Bipolar affective disorder, a common and chronic, severe and recurrent, mood and brain disorder that causes unusual shifts in mood, energy, activity levels and ability to carry out day-to-day tasks, is characterized by recurrent episodes of mania and depression [1-4]. Bipolar disorder (BD) remains among the 10 most disabling disorders according to the World Health Organization (WHO) causing a significant amount of disability [5]. Studies have been published before, estimated the prevalence of bipolar disorder in between 1% and 2% [2]. One of the important types of BD is bipolar I disorder with lifetime prevalence of 0.24% [6]. Although BD is often hard to be diagnosed due to its irregular patterns, it can be treated and patients life would be improve [1,7]. Bipolar disorder causes substantial psychosocial morbidity that frequently affects the patient's marriage, children, occupation, and other aspects of the patient's life [8]. Bipolar disorder is the sixth cause of disability in the United States and symptoms can damaged patients relationships, job and school performance, as a consequence, it is one of leading causes of disability which contributes to economic burden of this disorder to society [9,10]. Several studies have reported that 50% to 75% of patients have a recurrence within 4 to 5 years after the first admission [11]. Patients with long duration of illness and highly recurrent course show great impairment of global functioning and number of episodes has been found consistently associated with poor outcome [4]. An important feature of bipolar I disorder is the repetition of relapses over time and up to 90% of BD patients have at least one relapse in their lifetime [12-15]. Information about time to relapse and efficacy of some risk factors in preventing relapses will provide a valuable tool for planning and evaluating the health-outcome results of treatment [16]. There are some studies about relationship of factors and onset of BD but little is known about the factors that may precipitate relapse [10]. Few studies have assessed time to relapse, remission and recovery of patients, using Kaplan-Meier estimator or Cox regression, but most of them failed to account the repeated feature of relapses and correlation caused by recurrent events whit-in subjects [17-23]. In order to study time to relapses, with respect to repeated characteristic of bipolar disorder, we need recurrent event analyzing techniques of survival analysis. Survival analysis typically focuses on time to event data and recurrent event is a multivariate survival analysis in which event occurs more than once per subject over follow-up time such as hospital stays or heart attacks [24,25]. The most important issue in recurrent data is correlation among relapses of each patient. In fact, individuals have varied lifestyles, genetic traits, and experiences which influence the likelihood that they will succumb to disease but either cannot be measured and are unknown. These factors are called latent variable. As a result, some individuals are more prone to experience their next disease relapses more quickly than other individuals. This introduces heterogeneity across individuals and produces within-subject correlation in the timing of recurrent events within a given subject so, response rates can be homogeneous within individuals producing within-subject correlation in event times [26,27]. There is a need for a general and flexible model that simultaneously incorporates the effects of covariates, as well as the effect of latent or unobserved variables [28]. Frailty or random effects models incorporate heterogeneity into the estimator by making assumptions about the frailty distribution and incorporating it into the model estimates [29]. Bipolar I disorder is a recurrent disorder for the vast majority of patients, and hospitalization is normally used to control severe symptoms and the goal of treating bipolar I disorder is prevention of relapses/recurrences. In this study, we present a semi-parametric frailty model using penalized approach which accounts for time dependent covariates and heterogeneity in repeated relapses of bipolar I disorder in order to identify some factors that would participate relapses.

