Estimating Long-Term Crude Probability of Death among Young Breast Cancer Patients: A Bayesian Approach

Abstract

Aims and Background

Bayesian survival analysis was applied to assess the long-term survival and probability of death due to breast cancer (BC) in Girona, the Spanish region with the highest BC incidence.

Methods

A Bayesian autoregressive model was implemented to compare survival indicators between the periods 1985–1994 and 1995–2004. We assessed the long-term excess hazard of death, relative survival (RS), and crude probability of death due to BC (P_BC) up to 20 years after BC diagnosis, reporting the 95% credible intervals (CI) of these indicators.

Results

Patients diagnosed from 1995 onwards showed lower 20-year excess hazards of death than those diagnosed earlier (RS during 1985–1994: local stage: 76.6%; regional stage: 44.9%; RS during 1995–2004: local stage: 85.2%; regional stage: 57.0%). The P_BC after 20 years of BC diagnosis for patients diagnosed in 1995 and after might reach 14.4% (95% CI: 8.9%–21.2%) in local stage and 41.0% (95% CI: 36.1%–47.1%) in regional stage.

Conclusions

The method presented could be useful when dealing with population-based survival data from a small region. Better survival prospects were found in patients diagnosed after 1994, although we detected a non-decreasing long-term excess hazard of death, suggesting that these patients have higher mortality than the general population even 10 years after the diagnosis of BC.

Keywords

Bayesian Breast cancer Excess mortality Probability of death Relative survival

Introduction

In Europe, breast cancer (BC) is the most frequent cancer type and the first cause of cancer death among women (1). Recently, BC incidence rates have shown a downturn among Spanish women over 45, which is best explained by screening saturation (2). Improvements in 5-year BC survival have been detected in Spain (3) in parallel with a decline in BC mortality (4), clearly marked in women under 50 years of age (5, 6). In these age groups, life expectancy can lengthen by several decades, and these patients may hope for cure after 5 years from a BC diagnosis (7). However, several studies have reported an excess mortality into the second and even third decade after BC diagnosis (8 –19).

Given these complexities, relative survival analysis (20) is the preferred method in long-term cancer survival studies (21). The role of regional cancer registries in planning and evaluating national cancer plans is becoming increasingly recognized (1), prompting the development of new statistical methods to provide national estimates based on data from these registries (22, 23). We provided survival indicators of young BC patients based on the data of the population-based cancer registry of Girona (Spain), which has collected BC incidence data since the 1980s (24). Taking into account BC stage at diagnosis and the size of the study population, we assessed the long-term probability of death due to BC and other causes. We developed a Bayesian additive excess hazards model for this purpose.

Materials and Methods

Data Source

BC data were obtained from the population-based cancer registry of the Girona region, which covers a population of 354,526 women (2007 census) (25). Since its creation, the registry has collected data on invasive and in situ breast tumors, with case registration performed according to the European Network of Cancer Registries recommendations (26). The Girona Cancer Registry was set up in 1994 as a general cancer registry for the whole region; its data have been published in the Cancer Incidence in Five Continents series (27) and included in studies related to the estimation of cancer incidence and survival in Spanish and European populations (7). The most recent data (period 2003–2007) shows that 28% (380 of 1,350) of newly diagnosed cancer cases in women in Girona are due to BC (28). Each woman with BC in this region has been followed up to December 31, 2012.

Study Cohort

Only newly diagnosed women were included in the study cohort, and for each included woman, follow-up started at the date of the first primary BC diagnosis identified during 1985–2004. A total of 998 female patients aged 15–49 years and diagnosed with invasive primary BC with local and regional stage (codes 174 and C50 of the 9^th and 10^th editions of the International Classification of Diseases, ICD-9 and ICD-10, respectively) in the period 1985–2004 in Girona were included in the cohort. In this age group, there were between 17% and 20% of cases with unknown stage. Data from these cases were excluded from the analysis. In women with bilateral BC, only data from the first tumor were considered. Stage classification was based on the SEER classification system and program (29). In addition to the active and passive follow-up via hospitals, date of death was retrieved from the Catalan Mortality Registry and the National Death Index of the Spanish Ministry of Health. Passive follow-up was based on record linkage with the Catalan mortality registry, which covers the 4 Catalan provinces: Girona, Tarragona, Lleida and Barcelona. The follow-up times of those patients not found to be dead at the end of follow-up are considered as censored times.

