Interpretable machine learning reveals phosphorus-to-albumin ratio as a novel predictor of mortality and acute kidney injury in critically ill pancreatitis patients: a multi-center retrospective analysis

Abstract

Acute kidney injury (AKI), a common and severe complication of acute pancreatitis (AP), is amenable to early intervention. Phosphorus-to-albumin ratio (PAR) is a novel composite biomarker unexplored in AP largely. Data of ICU patients with AP were extracted from MIMIC-IV database and eICU–CRD, respectively. PAR’s link to prognosis and AKI in AP were analyzed via Kaplan–Meier curves, Cox regression, restricted cubic splines (RCS), and logistic regression. Subgroup analysis tested interactions. Key variables were identified using least absolute shrinkage and selection operator regression. AKI prediction models were built and evaluated via seven machine learning (ML) algorithms. Shapley additive explanations (SHAP) interpreted variable contributions. Survival analysis, Cox models, and RCS collectively demonstrated PAR, as a potential risk factor, is associated with 28-day and 1-year all-cause mortality in patients with AP. Logistic regression identified PAR as a risk factor for AKI development in AP. AKI-related clinical features including PAR were indentified. Seven ML models were constructed, among which the Light Gradient Boosting Machine (LightGBM) model achieved an area under receiver operating characteristic curve (AUROC) of 0.880 (95% CI 0.825–0.935) and area under precision–recall curve (AUPRC) of 0.944 in the test set, and an AUROC of 0.837 (95% CI 0.785–0.889) and AUPRC of 0.784 in the external validation set. SHAP analysis of the LightGBM model confirmed that higher PAR levels corresponded to a higher predicted probability of AKI. PAR is associated with prognosis and AKI of patients with AP. Integrating PAR with other key clinical features, our LightGBM model provides physicians with a streamlined and efficient tool for early AKI identification in high-risk AP patients.

Graphical abstract

Supplementary Information

The online version contains supplementary material available at 10.1186/s40001-026-03930-y.

Introduction

Acute pancreatitis (AP) represents a common digestive system disorder requiring hospitalization with incidence rising steadily in the past decades [1]. Epidemiological data indicate an annual mortality rate of 6.9–11.7 per million [2]. In patients who develop moderate severe AP (MSAP) or severe AP (SAP), the mortality escalates significantly to 30% or more [2]. The increased mortality in these critically ill patients is closely linked to organ failure and secondary infections [3, 4].

It is well-acknowledged that the timely detection of organ failure (involving the respiratory, cardiovascular, and renal systems) and the identification of sepsis are critical for accurate severity classification in AP patients [5, 6]. Therefore, accurate early prediction of acute kidney injury (AKI) in AP patients can alert clinicians to initiate timely interventions, thereby improving patient prognosis. Currently, there is no well-established prediction system widely accepted for forecasting AKI occurrence in this population. For practical clinical application, indicators in the prediction model for AKI should possess the advantages of simplicity, cost-effectiveness, reproducibility, and minimal invasiveness. Machine learning (ML), an emerging field, leverages sophisticated algorithms to integrate diverse candidate variables, enabling the construction of models that provide more precise predictions for disease diagnosis, treatment, and prognosis [7]. Previous studies have developed ML models to predict AKI in AP. However, these models often suffer from either a single-center design lacking robust external validation or the inclusion of excessive variables, compromising clinical utility [8–11]. Therefore, developing a reliable and practical prediction model for early identification of AKI in pancreatitis remains imperative.

As one of the serum electrolytes, higher phosphorus levels measured within 24 h after hospital admission have been associated with higher short-term mortality in pancreatitis when evaluated as a standalone predictor [12]. Previous studies have demonstrated that serum phosphorus levels rise rapidly following the onset of AKI, while restricting phosphorus intake can effectively improve its prognosis [13, 14]. Similarly, previous observational studies have demonstrated strong associations between composite indicators derived from albumin levels and both the severity and prognosis of AP [15, 16]. Moreover, hypoalbuminemia has been identified as an independent risk factor for AKI development in hospitalized patients [17–19]. The phosphorus-to-albumin ratio (PAR) is a recently proposed composite index with its increase simultaneously reflecting the underlying pathologies of hyperphosphatemia and hypoalbuminemia. A multicenter study demonstrated that as an independent prognostic indicator, PAR could also predict the occurrence of neurological sequelae in out-of-hospital cardiac arrest (OHCA) patients [20]. However, to date the prognostic relevance of PAR in patients with AP remains unexplored, and its predictive value for AKI occurrence in this population has yet to be elucidated.

In this study, we first downloaded and cleaned the clinical data of AP patients admitted to the intensive care unit (ICU) between 2008 and 2022 from the Medical Information Mart for Intensive Care IV version 3.0 (MIMIC-IV v3.0) database. Then we analyzed the predictive utility of PAR for the short-term and long-term all-cause mortality, and the occurrence of AKI. Subsequently, predictive models for AKI were developed using seven ML algorithms based on PAR and other key clinical features. Finally, the ideal models was selected and underwent external validation utilizing an independent cohort from the eICU Collaborative Research Database (eICU–CRD). This work aims to investigate the association between PAR and both clinical outcomes and AKI development in AP patients while comprehensively evaluating its predictive value.

Methods

Data source and ethnic consideration

The MIMIC-IV database is a publicly available, free critical care database containing clinically relevant data for all patients admitted to the Beth Israel Deaconess Medical Center (Boston, USA) between 2008 and 2022 [21]. Similarly, eICU–CRD is a publicly available, multicenter ICU database covering clinical data from patients within ICU at over 200 American hospitals during the period 2014–2015 [22]. In this study, Xuan Chen, the first author, has successfully completed the Collaborative Institutional Training Initiative (CITI) Program certification, passing the ethics-related examinations for ‘Conflict of Interest’ (certificate ID: 63153642) and ‘Research with Data or Samples Only’ (certificate ID: 63153641).

Criteria for population selection

In this study, patients with AP were screened out using the International Classification of Diseases, 9th Revision (ICD-9) code 577.0 or ICD-10 codes K85.0 to K85.92, regardless of whether the diagnosis was recorded as primary or other diagnosis. The patients meeting any of the following criteria were excluded: (1) age < 18 years; (2) Length of ICU stay < 24 h; (3) Multiple ICU admissions (only the first admission was retained); (4) Pre-existing end-stage renal disease, liver cirrhosis, or malignancy; (5) Absence of recorded serum phosphate or albumin levels within the first 24 h of ICU stay or absence of recorded serum phosphate or albumin levels within the first 24 h of ICU stay; (6) AKI happened before ICU admission or within 24 h after admission; (7) Patients with chronic pancreatitis or those who are not admitted to the ICU for the first time due to AP. The workflow of screening is illustrated in Fig. 1. Ultimately, 497 patients with AP were selected from the MIMIC-IV database, comprising the initial study cohort. This cohort was subsequently randomly divided into train set and test set at a 7:3 ratio for ML model development to predict the AKI occurrence. There were 235 patients with AP in the eICU–CRD who also met the inclusion criteria selected as the external validation cohort.

Fig. 1.

