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Abstract

Objective

In evaluating the efficacy of cancer screening programmes, sojourn time (duration of the preclinical detectable phase) and sensitivity of the screening test are the two key parameters. Studies suggest that in breast cancer screening, both parameters may vary depending on age at the time of screening, but few studies have examined other cancers. We expanded an existing probability model for periodic screening by performing simultaneous estimation of age group-dependent and sensitivity at preclinical onset time, and tested the expanded model using data from the Korean National Colorectal Cancer Screening Programme.

Methods

Simulation studies were conducted to assess the performance of the proposed probability model. The method was then applied to the analysis of 376,542 participants aged 50 or over who underwent fecal occult blood testing (FOBT) as part of the National Colorectal Cancer Screening Programme between 2004 and 2007. Age group-dependent mean sojourn time and screening sensitivity of FOBT for colorectal cancer were derived using maximum likelihood estimation.

Results

The method performed well in terms of bias, standard deviation, and coverage probability. National Colorectal Cancer Screening Programme data results indicated that the sensitivity of FOBT to detect colorectal cancer increases with age, while mean sojourn time decreases with age (approximately 4.3 years for participants aged 50–54, 3.9 years at age 55–59, 3.4 years at age 60–64, and 3.6 years at age 65–69, with corresponding sensitivity estimates around 41%, 47%, 45%, and 51%, respectively).

Conclusion

Simulation studies showed that the proposed stochastic model considering both mean sojourn time and sensitivity yields highly accurate results.

Keywords

Bias colorectal cancer screening faecal occult blood test sensitivity sojourn time

Introduction

The premise underlying cancer screening programmes is that effective detection methods allow diagnosis at an early stage, when the cancer is still curable. Through early-stage detection followed by appropriate treatment, screening programmes can potentially reduce cancer related morbidity and mortality. The key parameters in modelling the effectiveness of a screening test are test sensitivity (the probability that the screening test is positive, given that the individual is in the preclinical stage), and mean sojourn time (MST) (the average duration of the preclinical stage).¹ Studies have estimated these parameters through various modelling strategies.^2–4 Some studies^5–7 have focused on probabilistic models of the screening process and estimated the relevant quantities, while others^8,9 have sought an optimal screening schedule for breast cancer using a stochastic model based on the threshold method. Jiang et al.¹⁰ extended the Day and Walter⁵ Markov model and applied it to a cohort of women participating in the Canadian mammography screening programme. All these studies assumed that the screening sensitivity is constant, and that the sojourn time follows a parametric distribution with a constant mean. However, screening sensitivity and the sojourn time have been shown to vary by age, size of tumour, and cancer location.^11–13 Wu et al.¹⁴ have modelled age-dependent sensitivity and transition probability in breast cancer screening data, using a log-normal distribution for the transition probability and a logistic model for the sensitivity. Similarly, Cong et al.¹⁵ proposed a model-based approach incorporating age effects on screening sensitivity and sojourn time distribution. Numerous studies have simultaneously estimated sensitivity and MST for breast cancer screening, but few have investigated this in other cancers.¹⁶

We aimed to expand on the Shen and Zelen¹⁷ probability model for periodic cancer screening examination, by including age-group screening sensitivity and MST in the stochastic model, and constructing the full likelihood. Intensive simulation works were conducted to check the performance of proposed method in terms of bias, standard deviation, and coverage probability. The extended probability model was then tested in a real-world situation using data from the Korean National Colorectal Cancer Screening Programme (NCSP). In Korea colorectal is the third most common cancer and the fourth leading cause of death. Colorectal cancer incidence in Korea has continued to increase over the last few decades from 21.2 per 100,000 in 1999 to 38.6 per 100,000 in 2012.¹⁸ Because a long latency period is usually required for colorectal cancers to develop, early detection through screening can potentially improve patient survival. Several screening modalities exist for identifying colorectal cancer.¹⁹ The Korean NCSP provides an annual Fecal Occult Blood Test (FOBT) as the primary screening test for adults aged 50 and older. Using the proposed method, we evaluated the age group-dependent sensitivity and MST of FOBT for colorectal cancer.

