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Impact of Drug-Exposure Intensity and Duration of Therapy on the Emergence of Staphylococcus aureus Resistance to a Quinolone Antimicrobial

  1. V. H. Tam1,a,
  2. A. Louie1,
  3. T. R. Fritsche2,
  4. M. Deziel1,b,
  5. W. Liu1,
  6. D. L. Brown1,
  7. L. Deshpande2,
  8. R. Leary3,a,
  9. R. N. Jones2 and
  10. G. L. Drusano1
  1. 1Emerging Infections and Host Defense Laboratory, Ordway Research Institute, Albany, New York
  2. 2JMI Laboratories, North Liberty, Iowa
  3. 3University of California, San Diego, Supercomputer Center
  1. Reprints or correspondence: Dr. G. L. Drusano, Codirector, Ordway Research Institute, 150 New Scotland Ave., Albany, NY (GDrusano{at}ordwayresearch.org).

  • ↵a Present affiliations: Department of Clinical Sciences and Administration, University of Houston College of Pharmacy, Houston, Texas (V.H.T.); Pharsight Corporation, Cary, North Carolina (R.L.).

  • ↵b Deceased.

 
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Abstract

We have shown previously in animal model and in vitro systems that antimicrobial therapy intensity has a profound influence on subpopulations of resistant organisms. Little attention has been paid to the effect of therapy duration on resistant subpopulations. We examined the influence of therapy intensity (area under the concentration/time curve for 24 h:minimum inhibitory concentration [AUC24:MIC] ratio) and therapy duration on resistance emergence using an in vitro model of Staphylococcus aureus infection. AUC24:MIC ratios of ⩾100 were necessary to kill a substantial portion of the total population. Importantly, we demonstrated that therapy duration is a critical parameter. As the duration increased beyond 5 days, the intensity needed to suppress the antibiotic-resistant subpopulations increased, even when the initial bacterial kill was >4 log10 (cfu/mL). These findings were prospectively validated in an independent experiment in which exposures were calculated from the results of fitting a large mathematical model to all data simultaneously. All of the prospectively determined predictions were fulfilled in this validation experiment.

How much drug should we give to a patient to treat an infection, and for how long should we administer it? To begin to answer such questions, it is important to understand the goals of antimicrobial therapy. As a first premise, we would like to choose a dose of drug that will optimally inhibit or kill the infecting pathogen at the primary infection site without engendering undue toxicity. However, other goals of therapy beyond simply killing microorganisms are possible. We currently are in a crisis of antimicrobial resistance. We are rapidly losing agents for the treatment of infections to resistance not only in the hospital but also in the community. Consequently, it would be critical to identify doses and durations of therapy that would help suppress the emergence of resistant subpopulations.

Although we have learned much about anti-infective pharmacokinetics and pharmacodynamics from various rodent infection models, these models have inherent limitations, leaving certain critical questions unanswered. For instance, it is difficult to perform animal experiments over a time frame that mimics the duration of therapy seen in the clinic. Moreover, the pharmacokinetic profile of anti-infective agents is significantly different in rodents than in humans, and such differences can pose a translational problem. Consequently, little has been done experimentally to delineate what duration of therapy is required to reach our goal of killing microorganisms at the primary infection site while also suppressing the emergence of resistant sub-populations. One solution is the use of in vitro infection models, because these systems allow longer therapy durations to be studied using profiles that are far more similar to those in humans than is possible in rodents.

Our group has recently published an approach to the resistance problem in which a dose of a quinolone antimicrobial agent was identified that would suppress the amplification of the resistant mutant subpopulation present at the primary infection site in a nonneutropenic mouse-thigh Pseudomonas aeruginosa infection model [1]. We have since used an in vitro pharmacodynamic infection model to validate the previous findings from the mouse-thigh system [2]. Such a system allows delineation of the response of the bacterial populations to drug pressure (sensitive and resistant) and also allows the drug concentration/ time profile of patients to be simulated. In addition, there are no granulocytes or other immune functions present in the system, so that it is truly the effect of the drug that is being explicitly evaluated. This makes the findings emerging from such a system conservative. It was the aim of the present investigation to use our in vitro pharmacodynamic infection model to examine the impact of duration of therapy on these end points over a clinically relevant 10-day time frame.

