the absence of screening (movers) when over-detected cases (stayers) were excluded. However, whether the model assumption on over-detection is robust should be validated with a longer and different follow-up data.

Acknowledgements

This work was financially supported in part by grants from the Finnish Distinguished Professor (FiDiPro) Academy of Finland, Tampere University Hospital Research Fund, and the Taiwan National Science Council (Grant No. NSC 91-2320-B-002-215; NSC 94-2314-B-002-106; NSC 97-2314-B-002-019-MY3). The authors thank Drs. Jane Warwick and Adam Brentnall from the Wolfson Institute of Preventive Medicine, Queen Mary University of London for their help with technical English editing.

Conflict of interestThe authors declare no conflict of interest.

Appendix A: Piecewise method

Four piecewise estimates for the pre-clinical incidence rate λ₀(·) corresponding to four age groups are denoted by (r=1, 2, 3, 4): for ages 55 years or younger, for ages 56–59 years, for ages 60–63 years, and for ages 64 years or older. Four piecewise estimates for λ₁(·) corresponding to age groups are denoted by in a similar manner. The lower and upper limits of age in the r-th age group are denoted by L^r and U^r, respectively. To reduce the number of parameters involved in the estimation, we chose two piecewise parameters of λ₁(·), with the following assumptions: and , indicating that the mean sojourn time for the age group ≤55 years was equal to that for the age group 56–59 years and that the mean sojourn time for the age group 60–63 years was equal to that for the age group ≥64 years. As the mean sojourn time for prostate cancer is at least 5 years, according to the literature, such an assumption was reasonable and contributed to a parsimonious model.

We derived the transition probabilities and density functions used for formulating the likelihood function based on this piecewise method, as follows, using various detection methods and patient outcomes at follow-up.

• (i)
  First screen

The expression of P₀₀(a₁), the numerator in Eq. (7) and denominator in Eqs. (7) and (8), for subjects attending their first screen at 55 years (a₁=55) or at 67 years (a₁=67) of age are expressed as

• (25)

where r=2, 3, 4 correspond to the three age groups 56–59, 60–63, and ≥64 years, respectively.

We derived similar expressions for P₀₁(a₁)

• 

and

• (26)

Note that the value of 4 is the fixed interval between the lower and upper ages of each age group.

• (ii)
  Subsequent screens

For a subject attending the k-th and (k+1)-th screens at ages a_k and a_k+1, respectively, with the regular inter-screening interval denoted by x (=a_k+1−a_k), which is not longer than 4 years, the piecewise changing point is determined by c (=U^r−a_k). The corresponding Eqs. (9) and (10) are then expressed as

• (27)

and

• (28)

• (iii)
  Interval cancer

For interval cancer cases diagnosed at age a_kc after the k-th screen, the piecewise probability density for interval cancer is dependent on U^r, a_k, and a_kc.

• (a)
  When a_kc≤U^r

• (29)

where c=a_kc−a_k.

• (b)
  When a_kc>U^r

• (30)

where c₁=U^r−a_k, c₂=a_kc−U^r.

• (iv)
  Follow-up

Suppose that a screen-detected prostate cancer diagnosed at age a_k was tracked until age a_t. Again, we consider two scenarios:

• (a)
  When a_t≤U^r

The probability for subjects still alive is

• (31)

where c=a_t−a_k.

The probability density function for prostate cancer death is

• (32)

and the probability density function for other causes of death is

• (33)

• (b)
  When a_t>U^r

The probability for subjects still alive is

• (34)

the probability density function for prostate cancer death is

• (35)

and the probability density function for other causes of death is

• (36)

where c₁=U^r−a_k, c₂=a_t−U^r.

Appendix B

Following Eqs. (4), (5), (7), (8), and (16)–(22), the proportion of stayers in the remaining population at the second screen (k=2) is

• (37)

Regarding the likelihood of follow-up of prostate cancer cases, the proportions of stayers among asymptomatic cases detected at the first screen (π_C1) and at later screens (π_C2) are expressed as

• (38)

and

• (39)

respectively.

Appendix C

At the initial step (maximization step), by setting zero as the initial value of the proportion of stayers among screen-detected cases at the first screen, , the estimates of , , , and , were derived using empirical data by the maximum-likelihood approach through a set of mediator parameters such as , , and . At the second step (expectation step), we proposed an expectation equation to derive the estimate of as follows. Because subjects were randomized, the risk of being diagnosed as a prostate cancer case should be equal in the control and screen arms if no screening had taken place in the screen arm. Interval cases (n_IC) and prostate cancer cases among subjects who refused to attend screening (n_RE) would still be diagnosed if they were in the control arm. Additionally, during the 4-year inter-screening interval, a proportion (1−π_C1) of screen-detected cases (n_SD) would also be diagnosed due to symptoms or signs with a probability of P₁₂(4), i.e., the probability of transition from PCDP to CP during 4 years, if they had not been screened. Furthermore, to correct for contamination, a proportion (f) of the cancer cases were excluded from the prostate cancer cases in the control arm in the first 4 years after randomization, n_C. The value f was determined by comparing the T1c distribution in the control arm with that in the general population, as described above. Therefore, the equation can be written as

• (40)

The estimate of is derived by the solution of the above expectation equation. This completed the first iteration. In the second iteration, given , the estimates of , , , and , can be derived by using the maximum-likelihood approach, and then can be derived. The iteration procedures were repeated until the convergence criterion was achieved, namely the difference between two iterations of each parameter was less than 10⁻⁴.

Furthermore, let P denote the proportion of screen-detected cases among total cases in the study arm (n_SD/n_s). By re-parameterization of n_c/n_s (=K×P) as K-fold of P, the variance of π_C1 can be easily derived by using the delta method, assuming N_S, N_C, and f are constant, and P and K are independent.