Bayesian survival analysis of time-to-remission of prostate cancer and associated prognostic factors in a health-based institute, South-West Nigeria

Background: Prostate cancer (CaP) develops when healthy cells in the prostate gland change and grow out of place, forming a tumour. In Nigeria, this disease is on an upward trajectory, despite the availability of screening services. This study seeks to estimate the time-to-remission and to determine the prognostic factors for prostate cancer remission. Method: A retrospective analytical study design was employed, through the extraction of case �les of patients diagnosed and treated for CaP from January 2010 to December 2017 at the University College Hospital, Ibadan. The extracted data were further divided into two cohorts 2010 to 2014 and 2015 to 2017 to account for non-treatment months. Kaplan Meier method was used to estimate the time-to-remission. Bayesian parametric (Weibull) Accelerated Failure Time (AFT) model was used to determine the factors associated with time-to-remission. Results: The average age(SD) of the patients was 72(3.65) years, with peak incidence among those aged 70 – 79 years. Most CaP patients (87.3%) were diagnosed at stage IV, with many having metastasis to the spine. Among the patients, (33.6%) received chemotherapy and surgery. Patients from Northcentral part of Nigeria had the highest Median Time to Remission (MTR) of 3.7 months in the (2015 - 2017) cohort. The MTR for the 2010 - 2014 cohort was 2.9 months, 3.3 months for the 2015 - 2017 cohort while the overall MTR for the study was 3.2 months. Ages 60 – 69 years and 79 years and above in the 2010 – 2014 cohort decelerated time-to-remission by 30% (adjusted Time Ratio (aTR) = 1.3; 95% CrI (Credible Interval): 1.1 – 1.5) and 40% (aTR = 1.4; 95% CrI: 1.1 – 1.8) respectively. Time-to-remission was signi�cantly delayed by 40% (aTR = 1.4; 95% CrI: 1.1 – 1.7), and 230% (aTR = 3.3; 95% CrI: 2.1 – 4.9) for patients from the south-south and north-central respectively for the


Background
The Prostate is a walnut-sized organ located just between the bladder and the male reproductive organ, and it slowly grows larger to an average weight of 40 grams in ageing men [1,2].A prostate gland surrounds the urethra that empties urine from the bladder and also secretes prostate uid that protects sperm [3,4].These physiological functions may be compromised during various prostate diseases including prostatitis, benign prostatic hyperplasia, and cancer [5,6].Prostate cancer (CaP) develops when healthy cells in the prostate gland change and grow out of place, forming a mass called a tumour [4,7].Pathologic evidence suggests that neoplastic changes of the prostate epithelium develop early in a man's adult life, but do not become clinically evident or problematic until decades later.Some patients live out their lives with prostate cancer that remains stable for decades without treatment.In other cases, CaP grows aggressively, responds poorly to therapy, and causes death within a few years [8,9].According to Global Cancer Statistics 2018, lung cancer is the leading cause of cancer death (18.4% of the total cancer deaths), followed by female breast cancer (11.6%), and prostate cancer (7.1%) been third in the hierarchy [10].Prostate cancer is the second most common and fth-most aggressive neoplasm among men worldwide [2,10].About three-quarters of CaP cases worldwide occur in men aged 40 years and above with increasing incidence and morbidity in men of black African ancestry [11][12][13].In 2012, 1.1 million men were diagnosed with CaP and 70% of them (795,000 cases) were in developed countries [9,14].A CaP lifetime prevalence of one in seven men in the US and one in 25 globally have been reported [10,15].Available reports indicate that CaP accounts for approximately 20.3% and 4.8% of all male cancers in sub-Saharan Africa and North Africa, respectively [11,16].CaP is on the upward trajectory among Nigerian men and remains one of the most common cancers among men in this region [17].A CaP prevalence rate of 1046 per 100,000 men of age > 40years screened was reported in Lagos, Nigeria [18,19].Compared to African-American men, Nigerian men are 10 times more likely to have prostate cancer and 3.5 times more likely to die from it [18][19][20].
Environmental, clinic-pathological factors and most importantly treatment regimen, have been incriminated as the reason for geographic variations in incidence and mortality [9,14].Studies have reported that the main risk factors of CaP are age, inheritance known as race and/or family history, smoking, benign prostatic hypertrophy (BPH) and diet [2,[21][22][23][24][25]. CAP has been reported to be more prevalent among older people, those with poorer diet, having animal fat and/or dairy products or smokers.The chance of having prostate cancer rises rapidly after age 50 [21,22].About 60% of prostate cancer are found in men older than 65 years [22].In particular, the American Cancer Society a rmed that CaP develops more often in African-American and African-Caribbean men more than in men of other races in the world [22].Some studies have demonstrated an association between BPH and prostate cancer, while others have not [21,22].While some studies have explored stages at the presentation of CaP and explored associated risk factors in Nigeria using statistical frequentist approaches, we are not aware of any study that combined robust methods such as Bayesian statistics and survival analysis to assess prognostic factors of Cap in Nigerian.Hence, this study seeks to estimate the time to remission and to determine the prognostic factors for prostate cancer remission among people seeking care at a Nigerian tertiary health facility using a Bayesian survival analysis method.

