Estimation of natural history parameters of breast cancer based on non-randomized organized screening data: subsidiary analysis of effects of inter-screening interval, sensitivity, and attendance rate on reduction of advanced cancer

Abstract

Estimating the natural history parameters of breast cancer not only elucidates the disease progression but also make contributions to assessing the impact of inter-screening interval, sensitivity, and attendance rate on reducing advanced breast cancer. We applied three-state and five-state Markov models to data on a two-yearly routine mammography screening in Finland between 1988 and 2000. The mean sojourn time (MST) was computed from estimated transition parameters. Computer simulation was implemented to examine the effect of inter-screening interval, sensitivity, and attendance rate on reducing advanced breast cancers. In three-state model, the MST was 2.02 years, and the sensitivity for detecting preclinical breast cancer was 84.83%. In five-state model, the MST was 2.21 years for localized tumor and 0.82 year for non-localized tumor. Annual, biennial, and triennial screening programs can reduce 53, 37, and 28% of advanced cancer. The effectiveness of intensive screening with poor attendance is the same as that of infrequent screening with high attendance rate. We demonstrated how to estimate the natural history parameters using a service screening program and applied these parameters to assess the impact of inter-screening interval, sensitivity, and attendance rate on reducing advanced cancer. The proposed method makes contribution to further cost-effectiveness analysis. However, these findings had better be validated by using a further long-term follow-up data.

Appendix

Three-state Model

Let the stochastic process of disease natural history of breast cancer denoted by a random variable X(t), the outcome of which is defined by a state space W = {1,2,3}, where states 1, 2, and 3 stand for free of breast cancer, the PCDP, and clinical caner, respectively. In this three-state model, the transition probabilities from previous state (i) to current state (j) during time t can be represented by Table 6.

Table 6 Transition probabilities from the previous state to the current state during time t (P _ij (t))

The application of this transition matrix to data on breast cancer screening has been described in full elsewhere [10]. The detailed likelihood functions for estimating the natural history parameters are decomposed by round of screens and detection modes.

Prevalent Screen

Suppose women invited to first screen (prevalent screen) at age m, the probabilities of having negative screening result (P _{s1_1}) and positive screen-detected results (P _{s1_2}) using transition probabilities and sensitivity (sen) and specificity (spe) are written as follows.

$$ \begin{gathered} P_{{{\text{s}}1\_1}} \left( {\text{Probability of having negative screening result at first screen}} \right) \hfill \\ = \,{\text{Probability of being true negative}}\, + \,{\text{probability of being false negative}} \hfill \\ = \,{\frac{{ \, \left( \begin{gathered} {\text{Probability of free of cancer at age of entry }}m\, \times \,{\text{specificity}} \hfill \\ \, + \,{\text{probability of preclinical cancer at age of entry }}m\, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right)}}{{\left( {{\text{Probability of free of cancer}}\, + \,{\text{probability of preclincial cancer at age of entry }}m} \right)}}} \hfill \\ = \,{\frac{{P_{11} \left( m \right)\, \times \,{\text{spe}}\, + \,P_{12} \left( m \right) \times \left( {1 - \,{\text{sen}}} \right)}}{{P_{11} \left( m \right) + P_{12} \left( m \right)}}} \hfill \\ \end{gathered} $$

(1)

$$ \begin{gathered} P_{{{\text{s}}1\_2}} \left( {\text{Probability of having positive screen results at first screen}} \right) \hfill \\ = \,{\text{Probability of being false positive}}\, + \,{\text{probability of being true positive }} \hfill \\ = \,{\frac{{\left( \begin{gathered} {\text{Probability of free of cancer at age of entry }}m\, \times \,{\text{probability of being false positive }} \hfill \\ \, + \,{\text{probability of preclinical cancer at age of entry }}m\, \times \,{\text{sensitivity}} \hfill \\ \end{gathered} \right)}}{{{\text{(Probability of free of cancer}}\, + \,{\text{probability of preclincial cancer at age of entry }}m)}}} \hfill \\ \, = \,{\frac{{P_{11} \left( m \right) \times (1 - \,{\text{spe}}) + P_{12} \left( m \right)\, \times \,{\text{sen}}}}{{P_{11} \left( m \right) + P_{12} \left( m \right)}}} \hfill \\ \end{gathered} $$

(2)

Those women who have negative screening results at first screen are composed of those who are actually disease free (true negative) or misclassified (false negative), whereas those women who have positive screening results at first screen consist of who are actually disease (true positive) or misclassified (false positive).

