Sulperazon was purchased from Pfizer Pharmaceutical Co., Ltd. Cefoperazone sodium (chemical purity of 90.0%) and sulbactam sodium (chemical purity of 91.5%) were provided by WELMAN Pharmaceutical Co., Ltd.

Bacterial strains were investigated in antibacterial experiments in vitro (as shown in Table 1 and Supplemntary Table S1). The bacterial strains investigated included extended-spectrum beta-lactamases (ESBLs)-producing, ESBL-nonproducing, low and high cefoperazone-resistance Escherichia coli (E.coli); ESBL-producing, ESBL-nonproducing, low and high cefoperazone-resistance Klebsiella pneumoniae (K.pneumoniae).

Minimum inhibitory concentrations (MICs) were determined using the two-fold broth microdilution method described in the Clinical and Laboratory Standard Institute (CLSI) guidelines (CLSI, 2018). The inoculums were prepared by making a direct saline suspension of isolated colonies selected from an agar plate after being incubated for 18–24 h. Adjust the suspension to achieve turbidity equivalent to a 0.5 McFarland turbidity standard. Dilute the adjusted inoculum suspension in cation-adjusted Mueller-Hinton broth (CAMHB) so that each well contain approximately 5 × 10⁵ CFU/ml, finally. E. coli ATCC25922 was used as the quality control (QC) organism. The validation results indicated cefoperazone against the QC organism with MICs for E. coli ATCC25922 ranging from 0.12–0.5 mg/L. Bacteria were treated with cefoperazone at concentrations ranging from 0.008–256 mg/L.

In the in vitro antibacterial efficacy analysis, we examined a series of concentration combinations with the fixed ratio of cefoperazone and sulbactam. For a particular strain, the concentration combinations of cefoperazone and sulbactam that lead to the 90% inhibition of the tested bacterial isolates (MIC₉₀) were selected as the cutoffs for this concentration ratio. According to this criterion, the antibacterial effect of all the concentration combinations under this fixed ratio was classified into two categories: ≥MIC₉₀ and <MIC₉₀. No statistical test was needed for this classification. The limitation of this classification system was that it only worked for these concentration ratios tested in vitro assay. However, due to the different plasma protein binding behavior between cefoperazone and sulbactam, the free drug concentration ratios of cefoperazone and sulbactam would keep changing over time. Hence, logistic regression was developed based on the existing in vitro data to help classify any in vivo concentration combinations of cefoperazone and sulbactam into two categories: ≥MIC₉₀ and <MIC₉₀. The collected data were randomly divided into two groups, a training group and a test group, in a ratio of 7:3. Equations 1-7 were fit to the training group data and the model with the lowest Akaike information criterion (AIC) was selected. The selected model was then evaluated using the test group data to examine its prediction accuracy. log ( p 1 − p ) = β i n t e r c e p t + β c ⋅ C e f o (1) log ( p 1 − p ) = β i n t e r c e p t + β s ⋅ S u l b (2) log ( p 1 − p ) = β i n t e r c e p t + β c : s ⋅ C e f o ⋅ S u l b (3) log ( p 1 − p ) = β i n t e r c e p t + β c ⋅ C e f o + β s ⋅ S u l b (4) log ( p 1 − p ) = β i n t e r c e p t + β c ⋅ C e f o + β c : s ⋅ C e f o ⋅ S u l b (5) log ( p 1 − p ) = β i n t e r c e p t + β s ⋅ S u l b + β c : s ⋅ C e f o ⋅ S u l b (6) log ( p 1 − p ) = β i n t e r c e p t + β c ⋅ C e f o + β s ⋅ S u l b + β c : s ⋅ C e f o ⋅ S u l b (7) where p is the probability of different concentration combinations of cefoperazone sodium and sulbactam sodium greater than MIC₉₀; β_intercept is the intercept; β_c and β_s are the PK/PD correlation coefficient of cefoperazone sodium and sulbactam sodium, respectively; β_c:s is the interaction coefficient between cefoperazone and sulbactam. Cefo and Sulb represent the free drug concentrations of cefoperazone and sulbactam, respectively.

There are 9 young-adult male subjects with similar body mass index (BMI) included in the cefoperazone/sulbactam clinical phase I trial, and the sample size was relatively small. Demographic data from this trial are presented in Table 2. PK data from the clinical phase I study [clinical trial approval of cefoperazone sodium and sulbactam sodium (3:1)] was obtained from WELMAN Pharmaceutical Co., Ltd. Study participants received an intravenous infusion of cefoperazone sodium and sulbactam sodium (3:1) of 1, 2 and 4 g over a 30-min infusion period. Blood samples were collected prior to drug administration (0 h) and at 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 4, 5, 6, 7, 8 and 10 h after administration. The protein binding of cefoperazone and sulbactam to human plasma proteins were retrieved from the published literature, presented in Table 3 (Craig and Gerber, 1981; Rafailidis et al., 2007).