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Methods

Microorganism. A methicillin-susceptible Staphylococcus aureus strain (ATCC 25925) was used. S. aureus ATCC 25925 was stored in skim milk at −70°C. The isolate was subcultured on blood agar plates twice before each experiment. Testing of susceptibility to garenoxacin was determined in duplicate by the macrobroth dilution method described by the Clinical and Laboratory Standards Institute (CLSI; formerly known as the National Committee for Clinical Laboratory Standards) [3] in cation-adjusted Mueller-Hinton II broth. Susceptibility testing was repeated using 10% cation-adjusted Mueller-Hinton II broth plus 90% clarified pooled human serum. MBCs were determined by subculturing 100 μL from the tube onto cationadjusted Mueller-Hinton II plates and determining the concentration that reduced the inoculum by 99.9%.

Susceptibility testing methods for resistant mutants. Strains were tested by the CLSI broth microdilution method [4] using validated, commercially prepared panels (TREK Diagnostics) in cation-adjusted Mueller-Hinton II broth against 2 quinolone agents, ciprofloxacin (Sigma-Aldrich) and garenoxacin (gift from Schering-Plough). Norfloxacin, ciprofloxacin, and garenoxacin were subsequently tested in freshly prepared broth panels with and without known efflux-pump inhibitors (reserpine and phe-arg-β-napthylamide [PABN; Sigma-Aldrich]), which were incorporated into the test medium at a final concentration of 20 μg/mL. S. aureus ATCC 29213 was included as a quality control (QC) strain when susceptibility testing of the quinolone agents was performed; QC results for the 3 agents were within CLSI-specified ranges.

Polymerase chain reaction (PCR) amplification of the quinolone resistance—determining region (QRDR). All strains were analyzed for mutations in the QRDR. Amplification of the topoisomerase IV and gyrase A gene segments responsible for fluroquinolone-resistant phenotypes was done as described by Horii et al. [5]. The primer sets used included grlA-F (GATGAGGAGGAAATCTAG), grlA-R (GTTGGAAAATCGGACCTT), grlB-F (GACAATTGTCTAAATCACTTGTG), grlB-R (CATCAGTCATAATAATTACAC), gyrA-F (GCGATGAGTGTTATCGTTGCT), gyrA-R (CAGGACCTTCAATATCCTCC), gyrB-F (CAGCGTTAGATGTAGCAAGC), and gyrB-R (CGATTTTGTGATATCTTGCTTTCG). PCR products were cleaned using the QIAquick PCR Purification Kit (QIAGEN).

Sequencing of QRDR amplicons. Generated amplicons were sequenced using the Sanger-based dideoxy sequencing strategy, which involves the incorporation of fluorescent dye—labeled terminators into the sequencing reaction products. Sequences obtained were subjected to a National Center for Biotechnology Information BLAST search, to detect the presence of any mutations in the QRDR. QRDR sequences of S. aureus ATCC 25923 were used as a control.

Determination of garenoxacin protein binding. Pooled human serum was used. Garenoxacin was dissolved in serum in concentrations ranging from 0.2 to 20 mg/L. Protein binding was determined by an ultracentrifugation technique [6], using the high-performance liquid chromatography (HPLC) assay described below.

Hollow-fiber infection model. A schematic diagram of the hollow-fiber infection model is shown in figure 1. Garenoxacin was directly injected into the central reservoir over a period of 1 h once daily to achieve the peak concentration desired at 1 h into the dosing interval daily for 10 days. Methods specific to the operation of this model have been published previously in the Journal [2].

Figure 1.
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Figure 1.

Schematic diagram of the in-vitro hollow-fiber infection model.

