Introduction

The emergence of bacterial resistance to antibiotics is the major contributor to their reduced efficacies [1]. Given a growing number of clinical reports on the isolation of resistant pathogens combined with a weak antibiotic pipeline [2], the development of new compounds is currently aimed at the suppression and/or restriction of resistance [3, 4]. Given that the enrichment of resistant mutants with concomitant loss in pathogen susceptibility should be concentration-dependent, concentration-resistance relationships are the methodological basis on which so-called “anti-mutant” antibiotic dosing regimens can be designed [5]. Such a relationship was first established in an in vitro study with fluoroquinolone-exposed Staphylococcus aureus using a dynamic model that simulates human antibiotic pharmacokinetics [6]. The loss in susceptibility of S. aureus occurred at intermediate but not at lower or higher antibiotic concentrations thereby exhibiting a bell-shaped relationship between bacterial resistance and the ratio of 24-h area under the concentration-time curve (AUC24) to the MIC. This specific pattern of the AUC24/MIC relationships with changing susceptibility of antibiotic-exposed S. aureus appeared to be consistent with the mutant selection window (MSW) hypothesis [7]. Since the MSW hypothesis predicts the enrichment of resistant mutants at antibiotic concentrations above the MIC, but below the mutant prevention concentration (MPC) the concentration-resistance relationship can be described by an extremum-containing function. This also has been confirmed in further in vitro studies with fluoroquinolones [6, 8,9,10,11,12,13,14,15,16], glycopeptides and lipopeptides [17], and oxazolidinones [18] that demonstrate bell-shaped relationships between the amplification of resistant mutants or loss in susceptibility of antibiotic-exposed bacteria and AUC24 or AUC24/MIC. It was this ratio that allows prediction of strain-independent resistance thresholds, i.e., the “anti-mutant” AUC24/MIC ratios, which were surprisingly less variable than the respective AUC24/MPC ratios among fluoroquinolone-exposed Gram-negative bacteria [12, 15, 16] but not such exposed Gram-positive bacteria [19].

While AUC24/MIC is commonly used to predict the emergence of bacterial resistance, the predictive potential of another parameter more closely linked with the MSW hypothesis—the time during which antibiotic concentrations are inside the MSW (TMSW) remained uncertain until recently. Some studies have reported sigmoid relationships between TMSW and the MIC elevations [6, 17, 20] or the enrichment of resistant mutants [17, 21, 22], whereas other studies [23,24,25,26,27,28] have not reported links between TMSW and the emergence of resistance. Given the pharmacokinetic profile-dependent TMSW-resistance relationships [20], conclusions about the low predictive power of the TMSW drawn in some of the latter studies could have resulted from unjustified pooling of data obtained with different modes of antibiotic administration [23, 24], different dosing frequencies [27], and different half-lives of the same antibiotic [25]. However, even in those cases where TMSW-resistance relationships were established, mutant enrichment better correlates with AUC24/MIC than the TMSW [6, 13, 17, 29]. The explanation for this variability was found only recently in our in vitro study with ciprofloxacin-exposed Escherichia coli [13]. When simulating ciprofloxacin pharmacokinetics over a wide range of the AUC24/MIC ratio, TMSW-resistance curves split into two portions, one for antibiotic concentrations below the MPC (T>MPC = 0) and another for concentrations consistently above the MPC (T>MPC >0), exhibiting a hysteresis loop. Based on the separate data sets, the enrichment of resistant E. coli correlates better with TMSW than for the entire data set ignoring T>MPC.

To verify the same approach as applied to linezolid that has not been studied in this aspect, three methicillin-resistant S. aureus strains with different MPC/MIC ratios were exposed to linezolid pharmacokinetics over a wide range of the AUC24/MIC ratio in five-day treatments simulated in an in vitro dynamic model. To ensure the presence of resistant cells at the start of simulated treatments, a mixed inoculum of the parent S. aureus strains and their linezolid-resistant mutants was used as described elsewhere [18].

