Mechanism-Based Pharmacodynamic Models of Fluoroquinolone Resistance in Staphylococcus aureus

doi:10.1128/AAC.00736-05

Antimicrob Agents Chemother. 2006 September; 50(9): 2957–2965.

PMCID: PMC1563538

Mechanism-Based Pharmacodynamic Models of Fluoroquinolone Resistance in Staphylococcus aureus

Philip Chung,¹ Patrick J. McNamara,¹ Jeffrey J. Campion,² and Martin E. Evans^2,³^*

Department of Pharmaceutical Sciences, College of Pharmacy,¹ Division of Infectious Diseases, Department of Internal Medicine, College of Medicine, University of Kentucky,² Division of Infectious Diseases, Department of Internal Medicine, Veterans Affairs Medical Center, Lexington, Kentucky³

^*Corresponding author. Mailing address: Division of Infectious Diseases, Department of Internal Medicine, Room B415, Veterans Affairs Medical Center, 1101 Veterans Drive, Lexington, KY 40502. Phone: (859) 381-5960. Fax: (859) 381-5840. E-mail: Martin.Evans/at/uky.edu.

Received June 9, 2005; Revised September 17, 2005; Accepted June 23, 2006.

This article has been cited by other articles in PMC.

Abstract

Pharmacodynamic modeling from earlier experiments in which two ciprofloxacin-susceptible Staphylococcus aureus strains and their corresponding resistant grlA mutants were exposed to a series of ciprofloxacin (J. J. Campion, P. J. McNamara, and M. E. Evans, Antimicrob. Agents Chemother. 49:209-219, 2005) and levofloxacin (J. J. Campion et al., Antimicrob. Agents Chemother. 49:2189-2199, 2005) pharmacokinetic profiles in an in vitro system indicated that the subpopulation-specific estimated maximal killing rate constants were similar for both agents, suggesting a common mechanism of action. We propose two novel pharmacodynamic models that assign mechanisms of action to fluoroquinolones (growth inhibition or death stimulation) and compare the abilities of these models and two other maximum effect models (net effect and MIC based) to describe and predict the changes in the population dynamics observed during our previous in vitro system experiments with ciprofloxacin. A high correlation between predicted and observed viable counts was observed for all models, but the best fits, as assessed by diagnostic tests, and the most precise parameter estimates were obtained with the growth inhibition and net effect models. All models, except the death stimulation model, correctly predicted that resistant subpopulations would not emerge when a high-density culture was exposed to a high initial concentration designed to rapidly eradicate low-level-resistant grlA mutants. Additional experiments are necessary to elucidate which of the proposed mechanistic models best characterizes the antibacterial effects of fluoroquinolone antimicrobial agents.

Pharmacodynamic models may prove to be efficient and powerful tools for optimizing antimicrobial dosing to prevent the emergence of resistance if they can accurately predict changes in susceptible and resistant bacterial subpopulations over time as a function of antimicrobial concentration. Such models could potentially be used to narrow the range of in vitro, animal, or clinical trials required for drug development and approval.

We previously evaluated the abilities of three maximum effect (E_max) models of increasing complexity, designed for two subpopulations of bacteria with different susceptibilities, to describe and predict the evolution of resistance to ciprofloxacin in Staphylococcus aureus by using pharmacokinetic, viable count, subpopulation, and resistance mechanism data obtained from in vitro system experiments (5). A two-population model with unique growth and killing rate constants for the ciprofloxacin-susceptible and -resistant subpopulations best described the initial killing and subsequent regrowth patterns observed. The model correctly characterized the enrichment of subpopulations with low-level resistance in the parent cultures and confirmed the importance of resistant variants to the emergence of resistance by successfully predicting that resistant subpopulations would not emerge when a low-density culture, with a low probability of mutants, was exposed to a clinical dosing regimen or when a high-density culture, with a higher probability of mutants, was exposed to a transient high initial concentration designed to rapidly eradicate low-level-resistant grlA mutants (5).