Exposure-response studies. S. aureus ATCC 25925 was stored, subcultured, and tested for susceptibility to garenoxacin as described above (see Microorganism). The bacterial inoculum was prepared using 3 medium-sized colonies grown overnight in Mueller-Hinton II broth at 35°C. Hollow-fiber systems were maintained at 35°C in a humidified incubator. Nine milliliters of bacterial culture in late-log-phase growth (2.0 × 108 cfu/mL) was infused into 7 hollow-fiber cartridges, 1 for nominal drug exposure (area under the concentration/time curve for 24 h [AUC24]:MIC ratio of 0 [control]) and 1 each for 6 drug regimens designed to achieve AUC24:MIC values between ∼40 and ∼600. These regimens simulated steady-state human pharmacokinetics of different regimens of unbound garenoxacin, given once every 24 h (terminal half-life, 12.8 h; clearance, 6 L/h) [7]. Experimentally attained garenoxacin exposures (central compartment) were based on drug concentrations quantified by a validated HPLC method [2] at 1.5, 4, 7, 11, 24, 25.5, 28, 35, and 48 h. At 0, 24, 48, 96, 144, 192, and 240 h, samples of the bacterial cultures were obtained, centrifuged at 3200 g for 15 min, resuspended in normal saline to minimize the drug carryover effect, and diluted 10-fold serially. The serially diluted samples were quantitatively cultured on drug-free Mueller-Hinton II agar plates to enumerate the total bacterial population. The garenoxacin-resistant bacterial subpopulation was quantified by culturing on media plates supplemented with garenoxacin at a concentration 3 times the baseline MIC. The media plates were incubated at 35°C for 24 and 72 h to evaluate the impact on the total and antibiotic-resistant subpopulation, respectively. MIC values were determined from the garenoxacinresistant subpopulation at the end of the experiment to confirm resistance.

Modeling methods. Modeling methods are shown in the appendix.

Prospective validation experiment. Using the data from the analysis of the initial experiments, we tested the following hypotheses prospectively: (1) the high-intensity regimen will drive down the colony counts by ∼5–6 log10 (cfu/mL) by day 5 of therapy; (2) the low-intensity regimen will be similar to the high-intensity regimen for 4 days with regard to total colony counts; (3) the high-intensity regimen will suppress resistance for the full 10 days of therapy; and (4) the low-intensity regimen will allow breakthrough growth of resistant microorganisms after day 4.

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Results

MIC:MBC ratio for garenoxacin. The MIC:MBC ratio was 0.04:0.08 mg/L when determined in Mueller-Hinton II broth. Determination of the MIC:MBC ratio in 90% human serum (heat inactivated) increased the values by 10-fold (0.04:0.08 mg/L to 0.4:0.8 mg/L).

Protein binding. Protein binding averaged 90.4% (SD, 0.47%) over the range of 0.2 to 20 mg/L.

Impact of escalating garenoxacin exposures on the total and resistant bacterial populations. The changes with time in the number of colony-forming units per milliliter for both the total population and the resistant (to at least 3 times the baseline MIC) subpopulation over the 10 days of the experiment are displayed in figure 2A–2G. The number of mutants as a percentage of the total population is relatively constant in the control arm, because no selective pressure is being placed on the population. Relatively low degrees of drug intensity (AUC24: MIC ratio of 41–79; figure 2B–2D) allow rapid amplification of the resistant subpopulation, with near complete replacement of the total population by garenoxacin-resistant microorganisms. When the AUC24:MIC ratio exceeds 100 (figure 2E), the resistant population is amplified but has not fully replaced the sensitive population. It is important to note that, early on during the course of therapy (up to hour 96), the resistant subpopulation is controlled by the intensity of therapy. Only later do we see the amplification of the resistant subpopulation. Higher intensity levels (AUC24:MIC ratios of 386 [figure 2F] and 604 [figure 2G]) completely suppress the total and resistant populations for the 10-day duration of the experiment.

Figure 2.
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Figure 2.

Effect of varying garenoxacin exposures on the total population and the resistant subpopulation of Staphylococcus aureus. Panel A shows the control. Panels B–G demonstrate increasing area under the concentration/time curve for 24 h (AUC24):MIC ratios (41–604) for garenoxacin.

