Novel Modeling Framework To Guide Design of Optimal Dosing Strategies for β-Lactamase Inhibitors

Abstract

The scarcity of new antibiotics against drug-resistant bacteria has led to the development of inhibitors targeting specific resistance mechanisms, which aim to restore the effectiveness of existing agents. However, there are few guidelines for the optimal dosing of inhibitors. Extending the utility of mathematical modeling, which has been used as a decision support tool for antibiotic dosing regimen design, we developed a novel mathematical modeling framework to guide optimal dosing strategies for a beta-lactamase inhibitor. To illustrate our approach, MK-7655 was used in combination with imipenem against a clinical isolate of Klebsiella pneumoniae known to produce KPC-2. A theoretical concept capturing fluctuating susceptibility over time was used to define a novel pharmacodynamic index (time above instantaneous MIC [T>MIC_i]). The MK-7655 concentration-dependent MIC reduction was characterized by using a modified sigmoid maximum effect (E_max)-type model. Various dosing regimens of MK-7655 were simulated to achieve escalating T>MIC_i values in the presence of a clinical dose of imipenem (500 mg every 6 h). The effectiveness of these dosing exposures was subsequently validated by using a hollow-fiber infection model (HFIM). An apparent trend in the bacterial response was observed in the HFIM with increasing T>MIC_i values. In addition, different dosing regimens of MK-7655 achieving a similar T>MIC_i (69%) resulted in comparable bacterial killing over 48 h. The proposed framework was reasonable in predicting the in vitro activity of a novel beta-lactamase inhibitor, and its utility warrants further investigations.

INTRODUCTION

Bacterial resistance to antimicrobial agents has been rising at an alarming rate and may result in many common infections becoming untreatable in the future. The cost of treatment as well as the risk of mortality will increase with resistance (4). This has led to an urgent demand for new molecular entities targeting novel metabolic pathways. However, the development of new drugs is a long, nontrivial process which has not been able to meet the current demand (8). An alternative is to restore the effectiveness of existing drugs. A viable approach toward that end is the development of inhibitors designed to target a specific resistance mechanism(s). For instance, it has long been known that resistance mediated by the production of beta-lactamases could be tackled by an inhibitor which inhibits the function of the beta-lactamases (i.e., drug hydrolysis). Similarly, efflux pump inhibitors could be used against bacteria that overexpress efflux pumps to extrude drugs from the bacterial cells, thereby decreasing intracellular drug concentrations.

Despite the fact that inhibitors have been clinically available for a long time, optimal dosing strategies for inhibitors are not well established. Pharmacokinetic and pharmacodynamic (PK/PD) indices such as the area under the concentration-time curve over 24 h at steady state divided by the MIC (AUC/MIC ratio), the maximum concentration of drug in serum divided by the MIC (C_max/MIC ratio), and time above MIC (T>MIC) have been used widely to guide the optimal dosing of antibiotics (1). However, such indices may not be directly applicable to inhibitors, since these inhibitors themselves have relatively weak to no intrinsic antimicrobial activity (3) and are generally administered in combination with an antimicrobial agent. As in the case of a single antimicrobial agent, variables such as dose, dosing interval, and intersubject pharmacokinetic differences make the process of determining optimal dosing regimens for drug-inhibitor combinations nontrivial. A comprehensive assessment of all possible dosing strategies is impractical in preclinical and clinical investigations. On the other hand, the full potential of these new inhibitor candidates may not be realized with empirical selection.

Mathematical modeling and computer simulation could greatly reduce the amount of experimental work involved in the design of optimal dosing regimens, simultaneously allowing a comprehensive evaluation of numerous dosing strategies for drug-inhibitor combinations. It can be used as a decision support tool in guiding the dosing strategy for the combination (10). Starting with feasible dosing strategies, modeling can produce a short list of the most promising strategies, which can then be evaluated experimentally. In this study, we propose such a modeling framework and apply it to guide the design of optimal dosing strategies for a beta-lactamase inhibitor used in combination with an antimicrobial agent. To demonstrate our approach, a novel beta-lactamase inhibitor, MK-7655, was used in combination with imipenem against a clinical strain of Klebsiella pneumoniae.

