Prediction of in vivo and in vitro infection model results using a semimechanistic model of avibactam and aztreonam combination against multidrug resistant organisms.

The combination of aztreonam-avibactam is active against multidrug-resistant Enterobacteriaceae that express metallo-β-lactamases. A complex synergistic interaction exists between aztreonam and avibactam bactericidal activities that have not been quantitatively explored. A two-state semimechanistic pharmacokinetic/pharmacodynamic (PK/PD) logistic growth model was developed to account for antimicrobial activities in the combination of bacteria-mediated degradation of aztreonam and the inhibition of aztreonam degradation by avibactam. The model predicted that changing regimens of 2 g aztreonam plus 0.375 and 0.6 g avibactam as a 1-hour infusion were qualitatively similar to that observed from in vivo murine thigh infection and hollow-fiber infection models previously reported in the literature with 24-hour log kill ≥1. The current approach to characterize the effect of avibactam in enhancing aztreonam activity from time-kill study was accomplished by shifting the half-maximal effective concentration (EC50 ) of aztreonam in increasing avibactam concentration using a nonlinear equation as a function of avibactam concentration, providing a framework for translational predictions.

WHAT IS THE CURRENT KNOWLEDGE ON THE TOPIC?

☑ In vitro and in vivo infection models have shown efficacy of aztreonam‐avibactam combination against Enterobacteriaceae harboring metallo‐β‐lactamases.

WHAT QUESTION DID THIS STUDY ADDRESS?

☑ How preclinical information of a drug combination efficacy can be described using a semimechanistic system model and how well does a mathematical model compare to animal models of infection in predicting bacterial response to human dosing regimens of aztreonam‐avibactam.

WHAT THIS STUDY ADDS TO OUR KNOWLEDGE

☑ This study indicates that it is feasible to determine the 24‐hour bacterial response to clinical dosing regimens using in vitro susceptibility information from the checkerboard assay that evaluates susceptibility of bacteria to the combination antibiotics.

HOW MIGHT THIS CHANGE DRUG DISCOVERY, DEVELOPMENT, AND/OR THERAPEUTICS?

☑ This model prospectively defines dosing regimens that can be effective against microbial infection in an in vivo setting, as it provides testable predictions for expected bacterial response to aztreonam‐avibactam regimens.

The current shortage of antibiotic options against multidrug‐resistant bacteria, and, in particular, the rise of bacterial isolates that are resistant to all available single‐agent therapies and some β‐lactam (BL)/β‐lactamase inhibitor combination is prompting a global health crisis. The older generation of BL/β‐lactamase inhibitor combinations, such as ticarcillin/clavulanate, amoxicillin/clavulanate, ampicillin/sulbactam, and piperacillin/tazobactam have coverage against only class A β‐lactamase enzymes and are ineffective against class B, C, and D β‐lactamases.1, 2, 3 Bacterial isolates that expressed the extended spectrum β‐lactamases, Klebsiella pneumoniae carbapenemase, and AmpC β‐lactamases are resistant to these combination chemotherapies.1, 3 Bacteria carrying these β‐lactamases present a challenge to treat secondary antimicrobial resistance against β‐lactamase inhibitors.

Ceftazidime/avibactam combination was approved by the US Food and Drug Administration in 2015 for the treatment against complicated intra‐abdominal and complicated urinary tract infections,4 and in 2016 by the European Medicines Agency for complicated intra‐abdominal, complicated urinary tract infection, hospital and ventilator‐acquired pneumonia, and aerobic gram‐negative infections with limited treatment options.5 Avibactam has a much larger range of coverage against β‐lactamases, encompassing enzymes belonging to class A, C, and even some D, including extended spectrum β‐lactamases, serine‐based carbapenemases, and Klebsiella pneumoniae carbapenemases.6, 7, 8 The inhibition of β‐lactamases by avibactam occurs by the formation of a highly stable covalent bond with the enzyme's active serine; the covalent adduct formation inactivates the β‐lactamases.7 Avibactam has no measureable activity against metallo‐β‐lactamases.9, 10 Given that aztreonam is impervious to hydrolysis by metallo‐β‐lactamases, combination with avibactam presents a strategic approach to the treatment of infections caused by isolates with multiple β‐lactamase profiles, including those containing class B β‐lactamases.5 The time‐dependent pharmacodynamic (PD) indices (fT>MIC; fT>CT) for aztreonam/avibactam were 50% fT>MIC for aztreonam and 50% fT>CT at 2–2.5 mg/L threshold avibactam concentration (CT) against highly resistant K. pneumoniae and Escherichia coli.11, 12 These target PD indices for both aztreonam and avibactam are consistent with the humanized dosing of 2 g/0.375 g and 2 g/0.6 g aztreonam/avibactam against metallo‐β‐lactamase‐producing gram‐negative organisms in the mouse thigh infection model.