Garenoxacin MIC values from microorganisms recovered from resistance plates. We examined 20 isolates from the resistance plates from differing study regimens on different days. In all instances, MIC values exceeded the baseline value by 2–8-fold (data not shown). When MIC values for the hydrophilic fluoroquinolone ciprofloxacin were determined from a subset of these isolates with and without the efflux-pump inhibitors reserpine and PABN, the MIC values decreased by 2–4-fold with the inhibitor. This indicates that efflux-pump overexpression may have a role to play in the emergence of resistance. Sequencing via the QRDR for gyrA and grlA targets did not reveal any mutations.

Modeling the drug effect on the populations. Given that the regimens studied controlled the infection for a short time span, exposure-response relationships were evaluated over the first 2 days of therapy to determine the breakpoint exposure that would suppress the resistant subpopulation for that time frame. The parameter estimates are displayed in table 1. As evidenced by the predicted-observed plot after the Bayesian step for the drug concentrations (figure 3A) and the predictedobserved plots for the total population and the resistant subpopulation (figure 3B and 3C, respectively), the model fit the data well. In all of the regressions, the slope approached 1.0, and the y-intercepts were small. All regressions had r2 values exceeding 0.97. On the basis of these data, an AUC24:MIC ratio of 101 is required for suppression of amplification of the resistant subpopulation for 2 days, with breakthrough after day 4.

Figure 3.
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Figure 3.

Observed-predicted plots obtained after the Bayesian step from the population model for drug concentrations (A), the total organism population (B), and the resistant subpopulation (C), using only the first 2 days of therapy for the model calculations.

Fitting the model to the data for the full 10 days of therapy resulted in the parameter values displayed in table 2. The predicted-observed plots after the Bayesian step for drug concentrations, total population values, and resistant subpopulation values are displayed in figure 4A–4C. As for the 2-day model, all of the slopes approached 1.0, and the y-intercepts were small. The r2 values all exceeded 0.91. An AUC24:MIC ratio of 279 was calculated as being necessary for suppression of amplification of the resistant subpopulation.

Figure 4.
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Figure 4.

Observed-predicted plots obtained after the Bayesian step from the population model for drug concentrations (A), the total organism population (B), and the resistant subpopulation (C), using all 10 days of therapy for the model calculations.

Prospective validation experiment. The validation experiment consisted of 3 hollow-fiber trials, one in which the intensity of therapy was high (AUC24:MIC ratio of 280), one in which it was low (AUC24:MIC ratio of 100), and one that was a no-treatment control. Furthermore, we wished for amplification of the resistant population to begin around the middle of a 10-day course of therapy. The results are displayed in figure 5A and 5B. The no-treatment control has little change in the resistant population after day 2, because there is no selective pressure. As predicted, the total kill (sensitive plus resistant subpopulations) is impressive for both regimens for the first 5 days. After 5 days, the low-intensity regimen fails, whereas the high-intensity regimen continues to kill. As seen in figure 5B, although resistant subpopulation amplification with the lowintensity regimen begins by day 4, the high-intensity regimen reduces the resistant population to nondetectability by day 4.

Figure 5.
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Figure 5.

Prospective validation experiment. The modeled data were used to set hypotheses regarding the differential impact of a regimen with an area under the concentration/time curve for 24 h (AUC24):MIC ratio of 100 vs. a more-intense regimen with an AUC24:MIC ratio of 280 over a clinically relevant 10-day period relative to a no-treatment control. Panel A demonstrates the differential impact of the regimens on the total organism population. Panel B demonstrates the impact on the resistant subpopulation over time.

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Discussion

When antimicrobial agents are administered for infections, it is important to identify a dose (therapeutic intensity) that is sufficient to kill the microorganism at the primary infection site at a near maximal rate and to identify a therapy duration that is sufficiently long to drive the total population to near extinction. It is also key to have this exposure be sufficient to suppress the emergence of resistance.

