Bayesian joint modelling of multiple longitudinal biomarkers and survival time of cardiac patients under clinical follow-up

Abstract

Introduction

Cardiac disease (CD) is a leading cause of death worldwide. Longitudinal studies often involve multiple patient biomarkers measured over time. Simultaneous monitoring of these biomarkers alongside time-to-death outcomes is crucial for understanding disease progression and informing clinical decision-making.

Methods

This study apply Bayesian joint model (BJM) to analyze multiple longitudinal biomarkers alongside time-to-death data and identify factors influencing the survival of cardiac patients. The data comes from Cardiac Center-Ethiopia, which comprises 323 children diagnosed with cardiac disease. The data contains biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) alongside time-to-death outcomes.

Results

The Bayesian Joint Model with current value and slope association structures provided the best fit for the data. The findings indicated that both the current levels and the rate of change in biomarkers were significantly associated with patient survival. Lower biomarker levels were linked to a higher probability of survival, whereas elevated levels were associated with an increased risk of mortality. Factors such as low oxygen saturation, uncorrected cardiac surgery, reduced ejection fraction, and lower hemoglobin levels negatively impacted biomarker profiles and shortened survival time. Additionally, patients with congenital heart conditions and those experiencing undernutrition exhibited lower survival probabilities.

Conclusions

The findings underscore the crucial role of identifying and utilizing biomarkers to improve survival of patients. We recommend the use of BJM with current value and slope association structures for analyzing longitudinal and time-to-event data to identify factors influencing the survival of patients.

Introduction

Cardiac disease (CD) is the leading cause of death and a significant contributor to disability and morbidity worldwide [1], CD includes congenital and acquired cardiac conditions affecting infants, children, and adolescents [2]. Early diagnosis and management are important for improving cardiac outcomes and reducing morbidity and mortality [3]. The identification and utilization of biomarkers in children are also vital for early diagnosis and risk management [4].

The survival of children with cardiac disease is critically dependent on medical and surgical interventions [2]. Pediatric professionals assess the health status of children using biomarkers such as oxygen saturation (SpO₂), pulse rate, and blood pressure. Monitoring these biomarkers facilitates the early detection of potential health issues, allowing for timely intervention and improved management of cardiac conditions and other related health problems [4, 5].

Accurate and continuous monitoring of biomarkers is vital for several reasons, including the assessment of cardiac function, evaluation of disease progression, guidance for clinical interventions, and early detection of complications [4, 5]. It helps clinicians assess disease severity and guide treatment decisions, such as determining the appropriate timing for surgical interventions [6].

The cardiac condition of children is indicated by longitudinal biomarkers and may affect their survival [4]. Longitudinal and survival data are expected to be correlated. The primary focus here is the analysis of the association between longitudinal biomarkers and children’s survival time.

To achieve this objective, joint models for longitudinal and survival data have been employed. This approach is appropriate when longitudinal biomarkers are correlated with survival data [7, 8]. The structure of a joint model for longitudinal and survival data consists of three components: (i) a model for time-to-event data, (ii) a model for longitudinal data, and (iii) a framework linking components (i) and (ii) through various association structures [7–9].

Most joint model formulations focus on the simple association between longitudinal and survival data, and extensions have not been extensively studied. Thus far, the most common association structure involves shared parameters to capture the relationship between longitudinal and survival data [10, 11]. However, the joint modelling approach can further incorporate different association structures, including current value, current value and slope, and shared random effects [12–14]. The aim of this paper is to apply joint modelling of longitudinal biomarkers and survival data using different association structures, with the goal of identifying key factors affecting the survival of pediatric cardiac patients under follow-up.

Research method

Study design

An observational study design was employed [16], allowing researchers to investigate the effects of certain factors without manipulating exposure. A retrospective cohort study design was used, in which the cohort is identified from past records and outcomes are measured moving forward from that point. This design was chosen due to the high cost of conducting a prospective study [17]. Inclusion and exclusion criteria were applied, and efforts were made to minimize loss to follow-up in order to reduce potential selection bias.

Study population and sampling

The study population consists of cardiac patients under follow-up. Patients under the age of 18 years were recruited for the study. The study includes patients followed from September 2021 to August 2022. A simple random sampling technique [18], along with a checklist, was used to select representative samples. The sample includes patients who attended at least three follow-up visits. Patients with fewer than three follow-ups or with missing information were excluded from the study.

Sample size determination

Before data collection begins, determining the sample size is a crucial step. The sample size was determined based on financial affordability, time constraints, and the data analysis method [18]. When determining the sample size , it must meet, at a minimum, the following criteria: optimum predictor effect, small absolute difference, and precise estimation [19]. A sample size that satisfies all three criteria provides the optimal values required for the study [19].

We designed to fit a model with time-to-event outcomes, we ensure the sample size was adequate in terms of the number of participants () and events () relative to the number of predictor parameters (). We assume there are potential predictors for inclusion in the model, and for suitable choice ofand based on previous study, the required sample can be determined [19, 20]. We need to include predictor used in the model with targeted expected shrinkage of 0.90, and the required sample size can be obtained by:

1

Therefore the minimum sample size required for the study is 323 participants.