Methods

Notation and previous models

According to Zelen and Feinleib,² the natural history of the disease for an individual develops by progressing through three states $S_{0} \to S_{p} \to S_{c}$ . In the disease-free state ( $S_{0}$ ), the disease cannot be detected. In the preclinical disease state ( $S_{p}$ ), an asymptomatic individual unknowingly has disease that the screening examination is designed to detect. In the clinical disease state ( $S_{c}$ ), disease manifests itself in clinical symptoms. The ultimate goal of screening is to detect the cancer in the preclinical state $S_{p}$ . Our notations closely follow those of Lee and Zelen.⁹ Let $q (t)$ be the probability density function of the sojourn time in $S_{p}$ . Then $Q (t) = \int_{t}^{\infty} q (x) d x$ denotes the survival distribution of the sojourn time in $S_{p}$ . Define $w (t) d t$ as the probability of transition from $S_{0}$ to $S_{p}$ during $(t, t + d t)$ and $I (t) d t$ as the probability of transition from $S_{p}$ to $S_{c}$ during $(t, t + d t)$ . The relationship among $I (t)$ , $w (t)$ and $q (t)$ can be expressed as

I (t) = \int_{0}^{t} w (x) q (t - x) d x

As Lee and Zelen⁹ have suggested, $w (x)$ can be calculated recursively if the incidence of age groups and the sojourn time distribution is known.

Let $t_{0} < t_{1} < \dots < t_{K - 1} (< T)$ represent K ordered screening examination times. Let T denote the follow-up time past the time of the last examination and define the $k th$ screening interval as $[t_{k - 1}, t_{k})$ for $k = 1, \dots, K$ where $t_{K} = T$ and the length of the $k th$ interval as $Δ_{k} = t_{k} - t_{k - 1}$ . For annual examinations, $t_{k} = t_{0} + k$ . Define the $k th$ generation of individuals as those people who enter $S_{p}$ during the $k th$ screening interval. The $0 th$ generation consists of all people who entered $S_{p}$ before the initial screening examination $t_{- 1} \equiv 0$ .

Suppose that each participant’s age of entry to the study is divided into g groups. The jth age interval be defined as $A_{j} = [x_{j - 1}, x_{j})$ for $j = 1, 2, \dots, g$ , where $x_{0} = 0$ and $x_{g} = \infty$ . We assume $Δ_{k} \leq δ_{j}$ where $δ_{j}$ denotes the length of the jth age interval. Let $t_{0} (j)$ denote the age of an individual in jth age group at entry (i.e. $t_{0} (j) \in A_{j}$ ) and $t_{k - 1}$ (j) denote the age at kth screening examination in jth age group at entry. Also, $β_{k - 1} (j)$ represents the screening sensitivity at age $t_{k - 1} (j)$ .

In previous models, Shen and Zelen¹⁷ defined $D_{k} (j)$ as the probability of being detected in S_p at the kth examination given at $t_{k - 1} (j)$ and $I_{k} (t | j)$ as the probability that the individual is incident at age t after k screening examinations, where $t_{k - 1} (j) < t < t_{k} (j)$ . Let $β$ be the constant screening sensitivity and μ be the MST. Define probability density function of the forward recurrence time for the prevalent cases at time $t_{0}$ by $q_{0} (t) .$ Then, $q_{0} (t) = \frac{Q (t)}{μ}$ , where $μ = \int_{0}^{\infty} Q (t) d t .$ Let $Q_{0} (t) = \int_{t}^{\infty} q_{0} (x) d x$ . The functional forms of $D_{k} (j)$ was derived as