Workflows for screening patients with AP from MIMIC-IV database and eICU–CRD. AP: acute pancreatitis

Data extraction

The first PAR within 24 h after ICU admission was the primary variable of interest in this study. The value of PAR was obtained by dividing serum phosphate level (mg/L) by the albumin concentration (g/L). To minimize subsequent treatment-related confounding, the first phosphorus level and serum albumin measured within 24 h after ICU admission were extracted and used for calculating the PAR value. All variables were retrieved using structured query language (SQL) within the software PostgreSQL (version 16.0). The SQL script codes needed for extracting data were obtained from the GitHub repository located at https://github.com/MIT-LCP/mimic-code/ and https://github.com/mit-lcp/eicu-code, respectively. For all dynamic variables, such as mechanical ventilation (MV), sepsis, white blood cell (WBC), and creatinine, only the first recorded values within 24 h of ICU admission were included. The details of extracted variables could be found in the Table 1. For the MIMIC-IV cohort, the extracted data comprised five main components: demographic data, laboratory results, comorbidities, clinical interventions, and clinical outcomes. For the eICU–CRD cohort, the extracted data included AKI diagnosis and the other relevant clinical features as input of models: urine output within 24 h after admission (Urineoutput_24hr), MV within 24 h after admission, the first PAR following ICU admission, weight at admission, sepsis, the first WBC following ICU admission, the first creatinine following ICU admission, and requirements for vasoactive drug use within 24 h of admission (Vasoactives_24hr).

Table 1.