Non-linear mixed effects modelling was performed using NONMEM 7 ™ (version VII, level 3; ICON Development Solutions, Ellicott City, MD, United States) using the first order conditional estimation with interaction (FOCEI) method (Keizer et al., 2013). Model evaluation was based on the objective function value (OFV), Akaike information criterion (AIC), the precision of parameter estimates, and the goodness-of-fit plots. Model diagnostic plots were performed using the Xpose4 package in R (version 3.5.3) (Jonsson and Karlsson, 1999). A visual predictive check (VPC) was performed with 1,000 simulations using PsN (version 4.8.0) to evaluate the ability of the model to describe the observed data (Lindbom et al., 2005).

The random-effects model included inter-individual random effects and residual random effects. An exponential model (Equation 8) was used to describe inter-individual variation, shown as follows: P i = P pop ⋅ e η i (8) where P_i is the PK parameter of each individual, P_pop is the PK parameter of the population and η_i represents the inter individual variation which follows a logarithmic normal distribution. A combined error model (Equation 9) was used to describe residual error: C obs = C pred ⋅ ( 1 + ε 1 ) + ε 2 (9) where, C_obs represents the observed values, C_pred represents the population predicted values and, ε₁ and ε₂ represent additive and proportional residual error, respectively.

Continuous fixed effects factors, such as biochemical indicators, were added to the PK model in the manner of a power function, as shown in Equation 10: P i = P pop ⋅ ( COV i COV tv ) θ COV ⋅ e η i (10) where, θ_cov is the influence coefficient of covariates, COV_tv and COV_i represent the population and individual values of covariates, respectively.

Covariates were tested for significance in the model using forward addition and backward elimination. Model selection was made on the basis of a Log-Likelihood ratio test at an acceptance p-value of 0.05 (ΔOFV = -3.84) in the forward step and 0.01 (ΔOFV = 6.63) in the backward step.

There were four steps in predicting the clinical efficacy of the cefoperazone/sulbactam combination via Monte Carlo simulation. First, total plasma concentration-time curves were simulated for each dosing regimen using the developed human Pop-PK model. The dosing regimens are listed in Table 4, designed according to the proposed clinical dosage for cefoperazone/sulbactam combination (3:1) and sulperazon (based on its package insert). The total plasma concentration was then converted into free concentration via the corresponding protein binding ratio. Since cefoperazone exhibited a time-dependent antibacterial effect, the fraction of time that the free drug combination concentration exceeded the MIC₉₀ within a dosing interval (fT_>MIC) was utilized as the PK/PD index (Crandon and Nicolau, 2011). For each patient, fT_>MIC was determined using Equation 11 as follows: f T > M I C ( % ) = ∑ i = 1 n f ( Cefo u i ,     Sulb u i ) n ⋅ 100 % (11) Where, n is the total sampling number; i indicates ith sampling point; Cefo u i and Sulb u i are the free drug concentrations of cefoperazone and sulbactam at ith sampling point. The developed logistic regression model was used to predict whether the concentration combination could achieve MIC for each pathogen (detailed in Table 5). The return value is 1 when this concentration combination is greater than MIC, or else return 0.

Lastly, the probability of target attainment (PTA) of various fT_>MIC targets (ranging from 0 to 100%) at steady state was calculated for each dosing regimen against different bacteria using Equations 11, 12. PTA ( % ) = ∑ j = 1 m f ( f T > M I C j ) m ⋅ 100 % (12) where, m is the total number of subjects; j denotes jth individual; fT_>MICj is the corresponding fT_>MIC for jth individual. The following logical equation was used to determine whether fT_>MIC was greater than the target value. A return value of 1 indicated that fT_>MIC was greater than the target value, or else return value is 0. f ( x ) = { 1 ,     x ≥ t a r g e t 0 ,     x < t a r g e t

For β-lactam antibiotics, therapeutic effectiveness is recognized to be achieved when fT_>MIC ≥ 50% for mild infections and, a significant bacteriostasis effect achieved when fT_>MIC ≥ 70% for the treatment of severe bacterial infection (Abdul-Aziz et al., 2015). Hence, for the purposes of this analysis, these two targets were chosen to define clinical efficacy.

