Population Pharmacokinetics and Model-Informed Precision Dosing of Colistin Sulfate in Critically Ill Patients with Carbapenem-Resistant Gram-Negative Infections: A Prospective Pharmacokinetic-Pharmacodynamic Study

Abstract

Background

Independently evaluated population pharmacokinetic models and clinically derived pharmacokinetic/pharmacodynamic targets for colistin sulfate remain limited. We aimed to develop and evaluate a model, explore an efficacy threshold, and construct a research model-informed precision dosing calculator.

Methods

This prospective study included 55 critically ill patients with microbiologically confirmed infection caused by carbapenem-resistant Gram-negative bacteria (CR-GNB) for model development and 15 with suspected CR-GNB infection without microbiological confirmation for independent evaluation of concentration predictions. Plasma concentrations were measured by validated ultra-performance liquid chromatography-tandem mass spectrometry. Modeling used nonlinear mixed-effects modeling software (NONMEM). Logistic regression and receiver operating characteristic (ROC) analysis assessed efficacy predictors and the exploratory threshold. Monte Carlo simulations evaluated probability of target attainment (PTA).

Results

A two-compartment model with creatinine clearance (CrCL) as a covariate on clearance (CL) best described the data (typical CL = 6.67 × (CrCL/76.9)^0.648 L/h). Independent evaluation yielded a relative root mean square error of 37.4%, with 53.7% of predictions within ±30% of observations. Clinical efficacy was achieved in 24/55 patients. The ratio of 24-hour steady-state area under the concentration-time curve (AUC_ss,24h) to minimum inhibitory concentration (MIC) and the Acute Physiology and Chronic Health Evaluation II (APACHE II) score were independently associated with efficacy (odds ratios per one-unit increase, 1.04 and 0.81, respectively). The exploratory AUC_ss,24h/MIC cut-off was 40.53 (area under the ROC curve, 0.712). At MIC ≥ 1 mg/L, no evaluated monotherapy regimen achieved PTA ≥ 90% in patients with normal or augmented renal function.

Conclusion

This study provides a pharmacokinetic model with preliminary independent predictive evaluation, an exploratory exposure threshold, and an open-access research dosing calculator. Prospective multicenter validation is required before clinical implementation.

Graphical Abstract

Introduction

Carbapenem-resistant Gram-negative bacteria (CR-GNB) represent one of the most urgent global public-health threats, ranked by the World Health Organization among the highest-priority pathogens for which new therapeutic strategies are critically needed.1 Critically ill patients in the intensive care unit (ICU) are particularly vulnerable, owing to severe underlying illness, immunosuppression, and frequent invasive procedures, and CR-GNB infections in this setting carry attributable mortality rates as high as 60%.2,3 The polymyxins remain an indispensable last-resort option against these organisms. Among the available formulations, colistin sulfate differs pharmacologically from colistimethate sodium (CMS): whereas CMS is an inactive prodrug requiring in-vivo hydrolysis to liberate active colistin, colistin sulfate is administered in its directly active form, conferring more rapid onset and more predictable pharmacokinetics.4,5

Despite this advantage, the clinical use of polymyxins is constrained by a narrow therapeutic window, pronounced inter-individual pharmacokinetic variability, and a high incidence of nephrotoxicity.4 In critically ill patients, rapid fluctuations in renal function, fluid resuscitation, capillary leak, and hypoalbuminemia substantially amplify pharmacokinetic variability, such that fixed dosing regimens frequently fail to achieve the intended pharmacokinetic/pharmacodynamic (PK/PD) exposure.6 Underexposure promotes treatment failure and resistance, whereas overexposure increases nephrotoxic risk. Population pharmacokinetic (PopPK) modeling provides a quantitative framework to characterize this variability, identify the covariates governing drug disposition, and support individualized dose optimization through Bayesian forecasting and Monte Carlo simulation.7–9

For the polymyxins, the ratio of the 24-hour steady-state area under the concentration-time curve to the minimum inhibitory concentration (AUC_ss,24h/MIC) is the principal PK/PD index of efficacy, and international consensus guidance recommends a total-drug AUC_ss,24h target of approximately 50 mg·h/L for colistin after CMS administration, with a similar target for polymyxin B.4 However, although colistin and polymyxin B differ by only a single amino-acid residue, they diverge in pharmacokinetics, protein binding, and nephrotoxicity, so their target values and dosing recommendations cannot be assumed to be interchangeable with those of colistin sulfate.4 Available clinical evidence for intravenous colistin sulfate consists mainly of observational cohorts and small PopPK studies, and dosing practice relies on the product label, renal function, organism susceptibility, clinical response, and therapeutic drug monitoring (TDM) where available.5,10–13 PopPK studies of colistin sulfate remain few and largely comprise small, single-center analyses. Independent predictive evaluation is limited, no clinical exposure threshold has been established, and the relative contributions of drug exposure and host factors to outcome remain poorly defined.5,10–12

We therefore conducted a prospective study in CR-GNB-infected ICU patients with four aims: (i) to develop a PopPK model for intravenous colistin sulfate and evaluate its predictive performance in a separate clinical cohort; (ii) to identify independent predictors of clinical efficacy and derive an exploratory PK/PD threshold; (iii) to evaluate the probability of target attainment across dosing regimens and renal-function strata by Monte Carlo simulation; and (iv) to integrate these elements into an openly accessible research model-informed precision dosing (MIPD) calculator.

