Application of a Non-Mixture Cure Rate Model for Analyzing Survival of Patients with Breast Cancer

Medical researchers are largely interested in studying the effect of multi-explanatory variables on the time to occurrence of a specific event; the event can be death, relapse of a specific event, divorce, recurrence of a disease, and so forth. When the analysis of survival data is mainly aimed to estimate the effect of a covariate, the Cox’s proportional hazard model as a semi-parametric approach is the most frequently-used method (Yu, 2008). Typically when using models for survival data, if the period of follow-up is sufficiently long, all patients will be eventually susceptible to experience the event under the study (Cox, 1972). One of the most important characteristics of survival data is the presence of censored observation, that are actually either lost during the followup or are the cases who do not experience the event under the study during the period of study. Also, for some outcomes, there may be some cases who never face the event of interest (Lambert, 2007). As a result of significant progresses made in treatment of many types of cancers during the last few decades, there


Introduction
Medical researchers are largely interested in studying the effect of multi-explanatory variables on the time to occurrence of a specific event; the event can be death, relapse of a specific event, divorce, recurrence of a disease, and so forth. When the analysis of survival data is mainly aimed to estimate the effect of a covariate, the Cox's proportional hazard model as a semi-parametric approach is the most frequently-used method (Yu, 2008). Typically when using models for survival data, if the period of follow-up is sufficiently long, all patients will be eventually susceptible to experience the event under the study (Cox, 1972). One of the most important characteristics of survival data is the presence of censored observation, that are actually either lost during the followup or are the cases who do not experience the event under the study during the period of study. Also, for some outcomes, there may be some cases who never face the event of interest (Lambert, 2007).
As a result of significant progresses made in treatment of many types of cancers during the last few decades, there Ahmad Reza Baghestani 1 , Sahar Saeedi Moghaddam 2, 1 *, Hamid Alavi Majd 1 , Mohammad Esmaeil Akbari 2 , Nahid Nafissi 2 , Kimiya Gohari 1 have been an increased number of patients who do not experience the event under study Rondeau et al., 2013). The cases that never experience the event of interest are referred to as cured, immune, or nonsusceptible. For example, researchers may be interested in analyzing the rejection of an organ transplanted. Many cases may never reject transplanted organ; therefore, a cured fraction of the population exists. With present of cure fraction in data, standard survival analysis methods such as Cox's proportional hazard model or log-rank test may not be appropriate because they do not account for the possibility of cure (Arano et al., 2010). Cure rate models (also known as long-term survival models) which are a special type of survival analysis models were developed to be used in such a situation.
Cure fraction can be of importance for patients, clinicians, and policy makers, and it may give more valuable insights into time trends in survival analysis of cancer patients (Andersson et al., 2011). Due to significant progress in therapies in recent years, there has been an increased focus on analyzing survival data via these type of models in all fields (Corbière and Joly, 2007;Ortega et al., 2014). Moreover, these survival models are known as split population models in the field of economists and other social scientists (Schmidt and Witte, 1989). Cure rate models are applicable for various types of cancers such as prostate, breast, melanoma, non-Hodgkin's lymphoma, leukemia, and head and neck cancer, in which a significant proportion of cases are immune of the event under study (Chen et al., 1999;Ortega et al., 2008). When the survival curve has a stable plateau at the end of the study it is recommended to use cure rate models for analyzing the survival data (Kim et al., 2013). Further, in the absence of cure fraction, cure rate models will be changed to standard survival analysis models (Yu, 2008).
The aim of the current study was to investigate the effects of clinical, pathological, and biological characteristics of patients on survival of patients with breast cancer using non-mixture cure rate model.

Materials and Methods
We studied 438 female patients with breast cancer who were visited and treated at Cancer Research Center in Shahid Beheshti University of Medical Sciences, Tehran, Iran. These patients were diagnosed from 1992 to 2012 and followed up until October 2014. We had to exclude some patients because of their incomplete information. The patients or patients' family members were contacted via phone calls to confirm whether the patients are still alive or not. Deaths due to breast cancer were regarded as failure and survival time was calculated as the time interval between date of cancer diagnosis and date of death due to breast cancer (for those who died) and date of the last follow-up (for those who were alive).
There are two major approaches in cure rate models. The oldest one which was introduced by Boag in 1949 is the mixture cure rate model, also known as standard cure rate model (Boag, 1949). In this model we assume that a certain fraction of the population are cured and the remaining are not cured (Maller and Zhou, 1996). In order to model cure fraction, parametric and non-parametric methods are used in this approach (Achcar et al., 2012). Second approach which was introduced by Chen et al in 1999 is the non-mixture cure rate model (Chen et al., 1999), also known as bounded cumulative hazard model and promotion time cure model (Tsodikov et al., 2003;Tournoud and Ecochard, 2008). As one of the advantages of the non-mixture cure rate model, it has a proportional hazards model . In this approach it is assumed that an individual is left with Ni metastaticcomponent cancer cells after treatment (Yakovlev et al., 1996;Tsodikov et al., 2003). In this study we assumed that Ni had a Poisson distribution.
For data analysis, clinical, pathological, and biological characteristics of patients were assessed in the cure model. A Weibull distribution was proposed for the survival time. The data were analyzed using STATA version 14 and the figure was prepared by R version 3.1.2. Quantitative results were expressed as mean ± standard deviation. The significance level was set at P ≤ 0.05.