Materials and methods

Antimicrobial agent and bacterial strains

Linezolid powder was kindly provided by Pfizer Inc. Three methicillin-resistant S. aureus strains including clinical isolates 479 and 688 and a well-characterized strain Mu50 (ATСС 700699) [30] and their previously selected linezolid-resistant mutants [18] were used in the study. The MIC of linezolid was 2 μg ml−1 for all three parent strains and 8 μg ml−1 for the resistant mutants. The MPCs of linezolid against S. aureus 479, S. aureus 688, and S. aureus ATCC 700699 regardless of the presence or absence of resistant mutants (mutation frequency 10−8) were 5, 6, and 10 μg ml−1, respectively [18].

Simulated pharmacokinetics and in vitro dynamic model

A series of monoexponential profiles that mimic twice-daily dosing of linezolid with a half-life of 6 h, in accordance with values reported in humans [31], was simulated for five consecutive days. The profiles were designed to provide ratios of AUC24/MIC from 7.5 to 240 h with a stepwise two-fold increase. With each S. aureus strain this range covers the clinically attainable AUC24/MIC ratio of ca. 120 h (AUC24 = 228 mg h l−1 divided by MIC = 2 μg ml−1) [32].

A previously described dynamic model was used in the study [33]. Briefly, the model consisted of two connected flasks: one containing fresh Mueller-Hinton broth (MHB) and the other with a magnetic stirrer, a central unit, with the same broth containing a bacterial culture plus antibiotic (killing/regrowth experiments). Peristaltic pumps circulated fresh nutrient medium to the flasks and from the central 110-ml unit (initial 100 ml volume corrected by including additional 10 ml volume of the sampling system tubes) at a flow rate of 12.7 ml h−1. Antibiotic dosing and specimen sampling of the central unit of the dynamic model were processed automatically using computer-assisted systems that provided sampling of each specimen from a separate port. The concordance between measured and designed linezolid concentrations has been reported elsewhere [18, 34].

The system was filled with sterile MHB and placed in an incubator at 37 °C. The central unit was inoculated with an 18-h culture of S. aureus. After a short incubation, the resulting exponentially growing cultures of linezolid-susceptible cells reached 108 cfu ml−1 (1010 cfus per a 100 ml central unit) and 1 ml of a bacterial suspension of 102 cfu of resistant mutants was added to the central unit resulting in mutant content of one cell per 108 cfu of susceptible cells in 1 ml MHB to achieve a mutation frequency of 10−8. A mixed inoculum of the parental cells and the resistant mutants was then exposed to linezolid administered as a bolus. The duration of each experiment was 120 h.

Population analysis

To determine viable counts of linezolid-susceptible and linezolid-resistant S. aureus, the central unit of the model was multiply sampled throughout the observation period (120 h), and the samples were plated on Mueller-Hinton agar (MHA) without antibiotic and with 4× MIC of linezolid. The inoculated plates were incubated for up to 72 h at 37 °C and screened visually for growth. To minimize antibiotic carryover, samples were serially ≥10-fold diluted as appropriate and 100 µl was plated evenly onto MHA plates, which were incubated at 37 °C for 24 h. The lower limit of accurate detection was 2 × 103 cfu ml−1 (equivalent to 20 colonies of linezolid-susceptible plus linezolid-resistant cells per plate) and 102 cfu ml−1 (equivalent to at least one colony of linezolid-resistant cells per plate).

Based on population analysis data, areas under the bacterial mutant concentration-time curves AUBCMs [19] were determined from the beginning of treatment to 120 h and were corrected for the area under the lower limit of quantification over the same time interval.

Relationships between AUBCM and MIC-related and MPC-related pharmacokinetic variables

AUBCMs determined with individual S. aureus strains in each simulated treatment were plotted against four MIC-related and MPC-related pharmacokinetic variables: TMSW, T>MPC, and AUC24/MIC and AUC24/MPC ratios.