In subsequent in vitro system experiments in which the same S. aureus strains were exposed to ciprofloxacin, gatifloxacin, and garenoxacin (J. J. Campion and M. E. Evans, Abstr. 11th Int. Symp. Staphylococci Staphylococcal Infect., abstr. AR-02, 2004) and levofloxacin (3), we also found that a two-population model with unique growth and killing rate constants for fluoroquinolone-susceptible and -resistant subpopulations best described the changes in population dynamics. Model parameter estimates for the net bacterial growth (g) and maximum fluoroquinolone killing (k) rate constants were lower for the resistant (R) than for the susceptible (S) subpopulations but were similar among fluoroquinolones for each subpopulation. In contrast, the estimated 50% effective concentrations (EC₅₀) (concentration at which 50% of maximal bacterial killing occurred) for S and R subpopulations varied among the fluoroquinolones, suggesting that the differences in killing activity at comparable concentrations were mainly due to disparities in the EC₅₀. The parameter estimates obtained from modeling also suggested that at concentrations well in excess of the EC₅₀ the killing of each subpopulation would occur at similar maximal rates for each fluoroquinolone. These predictions were in concordance with the in vitro system data showing that high drug concentrations of each fluoroquinolone produced similar maximal rates of killing for a given subpopulation. Others have shown in time-kill studies that the maximal killing rates of fluoroquinolones such as ciprofloxacin, levofloxacin, gatifloxacin, garenoxacin, and trovafloxacin tend to converge over the first 2 h of drug exposure (8, 10, 11).

Our pharmacodynamic models and those used by others (12, 14, 17) to describe the effect of fluoroquinolones on bacteria are net effect models. These models use a net growth rate constant to characterize the growth of bacteria. This net growth rate constant represents the difference between the rate of bacterial cell division and the rate of bacterial death due to all causes other than the antimicrobial agent. The pharmacodynamic models described above are also not based upon known mechanisms of drug action. Current understanding of the effects of fluoroquinolone antimicrobials on bacteria indicates that these agents may inhibit cell division and stimulate cell death (13, 16, 22, 25). If this was the case, we reasoned that more-complex models similar in structure to some of the indirect response models proposed by Jusko, Dayneka, and coworkers (6, 15) might be more appropriate than the net effect model. Indirect response models have been used to describe the delayed pharmacologic effects of methylprednisolone on T-lymphocyte trafficking (7), warfarin on the coagulation cascade (15), and aldose reductase inhibitors on the glycosylation of hemoglobin (24). One indirect response model proposed by Jusko and coworkers described the reduction of a baseline response as a result of inhibition of an input function. This might correspond to inhibition of the bacterial cell division rate by ciprofloxacin (growth inhibition model). If fluoroquinolones completely inhibit cell division, such a model would reveal a common, natural death rate of the bacteria. Another indirect response model proposed by Jusko and colleagues described the decline in baseline response as a result of stimulation of the output function. This might be analogous to stimulation of the natural cell death rate by ciprofloxacin (death stimulation model). Meagher and coworkers have recently proposed a death stimulation model based on the MIC (18).

In this paper, we compare the performances of our previously reported net effect model (5), the MIC-based model reported by Meagher and coworkers (18), and two novel mechanistic models, analogous to those proposed by Dayneka and coworkers (6), where ciprofloxacin either indirectly inhibits cell division or stimulates the natural death process. We decided to evaluate these models using our large database from in vitro system experiments with ciprofloxacin and two S. aureus strains (4). To our knowledge, there have been few reports comparing the goodness-of-fit and predictive performance of pharmacodynamic antimicrobial models by using a single data set (3, 5, 20).

MATERIALS AND METHODS

Bacterial viable count data for pharmacodynamic modeling.

Pharmacodynamic modeling was performed using data from previously reported in vitro hollow-fiber system experiments with two genetically unrelated ciprofloxacin-susceptible clinical isolates of methicillin-resistant Staphylococcus aureus (MRSA) (MRSA 8043 and MRSA 8282; ciprofloxacin MIC of 0.5 μg/ml) (4). The inoculum in all experiments was ~1 × 10⁷ CFU/ml (4). The bacteria were grown in the in vitro system without the antimicrobial agent (growth control) and exposed to two simulated clinical (intravenous ciprofloxacin clinical dosing regimens of 400 mg given every 8 h and 12 h) and three experimental (continuous infusion of 12.5 mg/h and intermittent 1-h infusions of 750 mg and 2,800 mg given every 12 h) pharmacokinetic profiles in the in vitro system over 96 h. These regimens provided a wide range of concentration-time exposures that resulted in different rates of bacterial killing and, in some cases, selection of resistant subpopulations (4). In addition, the most resistant spontaneously appearing single-step grlA mutants in the starting cultures (MRSA 8043C0-1, an S80Y mutant with a ciprofloxacin MIC of 4 μg/ml, and MRSA 8282C0-1, an A116P mutant with a ciprofloxacin MIC of 2 μg/ml) were isolated and tested in the in vitro system using an inoculum of ~1 × 10⁷ CFU/ml to provide additional independent information about growth (no antimicrobial agent) and killing (6,250 mg every 12 h) of the most resistant subpopulations over the first 24 h of drug exposure (5). All experiments were performed in duplicate for each of the dosing regimens and growth controls. Data from the 12 experiments with MRSA 8043, 11 experiments with MRSA 8282 (one experiment with MRSA 8282, 750 mg every 12 h, was excluded because no resistant subpopulations were detected, as described previously [5]), 4 experiments with MRSA 8043C0-1, and 4 experiments with MRSA 8282C0-1 constituted the database for model fitting.