The in vitro activity of combinations of cefoperazone sodium and sulbactam sodium (3:1, 2:1 and 1:1) is shown in Table 1. The MIC₅₀ and MIC₉₀ of cefoperazone/sulbactam (1:1) were 1-4 folds lower than that of combinations in the ratios of 2:1 and 3:1 for the ESBL⁺ E.coli and ESBL⁺ K.pneumoniae. The breakpoints of cefoperazone against Enterobacterales are ≤16 mg/L (susceptible), 32 mg/L (medium) and ≥64 mg/L (resistant), which were published by Clinical and Laboratory Standards Institute (CLSI) (CLSI, 2021). ESBLs positive strains can be divided into high cefoperazone-resistance strains (MIC≥256 mg/L) and low cefoperazone-resistance strains (MIC ranged between 32 mg/L-128 mg/L) according to the breakpoints of cefoperazone. The different concentration combination of cefoperazone/sulbactam displayed similar activity against cefoperazone-resistance strains.

A logistic regression model was developed to link the cefoperazone/sulbactam concentration combinations to the antibacterial effect for each tested bacterium (Table 5). The predictive accuracy of the final models in the testing dataset was all greater than 90%. The established models indicated that cefoperazone and sulbactam play synergistic roles in the inhibition of most tested bacterium. For the high cefoperazone-resistance and ESBLs + bacterium, the activity of cefoperazone alone was insignificant as that was observed in low cefoperazone-resistance and ESBLs-bacterium.

The PK profiling of cefoperazone in humans was described by a two-compartment model (Eq. 13 to Eq. 14), CL = 5.34 ⋅ e η CL ( L / h ) (13) V 1 = 8.23 ⋅ e η VC ( L ) (14) Q = 3.54 ( L / h ) V 2 = 3.55 ( L )

The PK properties of sulbactam in humans were also profiled by a two-compartment model (Eq. 15 to Eq. 16), CL = 8.89 ⋅ e η CL ( L / h ) (15) V 1 = 10.1 ⋅ e η VC ( L ) (16) Q = 6.24 ( L / h ) V 2 = 3.57 ( L )

No significant covariate was identified in the stepwise covariate search. As shown in Table 6, all the model parameters were precisely estimated with relative standard error below 30%. The goodness-of-fit plots for both models are shown in Figures 1, 2, respectively. The observed values versus the population and individual predicted values were closely distributed around the line of identity. The conditional weighted residuals were randomly and homogenously distributed around 0. As shown in the VPCs for cefoperazone and sulbactam, in Figures 3, 4, the models adequately describe the observed plasma concentrations.

Investigation of clinical efficacy of cefoperazone/sulbactam combination against bacteria via Monte Carlo simulation

The concentration-time profiles of cefoperazone/sulbactam combinations under different dosing regimens were simulated (as shown in Supplementary Figure S1–S24). The PTA of cefoperazone/sulbactam combinations in the ratios of 3:1 and 2:1 following single- and multiple-dose against eight bacteria are shown in Supplementary Figure S25–S40 and Supplementary Table S2. The PTA = 90% was used as the cutoff to determine whether a dosing regimen presented good efficacy or not. All the dosing regimens investigated exhibited good antibacterial activity against ESBLs⁻ E. coli and ESBLs⁻ K. pneumoniae.

Based on the simulation results following single-dose, when a target of fT_>MIC at either 50% or 70% was set, the PTA was 100 and 47% with the regimen of cefoperazone/sulbactam (5.33 g, 4 g:1.33 g) for the ESBLs⁻ E. coli, respectively. Under a dosing regimen of cefoperazone/sulbactam (6 g, 4 g:2 g), the PTA achieved 100 and 53% with fT_>MIC target equal to 50 and 70%, which was similar as cefoperazone/sulbactam (5.33 g, 4 g:1.33 g). For the ESBLs⁻ K. pneumoniae, the regimen of cefoperazone/sulbactam (5.33 g, 4 g:1.33 g) achieved the same PTA (100 and 99%) as cefoperazone/sulbactam (6 g, 4 g:2 g) when targeting fT_>MIC = 50 and 70%. However, all the designed single dosing regimens provided poor antibacterial effect against ESBLs⁺ E. coli and ESBLs⁺ K. pneumoniae.

From the simulation of multiple administration, the regimens of cefoperazone/sulbactam (4 g, TID, 3 g:1 g) and cefoperazone/sulbactam (6 g, BID, 4 g:2 g) achieved 97 and 39% of PTA (target: fT_>MIC = 70%) against ESBLs⁻ E. coli, respectively. For the ESBLs⁻ K. pneumoniae, the above regimens can achieve 100 and 97% of PTA (target: fT_>MIC = 70%). Similar to the results of single administration, all the designed multiple dosing regimens provided poor antibacterial effect against ESBLs⁺ E. coli and ESBLs⁺ K. pneumoniae.