$D_{1} (j) = β P_{0} (t_{0} (j)), for k = 1$ , and $for k > 1,$

D_{k} (j) = β {{(1 - β)}^{k - 1} P_{0} (t_{0} (j)) Q_{0} (t_{k - 1} - t_{0}) + \sum_{l = 1}^{k - 1} {(1 - β)}^{k - l - 1} \int_{t_{l - 1}}^{t_{l}} w (x) Q (t_{k - 1} - x) d x}

Also, similarly $I_{k} (t | j)$ was obtained as follows

\begin{array}{l} I_{k} (t | j) = {(1 - β)}^{k} P_{0} (t_{0} (j)) q_{0} (t - t_{0}) \\ + ϕ (k - 1) \sum_{i = 1}^{k - 1} {(1 - β)}^{k - i} \int_{t_{i - 1}}^{t_{i}} w (x) q (t - x) d x \\ + \int_{t_{k - 1}}^{t} w (x) q (t - x) d x \end{array}

where

ϕ (x) = 1

if x > 0 and 0 otherwise. The probability of being incident in the screening interval (

t_{k - 1}, t_{k}

) after k examinations is

I_{k} (j) = \int_{t_{k - 1}}^{t_{k}} I_{k} (t | j) d t

. They also derived the prevalence of the jth age group at entry

P_{0} (t_{0})

as follows

\begin{matrix} P_{0} (t_{0}) = μ [\sum_{l = 1}^{j - 1} w_{l} {Q_{0} (t_{0} - x_{l}) - Q_{0} (t_{0} - x_{l - 1})} \\ + w_{j} {1 - Q_{0} (t_{0} - x_{j - 1})] \end{matrix}

Proposed model

In the previous work by Shen and Zelen,¹⁷ sojourn time distribution and screening sensitivity were assumed to be the same for all age groups. Cong et al.¹⁵ considered modelling age at detection on the sojourn time distribution. However, our inference of model parameters was based on age at preclinical onset and screening age for MST and screening sensitivity, respectively. We parameterised age group-dependent MST and screening sensitivity in the periodic screening examination stochastic model. Let $μ_{j}$ be the MST for the jth age group at the time of preclinical onset and $β_{j}$ be the screening sensitivity for the jth age group at the time of screening test $(j = 1, \dots,$ g). When extending components of $D_{k} (j)$ and $I_{k} (t | j)$ in the previous model,¹⁷ the contribution of detected cases after study entry and interval cases between $t_{0} and t_{K - 1}$ can be simply obtained by including age group dependent parameters for the MST and screening sensitivity to the likelihood. However, for the prevalent cases at time of entry, it must be expressed as the sum of the function of the different forward recurrent time for age group-dependent sojourn time from entry time to clinical onset time. A detailed description of our proposed model is given in Appendix 1, Supplemental material.

For the jth age group, the observed data are recorded as ${(n_{k} (j), s_{k} (j), r_{k} (j)), k = 1, 2, \dots, K}$ , which denote the number of individuals examined at $t_{k - 1} (j)$ , the number of screening detected cases at $t_{k - 1} (j)$ and the number of interval cases in the screening interval $(t_{k - 1} (j), t_{k} (j))$ , respectively. Then, the full likelihood function is derived as follows

\begin{array}{l} L (μ, β, w) = \prod_{j = 1}^{g} \prod_{k = 1}^{K} {D_{k} (j) s k (j^{)} I_{k} (j) r_{k} (j^{)} \\ \times (1 - D_{k} (j) - I_{k} (j)) (n_{k} (j) - s_{k} (j) - r_{k} (j))} \end{array}

where

μ = (μ_{1}, \dots, μ_{g})

β = (β_{1}, \dots, β_{g})

w = (w_{1}, \dots, w_{g})

We used the maximum likelihood estimates from the full likelihood function for estimation of the unknown parameters $(μ, β, w)$ . Zelen and Feinleib² and Day and Walter⁵ showed that the exponential distribution was the best fit among a variety of distributions. Therefore, we assumed an exponential distribution for the duration of the preclinical screening-detectable period.