Baseline characteristics of MIMIC-IV cohort

Characteristic	Overall	PAR.Q1 (< 0.75)	PAR.Q2 (0.75–1.10)	PAR.Q3 (1.10–1.52)	PAR.Q4 (≥ 1.52)	Statistics	P value
N	497	123	126	123	125
Demographics
Age (year)	58.00 (45.00–71.00)	50.00 (39.00–63.00)	58.00 (47.25–74.00)	63.00 (46.00–75.00)	61.00 (51.00–73.00)	19.904	< 0.001
Gender						5.461	0.141
Female (n, %)	224 (45.07%)	61 (49.59%)	46 (36.51%)	56 (45.53%)	61 (48.80%)
Male (n, %)	273 (54.93%)	62 (50.41%)	80 (63.49%)	67 (54.47%)	64 (51.20%)
Race						2.739	0.434
Caucasian (n, %)	306 (61.57%)	69 (56.10%)	80 (63.49%)	81 (65.85%)	76 (60.80%)
Other (n, %)	191 (38.43%)	54 (43.90%)	46 (36.51%)	42 (34.15%)	49 (39.20%)
Laboratory parameters (first test after ICU admission)
PAR	1.10 (0.75–1.52)	0.56 (0.45–0.67)	0.89 (0.81–1.00)	1.31 (1.18–1.41)	2.14 (1.74–2.67)	465.001	< 0.001
AG (mmol/L)	15.00 (13.00–18.00)	15.00 (12.00–18.00)	15.00 (13.00–17.00)	15.00 (12.00–18.00)	16.00 (14.00–21.00)	15.327	0.002
Alb (g/dL)	2.90 (2.40–3.30)	3.10 (2.80–3.50)	3.20 (2.80–3.60)	2.80 (2.50–3.20)	2.30 (2.10–2.70)	124.57	< 0.001
Phosphate (mg/dL)	3.20 (2.20–4.10)	1.60 (1.30–2.10)	2.90 (2.40–3.20)	3.70 (3.20–4.10)	5.10 (4.10–6.40)	387.609	< 0.001
Calcium (mg/dL)	7.90 (7.30–8.50)	7.80 (7.20–8.40)	8.00 (7.32–8.50)	7.90 (7.40–8.70)	7.80 (7.00–8.30)	5.505	0.138
Chloride (mmol/L)	105.00 (100.00–109.00)	104.00 (100.00–110.00)	105.00 (102.00–109.00)	105.00 (101.00–110.00)	103.00 (98.00–109.00)	10.302	0.016
Creatinine (mg/dL)	1.10 (0.70–1.90)	0.90 (0.60–1.10)	0.95 (0.70–1.30)	1.20 (0.70–2.15)	2.20 (1.00–4.20)	68.476	< 0.001
Glucose (mg/dL)	141.00 (105.00–189.00)	138.00 (105.50–201.50)	140.00 (112.00–190.75)	135.00 (105.00–183.50)	154.00 (104.00–194.00)	1.389	0.708
Hemoglobin (mg/dL)	11.30 (9.70–13.10)	11.90 (10.65–13.30)	12.00 (9.95–13.70)	11.20 (9.75–12.85)	10.20 (8.50–12.10)	37.533	< 0.001
Hematocrit (%)	34.50 (29.60–39.50)	34.80 (32.15–39.70)	35.75 (30.35–41.00)	35.10 (29.65–39.15)	30.90 (27.00–36.70)	26.076	< 0.001
Potassium (mmol/L)	4.10 (3.60–4.60)	3.80 (3.40–4.20)	4.10 (3.60–4.47)	4.10 (3.65–4.60)	4.40 (3.90–5.10)	51.273	< 0.001
Sodium (mmol/L)	138.00 (135.00–141.00)	138.00 (135.00–141.00)	139.00 (137.00–141.00)	138.00 (135.50–142.00)	137.00 (133.00–141.00)	11.425	0.01
Platelet (10^9/mL)	199.00 (140.00–277.00)	174.00 (133.00–232.50)	201.00 (141.50–288.75)	198.00 (142.00–268.50)	233.00 (162.00–314.00)	19.821	< 0.001
RBC (10^12/mL)	3.73 (3.21–4.30)	3.80 (3.37–4.31)	3.96 (3.32–4.42)	3.78 (3.29–4.30)	3.31 (2.97–3.97)	23.964	< 0.001
WBC (10^9/mL)	13.50 (9.40–19.00)	11.20 (8.15–15.40)	13.05 (9.38–19.25)	14.60 (10.65–19.15)	15.40 (10.60–21.90)	21.265	< 0.001
ALP (U/mL)	95.00 (65.00–163.00)	77.00 (60.00–125.50)	95.00 (64.00–162.75)	95.00 (68.50–201.50)	111.00 (71.00–170.00)	13.165	0.004
ALT (U/mL)	48.50 (24.00–146.00)	48.50 (20.00–148.00)	81.50 (35.25–174.75)	47.00 (25.00–146.50)	40.00 (20.00–95.00)	9.514	0.023
AST (U/mL)	70.00 (34.00–149.00)	65.00 (32.50–130.00)	80.50 (38.75–159.00)	63.00 (33.50–178.50)	69.00 (31.00–149.00)	1.329	0.722
BUN (mg/dL)	21.00 (12.00–35.00)	14.00 (9.00–22.50)	17.00 (11.00–26.00)	23.00 (13.50–34.00)	38.00 (20.00–59.00)	93.3	< 0.001
TB (mg/dL)	0.90 (0.60–2.20)	0.90 (0.60–2.10)	0.90 (0.60–2.80)	0.90 (0.50–2.20)	0.80 (0.50–1.90)	2.47	0.481
PT (s)	14.40 (13.00–16.60)	14.10 (12.70–16.20)	14.25 (12.90–15.50)	14.30 (12.95–15.95)	15.20 (13.40–19.00)	14.632	0.002
Scale scores
SOFA	1.00 (0.00–3.00)	1.00 (0.00–3.00)	0.00 (0.00–2.00)	1.00 (0.00–3.00)	2.00 (0.00–5.00)	28.729	< 0.001
SIRS	3.00 (3.00–4.00)	3.00 (2.00–3.00)	3.00 (3.00–4.00)	3.00 (3.00–4.00)	3.00 (3.00–4.00)	9.224	0.026
SAPSII	35.00 (25.00–47.00)	30.00 (19.00–36.50)	30.00 (21.25–41.75)	39.00 (28.00–49.50)	45.00 (35.00–58.00)	75.099	< 0.001
APSIII	57.00 (40.00–84.00)	45.00 (34.00–66.50)	47.00 (32.00–66.50)	59.00 (42.50–85.50)	83.00 (62.00–102.00)	88.949	< 0.001
GCS	15.00 (15.00–15.00)	15.00 (15.00–15.00)	15.00 (15.00–15.00)	15.00 (15.00–15.00)	15.00 (15.00–15.00)	3.839	0.279
Vital signs at ICU admission
HR (bpm)	104.00 (88.00–118.00)	108.00 (93.50–122.50)	103.50 (87.00–121.75)	100.00 (83.00–114.00)	105.00 (89.00–118.00)	8.307	0.04
SBP (mmHg)	128.52 (110.00–144.00)	133.00 (121.50–151.00)	133.00 (117.25–150.75)	124.00 (107.00–140.50)	118.00 (101.00–137.00)	34.37	< 0.001
DBP (mmHg)	73.00 (61.00–85.00)	80.00 (69.00–92.50)	75.50 (64.00–86.75)	70.00 (57.50–84.00)	65.00 (56.00–75.00)	40.828	< 0.001
MBP (mmHg)	87.00 (74.00–99.00)	91.00 (81.50–105.50)	90.00 (77.00–100.75)	83.00 (71.00–99.00)	81.00 (69.00–92.00)	33.584	< 0.001
RR (bpm)	22.00 (18.00–26.00)	22.00 (18.00–25.50)	21.00 (18.00–25.75)	22.00 (17.00–25.00)	22.00 (18.00–27.00)	1.683	0.641
SpO2 (%)	96.00 (94.00–99.00)	96.00 (94.00–98.50)	96.00 (94.00–98.00)	96.00 (93.50–99.00)	98.00 (95.00–100.00)	8.373	0.039
Temperature (°C)	36.89 (36.56–37.39)	37.00 (36.72–37.61)	37.00 (36.57–37.38)	36.89 (36.50–37.44)	36.72 (36.39–37.06)	23.832	< 0.001
Urineoutput_24hr (mL)	1537.50 (875.00–2455.00)	2105.00 (1302.50–3097.50)	1754.00 (1176.25–2526.25)	1452.00 (808.50–2197.00)	1083.00 (385.00–1752.00)	45.814	< 0.001
Weight (kg)	82.10 (70.70–99.00)	79.00 (68.10–93.70)	82.40 (72.60–101.47)	82.70 (73.10–99.75)	82.10 (71.00–94.40)	4.375	0.224
Comorbidities
AF (n, %)	101 (20.32%)	14 (11.38%)	30 (23.81%)	35 (28.46%)	22 (17.60%)	12.614	0.006
AKI (n, %)	341 (68.61%)	65 (52.85%)	79 (62.70%)	89 (72.36%)	108 (86.40%)	35.41	< 0.001
Diabetes (n, %)	162 (32.60%)	39 (31.71%)	47 (37.30%)	37 (30.08%)	39 (31.20%)	1.779	0.62
HF (n, %)	83 (16.70%)	7 (5.69%)	24 (19.05%)	28 (22.76%)	24 (19.20%)	15.028	0.002
Hypertension (n, %)	240 (48.29%)	66 (53.66%)	66 (52.38%)	56 (45.53%)	52 (41.60%)	4.88	0.181
Obesity (n, %)	79 (15.90%)	15 (12.20%)	21 (16.67%)	21 (17.07%)	22 (17.60%)	1.715	0.634
RF (n, %)	231 (46.48%)	35 (28.46%)	54 (42.86%)	70 (56.91%)	72 (57.60%)	28.322	< 0.001
Sepsis (n, %)	351 (70.62%)	69 (56.10%)	76 (60.32%)	101 (82.11%)	105 (84.00%)	37.569	< 0.001
Treatments
CRRT (n, %)	20 (4.02%)	0 (0.00%)	3 (2.38%)	4 (3.25%)	13 (10.40%)	19.385	< 0.001
Betablocker (n, %)	199 (40.04%)	43 (34.96%)	50 (39.68%)	54 (43.90%)	52 (41.60%)	2.22	0.528
ERCP (n, %)	55 (11.07%)	13 (10.57%)	17 (13.49%)	17 (13.82%)	8 (6.40%)	4.498	0.212
Fibrates (n, %)	24 (4.83%)	13 (10.57%)	6 (4.76%)	2 (1.63%)	3 (2.40%)	13.17	0.004
MV (n, %)	279 (56.14%)	49 (39.84%)	61 (48.41%)	74 (60.16%)	95 (76.00%)	37.162	< 0.001
Metformin (n, %)	16 (3.22%)	6 (4.88%)	7 (5.56%)	1 (0.81%)	2 (1.60%)	6.631	0.085
Octreotide (n, %)	28 (5.63%)	4 (3.25%)	4 (3.17%)	7 (5.69%)	13 (10.40%)	8.088	0.044
Statins (n, %)	112 (22.54%)	31 (25.20%)	26 (20.63%)	35 (28.46%)	20 (16.00%)	6.29	0.098
Vasoactives_24hr (n, %)	142 (28.57%)	17 (13.82%)	25 (19.84%)	39 (31.71%)	61 (48.80%)	43.474	< 0.001
Clinical outcomes
LOS ICU (day)	3.69 (1.96–9.25)	2.86 (1.77–5.71)	3.68 (2.05–6.81)	4.38 (2.19–13.57)	5.97 (2.31–12.70)	15.559	0.001
LOS Hospital (day)	13.19 (7.28–23.37)	9.05 (5.76–18.29)	11.02 (6.56–20.82)	15.76 (9.22–23.94)	17.98 (9.92–32.05)	32.042	< 0.001
In-hospital mortality (%)	65 (13.08%)	2 (1.63%)	10 (7.94%)	21 (17.07%)	32 (25.60%)	36.088	< 0.001
28-day mortality (n, %)	58 (11.67%)	3 (2.44%)	8 (6.35%)	16 (13.01%)	31 (24.80%)	34.747	< 0.001
365-day mortality (n, %)	114 (22.94%)	13 (10.57%)	22 (17.46%)	33 (26.83%)	46 (36.80%)	27.427	< 0.001