Participants and procedure

In a retrospective longitudinal study data of 526 bipolar I patients with at least one relapse, who were hospitalized at Razi Psychiatric hospital in Tehran between 1993 and 2011, were collected from their medical records. The first relapse of each patient is considered as entry in to study. Patients relapse has been followed up during this past 10 years. In survival analysis notation, recurrent event was defined as observing bipolar I disorder symptoms in the first re-hospitalization after discharge and following re-hospitalizations. Patients who had been admitted to other hospitals before or had relapses after being discharged from their first hospitalization were excluded from study because we suppose that hospitalization had happened immediately after relapse. The time to recurrent relapses were considered in months. We use the gap time timescale that is most often used timescale in survival analysis: after an event, the subject starts again at time 0 and the time to the next event corresponds to the number of month that it takes to experience the next event [30]. We assessed number of month patients remained to be re-hospitalized as time to event. Right censoring occurs when a subject leaves the study before an event occurs, or the study ends before the event has occurred [31]. Available data fields included: gender, marital status, family history of bipolar disorder (FH), substance abusing, Regular fluctuation (RF), negative life events (NLE) and types of episode. Patients with abusing any kind of drugs other than cigarettes have the positive substance abuse. One of our limitations in this study is, there are some conditions that have been changed in patient life which we could not record them all during this past 10 years. We just consider patients conditions including: marital status, FH, substance abuse and NLE are not changed during this past 10 years. So any change which noticed in each patient condition, by the view of these 4 factors, they are considered to be out of our study in order to have stability in conditions and avoid too many time varying coefficients. There were cases of relapses in some patients during the study that an individual has the mood fluctuations of bipolar disorder but the symptoms do not meet the criteria for any of the other subtypes or the mood changes are abrupt and irregular [32]. RF factor (Regular Fluctuation) was created for this situations. If a patient in one of clients, was cyclothymia, NOS or schizoaffective then RF is negative. NLE and types of episodes defined according to the Diagnostic and Statistical Manual of Mental Disorders book (DSMIV). NLE include stressful life events that often require sweeping readjustments in a person's life such as; death of a loved one, job loss, major illness or injury. There are 3 types of episode in bipolar I disorder (mania, depression, mixed) which could vary in each recurrence of episodes, so we should recognize it as a time varying covariate [33].

Statistical analysis

In many epidemiological and medical studies, the outcome variable of interest is a recurrent event [17]. In these studies each subject is at risk of experiencing repeated events and investigators are interested in evaluating the effects of covariates on the recurrent event times and predicting the developments of future events [34]. An important issue in recurrent events approach is to consider dependency of observations per subjects, called frailty or latent random variable, which is unobservable but influence the model with the meaning that all sampled individuals into the study are not subject in principle under the same risk of disease recurrence [35]. Frailty model that is useful approach to accommodate dependency of the recurrent event times within the same subject has many uses in survival analysis. Frailty models are extensions of well-known Cox (1972) proportional hazards model and are used when proportional assumption holds not true [36,37]. A very common situation in survival analysis is clustered or repeated data. The dependence usually arises because of the recurrence of an event for the same individual. Frailty is a multivariate approach to account the presence of the correlation between the recurrent survival times [38]. In statistical terms, frailty model is a multivariate random effect model for time-to-event data and each individuals repeated relapses are assumed to share the same frailty which is why this model is called shared frailty model [39]. Statistical analysis is complicated by presence of time-dependent covariates, factors that may vary in each relapse of same individual like types of episode in our study [37]. Semiparametric penalized frailty model is a recurrent frailty model that is suitable when there are time-varying covariates. The advantage of this model is that it can easily evaluate frailty parameter and time varying covariates. R3.2.0 software was used to analyze data. In this software with default packages, results could evaluate without problem.

Frequencies of all time-independent variables which are fix in each relapses of one individual are shown in Table 1. In studied sample 64.8% (341) of patients were males, 60.4% (318) were single and 58.2%% (306) with-out family history of bipolar disorder. 56.4% (297) of patients did not experience negative life events, 52.4% (276) have substance abuse and 60.0% (316) have not RF (Table 1). The type of episodes that may vary in each relapses of same individual is our only time-dependent variable that is shown in Table 2. Entirely 2889 relapses were registered for 526 patients. 2519 of these were for BP I and 391were for NOS or cyclothymia or schizoaffective. Maximum number of recurrence was 24 but 98.01% of relapses happened up to 9^th recurrenc.77.25% of relapses were in mania phase, 3.65% were mixed and 19.09% were depressed (Table 2).

We used semi-parametric penalized frailty model to identify which of these 7 factors could participate next relapse. Usually, it is assumed that the frailty follows a gamma distribution with mean 1 and unknown variance θ. The value θ=0 corresponded to independence and a high value of θ should preferably correspond to a high correlation between the survival times [40]. Results of this analysis are shown in Table 3. Variance of frailty parameter (θ=0.287) is highly significant (p<0.001), suggesting present of heterogeneity among data. In other words in this sample there is correlation between repeated relapses of each patients and with-out considering this frailty of each patients result would be incorrect. Among risk factors, substance abuse (p=0.041), RF (p=0.002) and marital status (0.009) were significant, suggesting these factors have effects to the time of next relapse of patients (Table 3).

