Pharmacometrics in Bacterial Infections

Abstract

Pharmacometrics plays an important role in rational dosing design of antimicrobial therapy by defining the relationship between dose, dosing interval, drug concentration, bacterial response, infection outcome, and toxicity, as well as the variability surrounding these variables. Modeling and simulation is becoming more extensively used to characterize antimicrobial strategies to combat emergence of drug-resistant bacteria, which is imposing a high cost on patient survivability and the overall health-care system. In this chapter, we discuss how pharmacometric approaches are being utilized to tackle these challenges. The approaches in designing treatment strategies using minimum inhibitory concentrations and in vitro time course of bacterial killing are explained to provide a thorough overview of the current state-of-the-art exploration of antimicrobial pharmacokinetic–pharmacodynamic (PKPD) modeling and simulation. In turn, the information from in vitro and animal studies is connected with clinical outcome for the purpose of determining key information that can improve therapeutic success in the clinic. This chapter also discusses the strategies for optimized dosing regimens and combination therapies that are currently explored to tackle the challenges posed by development of drug-resistant infections.

8.1 Introduction

The issue of bacterial resistance to antimicrobial agents, which is evident by a diminishing therapeutic value of many commercially available antimicrobials, has reached an alarming height of imminent danger to the general population. The rapid emergence of bacterial resistance to antimicrobial agents has rendered many of the commercially available antibiotics useless at the clinically tolerable dose. The resistance to treatment has also rapidly increased hospital mortality due to opportunistic infections (De Kraker et al. 2011). To combat this crisis, there are two options: (1) the development of new antimicrobials, which requires developing a new class of drugs and/or (2) preserving the value of existing ones by tackling bacterial resistance mechanisms. The first option can prove extremely costly and lengthy; the pharmaceutical industry has no incentive in developing new class of antibiotics due to a small return on the investment. The second option can be addressed in two ways: first, development of new drugs that counter the resistance mechanisms in bacteria for example the use of β-lactamase inhibitor in combination with a β-lactam agent; and second, optimizing treatment of existing antibiotics. The need for an optimized treatment, whether it is for new or existing drugs, in order to limit the chance of bacterial resistance, has prompted the use of quantitative approaches to guide dosing regimens.

The application of pharmacokinetic–pharmacodynamic (PKPD) modeling and simulation has been proven useful in the selection of dosing regimens that overcome resistance development and achieve the desired clinical outcome (Drusano 2004). Even with a good track record of optimizing antimicrobial dosages, however, the PKPD model and simulation is still underutilized in managing bacterial infections. This chapter focuses on the available PKPD models that were derived from in vitro, animal, and clinical data. The discussion separates two major modeling approaches applied to antimicrobial PKPD, namely, the minimum inhibitory concentration (MIC)-based and the in vitro time-course-based approaches. We explore how each approach handles monotherapy and combination therapy , as well as in the context of emergence of drug-resistant infections .

8.2 MIC-based Approaches

The PKPD properties of antibiotics to guide dosing schedules were conceived as early as the 1950s by Eagle who demonstrated the time-dependent nature of penicillin antibacterial activity, the concentration-dependent pattern for streptomycin and bacitracin, and a characteristic mixture of both patterns for tetracyclines (Eagle et al. 1950a, b; 1953a, b). With this knowledge, Eagle suggested that the efficacious way to administer penicillin was to give continuous infusion and regimens that gave the highest peak concentrations, such as an intravenous bolus, and would provide an effective cure for drugs that are concentration dependent (Eagle et al. 1950a) .

It was not until much later, when Craig rediscovered and expanded the PKPD concepts in antimicrobial therapy using rodent studies (Craig 1998), that the PKPD relationship of new antibiotics was evaluated routinely. This information provides the basis for deciding the dose and dosing interval of antimicrobial agents, as well as determination of susceptibility breakpoints.

This first part of characterizing the PKPD properties of antibiotics is generally classified as the MIC -based approaches. This section will discuss how the MIC-based approaches are utilized to optimize dosing strategies, as well as their limitations .