PAR, phosphorus-to-albumin ratio; Alb, albumin; AG, anion gap; RBC, red blood cell; WBC, white blood cell; ALP, alkaline phosphatase; ALT, alanine aminotransferase; AST, aspartate aminotransferase; BUN, blood urea nitrogen; TB, total bilirubin; PT, prothrombin time; SOFA, Sequential Organ Failure Assessment; SIRS, Systemic Inflammatory Response Syndrome; SAPSII, Simplified Acute Physiology Score; APSIII, Acute Physiology Score III; GCS, Glasgow Coma Scale; HR, heart rate; SBP, systolic blood pressure; DBP, diastolic blood pressure; MBP, mean blood pressure; RR, respiratory rate; SpO₂, peripheral oxygen saturation; AF, atrial fibrillation; AKI, acute kidney injury; HF, heart failure; RF, respiratory failure; CRRT, continuous renal replacement treatment; ERCP, Endoscopic Retrograde Cholangiopancreatography; MV, mechanical ventilation; LOS ICU, length of ICU stay; LOS Hospital, length of hospital stay

Handling of missing data and outlier

Features with missing values more than 30% were excluded directly, including amylase, high density lipoprotein, height, lactate, low density lipoprotein, lipase, absolute count of lymphocytes, absolute count of neutrophils, total cholesterol, total Triglyceride. For variables with missing values less than 5%, the median or mean was used for replacing the missing values according to the result of Kolmogorov–Smirnov test (normality test). For the only one variable—prothrombin time (PT)—with missing values between 5 and 30%, multiple imputation (MI) was conducted to handle the missing data. In this step, imputed data sets (m = 50) were generated via the ‘mice’ package (version 3.18.0), which were sufficient to ensure stable estimates given the fraction of missing information. The imputation model was specified to be inclusive, incorporating: (1) the outcome variable (AKI) and (2) and all baseline variables intended for use in the logistic regression models (including PAR and planned covariates such as age, gender, ethnicity, laboratory results, and vital signs after ICU admission) to strengthen the plausibility of the missing at random assumption. Predictive mean matching was selected as the appropriate interpolation method as PT is a continuous variable.

Crucially, the imputation procedure respected the temporal ordering of the outcome in this prognostic study. When imputing PT, the imputation model was restricted to only use information that would have been available prior to or at the time of AKI diagnosis. The variables such as ICU length of stay, hospital length of stay, and survival time were excluded from the model. This was implemented by carefully defining the predictor matrix, preventing any post-AKI information from informing the imputation of pre-AKI measures. Convergence of the imputation algorithm was assessed after maxit = 20 iterations. We examined trace plots of the imputed values across iterations for key parameters (Additional file 1: Fig. S1) and computed the potential scale reduction factor (PSRF). The trace plots showed good mixing of chains with no discernible trends, and all PSRF values were below 1.05, indicating satisfactory convergence (Additional file 2: Table S1).

Outliers were handled using Winsorization method with the 99th and 1 st percentiles as cutoffs. All procedures described above were performed using software R (version 4.4.3).

Endpoint events

The occurrence of AKI was the primary outcome in this study. To diagnose AKI, we followed the guidelines outlined in the 2012 kidney disease: improving global outcomes (KDIGO) guidelines [23]. The diagnostic criteria were as follows: an increase in serum creatinine level of ≥ 26.5 μmol/L (0.3 mg/dl); or an increase in creatinine to ≥ 50% above the baseline value; or a urinary output less than 0.5 mL/kg/h for more than 6 h. Importantly, the baseline creatinine—a key parameter in this diagnostic framework—was defined as the lowest creatinine value observed within the previous week. The initiation of acute renal replacement therapy during the ICU stay was considered as Stage 3 AKI. In this study, the diagnosis data of AKI were primarily extracted from the built-in ‘KDIGO_Stage’ derivative table in the MIMIC-IV database. As this table has integrated both serum creatinine and urine output data and is readily usable, no additional handling of missing values for creatinine or urine output was required. AP-associated AKI was defined as a KIDGO stage > 0 within 7 days and after 24 h of ICU admission (cases with AKI before ICU admission or within 24 h after admission was excluded). Hence, all the PAR values were obtained based on the laboratory data before the happening of AP-associated AKI. In addition, all-cause mortality at 28 days (d), 1-year (y) and in-hospital mortality were set to be the secondary endpoint events during analysis.

Model development and validation

(1) Feature selection: A nested cross-validation workflow was employed to assess the model’s generalization performance while conducting feature selection. Specifically, we randomly partitioned the data set into five subsets of similar size (outer folds). In each iteration, four subsets were used as the training set, and the remaining subset served as the test set. Within the training set, we further applied fivefold cross-validation (inner folds) to tune the hyperparameter C (the inverse of regularization strength) of the least absolute shrinkage and selection operator (LASSO) regression model. Using grid search over a predefined range of C values (e.g., 0.001, 0.01, 0.1, 1, 10, 100), we identified the optimal C value that maximized the average area under receiver operating characteristic curve (AUROC) of the inner cross-validation. Subsequently, the entire training set and the optimal C value were used to retrain the LASSO regression model, yielding the final feature subset (i.e., features with non-zero coefficients) and model parameters. To minimize overfitting and ensure model parsimony, we selected the lambda value that was closest to the one yielding the minimum mean squared error (lambda.1se) during the LASSO regression process. Finally, the model performance was evaluated on the test set. With details shown in Additional file 2: Table S2, this process was repeated five times, each time with a different subset as the test set, and the performance metrics (such as AUROC, sensitivity, and specificity) from the five test sets were aggregated to estimate the model’s generalization performance. (2) Data preprocessing: The MIMIC-IV cohort was split into training and test sets in a 7:3 ratio, stratified on the outcome of AKI. To avoid information leakage in model construction, normalization and outlier handling were conducted after data set splitting. And then, class imbalance in the AKI outcome was addressed using the Synthetic Minority Over-sampling Technique (SMOTE), implemented via the ‘smotefamily’ package in R [24]. (3) Model construction, validation and comparison: after the rigorous process of feature selection and data preprocessing, a total of eight features were included for further analysis. A total of seven machine learning algorithms were selected for constructing AKI prediction models to ensure comprehensive model comparison: adaptive boosting (AdaBoost), categorical boosting (CatBoost), light gradient boosting machine (LightGBM), gradient boosting machine (GBM), extreme gradient boosting (XGBoost), and random forest were chosen for their proven excellence in handling complex nonlinear relationships, imbalanced classification, and feature interactions in clinical data; the GNB algorithm, which assumes conditional independence among features, offers high computational efficiency and serves as a suitable baseline model. As shown in the Additional file 2: Table S3, the parameter tuning ranges for each algorithm were determined through preliminary exploratory analysis. In this study, the hyperparameters of all models were optimized via grid search combined with fivefold cross-validation. Nested cross-validation was adopted to strictly separate the model development and performance evaluation phases: the outer loop, implemented via fivefold cross-validation, was used to assess the model’s generalization performance. Within each outer training fold, the inner loop (for hyperparameter tuning) further employed fivefold cross-validation combined with grid search to select the optimal hyperparameter combination. And then, each model was utilized to predict AKI occurrence after 24 h of ICU stay in both training set and testing set. Subsequently, models were externally validated in eCIU–CRD cohort to evaluate the generalization ability. Model performance was rigorously assessed using the receiver operating characteristic curve (ROC) curve and precision–recall (PR) curve, with metrics including AUROC, area under PR curve (AUPRC), F1-score, and Youden's index calculated. To improve the robustness of these metrics, Bootstrap method was performed, with number of iterations set to be 1000. Accuracy and robustness of the models were assessed using calibration curve, while the clinical net benefit was evaluated through decision curve analysis (DCA) curve. (4) Model interpretation: finally, the LightGBM model was selected for interpretational analysis due to its best performance. The Shapley additive explanations (SHAP) were conducted to ascertain and visualize the contribution of individual features in LightGBM model via ‘kernelshap’ and ‘shapviz’ packages.