As there are several published in vitro and in vivo studies that characterize the antimicrobial activities of aztreonam/avibactam,11, 12, 13 no mathematical model has yet been developed to describe the actions of combined chemotherapies. Additionally, the PD indices are based merely on summary endpoints of pharmacokinetic (PK) parameters and point estimates of the effect, wherein many interactions between drug concentration and bacteria responses are lost.14 A better approach should consider connecting the full time‐course of the combination to the dose‐dependent bacterial response using a semimechanistic PK/PD model to guide dose selection in the clinic.15, 16, 17

This study aimed to build a semimechanistic PK/PD model of aztreonam/avibactam with the following steps: (1) quantify the time course of growth and killing of four highly drug‐resistant clinical isolates in response to aztreonam/avibactam combination using a semimechanistic model13, 20; (2) qualify this model by simulating the PK profiles of humanized dosing in in vitro and in vivo infection models, as well as the corresponding bacterial response using the developed model. An overview of antimicrobial PK/PD is furnished in the Supplementary Material to explain the PK/PD approach in antimicrobial therapy, strategy, and significance of the current modeling approach.

MATERIALS AND METHODS

Bacteria strains and time‐kill kinetic studies

The data source for the mathematical model development came from previously reported constant‐concentration (static) time‐kill kinetic studies of avibactam and aztreonam combinations against four multidrug‐resistant clinical isolates: Escherichia coli ARC3600 (NDM‐1, CMY‐6, and OXA‐1), ARC3807 (NDM‐1, SHV‐12, TEM‐1, OXA‐9, and CMY‐42), Klebsiella pneumoniae ARC3802 (NDM‐1, SHV‐2a, SHV‐11, CTX‐M‐15, and TEM‐1), and Pseudomonas aeruginosa ARC3928 (OXA‐2 and GES‐1).13 The minimum inhibitory concentrations (MICs) of aztreonam alone against these isolates were 32, 2,048, 128, and 512 mg/L, respectively, and the MICs of aztreonam‐avibactam (avibactam at 4 mg/L) were 0.5, 4, 0.25, and 32 mg/L.13 The static time‐kill study concentration of aztreonam and avibactam investigated as both single‐agent and combination ranged from 0.015–8,192 mg/L and from 1–64 mg/L, respectively, in twofold increments. The degradation of both aztreonam and avibactam in the time‐kill studies were monitored during the static time‐kill study. The PD hollow‐fiber infection model studies of aztreonam‐avibactam against K. pneumoniae ARC3802 came from Singh et al.11; the neutropenic and immunocompetent mouse thigh infection models of human‐simulated drug concentrations were previously reported by Crandon and Nicolau.12

In vitro pharmacodynamic model

The schematic representation of the effect of aztreonam and avibactam on bacterial density dynamics, as well as the degradation of aztreonam due to β‐lactamase activities of the bacteria, is shown in Supplementary Material Figure S1. The core PK/PD model to describe the in vitro bacterial density changes came from the logistic growth model,18, 19 and the structural model for the bacterial population included two compartments representing two subpopulations of bacteria: P1 (actively growing and responding to aztreonam concentration as affected by the concentration of avibactam) and P2 (resting or dormant and not responding to changes in aztreonam or avibactam concentrations). Aztreonam degradation that is dependent on the bacteria density was also modeled mathematically as a function of the density of P1, assuming that P1 ≫ P2. The majority of bacteria was assumed to be in the P1 state at the start of the experiment. The initial condition for the secondary state P2 was 1/107 of P1.20 The logistic growth model was modified to incorporate bactericidal effect of both aztreonam and avibactam, which were described separately by two sigmoidal maximum effect (Emax) models.