Study design
A retrospective analytical study design was employed.This was done through a review and extraction of hospital records.The case les of all patients diagnosed and treated for prostate cancer from January 2010 to December 2017 at the University College Hospital (UCH) Ibadan were be included in the study.

Study setting
The study site was the Department of Radiation Oncology, University College Hospital, Ibadan, Nigeria.The facility serves as a referral centre for secondary health facilities in the southern part of Nigeria.It is one of the leading facilities for the management of different types of cancer in Nigeria [26].The hospital has over 500-bed spaces and has been used for the study of cancer survivorship in Nigeria [26].

Study Population
This study consists of all Prostate cancer patients who were managed for prostate cancer at the University College Hospital, Ibadan from January 2010 to December 2017.

Data Collection Procedure
A proforma was used to extract relevant data from patients' records, with the assistance of medical record clerks.The extracted data were divided into two cohorts 2010 -2014 and 2015 -2017 to account for non-treatment months.A total of 354 patients with 90% of required records was extracted.110 experienced remission, while 54 were censored in the 2010-2014 cohort.Also, in the 2015-2017 cohort, 115 patients experienced remission, while 75 were censored; with a follow-up time of 15 months observed in each cohort.To ensure data reliability, the extracted records were cross-checked by a radiotherapist.All methods were carried out in accordance with relevant guidelines and regulations.

Study Variables
Dependent variable: The dependent variable was the time to remission measured in months from time of presentation to time patient was last seen.A patient or an individual is declared to have experienced remission when they become prostate cancer-free or have no residual disease on medical examination after completion of treatment.Meanwhile, patients that were either still under follow-up, lost to follow-up, death from CaP without having remission or other diseases were censored.Independent variables: The independent variables for this study include patients' socio-demographics and clinical characteristics.

Data Analysis
Descriptive statistics such as frequency, proportions, the mean and standard deviation were used to describe the patient's socio-demographic characteristics, clinical characteristics and treatment modality.Kaplan-Meier method was used to estimate the time-to-remission.Bayesian parametric (Weibull) Accelerated Failure Time (AFT) model was used to determine the factors associated with time-to-remission.This approach has been reported to perform better than the frequentist approach and the Cox semi-parametric models in the literature [27,28].STATA version 15 was employed in the analysis.All statistical tests were two-tailed with a 5% signi cance level.

Estimation of time-to-remission
Due to the presence of right-censored data, the Kaplan-Meier method was used to assess survivorship in this study.Suppose we have p distinct survival times arranged in increasing order, say t 1 < t 2 <. .. < t p. Wherein, at time t i there are n i subjects who are said to be at risk i.e. they survived up to this time and were not censored.The number of subjects who failed at time t i is denoted by d i .Consider the recursive equation Where t 0 = 0 and S 0 = 1 Therefore, the survival function is given as The survival time t is the period (months) to remission, from time of diagnosis to time last treated/seen, since all patients are at risk of this event.The censoring index for patients who experienced remission is "1" whereas patients who are is still undergoing treatment, died from CaP or a disease other than CaP, patients who can no longer be observed (Loss-to-followup), are declared censored and coded as "0".

Bayesian Parametric (Weibull distribution) Accelerated Failure Time model
Bayesian analysis generates conclusions based on the synthesis of new information from the observed data and previous knowledge or external evidence.The Weibull model is one of the accelerated failure time models that can be used to t survival data under the Bayesian framework.It was adopted in this study as it tted the data more than other parametric models.The accelerated failure time implies a deceleration of time or, in other terms, an increment in the expected waiting time for an event of interest to occur.
Let denote the survival time of the i th patient.Assuming that has a Weibull distribution with space and shape parameters and having a density function of the form: 0< <∞.