Additionally, the effectiveness of a screening programme may not only be affected by initial screening participation rate, but also by adherence during consecutive screening rounds. Therefore, we considered the participation rates for sequential screening rounds when simulating our model. It has been reported that adherence to screening follow-up is less than 50% after the first screening examination.²⁰ Thus, let $r_{F_{k}}$ be the proportion of loss to follow-up in the $k th$ screening examination, in contrast to the $(k - 1) th$ screening examination. The total number of individuals $n_{k}$ examined at the $k th$ screening is given by

n_{k} = n_{k - 1} \times (1 - r_{F_{k}}) for k = 1, \dots, K

If the participation in the following screening is equal, put $r_{F_{k}} = r_{F}$ and $n_{k} = n_{0} \times {(1 - r_{F})}^{k}$ .

Simulation procedure

We conducted a series of simulation studies to assess the bias, mean squared error, and coverage probabilities for the estimated parameters. Our model includes parameters $θ = (μ_{1}, \dots, μ_{g}, β_{1}, \dots, β_{g}, w_{1}, \dots, w_{g})$ . Some parameters related to the distribution of sojourn time and screening sensitivity are not of immediate interest. For example, if the screening programme is targeted at those aged 50 or over, the parameters of MST and sensitivity for those aged less than 50 are considered as nuisance. Let d denote the number of age groups not subject to screening examination. Then the number of target age groups for screening programme is m= g−d. In the simulation and real data application, we set m = 4 and assumed the nuisance parameters related to MST and sensitivity be set as known values.

The simulation consisted of three steps. Step 1: We assume that there are about 1,000,000 individuals who were born in the same year. The individual’s age at initiation of the screening programme was generated uniformly from ages 50 to 69. Four equal screening intervals were used, and the length of screening interval was selected as one year $(K = 4, Δ_{k} = 1)$ . Step 2: The age-dependent screening data based on input values of $θ$ were generated for age groups of m = 4 (50–54, 55–59, 60–64, 65–69). The transition probability from the disease-free state $S_{0}$ into the preclinical state $S_{p}$ was generated by the step functions $w$ using the CRC incidence data from the Korean Central Cancer Registry in 2003. Then the incident time for people in the preclinical state at age t was generated uniformly in $(t, t + 1)$ . For input values, ${μ_{1}, \dots, μ_{m}, β_{1}, \dots, β_{m}}$ , we considered two scenarios: 1) MST was generated from an exponential distribution with $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (3, 3, 3, 3)$ and screening test sensitivity $(β_{1}, β_{2}, β_{3}, β_{4}) =_{(0.5, 0.5, 0.5, 0.5)} and$ 2) MST $μ_{k}$ ’s and screening test sensitivity $β_{k}' s$ were $(2, 3, 4, 5)$ and $(0.55, 0.50, 0.45, 0.40)$ , respectively. Age groups not subject to screening were assigned known true values for parameters of MST and sensitivity. For each scenario, adherence to annual follow-up screening was assumed to be 100% and 50% $(r_{F_{k}} = 0$ and $r_{F_{k}} = 0.5$ for $k = 1, \dots, K - 1$ ). Step 3: The initial estimation of $w^{1}$ = ${w_{1}^{1}, w_{2}^{1}, w_{3}^{1}, w_{4}^{1}}$ was calculated from the incidence data with given initial values of $μ^{0} = {μ_{1}^{0}, μ_{2}^{0} \dots, μ_{m}^{0}}$ . Each value of $θ = {μ_{1,} μ_{2,} μ_{3,} μ_{4}, β_{1}, β_{2}, β_{3}, β_{4}}$ was updated from maximum likelihood estimates using Newton–Raphson algorithm by replacing w by $w^{0} - w^{1}$ . These procedures were repeated until the estimated quantities reach convergence. To obtain bootstrapped standard error of $θ,$ we constructed 100 replications of data. The sample mean, bias, MSE, coverage probability, and standard deviation of the MLE of $θ$ from 100 simulations are listed. The simulation program was written in SAS software (ver. 9.4; SAS Institute Inc., Cary NC, USA) and data were analysed using the nonlinear programing procedure PROC NLP in SAS statistical software.