The modified logistic growth model was formulated as follows:

The description of model parameters and symbols can be found in Table 1 and in the Supplementary Materials. The fitness of P1 is assumed to be greater than P2 by setting kgrowth,2 to 1/20 of kgrowth,1 estimated value.21 The initial condition of the resting state was 0.1 log10 cfu/mL. The bactericidal effect of aztreonam was described by a sigmoidal Emax model for all bacteria isolates. The synergistic effect of avibactam was characterized such that increasing avibactam concentrations decrease the half‐maximal effective concentration (EC50) of aztreonam. The decreasing aztreonam EC50 was described by a bi‐exponential function of avibactam concentration on the aztreonam EC50. Another sigmoidal Emax model was used to describe the antimicrobial effect of avibactam, which was only observed in E. coli ARC3600 and ARC3807. I is the delay function to allow the model to bend to the shape of the bacteria during the initial decline and also the regrowth of bacteria at the later timepoints was characterized such that:

Table 1

Parameter estimates for the bacterial population dynamic model developed from four clinical isolates of multidrug resistant Enterobacteriaceae and Pseudomonas aeruginosa time‐kill kinetics and aztreonam degradation

		Parameter ± SE
Parameter	Description	E. coli ARC3600	E. coli ARC3807	K. pneumoniae ARC3802	P. aeruginosa ARC3928
Nmax, log10 cfu/mL	Maximum achievable carrying capacity in the system	10.2 ± 0.175	10.0 ± 0.263	10.0 ± 0.198	11.6 ± 1.55
kgrowth,1, h−1	Bacterial growth rate constant associated with log10 of active population	1.28 ± 0.223	0.984 ± 0.0887	0.461 ± 0.0249	0.113 ± 0.0461
Emax,ATM, h−1	Maximum kill rate constant due to aztreonam	0.266 ± 0.0318	0.287 ± 0.00926	0.313 ± 0.0059	0.142 ± 0.0293
A, mg/L	First parameter of bi‐exponential function to characterize aztreonam EC50 in monotherapy	52.8 ± 7.85	2560 ± 474	213 ± 9.0	320 ± 28.3
B, mg/L	Second parameter of bi‐exponential function to characterize aztreonam EC50 in monotherapy	2.36 ± 0.549	31.4 ± 14.0	0.321 ± 0.0318	27.4 ± 9.21
α, L/mg	Exponential constant associated with A parameter that describes the relationship between avibactam concentration and potency of aztreonam	14.9 ± 2.64	5.06 ± 0.5	7.63 ± 0.298	1.32 ± 0.140
β, L/mg	Exponential constant associated with B parameter that describes the relationship between avibactam concentration and potency of aztreonam	0.480 ± 0.0409	0.401 ± 0.0633	0.174 ± 0.0149	0.0642 ± 0.0425
γ	Hill coefficient that determined the steepness of the slope of the sigmoidal Emax curve associated with avibactam enhancement of aztreonam potency	1.95 ± 0.509	1.30 ± 0.0555	2.98 ± 0.0912	2.79 ± 0.999
δ, h−1	Exponential constant of the delay function to retard growth of active population	0.0569a fix 0.0119b fix 0.0160c fix	0.0296a fix 0.0331b fix 0.0232c fix	0.132 fix	0.270 fix
k12, h−1	Rate constant for the conversion of bacterial cells from active to resting states	0.005 fix	0.005 fix	0.005 fix	0.005 fix
Emax,AVI, h−1	Maximum kill rate constant due to avibactam	0.262 ± 0.0355	0.180 ± 0.0156	–
EC50,AVI, mg/L	Concentration of avibactam that produces 50% of Emax,AVI	14.2 ± 0.540	26.6 ± 3.02	–
Φ	Hill coefficient that determined the steepness of the slope of the sigmoidal Emax curve associated with avibactam bactericidal effect	5.71 ± 0.822	1.98 ± 0.0477	–
Degmax, h−1	Maximum degradation rate constant of aztreonam	–	–	0.190 ± 0.00677	0.406 ± 0.00774
Km, log10 cfu/mL	Log10‐transformed cfu number density that yielded 50% of the maximum degradation rate	–	–	7.5 fix	6.87 fix
φ	Hill coefficient that determined the slope of sigmoidal Emax model for aztreonam degradation	–	–	3.59 ± 0.11	4.15 ± 0.662

Emax, maximum effect.

Avibactam only; baztreonam only; cboth aztreonam and avibactam present.