Pharmacodynamic modeling.

Four two-population E_max pharmacodynamic models were fitted to the bacterial viable count-versus-time data obtained from the in vitro system experiments. These included a net effect model (5), an MIC-based model (18), and two mechanistic models that described indirect responses to ciprofloxacin (growth inhibition or death stimulation). In the net effect model, killing of bacteria by the antimicrobial agent was assumed to occur according to a simple E_max pharmacodynamic relationship, where the killing rate is a saturating function of antimicrobial concentration (5). Changes in S and R over time were described by the following equations:

where S and R are the sizes of susceptible and resistant subpopulations at time t, g is the net growth rate constant (g = b − x, where b is the rate of cell division and x is the death rate due to all causes other than the antimicrobial), N_max is the carrying capacity of the in vitro system, k is the apparent maximal bacterial killing rate constant, EC₅₀ is the ciprofloxacin concentration at which 50% of the maximal bacterial killing occurs, and C is the ciprofloxacin concentration at time t. This model used unique net growth and antibiotic-associated killing rate constants for the S and R subpopulations (5).

The MIC-based model was developed to describe the viable count-versus-time data from in vitro system experiments with Escherichia coli exposed to simulated human ciprofloxacin pharmacokinetic profiles (18). In this model, the antimicrobial enhances the natural bacterial death rate according to the following equations:

where VG_max is the maximal velocity of bacterial growth, CFU_m is the number of CFU at which the rate of replication is half maximal, K_d is the natural death rate (in the absence of antimicrobial), E_max is the maximal stimulatory effect of the antimicrobial on the death rate, C is the antibiotic concentration, SIT_m is the median effect value for C/MIC (the value at which the drug effect is half maximal), and H is Hill's constant. This model assumes that the S and R subpopulations have the same parameter values with the exception of the SIT_m.

The growth inhibition model assumed that the antimicrobial agent reduced the cell division rate (b) of the bacterial populations. This model was based on the indirect response model proposed by Dayneka and coworkers where the drug inhibits factors that control production of a response (model I) (6). In our model, we modified the input function from a zero-order to a first-order process and assumed that the S and R subpopulations had unique cell division (b) and natural death (x) rates. In addition, the maximal inhibitory effect of ciprofloxacin was assumed to be complete inhibition of cell division (I_max = 1) according to a simple inhibitory E_max model. Changes in the S and R subpopulations were described by the following equations:

The growth inhibition model assumed that at antimicrobial concentrations well in excess of the EC₅₀ the killing of each subpopulation would occur at a rate corresponding to x provided that the total density (or number) of bacteria is not at or near N_max (see Appendix).

The death stimulation model assumed that the antimicrobial agent accelerated the natural cell death rate (x) of the bacterial populations according to a simple E_max model. This model was based on the indirect response model proposed by Dayneka and coworkers where the drug stimulates factors that control the output or degradation of a response (model IV) (6) with modification of the input function from a zero-order to a first-order process. Changes in the S and R subpopulations were described by the following equations:

The death stimulation model assumed that at antimicrobial concentrations well in excess of the EC₅₀ the killing of each subpopulation would occur at a rate corresponding to x(1 + E_max) − b provided that the total bacterial count is not at or near N_max (see Appendix).

All four pharmacodynamic models made other simplifying assumptions. First, the growth of S and R subpopulations was influenced solely by the total number of each subpopulation in relation to N_max. Second, there was no interaction between S and R subpopulations. Third, mutation of bacteria from S to R subpopulations and vice versa was negligible, and fourth, antimicrobial concentrations were uniform throughout the peripheral (bacterial) compartment of the in vitro hollow-fiber system (i.e., there was no spatial heterogeneity in C).