Choosing the correct therapeutic intensity and duration of therapy can meet the twin goals of favorable microbiological outcome and suppression of resistance. Dose and duration choice will not be helpful in all instances of prevention of resistance. When DNA is transferred by plasmids, transposons, or transformation, it is unlikely that dose and duration choice will have an important impact. This is also true when horizontal transmission of resistant microorganisms occurs, although it should be pointed out that, in those circumstances in which dose and duration choice is helpful, resistance suppression means that there will be no resistant microorganisms to transmit horizontally.

Proper choice of dose and duration of therapy will be maximally important when there is a point mutation that results in the production of a resistant mutant subpopulation. This idea has been promulgated elegantly by K. Drlica's laboratory [10]. It generally occurs when the size of the bacterial population exceeds the inverse of the mutational frequency to resistance. For quinolones, target site mutations and/or effluxpump overexpression occurs with a frequency of ∼1/106–1/108. Patients with severe multilobar pneumonia frequently will have total microorganism burdens exceeding the inverse of these numbers, indicating that a small preexistent resistant population will be present before the administration of therapy.

Our group has recently shown that is possible to choose a dose to suppress the emergence of the resistant subpopulation in a mouse-thigh infection model in which P. aeruginosa was the infecting pathogen [1]. Part of this work was a successful prospective validation study that lasted 72 h and that demonstrated that doses respectively chosen to maximally amplify and suppress the resistant mutant population did, indeed, perform as expected. This study, although important, demonstrated some of the limitations of the mouse-thigh infection model. It is difficult to continue therapy in this model for durations that are relevant to the therapeutic regimens that are prevalent in the clinic. For this reason, we chose to use our in vitro pharmacodynamic infection model to examine the issues of therapy intensity and duration.

We chose S. aureus as the infecting pathogen because of its clinical relevance. Likewise, the 10-day therapy duration was chosen because, in the clinic, therapy duration is most often 10–14 days, particularly for severe pneumonias caused by this pathogen.

Examination of the effect of the different regimens makes plain that the duration of therapy has a major impact on the outcome as measured by the ability to suppress the resistant microorganism subpopulation. If one administers a dose that will result in an AUC24:MIC ratio of 114 (figure 2E), a slightly greater than 5 log10 (cfu/mL) decline in the population results, if the administration is done for only 96 h. This exposure allows control of the resistant mutant subpopulation if therapy is limited to this duration. However, as can also be seen in figure 2E, administration for an extra 24 h and beyond allows the emergence of resistance.

Most excellent studies using the hollow-fiber infection model have examined the effect of therapy for only 48–96 h [11–14]. Consequently, we wished to examine the impact that such a limitation of therapy duration would have on the conclusions drawn. We applied a mathematical model to the data from the first 2 days of therapy. The results were that the model fit the data well (figure 3). Using the data shown in table 1, we were able to obtain Bayesian estimates of the parameters for the individual regimens and to use these values to identify the AUC24:MIC ratio that would suppress the resistant subpopulation over a period of 48 h. This value was 101.

Table 1.
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Table 1.

Population mean parameter estimates for the pharmacodynamic model using only the first 2 days of therapy.

Table 2.
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Table 2.

Population mean parameter estimates for the pharmacodynamic model using all 10 days of therapy.

Repeating this analysis but using data from the full 10 days of therapy demonstrated again that the model fit the data well (figure 4). However, the drug exposure required to suppress resistance for 10 days was an AUC24:MIC ratio of 279, almost a 3-fold increase.

Clearly, duration of therapy has a major negative impact on the probability of the emergence of resistance by increasing the likelihood of amplification of the resistant subpopulation. If a mutation is preexistent, then this will amplify over time as a function of drug pressure and of the change in the MIC provided by the mutation.

In the case of quinolones, another possible explanation is that these agents interfere with DNA replication and cause surviving staphylococci to undergo error-prone replication. The longer the therapy duration, the greater the number of rounds of error-prone replication that take place. Alternatively, effluxpump overexpression (by induction) may account for some of the emergence of resistance. Finally, there may be interaction, with early induction of efflux pumps providing a small survivorship advantage and more rounds of replication. A mutation can then arise from error-prone replication mechanisms.