Application to the data from the Korean NCSP

For illustration, we evaluated our model performance using colorectal cancer screening data from Korea. The NCSP recommends annual FOBT for people aged 50 or older. Study participants were those who received an FOBT through the NCSP between 2004 and 2007. We identified study participants who were diagnosed with cancer between January 2004 and December 2008, through data linkage with the Korea National Cancer Incidence Database of the Korea Central Cancer Registry. Codes C18 (colon) and C19–20 (rectal) in the International Classification of Diseases, 10th edition, were used to identify colorectal cancer cases. Cases with code C17 (small intestine) and C21 (anal) were excluded from analysis. There were four annual screening examinations $(k = 1, 2, 3, 4)$ and the cohort was of people whose age at study entry was $t_{0} (j) (= 50, 51, \dots, 69)$ . The number of subjects at first examination in the NCSP included 127,247 participants aged 50–54, 86,859 participants aged 55–59, 107,531 participants aged 60–64, and 54,905 participants aged 65–69. The initial values for the MST and the screening sensitivity were set to $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (4, 3.5, 3, 2.5)$ and $(β_{1}, β_{2}, β_{3}, β_{4}) = (0.40, 0.45, 0.50, 0.55)$ , respectively, based on the finding of previous studies^16,21,22 and the NCSP report data.²³ Age groups not subject to screening were also assumed to be known values based on findings of previous studies^16,21,22 and the NCSP report data.²³

Incidence data for colorectal cancer

Data for the age-specific incidence of colorectal cancer were obtained from the Korea Central Cancer Registry 2003. This is a nationwide hospital-based cancer registry, which was initiated by the Ministry of Health and Welfare in 1980. Incidence rates were summarised into five-year age intervals: 0–4 years through 80–84 years, and 85 or older (Figure 1).²⁴

Figure 1.

Age-specific colorectal cancer incidence in Korea.

Results

The results obtained from 100 replications are summarised in Table 1 for $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (3, 3, 3, 3)$ and $(β_{1}, β_{2}, β_{3}, β_{4}) = (0.5, 0.5, 0.5, 0.5)$ and Table 2 for $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (2, 3, 4, 5)$ and $(β_{1}, β_{2}, β_{3}, β_{4}) = (0.55, 0.50, 0.45, 0.4)$ . Models are evaluated based on the average estimate (Estimate), average bias (Bias), the empirical standard deviation (SD), the mean value of the estimated standard errors (MSE), and the coverage probability (CP) of the nominal 95% confidence intervals. Results of the simulations indicate that the proposed model yields reliable parameter estimates. As shown in Table 2, maximum likelihood estimates of the age group-dependent model were consistent with the true parameters. Biases were small, and most of the coverage probabilities were around the nominal level of 0.95. The magnitude of SDs and MSEs of the estimates were larger in the model, which assumed 50% adherence to screening follow-up, compared with the model assuming 100% adherence, but CPs of parameters did not differ between the two models.

Table 1.

Simulation result for $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (3, 3, 3, 3)$ and $(β_{1}, β_{2}, β_{3}, β_{4}) = (0.5, 0.5, 0.5, 0.5)$ (100 replicates).