Sensitivity analysis and subgroup analysis

To interrogate the robustness of the finding that elevated PAR augments AKI risk, sensitivity analyses were conducted within pertinent clinical subgroups. These specifically included patients with concomitant liver disease, those who received albumin infusion prior to ICU admission, those with sepsis, and those with malnutrition or cachexia. Subgroup analysis were performed to assess heterogeneity in the association between AKI risk associated and elevated PAR across different populations, with outcomes visualized through forest plots.

Statistical analysis

The sample size required for developing a two-class prediction model was calculated via a formula provided by a previous study [25]:

$$n=\text{exp}\left(\frac{-0.508+0.259\text{ln}\left(\phi \right)+0.504\text{ln}\left(P\right)-\text{ln}\left(\text{MAPE}\right)}{0.544}\right),$$

where φ is the anticipated outcome proportion, P is the number of candidate predictor parameters, and MAPE denotes the mean absolute prediction error. With the AP-associated AKI proportion of 0.20 [26], MAPE set at 0.05 and eight potential predictive features, a minimum sample size of 309 was estimated. The normality of continuous variables was assessed through Kolmogorov–Smirnov test. Mean ± standard deviation (SD) was used for describing the distributions of normally distributed continuous variables, median (interquartile range) for non-normally distributed continuous ones and numbers (%) for the categorical ones. Student’s T test, Mann–Whitney U test, and one-way ANOVA test were performed to compare continuous variables, while Pearson’s χ² test and Fisher’s test were used for comparing categorical variables. Survival curves were constructed through the Kaplan–Meier (KM) method, with p values calculated via the log-rank test. Logistic regression analysis was employed to assess whether PAR constitutes a risk factor for both AKI development among ICU patients with AP. Restricted cubic splines (RCS) were utilized to assess potential nonlinear relationships between PAR levels and survival time. AUROCs and AUPRCs were derived from the ROC and PR curves, respectively. Sensitivity, specificity, F1-score, and Youden’s index were calculated using the confusion matrix of model predictions. SHAP values were computed via the ‘shapviz’ package. All statistical procedures described above were executed in the software R (version 4.4.3) utilizing pertinent computational packages.

Results

Baseline characteristics

There were 497 ICU patients with AP recruited from the MIMIC-IV database as the primary cohort and a total of 235 patients with AP recruited from eICU–CRD as the external validation cohort, with detailed workflows illustrated in Fig. 1. Given the relatively scarce previous research on PAR, and although its clinical significance remains unclear, we exploratively categorized PAR into four levels based on quartiles for subsequent analysis. Hence, in the MIMIC-IV cohort, PAR was categorized into four levels: Q1 (< 0.75), Q2 (0.75–1.10), Q3 (1.10–1.52), and Q4 (≥ 1.52). The baseline characteristics are presented in Table 1. Compared to the low PAR level groups (Q1 and Q2), the high PAR level groups (Q3 and Q4) exhibited higher incidence of AKI, in-hospital mortality, 28-day mortality, 1-year mortality, as well as longer ICU length of stay and total hospital length of stay.

Association analysis

The survival curves (Fig. 2B, C) demonstrated that both short-term (28-day) and long-term (1-year) survival rates progressively decreased as PAR levels increased among patients (p < 0.0001). RCS (Fig. 2D, E) indicated there existed no nonlinear relationship between PAR and the risk of all-cause mortality, whether in short term (p for Non linearity = 0.056) or long term (p for Non linearity = 0.057). Patients were further stratified based on the occurrence of AKI during their ICU stay and analyzed separately. Survival curves revealed that in the subgroup without AKI, both short-term (p = 0.620) and long-term mortality (p = 0.820) remained largely unchanged with PAR levels increasing (Fig. 2F, H). However, in the subgroup accompanied with AKI (Fig. 2G, I), both short-term (p < 0.0001) and long-term (p < 0.0001) survival rates progressively declined as PAR levels elevated. As shown in Additional file 1: Fig. S2, the event-time distribution histograms for AKI and 7-day mortality indicate that AKI in the enrolled patients mainly occurred between 24 and 48 h after admission, during which a concurrent peak in mortality was also observed. As shown in Additional file 1: Fig. S3, the event-time distribution histograms for AKI and AKI stage across PAR strata indicate that AKI in the enrolled patients occurred between 24 and 48 h after ICU admission tended to have higher PAR values. Besides, stage 2 and stage 3 AKI also tended to have higher PAR values. In the multivariable Cox proportional hazards regression model, anion gap (AG), Acute Physiology Score III (APSIII), creatinine, glucose, potassium, sodium, platelet, WBC, blood urea nitrogen (BUN), diastolic blood pressure (DBP), Glasgow Coma Scale (GCS) score, respiratory rate (RR), systolic blood pressure (SBP), Sequential Organ Failure Assessment (SOFA) score, ‌peripheral capillary oxygen saturation (‌SpO2), total bilirubin (TB), temperature, Urineoutput_24hr, atrial fibrillation (AF), AKI, continuous renal replacement therapy (CRRT), heart failure (HF), MV, respiratory failure (RF), sepsis, Vasoactives_24hr, age, ethnicity, and gender were adjusted as confounders, as they had shown statistical significance in the univariable Cox models. In the multivariable logistic regression analyses for in-hospital mortality and AKI, the SOFA score, sepsis, AF, HF, RR, BUN, creatinine, calcium, age, ethnicity, and gender were adjusted as confounders, as they had either shown statistical significance in the univariable logistic regression analyses or were included based on their established role as confounders in previous literature [10, 11]. The results presented in Table 2 demonstrated that elevated PAR remained a risk factor for AKI occurrence and all-cause mortality after adjustment for potential confounders. As shown in Additional file 2: Table S4, PAR was included as a categorical variable in the aforementioned Cox and logistic regression analyses. The Cox analysis indicated that, after adjusting for all potential confounders, PAR at the Q4 level (≥ 1.52) remained a risk factor for 28‑day mortality (p for trend = 0.003, p = 0.003, HR = 3.79 [1.57–9.16]), whereas neither Q3 (1.10–1.52) nor Q4 levels of PAR were risk factors for 1‑year mortality (p for trend = 0.051, Q3: p = 0.222, Q4: p = 0.052). The logistic analysis revealed that, after adjusting for all potential confounders, PAR at Q3 and Q4 levels were risk factors for in‑hospital mortality (p for trend = 0.002, Q3: p = 0.012, OR = 2.78 [1.25–6.17], Q4: p = 0.002, OR = 3.78 [1.65–8.66]), while only PAR at the Q4 level was a risk factor for AKI occurring after 24 h of ICU admission (p for trend = 0.016, Q4: p = 0.016, OR = 2.33 [1.17–4.63]). As shown in Additional file 2: Table S5, in all sensitivity analyses—which sequentially excluded patients with liver disease (N = 104), pre-ICU albumin infusion (N = 7), sepsis (N = 351), and malnutrition/cachexia (N = 94)—the association between elevated PAR and an increased risk of AKI remained consistent, confirming the robustness of the primary finding. Subgroup analysis (Fig. 2J) indicated that no significant heterogeneity was observed in the association between elevated PAR and increased AKI risk across most subgroups, except when comparing patients with diabetes to those without diabetes. Although elevated PAR was associated with increased AKI risk in both diabetic and non-diabetic patients, the increase of AKI risk was greater among AP patients accompanied with diabetes (OR: 8.78 [3.45–22.38] vs. 2.18 [1.42–3.36], p for interaction = 0.008). In addition, although no significant difference was observed on the effect of elevated PAR increasing AKI risk across AP patients of different etiologic types (p for interaction = 0.85), it failed to reach statistical significance within the subgroup of biliary pancreatitis patients (OR: 13.41 [0.07–2502.40], p = 0.337).