Statistical Methods

A full description of the statistical methods is provided in the supplementary material (available online at www.tumorijournal.com). In brief, survival analyses were carried out by modeling the excess hazard of death, which provides a measure of the number of deaths that exceed expectations for the study population. Under the additive model, the observed hazard of death $λ_{t}^{o}$ at the annual interval t for persons diagnosed with cancer has been modeled as the sum of the expected hazard of death $λ_{t}^{E}$ and the excess hazard of death due to a diagnosis of cancer $λ_{t}^{M}$ , therefore, $λ_{t}^{o} = λ_{t}^{M} + λ_{t}^{E}, t \in {1, \dots, 20}$ ,

Accordingly, for the additive excess hazard modeling we can assume piecewise constant hazards that may imply a Poisson process for the number of deaths in each time interval (30, 31). The cumulative relative survival (RS) can be estimated from the cumulative excess hazard of mortality, $Λ_{M} (T) = \sum_{t \leq T} (λ_{t}^{o} - λ_{t}^{E}), as R S (T) = e^{- Λ_{M} (T)} = \frac{S_{o} (T)}{S_{E} (T)} .$ .

Since the excess hazard rates could be zero or even negative in the long term, our model has been adapted accordingly to Bayesian survival analysis (4). Therefore, the time trend of the excess hazard rates was smoothed through a first-order autoregressive (AR1) prior distribution, and then, making use of the smoothed estimates of the excess hazard rates, we derived survival indicators of BC up to 20 years after diagnosis. Since the model considers an AR1 where each excess hazard depends on the previous one, we have to assume a prior estimate ${\hat{λ}}_{1}^{M}$ for the first excess hazard $λ_{1}^{M}$ , that is, the excess hazard observed at the end of the first year of follow-up. In this case we considered a constant value, ${\hat{λ}}_{1}^{M} = \frac{D_{1} - E_{1}}{Y_{1}}$ , where $\frac{D_{1} - E_{1}}{Y_{1}}$ is the observed excess hazard, D₁are the observed deaths during the first annual interval, the expected deaths during the first annual interval, D₁ and the person-years observed during the first annual interval. We did not consider ${\hat{λ}}_{1}^{M} = 0$ because cancer patients have a higher risk of death than the general population during the first year of follow-up.

We provided the 95% credible intervals (95% CI) of the excess hazard, relative survival and probabilities of death due to BC (P_BC), these last constituting a quantity of clinical interest in long-term follow-up (32).

Making use of the Bayesian autoregressive approach, we could compare the long-term survival estimates of patients diagnosed between 1985 and 1994 with those of patients diagnosed between 1995 and 2004. Since the maximum follow-up for the period 1995–2004 was 18 years (patients diagnosed in 1995 could be followed up to 2012), the 19-year and 20-year survival for these patients was estimated using the 19-year and 20-year survival of patients diagnosed between 1985 and 2004. Since our interest was in estimating up to 20-year excess hazards, we used the observed excess hazards for the 19^th and 20^th annual interval of the whole 1985–2004 cohort as estimates of the observed hazards beyond the 18^th annual interval for the cohort diagnosed between 1995 and 2004. From these excess hazards we estimated 19- and 20-year RS. Survival indicators between periods were compared making use of Markov chain Monte-Carlo (MCMC) sampling and the matrix of posterior samples obtained. Statistical analyses were carried out with R (https://cran.r-project.org/) and WinBUGS (33). We used an MCMC run of 10,000 and discarded the first 2,000 values of the chain as a burn-in process. Therefore, our estimates were based on 8,000 values of the chain. However, we observed that convergence could also be easily found running an MCMC run of 5,000, discarding the first 2,000 values, and keeping the remaining 3,000. Additional tables and the R and WinBUGS code to reproduce the statistical analyses are provided in the supplementary material (available online at www.tumorijournal.com).