To ensure bacterial strain-independent prediction of the AUBCM, combined data on the three S. aureus strains versus AUC24/MIC or AUC24/MPC and TMSW or T>MPC were fitted with a modified Gaussian function:

$$Y = Y_0 + a\,{\rm exp}\left[ {0.5\left( {\left| {x - x_0} \right|/b} \right)^c} \right],$$
(1)

where Y is AUBCM, x is log (AUC24/MIC) or log (AUC24/MPC), Y0 is the minimal value of Y, x0 is log (AUC24/MIC) or (AUC24/MPC) that corresponds to the maximal value of Y, and a, b and c are parameters, and a sigmoid function:

$$Y = Y_0 + a/\left\{ {1 + {\rm exp}\left[ { - \left( {x-x_0} \right)/b} \right]} \right\},$$
(2)

where Y is AUBCM, x is TMSW or T>MPC, Y0 and a are the minimal and maximal values of the AUBCM, respectively, x0 is x corresponding to a/2, and b is a parameter reflecting sigmoidicity.

All calculations were performed using SigmaPlot 12 software.

Results

In most simulated treatments, bacterial regrowth followed the initial decrease in density of the total population of linezolid-susceptible and linezolid-resistant S. aureus grown on antibiotic-free plates. At the intermediate AUC24/MIC ratios (30–60 h), when mutants resistant to 4 × MIC of linezolid were enriched most intensively, their post-treatment numbers approached the sum of linezolid-susceptible and linezolid-resistant staphylococci. Typical time courses of viable counts observed for example with linezolid-exposed S. aureus 688 at one of the simulated AUC24/MIC ratios are shown in Fig. 1.

Fig. 1
figure 1

Simulated pharmacokinetics of linezolid and time courses of surviving cells of S. aureus 688 grown on antibiotic-free plates (0 × MIC) and on plates containing 4 × MIC of linezolid. Simulated AUC24/MIC ratio was 60 h. Antibiotic dosing is indicated by arrows. MSW is marked by the shaded area

A more detailed presentation of resistance data obtained with S. aureus 479, S. aureus 688, and S. aureus ATCC 700699 (Fig. 2) highlights TMSW-dependent and AUC24/MIC-dependent enrichment of linezolid-resistant mutants. At the smaller AUC24/MIC ratios (7.5 and 15 h) when linezolid concentrations were below the MIC (TMSW 0) or above the MIC for only 10% of the dosing interval, resistant mutants of S. aureus 479 and S. aureus ATCC 700699 were not enriched. With S. aureus 688 moderate mutant enrichment occurred only at the end of treatment. The most pronounced amplification of resistant staphylococci was observed at the AUC24/MIC ratios of 30 (TMSW 59% for all strains) and 60 h (TMSW 52, 65, and 99% for S. aureus 479, S. aureus 688 and S. aureus ATCC 700699, respectively). At these AUC24/MIC ratios the number of resistant S. aureus ATCC 700699 mutants elevated significantly after 48 h from the start of the treatment and reached 8 log cfu ml−1 by 120 h. Substantial, but less than observed with S. aureus ATCC 700699 increase in the resistant S. aureus 688 cells was seen after 48 and 72 h at AUC24/MIC ratios of 30 and 60 h, respectively, a day earlier than with S. aureus 479. Weaker and later growth was observed with S. aureus 479 resistant mutants. At the AUC24/MIC ratio of 30 h their numbers were comparable to other strains only at the end of the observation period; at the AUC24/MIC of 60 h maximal numbers with S. aureus 479 mutants were 1.5-fold lower than with S. aureus 688 and S. aureus ATCC 700699. There was no enrichment with resistant mutants at the highest simulated AUC24/MIC ratio of 240 h (TMSW 0% for S. aureus 479 and 688 or 4% for S. aureus ATCC 700699), while at the clinically achievable ratio of ca. 120 h moderate enrichment did occur with S. aureus ATCC 700699 (TMSW 53%) but not S. aureus 479 and 688 (TMSW 6 and 16%, respectively). Thus, the enrichment of resistant mutants of the studied S. aureus strains was AUC24/MIC-dependent.