Materials and Methods

Study Design and Patients

This prospective observational study was conducted in the ICU of a tertiary teaching hospital and comprised a model-development phase and an independent PK predictive-evaluation phase between May 2023 and September 2025. The modeling cohort enrolled 55 patients with microbiologically confirmed CR-GNB infection who received intravenous colistin sulfate. The independent evaluation cohort comprised 15 additional patients who were clinically suspected of having CR-GNB infection but lacked microbiological confirmation. This separate cohort was used only to evaluate concentration prediction. Because microbiological confirmation and MIC data were unavailable, it was not used to evaluate the exposure-response model or the PK/PD threshold. The protocol was approved by the institutional ethics committee (approval no. 2023-K131-01), and all patients or their legal representatives provided written informed consent. The study was conducted in accordance with the Declaration of Helsinki and registered at the Chinese Clinical Trial Registry (www.chictr.org.cn) (trial registration number ChiCTR2300072191).

Eligible patients were aged ≥18 years, received intravenous colistin sulfate for ≥48 h, and had at least one valid plasma concentration obtained during TDM. Patients were excluded if they had critical missing data, were pregnant or lactating, or received concurrent continuous renal replacement therapy or extracorporeal membrane oxygenation. Body weight was missing in 9% of patients. Missing body weight values were handled by single imputation using the median body weight of the relevant analysis population.

Drug Administration and Blood Sampling

The dosing regimen for colistin sulfate (Shanghai Shangyao Xinya Pharmaceutical, China) was determined by the treating physician according to infection severity, renal function, and susceptibility results. In accordance with the prescribing information, the conventional regimen was 1.0–1.5 million units (MU; 1 MU = 44 mg) daily, in two or three intravenous infusions over 0.5–2.0 h each.13 For pulmonary infection, adjunctive aerosolized colistin sulfate (0.25 MU every 12 h) could be added at the clinician’s discretion. After steady state (≥48 h), the first TDM sampling comprised a trough sample (within 30 min before the next dose) and four to six post-dose samples (principally at 0.5, 1, 2, 4, and 6 h). Samples were centrifuged at 4000 rpm for 5 min and plasma stored at −80 °C.

Bioanalytical Method

Plasma colistin concentrations were determined by validated ultra-performance liquid chromatography-tandem mass spectrometry (UPLC-MS/MS) (DMT 9600BG; Demai Instruments, China) using a previously published method.14 Separation used a K9 HILIC column (4.6 mm × 150 mm, 2.5 μm) at 40 °C, with 0.1% formic acid in water (A) and 0.1% formic acid in acetonitrile (B) at 0.3 mL/min over a 10-min gradient, and detection by electrospray ionization in positive multiple-reaction-monitoring mode. Colistin A and B were quantified separately and summed to give the total colistin concentration. The lower limits of quantification were 0.027 mg/L (colistin A) and 0.053 mg/L (colistin B); quality-control relative standard deviations were 0–10.8% with accuracy of −10.4% to 10.1%. Implausible values from documented sampling or analytical errors, and values below the limit of quantification, were excluded.

Population Pharmacokinetic Modeling

PopPK analysis used nonlinear mixed-effects modeling (NONMEM version 7.4; Icon Development Solutions, USA) with Pirana (2.9.7) and R (4.2.1); diagnostics used Xpose (4.3.2) and ggplot2. Parameters were estimated by first-order conditional estimation with interaction. One- and two-compartment models were compared. The two-compartment model used the ADVAN3 TRANS4 subroutine with clearance (CL), central (V₁) and peripheral (V₂) volumes, and inter-compartmental clearance (Q). Inter-individual variability (IIV) was described by an exponential model and residual variability by additive, proportional, or combined error structures. Candidate covariates included demographics, biochemical indices, severity scores (Acute Physiology and Chronic Health Evaluation II [APACHE II] and Sequential Organ Failure Assessment [SOFA]), hematological indices, and concomitant medications. Covariate selection used stepwise forward inclusion (Δobjective function value (OFV) >3.84, P < 0.05) and backward elimination (ΔOFV >6.63, P < 0.01).

Model Evaluation

Internal evaluation comprised goodness-of-fit plots, non-parametric bootstrap (1000 replicates), and a prediction-corrected visual predictive check (pcVPC). For independent PK predictive evaluation in the separate 15-patient cohort, the final fixed-effect parameters were retained. Variance parameters and individual random effects were estimated for diagnostic purposes. The reported predictive metrics were calculated from population predictions (PRED), without using post hoc individual predictions (IPRED) or empirical Bayes estimates. Predictive performance was quantified by mean error (ME) and mean prediction error (MPE), with a 95% confidence interval (CI) of the ME spanning zero indicating no significant systematic bias,15 by root mean square error (RMSE) and relative RMSE (rRMSE; <50% considered acceptable),16 and by F20 and F30, the proportions of predictions within ±20% and ±30% of observations (F20 ≥35% and F30 ≥50% considered acceptable).17 All 95% CIs were obtained by 1000-replicate bootstrap. Because the independent cohort lacked confirmed isolates and MIC data, this evaluation was limited to PK concentration prediction and did not constitute validation of the clinical exposure-response model or threshold.

Clinical Outcomes and PK/PD Index Calculation

Individual pharmacokinetic parameters for the 55 modeling-cohort patients were derived by empirical Bayes estimation, yielding AUC_ss,24h, C_ss,min, steady-state peak concentration (C_ss,max), and the corresponding AUC_ss,24h/MIC, C_ss,min/MIC, and C_ss,max/MIC ratios. For patients with multiple confirmed pathogens, the highest MIC was used as a conservative denominator representing the least susceptible isolate. The primary outcome was clinical efficacy, assessed at the end of treatment from the medical record. Patients were classified as having achieved clinical efficacy when the treating team documented improvement in infection, with supporting indicators considered in the assessment including ventilator weaning, reduced or discontinued vasopressors, temperature ≤38 °C, white-cell count ≤10 × 10⁹/L, and procalcitonin <0.05 ng/mL or a decrease >80%. Other patients were classified as treatment failures. Thus, this was a composite clinical classification rather than a weighted score. The exposure-response and PK/PD analyses were restricted to the 55 microbiologically confirmed patients in the modeling cohort. Secondary outcomes were microbiological eradication (negative follow-up culture from the same site during therapy), 30-day all-cause mortality from treatment initiation, and acute kidney injury (AKI) according to Kidney Disease: Improving Global Outcomes (KDIGO) criteria (serum creatinine rise ≥1.5 times baseline or ≥26.5 μmol/L within 48 h), assessed from treatment initiation to 48 h after discontinuation.12,13