Fig. 2.

Association analysis for PAR’s link to prognosis and AKI development in AP. A Distribution histogram of PAR with cutoffs. B, C Kaplan–Meier survival curves for all-cause mortality at 28 days and 1-year. D–F RCS for the effects of PAR on all-cause mortality at 28 days and 1 year. F–I Survival curves for all-cause mortality at 28 days and 1 year in AKI group and without AKI group, respectively. J Forest plot for the subgroup analysis, with AKI as outcome. PAR, phosphorus-to-albumin ratio; AKI, acute kidney injury; AP, acute pancreatitis; RCS, restricted cubic splines

Table 2.

Cox analysis and logistic analysis of PAR

Condition	Model 1	Model 2	Model 3
Cox analysis for 28-day mortality
HR (95% CI)	1.91 (1.57–2.32)	2.06 (1.65–2.58)	1.95 (1.30–2.92)
P value	< 0.001	< 0.001	< 0.001
Cox analysis for 1-year mortality
HR (95% CI)	1.66 (1.41–1.95)	1.73 (1.45–2.06)	1.34 (1.01–1.77)
P value	< 0.001	< 0.001	0.041
Logistic analysis for in-hospital mortality
OR (95% CI)	2.28 (1.70–3.05)	2.25 (1.66–3.05)	1.89 (1.31–2.71)
P value	< 0.001	< 0.001	< 0.001
Logistic analysis for AKI
OR (95% CI)	2.83 (1.96–4.09)	2.68 (1.86–3.87)	1.76 (1.13–2.74)
P value	< 0.001	< 0.001	0.012

All model 2 were adjusted for age, ethnicity, and gender. The model 3 for 28-day and 1-year mortality were adjusted for AG, APSIII score, creatinine, glucose, potassium, sodium, platelet, WBC, BUN, DBP, GCS score, RR, SBP, SOFA score, SpO₂, TB, temperature, Urineoutput_24hr, AF, AKI, CRRT, HF, MV, RF, sepsis, Vasoactives_24hr, age, ethnicity, and gender. The Model 3 for in-hospital mortality and AKI were adjusted for SOFA score, sepsis, AF, HF, RR, BUN, creatinine, calcium, age, ethnicity, and gender. AG, anion gap; APSIII, Acute Physiology Score III; WBC, white blood cell; BUN, blood urea nitrogen; DBP, diastolic blood pressure; GCS, Glasgow Coma Scale; RR, respiratory rate; SBP, systolic blood pressure; SOFA, Sequential Organ Failure Assessment; SpO₂, peripheral oxygen saturation; TB, total bilirubin; AF, atrial fibrillation; AKI, acute kidney injury; CRRT, continuous renal replacement treatment; HF, heart failure; MV, mechanical ventilation; RF, respiratory failure

Development and evaluation of AKI predictive models

The above results suggested that PAR might serve as a potential biomarker for predicting AKI development in patients. Given the critical importance of early AKI identification for improving prognosis in patients with AP, subsequent research focused on the predictive value of PAR for AKI occurrence in this population. AKI prediction models were developed using seven ML algorithms by integrating PAR with other significant features. First, a nested cross-validation based on LASSO regression was employed to screen potential key features. As shown in Fig. 3A, B and Additional file 2: Table S2, the LASSO model achieved stable AUROC, sensitivity, and specificity across the process of cross-validation. Figure 3C illustrates that within this nested cross‑validation framework, LASSO regression identified 10 stable predictors (selected in at least 80% of the outer folds), which include: Urineoutput_24hr, MV, Vasoactives_24hr, sepsis, creatinine, weight, RF, WBC, obesity, and PAR. However, due to the underlying collinearity exhibited in the logistic regression between RF and MV, and between obesity and weight, RF and Obesity with smaller frequencies were excluded in the subsequent analysis. The remaining eight core features were retained for ML model training.

Fig. 3.

Clinical features selection and ML models development. A–C Lasso regression to screen potentially important features in AP with AKI. C Boruta algorithm ranking of the potentially important features. D, E ROC curves of the seven ML models in train set and test set. F, G PR curves of the seven ML models in train set and test set. H Calibration curves of the seven ML models in the test set. I DCA curves of the seven ML models in the test set. ML, machine learning; ROC curve, receiver operating characteristic curve; AKI, acute kidney injury; PR curve, precision–recall curve; DCA, decision curve analysis

The MIMIC-IV cohort was randomly divided into training and test sets at a 7:3 ratio, with both sets demonstrating comparable baseline characteristics (Additional file 2: Table S6). To develop and identify the optimal two-class models, seven distinct machine learning algorithms were employed. Key metrics for evaluating the performance of models included the AUROC, AUPRC, Brier score, calibration curve, and DCA curve. As illustrated in Fig. 3D–G, all models except AdaBoost model achieved AUROC values > 0.8 in the test set, which were comparable to those of training set. All models attained AUPRC values > 0.8 in the test set, which indicated that the models demonstrated robust discriminative power for predicting AKI in patients within the MIMIC-IV cohort. Calibration curves (Fig. 3H) revealed that the LightGBM, GBM, Random Forest, and XGBoost models exhibited calibration curves closely aligned with the ideal reference line, accompanied with Brier scores < 0.2. These results indicate favorable stability and predictive accuracy in the test set. Correspondingly, in the DCA curves (Fig. 3I), the net benefit curves for LightGBM, GBM, Random Forest, and XGBoost models consistently surpassed the ‘Treat all’ threshold line when threshold > 0.2, demonstrating the clinical benefits within the test cohort.

External validation of the models

To evaluate the generalization ability of the prediction models, the performance of them were assessed in the eICU–CRD cohort (external validation set), with baseline characteristics of the data set presented in Additional file 2: Table S7. As shown in Fig. 4A, B, the LightGBM, GNB, RandomForest, and XGBoost models achieved AUROC values > 0.8 and AUPRC values > 0.7 in the external validation set. As illustrated by Fig. 4C, D, ROC curves and PR curves of the four models further indicated that these models exhibited considerable predictive classification abilities. Among these models, LightGBM achieved the highest AUROC of 0.837 (95% CI 0.785–0.889) and the second highest AUPRC of 0.784, establishing it as the best-performing model among its counterparts. The detailed metrics of the ML models are presented in Table 3.