Each model was simultaneously fit to log-transformed colony count data from all 16 in vitro hollow-fiber model experiments with MRSA 8043 and MRSA 8043C0-1 and then again to viable count data from all 15 experiments with MRSA 8282 and MRSA 8282C0-1 by use of a nonlinear regression program (WinNonlin version 4; Pharsight Corp., Mountain View, Calif.). The size of the susceptible subpopulation at time zero (S₀) in all models was fixed as the actual viable count observed at the start of each in vitro system experiment. The size of the resistant subpopulation at 0 h (R₀) was included as a parameter in all models because of the difficulty in precisely quantifying the small numbers of spontaneously appearing resistant variants in the starting cultures by use of traditional microbiological techniques. Initial parameter estimates for modeling were obtained using an approach outlined previously (5). R₀ was assumed to be 1 CFU/ml in all experiments with MRSA 8043 and MRSA 8282, based on the estimated frequency of resistant subpopulations in the starting cultures (4). For experiments with MRSA 8043C0-1 and MRSA 8282C0-1, R₀ was set as the calculated viable count at the beginning of the experiment.

Predicted ciprofloxacin concentrations (C) were linked to the pharmacodynamic models with pharmacokinetic parameter estimates obtained by modeling concentration-time profiles (area under the concentration-time curve [AUC]) from all in vitro system experiments using a one-compartment pharmacokinetic model (5). The one-compartment model accurately characterized the ciprofloxacin concentration-time profiles observed in the central compartment of the in vitro system (r = 0.957 to 0.999) and the exposure intensities in the peripheral compartment (AUC_peripheral/AUC_central = 97 to 105%) (4).

Least-square minimization was performed using the Nelder-Mead algorithm with up to 3,000 iterations when necessary (21). Convergence of each model was achieved when the relative change of the residual sum of squares was less than 0.0001. The goodness of the model fits was assessed by performing residual analysis and determining correlation coefficients (r) (9). The runs test was used to determine whether observed viable counts deviated systematically from model-predicted values. The pharmacodynamic model of best fit was discerned by the second-order Akaike information criteria (AIC_c) (2).

Model predictive performance.

The ability of each model to predict the observed results of the in vitro system experiments was tested in two ways. In the first, the viable count data for one set of dosing regimen experiments were excluded and each model was fitted to the remaining data. The resulting parameter estimates were then used to predict viable counts that would be observed during experiments with the excluded dosing regimen, and the correlation (r) between model-predicted and observed viable counts was calculated. This process was repeated using all possible permutations of excluded regimens for each strain.

In the second method, each model and its respective parameter estimates were evaluated for their ability to predict the observed results of two additional in vitro system experiments reported previously (5). In one of these experiments, inocula equivalent to the mean viable counts of all in vitro system experiments (~1 × 10⁷ CFU/ml) for each strain were exposed to a transient high initial ciprofloxacin concentration produced by a large dose (equivalent to a clinical dose of 6,250 mg) followed by concentrations achieved with the clinically recommended regimen of 400 mg given every 12 h. Previous mathematical modeling using the net effect model (see the first set of equations in “Pharmacodynamic modeling” above) suggested that the initial high dose would eradicate the small number of R bacteria usually present in an inoculum of this size and the subsequent clinical dosing regimen would be sufficient to extinguish the remaining majority S subpopulation (5). In these experiments, colony counts fell to the reliable limit of detection (~10² CFU/ml) by 24 h for MRSA 8043 and by 48 h for MRSA 8282 and remained there for the remainder of the 96-h experiments (5). In the second in vitro system experiment, a smaller inoculum (~1 × 10⁵ CFU/ml) of each strain was exposed to concentrations typical of a clinical ciprofloxacin dosing regimen of 400 mg every 12 h. No resistant subpopulations were detected in the inoculum. Under these conditions, the two-population net effect model collapsed to a one-population model (S bacteria only) and the simplified model predicted that the clinical ciprofloxacin regimen would eradicate the bacteria (5). These predictions were corroborated by in vitro system experiments (5).

Simulations with the four models were done using strain-specific mean parameter estimates for each model with numerical integration software (Stella version 8.0; ISEE Systems, Hanover, N.H.).

RESULTS

Pharmacodynamic modeling.