8.2.1 In Vitro Susceptibility Tests

The MIC has been the primary tool for determining bacterial susceptibility to an antibiotic. This test is carried out by either an agar diffusion or broth dilution; both methods are most commonly used for MIC determination, since they are easy to perform (Jorgensen and Ferraro 2009). In the agar diffusion method, the bacterial culture is spread uniformly across the agar plate and then grown overnight; a rectangular strip impregnated with a gradient amount of drug is laid on top of the agar plate. This test is commonly known as the Epsilometer test (or Etest). Because MICs are typically based on twofold dilution, the drug concentrations on the Etest strip also increases exponentially. In post-24-h incubation, an elliptical zone of bacteria-free area resulted along the strip where the drug concentrations were sufficient against the specific bacteria. The point at which the bacteria-free ellipse intersects with the Etest strip is the MIC . Older agar diffusion test utilizes disk diffusion wherein circular wafers impregnated with fixed concentrations of antibiotic are placed on a plate full of bacteria.

The broth dilution method utilizes a liquid medium usually Mueller-Hinton broth inoculated with specific bacterial colony forming unit (5 × 10⁵ CFU/mL) and specific drug concentration is pre-added in twofold dilutions (e.g., 0.125, 0.25, 0.5,1, 2, 4, 8,…μg/mL). The mixture is incubated for 24 h at 37 °C. The lowest drug concentration of antibiotic allowing no visible bacterial growth in the media is the MIC. Positive controls containing only the bacteria and negative controls containing only MHB are observed simultaneously. Because the MIC is determined by visual inspection, it does not necessarily mean that there are no bacteria remaining in the media. Rather in most cases, the bacteria level is below a CFU size that is detectable by the human eye (≤ 10⁶ CFU/mL). For large MIC values (MIC ≥ 100 μg/mL), it is advisable to evaluate susceptibility using a linear increase (e.g., 100, 200, 300,…µg/mL) than an exponential increase (i.e., twofold) in drug concentration.

The macrodilution and microdilution methods differ in the volume of the media wherein macrodilution method is often between 1 and 2 mL and microdilution is ≤ 500 μL. The bacteria inoculum should be the same for both methods. The volume of bacteria solution to add to the mixture should be adjusted to achieve a final inoculum of 5 × 10⁵ CFU/mL.

For the past several decades, the MIC has been used extensively to define the susceptibility of a specific bacterial species or strain to an antibiotic agent. In the hospital setting where multiple strains of a specific bacterial species are available, MIC₅₀ and MIC₉₀, representing the concentration of the antimicrobial agents wherein 50 and 90 % of the bacterial population do not show visible growth after 24-h incubation, are often reported (Walkty et al. 2011). The ease of use, rapid turnover of results, and cost effectiveness have made the MIC approach the testing of choice in the clinical setting.

8.2.2 PKPD Indices

The current approach in the treatment of microbial infection in the clinic is primarily based on the relationships between drug exposure and MIC (Drusano 2004; Schmidt et al. 2008). The three standard PKPD indices are fT > MIC, fC _max/MIC, and fAUC/MIC. The duration of time in the 24-h period wherein the drug concentration is above the MIC is fT > MIC. The percentage of time above MIC over the 24-h period is often used instead (% fT > MIC). AUC refers to the area under the drug concentration–time curve over the 24-h period and C _max is the peak drug concentration. The prefix f refers to the free drug concentration. The indices are based on the free and unbound drug concentration, as only the unbound drug can exert its pharmacological effect. If the relationship is time dependent , the dosing strategy is simply to maintain the free drug concentrations above the MIC value for an extended period of time. On the other hand, if the efficacy is concentration dependent , the goal is to attain sufficient peak drug concentrations or drug exposure above MIC (Mueller et al. 2004). The β-lactams are commonly associated with the term “concentration-independent kill” or “time-dependent kill.” That is because the efficacy of the β-lactams is associated with the time that the free drug concentration of these agents is maintained above MIC . The quinolones and aminoglycosides, on the other hand, are “concentration dependent” in their effects. Whether these agents are above MIC for an extended period or not do not seem to have a significant impact on the observed antimicrobial effect but rather the magnitude of the peak concentration is associated with a more efficient bacterial kill. The third type of antibiotics which include azithromycin and vancomycin is not concentration dependent but their efficacy is linked to the fAUC/MIC ratio (Drusano et al. 2004; Rybak et al. 2009a, b). The 24-h exposure, measured by the AUC-to-MIC (fAUC/MIC) ratio, is related to the observed effect. The action of many antimicrobial agents has generally been classified based on these PKPD indices.