(This work was presented in part at the 51st Interscience Conference on Antimicrobial Agents and Chemotherapy, Chicago, IL, 17 to 20 September 2011 [1a].)

MATERIALS AND METHODS

Microorganism.

A Klebsiella pneumoniae carbapenemase (KPC-2)-producing clinical isolate of Klebsiella pneumoniae (KP6339/CL6339) was provided by Merck (Whitehouse Station, NJ) and used in the study. Details of molecular confirmations were reported previously (7). The bacterium was stored at −70°C in Protect storage vials (Key Scientific Products, Round Rock, TX). Fresh isolates were subcultured twice on 5% blood agar plates (Hardy Diagnostics, Santa Maria, CA) for 24 h at 35°C prior to each experiment.

Antimicrobial agent and inhibitor.

Imipenem was used in combination with an experimental beta-lactamase inhibitor, MK-7655. Both imipenem and MK-7655 were obtained from Merck (Whitehouse Station, NJ). The first-order elimination half-life for both imipenem (6) and MK-7655 (data on file) in healthy volunteers was approximately 1 to 1.5 h.

Susceptibility studies.

The susceptibility of KP6339 to imipenem was assessed in the presence of escalating concentrations of MK-7655 (0 to 32 mg/liter) in 2-fold increments, using a modified broth dilution method (2). The MIC was defined as the minimum drug concentration which resulted in no visible growth after incubation for 24 h at 35°C. All MIC experiments were conducted in triplicate and repeated at least once on a separate day.

Mathematical modeling.

The dependency of the MIC reduction on the inhibitor concentration was characterized using a modified sigmoid maximum effect (E_max)-type model (5), as follows:

$${\log }_2\left(MIC\right)={\log }_2\left(MI{C}_0\right)-I_{\max }\frac{I^H}{I^H+{{{\displaystyle I}}_{50}}^H}$$

where MIC is the MIC in the presence of inhibitor, MIC₀ is the intrinsic MIC, I is the inhibitor concentration, I_max is the maximum inhibitor effect, H is the sigmoidicity coefficient, and I₅₀ is the inhibitor concentration for 50% of the maximum inhibitory effect.

The logarithm of the MIC taken to base 2 is a common approach to reduce overweighting in fitting a heteroscedastic data set. The best-fit parameter estimates were then used to simulate an instantaneous MIC (MIC_i) profile as a function of the MK-7655 concentration. Conceptually, the MIC_i could be thought of as a measure of imipenem susceptibility when the MK-7655 concentration fluctuates with time. The instantaneous MIC profile was then superimposed on a clinically achievable imipenem serum concentration profile (corresponding to a clinical dose of 500 mg every 6 h), as shown in Fig. 1. The time above the MIC_i (T>MIC_i) was assessed quantitatively. When both the imipenem concentration and MIC_i fluctuated within their respective dosing intervals, T>MIC_i was the total time when the drug concentration was higher than the MIC_i and is reported as a percentage of the dosing interval. In addition, the average MIC_i was computed by integrating the area under the instantaneous MIC profile over 24 h and divided by 24. The AUC_0-24/average MICi ratio was computed as the imipenem AUC_0-24 divided by the average MIC_i. The effects of different magnitudes of T>MIC_i and the AUC_0-24/average MICi ratio were assessed subsequently for different inhibitor dosing strategies (dose and dosing intervals).

Fig 1.

Different concentration-time profiles. (A) Imipenem concentrations resulting from a clinical dose of 500 mg every 6 h. (B) Typical instantaneous MIC (MIC_i) profile with fluctuating MK-7655 concentrations. (C) Imipenem concentrations superimposed with MIC_i values.

Experimental validation.