Table 1: Frequency of risk factors in patients with bipolar I disorder.

Table 2: Frequency of recurrences in 3 types of episodes.

Table 3: Parameter estimation in semi-parametric penalized frailty model.

Even though frailty is an un-measurable factor but if we have density function of frailty we can have information around heterogeneity of individuals. Shape of shared gamma frailty distribution of our sample is shown in Chart 1. It is a gamma function with mean=0 and variance=0.287. Quartiles are 0.609, 0.906 and 1.287 respectively. We can see that the frailty among patients in our study is not same (Chart 1). Overall hazard function show that hazard increased by time generally. It means with-out considering any covariates in study hazard of next relapse increased by time (Chart 2).

Chart 1: Distribution of frailty.

Chart 2: Overall hazard function over time.

About 77.25% of episodes were manic, 19.09% depressed and 3.65% mixed (Table 2); these results were close to those of the previews study which reported 74% of index episodes to be mania, 21% to be depressed, and 5% to be mixed [16]. Clearly, unobserved patients characteristics have a very substantial effect on survival so the result will be bias whit out considering individuals heterogeneity. Heterogeneity is considered in terms of frailty and variance of frailty (theta) measures the variability of frailties. If θ differs from zero indicating the presence of unobserved heterogeneity [41], so in the significant level of .05 the frailty should be considered in the model (Table 3). From the shape of frailty distribution (Q1/Q2=.672 and Q3/Q2=1/420) we see that patients with frailty at Q1 have 33% lower risk, and patients with frailty at Q3 have 42% higher risk, than patients with median frailty (Chart 1). However the interpretation of the hazard ratios in the frailty model is slightly different from the other models and the hazard ratios are compared in the given same value of frailty [42]. In semi-parametric penalized frailty model substance abuse have significant effect on hazard rate of relapses and patients with equal frailty who use substance have 0.34 times more chance to have next relapse. Substance abuse may prolong bipolar symptoms, and the behavioral control problems associated with mania can result in a person drinking too much [1] this results support the suggestion that individuals with bipolar affective disorder complicated by substance abuse may have more hospitalizations [43]. Marital status is another effective factor. Single patients have 0.37 times more chance to have next relapse compared to married patients at the same level of frailty although in other studies with cox models, which needs initial assumption that always not hold, marital status was not significant [44]. There is relationship between marital status and substance abuse with onset of bipolar disorder, but our study suggested that these two factors could postpone the time of next relapses. RF factor is also significant. Patients have not RF have 0.45 times more chance to experience next relapse than who have RF. It implies that there should be more attention in treatment procedure and diagnostic criteria in confront of patients with irregular pattern. We found that type of episodes and sex have no influence on time to relapses in bipolar I disorder patients. Recent study with using Cox regression model in the 12-month followup period study, has reported no significant differences in recovery from hospitalization between patients with mania compared with mixed bipolar disorder and another study found that bipolar women experienced an equal greater risk of recurrence than men [45]. Negative life event has not significant effect on outcome although most studies with-out considering frailty have found that negative life events precede episodes of bipolar individuals [46]. It shows that when we consider individuals heterogeneity in model this factor lose its efficiency. It seems negative life events have not direct effect on hazard of next relapse but it depends on each individual to how handle it. Family history has not significant effect on time to relapse although have effect on incident of disorder and children with a parent or sibling who has bipolar disorder are much more likely to develop the illness, compared with children who do not have a family history of bipolar disorder [1]. Single patients with substance abuse and with-out irregular fluctuation have more chance to experience next relapse earlier.

Substance abuse, RF and marital status are important risk factors influencing hazard of time to relapses. Having a good relationship with the spouse and relinquishing addiction could help to postpone the next relapses. Significance of RF factor suggesting further studies around borderlines and treatments of types of bipolar disorder. The course of bipolar I disorder seems to be progressive in nature irrespective of gender, type of disorder, FH and NLE. More studies are required to clear out the effect of other covariates such as place of residence (urban vs. rural), education level, economic status of patients on time to relapses with this model.

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