The degradation of aztreonam during the time‐kill experiment in K. pneumoniae ARC3802 and P. aeruginosa 3928 was best described by a second‐order nonlinear function dependent on both aztreonam concentration and bacterial density of P1 (Eq. (4)):

Qualitatively, Km was the value of P1 at which rapid aztreonam degradation commenced. The φ parameter determined the shape of the function. When φ is a large value (>3), the Hill function behaved as a switch over the range of values of P1 from just below Km to just above Km.

As shown in Eq. (4), the degradation of aztreonam during the time‐kill experiment was dependent on both the aztreonam concentration and the bacteria density in P1 state. When avibactam was present in the system, no degradation was observed. Consequently, Degmax was set to 0 in the presence of avibactam.

Given that the progeny was derived from a single inoculum and was assumed to be genetically identical, as long as they are from the same strain, as well as the bacteria in all drug combinations were from the same colony, the bacteria response to various combinations of drug concentrations was assumed to be reproducible with minimal variability. No interconcentration variability was incorporated to any of the model parameters.

The residual variability that described the difference between the observed and model‐predicted value was an additive model. The data used for model development were based on the log10‐transformed data and the residual errors conform to a normal distribution after transformation.

Model qualification

The external data for model qualification were extracted from literature sources. The study of Crandon and Nicolau12 evaluated human‐simulated aztreonam‐avibactam doses in murine thigh infection models against 14 NDM‐expressing Enterobacteriaceae and 13 multidrug‐resistant P. aeruginosa. The range of MIC of aztreonam alone against these microorganisms ranged from 8 to >256 mg/L and the MICs of aztreonam‐avibactam combination (with 4 mg/L avibactam) were between 0.13 and 32 mg/L. The human‐simulated dose equivalent of 2 g aztreonam and 0.375 or 0.6 g avibactam every 6 hours as a 1‐hour infusion was administered to both neutropenic and immunocompetent mice. Aztreonam exposures in the mouse infection models were derived from a previously published population PK model in patients with cystic fibrosis,22 whereas avibactam exposures were simulated from a population PK model of ceftazidime‐avibactam combination.23 The typical human free‐drug concentration‐time profiles of aztreonam and avibactam in administration of 2 g aztreonam alone and in combination with 0.375 or 0.6 g avibactam were used as PK input for the PD model. The simulations also included control groups, wherein no drug was administered. The initial inoculum was 6.5 log10 cfu/mL.

The second data source investigated constant 4 mg/L avibactam concentration with dynamically changing aztreonam concentrations against K. pneumoniae ARC3802 (reported MIC aztreonam alone/combination at 4 mg/L avibactam: 256 mg/L/0.5 mg/L) in the hollow‐fiber infection model (HFIM).11 The PK model for aztreonam concentration in HFIM was a one‐compartment intravenous bolus starting from specific maximum free aztreonam concentrations (fCmax) with a drug elimination half‐life of 2 hours. The simulation of the PK model of aztreonam assumed a one‐compartment model with a volume of distribution (V) of 1 L and clearance of 0.347 L/h, which is equivalent to an elimination rate constant (ke) of the natural logarithm of 2 divided by the half‐life of 2 hours, because ke = CL/V. The regimens of aztreonam with constant 4 mg/L avibactam and as monotherapy are described in the Supplementary Material. The evaluation was based on comparison of 24‐hour bacteria density in log10 cfu/mL between model prediction and HFIM, assuming an initial inoculum size of 6.5 log10 cfu/mL.

Predictability of minimum inhibitory concentration‐fitted aztreonam half‐maximal effective concentration function on the 24‐hour bacteria density

A bi‐exponential model (Eq. (5)) was fitted to the MIC of aztreonam with 0–8 mg/L avibactam against E. coli ARC3807 and K. pneumoniae ARC3802:

The parameter estimates A, B, α, and β obtained from the model fit were substituted into Eq. (1) to determine the time‐course of bacterial population dynamics in response to the same static concentrations of the combination previously evaluated in the time‐kill kinetic studies,13 assuming a starting inoculum size of 6.3 log10 cfu/mL, which is similar to the condition in the time‐kill study.

The 24‐hour bacteria density in log10 cfu/mL from the prediction using the MIC‐fitted bi‐exponential model substituted into Eq. (1) was then compared to the same target metrics determined by the PD model developed from the time‐kill kinetic information.