Data

Data were obtained in both primary and secondary forms. However, due to the absence of complete information in the secondary data, additional data were collected from patients’ medical charts. In Ethiopia, there is a standardized patient chart form developed in accordance with World Health Organization (WHO) standards. Data were manually extracted from these charts using a structured questionnaire. The questionnaire was pre-tested on randomly selected patients’ charts to ensure data quality. Data collection began after obtaining ethical clearance from Kotebe University of Education and permission from the Cardiac Center–Ethiopia. The data collection protocols strictly adhered to confidentiality standards to ensure the protection of sensitive information. Data were collected by trained nurses.

Missing data in longitudinal studies is a common challenge arising from factors such as participant dropouts or intermittent data collection issues [21]. It can be classified as Missing Completely at Random (MCAR), Missing at Random (MAR), or Missing Not at Random (MNAR). In this study, the missing data are considered to be MCAR, meaning they are independent of both observed and unobserved data [21].

Variables in the study

The study variables include the following response variables: longitudinal measurements of systolic blood pressure (SBP), diastolic blood pressure (DBP), pulse rate (PR), and time to death due to cardiac disease. The biomarkers were measured repeatedly during patient follow-ups, starting from the time of diagnosis until death, loss to follow-up, or ongoing follow-up. The predictor variables considered are listed in the diagram in Fig. 1.

Fig. 1.

Study variables and their assumed relationships

The bayesian joint model

In this section, we define the Bayesian Joint Models (BJM) and the modelling approach for multiple longitudinal biomarkers and a single time-to-event variable with censoring. The model consists of three components: (i) a linear mixed sub-model for the longitudinal process, (ii) a survival sub-model for the time-to-event process, and (iii) a parameterization component that links both processes [7, 22]. The association structures explored in this study include joint modelling approaches using dynamic random effects (intercept and slope values) and shared effects [12–14].

Longitudinal Sub-model

The longitudinal data are repeated measurements on a specific response from a patient taken over a period of time. These data taken from a patient are correlated. Each patient can have subject-specific pattern over time [23, 25]. Three biomarkers data are considered.

Let be the observations on systolic blood pressure (SBP), be observations on diastolic blood pressure (DBP), and be observations on pulse rate (PR) for patient at visiting time . The linear mixed effects models for the three biomarker responses as adopted from [23, 24, 26] are defined by:

2

where denotes the design matrix consisting of predictors of the respective response and a vector of 1’s for constant intercept, combined with vector of fixed effects . For the random component, denotes the design matrix consisting of predictors combined with vector of random effects . The number and type of predictors can be different for the each response may be different. All the random errors and random effects in each of the linear mixed model (2) are assumed to be pairwise independent and each having normal distribution. Moreover, some of the predictors can be time varying.

Survival Sub-model

Survival model refers to statistical methods for analyzing time-to-event data [27]. An event time is the time elapsed up to the occurrence of an event of interest such as death or disease diagnostic [27, 28]. There are different models for time-to-event data. The most popular is the semi-parametric Cox proportional hazard model [27], which does not require distributional assumption for hazard function and covariates are assumed to have a multiplicative effect on the hazard of an event [27].

Let denote the failure time, is the censoring time and is the observed time for subject. The Cox PH model can be written as:

3

where is the baseline hazard function at time and is the vector of predictor variables with corresponding regression coefficients .

The joint model formally associates the longitudinal and survival sub-models through association parameters [12–14, 29]. To explain the association between the multiple longitudinal outcomes and time-to-event, we assume that the distribution of , survival time for the subject, depends on subject-specific random effect and/or linear predictor. Then hazard model becomes:

4

where denotes the history of the underlying longitudinal mean process up to time for patient , as defined in Eq. (2). denotes the baseline hazard rate, and is a vector of covariates combined with fixed coefficients in the Cox regression in Eq. (3). The vector relates components of longitudinal outcomes to predictor of survival outcomes [12–14, 29]. It quantifies the association between the longitudinal and survival processes.

Association Structure.

The function in Eq. (4) indicate that the hazard of an event may be associated with the biomarkers and which lead to different forms of association between the longitudinal biomarkers and time-to-event outcomes.

• First one is the current value association in which longitudinal measure is predictive of the hazard of event at the same time .

• Second one is current value plus slope association . This extends the first structure by adding the slope term at time .

• Third one is the shared random effects association , which includes only the random effects from the longitudinal sub-model [12–14].

Here the common association structures between longitudinal and time-to-event outcome are presented as follows:

1. Association structure takes into account current value of longitudinal biomarkers at time t is defined as

   5

    So, the time-to-event sub-model is rewritten as

   6

2. Association structure includes both current or true value of longitudinal biomarkers and slopes of true biomarkers trajectory at time t is defined as

   7

   So, the time-to-event sub-model is rewritten as

   8

3. Association structure takes into account random time trend or shared random effects is defined as

   9

   So, the time-to-event sub-model is rewritten as

   10

Here represents a parameter for the current true value , associating the longitudinal biomarker and the hazard of the event at time , and is a parameter for the slope value , associating the slope of the longitudinal biomarker and the hazard of the event at time .

Likelihood function and posterior distribution

Likelihood function

We assume that observations of the longitudinal outcomes and the survival time are conditionally independent given the random effects and parameters of the longitudinal sub-model, and also the random effects and parameters of the survival sub-model. The random effects are and . The joint likelihood functions [30, 31] can be written as products of individual functions in two forms:

11

12

Prior distribution

Weakly informative joint prior distribution of the parameters was considered. Fixed effects parameters and are assumed to be independently and identically normally distributed with mean zero and large variance 1000. The association parameters , are each assumed to have normal distribution with mean zero and variance 100 and variance component have .