The survival function of is given by
The hazard function is given by The likelihood function of the unknown parameters given the data can be written as: = [29][30][31][32] where is an indicator variable taking value 1 if is the failure time (i.e time-to-remission) and 0 if is right censored (loss-follow-up, death, still under follow up).
To incorporate covariates we, therefore, write where and are p x 1 vector of covariates and regression coe cients respectively.
Assuming a normal prior for , then the joint posterior is given by where D = (n, t, X, ) denote the observed data for the regression model and X is an n x p matrix of covariates with the i th row as and The "bayes", "streg", and "tr" packages in STATA 15 was used to t the Bayesian AFT model using the random-walk Metropolis-Hastings Monte Carlo Markov Chain (MCMC) sampling method for simulating the results of the parameters.
For each covariate in the model, a Time Ratio (TR) > 1 implies that an individual experienced the event at a later timing, TR = 1 suggest no difference in the timings, while TR < 1 suggest an earlier timing.
The Metropolis-Hastings algorithm is given as follows: Let be a proposal density which is also termed as a candidate generating density such that .Also, let denote the uniform distribution over (0,1).Then, a general version of the Metropolis-Hastings algorithm for sampling from the posterior distribution P(θ|x) can be described as follows: Step 0. Choose an arbitrary starting point and set i = 0.
Step 1. Generate a candidate point from and u from Step 2. Set if u ≤ a and otherwise, where the acceptance probability is given by Step 3. Set , and go to Step 1.
The iteration steps from 0 through 3 is referred to as an MH update.By design, any Markov chain simulated using this MH algorithm is guaranteed to have the posterior distribution as its stationary distribution.
Criteria's for measuring the e ciency of MCMC are the acceptance rate of the chain and the degree of autocorrelation in the generated sample.When the acceptance rate is close to "0" then most of the proposals are rejected, which means that the chain failed to explore regions of the appreciable posterior probability.The cut-off for acceptance rate is 0.234 for a multivariate posterior and 0.45 for a univariate posterior.The -2loglikelihood, Bayesian Information Criterion and Deviance Information Criterion (DIC) were also used to evaluate model performance.

Distribution of prostate cancer (CaP) patients by sociodemographic characteristics
Three hundred and fty-four patients were aged 50 -104 years, with an average age of 72years and a standard deviation of 9.52.  1.  3).
Among patients with no family history of prostate cancer, time-to-remission was 3.1 months, while it was 2.9 months for patients in the well-differentiated Gleason group and 3.3 months for patients in the poorly differentiated Gleason group.Time-to-remission was 2.3 months for patients diagnosed at stage I and 3.3 months for patients diagnosed at stage II and IV respectively.