Identification of Efficacy Predictors and PK/PD Target

Candidate predictors were screened by univariable logistic regression (P < 0.10). To reduce overfitting, we assessed collinearity and retained one representative variable from each collinear set. The remaining covariates were tested individually, and those changing the odds ratio of a core predictor by more than 10% were retained as confounders.18 There were 24 clinical-efficacy events. The final model was limited to two predictors, corresponding to 12 events per predictor. Final-model selection followed the Akaike information criterion (AIC) and the events-per-variable principle, with multicollinearity checked by the variance inflation factor (<5 considered acceptable). Discrimination was assessed by receiver operating characteristic (ROC) analysis, with the 95% CI for the area under the ROC curve (AUROC) calculated using the DeLong method.19 The exploratory cut-off was selected using the maximum Youden index. Its uncertainty was assessed by a stratified non-parametric bootstrap with 10,000 resamples, and the 2.5th and 97.5th percentiles were reported as the 95% CI. Because the cut-off was derived and assessed in the same cohort, it was treated as exploratory rather than clinically validated.

Exploratory Machine-Learning Sensitivity Analysis

As an exploratory sensitivity analysis, three ensemble-learning algorithms (extreme gradient boosting [XGBoost], gradient boosting machine, and random forest) were applied using the variables retained after screening.20 Given the modest sample, leave-one-out cross-validation (LOOCV) was used in preference to k-fold cross-validation,21–23 with ROC 95% CIs obtained by 1000-replicate bootstrap. Model interpretability was assessed using SHapley Additive exPlanations (SHAP),24 generating beeswarm, feature-importance, and interaction plots, and decision-curve analysis evaluated net benefit.25 These analyses were intended to assess consistency with the parsimonious logistic model and were not considered independent validation.

Monte Carlo Simulation

Using the final model, Monte Carlo simulation (2000 virtual patients per scenario) evaluated the probability of target attainment (PTA) across four creatinine clearance (CrCL) strata (10–30, 30–80, 80–130, 130–150 mL/min), six regimens (0.5, 0.75, 1 MU every 12 or 8 h), and five MIC values (0.25–4 mg/L), using the study PK/PD target. PTA ≥90% was regarded as adequate.26 Additionally, dose-by-dose AUC_τ accumulation with versus without first-dose doubling was simulated to compare the time to steady state.

Research MIPD Calculator

An open-access research MIPD calculator was developed in Python (3.12) with Streamlit. Its engine integrates the final PopPK model, the exploratory study-derived PK/PD threshold, and the multivariable logistic-regression efficacy model. The calculator was developed for research demonstration and has not been prospectively evaluated as a clinical decision-support tool.

Statistical Analysis

Continuous variables are presented as mean ± standard deviation and categorical variables as number (percentage). Between-group comparisons used the independent-samples t test or Wilcoxon rank-sum test, and the Pearson χ² test or Fisher exact test. Logistic regression and descriptive statistics used R (4.2.1); machine learning, LOOCV, SHAP, and Monte Carlo simulation used Python (3.12) with scikit-learn, xgboost, shap, NumPy, and SciPy. A two-sided P < 0.05 was considered significant.

Results

Patient Characteristics

Seventy patients were enrolled: 55 in the modeling cohort (277 valid plasma concentrations) and 15 in the independent PK evaluation cohort (82 concentrations). In the modeling cohort, the mean age was 59 ± 15 years, 81.8% (45/55) were male, and the mean weight was 65 ± 11 kg. Renal function was relatively preserved overall (mean CrCL 90 ± 54 mL/min), while severity scores indicated critical illness (APACHE II 23.7 ± 5.7; SOFA 7.8 ± 3.0). Pulmonary infection was the most common site (92.7%), followed by bloodstream infection (40.0%), with multisite infection in 50.9%. The predominant pathogen was Acinetobacter baumannii (78.2%), followed by Pseudomonas aeruginosa (38.2%) and Klebsiella pneumoniae (25.5%). The most frequently co-administered antimicrobials were meropenem (40.0%), imipenem/cilastatin (32.7%), and tigecycline (27.3%). Measured baseline and dosing variables did not differ significantly between cohorts (all P > 0.05) (Table 1). Nevertheless, the cohorts differed by design because microbiological confirmation and MIC data were required only in the modeling cohort.

Table 1.