An in vitro hollow-fiber infection model (HFIM) was used to validate model predictions for clinically achievable concentrations of imipenem–MK-7655. The details of the HFIM setup were described elsewhere previously (9). An imipenem concentration profile equivalent to 500 mg every 6 h was maintained in the background; the dosing interval and C_max (40 mg/liter) of imipenem were kept the same for all experiments. A series of HFIM experiments were performed with different C_max values and dosing intervals of MK-7655, corresponding to escalating predicted T>MIC_i values (0%, 45%, 69%, and 99%) and AUC_0-24/average MICi values, as shown in Table 1.

Table 1.

T >MIC_i values for different MK-7655 dosing strategies

MK-7655 Cmax (mg/liter)	Dosing interval (h)	MK-7655 AUC0-24 (mg · h/liter)	T >MICi (%)	Imipenem AUC0-24a/avg MICi ratio
0		0	0	5.3
2	6	17.3	45	20.6
6	6	51.9	69	76.7
20	12	86.6	69	26.5
20	6	173.1	99	329.0

^aAll regimens had an identical imipenem concentration-time profile (AUC_0-24 = 338.9 mg · h/liter).

A culture of KP6339 grown overnight was inoculated into prewarmed cation-adjusted Mueller-Hinton broth (Ca-MHB) (BBL, Sparks, MD) and incubated further at 35°C until log-phase growth was achieved. The bacterial suspension was diluted to approximately 1 × 10⁵ CFU/ml with Ca-MHB based on the absorbance at 630 nm. Twenty milliliters of the diluted suspension was used in each experiment. At the start of each dosing interval, imipenem and MK-7655 were administered as infusions over 30 min. All experiments were performed for 48 h, and serial samples were obtained in duplicate (0, 6, 12, 24, and 48 h) to determine the viable bacterial burden. Bacterial samples were centrifuged and washed once before plating (50 μl) onto drug-free Mueller-Hinton agar (MHA) plates (BD Diagnostics, Sparks, MD). The MHA plates were incubated for 24 h at 35°C, and CFU were visually enumerated. To ascertain the simulated pharmacokinetic profiles of imipenem–MK-7655, samples were withdrawn from the circulatory loop of the system and assayed in selected experiments. Subsequently, a one-compartment model was fit to the concentration-time profiles of both imipenem and MK-7655.

RESULTS

Susceptibility studies and mathematical modeling.

In the absence of MK-7655, the imipenem MIC for KP6339 was 64 mg/liter. An MK-7655 concentration-dependent decrease in the imipenem MIC was observed, which was well characterized by using parameter estimates shown in Fig. 2. Different MK-7655 dosing regimens were simulated. The corresponding T>MIC_i values are shown in Table 1 and Fig. 3. Additionally, two different dosing regimens of MK-7655 resulting in similar T>MIC_i values (69%) were also identified.

Fig 2.

Model fit to experimental MIC data. Open circles represent imipenem MICs in the presence of different MK-7655 concentrations. The solid line represents the model fit with best-fit parameters (95% confidence intervals): I_max = 7.10 mg/liter (6.70 to 7.50 mg/liter), H = 1.34 (1.00 to 1.68), and I₅₀ = 0.94 mg/liter (0.75 to 1.13 mg/liter). When I approaches ∞, the MIC approaches 0.47 mg/liter.

Fig 3.

Superimposed imipenem concentration (solid line) and different MIC_i (dotted line) profiles. (A) T>MIC_i of 0%. (B) T>MIC_i of 45%. (C) T>MIC_i of 69%. (D) T>MIC_i of 69%. (E) T>MIC_i of 99%. q6h, every 6 h.

Experimental validation.

The results of the HFIM experiments were in reasonable agreement with the mathematical model predictions. An apparent trend in the bacterial response was observed as the T>MIC_i increased, as shown in Fig. 4A. For the isolate investigated, a T>MIC_i greater than 69% was needed to suppress the bacterial population over time. Two experiments were performed with different MK-7655 dosing regimens, resulting in similar T>MIC_i values. As shown in Fig. 4B, the bacterial responses in these experiments were comparable and within experimental errors. However, a trend in the bacterial response was not observed with increasing AUC_0-24/average MIC_i ratios.