The determination of which indices best characterize the drug action is based on fitting a sigmoidal E _max model to the PD endpoint such as the bacterial log₁₀ CFU/mL at 24 h or the log change in CFU/mL against the three PK/PD indices (Dudhani et al. 2010). The PD endpoint is often taken from animal studies wherein several live mouse thighs or lungs were injected with specific bacteria with predetermined MIC . The mice were then administered antibiotic at different drug doses and regimens in dose fractionation studies. The pharmacokinetic parameters (fT > MIC, fC _max/MIC, fAUC/MIC) were then determined for each animal. At the end of the experiments, the bacterial CFU/mL was determined from the tissues injected with bacteria. The 24-h log₁₀ CFU/thigh against the PKPD indices was used to evaluate which PKPD index best characterizes the activity of the specific antimicrobial agent being tested (Dudhani et al. 2010). The relationship between the PD endpoints and PKPD indices are plotted and the best fits for the relationships were determined by R ² (coefficient of determination). It was suggested that the PKPD index determined in mice could be extrapolated to clinical efficacy (Ambrose et al. 2007). Many of the current dosing regimens in the clinic were based on the PKPD indices determined from animal studies. Vancomycin dosing regimens, for example, were determined based on the target of AUC/MIC ratio of approximately 325 in treating ventilator-associated Staphylococcus aureus pneumonia (Moise-Broder et al. 2004a, b; Sakoulas et al. 2004; Rybak et al. 2009b). The vancomycin nomogram was designed to achieve a target trough concentration of 15–20 mg/L (Kullar et al. 2011).

Drusano provided an explanation of how the shape of the drug profile affects the type of cell kill for drugs that are concentration dependent versus those that are time dependent (Drusano 2004; Jumbe and Drusano 2011). The rate of kill in concentration-dependent drugs is different at each segment of the concentration–time profile and the total number of organisms killed can be approximated as an expectation which is the summation of the kill rate and time period over the specific kill rate. For time-dependent drugs, the kill rate is constant and the total cell kill is the rate constant multiplied by the time period that the drug concentration is above the MIC.

8.2.3 Probability of Target Attainment and Clinical Breakpoints

The probability of target attainment (PTA) is often determined from simulation of 1000–10,000 individual drug concentration–time profiles using a population pharmacokinetic model and the proportion of the population above a specific target (Drusano et al. 2001; de Kock et al. 2014). The simulation generates a distribution of PKPD index (e.g., fAUC₂₄/MIC) which becomes the basis for determining the likelihood of achieving a certain target attainment. The fAUC₂₄/MIC will be used as an example because it is easier to generate than fT > MIC or fC _max/MIC, as AUC can be estimated by integrating the population-PK model or estimated from the clearance values without running secondary pharmacokinetic analysis (e.g., noncompartmental analysis) of the generated profiles. The PTA is determined as the proportion of simulated individual profiles that are above a specific target, such as fAUC₂₄/MIC, to achieve greater than or equal to 2 log₁₀ kill from animal studies, for a range of increasing MIC values and is usually evaluated using several dosing regimens in dose fractionation studies. In the study of tigecycline against E. coli, Ambrose et al. (2009) determined the potential tigecycline–Enterobacteriaceae susceptibility using both PTA and clinical response expectation as responses. The steady-state AUC₂₄ was simulated from the distribution of clearance parameter from a population pharmacokinetic model of tigecycline. In the example in Fig. 8.1, PTA is plotted as a function of MIC, represented by triangle symbols. The clinical response expectation versus MIC, represented by circular symbol, was determined from a logistic regression model that describes the PKPD relationship for efficacy in patients with complicated intra-abdominal infections (Meagher et al. 2007; Passarell et al. 2008). As shown in their study, the two metrics, namely PTA and clinical response expectation, may not necessarily correlate with each other. However, both metrics indicate a trend towards less favorable outcome with increasing MIC.