Demographic and Baseline Clinical Characteristics of the Modeling and Independent PK Evaluation Cohorts

Characteristic	Total (N=70)	Modeling (n=55)	Validation (n=15)	P
Demographics
Male sex, n (%)	54 (77.1)	45 (81.8)	9 (60.0)	0.091
Age, years	59 ± 15	59 ± 15	57 ± 16	0.864
Weight, kg	65 ± 12	65 ± 11	64 ± 14	0.840
Severity
APACHE II	24.0 ± 6.0	23.7 ± 5.7	25.1 ± 7.3	0.455
SOFA	8.0 ± 3.0	7.8 ± 3.0	8.8 ± 2.9	0.096
Comorbidities
Hypertension, n (%)	31 (44.3)	23 (41.8)	8 (53.3)	0.426
Diabetes, n (%)	26 (37.1)	22 (40.0)	4 (26.7)	0.343
Heart failure, n (%)	15 (21.4)	10 (18.2)	5 (33.3)	0.286
Chronic lung disease, n (%)	9 (12.9)	9 (16.4)	0 (0.0)	0.189
Chronic kidney disease, n (%)	7 (10.0)	7 (12.7)	0 (0.0)	0.333
Chronic liver disease, n (%)	10 (14.5)	9 (16.4)	1 (7.1)	0.674
Solid tumor, n (%)	14 (20.0%)	10 (18.2%)	4 (26.7%)	0.480
Hematologic malignancies, n (%)	4 (5.7%)	4 (7.3%)	0 (0.0%)	0.571
Laboratory findings
ALT (U/L)	63 ± 74	67 ± 80	45 ± 43	0.637
AST (U/L)	77 ± 107	74 ± 91	85 ± 158	0.758
TBIL (μmol/L)	30 ± 48	27 ± 42	41 ± 67	0.252
DBIL (μmol/L)	22 ± 39	18 ± 32	35 ± 58	0.074
ALB (g/L)	31.5 ± 4.7	31.3 ± 5.0	32.3 ± 3.6	0.586
BUN (mmol/L)	16 ± 10	16 ± 10	17 ± 10	0.753
SCR (μmol/L)	98 ± 71	98 ± 75	96 ± 56	0.678
CrCL (mL/min)	88 ± 51	90 ± 54	81 ± 40	0.710
CRP (mg/L)	83 ± 59	85 ± 62	75 ± 49	0.621
PCT (ng/mL)	5.2 ± 15.2	5.3 ± 16.5	4.4 ± 9.2	0.869
HGB (g/L)	78 ± 19	80 ± 19	70 ± 15	0.103
WBC (×109/L)	11.1 ± 6.7	10.6 ± 5.5	13.0 ± 10.1	0.802
PLT (×109/L)	232 ± 159	227 ± 159	251 ± 164	0.737
Concomitant treatment
Vasoactive agents, n (%)	46 (65.7)	36 (65.5)	10 (66.7)	0.930
Mechanical ventilation, n (%)	50 (71.4)	37 (67.3)	13 (86.7)	0.202
Aerosolized colistin sulfate, n (%)	33 (47.1)	28 (50.9)	5 (33.3)	0.227
Infection site
Pulmonary, n (%)	64 (91.4)	51 (92.7)	13 (86.7)	0.602
Bloodstream, n (%)	31 (44.3)	22 (40.0)	9 (60.0)	0.167
Abdomen, n (%)	7 (10.0)	6 (10.9)	1 (6.7)	>0.999
Urinary tract, n (%)	8 (11.4)	6 (10.9)	2 (13.3)	>0.999
Skin and soft tissue, n (%)	3 (4.3)	2 (3.6)	1 (6.7)	0.521
Multisite (≥2), n (%)	38 (54.3)	28 (50.9)	10 (66.7)	0.278
Pathogen (modeling cohort)
A. baumannii, n (%)	43 (61.4)	43 (78.2)	—	—
P. aeruginosa, n (%)	21 (30.0)	21 (38.2)	—	—
K. pneumoniae, n (%)	14 (20.0)	14 (25.5)	—	—
MIC
≤0.5, n (%)	37 (52.9%)	37 (67.3%)	—	—
1, n (%)	3 (4.3%)	3 (5.5%)	—	—
≥2, n (%)	15 (21.4%)	15 (27.3%)	—	—
Dosing regimen
Daily dose 1.5 MU, n (%)	51 (72.9)	40 (72.7)	11 (73.3)	0.763
Interval q8h, n (%)	43 (61.4)	32 (58.2)	11 (73.3)	0.285
Infusion 0.5–1 h, n (%)	40 (57.1)	32 (58.2)	8 (53.3)	0.737
Concomitant antibiotic
Meropenem, n (%)	29 (41.4)	22 (40.0)	7 (46.7)	0.642
Imipenem and cilastatin, n (%)	26 (37.1)	18 (32.7)	8 (53.3)	0.143
Tigecycline, n (%)	18 (25.7)	15 (27.3)	3 (20.0)	0.744
Cefoperazone and sulbactam, n (%)	16 (22.9)	11 (20.0)	5 (33.3)	0.308
Piperacillin and tazobactam, n (%)	8 (11.4)	7 (12.7)	1 (6.7)	>0.999
Sulfamethoxazole and trimethoprim, n (%)	7 (10.0)	5 (9.1)	2 (13.3)	0.637
Ceftazidime and avibactam, n (%)	6 (8.6)	4 (7.3)	2 (13.3)	0.602
Amikacin, n (%)	5 (7.1)	5 (9.1)	0 (0.0)	0.577

Notes: Data are mean ± SD or n (%). P values: Wilcoxon rank-sum test (continuous) or Fisher exact test (categorical). --, not applicable.

Abbreviations: APACHE II, Acute Physiology and Chronic Health Evaluation II; SOFA, Sequential Organ Failure Assessment; ALT, alanine aminotransferase; AST, aspartate aminotransferase; TBIL, total bilirubin; DBIL, direct bilirubin; ALB, albumin; BUN, blood urea nitrogen; SCR, serum creatinine; CrCL, creatinine clearance; CRP, C-reactive protein; PCT, procalcitonin; HGB, hemoglobin; WBC, white blood cell count; PLT, platelet count; MIC, minimum inhibitory concentration; MU, million units.