Model Validations

An assessment of how well the model fitted the observed data is shown in supplementary Figure S1 (Comparison between observed and smoothed long-term annual excess hazard [% per 100 women-years] of death by stage and period. (A) Patients diagnosed with local-stage tumors during the period 1985–1994; (B) patients diagnosed with local-stage tumors during the period 1995–2004; (C) patients diagnosed with regional-stage tumors during the period 1985–1994; (D) patients diagnosed with regional-stage tumors during the period 1995–2004. Available online at www.tumorijournal.com). The out-of-sample predictive performance of the model through Monte Carlo cross-validation is presented in supplementary Figure S2 (Monte Carlo cross-validation. Difference between observed and predicted long-term annual excess hazard [% per 100 women-years] of death by stage. (A) Patients diagnosed with local-stage tumors during the period 1985–2004; (B) patients diagnosed with regional-stage tumors during the period 1985–2004. Available online at www.tumorijournal.com).

Results

Table I shows the age distribution and the number of patients at risk at the beginning of the study for each 5-year cohort from 1985 to 2004. There were no differences in age distribution among these cohorts. The number of patients in 2000–2004 was at least twice that of 1985–1989, and the increase in the number of cases was mainly found after 1994 in patients with localized diagnoses: there were 54 patients for the 1985–1989 cohort and 103 patients for the 1995–1999 cohort. However, this change was not found among patients diagnosed with regional-stage tumors. In our study we found that the proportion of deaths due to BC was 82.2% (268 deaths from BC out of 326 deaths).

Table I

Breast cancer patients younger than 50 years at diagnosis during the period 1985–2004 (n = 998) in Girona (Spain) according to stage and period of diagnosis. Age distribution and number of patients available after a certain year of follow-up

	Stage	Age distribution at diagnosis (years)				Number of patients available for the survival analysis after year
	Stage	Age distribution at diagnosis (years)				0		5		10		15		20
	Localized	Min	Q1	Median	Q3	N	D1	N	D2	N	D3	N	D4	N
Period	1985–1989	25	38	42	46	54	5	49	4	45	6	39	2	37
Period	1990–1994	27	40	42	46	59	0	57	6	51	3	48	3	22
Period	1995–1999	29	39	44	46	103	4	95	4	90	2	88^*	–	–
Period	2000–2004	29	40	45	47	111	3	103	4	no^**	–	–
	Regional
Period	1985–1989	28	39	43	47	109	38	65	15	50	6	43	5	37
Period	1990–1994	29	38	42	46	159	38	120	21	98	20	77	7	37
Period	1995–1999	26	38	43	46	169	31	136	22	113	5	109^*	–	–
Period	2000–2004	26	39	43	46	224	29	190	21	169^**	–	–	–	–

Min = minimum age; Q1 = first quartile; Q3 = third quartile; D1 = deaths within interval 0 to 5 years of follow-up; D2 = deaths within interval 5 to 10 years of follow-up; D3 = deaths within interval 10 to 15 years of follow-up; D4 = deaths within interval 15 to 20 years of follow-up.

For the cohort 1995–1999 follow-up was considered up to 13 years, since this was the maximum follow-up for patients diagnosed during 1999.

For the cohort 2000–2004 follow-up was considered up to 8 years, since this was the maximum follow-up for patients diagnosed in 2004.

Figure 1 monitors the smoothed annual excess hazard of death by stage at diagnosis, assessing the difference in excess hazard between periods. During the first 10 years of follow-up, we found that annual excess hazards among patients diagnosed at a localized stage in the period 1985–1994 (Fig. 1A) were slightly higher than those of patients diagnosed after 1994 (Fig. 1B) (the difference in excess hazards between 1985–1994 and 1995–2004 was slightly above 0, see Fig. 1C). On the other hand, during the first 7 years of follow-up there was a larger excess hazard among patients diagnosed at a regional stage during 1985–1994 (Fig. 1D) than in those diagnosed during 1995–2004 (Fig. 1E). However, these differences disappeared beyond 10 years of follow-up (Fig. 1F). Making use of these annual excess hazards, the RS was calculated.

