**DOI:**10.1128/AAC.48.12.4718-4724.2004

## ABSTRACT

The primary objectives of this analysis were to determine which pharmacokinetic model most accurately describes the elimination pathways for piperacillin in the presence of tazobactam through population pharmacokinetic modeling and to characterize its pharmacodynamic profile. Once the optimal pharmacokinetic model was identified, Monte Carlo simulation of 10,000 subjects with ADAPT II was performed to estimate the probability of attaining a target free-piperacillin concentration greater than the MIC for 50% of the dosing interval for 3.375 g every 6 h or every 4 h given as a 0.5-h infusion at each MIC between 0.25 and 32 μg/ml. In the population pharmacokinetic analysis, measurements of bias and precision, observed-predicted plots, and *r*^{2} values were highly acceptable for all three models and all three models were appropriate candidates for the Monte Carlo simulation evaluation. Visual comparison of the distribution of the piperacillin concentrations at the pharmacodynamic endpoint—h 3 concentrations of a 6-h dosing interval—between the simulated populations and raw data revealed that the linear model was most reflective of the raw data at the pharmacodynamic endpoint, and the linear model was therefore selected for the target attainment analysis. In the target attainment analysis, administration of 3 g of piperacillin every 6 h resulted in a robust target attainment rate that exceeded 95% for MICs of ≤8 mg/liter. The 4-h piperacillin administration interval had a superior pharmacodynamic profile and provided target attainment rates exceeding 95% for MICs of ≤16 mg/liter. This study indicates that piperacillin-tazobactam should have utility for empirical therapy of hospital-onset infections.

Antimicrobial pharmacodynamics is a term used to describe the relationship between drug exposure and antimicrobial activity. In recent years there have been tremendous strides in understanding the relationship between the pharmacodynamics of beta-lactams and treatment outcomes. Beta-lactams, in contrast to aminoglycosides and fluoroquinolones, exhibit little concentration-dependent bacterial killing (7-9, 14, 24, 25, 29, 30). This phenomenon was observed in early *Staphylococcus aureus* time-kill curve studies, which demonstrated that the rate of bacterial killing was not improved by increasing the concentration of penicillin (19). For beta-lactams, the time during which the free (unbound) serum drug concentration exceeds the MIC of the drug for the organism (*T* > MIC) appears to be the best predictor of outcomes (7-9, 24, 25). The serum free-beta-lactam concentration does not have to remain above the MIC for the entire dosing interval, and the fraction of the dosing interval during which it is required to remain above the MIC for the maximal antimicrobial effect varies for the different types of beta-lactams (7-9). Although the *T* > MIC varies for different drug-bacteria combinations, satisfactory outcomes (near-maximal bactericidal effect) are achieved when the free (unbound)-antibiotic concentration remains above the MIC for the organism for approximately 40 to 50% of the dosing interval for the penicillins (7-9, 24, 25). The precise endpoints may be slightly lower for penicillins compared to cephalosporins, and the endpoints for carbapenems are lower than those for penicillins (7, 8).

Piperacillin-tazobactam is an acylureido-penicillin-beta-lactamase inhibitor combination and is frequently used in the empirical treatment of hospital-acquired infections. Similar to other beta-lactam antibiotics, piperacillin-tazobactam exhibits time-dependent killing and the *T* > MIC appears to be the best outcome predictor (7, 8). Because a majority of infections are treated empirically, it is necessary to achieve a *T* > MIC equal to 50% of the dosing interval (50% *T* > MIC) against the most likely pathogens, including those with only moderate susceptibility. Although the pharmacokinetics (PK) of piperacillin-tazobactam are well described, its pharmacodynamic profile or ability to achieve 50% *T* > MIC in its targeted patient population—patients with hospital-related infections—against the array of MICs encountered in clinical practice has not been extensively characterized (6, 15-18, 20, 22, 27, 28).

There is also a debate about the PK model that most accurately describes the elimination of piperacillin. Although piperacillin PK have been most frequently described as linear, some authors have recently argued that piperacillin exhibits nonlinear PK and studies have documented evidence for fluctuating clearance with increasing concentrations (3-5, 34, 35). Piperacillin is primarily excreted through the kidneys by glomerular filtration. Piperacillin is also eliminated by active tubular secretion, and this active secretion process may be saturable.