All four pharmacodynamic models predicted that in the absence of ciprofloxacin both susceptible and resistant subpopulations of MRSA 8043 and MRSA 8282 would reach the carrying capacity of the in vitro system by 12 h after the start of the growth control experiments. The models also described the patterns of change in bacterial colony counts following ciprofloxacin exposure observed during the 96-h in vitro system experiments. Each model predicted that colony counts would initially decline and reach a nadir by 20 to 24 h. The models predicted that colony counts would subsequently rebound after the nadir and continue to increase throughout the 96-h experiments unless the carrying capacity of the in vitro system was reached. The correlation coefficients (r) between model-predicted and observed colony counts for the four models ranged from 0.962 to 0.998 for MRSA 8043 and MRSA 8043C0-1 and from 0.934 to 0.999 for MRSA 8282 and MRSA 8282C0-1. However, the net effect and growth inhibition models (Fig. (Fig.1)1) demonstrated the least bias between predicted and observed counts for experiments within the data set. Comparison of the goodness-of-fit of the four models using AIC_c indicated that the growth inhibition model was best, followed by the net effect model (Table (Table1).1). The growth inhibition model had the smallest second-order AIC_c value among the models. The net effect model, with simple AIC_c differences (Δ_i) of 3.43 for MRSA 8043 and MRSA 8043C0-1 and 7.46 for MRSA 8282 and MRSA 8282C0-1 compared to the growth inhibition model, also fit the data well. There was less support for the death stimulation and MIC-based models based on AIC_c (Δ_i ranging from 13.4 to 42.2 depending on model and strain). Among the four models, the net effect and growth inhibition models also resulted in parameter estimates of the highest precision, with coefficients of variation (CV) of <12% for all parameters except R₀ (Table (Table22).

FIG. 1.

Fit of the growth inhibition model to experimental data from in vitro hollow-fiber system experiments. Observed colony counts of MRSA 8043 (filled symbols, top panel), MRSA 8043C0-1 (open symbols, top panel), MRSA 8282 (filled symbols, bottom panel), (more ...)

TABLE 1.

Selection statistics for the pharmacodynamic models^a

TABLE 2.

Pharmacodynamic model parameter estimates

The net effect, growth inhibition, and death stimulation models evaluated herein were all simple E_max models rather than their more complex sigmoid E_max counterparts. Previous experience with the net effect model (5) and additional analyses performed for the growth inhibition and death stimulation models (data not shown) indicated that the inclusion of a sigmoidicity factor (Hill constant) was not warranted. When a sigmoidicity constant was incorporated into the models, the goodness-of-fit based on AIC_c improved for MRSA 8043 but not for MRSA 8282. However, there was less bias in the residuals with the simple E_max models than with the sigmoid E_max counterparts. The planar confidence intervals of the Hill factors for the sigmoid E_max models overlapped by a value of 1, suggesting that the addition of sigmoidicity constants to the E_max models was redundant (of note, the Hill coefficients of the MIC-based model were also close to unity [Table [Table2]2] and their planar confidence intervals encompassed this value). Furthermore, parameter precision, based on assessment of the mean CV of common parameter estimates, was actually better with the simple E_max versions of the models. Following the principle of parsimony, we decided that the simple E_max forms of the net effect, growth inhibition, and death stimulation models were better because they described the data as well as their sigmoid E_max counterparts, with fewer parameters.

Results of modeling using the net effect model were similar to those previously reported (5). Among the four models examined, this model displayed the least bias between model-predicted and observed viable counts for experiments within the data set. Observed colony counts were randomly distributed above and below the predicted colony counts, with the exception of the experiments with ciprofloxacin at 400 mg every 12 h with MRSA 8043 and with ciprofloxacin at 400 mg every 8 h with MRSA 8282. For these experiments, a systemic deviation in the observed colony counts above and below the predicted killing and regrowth curves was detected (P < 0.05, runs test). Coefficients of variation for parameter estimates were <11.2% for both bacterial strains, with the exception of R₀. Estimates of R₀ were less precise, with CV of 28.0% for MRSA 8043 and 35.4% for MRSA 8282. Strong positive correlations (r = 0.99 for MRSA 8043; r = 0.99 for MRSA 8282) were observed between g_S and k_S, whereas other model parameters were less well correlated (r = −0.84 to 0.67). Univariate 95% confidence intervals for g, k, and EC₅₀ for both strains of bacteria did not overlap (data not shown). Planar 95% confidence intervals, however, overlapped slightly for g and k of the two subpopulations (data not shown). The planar 95% confidence intervals were wider than the univariate 95% confidence intervals, as the latter take into consideration correlation among parameters.

The growth inhibition model exhibited slightly more bias in predictions than the net effect model. The growth inhibition model displayed the same systematic deviations (experiments with ciprofloxacin at 400 mg every 12 h with MRSA 8043 and with ciprofloxacin at 400 mg every 8 h with MRSA 8282) noted with the net effect model but also had significant bias (P < 0.05, runs test) in model-predicted versus observed colony counts for the 12.5-mg/h-infusion dosing regimen for MRSA 8043. The growth inhibition model had a parameter estimate precision comparable to that of the net effect model. The CV of the growth inhibition model parameter estimates ranged from 1.7% to 11.5%, except for estimates of R₀, which had CV of 27.2% for MRSA 8043 and 34.8% for MRSA 8282. Parameters of the growth inhibition model (including b_S, x_S, b_R, and x_R) were less strongly correlated (r = −0.75 to 0.78 for MRSA 8043 and MRSA 8043C0-1; r = −0.63 to 0.82 for MRSA 8282 and MRSA 8282C0-1) than those of the net effect model. No overlap in the univariate 95% confidence intervals for b, x, and EC₅₀ between the S and R subpopulations was observed. The planar 95% confidence intervals for x of the S and R subpopulations of both bacterial strains and for b of MRSA 8282/MRSA 8282C0-1 coincided.