	True	Estimate	Bias	SD	MSE	CP
Assuming 100% adherence to screening
$μ_{1}$	3	3.122	0.122	0.411	0.428	0.97
$μ_{2}$	3	3.210	0.210	0.327	0.335	0.96
$μ_{3}$	3	3.214	0.214	0.239	0.286	0.95
$μ_{4}$	3	3.168	0.168	0.235	0.230	0.91
$β_{1}$	0.5	0.504	0.004	0.040	0.042	0.96
$β_{2}$	0.5	0.495	−0.005	0.039	0.037	0.95
$β_{3}$	0.5	0.491	−0.009	0.028	0.031	0.94
$β_{4}$	0.5	0.497	−0.003	0.026	0.026	0.94
$w_{1}$	53.7	48.85		3.81
$w_{2}$	84.1	84.32		5.89
$w_{3}$	114.7	113.87		6.23
$w_{4}$	159.1	159.38		7.75
Assuming 50% adherence to screening
$μ_{1}$	3	3.118	0.118	0.567	0.562	0.98
$μ_{2}$	3	3.252	0.252	0.432	0.481	0.98
$μ_{3}$	3	3.288	0.288	0.372	0.417	0.97
$μ_{4}$	3	3.184	0.184	0.342	0.332	0.96
$β_{1}$	0.5	0.510	0.010	0.048	0.050	0.95
$β_{2}$	0.5	0.497	−0.003	0.049	0.050	0.93
$β_{3}$	0.5	0.490	−0.010	0.040	0.042	0.98
$β_{4}$	0.5	0.497	−0.003	0.036	0.036	0.94
$w_{1}$	53.7	48.25		5.20
$w_{2}$	84.1	85.02		6.40
$w_{3}$	114.7	113.79		8.37
$w_{4}$	159.1	159.59		10.89

Note: Estimate is the mean of the parameter estimates. Bias: mean of the parameter estimates minus the true value; SD: empirical standard deviation of the parameter estimates; MSE: mean value of the estimated standard errors; CP: coverage probability of the nominal 95% confidence intervals; $w_{j}$ : rate per 10⁵.

Table 2.

Simulation result for $(μ_{1}, μ_{2}, μ_{3}, μ_{4}) = (2, 3, 4, 5)$ and $(β_{1}, β_{2}, β_{3}, β_{4}) = (0.55, 0.50, 0.45, 0.4)$ (100 replicates).

	True	Estimate	Bias	SD	MSE	CP
Assuming 100% adherence to screening
$μ_{1}$	2	2.005	0.005	0.283	0.308	0.97
$μ_{2}$	3	3.078	0.078	0.343	0.369	0.97
$μ_{3}$	4	4.189	0.189	0.398	0.414	0.95
$μ_{4}$	5	5.354	0.354	0.479	0.417	0.84
$β_{1}$	0.55	0.564	0.014	0.056	0.058	0.97
$β_{2}$	0.50	0.502	0.002	0.042	0.043	0.97
$β_{3}$	0.45	0.443	−0.006	0.024	0.030	0.99
$β_{4}$	0.40	0.398	−0.002	0.018	0.021	0.96
$w_{1}$	53.7	53.74		3.42
$w_{2}$	84.1	84.15		5.88
$w_{3}$	114.7	114.76		8.49
$w_{4}$	159.1	159.23		11.89
Assuming 50% adherence to screening
$μ_{1}$	2	2.113	0.113	0.485	0.423	0.92
$μ_{2}$	3	3.114	0.114	0.514	0.562	0.98
$μ_{3}$	4	4.221	0.221	0.562	0.647	0.98
$μ_{4}$	5	5.192	0.192	0.619	0.617	0.96
$β_{1}$	0.55	0.553	0.003	0.073	0.067	0.93
$β_{2}$	0.50	0.500	0.000	0.061	0.060	0.94
$β_{3}$	0.45	0.445	−0.005	0.039	0.045	0.97
$β_{4}$	0.40	0.404	0.004	0.027	0.031	0.98
$w_{1}$	53.7	53.81		5.83
$w_{2}$	84.1	84.29		8.82
$w_{3}$	114.7	114.64		11.37
$w_{4}$	159.1	159.42		14.85