Posterior distribution

Bayesian estimation is defined as in [32] by posterior distribution as the product of the likelihood function and prior distribution with normalizing constant. The posterior distribution of the model parameters given the observed data, was derived under the assumptions that the longitudinal outcomes were independent of the event time. Conditional independence was assumed on random effects and parameters. The posterior distribution is given by:

13

where is the joint likelihood function, is the joint prior probability distribution, and is the normalizing constant.

Analytical solution of expression (13) is not available since it involves high dimensions and intractable expression. The Markov Chain Monte Carlo (MCMC) is used because of its flexibility in the implementation and availability of high computing power [29, 33].

Results and discussion

Exploratory data analysis

This study included 323 children with cardiac disease (CD). Among them, 182 (56.35%) were female, and 56.04% were under the age of five. The mean age of the patients was 7.73 years. The majority of the children, 215 (66.56%), resided in urban areas. Regarding the occupational status of the families, 43.65% of fathers and 30.03% of mothers were employed by the government. Approximately 74.82% of households had fewer than four family members (Table 1).

Table 1.

Socio-demographic characteristics of study participants under follow-up

Covariates	Attributes	No. of cases	(%100)	Survival Status
				Death (%)	Censored (%)
Gender	Boy Girls	141 182	43.65 56.35	17(12.06) 13(7.14)	124(87.94) 169(92.86)
Age (months)	< 24 24–59 60–143 >=144	52 129 92 50	16.10 39.94 28.48 15.48	8(15.38) 10(7.75) 7(7.61) 5(10.00)	44(84.62) 119(92.25) 85(92.39) 45(90.00)
Birth order	First Second and third Forth or more	123 159 41	38.08 49.23 12.69	8(6.50) 18(11.32) 4(9.76)	115(93.50) 141(88.68) 37(90.24)
Residence	Urban Rural	215 108	66.56 33.44	12(5.58) 18(16.67)	193(94.42) 100(83.33)
Father occupation	Employed Farmer Merchant Other	141 98 34 50	43.65 30.34 10.53 15.48	10(7.09) 14(14.29) 2(5.88) 4(8.00)	131(92.91) 84(85.71) 32(94.12) 46(92.00)
Mother occupation	Housewife Employed Merchant Other	182 97 20 24	56.35 30.03 6.19 7.43	21(11.54) 7(7.22) 0(0.00) 2(8.33)	161(88.46) 90(92.78) 20(100.00) 22(91.67)
Family history of CD	No Yes	296 27	91.64 8.36	26(8.78) 4(13.33)	270(91.22) 23(86.67)
Family size	< 4 >= 4	242 81	74.92 25.08	22(9.10) 8(9.88)	220(90.90) 73(90.12)

Of the 141 male and 182 female involved in the study, the death rate was higher in males (12.06%) compared to females (7.14%), and higher in urban residents (16.67%) than in rural residents (5.58%). Congenital heart disease (CHD) is more prevalent than acquired heart disease, with 71.21% of children diagnosed with CHD and 29.79% with acquired heart disease. Among the study participants, 16.10% had pulmonary hypertension, and 35.29% had a history of anemia. Undernutrition was identified in 59.29% of the children, with a higher prevalence in those with acquired heart disease (61.29%) than in those with congenital heart disease (58.29%). Of the study participants, only 30 (9.28%) experienced an event of interest (death), while the remaining 293 children were censored. Table 2 summarizes the descriptive statistics of the clinical covariates. There are 47 (14.55%) corrective surgeries were performed, and 276 (85.45%) underwent surgery during the study period. Moreover, 110 (34.05%) patients had no history of heart failure, 125 (38.69%) had mild heart failure, and the others had moderate-to-severe heart failure (Table 2).

Table 2.

Summary of clinical variables of the study participants under follow-up

Covariates	Attributes	Total	(%100)	Survival Status
				Death (%)	Censored (%)
Cardiac disease	Congenital Acquired	230 93	71.21 28.79	24(10.43) 6(6.45)	206(89.57) 87(93.55)
Chamber enlargements	Yes No	167 156	51.70 48.30	22(13.17) 8(5.13)	145(86.83) 148(94.87)
Pulmonary hypertension	Yes No	52 271	16.10 83.90	14(26.92) 16(5.90)	38(73.08) 255(94.10)
Corrective surgery	Yes No	47 276	14.55 85.45	3(6.38) 27(9.78)	44(93.62) 249(90.22)
Nutritional status	Under weight Healthy weight	192 131	59.40 40.60	27(14.06) 3(2.29)	165(85.94) 128(97.71)
ROSS/NYHA	I II III IV	110 125 62 26	34.05 38.69 19.19 8.05	2(1.82) 5(4.00) 9(14.52) 14(53.84)	80(98.18) 140(96.00) 57(85.48) 16(46.15)
Heart failure	No Mild Moderate to severe	110 125 88	34.05 38.69 27.24	2(1.82) 5(4.00) 23(26.14)	80(98.18) 140(96.00) 65(73.86)
Anemia	Yes No	114 209	35.29 64.71	14(12.28) 16(7.66)	100(87.72) 193(92.34)
Pneumonia	Yes No	58 265	17.96 82.04	9(15.52) 21(7.92)	49(84.48) 244(97.02)
Cardiomyopathy	Yes No	78 245	24.15 75.85	12(15.38) 18(7.35)	66(84.62) 227(92.65)

Pneumonia was identified in 17.96% of children, and anemia was also common in the study area, with a prevalence of 35.29%. Looking into the family history, 27(8.36%) had a history of cardiac disease, and 129 (39.96%) of the study participants were aged between (24–59) months followed by 92(28.48%) between (60–144) months.