Fig 4.

Observed mean bacterial burden over time with different MK-7655 dosing regimens, with increasing T>MIC_i exposures (A) and with similar T>MIC_i (69%) exposures (B).

DISCUSSION

The lack of antimicrobials with novel mechanisms against resistant bacteria has led to the development of inhibitors, with the aim of restoring the effectiveness of existing antimicrobials by targeting a specific resistance mechanism(s). Inhibitors intrinsically do not have significant antimicrobial activity and must be administered in combination with an antimicrobial agent. The intermittent dosing of an inhibitor typically results in a fluctuating susceptibility (MIC) of the target pathogen over time; the use of conventional PK/PD indices such as the AUC/MIC ratio, T>MIC, and the C_max/MIC ratio may not be directly applicable under these circumstances. Thus, the optimal dosing strategy for inhibitors may require an unprecedented modeling-and-simulation approach to address the additional system complexity.

We have previously characterized antimicrobial agent-bacterium interactions using a variety of modeling approaches, including conventional PK/PD indices (11, 12). Recognizing the common challenges and limitations, an extension of the mathematical modeling framework was proposed to guide the design of dosing regimens for drug-inhibitor combinations. The proposed modeling framework was based on the concept of fluctuating susceptibilities and was useful in predicting the in vitro activity of MK-7655 in combination with imipenem. With minimal modifications of a standard MIC measurement procedure, the experimental setup was relatively straightforward, and detailed knowledge of the mechanism of inhibitor action was not necessary. This pragmatic approach appeared to meet our study objective and provided us with valuable insights into a complex biological system. However, the modeling framework was not completely empirical, and useful information on the inhibition profiles could be indirectly captured irrespective of the enzyme type, expression level, or inhibitor affinity. The efficiency and expression levels of an enzyme could have direct consequences on bacterial susceptibility, which in turn would affect the intrinsic MIC (MIC₀). For example, the MIC₀ would be relatively low (e.g., a 2× to 4× elevation from the baseline level) for a bacterium poorly expressing a low-efficiency enzyme. On the other hand, for a highly efficient enzyme expressed at a high level, the corresponding MIC₀ would be much higher (e.g., a 64× to 128× elevation from the baseline level). Similarly, the potency and affinity of an enzyme inhibitor could be represented by the magnitudes of the I₅₀ and I_max values, respectively. For instance, the high potency and affinity of MK-7655 against the KPC-2 enzyme was reflected in a high I_max and a low I₅₀ value.

While the modeling framework was simple and efficient for characterizing the activity of MK-7655, there were limitations. First, the pharmacological effect of the inhibitor (prevention of drug hydrolysis by the enzyme) was not explicitly characterized in the model. The instantaneous MIC is a theoretical concept used to reflect fluctuating susceptibilities over time, which may not be easily verified experimentally. Second, the novel pharmacodynamic index T>MIC_i could be used as a surrogate for the conventional index T>MIC, when applying it to fluctuating susceptibilities. Consistent with our expectation, an apparent trend was observed in the bacterial response with increasing T>MIC_i values. However, additional investigations are required to define a robust threshold of T>MIC_i for optimal killing for a larger number of drug-inhibitor-pathogen combinations. Similarly, more comprehensive dose fractionation studies (and with longer durations) should be performed to ascertain if similar T>MIC_i values indeed correspond to similar bacterial responses. If validated, the proposed framework could be generalized for a variety of inhibition mechanisms and used to screen multiple inhibitor dosing strategies. The instantaneous MIC could be extended to define the C_max/MIC_i and AUC/MIC_i ratios as surrogates for other conventional pharmacodynamic indices, in analogy to related indices used for the assessment of antimicrobial agent effectiveness. The mathematical modeling framework could be used as a decision support tool to guide the design of inhibitor dosing regimens in both the developmental as well as clinical stages.

In summary, an alternative computational approach is proposed for the dosing strategy design of a beta-lactamase inhibitor. The results are promising, and further in vivo investigations are warranted.