The primary objectives of this investigation were to describe the PK of piperacillin in the presence of tazobactam and to characterize its pharmacodynamic profile. Through population PK modeling, the PK model that best characterizes its clearance was determined. Once the optimal PK model was identified, Monte Carlo simulation—using the mean parameter vector and full covariance matrix from the population model—was used to characterize the pharmacodynamic profile of currently recommended piperacillin-tazobactam dosing regimens against the array of MICs encountered clinically. Specifically, the Monte Carlo simulation profiled the likelihood of attaining a free-piperacillin 50% *T* > MIC at each MIC between 0.25 and 32 μg/ml for each regimen examined. We did not examine the pharmacodynamic profile of tazobactam because current doses of tazobactam in the piperacillin-tazobactam formulation have been shown to be sufficient for an antibacterial effect when the target is attaining a free-drug concentration exceeding the MIC for 50% of the dosing interval (31).

The second objective was to compare the piperacillin-tazobactam target attainment rates between hospitalized patients and healthy subjects. Monte Carlo simulations of antibiotic target attainment rates frequently use healthy-subject PK data, and this practice may provide a conservative estimate of antibiotic target attainment rates (20, 21). Piperacillin-tazobactam is primarily used in patients with hospital-related infections, and this patient population typically has a lower glomerular filtration rate than that of healthy subjects. Although it is advisable to use the PK data from the population of interest, the impact of using healthy-subject data to estimate target attainment rates for piperacillin in the presence of tazobactam has not been extensively described.

## MATERIALS AND METHODS

Population of hospitalized patients.The plasma concentration-time data for piperacillin-tazobactam in hospitalized patients were provided by Wyeth-Ayerst Research, and a summary of the studies can be found in Table 1. The study was restricted to hospitalized patients with clinical indications for piperacillin-tazobactam therapy. Healthy-subject data were not included in this analysis. The data were from open-label PK studies—two single-dose studies and two multiple-dose studies—and included subjects undergoing abdominal, thoracic, and colorectal surgery and hospitalized patients with neutropenia and bacterial infections. Of the 139 available patients, complete data were available for 128 patients, with 873 plasma drug concentrations.

Population PK modeling methods for hospitalized patients.All data were analyzed in a population PK model by using the nonparametric adaptive grid with adaptive γ (NPAG) program of Leary et al. (23). To determine which PK model most accurately describes the elimination pathways for piperacillin, models were parameterized as two-compartment models with zero-order infusion and with elimination from the central compartment modeled as a Michaelis-Menten (MM) process, as a MM process in parallel with first-order elimination, or as a first-order elimination process only. The models with zero-order input and MM with or without a parallel first-order elimination term were evaluated because of a recent study by Vinks et al. (35). Population PK modeling for the MM model and a parallel first-order-MM model was performed with the BIGNPAG program of Leary et al.

The general differential equations for the models are *dX*(1)/*dt* = *R*(*t*) − [((*V*_{max} · *V*/(*K*_{m} · *V* + *X*(1)) (− *k*_{el} + *k*_{cp}) · *X*(1) + *k*_{pc} · *X*(2)] and *dX*(2)/*dt* = *k*_{cp} · *X*(1) − *k*_{pc} · *X*(2), where *K*_{m} is the MM constant and the concentration in the central compartment for a half-maximal effect (milligrams per liter), *V*_{max} is the maximum elimination rate (milligrams per hour), *X*(1) is the amount of drug in the central compartment (milligrams), *X*(2) is the amount of drug in the peripheral compartment (milligrams), *k*_{el} is the first-order elimination rate constant (per hour), *k*_{cp} and *k*_{pc} are first-order intercompartmental transfer rate constants (per hour), *V* is the volume of the central compartment (liters), and *R*(*t*) is the time-delimited zero-order rate (piecewise input function) of drug input into the central compartment (milligrams per hour).