Although some regimens might be sufficient to suppress the amplification of a preexistent mutant subpopulation, borderline regimens will ultimately fail (figure 2E). With enough time, such isolates can have enough rounds of replication to pick up a resistance mechanism that will allow amplification of this subpopulation, even in the face of reasonably intensive therapy, as seen in figure 2E. The response may be to limit the duration of therapy. This needs to be done thoughtfully, as the major goal of therapy is to kill enough microorganisms to allow the patient's immune system to clear the infection and allow the patient's survival.

We wished to recapitulate our prior study designs [1, 2] that had a prospective validation step. Here, we examined a notreatment control, a regimen generating an AUC24 :MIC ratio of 100, and a regimen generating a nominal AUC24:MIC ratio of 280. The predictions from the model were that (1) the highintensity regimen will drive down the colony counts by ∼5–6 log10 (cfu/mL) by day 5 of therapy; (2) the low-intensity regimen will be similar to the high-intensity regimen for 4 days with regard to total colony counts; (3) the high-intensity regimen will suppress resistance for the full 10 days of therapy; and (4) the low-intensity regimen will allow breakthrough growth of resistant microorganisms after day 4. Figure 5 demonstrates that each of these predictions was validated in the prospective experiment. The decrement by day 4 was 4.14 and 4.18 log10 (cfu/mL) for the low- and high-intensity regimen, respectively. On days 5 and 7, the decrements were 5.58 and 4.71 (day 5) and 3.30 and 5.76 (day 7) log10 (cfu/mL), respectively. The 2 regimens were nearly identical with respect to total effect for the first 4–6 days. However, it was at this point in the experiment where the divergence occurred. By day 7, breakthrough growth of resistant mutants occurred in the low-intensity regimen, whereas the high-intensity regimen allowed continued control of the resistant subpopulation. Indeed, the resistant subpopulation was no longer able to be recovered by day 4 with the regimen that had an AUC24:MIC ratio of 279. Consequently, the intense regimen achieves the goal of reduction of the total population to a level that the immune system can deal with by day 5–7 and also suppresses the resistant subpopulation. These data indicate that clinical trials should be conducted to examine the impact of intense but foreshortened (∼5–7 days) durations of therapy on clinical outcome and the emergence of resistance.

Such prospective experimental validation is rare. That produced here is the only such experimental validation of which we are aware that has a duration that matches the clinical durations of therapy. The lessons that can be drawn are straightforward. Regimen intensity is important and differentially affects the susceptible and resistant subpopulations. The longer the duration of therapy, the more intense the therapeutic regimen needs to be to suppress the resistant mutant subpopulation. Because therapy needs to be long enough to clear the infection, therapy should hit hard (high-intensity therapy to obtain maximal reduction of the microorganisms) and should stop early (to prevent rounds of replication). This balance between a regimen being long enough for maximal kill and short enough to help suppress resistance is struck with an AUC24: MIC ratio of 280 and ∼5 days of therapy.

It should be recognized that this is an in vitro set of experiments. There was no immune system to help in the clearance of the microorganisms. Consequently, the results obtained are conservative. It will be important to test such high-intensity, short-duration regimens in the clinic. If successful, we can optimize the outcome for our ill patients and still preserve the susceptibility of pathogens to the agents in our therapeutic armamentarium.

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Footnotes

  • Potential conflicts of interest: none reported (garenoxacin is no longer owned by Bristol-Myers Squibb but is licensed to Schering-Plough).

  • Financial support: Bristol-Myers Squibb.

  • This article is dedicated to the memory and career of Dr. Theodore E.Woodward, a treasured mentor.