RESULTS

Pharmacodynamic model development from static time‐kill data

The concentration ranges of aztreonam and avibactam in static time‐kill kinetic studies13 were designed to provide optimized information for model development. The avibactam concentrations in twofold increments in the combination were 1–8 mg/L. Depending on the MIC of aztreonam at the respective avibactam concentrations, the drug concentration of aztreonam used in the static time‐kill kinetic studies ranged from 0.25‐fold to 4‐fold MIC. The bacteria responses at concentrations lower than 0.25‐fold MIC were similar to growth control, whereas bacteria responses at greater than 4‐fold MIC resulted in significant bacteria kill. The experimental design of the static time‐kill kinetic studies covered the necessary concentration range to observe bacteria kill and regrowth patterns. Significant aztreonam degradation was observed at ≤MIC at the respective avibactam concentrations in K. pneumoniae ARC3802 and P. aeruginosa ARC3928, but no degradation was observed in the two E. coli isolates.13

The time‐course of the bacterial population dynamics and aztreonam degradation during the course of time‐kill kinetic studies were modeled by a two‐state PK/PD model for the bacterial system that affects the drug degradation process. The model parameter estimates and SE of the estimates are summarized in Table 1. We utilized a nonlinear aztreonam EC50 function dependent on avibactam concentration for the sigmoidal Emax function in the PK/PD model to describe how avibactam conferred aztreonam efficacy. This empirical approach described well the increase in bacterial susceptibility toward aztreonam, as shown in representative Figures 1, 2, 3 for E. coli ARC3807, K. pneumoniae ARC3802, and P. aeruginosa ARC3928. The model fit of bacteria response for E. coli ARC3600 can be found in Supplementary Figure S2. This EC50 function was characterized by a bi‐exponential model to shift the EC50 of aztreonam kill to a lower value at each incremental avibactam concentration. The resulting EC50 values of aztreonam follow closely the MIC values at respective avibactam concentrations. For example, aztreonam MIC against E. coli ARC3807 decreased from 2,048 to 4 mg/L at 0 and 4 mg/L avibactam13 and aztreonam EC50 was predicted to decrease from 2,591 to 4.6 mg/L for the same range of avibactam concentration. The predicted aztreonam EC50 values against the other three isolates were also very close to the MIC values at respective avibactam concentrations.

Figure 1

Model‐prediction and observed static time‐kill curves of aztreonam and avibactam against E. coli ARC3807 over 24 hours. The points show the experimental data; color‐matched lines show the results of the pharmacokinetic/pharmacodynamic model described in the text.

Figure 2

Model‐prediction and observed static time‐kill curves of aztreonam and avibactam against K. pneumoniae ARC3802 over 24 hours (top) and aztreonam concentration remaining in the system during the time‐kill experiment (bottom). The points show the experimental data; color‐matched lines show the results of the pharmacokinetic/pharmacodynamic model described in the text.

Figure 3

Model‐prediction and observed static time‐kill curves of aztreonam and avibactam against P. aeruginosa ARC3928 over 24 hours (top) and aztreonam concentration remaining in the system during the time‐kill experiment (bottom). The points show the experimental data; color‐matched lines show the results of the pharmacokinetic/pharmacodynamic model described in the text.

A separate bactericidal effect of avibactam was incorporated to the model using another sigmoidal Emax model in the two E. coli isolates, wherein avibactam was shown to have activity. The EC50 values of avibactam were 14 and 25 mg/L (Table 1), which were close approximation of the MIC values of avibactam (16 mg/L) for both E. coli ARC3600 and ARC3807, respectively.13

The drug degradation was characterized by a nonlinear function that was dependent on both the active bacterial population and the remaining drug in the system for K. pneumoniae ARC3802 and P. aeruginosa ARC3928, as shown in the lower panel of Figures 2 and 3. The addition of avibactam resulted in no drug degradation of aztreonam in both isolates, even at the lowest concentrations of 1 and 2 mg/L avibactam in both isolates, respectively. The maximum degradation rate in the drug degradation model was set to 0 when avibactam was present in the system. Drug degradation model was not incorporated in the two E. coli isolates because no aztreonam degradation was observed in the time‐kill study.13

Models with successful minimization or convergence without boundary problems were accepted as the final model. Models were fitted separately to drug degradation and bacterial response information for the data from K. pneumoniae and P. aeruginosa isolates. A model that did not include the drug degradation process was initially developed for bacterial response data, assuming no drug degradation. In the second phase of model development, the parameters determined from the initial model that did not include drug degradation were fixed to obtain parameters for aztreonam degradation. The densities of the active population that resulted in half the maximum degradation were 7.5 and 6.87 log10 cfu/mL, respectively, indicating that aztreonam was efficiently removed even when the drug was just added. The 32 and 64 mg/L aztreonam panels in Figure 2 for K. pneumoniae ARC3802 show a very rapid decline in remaining aztreonam in the closed system. For P. aeruginosa ARC3928, the 128 to 512 mg/L aztreonam panels in Figure 3 show an effective drug removal even when bacteria density was decreasing.