The inverse of the estimated assay variance was used as the first estimate for weighting in the PK modeling throughout. Weighting was accomplished by making the assumption that total observation variance was proportional to assay variance. Assay variance was determined on a between-days basis. The analysis was performed with adaptive γ, a scalar that multiplies the polynomial described above and is optimized with each cycle to produce the best approximation to the homoscedastic assumption.

Upon attaining convergence, maximal a-posteriori probability (MAP) Bayesian estimates for each patient were obtained by using the population-of-one utility within NPAG and BIGNPAG. For each model, the mean, median, and modal values were used as measurements of the central tendency of the population parameter estimates and were evaluated in the MAP Bayesian analysis. Scatter plots were examined for individual patients and for the population as a whole. Goodness of fit was assessed by regression with an observed-predicted plot, coefficients of determination, and log-likelihood values. Predictive performance evaluation was based on weighted mean error and the bias-adjusted weighted mean squared error.

Monte Carlo simulation of hospitalized-patient data.The mean parameter vector and covariance matrix from the population PK models were embedded in the PRIOR subroutine of the ADAPT II package of the programs of D'Argenio and Schumitzky (ADAPT II User's Guide, Pharmacokinetics/Pharmacodynamics System Analysis Software; Biomedical Simulations Resource, Los Angeles, Calif.) (10, 11). The population simulation without process noise option was used. A 10,000-subject Monte Carlo simulation was performed for standard 0.5-h infusions of 3 g of piperacillin administered every 4 or 6 h. Both normal and log-normal distributions were evaluated. These were discriminated on the ability to recreate the original mean parameter values and corresponding standard deviations from the population analyses.

The parameter values from the optimal distributions were used to simulate steady-state concentrations and to generate plasma concentration-time curves for each dosing regimen. The PK data for piperacillin were adjusted for 30% protein binding of piperacillin to reflect unbound drug concentrations in the data analysis (2). For each regimen examined, the fraction of simulated subjects who achieved 50% *T* > MIC was calculated for the range of piperacillin MICs (in the presence of tazobactam) from 0.25 to 32 μg/ml. Systat for Windows (version 10.2) was used for all data transformations.

Population PK modeling and Monte Carlo simulation of healthy-subject data.The optimal PK model identified in the population PK analysis of hospitalized-patient data was used to characterize the PK and pharmacodynamic profile of piperacillin for healthy subjects. Population PK modeling and Monte Carlo simulation were performed as already described for hospitalized patients. The data were drawn from a previously published investigation of 12 healthy male subjects administered piperacillin-tazobactam at 3.375 g every 6 h and 4.5 g every 8 h. The subjects ranged in age from 23 to 40 years (mean, 25 years) and in weight from 60.4 to 96.3 kg (mean, 78.4 kg). Their creatinine clearance values ranged from 94.1 to 134.1 ml/min/1.73 m^{2} (28).

## RESULTS

Population PK modeling of hospitalized-patient data.The mean and median population parameter estimates and associated dispersions identified by the NPAG (linear clearance model) and BIGNPAG (MM PK models) programs for the three models are displayed in Table 2.

Evaluation of predictive performance of population PK modeling of hospitalized-patient data.Table 3 summarizes the log-likelihood values, coefficients of determination (*r*^{2} values), and predictive performance of the linear model and the two nonlinear models, determined with weighted mean error and the bias-adjusted weighted mean squared error.

The highest log likelihood was observed for the parallel first-order-MM model, and the lowest was noted for the linear model. Comparison of log-likelihood values for the three models by likelihood ratio test—twice the log-likelihood difference evaluated against a χ^{2} distribution with the appropriate number of degrees of freedom—revealed statistically significant differences between the parallel first-order-MM model and the linear model, the parallel first-order-MM model and the MM model, and the MM model and the linear model (*P* < 0.05). In contrast, the *r*^{2} and bias and precision measurements were better for the linear model relative to those of the MM model and the parallel first-order-MM model. The best-fit regression lines for the observed-predicted plots were highly satisfactory for all three models, but the slope and intercept of the regression line most closely approached a slope of 1 with a small *y* intercept for the linear model relative to the other models. The observed-predicted plot for the linear model showed a best-fit regression line of observed = 1.032 × predicted + 0.31. For the MM and parallel first-order-MM models, the observed-predicted plots showed best-fit regression lines of observed = 1.041 × predicted + 3.69 and observed = 1.053 × predicted + 3.20, respectively. Overall, the measurements of bias and precision, observed-predicted plots, and *r*^{2} values were highly acceptable for all three models and all three models were appropriate candidates for the Monte Carlo simulation evaluation.