Fig. 2
figure 2

Time courses of linezolid-resistant S. aureus subpopulations at different simulated AUC24/MIC ratios (boxed numbers) and TMSW values (bars). White squares and bars, S. aureus 479; gray circles and bars, S. aureus 688; black diamonds and bars, S. aureus ATCC 700699

With each S. aureus strain, TMSW plots of the AUBCM were qualitatively similar and they split into two parts (Fig. 3, left panel). The upper plots meet the condition of T>MPC = 0, and the lower plots—T>MPC > 0. This results in a hysteresis loop, more distinct with S. aureus 688 and, in particular, ATCC 700699 mutants than with S. aureus 479. Although AUBCM increased with an increase in TMSW regardless of whether linezolid concentrations were above the MPC or not, the upper plots predict greater AUBCMs than the lower plots at the same TMSW. Using TMSW of 50% of the dosing interval as a vertical intersecting line, the AUBCMs for S. aureus 688, S. aureus 479 and S. aureus ATCC 700699 were respectively 1.6-fold, 1.9-fold, and 2.5-fold greater when linezolid concentrations did not reach the MPC (T>MPC = 0) than at antibiotic concentrations which exceeded the MPC (T>MPC > 0).

Fig. 3
figure 3

TMSW and AUC24/MIC relationships with AUBCM. T>MPC values are shown in callouts. White squares, S. aureus 479; gray circles, S. aureus 688; black diamonds, S. aureus ATCC 700699

AUC24/MIC relationships with AUBCM were bell-shaped with each S. aureus strain (Fig. 3, right panel). The AUBCM increased with an increase in the AUC24/MIC, reaching a maximum, and then, at higher AUC24/MICs, AUBCM decreased to zero. AUC24/MPC relationships with AUBCM were similar. With each organism, the descending portion of the bell-shaped curve was associated with T>MPC > 0. Like TMSW plots, the patterns of the AUC24/MIC-AUBCM curves were similar for all three S. aureus strains, but these curves were strain-specific in terms of the absolute AUBCM values: smaller with S. aureus 479, intermediate with S. aureus 688 and larger with ATCC 700699. For example, the respective maximal AUBCM values (205, 379, and 431 log cfu h ml−1) observed at the same AUC24/MIC ratio (30 h) varied in a more than two-fold range. For this reason, when combining data obtained with different S. aureus strains, the AUBCMs were related to the maximal value observed with a given strain. As seen in Fig. 4a, b, Gaussian function (Eq. 1) fits the normalized AUBCM-AUC24/MIC, and AUBCM-AUC24/MPC data with high r2s (0.85 and 0.80, respectively) that are higher than for the non-normalized data (0.36 and 0.65, respectively).

Fig. 4
figure 4

AUC24/MIC, AUC24/MPC, TMSW, and T>MPC relationships with AUBCM (combined data on three S. aureus strains) fitted by Eq. (1): a Y0 = 0, a = 100.0, b = 0.2389, с = 2.392, x0 = 1.630; b Y0 = 0, a = 100.0, b = 0.2643, с = 3.518, x0 = 1.066) and Eq. (2): c Y0 = 0, x0 = 54.22, a = 89.35, b = 7.101; d Y0 = 0, x0 = 45.28, a = 89.02, b = −7.026). White squares, S. aureus 479; gray circles, S. aureus 688; black diamonds, S. aureus ATCC 700699

Using normalized AUBCM belonging to the lower plots shown on the left panel of Fig. 3 (T>MPC > 0), a strain-independent relationship between AUBCM and TMSW was established (Fig. 4c). A sigmoid function (Eq. 2) fits combined data with the three S. aureus strains with high r2 (0.99). For the points that meet the condition T>MPC > 0 the sum of TMSW and T>MPC equals 100% of the dosing interval, the T>MPC plot of the AUBCM (Fig. 4d) is a mirror image of the TMSW plot with the same r2. Thus, all four MIC-related and MPC-related pharmacokinetic variables are equally predictive of the emergence of staphylococcal resistance to linezolid.