Incident Screen

Owing to false negative cases missed at prevalent screen, the underlying population for the second screen after the prevalent screen consist of disease free (true negative) and asymptomatic women missed at first screen with the respective proportions denoted by tn(1) (true negative at first screen) and fn(1) (false negative at first screen), respectively. The formulae of these two probabilities are expressed as follows.

$$ \begin{gathered} {\text{tn}}\left( 1 \right) \, \left( {\text{Probability of being true negative at first screen}} \right) \hfill \\ = \,{\frac{{{\text{Probability of free of cancer at age of entry }}m\, \times \,{\text{specificity}}}}{{\left( \begin{gathered} {\text{Probability of free of cancer at age of entry }}m\, \times \,{\text{specificity}} \hfill \\ \, + \,{\text{probability of preclinical cancer at age of entry }}m\, \times \,{\text{probability of being missed }} \hfill \\ \end{gathered} \right)}}} \hfill \\ = \,{\frac{{P_{11} \left( m \right)\, \times \,{\text{spe}}}}{{P_{11} \left( m \right)\, \times \,{\text{spe}}\, + \,P_{12} \left( m \right)\, \times \,\left( {1\, - \,{\text{sen}}} \right)}}} \hfill \\ \end{gathered} $$

(3)

$$ \begin{gathered} {\text{fn}}\left( 1 \right)\left( {\text{Probability of being false negative at first screen}} \right) \hfill \\ = \,{\frac{{{\text{Probability of preclinical cancer at age of entry }}m\, \times \,{\text{probability of being missed }}}}{{\left( \begin{gathered} {\text{Probability of free of cancer at age of entry }}m\, \times \,{\text{specificity}} \hfill \\ \, + \,{\text{probability of preclinical cancer at age of entry }}m\, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right)}}} \hfill \\ = \,{\frac{{P_{12} \left( m \right) \times \left( {1 - \,{\text{sen}}} \right)}}{{P_{11} \left( m \right)\, \times \,{\text{spe}}\, + \,P_{12} \left( m \right) \times \left( {1 - \,{\text{sen}}} \right)}}} \hfill \\ \end{gathered} $$

(4)

Assume the false negative cases missed at first screen can be detected at the second screen, the probabilities of screen-negative women (P _{s2_1}) and screen-detected cases (P _{s2_2}) at the second screen and the probabilities of interval cancer (P _{s2_I}) diagnosed before second screen and refuser cases (P _{s2_R}) at second screen but diagnosed at time t after first screen are expressed as follows.

$$ \begin{gathered} P_{{{\text{s}}2\_1}} \left( {\text{Probability of having negative screening result at second round}} \right) \hfill \\ = \,\left( \begin{gathered} {\text{Probability of being true negative at first round}}\, \times \,{\text{probability of staying normal during time }}t \, \hfill \\ \, \, \times \,{\text{specificity}} \hfill \\ \end{gathered} \right) \hfill \\ + \,\left( \begin{gathered} {\text{probability of being true negative at first round}}\, \times \,{\text{transition probability from normal to preclinical during time }}t \hfill \\ \, \, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right) \hfill \\ = \,{\text{tn}}(1)\, \times \,P_{11} \left( t \right)\, \times \,{\text{spe}}\, + \,{\text{tn}}(1)\, \times \,P_{12} \left( t \right)\, \times \,\left( {1 - \,{\text{sen}}} \right) \hfill \\ \end{gathered} $$

(5)

$$ \begin{gathered} P_{{{\text{s}}2\_2}} \, \left( {\text{Probability of having positive result at second round}} \right) \hfill \\ = \left( \begin{gathered} {\text{Probability of being true negative at first round}}\, \times \,{\text{probability of staying normal during time }}t \hfill \\ \, \times \,{\text{probability of being false positive}} \hfill \\ \end{gathered} \right) \hfill \\ + \,\left( \begin{gathered} {\text{probability of being true negative at first round}}\, \times \,{\text{transition probability from normal to preclinical during time }}t \, \hfill \\ \, \times \,{\text{sensitivity}} \hfill \\ \end{gathered} \right) \hfill \\ + \,\left( {{\text{probability of being false negative at first round}}\, \times \,{\text{probability of staying preclinical state during time }}t} \right) \hfill \\ = \,{\text{tn}}(1)\, \times \,P_{11} \left( t \right)\, \times \,(1 - \,{\text{spe}}) + {\text{tn}}(1)\, \times \,P_{12} \left( t \right)\, \times \,{\text{sen}}\, + \,{\text{fn}}(1)\, \times \,P_{22} \left( t \right) \hfill \\ \end{gathered} $$