Monte Carlo simulation of hospitalized-patient data.We performed a 10,000-subject Monte Carlo simulation for the linear model, the MM model, and the parallel first-order-MM model. Log-normal distributions were selected for the population simulation for all three models on the basis of the ability to recapitulate the original mean parameter values and the corresponding standard deviations (Table 4). The Monte Carlo simulations recreated the original mean parameter values extremely well for both the linear model and the parallel first-order-MM model. For the MM model, considerable differences were observed between the original and simulated means and standard deviations of *V*_{max} and K_{21}.

To further assist in differentiating the models, we compared the distribution of piperacillin concentrations at the pharmacodynamic target—50% time point for a 6-h dosing interval (h 3 after initiation of infusion)—among the raw data (Fig. 1A), a 10,000-subject Monte Carlo simulation using the linear model (Fig. 1B), a 10,000-subject Monte Carlo simulation using the MM model (Fig. 1C), and a 10,000-subject Monte Carlo simulation using the parallel first-order-MM model (Fig. 1D). For the Monte Carlo simulations, the mean parameter values were used to simulate steady-state concentration-time curves for a piperacillin (in the presence of tazobactam) dose of 3 g administered every 6 h (0.5-h infusion) and all concentrations were corrected for an assumed protein binding of 30%. The raw data were scaled to a 3-g dose and also corrected for 30% protein binding (2). Visual comparison of the distribution of the piperacillin concentrations at the pharmacodynamic endpoint—concentrations at h 3 of a 6-h dosing interval—between the simulated populations and raw data revealed that the linear model was most reflective of the raw data at the pharmacodynamic endpoint, and the linear model was therefore selected for the target attainment analysis.

The results of target attainment analysis for the Monte Carlo simulation using the linear model are displayed in Fig. 2. Administration of 3 g of piperacillin at 6-h intervals resulted in a robust target attainment rate that exceeded 95% for MICs of ≤8 mg/liter and was 72.7% at 16 mg/liter. Piperacillin administration every 4 h (18 g/day) had a superior pharmacodynamic profile and provided target attainment rates exceeding 95% for MICs of ≤16 mg/liter.

Population PK modeling and Monte Carlo simulation of healthy-subject data.On the basis of the findings of the population PK analysis of hospitalized-patient data, a two-compartment linear model for piperacillin was used. The mean population parameter estimates and associated standard deviations identified by the NPAG program are displayed in Table 5. The overall fit of the model to the data was good. The observed-predicted plot showed a best-fit regression line of observed = 1.099 × predicted − 2.21. The *r*^{2} value was 0.946 for the 600 plasma samples (*P* < 0.001). The mean weighted error (measure of bias) was −0.088214, and the bias-adjusted weighted mean squared error (measure of precision) was 0.170119.

Monte Carlo simulation of healthy-subject data.A 10,000-subject Monte Carlo simulation using the linear model was performed. For a 6-h dosing interval (0.5-h infusion), 50% *T* > MIC attainment was >95% for MICs of ≤1 mg/liter (Fig. 3) and dropped precipitously with increasing MICs. For a 4-h dosing interval (0.5-h infusion), 50% *T* > MIC attainment exceeded 95% for MICs of ≤8 mg/liter and was 72% at 16 mg/liter. Compared to the simulated pharmacodynamic profile of hospitalized patients, the use of healthy-subject PK data underestimated the target attainment rate of 50% *T* > MIC for piperacillin.