Discussion

Using in vitro simulations of five-day treatments of linezolid-susceptible S. aureus supplemented by resistant mutants, their enrichment was shown to correlate with MIC-related and/or MPC-related pharmacokinetic variables. With each of the three studied strains TMSW relationships of the AUBCM had the form of hysteresis, with the upper portion of its loop corresponding to T>MPC = 0, whereas the lower portion to T>MPC > 0. Because the sigmoid rise in the AUBCM with an increase in TMSW was steeper at T>MPC = 0 than at T>MPC > 0, greater AUBCMs were observed at a given TMSW when linezolid concentrations were below the MPC than above the MPC. Recently, similar patterns have been reported in our study with ciprofloxacin-exposed E. coli [13]. These findings suggest the previously hypothesized idea of heterogeneity of the MSW that was tested by simulations of ciprofloxacin concentrations oscillating closer either to the MPC (“upper case”) or the MIC (“lower case”) at the same TMSW [35]. The AUBCM in the upper case was shown three times smaller than in the lower case for two strains of ciprofloxacin-exposed S. aureus.

A distinct T>MPC-dependent splitting of the AUBCM-TMSW curves makes combining data obtained at T>MPC = 0 and at T>MPC > 0 incorrect. It is no coincidence that Eq. (2) described AUBCM-TMSW data at T>MPC > 0 (Fig. 4c) much better than the combined data at T>MPC = 0 and at T>MPC > 0 (r2 0.99 versus r2 0.24). It is very likely that the incorrect combination of data obtained at T>MPC = 0 and at T>MPC > 0 might contribute to the underestimation of the true role of the TMSW as a predictor of the emergence of bacterial resistance. Apparently, this occurred in a resistance study with meropenem-exposed Acinetobacter baumannii: [28] with each of the studied strains, the TMSWs observed at the minimal antibiotic exposure met the condition T>MPC = 0, whereas the TMSWs at the maximal exposure met the condition T>MPC > 0.

As established in the present study, TMSW at T>MPC > 0 and T>MPC are equally predictive of the enrichment of resistant S. aureus: the shorter the TMSW or the longer the T>MPC, the less mutants. Therefore, T>MPC plot of the AUBCM (Fig. 4d) was a mirror image of the TMSW plot (Fig. 4c) with the same r2 (0.99); for the points that meet the condition T>MPC > 0 the sum of TMSW and T>MPC equals 100% of the dosing interval. However, this is true for antibiotics with long half-lives but not for shorter half-life agents such as beta-lactams. For example, even at the relatively high AUC24/MIC ratios that prevent the amplification of resistant Pseudomonas aeruginosa, doripenem trough concentrations were lower than the MIC [29], and the sum of TMSW and T>MPC was <100% of the dosing interval. In this light, TMSW and T>MPC are not interchangeable.

Along with TMSW and T>MPC, two other indices, AUC24/MIC and AUC24/MPC can be bacterial strain-independent predictors of the selection of linezolid-resistant S. aureus. Given the pronounced inter-strain variability in the maximal value of the AUBCM but not in the slopes of the ascending and descending portions of the AUC24/MIC- or AUC24/MPC-AUBCM curves (Fig. 3), to combine data obtained with individual S. aureus strains, the AUBCMs normalized by their maximal values were plotted against AUC24/MIC and AUC24/MPC (Fig. 4a, b). Both AUC24/MIC and AUC24/MPC relationships with the normalized AUBCM appeared to be strain-independent (r2 0.85 and 0.80, respectively), and in this sense they are equally predictive of resistant mutant enrichment. Because of the small variability in the MPC/MIC ratio for the studied strains from 2.5 (S. aureus 479), and 3 (S. aureus 688) to 5 (S. aureus ATCC 700699), AUC24/MIC and AUC24/MPC could not be discriminated by their predictive potentials.

Given the increasing prevalence of antibiotic resistant pathogens and the relative paucity of new antibiotics in development, optimal antibiotic therapy should consider the suppression of resistance [1]. In this light, searching for quantitative relationships between the enrichment of resistant mutants and MIC-related and MPC-related pharmacokinetic variables may be a basis for the development of “anti-mutant” antibiotic dosing. The establishment of dosing regimens that prevent or restrict mutant enrichment is critical for new antibacterial agents as well as for currently existing antibiotics.

Overall, the findings obtained in the present study support the MSW hypothesis [7] as applied to linezolid-resistant S. aureus.