Once the degradation model parameters were obtained, the degradation model was incorporated to the bacteria response model with the parameters from the drug degradation model fixed. The reported parameters in Table 1 are final estimates of the integrated model.

Model prediction of in vivo murine and in vitro hollow‐fiber infection models

The first part of external validation of the PD model of aztreonam‐avibactam activities in the four isolates was carried using in vivo efficacies data from murine thigh infection models, reported by Crandon and Nicolau.12 Their study used both immunocompetent and neutropenic mice whose thigh were inoculated with MDR P. aeruginosa; NDM‐containing Enterobactericeae studies were conducted only with neutropenic mice. In the current simulation exercise, the free drug concentration‐time profiles from the same dosing regimens were used to simulate bacteria response for the four isolates that were used in the PD model development. The human‐simulated aztreonam and avibactam free drug concentration profiles of 2 g aztreonam, 375 mg, or 600 mg avibactam every 6 hours as 1‐hour infusions are shown in Supplementary Figure S3.

Figure 4 shows the change in 24‐hour bacterial density from the initial inoculum size of ∼6.5 cfu/mL for the four bacterial isolates used for model development in the current study, under the “in silico” section. For comparison purposes, the bacterial responses from the in vivo thigh infection models are also shown in Figure 4. The maximum free aztreonam concentration after a 2 g dose in an hour infusion was ∼95.5 mg/L, which is still below the MIC in aztreonam alone against the isolates evaluated in our model, with the exception of E. coli ARC3600. As expected, the regimen containing aztreonam alone resulted in a net positive change at 24 hours (data not shown). The two aztreonam regimens containing avibactam resulted in at least 1 log10 kill at 24 hours, including P. aeruginosa ARC3928, wherein the decrease in aztreonam MIC in the presence of avibactam against this isolate was not as much as the other three isolates. No significant difference in efficacy between the two avibactam‐containing regimens (0.375 vs. 0.6 g) was observed, consistent with the in vivo efficacies in the murine infection models. Aztreonam MIC values with 4 mg/L avibactam against Enterobacteriaceae in this study are well below the breakpoints for susceptibility (MIC breakpoint ≤4 mg/L).24 Consequently, the log10 kill in the E. coli and K. pneumoniae isolates are well over 3 log10 units in the aztreonam‐avibactam combination. In contrast, aztreonam MIC in the presence of 4 mg/L avibactam against P. aeruginosa ARC3928 was 32 mg/L. The 24‐hour log10 kill in this isolate was 1 and 1.75 log10 unit in the 0.375 and 0.6 g avibactam with 2 g aztreonam combination, respectively, for this isolate.

Figure 4

Change in log10 bacteria density at 24 hours from model prediction (in silico) and Enterobacteriaceae in neutropenic mouse thigh infection model and P. aeruginosa in both neutropenic and immunocompetent mouse thigh infection model from Crandon & Nicolau.12 Model‐based responses to the Monte Carlo simulations of free drug concentration‐time profiles are represented in the “in silico” section. Observed bacterial responses in neutropenic and immunocompetent mouse thigh infection models against Enterobacteriaceae and P. aeruginosa. The minimum inhibitory concentration of aztreonam in 4 mg/L avibactam and aztreonam alone against the isolates is shown in brackets after the isolate number.

The second external validation was accomplished by comparison of 24 hours log10 bacteria density of K. pneumoniae ARC3802 from model prediction and from HFIM in response to various aztreonam regimens with constant 4 mg/L avibactam concentration. The aztreonam profiles were assumed to be identical to a one‐compartment model with intravenous bolus administration and starting with aztreonam concentration at the free maximum drug concentration (fCmax) listed in Singh et al.11 The model‐predicted 24‐hour bacterial densities, shown in Figure 5, were very close to the observed values from the HFIM for the q24‐hour and q6‐hour dosing regimens, as well as the monotherapy treatments (i.e., aztreonam alone or avibactam alone). A discrepancy between the observed and model predicted densities was found for the q12‐hour aztreonam regimen with constant 4 mg/L avibactam, wherein the model predicted a net 1.3 log10 kill units, whereas the HFIM resulted in a larger kill of 4.8 log10 units. The overall model prediction of the expected bacterial responses were consistent with those observed in both in vitro and in vivo aztreonam‐avibactam efficacy evaluation, even in the case of dynamically changing drug concentrations.