Analysis of survival data

The children were followed-up for 24 months. The mean survival time of children with cardiac disease aged 18 years and younger based on a sample of 323 patients was 16.75 months with a median and standard deviation of 17 and 5.95 month respectively.

The mean survival time was calculated to be 16.75 months, indicating that, on average, children survived close to 17 months after diagnosis or treatment initiation. The median survival time, which represents the midpoint of the data, was 17 months, meaning half of the children survived longer than this duration, while the other half had shorter survival times. Additionally, the standard deviation of 5.95 months highlights the variability among patients, showing that survival times often deviated by approximately six months from the average. This spread underscores the differences in individual outcomes, likely influenced by various factors. Such statistical information is crucial for healthcare planning, enabling medical teams to better understand typical survival durations, identify critical periods for intervention, and explore strategies to improve survival rates.

The graph of the Kaplan-Meier estimate of survival probability is shown in Fig. 2, which indicates a decreasing pattern of the survivorship function as expected. The plots show the pattern of one survivorship function lying above another, indicating that the groups defined by the lower curve had a lower survival probability than those defined by the upper curve. This clear visual differentiation highlights disparities in survival outcomes, which could be attributed to variations in patient characteristics, treatment approaches, or disease severity.

Fig. 2.

Kaplan Meier survival plots of children with cardiac disease under follow-up

Looking into survival probability of children; non-anemic have higher experience of survival probability than anemic and children resident in urban have higher survival probability than those resident in rural. Comparing survival probability based on NYHA/ROSS; the survival probability was lower for children in stage IV closely followed by stage III, in concordance with several other studies. Furthermore, pulmonary hypertension is also lowers the probability of survival in patients with cardiac diseases.

Models selection for survival data

We select an optimal time-to-event model. To do so, we selected the Cox Proportional Hazards (Cox PH) model [27] that relates the time with occurrence of the event (time-to-death) and it doesn’t depend on distributional assumptions [34]. It is considered a semi-parametric model because it does not rely on specific assumptions about the distribution of survival times, making it more flexible [35].

The proportional hazard assumption is key assumption in the Cox proportional hazard model [27]. The hazard ratio between the groups was constant over time. Testing this assumption is crucial because a violation could lead to biased results. To test this assumption, we conducted a global test with Schoenfeld residuals [36].

Variable selection was performed, and for all variables, the PH assumption was tested. The analysis showed that the tests for all covariates were not significant and the global test was also not statistically significant. Therefore, we can assume proportional hazards for variables; residence, corrective surgery, anemia, types of cardiac disease, economic stand, pulmonary hypertension, nutritional status, ejection fraction and hemoglobin levels as reported in Table 3. This suggests that the hazard ratios for these variables remain constant over time, supporting the validity of the Cox Proportional Hazards model for the study, with reliable and interpretable hazard ratio estimates.

Table 3.

Test of proportional hazard assumption

Covariates	Chi-square	p-value
Residence	0.129	0.720
Corrective surgery	2.364	0.271
Anemia	1.762	0.543
Types of Cardiac Disease	0.086	0.770
Economic status	1.342	0.452
Pulmonary hypertension	0.193	0.660
Nutritional status	0.044	0.830
Ejection fraction	1.415	0.230
Hemoglobin	2.041	0.461
Global	2.136	0.910

Analysis of longitudinal data

The data consists of three longitudinal outcomes: systolic blood pressure (SBP), diastolic blood pressure (DBP) and pulse rate (PR) which were followed retrospectively for 24 months. Before further analysis, the distributions of the data are checked. The normality assumption of the data was checked using histograms and Q-Q plots with corresponding Shapiro-Wilk test [37]. The result revealed that normality assumption is hold.

The Box-Cox transformation [38] is a statistical technique designed to stabilize variance and make data more closely follow a normal distribution, which is often a prerequisite for many statistical modelling [39]. This transformation is guided by a parameter called lambda , which typically ranges from − 5 to 5 and determines the specific mathematical form of the transformation [39]. Lambda determines the type and degree of transformation applied to the data. For instance, when λ equals 1, the data remains unchanged, while a λ of 0 corresponds to a logarithmic transformation [40]. Other values of λ result in various power transformations that can compress or emphasize data depending on its skewness [40]. The transformation is particularly useful in regression analysis, and parametric statistical tests, where it enhances the reliability and interpretability of results by ensuring assumptions of normality. However, the Box-Cox method has limitations, such as requiring data to be positive and non-zero. Overall, the Box-Cox transformation is a versatile tool for making statistical data analysis more robust and effective [40, 41]. Based on the graph depicted on Fig. 3 transformation is suggested only for systolic blood pressure.

Fig. 3.