Fig. 8.1

Probability of target attainment (PTA, open triangles) based on AUC_ss,24h/MIC ratio, clinical response expectation (open circles), and tigecycline MIC distribution (bars), showing a trend of decreasing PTA and median clinical response expectation in increasing MIC. (Image from Ambrose et al. 2009; used with permission)

A natural extension of the PTA is to categorize the antimicrobial activity of specific treatment against a microorganism population. The European Committee on Antimicrobial Susceptibility Testing (EUCAST) has provided definitions to categorize microorganisms’ antibiotic phenotype based on the quantitative antimicrobial susceptibility evaluation (Kahlmeter et al. 2003, 2006). Mouton et al. (2012) provided this categorical description:

A microorganism is defined as susceptible by a level of antimicrobial activity associated with a high likelihood of therapeutic success. A microorganism is categorized as susceptible by applying the appropriate breakpoints in a defined phenotypic test system. Conversely, resistance is defined as a high likelihood of therapeutic failure. Ideally, clinical breakpoints should therefore distinguish between patients that are likely or unlikely to respond to antimicrobial treatment .

The clinical breakpoints are determined from (1) statistical approach such as classification and regression tree (CART) analysis or multivariate logistic regression to look for a value of PKPD index that best differentiate failures and successes in treatment outcome and (2) probabilistic approach of PTA that considers the variability in patients’ pharmacokinetic and the MIC of the microorganism population. With the probabilistic approach of PTA, the microorganism with MIC values that result in the PKPD index value lower than the target are considered resistant, which translates to a lower probability of cure whereas those that result in a larger PKPD index values than a specific target are considered susceptible. This target value that separates the PKPD index for the two-microorganism phenotype is the clinical MIC breakpoint. It is noted that the clinical breakpoint may be dependent on the dosing regimen. A case is illustrated by Mouton et al. (2005, 2012) wherein the relationship between fT > MIC and MIC of ceftazidime for two different dosing regimens produces two separate and distinct PTA–MIC curves as shown in Fig. 8.2. Assuming that the target is 60 % fT > MIC, the dosing of 500 mg thrice daily and 1 g thrice daily resulted in breakpoints of 4 and 8 mg/L MIC.

Fig. 8.2

The percent of time that the free ceftazidime concentration is above MIC (% fT > MIC) for two dosing regimens of ceftazidime (1 g q8h vs. 500 mg q8h) against MIC to illustrate that clinical breakpoint is dependent on the dosing regimen. Arrows indicate that the pharmacodynamics target corresponding to 60 % fT > MIC is 4 and 8 mg/L for 500 mg q8h and 1 g q8h, respectively. (Image from Mouton et al. 2012; used with permission)

Ambrose et al. (2007) provided an excellent review to show how rodent studies translate to humans. The studies in rodent infection models showed that a total levofloxacin AUC₂₄:MIC value of 88 in immunosuppressed mice was associated with favorable microbiological response (Jumbe et al. 2003). Levofloxacin fAUC24:MIC value of 62 determined from patients with hospital-acquired pneumonia separates those patients with 90 and 43 % response to therapy (Drusano et al. 2004).The two studies show a good correlation between rodent studies and humans, given that the PKPD indices in animal studies are closely related to that in humans.

