Application of group LASSO regression based Bayesian networks in risk factors exploration and disease prediction for acute kidney injury in hospitalized patients with hematologic malignancies

During Oct. 1st, 2014 and Sept. 30th, 2015, a retrospective cohort study was conducted in Zhongshan Hospital of Fudan University, a tertiary hospital in eastern China. Patients who had a diagnosis of lymphoma, leukemia or multiple myeloma were enrolled as the study participants. Patients who hospitalized less than 24 h, underwent dialysis or renal replacement therapy (RRT) and lacked the repeated serum creatinine (SCr) tests were excluded from the final analysis [16, 17].

Patients’ demographic data, medical history, clinical diagnosis, anti-tumor treatment, biochemical tests, and other information were extracted from the hospital electronic medical records system and laboratory database. Baseline biochemical results refer to the first test within 24 h during hospitalization. We divided them into 3 parts: (1) Liver function: alanine aminotransferase (ALT), aspartate aminotransferase (AST) and total bilirubin (TBiL); (2) Renal function: SCr, the estimated glomerular filtration rate (eGFR) and serum uric acid (SUA); (3) Other: albumin, hemoglobin, white blood cell (WBC), serum sodium and potassium.

According to the KDIGO guideline in 2012 [18], AKI is defined as an absolute increase in SCr by ≥0.3 mg/dL within 48 h or ≥ 1.5-fold from the baseline within seven days. Since the urine output cannot be dated accurately, we only used the SCr changes for AKI diagnosis. The severity of AKI was divided into Stage 1: SCr increases ≥0.3 mg/dL or ≥ 1.5–fold to 1.9-fold baseline; Stage 2: SCr increases ≥2.9–3.0 fold baseline; Stage 3: SCr increases ≥3.0 fold baseline or ≥ 4.0 mg/dL, or the initiation of RRT [18]. According to the 10th revision of International Classification of Diseases (ICD-10), the hematologic malignancies in this study included lymphoma (C91-C95), leukemia (C81–85) and multiple myeloma (C90) [19]. Anti-tumor treatment was divided into autologous stem cell transplantation (ASCT), chemotherapy and untreated/palliative care. The baseline reference levels of serum sodium and potassium were 137~147 mmol/L and 3.5~5.3 mmol/L. Values below or above the reference level were defined as hypo−/hypernatremia and hypo−/hyperkalemia. The normal values of eGFR and SUA were set as ≥90 mL/min/1.73m² and ≤ 359 μmol/L, respectively. Anemia refers to hemoglobin < 115 g/L, and hypoalbuminemia refers to albumin < 35 g/L.

The absolute shrinkage and selection operator (LASSO) is a shrinkage method within least square method that enables to shrink estimation of continuous variables towards zero [20]. In order to handle the categorical variable, the Group LASSO (gLASSO) is extensively developed to perform the predefined grouping variable selection instead of single dummy variable selection. Assuming that we have J groups of categorical variables {G₁,G₂, …,G_j} and each of them had p₁,p₂, … p_j levels, the gLASSO estimator \( {\hat{\beta}}^{GrLasso} \) is presented as:

By adjusting penalty l₁ and l₂, the candidate variables can be selected in group level and remain invariant in group orthogonal transformation such as ridge regression. The coefficients in one group will either all be zero or all nonzero. The penalty functions of grLasso, grMCP, and grSCAD carry out group selection, while the gel and cMCP penalties carry out bi-level selection. The point estimation of fitted lambda (λ) along with the regularization path is selected according to AIC, BIC, or GCV criteria. Then, k-fold cross-validation for penalized gLASSO models is performed to plot a grid of values for the regularization parameter lambda (λ). The lambda.min refers to the optimal variable selection with the minimum cross-validation error. Compared with the logistic model, gLASSO performs better on multi-collinear or high-dimensional data.

The Bayesian networks (BNs) consists of two parts: a directed acyclic graph (DAG) and its subsequent conditional probability distribution (CPD). In the BNs, variables are graphically represented by the nodes X = {X_i, …, X_n} and the relationship between two nodes is connected by a unilateral arc. If the arc is going from X_i to X_i + 1, we defined the X_i as the parent node and X_i + 1 as the child node. CPD is acquired to quantify the probabilistic relationships between parent and child nodes. The global distribution factorization of X in BNs model could be specified as:

π (X_i) refers to the set of the X_i's parent nodes π (X_i)∈{X_i, …, X_n-1}, and the graphical separation refers to the conditional independence relationships between (X_i) and {X_i, …, X_i-1}. BNs modeling contained structure learning and parameter learning. The structure learning is acquired from data and can be traced to 3 algorithms: constraint-based, score-based and hybrid algorithms. Parameter learning refers to applying either maximum likelihood (ML) estimation or Bayesian estimation method to compute the CPD of nodes in the established network. BNs inference is achieved by computing the posterior probability of X in the presence of new evidence E. When E changes, conditional probability distributions of both parent and child nodes are also affected. There are two algorithms for BNs inference, logical sampling algorithm and likelihood weighting algorithm, and the latter has a lower variance.