Fig. 1

Monitoring long-term annual excess hazard (% per 100 women-years) of death by stage and period. (A) Patients diagnosed with local-stage tumors during the period 1985–1994; (B) patients diagnosed with local-stage tumors during the period 1995–2004; (C) difference in excess hazard between periods (reference 1995–2004) among patients with local-stage tumors; (D) patients diagnosed with regional-stage tumors during the period 1985–1994; (E) patients diagnosed with regional-stage tumors during the period 1995–2004; (F) difference in excess hazard between periods (reference 1995–2004) among patients with regional-stage tumors.

Table II compares RS between the periods 1985–1994 and 1995–2004. Patients diagnosed at a local stage in 1995–2004 had higher median RS than patients diagnosed in 1985–1994, and these differences in median RS slightly increased up to 20 years of follow-up. Differences in RS were clearly found during 5 to 10 years of follow-up when comparing survival in regional stage between periods of diagnosis, since the 95% CIs of RS estimates did not overlap. After 20 years of follow-up, the median RS of patients diagnosed at a regional stage during 1985–1994 reached 44.9%, whereas for patients diagnosed during 1995–2004 the median RS estimated was 57%.

Table II

Comparison of relative survival from breast cancer in Girona (Spain) among women younger than 50 years at diagnosis taking into account period and stage at diagnosis

	Stage	Relative survival (%) after a number of years since diagnosis of breast cancer
	Stage	1-year RS (95% CI)	5-year RS (95% CI)	10-year RS (95% CI)	15-year RS (95% CI)	20-year RS (95% CI)
	Localized
Period	1985–1994	99.9 (99.2–100)	95.1 (90.9–98.1)	87.6 (81.0–92.8)	81.6(73.5–88.6)	76.6 (67.6–84.7)
Period	1995–2004	99.9 (99.2–100)	97.0 (94.5–98.9)	92.5 (88.4–95.7)	89.1 (83.6–93.4)	85.2 (78.1–90.8)
	Regional
Period	1985–1994	96.4(94.4–98.0)	72.4 (66.5–77.7)	57.0 (50.0–63.8)	50.4 (43.2–57.9)	44.9 (43.2–57.9)
Period	1995–2004	97.4(96.4–98.3)	85.2 (81.9–88.1)	73.1(68.6–77.1)	64.4(59.3–69.2)	57.0(51.1–62.7)

“Year” is defined as the time after diagnosis. RS (95% CI) is the estimate of the relative survival and its 95% credible interval. Note that the 95% CI must be interpreted as the probability that the relative survival lies between the upper and lower limit of the credible interval is 95%. RS = relative survival; CI = credible interval.

Figure 2 depicts the temporal evolution of the estimated crude probabilities of death due to BC (supplementary material, page 5, equations 5 and 6) and their 95% CIs (supplementary material, page 6; Table S1 presents the numerical results. Table S1 - Crude probability of death due to breast cancer and due to other causes for women aged 50 and younger according to stage at diagnosis and stratified by period of diagnosis. Available online at www.tumorijournal.com). The crude probability of death due to BC showed 2 patterns depending on stage at diagnosis. There was a linear rise in patients diagnosed at a localized stage (Fig. 2A), whereas a logarithmic rise was detected in patients diagnosed at a regional stage (Fig. 2B). There were differences among patients with local disease stage when comparing the periods of diagnosis, although their 95% CIs overlapped (Fig. 2A). In these patients, the crude probability of death due to BC after 20 years of follow-up reached 23.2% among those diagnosed during 1985–1994 (95% CI: 15.2%–32.1%), whereas it may reach 14.4% (95% CI: 8.9%–21.2%) among those diagnosed in 1995–2004. On the other hand, differences in these probabilities were clearly found in patients diagnosed with regional-stage disease when periods of diagnosis were compared. The 95% CIs did not overlap (Fig. 2B), and the crude probability of death due to BC reached a median of 53.7% among patients diagnosed during 1985–1994 (95% CI: 46.4%–60.5%), whereas it may reach a median of 41.0% (95% CI: 36.1%–47.3%) among those diagnosed in 1995–2004. Finally, when we compared the probabilities of death due to other causes between the periods of diagnosis, these were below 10% after 20 years, and their 95% CIs overlapped during the whole study period (Fig. 2C and 2D).