Box-Cox plots of the SBP, DBP and PR data respectively

When assessing normality, we use both graph and formal test to check the distribution of the data. A histogram offers a first view; a bell-shaped and symmetric histogram is expected from normal distribution [42]. Complementing this, the Q-Q plot compares the data’s quantiles to those of a theoretical normal distribution, if the points lie close to reference line; it signifies normality, while systematic deviations suggest departures like skewness or atypical tail behavior [43]. Formal tests: the Shapiro-Wilk [44] and Kolmogorov-Smirnov (K-S) [45] test examines the null hypothesis that the data are normally distributed, where a high p-value (commonly above 0.05) indicates no significant deviation from normality.

Based on the collective evidence, the data appear to be normally distributed. The histogram displays a bell-shaped, symmetric distribution, and the Q-Q plot shows that the data points follow the reference line closely, and both (Figs. 4 and 5) visually suggest normality. In addition, both the Shapiro-Wilk and Kolmogorov-Smirnov (K-S) tests produced p-values, indicating that there is insufficient evidence to reject the null hypothesis of normality (Table 4). All these methods together support that the data are indeed normally distributed.

Fig. 4.

Histogram of the SBP, DBP and PR data respectively

Fig. 5.

Quantile-Quantile (Q-Q) plots of the SBP, DBP and PR data respectively

Table 4.

Test of normality of the SBP, DBP and PR data respectively

Data	Statistic	Estimate	Sign.
SBP	Shapiro-Wilk	0.939	0.497
	Kolmogorov-Smirnov (K-S)	0.037	0.378
DBP	Shapiro-Wilk	0.922	0.444
	Kolmogorov-Smirnov (K-S)	0.118	0.216
PR	Shapiro-Wilk	0.995	0.116
	Kolmogorov-Smirnov (K-S)	0.136	0.143

When study longitudinal data, detecting the differences between and/or within individuals overtime is important. The plot depicts in Fig. 6 reveals the variability and change in pulse rate over follow-up periods and shows that there was a shift in children pulse rate. It indicates that children begin with varying pulse rate and differ at different follow-up periods. The profile plot for diastolic blood pressure and pulse rate are given in the appendix.

Fig. 6.

Individual profiles plot of pulse rate data of the patients

Model selection for longitudinal data

We select models for the longitudinal data. We use of the linear mixed-effects model and test for six different models. The first model is intercept, the second is intercept and slope, the third is intercept and quadratic slope, and the fourth to sixth are cubic spline effects of time with different numbers of nodes. To do so, we fit models and compare them on the basis of their Bayesian Information Criterion (BIC), smaller value is preferable [46]. Accordingly, based on the analysis of data, the random intercept and slope model is selected for all three outcome variables (SBP, DBP, PR) reported in Table 5.

Table 5.

Model selection for longitudinal data

Models	SBP	DBP	Pulse
	BIC	BIC	BIC
Random intercept	5240.450	9959.483	1383.60
Random intercept and slope	5226.579	9945.963	1377.77
Random intercept and quadratic slope	5233.787	9952.760	1380.76
Splines with nodes 2	5233.388	9958.485	1383.48
Splines with nodes 3	5239.640	9965.619	1382.85
Splines with nodes 4	5244.776	9965.818	1389.64

The analysis confirms that the random intercept and slope model is the optimal choice for all three biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) (Table 5). This model effectively captures both the initial levels (intercepts) and rates of change (slopes) of these biomarkers over time, making it ideal for understanding their dynamic patterns and variability in longitudinal data.

Comparison of joint models

Finally, we combined the two sub-models in joint modelling. Linear mixed-effects model is used for longitudinal data and Cox PH model for survival data. We checked the sub-model quality to ensure that the joint model gives a reliable estimate. Indeed, joint model is done.

Model selection was done for the joint models. The results of the joint model fit with various association structures are given in Table 6. In practice, the association structures are unknown, we can compare different association structures and select the best one via model selection criteria. To do so, we rely on the Widely Applicable Information Criteria (WAIC), Leave-P-out Marginal Likelihood (LPML) and Deviance Information Criterion (DIC) [46, 47], and smaller values indicating better model fits to the data [46, 47]. Table 8 reports WAIC, DIC and WAIC values for the fitted joint models with different association structures.

Table 6.

Joint model comparison with different association structures

Association structures	DIC	WAIC	LPML
Current	29102.27	78236.23	−11256.21
Current + slope	28445.30	53507.77	−10823.49
Shared	30904.84	79418.98	−10970.03

Table 8.

Results of bayesian joint model with current value and slope association structures

Survival sub-model
Covariates		SD	HR (95% CI)
Reside: Urban	−1.428	0.146	(0.042, 1.209)
Corrective: Yes	−0.532	0.013	(0.399, 0.899)
Eco.: Average	−1.966	0.040	(0.028, 0.703)
: Above	−1.835	0.044	(0.040, 0.636)
CD: congenital	1.673	0.125	(1.428, 4.913)
Pulmonary: Yes	1.690	0.314	(1.161, 7.622)
Anemia: Yes	1.429	0.522	(1.401, 12.342)
BMI	1.577	0.393	(0.880, 9.776)
Ejection	−0.124	0.035	(0.820, 0.939)
Hemoglobin	−1.073	0.027	(0.168, 0.696)
(SBP)	0.021	0.010	(1.001, 1.045)
(SBP)	0.034	0.042	(1.012, 1.057)
(DBP)	0.167	0.074	(1.046, 1.339)
(DBP)	0.231	0.177	(0.685, 2.321)
(Pulse Rate)	0.001	0.014	(0.974, 1.027)
(Pulse Rate)	0.021	0.010	(1.008, 1.049)

Based on the criteria, the evidence is strong to prefer the joint model with current value and slope association structures, which includes the current value and the current rate of change longitudinal biomarkers (SBP, DBP, and PR) as predictors of the risk of death (Table 6).