Total number of participants, detected cases, and interval cancers according to age at first screening examination in the Korean NCSP data are shown in Table 3. Percentage of screen-detected cases and interval cases varied from 0.014 to 0.075 and 0.028 to 0.083, respectively. The number of participants who were subsequently screened was relatively small (Table 3). Table 4 illustrates age group-dependent estimates of FOBT sensitivity and MST for colorectal cancer from the NCSP data. MSTs were approximately 4.3 years for participants aged 50–54, 3.9 years for those aged 55–59, 3.4 years for age 60–64, and 3.6 years for age 65–69. Estimates of sensitivity were around 41%, 47%, 45%, and 51% for people aged 50–54, 55–59, 60–64, and 65–69 years, respectively. Based on the results of the proposed model, the MST for those in their 50s was generally longer than for people in their 60s, and the screening sensitivity of FOBT for participants in their 50s was relatively lower than for the older age group. The estimated transition probabilities were $({\hat{w}}_{1,} {\hat{w}}_{2,} {\hat{w}}_{3,} {\hat{w}}_{4}) = (16.54, 20.53, 25.58, 39.84)$ per 100,000 person-years and the standard deviations for the estimated parameters from 100 bootstrap replicates were $(13.22, 12.11, 15.85, 18.71)$ per 100,000 person-years. Estimates of transition probability increased with age, and they have affected the CRC incidence rate in Korea.

Table 3.

Total number of participants, detected cases and interval cancers according to age at first screening examination, National Cancer Screening Program 2004–2007.

	Participants	Detected cases	Interval cases
First round [t₀, t₁)
50–54	127,247	51 (0.040)	38 (0.030)
55–59	86,859	69 (0.079)	54 (0.062)
60–64	107,531	92 (0.086)	79 (0.073)
65–69	54,905	72 (0.131)	48 (0.087)
Total	376,542	284 (0.075)	219 (0.058)
Second round [t₁, t₂)
50–54	7989	0 (0.000)	4 (0.050)
55–59	9543	5 (0.052)	6 (0.063)
60–64	9580	12 (0.125)	4 (0.042)
65–69	6763	7 (0.104)	7 (0.104)
Total	33,875	24 (0.071)	21 (0.062)
Third round [t₂, t₃)
50–54	2534	1 (0.039)	2 (0.079)
55–59	3707	3 (0.081)	1 (0.027)
60–64	3990	4 (0.100)	3 (0.075)
65–69	3054	2 (0.065)	5 (0.164)
Total	13,285	10 (0.075)	11 (0.083)
Fourth round [t₃, t₄^a)
50–54	905	0 (0.000)	1 (0.110)
55–59	2166	0 (0.000)	0 (0.000)
60–64	2006	0 (0.000)	1 (0.050)
65–69	2006	1 (0.050)	0 (0.000)
Total	7083	1 (0.014)	2 (0.028)

Unit: N (%).

^at₄: t₃ + Additional one-year follow-up.

Table 4.

Results of the proposed model for CRC data from NCSP.

Age (years)	Total^a number observed	Parameters
Age (years)	Total^a number observed		MST	Sensitivity
50–54	127,247	Estimate (95% CI)	4.317 (2.684–5.949)	0.411 (0.304–0.516)
		SE of Estimate	0.833	0.054
55–59	86,859	Estimate (95% CI)	3.903 (2.644–5.161)	0.469 (0.360–0.560)
		SE of Estimate	0.642	0.051
60–64	107,531	Estimate (95% CI)	3.419 (2.251–4.587)	0.447 (0.365–0.534)
		SE of Estimate	0.596	0.043
65–69	54,905	Estimate (95% CI)	3.567 (1.897–5.237)	0.508 (0.400–0.619)
		SE of Estimate	0.852	0.056

^aTotal number observed is the number of subjects at first examination.