  • Received October 12, 2006.
  • Accepted January 6, 2007.
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Appendix

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Modeling of the Exposure-Response Relationship of the Total Bacterial Population and the Resistant Bacterial Subpopulation to Drug Pressure

To mathematically determine the minimal drug exposure necessary to suppress the emergence of resistance, 3 simultaneous, parallel, inhomogeneous differential equations (shown below) were used to describe the time course of garenoxacin concentrations and the total and resistant subpopulations. Multiple drug-treatment regimens as well as a control regimen were simultaneously comodeled in a population sense by use of the population modeling program BigNPAG [8]. Bayesian estimates were generated for each regimen. The parameter estimates from the Bayesian best-fit model were then incorporated into the subroutines of ADAPT II [9], and the biological responses of the susceptible and resistant bacterial subpopulations under escalating drug exposure conditions were calculated.

A mathematical model was constructed to describe the exposure- response relationships of a mixed garenoxacin-sensitive and -resistant bacterial population in a hollow-fiber infection model under differing amounts of antimicrobial pressure. The mass-balance equations (3 parallel, first-order, inhomogeneous differential equations) that describe the subpopulations of interest (garenoxacin sensitive and resistant) are described in equations (1)–(3).

Formula Formula Formula Formula Formula

Equation (1) describes drug pharmacokinetics in the hollow-fiber system (a standard 1-compartment open model with zeroorder input and first-order elimination). X1 is the amount of drug in the central compartment; R(1) is the zero-order, time-delimited drug-infusion rate into the central compartment (milligrams per hour); SCL is the rate of clearance of drug from the central compartment (liters per hour); and Vc is the volume of the central compartment.

Equations (2) and (3) describe the rates of change of the garenoxacin-sensitive and -resistant subpopulations, respectively, over time. The model equations for describing the rate of change of the numbers of microorganisms in the sensitive and resistant bacterial subpopulations were developed on the basis of the in vitro observation that bacteria in the hollowfiber system are in logarithmic growth phase in the absence of drug and exhibit an exponential density-limited growth rate (eq. [4]). There is one equation to describe the sensitive bacterial subpopulation (eq. [2]) and one for the resistant subpopulation (eq. [3]). In each, first-order growth was assumed up to a density limit. Each subpopulation has an independent growth rate constant (for the sensitive subpopulation, Kgmax-S; for the resistant subpopulation, Kgmax-R). As themicroorganisms approach maximal bacterial density, they approach stationary phase. This is accomplished by multiplying the first-order growth terms by E (a logistic growth term; eq. [4]). The maximal bacterial density (POPmax) is identified as part of the estimation process. Most of the information for identifying this parameter is derived from the bacterial growth in the control group.

Equations (2) and (3) also allow the antibacterial effect of the different drug doses administered to be modeled. For both sensitive and resistant subpopulations, there is an independent effect of the drug dose on the 2 populations, one mediated through equation (2) (the sensitive subpopulation) and one through equation (3) (the resistant subpopulation). There is a maximal kill rate that the drug can induce for each population (Kkmax-S and Kkmax—R). The killing effect of the drug was modeled as a saturable kinetic event (M; eq. [5]; M is separate for the sensitive [MS] and resistant [MR] subpopulations) that relates the kill rate to drug concentration, where H is the sigmoidicity constant and EC50 (milligrams per milliliter) is the drug concentration needed to achieve 50% of the maximal kill rate. Separate H and EC50 terms are provided for the sensitive and resistant subpopulations. The drug effect observed is a balance between growth and death induced by the drug concentrations achieved. NS and NR are the number of colony-forming units per milliliter for the sensitive and resistant subpopulations, respectively.

Other model forms were evaluated for equations (2) and (3) that had growth rate constants that were drug-concentration dependent. Evaluation of these larger models did not demonstrate improved model performance. Consequently, under the rule of parsimony, the simpler model was used. Measured outputs were changes in total population (NS + NR) and resistant mutant (NR) densities, where the sensitive and resistant subpopulations were enumerated on drug-free plates and plates containing 3 times the MIC of the drug, respectively.

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  1. J Infect Dis. (2007) 195 (12): 1818-1827. doi: 10.1086/518003
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  1. March 1, 2012 205 (5)
  1. Journal of Infectious Diseases
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