## DISCUSSION

Piperacillin-tazobactam is frequently used in the empirical treatment of hospital-related infections. Because it is used empirically, it is important to achieve 50% *T* > MIC (maximal bactericidal activity) to cover the most likely pathogens, including those with only moderate susceptibility. Clinicians frequently rely on National Committee for Clinical Laboratory Standards (NCCLS)-based susceptibility patterns within their institutions to determine the utility of an antibiotic as an empirical agent. When the NCCLS breakpoints for piperacillin-tazobactam were established, the understanding of the relationship between antimicrobial exposure and the MICs of beta-lactams was not fully elucidated and this information was not used in the NCCLS susceptibility breakpoint decision-making process. The ability of piperacillin-tazobactam, therefore, to achieve 50% *T* > MIC in its targeted population for the range of MICs deemed to indicate susceptibility by NCCLS standards was not characterized until the present study.

Monte Carlo simulation was used to estimate target attainment rates for piperacillin in the presence of tazobactam. Monte Carlo simulation has been integrated with accepted pharmacodynamic models to compare antimicrobials and determine their abilities to achieve critical dynamic endpoints (1, 12, 13, 15, 20-22, 32). It is a mathematical technique in which parameter values are randomly drawn from a multivariate distribution. These randomly selected values allow the characterization of drug concentration-time profiles for a large number of simulated subjects. In essence, Monte Carlo simulation is a technique that incorporates the variability in PK among potential patients (variability between patients) when predicting antibiotic exposures and allows calculation of the probability of obtaining a critical target exposure for the range of possible MICs.

The Monte Carlo simulations demonstrated that piperacillin-tazobactam is a suitable agent for the empirical treatment of hospital-acquired infections. Standard piperacillin-tazobactam doses administered every 6 h provided high target attainment rates for MICs of ≤8 (piperacillin) and 4 (tazobactam) mg/liter when hospitalized-patient data were used. The population utility of this dose and schedule(s) can be ascertained by taking an expectation (weighted average) over the MIC distribution for target pathogens (*Escherichia coli*, *Klebsiella pneumoniae*, etc.). In clinical situations in which the MICs are expected to be ≥16 (piperacillin) and 4 (tazobactam) mg/liter, the results of the Monte Carlo simulations indicate that is preferable to administer piperacillin-tazobactam every 4 h to ensure a higher probability of achieving 50% *T* > MIC—the dynamic endpoint of maximal bactericidal activity. It is important to note that the Monte Carlo results indicate that the probability of achieving 50% *T* > MIC for piperacillin-tazobactam every 4 h is >95% for MICs of ≤16 (piperacillin) and 4 (tazobactam) mg/liter and is <80% for higher MICs. In situations in which the MICs are suspected or known to be ≥32 (piperacillin) and 4 (tazobactam) mg/liter, caution should be exercised when using piperacillin-tazobactam.

The linear-clearance model was used in the Monte Carlo simulation. There is a debate about the PK model that best describes the clearance of piperacillin in the presence of tazobactam (3-5, 34, 35). The measurements of bias and precision and the *r*^{2} values indicate that all three models are appropriate candidates for the Monte Carlo simulation evaluation. However, it is also clear that the best model by log likelihood is the parallel first-order-MM model. This is not surprising, given that the parallel first-order-MM model is an expansion of the MM and linear models and log-likelihood values improve with addition of terms to the model.

The goal of this study, however, was to determine the probability of obtaining 50% *T* > MIC for standard piperacillin-tazobactam dosing regimens against the array of MICs encountered in clinical practice. Visual comparison of the distribution of the piperacillin concentrations at the pharmacodynamic endpoint—the midpoint of a 6-h dosing interval (i.e., h 3)—between the simulated populations and raw data revealed that the linear model was most reflective of the raw data at the pharmacodynamic endpoint. We felt that the linear model, therefore, was the most appropriate model for the Monte Carlo simulation, in which the goal was to elucidate the pharmacodynamic adequacy of currently approved piperacillin-tazobactam dosing regimens. Compared to the raw data, both MM models produced a distribution of simulated h 3 concentration data that differed substantially from the shape of the concentration distribution of the raw data for h 3. Specifically, both MM models overestimated the clearance rate of piperacillin and provided conservative estimates of piperacillin concentrations at 50% of the 6-h dosing interval.