Fig. 4.

External validation of the ML models. A AUROC of the seven ML models in the external validation set. B AUPRC of the seven ML models in the external validation set. C ROC curves of the LightGBM, XGBoost, RandomForest, and GNB models in the three datasets. D PR curves of the LightGBM, XGBoost, RandomForest, and GNB models in the three datasets. AUROC, area under receiver operating characteristic curve; AUPRC, area under precision–recall curve; ML, machine learning; DCA, decision curve analysis; LightGBM, light gradient boosting machine; XGBoost, extreme gradient boosting; GNB, Gaussian Naive Bayes

Table 3.

Key metrics of models calculated by bootstrap method (n = 1000)

Models	AUC (95% CI)	Precision (95% CI)	F1 score (95% CI)	Recall (95% CI)	Accuracy (95% CI)
Test set
AdaBoost	0.721 (0.626–0.806)	0.790 (0.705–0.872)	0.767 (0.692–0.824)	0.746 (0.663–0.827)	0.689 (0.608–0.757)
CatBoost	0.874 (0.811–0.929)	0.877 (0.806–0.938)	0.827 (0.766–0.880)	0.784 (0.703–0.861)	0.776 (0.709–0.838)
GBM	0.882 (0.822–0.936)	0.885 (0.817–0.946)	0.869 (0.811–0.918)	0.853 (0.779–0.922)	0.823 (0.757–0.885)
GNB	0.821 (0.741–0.896)	0.863 (0.783–0.931)	0.770 (0.693–0.832)	0.696 (0.602–0.788)	0.715 (0.635–0.791)
LightGBM	0.878 (0.814–0.931)	0.883 (0.812–0.946)	0.857 (0.796–0.907)	0.834 (0.755–0.900)	0.810 (0.743–0.872)
RandomForest	0.859 (0.790–0.917)	0.861 (0.788–0.929)	0.831 (0.770–0.883)	0.804 (0.729–0.875)	0.776 (0.703–0.845)
XGBoost	0.877 (0.812–0.930)	0.862 (0.780–0.930)	0.805 (0.738–0.862)	0.756 (0.673–0.838)	0.749 (0.676–0.818)
External validation set
AdaBoost	0.725 (0.662–0.782)	0.504 (0.433–0.579)	0.640 (0.573–0.703)	0.880 (0.812–0.939)	0.576 (0.511–0.638)
CatBoost	0.708 (0.643–0.775)	0.578 (0.500–0.662)	0.672 (0.605–0.738)	0.803 (0.725–0.876)	0.664 (0.604–0.728)
GBM	0.783 (0.723–0.840)	0.516 (0.446–0.591)	0.666 (0.606–0.729)	0.940 (0.890–0.981)	0.595 (0.536–0.660)
GNB	0.807 (0.749–0.861)	0.689 (0.598–0.780)	0.675 (0.596–0.745)	0.663 (0.573–0.757)	0.727 (0.668–0.779)
LightGBM	0.836 (0.783–0.888)	0.525 (0.455–0.599)	0.678 (0.616–0.740)	0.961 (0.922–0.991)	0.609 (0.549–0.672)
RandomForest	0.824 (0.771–0.874)	0.521 (0.453–0.593)	0.677 (0.618–0.739)	0.970 (0.933–1.000)	0.604 (0.540–0.668)
XGBoost	0.819 (0.764–0.871)	0.546 (0.477–0.623)	0.688 (0.627–0.754)	0.931 (0.879–0.973)	0.638 (0.579–0.702)

AdaBoost, adaptive boosting; CatBoost, categorical boosting; LightGBM, light gradient boosting machine; GBM, gradient boosting machine; GNB, Gaussian Naive Bayes; XGBoost, extreme gradient boosting

SHAP analysis of the best-performing model

The above analysis demonstrated that the LightGBM model was the best-performing and most stable predictive model in this study. To elucidate the relationship between individual variables and the model’s predicted output, as well as their respective contributions to predictions, SHAP was performed on the LightGBM model. As illustrated in Fig. 5A, B, the contributions of individual variables to SHAP values were similar in both the test set and the external validation set. Notably, higher PAR values were associated with higher final SHAP values. Dependence plots further revealed (Fig. 5C, D) that SHAP values were positively correlated with weight, creatinine, WBC, PAR, sepsis, Vasoactives_24hr, and MV. Conversely, SHAP values were negatively correlated with Urineoutput_24hr. These correlations remained consistent across both the test set and the external validation cohort.

Fig. 5.

SHAP analysis of the best-performing LightGBM model. A Swarm plot of LightGBM performance on the test set. B Swarm plot of LightGBM performance on the external validation set. C Partial dependence plot for LightGBM on the test set. D Partial dependence plot for LightGBM on the external validation set. SHAP, shapley additive explanations; LightGBM, light gradient boosting machine

Discussion

PAR is a recently proposed index that concurrently reflects changes in both albumin and serum phosphate levels, which are both pathophysiological indicators associated with the prognosis of AP [12, 27]. In this study, we first analyzed the association between the first PAR index measured following ICU admission and survival outcomes in patients with AP. Baseline characteristic analysis and survival curves demonstrated that elevated PAR levels correlated with increased short- and long-term mortality in this population, indicating a potential relationship between high PAR and adverse prognosis. Further Cox regression analyses and RCS analyses confirmed that, even after adjusting for potential confounders, PAR remains a risk factor for prognosis, exhibiting a linear dose–response mortality risk. When analyzed as a categorical variable and after adjusting for all potential confounders, PAR at Q4 (≥ 1.52) level was associated with a 3.79‑fold increase in the risk of 28‑day mortality, while PAR at Q3 (1.10–1.52) and Q4 (≥ 1.52) levels was associated with 2.78‑fold and 3.78‑fold increases in the risk of in‑hospital mortality, respectively. This suggests that PAR value may serve as an independent risk factor for short‑term mortality in critically ill patients with AP. In contrast, neither Q3 (1.10–1.52) nor Q4 (≥ 1.52) levels of PAR showed any impact on long‑term mortality after adjusting for all confounders, which may be attributed to the fact that the long‑term prognosis of critically ill AP patients is influenced by a broader range of subsequent therapeutic factors. From a pathophysiological perspective, hyperphosphatemia in AP may result from: necrotic pancreatic tissue releasing intracellular phosphorus-containing compounds (e.g., ATP, nucleic acids) into the bloodstream, metabolic acidosis facilitating phosphorus shift via phosphate buffer systems, and impairing phosphate excretion due to reduced renal perfusion caused by fluid extravasation [12]. Concurrently, systemic inflammation during pancreatitis induces increased vascular permeability, causing albumin leakage into interstitial spaces and consequent hypoalbuminemia [27, 28]. Both hyperphosphatemia and hypoalbuminemia are associated with adverse outcomes, thus mechanistically explaining the strong correlation between elevated PAR and all-cause mortality in this population.

