Where:

n: the number of observation

Y_i: the observed survival time

Y^+∞: an arbitrary and sufficiently large constant

X_i: model matrix

β: coefficients

w_i: the weight function

F: estimated locally Kaplan-Meier estimator

ρ_τ(u): the quantile loss function for τ^th quantile = u.{τ-I(u<0)}

However, because there is no analytical method for estimating the variance of estimated coefficients, the resampling (bootstrap) method is used.

The CQR was fitted using the prognosis factors significant in log-rank test; in conclusion, age, tumor size and number of involved LN were selected for predicting (conditional) OS quantiles. In addition, because of probably non-linear effect of age on OS distribution of BC patients, the age was included in the model of the second order. Package "survival" was used to conduct the Kaplan-Meier and log-rank test ²⁶ and function LCRQ proposed by Wang and Wang ²¹ was used to fit the CQR model.

Results

Patients' follow-up ranged from about 3 to 197 months and median follow-up was 54.13 months. By the end of the study, 185 (35.4%) patients experience the death and 122 (23.4%) patients experience the metastasis or recurrence. One-, three-, and five-year survival rates were 90%, 73% and 62.5%, respectively that shows a constant decline over study period (). Table 1 reports the patients and tumor characteristics and type of the received surgery. There was a crude relation between OS and age at diagnosis, tumor size, stage, metastasis status and number of involved LN (P<0.05).

As was mentioned in "statistical methods" subsection, the CQR model was used to explorer the multivariable relationship between prognosis factors and survival time (in month). Prognosis variables at diagnosis that showed a significant relationship with OS (Table 1), including standardized age (and age-squared), number of involved LN and tumor size were entered to the multivariable CQR model. Metastasis status and American Joint Committee on Cancer (AJCC) stage did not involve in CQR model, because the former showed the developed metastasis after the surgery and the latter had strong collinearity with number of involved LN and tumor size. The conditional median of survival time is:

$Q_{0.5}=66.8-1.46ag{e}^2-4.97std.age-1.33Tsize-0.78nLN$

Close

Figure 1. Plot of probability of surviving in BC patients after surgery.

Which in the above mentioned expression, std. age, age2, Tsize and nLN stand for standardized age, age-squared, tumor size and number of involved LN, respectively. Table 2 shows the estimated coefficients of 1, 2.5, 5, 10, 25, 50, 75, 90, 95, 97.5 and 99^th conditional quantile of survival times. Using the estimated CQR model presented in Table 2, quartile of survival time estimated for 18 different scenarios were calculated. These sceneries were defined using three different ages (40, 45 and 50 yr), two different number of involved LN (3 and 6) and three different tumor sizes (1.5, 3.5 and 6 cm). Table 3 shows the estimated conditional quartiles of time to death (in month) using the CQR for these eighteen scenarios. The higher age at diagnosis, the more number of involved LN and/or larger tumor size, led to decreasing the median and other quartile of OS distributions (Table 3).

Discussion

The main aim of this study was to investigate factors affecting the distribution of OS in patients with non-metastatic BC. For the above-mentioned aim, a distribution-free CQR model introduced by Wang and Wang ²¹ was used. The CQR model has many interesting features. This model is distribution-free but give complete information about distribution of time-to-event ^18,21 . In addition, the assumption of random censoring that is important in Cox proportional hazard model was relaxed in this model ^17,21 . Another feature of CQR is the direct interpretation of estimated effects in terms of change in quantile of survival time distribution ^17,21 . Our results revealed that, comparing to the studies in developed western countries ³ , patients participants in this study, were diagnosed at the later age (about one-third were diagnosed at age ≥ 50 yr), and later stages (about 40% were diagnosed with tumor size ≥ 2 cm, about 37% with number of involved LN ≥ 2 and about 53% at stage II/III). This maybe because of lack of awareness about the signs of BC, lack of participants in screening program and lack of (inadequateness) performing of regular breast self-examination ^3,27 . On the other hand, the late diagnosis of BC at advanced level could be cause to reduction in OS (Table 2). Unfortunately, in Iran, women have no sufficient knowledge about the breast cancer symptoms, breast self-examination, clinical examination and mammography and about 83% of women do not perform regular monthly breast self-examination ²⁷ . In addition to the late diagnosis, it seems that the socio-economics and race are important factors affecting –survival- of BC patients ^28,29 .

Age, tumor size, number of involved LN, stage of disease, progesterone receptor status and human epidermal receptor are important prognosis factors at diagnosis and, in addition, loco-regional metastasis is an important intermediate prognosis factor of unadjusted OS (Table 1). This result is consistent with another study as for the role of these risk factors ¹³ .

To the best of our knowledge, there are only limited studies addressing the distribution of OS in BC patients and its related factors ²¹ . The multivariable analysis of CQR showed that age at diagnosis, tumor size and number of involved LN are important factors in determination the distribution of patient's lifetime at diagnosis. Based on developed model, the median, first and 3^rd quartiles of OS decrease by increasing of age in all subgroups. Such conclusion is showed by number of involved LN and tumor size. This finding supports the important role of early detection of BC and is in concordance with other studies ¹³ .

On the other hand, an important finding that CQR model revealed is that the change in distribution of (log) survival time may not preserve the distribution family. Change in the distribution of other characteristics by the change of covariate has been reported earlier ³⁰ . Therefore, in this situation, application of parametric survival models may lead to invalid and biased conclusion.

In this study, there were some limitations on the data and the statistical method. Unfortunately, there are many non-responses in patients' medical records, especially in some important features including estrogen and progesterone receptors, as well as, Her-2 and p53. On the other hand, in the CQR that used in this study, all independent variables should be of quantitative type. Therefore, if there are one or more qualitative variables, a stratified analysis should be used.

Conclusions

This study confirms the importance of early detection of BC and warns on the possible changes in distribution family of survival time.

Acknowledgments

This article extracted from MSc dissertation supported by Deputy of Research and Technology of Hamadan University of Medical Sciences. Authors would like to thanks this deputy and would like to thanks Hamadan Mahdieh Oncology Center’s personnel for their corporation.

Conflict of interest statement

None declared.

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