Discussion
This study accessed the time-to-remission among people with prostate cancer and assessed the determinants of prostate cancer remission.We found that the incidence of CaP was highest among patients older than seventy-two years.This nding is corroborated by outcomes of an earlier study by Ifere et al. [33].CaP is known to be a disease of the aged with very few incidences observed among those below forty years, and very recurring as men progress in age.Advanced age is associated with an aggressive form of this disease [34] and a predictor for delayed remission as shown in this study.
The highest median time-to-remission in this study was observed among the CaP patients from the north-central region of Nigeria.This could be attributable to race and geographical variation and the distance patients had to travel to receive treatment.This is consistent with the literature [2,21,22].A further plausible explanation could be the lack of functional treatment facility, industrial action by health workers, thereby resulting in a patient presenting at a time where remission could be negatively affected.This opinion was also shared by Zhang et al. [35].Most patients either drink alcohol and/or smoke tobacco, these are known determinants for lung cancer, but it is not yet known if it could be a deterrent to CaP remission [36].
The majority of the diagnosis was done via DRE, closely followed by a biopsy, this is because DRE can indicate whether a prostate biopsy is recommended for a patient, especially in more aggressive cases [37].Being diagnosed with DRE shortened time-to-remission.This could be due to its ability to quickly detect the hardness of the prostate indicating the presence of CaP.DRE is known to be an independent marker for a better prognosis for CaP [38].However, this was not consistent with our ndings which showed that diagnosis via transrectal ultrasound, biopsy and a combination of DRE and biopsy were markers that decreased time-to-remission.
The absence of a parent or rst-degree male relative with a known case of CaP did impact the likelihood of experiencing remission among the CaP patients.Given the genetics and hereditary factors that come to play from one generation to another, it further highlights that there is every likelihood of this disease not recurring in another generation and is in agreement with an earlier nding [39].
A well-differentiated Gleason group was associated with better prognosis however moderately and poorly differentiated Gleason group was associated with delay time-to-remission.It was also observed that a large proportion of patients presented with either a moderately or poorly differentiated Gleason score.This could be due to a lack of adequate awareness of the importance of frequent medical check-ups and early presentation [40,41].As reported by Akinremi et al., most patients presented at an advanced stage [19].This has become a recurring theme among Nigerian CaP patients.This may be a case of negligence and lack of regular awareness on the part of policy-makers.While on the part of the patients, distance and cost of receiving care may be responsible.As shown in this study late presentation was also a contributing factor to long time-to-remission and patients in this category often experience recurrence [38].This is consistent with ndings in a Ghanaian study [42].This shows that early presentation particularly at stage I could shorten a patient's timeto-remission.
Most patients presented with metastasis especially in the spine and others with multiple metastases [19,43].It has been observed that before diagnosis, patients usually go through a series of assumptive diagnoses.Such that before the detection of such cancer, it has metastasized beyond the prostate region.Much more effort is needed particularly by policymakers to ensure early diagnosis and frequent screening practices.Paramount to remission is the management of CaP as most patients still experience residual and possible recurrence given the spread of this disease to advance areas, especially in the case of multiple metastases.
An elevated Prostate Speci c Antigen (PSA) level at diagnosis is usually associated with poor disease prognosis, although due to differences in technical performance and reference standards, PSA measurements from different labs are not necessarily comparable.However, approximately 98% of patients with metastatic CaP will have elevated PSA [44].In this study, PSA level did not affect treatment outcome.Observed among patients in this facility was the inconsistency of PSA levels during treatment.This could be due to a lack of proper care and lack of nance for consistent treatment on the part of patients, as some patients may miss a certain number of treatments before continuing treatment.Treatment via a combination therapy of chemotherapy, surgery and radiotherapy shortened time-to-remission and showed a better prognosis [45].This may not be unrelated to the unaffordability of treatments among most patients, meaning very few patients have access to treatments that are capable of effectively triggering remission.Patients who received radiotherapy alone also experienced remission.However only a few patients received this treatment regimen, this could be due to the unavailability of a radiotherapy machine, coupled with frequent breakdown and workers industrial action [46][47][48].
A limitation of this study lies in the secondary nature of the data extracted limited our choice of explanatory variables and the completeness of the data used.It would be necessary to access this data in a prospective longitudinal study.Although the tertiary hospital used in this study attracts patients from all over the country, the patients could have been more skewed to the Southwest, so our ndings on geographical differences should be interpreted with caution.Notwithstanding, we have used a robust statistical estimation procedure that enhances the reliability of our estimates while also taking into consideration the violation of the assumption of proportionality.

Conclusion
Prostate cancer still ranks among the second causes of cancer death among men.However, this trend could be deterred through awareness and early presentation which could mitigate against metastasis and progression to advanced stages.
This study showed geographical variation in time to remission as apparent in some European and North American studies.
We, therefore, conclude that central to the experience of remission in prostate cancer treatment are the age of the patient, family history of the disease, clinical factors such as the method of diagnosis, stage at diagnosis, site of metastasis and treatment received.Efforts should therefore be intensi ed to encourage regular prostate examination among men 40 years and above as well as an emphasis on early presentation.

Table 3
Table 4the result of the multivariate Bayesian Weibull Accelerated Failure Time model showed that CaP patients aged 60 -69 years and 79 years and above in the 2010 -2014 cohort decelerate time-to-remission by 30% (adjusted Time ) for the 2010 -2014, and 2015 -2017 cohorts and overall respectively compared to patients who were currently married.The time-to-remission decelerated more than twice for patients who were factory workers for the 2015 -2017 NB: CBC = can't be computed; Overall = 2010 -2017 NB: CBC = can't be computed; Overall = 2010 -2017 Bayesian Weibull Accelerated Failure Time model analysis In