(6)

$$ \begin{gathered} P_{{{\text{s}}2\_{\text{I}}}} \left( {\text{Probability of interval cancer between first and second round}} \right) \hfill \\ = \,\left( \begin{gathered} {\text{Probability of being true negative at first round }} \hfill \\ \, \times \,{\text{probability of surfacing to clinical state during the short time period }}\Updelta t{\text{ before the end of time }}t \hfill \\ \end{gathered} \right) \hfill \\ + \,\left( \begin{gathered} {\text{probability of being false negative at first round }} \hfill \\ \, \times \,{\text{probability of surfacing to clinical state during the short time period }}\Updelta t{\text{ before the end of time }}t \hfill \\ \end{gathered} \right) \hfill \\ = \,{\text{tn}}(1)\, \times \,\left[ {P_{11} \left( {t - \Updelta t} \right)\, \times \,P_{13} (\Updelta t)\, + \,P_{12} \left( {t - \Updelta t} \right)\, \times \,P_{23} (\Updelta t)} \right]\, + \,{\text{fn}}(1)\, \times \,\left[ {P_{22} \left( {t - \Updelta t} \right) \times P_{23} (\Updelta t)} \right] \hfill \\ \end{gathered} $$

(7)

$$ \begin{gathered} P_{{{\text{s}}2\_R}} \left( {\text{Probability of refuser cases after attending first round}} \right) \hfill \\ = \,\left( \begin{gathered} {\text{Probability of being true negative at first round }} \hfill \\ \, \times \,{\text{probability of surfacing to clinical state during the short time period }}\Updelta t{\text{ before the end of time }}t \hfill \\ \end{gathered} \right) \hfill \\ + \,\left( \begin{gathered} {\text{probability of being false negative at first round }} \hfill \\ \, \times \,{\text{probability of surfacing to clinical state during the short time period }}\Updelta t{\text{ before the end of time }}t \hfill \\ \end{gathered} \right) \hfill \\ = \,{\text{tn}}(1)\, \times \,\left[ {P_{11} \left( {t - \Updelta t} \right)\, \times \,P_{13} (\Updelta t)\, + \,P_{12} \left( {t - \Updelta t} \right)\, \times \,P_{23} (\Updelta t)} \right]\, + \,{\text{fn}}(1)\, \times \,\left[ {P_{22} \left( {t - \Updelta t} \right)\, \times \,P_{23} (\Updelta t)} \right] \hfill \\ \end{gathered} $$

(8)

The first components of the Eqs. 5–7 delineate the progression of true negative subjects after prevalent screen and the second components give delineate the progression of false negative cases missed at prevalent screen. Note that, in the above formulae, t is the time interval between the prevalent and the second screen, inter-screening interval, or time after first screen for the refuser, and $\Updelta t,$ say one month, which is an approximation to the instantaneous time (dt) used in the derivative of the probability density function. For example, the first component of the bracket on the right side of the Eq. 7, P ₁₁(t − ∆t) × P ₁₃(∆t) + P ₁₂(t − ∆t) × P ₂₃(∆t), which is called compound probability, is an approximation to dP ₁₃(t)/dt. The merit of using compound probability has been described in Duffy et al. study [14] and Kay [15] as it can accommodate the rapid and slow progression of breast cancer.

Similar to the prevalent screen, the underlying population for the third screen after the second screen round is also composed of two categories, disease free and asymptomatic women missed at second screen, with the respective probabilities of tn(2) and fn(2) with formulae as follows.

$$ \begin{gathered} {\text{tn}}\left( 2 \right)\left( {\text{Probability of being true negative at second round}} \right) \hfill \\ = {\frac{{\left( \begin{gathered} {\text{Probability of being true negative at first round}} \hfill \\ \, \, \times \,{\text{probability of staying normal during time }}t{\text{ between first and second round }}\, \times \,{\text{ specificity}} \hfill \\ \end{gathered} \right)}}{{\left[ \begin{gathered} \left( \begin{gathered} {\text{Probability of being true negative at first round}} \hfill \\ \, \times \,{\text{probability of staying normal during time }}t{\text{ between first and second round}}\, \times \,{\text{specificity}} \hfill \\ \end{gathered} \right) \hfill \\ \, + \left( \begin{gathered} {\text{Probability of being true negative at first round}} \hfill \\ \, \times \,{\text{probability of transition from normal to preclinical cancer during time }}t {\text{between first and second round }} \hfill \\ \, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right]}}} \hfill \\ = \,{\frac{{{\text{tn}}(1)\, \times \,P_{11} \left( t \right)\, \times \,{\text{spe}}}}{{{\text{tn}}(1)\, \times \,P_{11} \left( t \right)\, \times \,{\text{spe}}\, + \,{\text{tn}}(1)\, \times \,P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}} \hfill \\ \end{gathered} $$