Figure 5

Model‐predicted and observed K. pneumoniae ARC3802 24‐hour log10 cfu/mL (change in log10 cfu/mL from time 0) at 24 hours with starting inoculum of ∼6.5 log10 cfu/mL in response to various regimens of aztreonam and avibactam. Observed data were digitally extracted from Singh et al.11 Model‐based (gray) and observed (black) 24‐hour bacterial responses to the simulation of free changing drug concentration following mono‐exponential decline from fCmax starting values.

Prediction of 24‐hour bacteria density using minimum inhibitory concentration‐fitted aztreonam EC50 function

The utility of substituting the aztreonam EC50 function with a MIC‐fitted function was evaluated for the purpose of determining whether MIC‐avibactam relationship defined by a nonlinear equation can be swapped with the EC50 function determined from the time‐kill kinetic studies. This exercise was performed on both E. coli ARC3807 and K. pneumoniae ARC3802. The bi‐exponential models for the MIC values as a function of avibactam concentration are shown in Eqs. (6) and (7):

These two equations replaced the bi‐exponential model for aztreonam EC50 function in Eq. (1). The resulting bacteria density at 24 hours predicted using the substituted MIC‐derived bi‐exponential functions in Eqs. (6) and (7) were compared with the original model fitted values in Figure 6 for E. coli ARC3807 and K. pneumoniae ARC3802, respectively, assuming an initial inoculum size of 6.3 log10 cfu/mL. The MIC‐derived 24‐hour densities were very close to the original estimated densities from the PD model fit to the time‐kill kinetic data, indicating that one can use the MIC‐derived bi‐exponential equation to derive an expected bacterial response that is close to the time‐kill kinetic study results.

Figure 6

Comparison of the 24‐hour bacteria density (change in log10 cfu/mL from 0 hour) of E. coli ARC3807 and K. pneumoniae ARC3802 for substituted minimum inhibitory concentration (MIC)‐fitted aztreonam EC50 versus model‐fitted aztreonam EC50 from time‐kill kinetic study. Model‐based responses to the simulations of static‐free drug concentration using MIC‐fitted EC50 equation (gray) and the time‐kill model‐fitted EC50 equation (black) for the original concentrations of aztreonam and avibactam used in the time‐kill kinetic experiments.

DISCUSSION

Model development

The modeling approach developed in the present study was designed to efficiently characterize the PD action of two antibiotics in combination. The combined action was characterized by a shift in EC50 function, given the substantial increase in microorganism's susceptibility to aztreonam in increasing avibactam concentration. An empirical model was applied to describe the enhanced potency of aztreonam, as represented by the EC50 function within the sigmoidal Emax model, as a function of avibactam concentration. This approach contrasts our previous approach to characterize the synergistic effect of an aminoglycoside and a BL, wherein we utilized Loewe additivity model.15 The difference is that both aminoglycoside and BL have intrinsic antimicrobial activity, whereas avibactam has very limited activity against E. coli at high concentrations and has no intrinsic activity against K. pneumoniae and P. aerugionosa isolates.11, 13 The Loewe additivity model is better suited for evaluating synergy in combination antibiotics, wherein each drug has its own effect in a monotherapy setting. The current approach of shifting EC50 was also applied to the previous aminoglycoside and BL combination and described well the trend in bacteria killing over a 24‐hour time course (unpublished data).

The advantage of the current approach of using a mathematical function to shift the EC50 of the active BL agent by a partnering β‐lactamase inhibitor is that this function can be approximated by the MIC values at respective avibactam concentrations. Mouton and Vinks25 had previously shown that MIC and EC50 are correlated for the logistic growth model. Because aztreonam EC50 values were very close to its MIC values at respective avibactam concentration, the bi‐exponential EC50 function derived from the time‐kill kinetic study can be swapped with a bi‐exponential MIC function determined from the checkerboard assay. We have shown that the 24‐hour bacterial density can be predicted from a bi‐exponential MIC function at various combinations of aztreonam and avibactam concentrations in a virtual time‐kill kinetic study setting. The simple approach wherein aztreonam EC50 value can be approximated by its MIC against specific microorganisms at respective avibactam concentration allows for broader simulation of time‐course of bacteria response using MIC values at respective avibactam concentration, given that the checkerboard assay for MIC determination is less labor‐intensive than running a time‐kill kinetic study.