The death stimulation and MIC-based models also showed more bias in their predictions than the net effect model. These models had the same systematic deviations noted with the net effect model but in addition displayed significant bias (P < 0.05, runs test) for the growth control experiments with MRSA 8043 and, in the case of the MIC-based model, MRSA 8282. Parameter estimates of the death stimulation and MIC-based models were less precise than those of the net effect and growth inhibition models. CV were as high as 71.6% (x_S for MRSA 8282) for the death stimulation model and 39.9% (R₀ for MRSA 8282) for the MIC-based model. The use of a subpopulation-specific MIC for the S and R subpopulations and/or model parameters (other than SIT_m) did not improve the fit of the MIC-based model to the observed data (data not shown). In addition, strong positive correlations were observed between b_S and x_S (r = 0.91 for MRSA 8043; r = 0.99 for MRSA 8282) and b_R and x_R (r = 0.99 for MRSA 8043 and MRSA 8043C0-1; r = 1.00 for MRSA 8282 and MRSA 8282C0-1) of the death stimulation model and VG_max and K_d (r = 0.90 for MRSA 8043; r = 0.92 for MRSA 8282) of the MIC-based model.

Model predictive performance.

When the viable count data for one set of dosing regimen experiments were excluded and each model was fitted to the remaining data, parameter estimates differed by 4.51 to 68.9%, 1.73 to 46.6%, 11.0 to 103%, and 2.5 to 46.7% from the mean estimates obtained using the entire data set for the net effect, growth inhibition, death stimulation, and MIC-based models, respectively, depending on the parameter estimated and the dosing regimen excluded (data not shown). Estimates of R₀ displayed the greatest variability in all models except the death stimulation model. For this model, the biggest differences were observed for the estimates of equation M9

. In all cases, the variabilities in parameter estimates for each model were within the planar 95% confidence intervals of the mean estimates. When the resulting parameter estimates were used to predict viable counts that would be observed during experiments with the excluded dosing regimen, the correlation coefficients for the predicted and observed colony count data were similar among the four models regardless of which dosing regimen was excluded (r = 0.99). Thus, this approach was unable to differentiate between the predictive performances of the four pharmacodynamic models.

In the second test of predictive performance, all four models correctly predicted that resistant bacteria would not emerge when the lower inoculum (10⁵ CFU/ml) was exposed to the clinical ciprofloxacin dosing regimen (Fig. (Fig.2).2). Model predictions as to the times required for the lower inoculum exposed to the clinical dosing regimen to reach the reliable limit of detection were similar to the times observed during the in vitro system experiments (20 h for MRSA 8043; 24 h for MRSA 8282). The net effect, growth inhibition, and MIC-based models correctly predicted that resistant subpopulations would not be selected when the higher inoculum (10⁷ CFU/ml) was exposed to the experimental dosing regimen (Fig. (Fig.2).2). Predictions of these models were in agreement with results observed in the in vitro system experiments except that the models predicted that colony counts for the susceptible subpopulation would reach the limit of detection (10² CFU/ml) earlier than actually observed (~6 h earlier for MRSA 8043 and 24 h earlier for MRSA 8282). The death stimulation model incorrectly predicted that resistant subpopulations would emerge for MRSA 8282. This model predicted an initial decline in viable counts, largely due to killing of the susceptible subpopulation, with a subsequent growth of the resistant subpopulation by 32 h despite the high initial ciprofloxacin concentration.

FIG. 2.

Abilities of the net effect, MIC-based, growth inhibition, and death stimulation pharmacodynamic models to predict MRSA 8043 and MRSA 8282 viable count profiles for (i) an experimental ciprofloxacin dosage regimen (single 6,250-mg dose followed by 400 (more ...)

DISCUSSION

We developed two novel pharmacodynamic models that assign mechanisms of action to fluoroquinolone antimicrobial agents (growth inhibition or death stimulation) and compared the abilities of these models and two other maximum effect models—our previously described net effect model (5) and the MIC-based model reported by Meagher and colleagues (18)—to describe and predict the changes in bacterial population dynamics during in vitro system experiments where S. aureus was exposed to a series of simulated ciprofloxacin pharmacokinetic profiles (4, 5).