MST: mean sojourn time; SE: standard error; CI: confidence interval.

Discussion

In this study, we proposed a stochastic probability model for periodic cancer screening which considers simultaneously the age group-dependent MST and screening sensitivity. Simulations studies showed that the proposed model provides reliable estimates of the model parameters. To our knowledge, this is the first time that the NCSP data have been used directly to obtain age group-dependent estimates of MST and the screening sensitivity for FOBT. After applying our model to the NCSP data, we found that maximum likelihood estimates of the MST varied from 4.3 years for those aged 50–54 to 3.4 years for those aged 60–64. The estimate of sensitivity varied from 41% to 50%. This result is consistent with the previous estimates derived by Launoy et al.,¹³ who estimated MST to be 4.7 years (95% CI: 3.1–8.4), based on the analysis of French mass-screening programme for colorectal cancer using FOBT. Similarly, Wu et al.¹⁶ have estimated MST to be 4.08 years for males and 2.41 years for females, using Bayesian inference from the periodic probability model. This is an interesting finding that requires further investigation to determine whether the validity of the FOBT varies by gender. In the same study, screening sensitivity was found to increase with age. Gyrd-Hansen et al.²¹ estimated the sensitivity of FOBT at 62.1% and MST at about 2.1 years in the Danish randomised clinically controlled screening trial.

Variations in MST and sensitivity by age have previously been addressed by Prevost et al.,⁷ who analysed data from a FOBT-based colorectal cancer screening programme conducted in Calvados, France. Contrary to our research findings that have suggested a negative association between MST and age group, they found that MST increases with age: MSTs were approximately two years for those aged 45–54, three years for those aged 55–64, and six years for age those aged 65–74. Estimates of sensitivity were approximately 75%, 50%, and 40% for persons aged 45–54, 55–64, and 65–74 years, respectively. Possible explanation for the above discrepancy could be the difference of screening periods.

There are a number of limitations to our study. First, some nuisance parameters related to the distribution of sojourn time and screening sensitivity for those who were not subject to screening examination are set as known values. Second, previous studies have indicated that MST of colorectal cancer varies according to anatomic location of the cancer.^13,16,25,26 For example, in the study by Launoy et al.,¹³ the MST were 6.4 and 2.6 years for distal colon and rectal cancer, respectively, and the sensitivity for the FOBT was around 50% for both distal and rectal cancers. Using a Markov chain model to estimate MST and sensitivity of FOBT, Chiu et al.²⁶ observed MSTs of 2.06 and 1.36 years and sensitivities of 65.12% and 73.70 for localised and non-localised colorectal cancer, respectively. As information on specific cancer sites was not collected in the NCSP data, we were not able to explore site-specific differences in MST. Further studies should be undertaken to ascertain these findings in the Korean population. One further avenue for future research would be to determine the extent of over-diagnosis due to faecal testing.

Conclusion

In summary, simulation studies show that the proposed stochastic model considering both MST and sensitivity approach can accurately estimate model parameters. The estimated MST of colorectal cancer and sensitivity will help the NCSP determine the periodicity of CRC screening, and provide information on the effectiveness of FOBT in detecting colorectal cancer. Further research is needed to establish differences in MST by gender and anatomical locations, to develop appropriate screening regimes for colorectal cancer in Korea.

Supplemental Material

MSC790775 Supplemental material - Supplemental material for Estimating age group-dependent sensitivity and mean sojourn time in colorectal cancer screening

Supplemental material, MSC790775 Supplemental material for Estimating age group-dependent sensitivity and mean sojourn time in colorectal cancer screening by Na Young Sung, Jae Kwan Jun, Youn Nam Kim, Inkyung Jung, Sohee Park, Gyu Ri Kim and Chung Mo Nam in Journal of Medical Screening

Footnotes

Acknowledgement

We thank the National Cancer Center in Korea for providing access to data.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Supplemental material

Supplemental material is available for this article online.

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