Covariate Distribution and Base Model

Most candidate continuous covariates were approximately normally distributed, whereas inflammatory and hepatic markers were right-skewed. The concentration–time data declined in a clearly multi-exponential pattern, suggesting a two-compartment model (Supplementary Figure S1). In the correlation matrix, most covariate pairs had a Spearman coefficient <0.5; CrCL and serum creatinine were strongly correlated (r ≈ −0.7), as expected because CrCL is derived from serum creatinine, so only CrCL was carried forward (Supplementary Figure S2). The two-compartment model (ADVAN3 TRANS4) was markedly superior to the one-compartment model (ΔOFV = 124.8, df = 2, P < 0.001), and a proportional residual-error model was selected. IIV was supported only on CL; adding IIV on V₁, V₂, or Q gave near-zero estimates with excessive shrinkage (>40%). Exploratory plots showed a clear positive trend between the CL random effect and CrCL, with weak associations for severity scores, weight, and age (Supplementary Figure S3).

Covariate Model and Final Model

In forward inclusion, incorporating CrCL on CL as a power function produced the largest OFV reduction (ΔOFV = 33.481, P < 0.001), lowering the IIV of CL from 65.70% to 46.70% and thereby explaining approximately 28.9% of the inter-individual variability in CL. Weight reduced the OFV when added alone (ΔOFV = 5.786, P < 0.05) but contributed no independent effect once CrCL was included (P > 0.05), consistent with weight being implicitly captured by the Cockcroft–Gault equation. Backward elimination confirmed CrCL as the sole significant covariate (ΔOFV = 33.481 on removal, P < 0.01) (Supplementary Table S1). The final two-compartment model with first-order elimination was:

with V₁ = 0.57 L, V₂ = 58.70 L, and Q = 47.10 L/h, where η_CL represents the individual random effect on clearance and 76.9 mL/min is the modeling-cohort median CrCL. The positive exponent (0.648) indicates that clearance rises with renal function. Correspondingly, clearance falls and exposure rises as renal function declines.

Parameter Estimates and Internal Validation

Final-model parameters were estimated with good precision (relative standard error [RSE]: CL 6.7%, V₂ 4.2%, Q 6.3%, θ_CrCL 12.9%); the IIV of CL was 46.70% and residual variability 23.60%, both with acceptable shrinkage (Table 2). The bootstrap converged in >98% of 1000 replicates, and all point estimates lay within the bootstrap 95% CIs (eg, CL median 6.81 L/h [95% CI 5.77–7.92]; θ_CrCL 0.64 [0.42–0.93]) (Table 2 and Supplementary Figure S3). Compared with the base model, the final model showed improved goodness of fit, with individual predictions clustering closely around the line of identity and conditional weighted residuals distributed uniformly within ±2 without systematic trend (Figure 1). The pcVPC showed that the observed median and 5th/95th percentiles fell within their respective simulation-based intervals (Figure 2).

Table 2.

Population Pharmacokinetic Parameter Estimates from the Base and Final Models, with Bootstrap Results

Parameter	Base Est.	RSE	Final Est.	RSE	Boot 2.5%	Median	97.5%
CL (L/h)	6.73	9.7%	6.67	6.7%	5.77	6.81	7.92
V1 (L)	5.42	13.9%	0.57	15.8%	0.17	0.65	24.03
V2 (L)	59.40	8.4%	58.70	4.2%	48.24	59.17	74.65
Q (L/h)	52.60	21.9%	47.10	6.3%	28.49	49.72	91.98
θCrCL	—	—	0.648	12.9%	0.42	0.64	0.93
IIV-CL (%CV)	65.70	20.3%	46.70	19.0%	31.14	44.49	60.82
RSV (%CV)	23.60	8.7%	23.60	8.2%	18.97	23.02	27.02

Notes: Bootstrap 95% CI: 2.5th-97.5th percentiles.

Abbreviations: CL, clearance; V₁/V₂, central/peripheral volume of distribution; Q, inter-compartmental clearance; θ_CrCL, CrCL covariate exponent; IIV-CL, inter-individual variability of CL; RSV, residual variability; RSE, relative standard error.

Figure 1.

Goodness-of-fit plots for the final population pharmacokinetic model of colistin sulfate. (A) Individual predicted versus observed concentrations; (B) Population-predicted versus observed concentrations; (C) Conditional weighted residuals (CWRES) versus population predictions; (D) CWRES versus time after dose. Solid lines denote the line of identity (A and B) or the reference at zero (C and D).

Figure 2.

Prediction-corrected visual predictive check (pcVPC) for the final model. Solid and dashed lines represent the observed median and 5th/95th percentiles; shaded areas represent the corresponding 95% prediction intervals from 1000 simulated datasets.

Independent PK Predictive Evaluation

In the independent cohort (15 patients, 82 concentrations), ME was −54.68 ng/mL (95% CI −129.65 to 25.97), and the CI included zero. MPE was 12.62% (95% CI −0.01% to 26.98%) (Table 3 and Supplementary Figure S4). RMSE was 353.5 ng/mL and rRMSE was 37.38% (95% CI 31.13–44.83%). F30 was 53.66% (95% CI 42.68–64.63%) (Table 3). Although the point estimates met the prespecified rRMSE and F30 criteria, the small cohort and the F30 CI indicate substantial uncertainty.

Table 3.

Accuracy and Precision Metrics for the Independent PK Predictive Evaluation of the Final Model

Metric	Estimate	95% CI (Bootstrap)
Accuracy
ME (ng/mL)	−54.68	(−129.65 to 25.97)
MPE (%)	12.62	(−0.01 to 26.98)
Precision
RMSE (ng/mL)	353.5	(304.02 to 401.22)
rRMSE (%)	37.38	(31.13 to 44.83)
Clinical relevance
F20 (%)	43.90	(32.93 to 54.88)
F30 (%)	53.66	(42.68 to 64.63)

Notes: F20/F30, proportion of predictions within ±20%/±30% of observations. 95% CIs were obtained by 1000-replicate bootstrap.