All four models described the growth of ciprofloxacin-susceptible and -resistant bacteria observed in the absence of ciprofloxacin, the killing of susceptible subpopulations, and, in some cases, selection and growth of resistant subpopulations in the presence of ciprofloxacin during the in vitro system experiments. Overall, the growth inhibition and net effect models afforded the best combination of a good fit with minimal bias, high precision of parameter estimates, and good predictive performance. The correlation between model-predicted and observed viable counts for the growth inhibition and net effect models was strong and the residuals were small, with the observed counts randomly scattered about the model-predicted curves in most cases. Parameter estimates for the growth inhibition and net effect models were also the most precise among the four models evaluated. The goodness-of-fit as assessed by the AIC_c was best for the growth inhibition and net effect models, with the growth inhibition model having the lowest AIC_c. Although the correlation between model-predicted and observed viable counts was also high for the death stimulation and MIC-based models, these models exhibited more bias and had higher AIC_c values, and the resulting parameter estimates were less precise. Furthermore, several of the parameters for the all-important resistant subpopulations were highly correlated, making these models less desirable than the growth inhibition and net effect models. Estimates of the parameter R₀ were the least precise for all four models. It was set as a parameter to be estimated by the models because of an inability to precisely quantify the small numbers of resistant bacteria that spontaneously appeared in the inocula used in these experiments. The large difference in sizes of the susceptible (~10⁷ CFU/ml) and resistant (<10² CFU/ml) subpopulations made it more difficult for the least-square minimization algorithm to arrive at a precise solution for this parameter.

When the performances of the models were examined, all except the death stimulation model correctly predicted that resistant bacteria would not emerge in the high- and low-inoculum scenarios. The death stimulation model predicted selection of resistant subpopulations of MRSA 8282 upon challenge with one of the experimental dosing regimens (Fig. (Fig.2).2). This may have been due to the high estimate of R₀ for MRSA 8282 predicted by the death stimulation model compared to that predicted by the other models that were evaluated. It may also have been the result of the larger variation of parameter estimates for the death stimulation model, perhaps because of the increased number of parameters in the model (10 instead of 8 as in the other three models).

The MIC-based model did not describe the data as well as the other models examined, perhaps because it differed in several potentially important respects. First, it did not assign unique values for the growth (VG_max) or death (K_d) rates of ciprofloxacin-susceptible and -resistant subpopulations as did the other models. We found in our earlier work that the net growth and fluoroquinolone killing rates of the grlA mutant derivatives (MRSA 8043C0-1 and MRSA 8282C0-1) were slower than those of the wild-type parent strains (MRSA 8043 and MRSA 8282) (3, 4). Second, the term used in the MIC-based model to describe bounded growth in the in vitro system did not slow bacterial growth until the total population size exceeded VG_max. Based on our parameter estimates for the MIC-based model and the observed maximum bacterial densities in our in vitro system experiments, the growth rate of the bacteria will not approach zero when this growth-limiting factor is used. We found in our previous work that the net growth rates of the wild-type and grlA mutant strains started to slow as the bacterial population approached within 1 log₁₀ of the carrying capacity of the in vitro system and that this process was adequately described using a logistic growth expression (3, 4). Third, the MIC-based model relates pharmacokinetics to a discrete measure of antimicrobial effect, the MIC, which does not describe the concentration- and time-dependent effects of the antimicrobial agent. Instead, the MIC represents the net effect of an antimicrobial agent at a fixed concentration over a defined incubation period. This differs from exposure to an antimicrobial in the in vitro system or in humans, where drug concentrations and the patterns of bacterial growth and killing change with time. The MIC is also not equivalent to the concentration predicted by the pharmacodynamic models to produce no net growth or killing of bacteria. This concentration, termed the Z MIC, or stationary concentration, has been described for other net effect pharmacodynamic models (1, 17, 19, 23). Others have shown that the correlation between MIC and stationary concentration varies with the antimicrobial tested, making the MIC a poor pharmacodynamic parameter for characterization of the concentration-effect relationship of an antimicrobial agent (19).