Furthermore, elevation of the initial PAR upon ICU admission was associated with increased incidence of AKI in AP. Logistic regression analysis demonstrated that higher PAR was still a risk factor for AKI development after adjusting for age, sex, and ethnicity. Even as a categorical variable, a PAR level in Q4 (≥ 1.52) remained associated with a 2.33‑fold increased risk of developing AKI after 24 h of ICU stay, after adjusting for all confounders. It is well-established that hypoalbuminemia serves as a predictor of incidence and mortality of AKI for patients in systemic inflammatory states [18]. Pancreatitis-associated AKI stems partially from renal hypoperfusion, while hyperphosphatemia represents an early pathophysiological alteration in AKI development. This explains the close association between hyperphosphatemia and AKI in this population. Subgroup analysis suggested that elevated PAR confers a higher risk of AKI in AP accompanied with diabetes. Due to the direct toxicity of hyperglycemia, the chronic state of glomerular hyperfiltration, and elevated levels of inflammation and oxidative stress associated with diabetes, patients with diabetes exhibit heightened susceptibility to AKI [29, 30]. Subgroup analysis based on pancreatitis etiology did not reveal statistically significant associations for PAR in the biliary pancreatitis subgroup, which is likely attributable to the limited sample size of patients explicitly classified as having biliary pancreatitis in this study (only 39 patients).

ML algorithms were employed to construct practical prediction models. With LASSO regression to screen and rank AKI-related variables in patients with AP, PAR emerged as one of the top 10 important predictors, further substantiating its potential utility in AKI risk stratification. This nested cross-validated feature selection can reduce the risk of overfitting and double skewing to a certain extent. Based on MIMIC-IV cohort and integrating PAR with other pivotal features, seven prediction models were constructed, among which LightGBM, XGBoost, RandomForest, and GNB models exhibited robust generalization capabilities on external validation cohort, achieving AUROC > 0.8 and AUPRC > 0.7, indicating their promising clinical applicability. SHAP analysis revealed that while PAR contributed less to LightGBM model, the best-performing one, than other top features, it maintained meaningful predictive importance and exhibited a positive correlation with SHAP value. This finding aligns with the association between elevated PAR and AP-associated AKI. A previous study developed predictive models for AP-associated AKI using 49 clinical features, with its optimal GBM model achieving an AUROC of 0.867 on test set [10]. Although lacking external validation, it provides clinicians with a tool for identifying high-risk AKI patients. Another similar single-center study developed an AKI prediction model using 36 clinical features, with the final random forest model achieving an AUROC of 0.902 but still lacking external validation [11]. Compared to these approaches, this study demonstrates distinct advantages: first, the prediction framework was simplified using only eight clinically core features while innovatively incorporating the novel PAR index; Second, our LightGBM model, with best performance, not only achieved a superior AUROC of 0.880 in the test set but also maintained high AUROC of 0.837 in the eICU–CRD cohort, robustly confirming its generalization capability. Therefore, for critically ill patients with AP in the first 24 h after ICU admission, the AP‑associated AKI prediction model developed in this study—by utilizing patient data collected during this period—can effectively predict the risk of developing AKI within the subsequent 7 days. This may alert clinicians to maintain heightened vigilance for high‑risk patients, to monitor relevant indicators, such as urine output, serum creatinine, and blood urea nitrogen more closely, and to be prepared for the potential initiation of renal replacement therapy at any time.

This study has certain limitations. First and foremost, the prediction model developed in this work is static in nature. We employed several ML algorithms which predicts the binary outcome of AKI occurrence. As rightly pointed out, AKI is a highly dynamic process; its onset, progression, and recovery unfold over time. A static model inherently ignores this temporal dimension of the data. It treats all patient data up to the prediction window as a single snapshot, thereby losing critical information about the sequence, timing, and evolution of clinical events (e.g., changes in creatinine, urine output, or vasopressor doses). Consequently, our model cannot provide time-sensitive risk trajectories or differentiate between patients at risk for early vs. late-onset AKI, which are crucial nuances for clinical intervention.

Time-to-event analysis models, such as random survival forest or cox regression with time-varying covariates, are inherently more suited to capture this dynamicity. These models can incorporate longitudinal data, estimate the hazard of AKI at any given time, and identify predictors whose influence may wax or wane during the hospital stay. By not employing such a methodology, we have potentially oversimplified the clinical reality. Our model’s performance metrics (e.g., AUC) represent an aggregate measure that may mask its varying predictive utility at different timepoints.

Furthermore, a static model is less informative for guiding the timing of repeated clinical assessments or dynamic treatment decisions. We acknowledge this as a significant methodological limitation. Our primary focus in this initial investigation was on robust feature selection and establishing a strong baseline predictive performance using a widely interpretable model. The nested cross-validation framework was designed to mitigate overfitting within this static paradigm. However, we fully agree that future work must transition towards dynamic modeling. A logical next step is to apply survival analysis techniques to the same data set, which would allow us to model the instantaneous risk of AKI and potentially identify time-dependent predictors. This would not only enhance predictive accuracy but also align the model more closely with the progressive nature of the disease, ultimately increasing its clinical actionable value.

In addition, the relatively small sample sizes in our training, testing, and external validation sets necessitate confirmation in larger cohorts to establish more robust result. In addition, our model development did not incorporate other clinically significant factors, such as variables related to intra-abdominal hypertension and abdominal compartment syndrome, all of which may influence AKI development in this population. Incorporating these parameters in future iterations could substantially enhance predictive performance, constituting a key direction for our subsequent research.

Conclusion

This study revealed a strong association between the novel composite indicator PAR and both prognosis of patients with AP and the incidence of AKI in this population. PAR was identified as a risk factor for both long-term and short-term all-cause mortality, with a linear dose–response relationship observed between increasing PAR levels and mortality risk. Furthermore, elevated PAR levels independently increased the risk of AKI development in patients with AP. Using PAR and seven other clinical features, AKI prediction models were constructed with seven different machine learning algorithms, among which the LightGBM model demonstrated the highest clinical potential.

Author contributions

Xuan Chen: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Writing—original draft, Writing—review and editing. Boying Liu: Project administration, Supervision, Validation, Writing—review and editing. Gefei Wang: Project administration, Supervision, writing—review and editing.

Funding

The author(s) declare that there is not any financial support received for the research, authorship, and/or publication of this article.

Availability of data and materials

Datasets analyzed in this study are publicly available. This data can be found at: (https://mimic.mit.edu/) and (https://eicu-crd.mit.edu/), respectively. The datasets can be found in the Supplementary materials.

Declarations

Ethics approval and consent to participate

The studies involving MIMIC-IV database were reviewed and approved by the Institutional Review Board of the Massachusetts Institute of Technology and Beth Israel Deaconess Medical Center. The studies involving eICU–CRD were reviewed and approved by the Institutional Review Board of the Massachusetts Institute of Technology and the corresponding medical centers. Written informed consent for participation was not required for this study, because the personal information of the participants has been encrypted.

Competing interests

The authors declare no competing interests.

Generative AI statement

The authors declare that the creation of this manuscript did not involve any Gen AI.