Abbreviations: ME, mean error; MPE, mean prediction error; RMSE, root mean square error; rRMSE, relative RMSE.

Clinical Outcomes and Efficacy Predictors

Among the 55 modeling-cohort patients, 24 (43.6%) achieved clinical efficacy and 31 (56.4%) were treatment failures. For the secondary outcomes, 30-day all-cause mortality was 10.9% (6/55), microbiological eradication was achieved in 56.4% (31/55), and AKI occurred in 18.2% (10/55). In univariable logistic regression, variables associated with efficacy at P < 0.10 included APACHE II score (odds ratio [OR] 0.87; P = 0.018), AUC_ss,24h/MIC (OR 1.03; P = 0.026), diabetes mellitus, chronic pulmonary disease, C_ss,max/MIC, and MIC (Supplementary Table S2). After excluding collinear variables and confirming no material confounding (all assessed covariates changed the OR by <10%), the final model retained AUC_ss,24h/MIC and APACHE II score. With 24 efficacy events and two final predictors, the model had 12 events per predictor. AUC_ss,24h/MIC was an independent positive predictor (OR 1.04; 95% CI 1.01–1.06; P = 0.006), corresponding to a 4% increase in the odds of efficacy per one-unit increase. APACHE II score was an independent negative predictor (OR 0.81; 95% CI 0.69–0.94; P = 0.006), corresponding to a 19% decrease per one-point increase (AIC 63.7; variance inflation factor 1.20 for both) (Table 4). The model was:

Table 4.

Multivariable Logistic Regression of Clinical Efficacy (A) and Exploratory ROC Analysis of AUC_ss,24h/MIC for Predicting Clinical Efficacy (B)

(A) Multivariable logistic regression
Variable		N	OR	95% CI		P
AUCss,24h/MIC		55	1.04	1.01–1.06		0.006
APACHE II		55	0.81	0.69–0.94		0.006
(B) Exploratory ROC analysis
Index	AUROC	Cut-off	Youden	Se (%)	Sp (%)	PPV (%)	Acc (%)
AUCss,24h/MIC	0.712	40.53	0.367	62.5	74.2	65.2	69.1

Notes: The optimal cut-off was selected by the maximum Youden index. The AUROC 95% CI was calculated by the DeLong method, and the cut-off 95% CI was obtained from 10,000 stratified bootstrap resamples.

Abbreviations: OR, odds ratio; Se, sensitivity; Sp, specificity; PPV, positive predictive value; Acc, accuracy.

Exploratory PK/PD Threshold

Using AUC_ss,24h/MIC as a single continuous predictor, ROC analysis gave an AUROC of 0.712 (DeLong 95% CI 0.573–0.852), indicating moderate discrimination. The maximum Youden-index cut-off was 40.53 (bootstrap 95% CI 20.87–60.30), at which sensitivity was 62.5%, specificity was 74.2%, positive predictive value was 65.2%, negative predictive value was 71.9%, and accuracy was 69.1% (Table 4). This internally derived cut-off was used as an exploratory target for subsequent simulations. It was not validated against clinical outcomes in the independent cohort, which lacked microbiological confirmation and MIC data.

Exploratory Machine-Learning Sensitivity Analysis

On LOOCV, logistic regression had the numerically highest discrimination (AUROC 0.770; 95% CI 0.633–0.883), compared with random forest (0.711), XGBoost (0.696), and gradient boosting machine (0.672) (Supplementary Figure S5). Decision-curve analysis suggested comparable or greater net benefit for logistic regression across threshold probabilities of approximately 0.2–0.6 (Supplementary Figure S5). AUC_ss,24h/MIC and APACHE II score had the largest mean absolute SHAP values, with directions consistent with the logistic-regression findings (Supplementary Figure S6). The SHAP interaction plot showed a pattern of more positive SHAP values above the exploratory AUC_ss,24h/MIC cut-off (Supplementary Figure S7). Given the small sample and reuse of the same cohort, these machine-learning and SHAP findings were treated only as exploratory consistency checks and not as independent evidence that exposure offsets illness severity.

Monte Carlo Simulation

A first-dose loading strategy (first-dose doubling) produced near-steady-state exposure from the first administration across all regimens. With 0.75 MU every 12 h in a typical patient, the loading dose (1.5 MU) achieved a first-dose AUC_τ of 16.15 mg·h/L, exceeding the steady-state value (11.24 mg·h/L), whereas without a loading dose steady state was reached only by the third dose—a delay of approximately 24 h; comparable savings were seen across the other regimens (Supplementary Figure S8).

PTA for the exploratory target AUC_ss,24h/MIC ≥40.53 is shown across regimens, CrCL strata, and MIC values (Figure 3). In severe renal impairment (CrCL 10–30 mL/min), 0.5 MU every 12 h achieved PTAs of 88–99% at MIC ≤0.5 mg/L. At MIC = 1 mg/L, 0.75 MU every 8 h and 1 MU every 8 h achieved PTAs of 91% and 98%, respectively. In moderate impairment (CrCL 30–80 mL/min), 0.75 MU every 8 h achieved PTA 93–100% at MIC ≤0.5 mg/L, but no regimen reached PTA ≥90% at MIC ≥2 mg/L. In normal renal function (CrCL 80–130 mL/min), only 1 MU every 8 h achieved PTA 100% at MIC = 0.25 mg/L, and at MIC ≥1 mg/L no regimen exceeded 41%. In augmented renal clearance (CrCL 130–150 mL/min), no regimen exceeded 80% beyond MIC = 0.5 mg/L and all were ≤26% at MIC ≥1 mg/L. These simulations indicate a high risk of target nonattainment with the evaluated monotherapy regimens in patients with higher renal function and less susceptible pathogens. They do not establish a clinical benefit of combination therapy, which should be evaluated prospectively.