Results
Patients' characteristics are presented in Table 1. A total of 438 women with breast cancer were enrolled into this analysis. The mean of age at the time of diagnosis was 48.36 ± 10.90 years. Patients' age ranged from 22 to 84 years. The longest duration of follow-up was 253 months. A total of 75 patients (17.08%) died due to breast cancer during the study up to October 2014. The Kaplan-Meier observed curve, with 95% confidence interval, is illustrated in Figure 1. As the Figure shows, the curve reached a plateau at the right tail, so using the cure rate model led to more accurate results. The prognostic variables included in the model were age at diagnosis, marital status, history of abortion, history of breastfeeding, family history of breast cancer, type of surgery, number of metastatic lymph nodes, tumor size, stage, histologic grade, estrogen receptor, progesterone receptor, and lymphovascular invasion. According to the results which are presented in Table 2, number of metastatic lymph nodes and histologic grade were significant. In addition cure fraction was estimated to be 58%.

Discussion
One of the appropriate and useful ways to investigate any progresses in cancer therapy is to monitor trends in survival of patients over time (Bakar et al., 2008). The time from diagnosis until occurrence of a specific event under study is the most valuable measure of care for patients, clinicians, and even policy makers (Andersson et al., 2011). Therapies for many types of diseases especially cancers have been improved significantly in the recent years, so we are facing an increase in the proportion of patients who do not experience the event under study, i.e. those who are not susceptible to the occurrence of the event under study (Asano et al., 2014). The proportion of such observations is known as cure fraction. In addition, survival analysis via considering cure fraction is called cure rate models or long-term survival models and they perform a considerable function in survival analysis (Ortega et al., 2014).
For the past 20 years or more, several studies have been conducted on cure rate models and they have been well developed in the statistical literature (Rama et al., 2010;Achcar et al., 2012;Othus et al., 2012;Akhlaghi et al., 2013;Yu et al., 2013;Rahimzadeh et al., 2014), but the models are not common in all fields. More specifically, most of them are focused on cancers-related subjects. For cancers in which some cases may have long-term survival, cure rate models can be an interesting approach to characterize and study the survival of patients. It is worth mentioning that cure rate models are studied because of their optimal properties which may better explain the responses to the variables at different times (Akhlaghi et al., 2013). There are two general types of cure models, mixture and non-mixture. Mixture cure rate models may fit some data better than non-mixture and vice versa (Othus et al., 2012), however in this report we only study the non-mixture cure rate model. Non-mixture cure rate model can explain remaining cancer cells after therapy by using latent variable (Rahimzadeh et al., 2014).
Breast cancer is the first common cancer and the third leading cause of death among Iranian women (Sadjadi et al., 2005); in addition, it's mortality is still lower than that in Western industrialized countries (Taghavi et al., 2012). The aim of this study was to find factors that may have  an effect on the survival of patients with breast cancer including clinical, pathological, and biological variables. Based on our findings, the number of metastatic lymph nodes and histologic grade were significant. In our study a Weibull distribution was proposed for survival time but it was not restricted to Weibull and some other lifetime distributions could be also considered (Borges et al., 2012); However, Jafari-Koshki's study indicated that cure model with Weibull distribution had the smallest DIC in their report and fitted the data best (Jafari-Koshki et al., 2014). In our study we used logistic link function; as a matter of fact, determining the link function is important because it leads to different assumptions about the joint effect of covariates .
Most of the studies on cure models are conducted on prostate or breast cancers, but Sposto used mixture and non-mixture cure models to analyze children's cancer group data (Sposto, 2002). The majority of studies investigating the factors affecting the survival of patients with breast cancer mainly used Log-rank test and Cox analysis. Rahimzadeh's study indicated that in the survival analysis of breast cancer patients with cure rate model, metastasis and stage had a significant effect (Rahimzadeh et al., 2014). In this study data for some clinical, pathological, and biological variables were missing and some were recorded in a wrong way so that they had become useless. In addition, our study was carried out based on the data collected from a single institution; large sample size is one of the desirable properties required for cure models, and can reduce the problems related to convergence and fit of the model (Jafari-Koshki et al., 2014). Nevertheless, in our study the model converged after a few iterations. It is recommended to conduct further studies by assuming other distributions rather than Poisson for Ni and Weibull for lifetime.