(9)

$$ \begin{gathered} {\text{fn}}(2)\left( {\text{Probability of being false negative at second screen}} \right) \hfill \\ = \,{\frac{\left( \begin{gathered} {\text{Probability of true negative at first round}} \hfill \\ \, \times \,{\text{probability of transition from normal to preclinical cancer during time }}t {\text{between first and second round }} \hfill \\ \, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right)}{{\left[ \begin{gathered} \left( \begin{gathered} {\text{Probability of being true negative at first round}} \hfill \\ \, \times \,{\text{probability of staying normal during time }}t{\text{ between first and second round}}\, \times \,{\text{specificity}} \hfill \\ \end{gathered} \right) \hfill \\ \, + \left( \begin{gathered} {\text{Probability of true negative at first round}} \hfill \\ \, \times \,{\text{probability of transition from normal to preclinical cancer during time }}t {\text{between first and second round }} \hfill \\ \, \times \,{\text{probability of being missed}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right]}}} \hfill \\ = {\frac{{{\text{tn}}(1) \times P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}{{{\text{tn}}(1) \times P_{11} \left( t \right)\, \times \,{\text{spe}}\, + \,{\text{tn}}(1) \times P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}} \hfill \\ \end{gathered} $$

(10)

The above formulae can be simplified as

$$ {\text{tn}}(2)\, = \,{\frac{{P_{11} \left( t \right)\, \times \,{\text{spe}}}}{{P_{11} \left( t \right)\, \times \,{\text{spe}}\, + \,P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}} $$

(11)

$$ {\text{fn}}(2) = {\frac{{P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}{{P_{11} \left( t \right)\, \times \,{\text{spe}}\, + P_{12} \left( t \right) \times \left( {1\, - \,{\text{sen}}} \right)}}}. $$

(12)

The likelihood function for screen-detected cases, screen-negative women, and interval cancer at the third screen round can be derived in a similar manner by the incorporation of tn(2) and fn(2) into the likelihood function.

The Refuser Group

The probability of developing breast cancer for those who never come to screen $\left( {P_{\text{NA}} } \right)$ is expressed as follows. In the following formula, m represents the age of the first invitation to the screening program, t ₂ represents the time period from the first invitation to the year of the diagnosis of breast cancer.

$$ \begin{gathered} P_{\text{NA}} ({\text{Probability of developing breast cancer for those who never come to screen}}) \hfill \\ = {\frac{{\left[ \begin{gathered} \left( \begin{gathered} {\text{Probability of free of cancer at age of first invitation }}m \hfill \\ \times {\text{ probability of surfacing to clinical state during the short time period }}\Updelta t {\text{ before the end of time }}t2 \hfill \\ \end{gathered} \right) \hfill \\ + \left( \begin{gathered} {\text{probability of preclinical cancer at age of first invitation }}m \hfill \\ \times{\text{ probability of surfacing to clinical state during the short time period }}\Updelta t {\text{ before the end of time }}t2 \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right]}}{({{\text{Probability of free of cancer + probability of preclinical cancer at age of entry }}m})}} \hfill \\ = \,{\frac{{\left[ {P_{11} \left( m \right) \times P_{11} \left( {t_{2} - \Updelta t} \right) \times P_{13} (\Updelta t) + P_{11} \left( m \right) \times P_{12} \left( {t_{2} - \Updelta t} \right) \times P_{23} (\Updelta t) + P_{12} \left( m \right) \times P_{22} \left( {t_{2} - \Updelta t} \right) \times P_{23} (\Updelta t)} \right]}}{{\left[ {P_{11} \left( m \right) + P_{12} \left( m \right)} \right]}}} \hfill \\ \end{gathered} $$

(13)

The Eq. 13 is also applied to PSC (see Table 1).

Five-state model

The likelihood functions using for the five-state model are developed in the same way as the three-state model. In the five-state model, the state space is changed as W = {1,2,3,4,5}, where states 1, 2, 3, 4, and 5 denote free of breast cancer, preclinical localized cancer, preclinical non-localized cancer, clinical localized cancer, and clinical non-localized cancer, respectively. The transition probabilities from state i to state j during time t can be represented by Table 7.

Table 7 Transition probability from previous state to current state during time t (P _ij (t))

As in three-state model, the likelihood functions for the estimation of parameters are also decomposed by rounds of screen and detection modes. The details of likelihood functions are given with supplemental files and available online (http://homepage.ntu.edu.tw/~ntucbsc/tony_e.htm).