It is not clear whether the growth inhibition or death stimulation model most accurately characterizes what actually occurs when bacteria are exposed to ciprofloxacin. It is known that quinolones target DNA gyrase and topoisomerase IV in bacteria (13). DNA gyrase relieves topological stress ahead of the replication fork by maintaining negative supercoiling, and topoisomerase IV unlinks newly replicated DNA, allowing proper chromosome and plasmid segregation (16). Both enzymes create transient double-stranded breaks in the helix, catalyze passage of DNA strands through each other, and religate the break. Quinolones bind to single-stranded DNA within the topoisomerase-DNA complex and stop DNA synthesis by stabilizing a reversible intermediate, the cleavable complex, in which the topoisomerase is covalently bound to DNA. The ternary complex blocks passage of RNA polymerase, thereby terminating transcription (25). In this context, quinolone activity could be described by a pharmacodynamic model in which the agent inhibits cell division. If quinolones completely inhibit bacterial cell division, it would follow that the common maximal bacterial killing rates observed in our experiments with S. aureus and four quinolones merely represented the natural death rate of the cells that would occur without exposure to the antimicrobials (see Appendix). The ternary complex may also present a physical barrier to the replication fork. Collision of the replication fork with the ternary complex may result in formation of a nonreversible DNA lesion and ultimately the release of cytotoxic free ends of DNA (13, 16, 22). In this context, quinolone activity could be described by a death stimulation model in which the antimicrobial stimulates a natural death process. If this is the case, it would follow that the common maximal bacterial killing rates observed were due not only to similar natural death rates but also to similar cell division rates and maximal stimulatory effects among all of the quinolones (see Appendix).

The interactions between antimicrobial agents and bacteria are often very complex, and it is unlikely that modeling alone can distinguish among plausible mechanisms of action. This is due, in part, to the nature of the data used for modeling, which are often limited to pharmacokinetics and an effect measure, such as viable bacterial counts. An additional complication is the often greater number of parameters required for mechanistic models than for their empirical (descriptive) counterparts. Moreover, it is possible that many mechanistic models, including ours, are interrelated and not completely unique. Additional knowledge about the mechanism of action of quinolones will help to identify a model with the most biological relevance. However, the ultimate value of any of these models will be determined by evaluating their predictions under varied experimental conditions which directly test whether quinolone antimicrobial agents inhibit cell division or stimulate cell death. At present, however, it is difficult if not impossible to discern experimentally whether the observed antibacterial effect of quinolones is due to inhibition of cell division or stimulation of a natural cell death process. Rigorously validated models that accurately characterize the mechanisms of action of antimicrobials may ultimately prove valuable for optimizing dosing of old and newly introduced agents to prevent the emergence of resistance.

Acknowledgments

This work was supported by grant GM066072 (to M.E.E.) from the National Institutes of Health. Philip Chung was supported by the University of Kentucky Research Challenge Trust and a training grant from the National Institutes of Health (K30 HL04163).

APPENDIX

In previous in vitro system experiments in which two fluoroquinolone-susceptible S. aureus strains and their corresponding low-level-resistant grlA mutants were exposed to a series of ciprofloxacin (4), levofloxacin (3), gatifloxacin, and garenoxacin (J. J. Campion and M. E. Evans, Abstr. 11th Int. Symp. Staphylococci Staphylococcal Infect., abstr. AR-02, 2004) pharmacokinetic profiles, bacterial killing appeared to reach a common, maximal rate with increasing concentrations of each fluoroquinolone. If it is assumed that the total bacterial population is not at the carrying capacity of the in vitro system [(S + R) [double less-than sign]

N_max] and that fluoroquinolone concentrations far exceed the concentration at which the rate of bacterial killing is half maximal (C [dbl greater-than sign]

EC₅₀), the terminal phase of the kill curves for the susceptible subpopulation (S) can be simplified to the following differential equations for the growth inhibition and death stimulation models.

Growth inhibition model.

Assume that (S + R) [double less-than sign]

N_max and that C [dbl greater-than sign]

so that the terminal phase of the kill curve will be described by

If the terminal killing curves for all fluoroquinolone antimicrobial agents collapse to the same maximal apparent killing rate, the growth inhibition model suggests that this would be due to a common natural death rate constant (x_S).

Death stimulation model.

Assume that (S + R) [double less-than sign]

N_max and that C [dbl greater-than sign]

so that the terminal phase of the kill curve will be described by

If the terminal killing curves for all fluoroquinolone antimicrobial agents collapse to the same maximal apparent killing rate, the death stimulation model suggests that this would be due not only to a common natural death rate constant (x_S) but also to a common cell division rate constant (b_S) and that the maximal stimulatory effects ( equation M18

) of each fluoroquinolone on cell death would also need to be identical among all of the agents.

Analogous simplified differential equations for each model can be derived for the resistant (R) subpopulation.

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