Fig. 2

Comparison of crude probability of death due to breast cancer (A) local stage; (B) regional stage and due to other causes (C) local stage; (D) regional stage) taking into account the period of diagnosis.

Discussion

Bayesian inference was applied to monitor the development and changes in BC survival through different periods and cohorts, showing that excess hazards in the first 10 years after diagnosis improved for patients diagnosed more recently, a result more marked among patients with regional disease stage. Patients diagnosed after 1994 showed large differences in the 20-year excess hazards of death when compared to those diagnosed earlier (RS 1985–1994: local: 76.6%; regional: 44.9%; RS 1995–2004: local: 85.2%; regional: 57.0%). In the most recent period, we estimated that the crude probability of death due to BC after 20 years of diagnosis might reach 14.4% (95% CI: 8.9%–21.2%) among patients with localized stage and 41.0% (95% CI: 36.1%–47.1%) among patients with regional stage.

We developed a Bayesian model to smooth the excess hazard rates based on Poisson distribution and then used them to provide RS estimates. Our modeling was oriented to deal with relatively small cohorts of patients. In survival analysis, Bayesian inference of hazard rates has been proven to perform well in such situations and in other types of analysis, such as those with the aim to detect the shape of the cumulative (hazard) incidence function when sample size is small (34), and to our knowledge it is novel in population-based survival analysis. The prior distribution used in our modeling, along with the autoregressive temporal structure on excess hazard rates, can be easily adapted to other situations. Further improvement on these classes of modeling must be oriented towards inclusion of important prognostic factors such as histological grade and treatment (unfortunately not available in our study) as model covariates (23). The use of other prior temporal structures such as second-order ones (4) for other cancer sites must take into account that first-order differences assumes a constant rate of increase or decrease, whereas second order assumes a constant absolute change per unit of time (4). We combined cohorts of different periods in order to reflect the impact of very recent changes in BC management on long-term survival and to provide long-term estimates of survival for the most recent cohorts of patients. The potential impact of recent advances in the diagnosis and treatment of the most recent BC incident cases could be an explanation of the observed trends, since the cohort of women who contributed to survival estimates for the 18^th to 20^th years of follow-up, women diagnosed before 1995, were diagnosed in a period where major changes in treatment and management of BC took place (4, 5).

An advantage of using the method is related to small sample size. The Bayesian method presented here allows estimating excess hazards and their variability where other methods may fail. This is crucial where one may observe zero counts within an annual interval. The method could be useful for registries covering small areas, since cancer counts are also small, and precision in the estimates of excess hazards is an issue. Estimation through generalized linear models and maximum likelihood could be an alternative, but standard errors of the parameters might not be properly estimated. A major disadvantage is the use of prior distributions due to the Bayesian framework. Sensitivity analysis of the priors and assessment of the out-of-sample predictive performance of the model should be carried out.