Joint models that incorporate both the current values and rate of change (slope) of biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) are powerful tools for predicting health outcomes. The current value of a biomarker reflects its level at a current time and provides insight into current risks, whereas the slope indicates how the biomarker is changing over time, offering a view of the individual’s health trajectory. By combining these two aspects, joint models can account for both current conditions and long-term trends, thereby enhancing the predictive accuracy.

Bayesian joint model analysis

The results from Sect. 4.2 and 4.3 are jointly combines in this section. The results of Bayesian joint model analysis are reported in Tables 7 and 8. The analysis shows that birth order, corrective surgery, pulmonary hypertension, weight, hemoglobin, and oxygen saturation significantly associated with pulse rate of cardiac patients. However, under nutrition, pulmonary hypertension, and cardiomyopathy contribute to an increase in systolic blood pressure (SBP).

Table 7.

Results of bayesian joint model with current value and slope association structures

	Longitudinal sub-model: SBP			Longitudinal sub-model: DBP
Covariates		SD	95% CI		SD	95% CI
Intercept	112.191	1.249	(60.33,164.30)	57.308	0.325	(42.36,72.36)
Obs. Time	1.399	0.436	(0.585, 2.244)	0.316	0.004	(0.123, 0.530)
Age	−1.848	0.643	(−3.560,−0.526)	−0.343	0.241	(−1.00, 0.333)
Weight	1.227	0.112	(−0.970, 3.370)	0.426	0.194	(0.043, 0.805)
Cardiac: Congenital	3.066	0.596	(0.899, 5.233)	0.398	0.033	(−1.652, 2.443)
Surgery: Yes	−1.747	0.499	(−3.122,−0.375)	−0.168	0.218	(−0.322,−0.022)
BMI	−0.354	0.221	(−0.702,−0.008)	−0.243	0.232	(−0.504,−0.028)
ROSS-II	0.878	0.108	(−2.412,4.163)	0.298	0.468	(−1.207, 3.808)
ROSS-III	1.047	0.236	(−3.106,5.197)	0.238	0.264	(0.081, 1.262)
ROSS-IV	1.665	0.393	(0.344,3.984)	1.032	0.360	(0.541, 1.548)
Pulmonary: Yes	3.168	0.203	(2.753, 4.182)	1.140	0.117	(0.598, 1.805)
Cardiomopthy: Yes	1.241	0.056	(0.743,1.702)	2.563	0.042	(1.098, 4.113)
Ejection	−0.194	0.038	(−0.357,−0.077)	−0.014	0.043	(−0.105, 0.072)
	Longitudinal sub-model: Pulse Rate
Covariates		SD	95% CI
Intercept	170.63	1.014	(149.63,192.62)
Obs. Time	0.123	0.124	(−0.119, 0.356)
Birth order	2.514	0.053	(1.044, 3.993)
CD: Congenital	−1.956	0.238	(−6.246, 2.411)
Surgery: Yes	−7.930	0.545	(−12.75, −2.813)
Pulmonary: Yes	0.739	0.496	(0.407, 1.071)
Anemia: Yes	2.574	0.205	(−1.307, 6.514)
Weight	−0.875	0.125	(−1.117, −0.631)
Ejection	−0.140	0.094	(−0.326, 0.042)
Hemoglobin	−1.466	0.286	(−2.633, −0.327)
SpO2	−0.210	0.081	(−0.383, −0.073)

Furthermore, weight loss, uncorrected surgery, undernutrition, pulmonary hypertension, and cardiomyopathy are significantly associated with higher diastolic blood pressure (DBP) in cardiac patients. The result also indicates the patients with pulmonary hypertension and those who have not undergone corrective surgery show higher systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR). Moreover, patients in the late clinical stage show significant increases in both SBP and DBP (Table 7).

The results from the survival sub-model reveal that corrective surgery, pulmonary hypertension, anemia, ejection fraction, hemoglobin levels, and congenital cardiac disease are significantly associated with survival of patients. Corrective surgery improves outcomes, while pulmonary hypertension, anemia, low hemoglobin, and reduced ejection fraction decrease survival probabilities. Congenital cardiac disease significantly lowers survival probability, and uncorrected surgeries further worsen outcomes (Table 8).

Based on the result reported in the Table 8, we found the longitudinal biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) as significant predictors of survival of cardiac patients. The association parameters and represent and quantify the association between these biomarkers and survival outcome (time-to-death). Specifically, represents the association of the current values of SBP, DBP, and PR with survival outcome, highlighting the current risk by those markers. On the other hand, capture the impact of the rate of change (or slope) of these biomarkers over time, reflecting how trends or worsening patterns influence survival outcome. Together, these parameters emphasize the dynamic interplay of both current and changes in biomarkers, which are crucial in understanding and predicting the survival probabilities of cardiac patients. This knowledge can guide clinicians in developing more effective monitoring and intervention strategies.