Pearson chi-square test was used to compare the distribution differences of categorical variables and Cochran-Mantel-Haenszel (CMH) test was used for ordinal variables. The crude odds ratios (cOR) and its 95% confidence interval (CI) were calculated to quantify the association between factors and AKI. The analysis was run on IBM SPSS 22.0 (IBM Corp., Armonk, NY, USA), and the threshold of type I error (α) was set to 0.05. The process of variable selection in gLASSO regression was as follows: ① category variables were decomposed into dummy variables and their group label was assigned into another parallel dataset; ② the dummy and group datasets were analyzed in “grpreg” packages of R program 3.6.0 (R core team); ③ grLasso penalty and BIC criteria were used to estimate the fitted lambda (λ); ④ 10-fold cross-validation was performed to screen the optimal variable selection with the minimum cross-validation error. Then, the selected preditors further created a Bayesian network in “bnlearn” packages in the R program. The tabu-search algorithm was chosen to establish the BNs structure, and the ML method was used to acquire the CPD parameters. The area under the receiver operating characteristic curve (AUC) was applied to assess the prediction ability of the BNs model. A 10-fold cross-validation was also performed for internal validation and reducing the overfitting bias. The model diagram was drawn in Netica 5.18 (Norsys Software Corp., Vancouver, BC, Canada). Weka 3.8.0 (Waikato Environment for Knowledge Analysis, the University of Waikato, New Zealand) was used for model estimation.

A total of 370 (15.4%) episodes were diagnosed with AKI during hospitalization. Of them, 308(12.9%), 41(1.7%) and 21(0.9%) patients were located in AKI Stage 1, 2 and 3, respectively. Twenty patients require RRT. Stratified by HM category, the incidence of AKI in patients with multiple myeloma (24.1%) and leukemia (23.9%) was higher than that of lymphoma (13.4%).

As shown in Table 1, patients under 29 years old had the highest risk of AKI (cOR: 2.16). The AKI incidence was higher in female patients than in the male (18.2% vs. 13.4%). Pre-existing diabetes increased the likelihood of AKI, while such a correlation was not found in patients with hypertension. In comparison to untreated/palliative care, patients receiving ASCT and chemical treatment were more vulnerable to develop AKI (cOR: 4.37 and 2.24 respectively). Liver and renal dysfunction were also found to have a significant association with AKI. Patients with abnormal ALT, AST and SCr values on admission were more likely to develop AKI; insufficient eGFR and increased SUA level also increased the probability of AKI. Patients with initial anemia and hypoalbuminemia had a 2.72 fold and 3.85 fold increased risk of AKI.

The tuning parameter (λ) was specified in gLASSO regression by using 10-fold cross-validation in Fig. 1a. The optimal λ value was highlighted by the vertical lines with a minimizing cross-validation error. When log (λ) was equal to − 4.529, eleven of the initial nineteen variables were selected, including age, gender, diabetes, HM category, anti-tumor treatment, hemoglobin, SCr, eGFR, SUA, serum sodium and potassium levels. Figure 1b presented the gLASSO coefficient (\( \hat{\beta} \)) profiles of candidate variables. When the gLASSO model met BIC criteria(λ = 0.00896), the same predictors and their nonzero coefficients were identified.

Though BNs analysis, we delineated the probabilistic dependencies between HM-related AKI and its preditors in a complex network (Fig. 2). It was observed that age, hemoglobin, eGFR, serum sodium and potassium created direct connections with AKI, while other variables were related to AKI indirectly. For instance, HM category and anti-tumor treatment indirectly linked with AKI via hemoglobin and eGFR, and diabetes had connected with AKI by affecting eGFR level. Moreover, the relationship between covariates can also be given in the network. Hemoglobin was related to gender, HM category and anti-tumor treatment; eGFR was influenced by age, diabetes, HM category, SCR and SUA level. Table 2 manifested the CPD table of AKI, quantifying the relationship between AKI and its parent nodes of eGFR, hemoglobin and serum sodium. Patients whose eGFR < 59 mL/min per 1.73 m² together with anemia and hyponatremia shared the highest AKI incidence (78.5%). In a similar situation but hypernatremia, the probability of AKI was estimated to be 68.3%. In contrast, patients with normal eGFR, hemoglobin and sodium level had the lowest rate (5.2%).

As shown in Fig. 3, the AUC value of BNs model was 0.835 (95% CI: 0.812 to 0.858), which was higher than that of the logistic score model (AUC = 0.763). In 10-fold cross-validation, the AUC maintained at the level of 0.812 (95% CI: 0.787 to 0.837). By using the Mantel-Haenszel test, no statistically significant difference in predictive accuracy was found between initial and cross-validation datasets (p = 0.298). According to the patients’ demographics and limited available clinical records, BNs could infer the individual probability of AKI occurrence during hospitalization. For instance, when anemia, hyperuricemia, and hyponatremia were initially found on admission in patients with leukemia, the expected probability of AKI was estimated to be 53.8% based on the prior information of BNs. However, once these biochemical indicators were corrected to the normal level in time, the risk of AKI can be reduced to 9.9% (Fig. 4).