An important question with regard to the limitations in our modeling is whether the additive hazards model is adequate when the competing causes of death, BC and all other causes, are independent of each other (30). Since these patients are young patients diagnosed with BC, it is expected that most of the causes of death are BC. We found that the proportion of deaths due to BC, based on death certificates, was 82.2%. We have to assume independence is reasonable, but there is no way to test this with our data. If those causes were not independent, the survival indicators presented – which are based on excess hazards – might be overestimated. Therefore, we would assign higher survival than the actual survival to those patients. Another limitation is in fitting the additive hazards model. We considered estimates of the expected numbers of deaths due to causes other than BC from estimated life tables for Girona, and we did not consider uncertainty in these estimates. This is a potential issue in the modeling that could affect long-term survival estimates, and this could be considered as future research work in this type of analysis. There are 2 limitations to the modeling. First, evidence suggests that misclassification disproportionately affects older women and those diagnosed with in situ, metastatic, or unknown-stage disease, which is not the case in our study (35). We excluded in situ cases and metastatic ones. However, it may affect those cases of unknown stage, which have not been included in the present study. In addition, a major issue with staging revision is how patients diagnosed under the new system compare with similarly staged patients under the previous system. Second, it has been shown that patients diagnosed in stage II according to the 1988 staging criteria might be classified as stage III under the revised staging system. Then, for patients in stage III, this may result in improvements in overall survival (29). This effect is termed “stage migration” or the Will Rogers phenomenon. In our study, we used the SEER modified AJCC stage third edition 1998–2003 (http://seer.cancer.gov/seerstat/variables/seer/ajcc-stage/3rd.html) in order to minimize these changes, and this must be considered as a limitation when interpreting changes in survival between periods. Table S2 summarizes the coding procedure used (Table S2 - Staging criteria based on the SEER Program Coding and Staging Manual 2014. Available online at www.tumorijournal.com).

Monitoring the RS over 20 years, we detected changes showing better survival prospects for young patients diagnosed after 1994, among whom the major contributors to the excess hazard was BC mortality. In our study we found that the proportion of deaths due to BC was 82.2% (268 deaths from BC out of 326 deaths). However, in this result, the use of the cause of death reported in the death certificate is a potential concern due to distant metastases (36). We provided the crude probability of death due to cancer and other causes making use of the RS estimates (37). An advantage of RS is that it provides a measure of mortality associated with a particular disease, without the need for information on the cause of death. A recent study has found that the proportion of BC deaths could be close to 95% among women under 45 years of age at diagnosis and 82% among women aged 45–54 years (36). In this regard, evidence suggests that the long-term probability of death from BC in women younger than 50 years at diagnosis lies between 10% and 20% in women diagnosed with local-stage tumors and between 40% and 50% in women diagnosed with regional-stage tumors (38).

The women in our study diagnosed after 1994 would have benefited from more recent advances in treatment and management of BC (4), since advances in high-resolution imaging have led to a reclassification of the most aggressive localized cancers from regional cancers, decreasing the lethality of regional classified cancers (39).

We found a non-decreasing annual excess hazard of death independent of stage at diagnosis, suggesting that the risk of dying from BC does not decrease even after 10 years (7, 9 –11, 15, 19). Recent studies carried out in the Netherlands (15) and Italy (19) have shown that the long-term RS of BC patients has improved, although a significant 5% excess mortality might exist up to 15 years or more after diagnosis, even for patients with local-stage tumors (15). In many of these studies, second tumors, late recurrences among patients with positive hormone receptor status, and late side effects of BC treatment could be attributed to new BC (6, 7, 9 –11, 40).

We have developed a Bayesian modeling approach for the long-term excess hazard of death, which allowed us to assess the probabilities of death due to BC and other causes up to 20 years after a diagnosis of BC. In the age group considered, the most recent estimate of the crude probability of death due to BC after 20 years of diagnosis might reach 14.4% among patients with localized stage and 41.0% among patients with regional stage. Since the most frequent cause of death associated with late mortality was BC, our study endorses the need for further research to understand the benefits of monitoring survival and long-term cause-specific death of BC patients.

Footnotes

Acknowledgment

We would like to thank the reviewers for their careful review and constructive comments.

Financial support: This study has been funded by Instituto de Salud Carlos III through the Project PI14/01041, co-funded by European Regional Development fund/European Social fund: “investing in your future”. This work was also supported within Red Temática de Investigación en Cáncer (cofinanciado por Fondos FEDER: Una manera de hacer Europa) - (RD012/0036/0053).

Conflict of interest: All authors declare that they have no conflict of interest.

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