Evaluating model against external in vitro data

The current approach to model qualification was carried out by examining the predictions of the model applied to external data sources. These sources were: in vitro constant concentration time‐kill data13; and in vitro varying concentration and in vivo varying concentration 24‐hour bacterial response data.11, 12

The model described well the bacteria responses in both in vitro nonvarying and varying concentrations of aztreonam and avibactam. The HFIM with K. pneumoniae ARC380211 utilized identical bacteria isolate as the one used in the nonvarying concentration time‐kill kinetic study.13 The similarity between the model‐predicted 24‐hour bacteria response and the observed response in the varying concentration infection model indicates that the model is sufficiently robust to predict the bacteria response in varying‐concentration scenarios. The model also captured the time‐dependent nature of aztreonam bactericidal activity, as it predicts that more frequent dosing in the q6‐hour scenario provides a better kill than the once or twice‐daily frequency (Figure 5).

Testing model predictability against external in vivo data

The simulated bacteria response of the four clinical isolates over 24 hours to varying concentrations of aztreonam and avibactam were compared to bacterial responses to experimentally simulated patient exposures in the mouse thigh infection model (Figure 4). The killing predicted by the model for 2 g aztreonam with 375 or 600 mg avibactam every 6 hours were qualitatively similar to the mouse data for other isolates tested with similar humanized dosing regimens. There was no marked difference in log kill between 375 and 600 mg avibactam with 2 g aztreonam q6‐hour, as shown both by the model and the animal data. The model‐predicted killings were much larger than that in the animal model with the exception of E. coli ARC3807 and P. aeruginosa ARC3928. In the two isolates wherein aztreonam MIC values in combination with 4 mg/L avibactam were less than 1 mg/L, the net kill was predicted to be much greater as expected because these MIC values are markedly lower than the clinical breakpoints.24 The model‐predicted killing in P. aeruginosa ARC3928 was comparable to the animal data when aztreonam was administered along with avibactam. The fold decrease in MIC of aztreonam with 4 mg/L avibactam against P. aeruginosa ARC3928 isolate was much larger than the decrease observed in the isolates injected in the animals, compared to aztreonam alone. A stark difference in killing was predicted by the model between aztreonam with avibactam and aztreonam alone. The killing of the majority of P. aeruginosa isolates in the murine infection model12 were not different between aztreonam with avibactam and aztreonam alone because there was no change in MIC against these isolates whether avibactam was added or not. The MIC values in these isolates were ≤32 mg/L, which is identical to the aztreonam MIC with avibactam against P. aeruginosa ARC3928. This aztreonam MIC is higher than the breakpoint for P. aeruginosa intermediate susceptibility criteria for standard 0.5 hour infusion24 and a net kill was still observed in the animal data and was also predicted by the model using a 1‐hour infusion regimen. It has been documented that longer infusion durations for time‐dependent antimicrobial agents do shift the probability of target attainment and susceptibility profiles,26 as longer infusion duration can increase the time that drug concentrations remained above the MIC.27 The overall model‐predicted responses in P. aeruginosa to aztreonam and avibactam regimens were quantitatively similar to the animal data.

The current model described well the effect of aztreonam‐avibactam combination on bacterial population dynamics and aztreonam degradation in the absence of avibactam. Its predictive performance was evaluated by simulations of both in vitro and in vivo infection models of wide ranging resistance levels of Enterobacteriaceae and P. aeruginosa. The current approach of using external in vitro and in vivo infection data for model qualification is suitably relevant for evaluating model predictability. This mathematical model is robust and can be easily extended to predict behaviors of other isolates to different human‐simulated regimens of aztreonam‐avibactam combination. The current model is a useful translational tool for evaluation of other dosing regimens, which were not evaluated in this study, and can potentially serve as a guide for the design of human trials of aztreonam‐avibactam against multidrug‐resistant Enterobacteriaceae and P. aeruginosa.

Supporting Information S1

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Supporting Information S2

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Supporting Information S3

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Supporting Information S4

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Supporting Information S5

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