The results for systolic blood pressure (SBP) indicate the significance of both its current value and rate of change as predictors of survival of cardiac patients. The association parameter , with a hazard ratio (HR) of 1.021 (95% CI: 1.001, 1.045), indicate that for every unit increase in the current value of SBP, the hazard of death increases by approximately 2.1%. Similarly, the parameter , with an HR of 1.035 (95% CI: 1.012, 1.057), indicates that for unit increase in the rate of change of SBP, the hazard of death rises by about 3.5%. These findings suggest that both the current value of SBP and change over time are critical in assessing the survival probabilities of cardiac patients. The dual importance of these predictors underscores the value of continuous monitoring and analysis in clinical practice for managing patient outcomes effectively (Table 8).

The findings for diastolic blood pressure (DBP) highlight the greater significance of its current value compared to the rate of change as a predictor of survival of cardiac patients. The association parameter , with a hazard ratio (HR) of 1.182 (95% CI: 1.046, 1.339), indicates that for unit increase in DBP’s current value is associated with an approximately 18.2% increase in the hazard of death. Meanwhile, the parameter , with an HR of 1.260 (95% CI: 0.685, 2.321), reflects the impact of DBP’s rate of change, though its predictive strength appears less pronounced and the confidence interval is wider. This suggests that, at a given time point, the current DBP level holds more relevance in survival outcomes than its change over time. These results underscore the importance of timely clinical monitoring and management of diastolic blood pressure levels in cardiac patients, as the current values offer more actionable insights into their survival probability (Table 8).

However, the analysis of pulse rate (PR) reveals insightful findings regarding its association with survival outcome. The association parameter with a hazard ratio (HR) of 1.001 (CI: 1.046, 1.339), suggests an effect of the current value of PR on the hazard of death. On the other hand, 021, with an HR of 1.021 (CI: 1.008, 1.049), highlights the effect of the rate of change of PR on survival outcomes, showing that even small increases in PR trends over time can increase hazard of death. Importantly, the lower values of PR are associated with a higher probability of survival or reduced risk of death, indicates that maintaining lower and stable pulse rates could improve survival outcomes (Table 8).

The correlation between the intercepts and slopes of subject-specific random effects of longitudinal biomarkers was examined in the joint model. The correlations presented in Table 9 offer crucial insights into the relationships between the intercepts and slopes of longitudinal biomarkers. The negative correlation between the slope of systolic blood pressure (SBP) and its intercept () suggests that individuals starting with higher SBP levels tend to experience slower increases (or even decreases) in their SBP over time. On the other hand, the positive correlation between the slopes of SBP and diastolic blood pressure (DBP) () indicates that the rates of change in these two measures are aligned, where individuals experiencing an increase in SBP are likely to see a concurrent rise in DBP, signaling a change in overall blood pressure. Furthermore, the moderate inverse correlation between the slope and intercept of DBP () demonstrates that higher initial DBP levels are associated with slower rates of increase in DBP. The negative correlation between the slope of SBP and pulse rate () implies that as SBP increases more gradually, pulse rate trends may stabilize or decrease, suggesting a dynamic interplay between vascular pressure and cardiac response. These correlations provide valuable insights into the interrelationships between biomarkers and underscore their importance in understanding patient-specific trajectories for better clinical management.

Table 9.

The correlation matrix of the random effects of the joint model

Random effects		SBP		DBP		PR
		Intercept	Slopes	Intercept	Slopes	Intercept	Slopes
SBP	Intercept	1	-----	-----	-----	-----	-----
	Slopes	−0.714	1	-----	-----	-----	-----
DBP	Intercept	0.134	−0.106	1	-----	-----	-----
	Slopes	−0.116	0.404	−0.585	1	-----	-----
PR	Intercept	0.458	−0.438	0.035	−0.006	1	-----
	Slopes	−0.366	0.429	−0.085	−0.016	−0.695	1

Assessment of convergence

Convergence assessment plays a crucial role in the analysis of joint models, as it ensures that the model has adequately converged and that the estimated parameters are stable and reliable. Convergence of the Markov Chain Monte Carlo (MCMC) process was monitored using trace plots. Figure 7 provides a visual representation of the parameter estimates. By examining the plots, we observed patterns indicating convergence, with the chains stabilizing around a common point, signifying the convergence of the estimated values. The trace plots also indicate stable mixing of chains, with no evidence of divergent transitions.

Fig. 7.

Trace plots for estimate of parameters

Prediction for longitudinal and survival data using bayesian joint models

The results of prediction of longitudinal and survival data using the Bayesian joint model are illustrated in Fig. 8. Our aim was to estimate the probability of an event occurring at a certain point after the last observation, taking into account that the individual is censored immediately afterward. The figure presents two types of predictions: the mean trajectory of biomarkers, represented by the dashed line, with a 95% confidence interval (gray region), and the hazard probability, represented by the solid line, with a 95% confidence interval (red region). The mean trajectory of biomarkers provides insights into the expected progression of systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) over time for cardiac patients. The hazard probability offers an estimate of the risk associated with these changes and helps us understand the potential outcomes for these patients.

Fig. 8.