Figure 3.

Probability of target attainment (PTA) for the exploratory AUC_ss,24h/MIC target of ≥40.53 across dosing regimens, creatinine clearance (CrCL) strata, and MIC values, based on Monte Carlo simulation (2000 virtual patients per scenario). (A) CrCL 10–30; (B) 30–80; (C) 80–130; (D) 130–150 mL/min.

Abbreviations: MIC, minimum inhibitory concentration; MU, million units.

Research MIPD Calculator

The calculator presents a three-panel interface: patient inputs (CrCL, MIC, APACHE II score) on the left; model-derived CL, required daily dose, recommended regimen, and loading-dose information in the center; and a visualization comparing AUC_ss,24h/MIC attainment with and without first-dose doubling on the right (Figure 4). For an illustrative patient (CrCL = 50 mL/min, MIC = 1.0 mg/L, APACHE II = 15), the tool predicted a CL of 5.05 L/h and a required daily dose of 2.05 MU, recommended 1.0 MU every 12 h with first-dose doubling to 2.0 MU, and estimated a model-predicted clinical-efficacy probability of 84.8%. This output is illustrative and has not been prospectively validated as a treatment recommendation.

Figure 4.

Interface of the open-access research model-informed precision dosing (MIPD) calculator for colistin sulfate (https://gxmu-colistinsulfate-mipd.streamlit.app/). The three-panel layout comprises patient input (left), model-derived clearance, regimen information, and model-predicted efficacy probability (center), and visualization of AUC_ss,24h attainment with and without first-dose doubling (right). The interface is for research demonstration and has not been prospectively validated for clinical decision-making.

Discussion

In our study, a two-compartment model best described colistin sulfate disposition, consistent with recent reports.11,12 CrCL was identified as a significant covariate on clearance, in agreement with most previous colistin sulfate PopPK analyses.5,11,12 The typical clearance in our model was 6.67 L/h, higher than previously reported clearance estimates, including approximately 1.5–1.74 L/h in Yu et al,5 1.50 L/h in Xie et al,12 and 2.66 L/h in Sun et al.11 This difference may partly reflect differences in renal function, sampling design, model structure, and population characteristics. For example, our cohort had a higher CrCL than the cohort of Sun et al (median 51.9 mL/min; tvCL 2.66 L/h),11 consistent with the positive CrCL-clearance relationship. The positive CrCL exponent in the final model indicates that clearance increases nonlinearly with renal function. For example, a two-fold increase in CrCL would increase clearance by approximately 57%, helping to explain the lower exposure and reduced target attainment predicted in patients with normal renal function or augmented renal clearance. These findings support CrCL-guided dose individualization and early TDM in this population. Although CrCL reduced the IIV of clearance from 65.70% to 46.70%, nearly half of the variability remained unexplained. The relatively imprecise V₁ estimate, as reflected by its wide bootstrap CI, is a recognized limitation when early post-dose sampling is incomplete.27,28

The separate 15-patient cohort provided a preliminary evaluation of PK predictive performance. The ME confidence interval included zero, and the point estimates for rRMSE (37.38%) and F30 (53.66%) met the prespecified criteria.15–17 The inclusion of an independent evaluation cohort and a full set of quantitative performance metrics is important for assessing whether a PopPK model can support model-informed precision dosing in clinical practice.16,29 These results should not be interpreted as full external clinical validation. The cohort was small, patients lacked microbiological confirmation, and the analysis evaluated concentration prediction but not the exposure-response model or PK/PD threshold. Evaluation in larger, microbiologically confirmed, multicenter cohorts is required before the model can be considered for clinical MIPD implementation.

The exploratory PK/PD cut-off identified here (AUC_ss,24h/MIC ≥40.53) is lower than the 66.9 reported for polymyxin B, although differences in polymyxin pharmacokinetics warrant caution when comparing these thresholds.4,30 Plausible explanations include differences in formulation, protein binding, pathogen distribution, and study population.31 However, the cut-off was derived from only 55 patients with 24 efficacy events, its discrimination was moderate (AUROC 0.712; 95% CI 0.573–0.852), and its bootstrap 95% CI was wide (20.87–60.30). Exposure therefore does not fully determine outcome in critically ill patients, for whom host immune status, comorbidity burden, disease severity, and infection site also contribute.32,33 This is reflected in the independent association of APACHE II score with efficacy (OR 0.81 per point; P = 0.006).34,35 The cut-off was derived from univariable ROC analysis rather than the adjusted model, although AUC_ss,24h/MIC remained significant as a continuous variable after adjustment. The machine-learning analyses were exploratory. Their agreement on the relative importance of AUC_ss,24h/MIC and APACHE II supports internal consistency but does not provide independent validation.36,37

The simulations offer several hypothesis-generating clinical implications. First-dose doubling moved exposure closer to steady state from the first administration, consistent with loading-dose principles in polymyxin guidance.4,38 CrCL strongly influenced target attainment. Patients with reduced renal function had greater predicted exposure at a given dose, whereas those with normal or augmented renal function had a high risk of target nonattainment. At MIC ≥1 mg/L, PTA fell sharply across the evaluated regimens. This finding suggests that monotherapy may be insufficient in some patients, but the study did not compare treatment strategies and cannot establish that combination therapy improves outcomes.4,39 The research MIPD calculator integrates the PopPK model, exploratory threshold, and efficacy model into an interpretable interface.40,41 It has not undergone prospective impact evaluation and should not replace susceptibility results, TDM, source control, or clinical judgment.