For the comparison of the raw and population data, the raw data were scaled to a 3-g dose. By scaling the raw data, there is an assumption that the rate of excretion and the concentration vary in direct proportion. This is contrary to MM kinetics, where drug clearance changes in a nonlinear fashion with increasing concentrations, most notably at concentrations exceeding the *K*_{m} (26). Most (*n* = 115) of the patients were scaled from 2 to 3 g (Table 1). If the MM assumption was essential, scaling would only provide a conservative estimate of piperacillin clearance and concentrations would actually be much higher because of the decreased clearance with escalating drug input. As previously mentioned, the range of piperacillin concentrations at the pharmacodynamic target—the midpoint of a 6-h dosing interval—is much greater for the actual data than the range of concentrations for the simulated MM models.

Furthermore, only 24 of the 138 patients had a single piperacillin concentration that exceeded the *K*_{m} of the MM model and only 14 patients had a single piperacillin concentration that exceeded the *K*_{m} of the parallel first-order-MM model. This indicates that serum piperacillin concentrations do not frequently exceed the *K*_{m} values and the contribution of the MM portion of clearance would be expected to be pseudolinear in the patient population we examined. This reason and the better measurements of bias and precision drove our decision to use the simpler linear model for the Monte Carlo simulation.

While Vinks and colleagues published an MM model for this drug, there are several reasons for the dissimilar findings (35). First, their patient population was a specialized population (cystic fibrosis patients) that may have different *K*_{m} and *V*_{max} values for the anion-selective pumps in their renal tubules. Wang et al. found that the renal clearance of ticarcillin, a related acylureido-penicillin derivative, was enhanced in cystic fibrosis patients because of the greater affinity of the renal secretory system for these drugs (36). This notion is further supported by the different *K*_{m} and *V*_{max} values observed between studies. In the present study, the observed mean *K*_{m} and *V*_{max} values for the MM model were 307.03 ± 147.4 and 812.75 ± 801.64, respectively, which are substantially different from the observed mean *K*_{m} and *V*_{max} values (92.3 ± 85.6 and 2,284 ± 1,046) in the study of Vinks et al. Second, the observed differences in the clearance models between studies may be secondary to the different infusion methods (intermittent versus continuous). In the study of Vinks et al., it was only with a continuous-infusion mode of administration that the benefits of the MM model were manifested (35). The nonlinear behavior of piperacillin in non-cystic fibrosis patients, therefore, may not become as evident with intermittent infusion as with continuous infusion, especially with continuous-infusion steady-state concentrations exceeding the *K*_{m}. Examination of the clearance of continuous-infusion piperacillin in non-cystic fibrosis patients is needed to clarify this issue.

Comparison of the target attainment rates of hospitalized patients and healthy subjects revealed that the healthy-subject data underestimated target attainment rates for piperacillin and illustrates the importance of using PK data from the target patient population when possible. Frequently, Monte Carlo simulations rely on PK data from healthy subjects because of the dearth of data on the population of interest (1, 20-22). Even the NCCLS frequently relies on healthy-subject data to assess the viability of MIC breakpoints. When possible, PK data should be derived from the population of interest (1, 12, 22, 30, 32, 33).

In summary, 3.375 g of piperacillin-tazobactam given intravenously every 6 h performed well, achieving excellent target attainment rates for MICs of ≤8 (piperacillin) and 4 (tazobactam) mg/liter. Use of 3.3375 g of piperacillin-tazobactam every 4 h extended the range to 16 (piperacillin) and 4 (tazobactam) mg/liter, respectively, indicating that piperacillin-tazobactam should have utility for empirical therapy of hospital-onset infections.

## ACKNOWLEDGMENTS

This study was supported in part by Wyeth Research.

This study also greatly benefited from the gracious critical reviews of Johan W. Mouton and Alexander A. Vinks.

## FOOTNOTES

- Received 17 June 2004.
- Returned for modification 8 August 2004.
- Accepted 8 September 2004.

- American Society for Microbiology