Predictions of longitudinal and survival data using Bayesian joint models

Based on the analysis, individual are predicted to live longer than what occurred in reality as cumulative risk is slowly growing. This can be justified by small percentage of death in the data, and resulting in slow growing of cumulative risks in individuals.

Discussion

In this study, we employed a Bayesian joint model (BJM) to analyze multiple longitudinal biomarkers alongside a time-to-event outcome. While traditional joint models typically address a single longitudinal and time-to-event outcome [48, 49], longitudinal studies often involve patients with multiple biomarkers measured over time. Simultaneously monitoring these biomarkers and time-to-event outcomes is crucial for understanding disease progression and informing clinical decisions. In this paper, we propose a Bayesian joint model that integrates multiple longitudinal biomarkers, specifically systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR), with survival data [29, 33].

The longitudinal and survival outcomes are linked through various association structures [12–14]. In this paper, the linear mixed effects sub-model for longitudinal data and the Cox proportional hazards (PH) sub-model for survival data were linked through current value, current value and slope, and shared random effect association structures. The results indicate that the Bayesian joint model (BJM) with current value and slope association structures offers more efficient and accurate parameter estimates compared to existing association structures. Consequently, we report the findings of the BJM using the current value and slope association structures.

The results indicate that the biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR) are significantly associated with the survival of cardiac patients. This suggests that these biomarkers can be used to monitor the likelihood to experience an event. An increase in these biomarkers corresponds to a decrease survival of patient and aligns with the study by [50, 51]. Additionally, under nutrition was found to reduce survival time [51], aligning with our findings. Moreover, corrective surgery is significantly reduces SBP and pulse rate [52, 53], thereby enhancing the survival probability of cardiac patients.

As expected, the survival of children varies with the severity of disease, and survival is higher for children with no history of pulmonary hypertension concordance with result in [54]. The survival of children with pulmonary hypertension is 41.94% lower compared to no history of pulmonary hypertension. This rate is lower in other studies, which range from 16 to 25% [53, 55].

Hemoglobin and ejection fraction are also factors for the survival of children. Hemoglobin at measurement is significantly decrease the pulse rate by 1.466 and in line with the study by [54, 55]. Children who have corrective surgery have improved biomarkers and results in higher survival probability [53, 55]. Additionally, anemia can lowers survival of children by 17.45% and concordance with the result in [53].

Conclusions

This study introduced a Bayesian joint model to analyze multiple longitudinal biomarkers (SBP, DBP, and PR) alongside time-to-death data among cardiac child patients. The longitudinal and survival sub-models were linked through various association structures, including current value, current value and slope, and shared parameter frameworks. Model comparison using the Widely Applicable Information Criterion (WAIC), Leave-P-out Marginal Likelihood (LPML), and Deviance Information Criterion (DIC) indicated that the Bayesian joint model with the current value and slope association structure provided the best fit to the data.

The results indicate that both the current values and the rate of change in the biomarkers; systolic blood pressure (SBP), diastolic blood pressure (DBP), and pulse rate (PR), were significantly associated with the survival of cardiac patients. Patients with lower or steeper trajectories of these biomarkers have a higher probability of survival, while elevated biomarkers are associated with an increased risk of mortality.

Factors such as low oxygen saturation, uncorrected surgery, declining ejection fraction, and lower hemoglobin levels negatively affect the biomarker responses, and thereby shortening survival time. Congenital cardiac patients and those suffering from under nutrition exhibit lower survival probabilities. These findings underscore the crucial role of identifying and utilizing biomarkers to improve survival of cardiac patients. We recommend use of the Bayesian joint model (BJM) with current value and slope association structures for analyzing multiple longitudinal and time-to-event data to identify the factors influencing the survival of children with cardiac disease. Future research can also extend these models by using different types of association structures, and inclusion of more covariates.

Rationale and Limitations of the Study

The study employed a Bayesian framework to jointly analyze longitudinal biomarkers (SBP, DBP, PR) and survival outcome (time-to-death) data, capturing the dynamic interplay between patients’ health markers and survival outcomes. Limitation of this study were, eventhough key clinical variables were considered, the study acknowledges the potential for improved accuracy with more covariates, such as medication adherence, socioeconomic status, or genetic factors. The results may not generalize across different populations or types of cardiac diseases, as the model was tailored to a specific cohort.

Abbreviations

AIC

Akaike’s information criteria

BIC

Bayesian information criteria

BJM

Bayesian joint model

BMI

Body mass index

CAIC

Consistent akaike’s information criteria

DBP

Diastolic blood pressure

DIC

Deviance information criterion

HQIC

Hannan–quinn information criteria

K-S

Kolmogorov-smirnov statistics

LMM

Linear mixed effect model

LPML

Leave-p-out marginal likelihood

MLE

Maximum likelihood estimation

PR

Pulse rate

SBP

Systolic blood pressure

WAIC

Widely applicable information criteria

Authors’ contributions

The study was designed by TN and AT. Data analysis and initial drafting of the manuscript were carried out by TA and AT. The manuscript was revised by TA, AT and MM. All authors have read and approved for the final manuscript.

Funding

No funding.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval and consent to participate

Ethical approval to undertake the studies was obtained from the Cardiac Center-Ethiopia with reference number (ref. no: chfe/9815/17/2024). Informed consent was obtained from all participants. The study strictly adheres to the principles outlined in the Declaration of Helsinki.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.