In this cohort, 30-day mortality was 10.9%, microbiological eradication was 56.4%, and treatment-emergent AKI occurred in 18.2% of patients. These outcomes fall within the broad range reported in previous polymyxin-treated CR-GNB cohorts.13,30,34,42 However, differences in infection site, pathogen distribution, disease severity, renal function, combination therapy, and AKI definitions limit direct cross-study comparison. For this reason, the adjusted exposure-response association with clinical efficacy provides the stronger evidence for the clinical relevance of AUC_ss,24h/MIC in this study.

This study has several limitations. First, the modeling cohort was modest (n = 55), with 24 efficacy events. Although the final logistic model was limited to two predictors, statistical power, coefficient stability, and parameter precision remain constrained. Second, the independent cohort included only 15 patients with clinically suspected without microbiological confirmation. It provided a preliminary evaluation of PK concentration prediction but could not validate the exposure-response model or the PK/PD threshold. Third, the AUC_ss,24h/MIC cut-off was derived and evaluated in the same cohort, showed only moderate discrimination, and had a wide bootstrap confidence interval. It should therefore be considered exploratory. Fourth, total rather than free plasma concentrations were measured, and variable protein binding may affect exposure-response relationships. Fifth, the composite efficacy endpoint included subjective clinical documentation and was not independently adjudicated. Mixed infections and use of the highest MIC further complicated PK/PD attribution. Finally, the machine-learning analyses and MIPD calculator are hypothesis-generating, and the study did not directly compare monotherapy with combination therapy. Larger prospective multicenter studies with microbiological confirmation, prespecified outcomes, free-drug measurements, and external clinical validation are needed.

Conclusions

This study developed a two-compartment PopPK model for intravenous colistin sulfate and provided preliminary evaluation of PK predictive performance in a separate 15-patient cohort. CrCL was the main determinant of clearance. An empirically derived AUC_ss,24h/MIC cut-off of 40.53 was associated with clinical efficacy, but it remains exploratory because it was derived from a small cohort and showed moderate discrimination and wide uncertainty. Simulations identified a high risk of target nonattainment in patients with normal or augmented renal function at MIC ≥1 mg/L, and the findings were incorporated into a research MIPD calculator. The results support further evaluation of CrCL-guided dosing and early TDM, but they do not establish the benefit of combination therapy or the clinical utility of the calculator. Prospective multicenter validation is required.

Funding Statement

This research was funded by the Self-funded Scientific Research Project of the Western Medicine Category of the Health Commission of Guangxi, grant numbers Z-A20230554 and Z-A20241869.

Abbreviations

AIC, Akaike information criterion; AKI, acute kidney injury; ALB, albumin; ALT, alanine aminotransferase; APACHE II, Acute Physiology and Chronic Health Evaluation II; AST, aspartate aminotransferase; AUC_τ, area under the concentration-time curve over a dosing interval; AUC_ss,24h, 24-hour steady-state area under the concentration-time curve; AUC_ss,24h/MIC, ratio of 24-hour steady-state area under the concentration-time curve to minimum inhibitory concentration; AUROC, area under the receiver operating characteristic curve; BUN, blood urea nitrogen; CI, confidence interval; CL, clearance; CMS, colistimethate sodium; CR-GNB, carbapenem-resistant Gram-negative bacteria; CRP, C-reactive protein; CrCL, creatinine clearance; C_ss,max, steady-state peak concentration; C_ss,min, steady-state trough concentration; CWRES, conditional weighted residuals; DBIL, direct bilirubin; F20/F30, proportions of predictions within ±20% and ±30% of observed concentrations; HGB, hemoglobin; ICU, intensive care unit; IIV, inter-individual variability; KDIGO, Kidney Disease: Improving Global Outcomes; LOOCV, leave-one-out cross-validation; ME, mean error; MIC, minimum inhibitory concentration; MIPD, model-informed precision dosing; MPE, mean prediction error; MU, million units; NONMEM, nonlinear mixed-effects modeling; OFV, objective function value; OR, odds ratio; pcVPC, prediction-corrected visual predictive check; PCT, procalcitonin; PK/PD, pharmacokinetic/pharmacodynamic; PLT, platelet count; PopPK, population pharmacokinetic; PPV, positive predictive value; PTA, probability of target attainment; Q, inter-compartmental clearance; RMSE, root mean square error; ROC, receiver operating characteristic; rRMSE, relative root mean square error; RSE, relative standard error; RSV, residual variability; SCR, serum creatinine; SD, standard deviation; Se, sensitivity; SHAP, SHapley Additive exPlanations; SOFA, Sequential Organ Failure Assessment; Sp, specificity; TBIL, total bilirubin; TDM, therapeutic drug monitoring; UPLC-MS/MS, ultra-performance liquid chromatography-tandem mass spectrometry; V₁, central volume of distribution; V₂, peripheral volume of distribution; WBC, white blood cell count; XGBoost, extreme gradient boosting.

Data Sharing Statement

The data presented in this study are available on reasonable request from Taotao Liu, one of the corresponding authors, subject to patient privacy and institutional ethics restrictions. The model-informed precision dosing calculator is available at https://gxmu-colistinsulfate-mipd.streamlit.app/.

Ethics Approval and Informed Consent

The study was conducted in accordance with the Declaration of Helsinki and approved by the Ethics Committee of The First Affiliated Hospital of Guangxi Medical University (protocol code 2023-K131-01, date of approval: May 2023). Written informed consent was obtained from all patients involved in the study or their legal representatives.

Author Contributions

All authors made a significant contribution to the work reported, whether that is in the conception, study design, execution, acquisition of data, analysis and interpretation, or in all these areas; took part in drafting, revising or critically reviewing the article; gave final approval of the version to be published; have agreed on the journal to which the article has been submitted; and agree to be accountable for all aspects of the work.

Disclosure

The authors declare no conflicts of interest in this work.