8.2.4 Limitations of the MIC-based Approach

Though this approach has been used to guide dosing for various classes of antimicrobials, several shortfalls of the MIC-guided approach are discussed here . The determination of the MIC of the patient’s infection often takes several days following the initial treatment and consequently the drug concentration and duration may not be optimal without the prior knowledge of the MIC, particularly when using the relationship between pharmacokinetic properties of the drug and the MIC-based PKPD indices to guide treatment. This simplification of dosing scheme is believed to potentially lead to treatment failure and may foster the emergence of resistant bacterial populations (Hoffman and Stepensky 1999).The utility of MIC assumes that this value is stationary. This is not the case because MIC within a bacteria species can change. When bacteria are exposed to a low concentration of drug, which is not enough to eradicate them, the bacteria will acquire resistance, resulting in a shift towards a higher MIC level (Tam et al. 2007a). Also depending on the species and strain of bacteria, the MIC may not be consistent across the species and strains. This scenario renders a “nonstationary” MIC. With an increasing rate of treatment failure, MIC is more likely to be changing over time due to development of resistance. For example, the AmpC β-lactamase is induced when exposed to low β-lactam concentration. The ampC expression is repressed by three AmpD homologues, including the previously described AmpD protein (Langaee et al. 1998, 2000) plus two additional proteins AmpDh2 and AmpDh3 (Juan et al. 2006). The two additional homologues are responsible for the stepwise ampC upregulation that results in hyper-expression of cephalosporinase and high level of β-lactam resistance (Juan et al. 2006) .

The rate of bactericidal activity or bacteriostatic effect with different drug concentrations cannot be determined from the MIC approach. Several killing patterns can converge to the same MIC value when only the 24-h time point is measured. Relying on a “snapshot” view of MIC for defining the PKPD relationship for the entire treatment duration can be misleading.

Because only unbound drug concentration can exert its pharmacological effect, ignoring protein binding and tissue distribution of the drug can have serious implication in extrapolating in vitro efficacy results to human studies. When the drug enters the blood stream, it can bind to proteins such as albumin, α-, β-, or γ-globulins, α1-acid glycoprotein, lipoproteins, and/or erythrocytes (Dasgupta 2007; Treyaprasert et al. 2007; Mouton et al. 2008). The percentage of drug binding can be constant (linear) or nonconstant (nonlinear) depending on the drug concentration; characterization of protein binding across a range of drug concentrations can provide important information on its protein binding properties. Since anti-infective drugs need to get to the infection sites, for example, in skin infection, the unbound fraction of drug crosses the membrane to the infected tissue. Microdialysis procedures have been used to determine the free fraction of drug at a specific tissue (e.g., adipose and skin; Li et al. 2006). It is important to consider protein binding when translating in vitro results to the clinic .

8.2.5 Resistance Problem in Antimicrobial Therapy

The number of new bacterial strains with more efficient resistance mechanisms has emerged over the past decade . From the year 2000 through 2004, the percentages of methicillin susceptible and resistant S. aureus (MSSA and MRSA) isolates with vancomycin MIC of 1 μg/mL increased from 40 to > 70 % and from 10 to > 60 %, respectively (Wang et al. 2006). Within the span of 5 years, S. aureus clinical isolates have evolved towards decreasing vancomycin susceptibility. The resistance problem is not isolated to just one class of antimicrobials. Various newly discovered β-lactamases can rapidly inactivate β-lactams and some β-lactamases such as the variant of TEM-1 are resistant to β-lactamase inhibitors, for example clavulanic acid (Sideraki et al. 2001).

There are experimental evidences that the efflux pump upregulation is a first-line defense for microorganisms when challenged with antimicrobial agents (Jumbe et al. 2006; Louie et al. 2007; Drusano et al. 2009). The MexCD-OprJ, not typically expressed under noninduced conditions, exports fluoroquinolones and a number of β-lactams (Poole et al. 1996; Masuda et al. 2000b). MexXY-OprM contributes to resistance to fluoroquinolone, aminoglycoside, and some β-lactam (Aires et al. 1999; Mine et al. 1999; Masuda et al. 2000b; Sobel et al. 2003); this efflux pump is induced by tetracycline and aminoglycosides (Aires et al. 1999; Mine et al. 1999; Masuda et al. 2000a). β-Lactam resistance in clinical isolates of Pseudomonas aeruginosa has been shown to interplay between diminished production of OprD (an outer membrane protein that regulates the entry of carbapenems) and an increased AmpC β-lactamase activity (Quale et al. 2006) .

The choice of dosing regimen affects the extent of resistance development. Tam et al. (2007a) demonstrated that the relationship between quinolone exposure and resistance amplification is characterized by an inverted U-shaped curve. This indicates that development of resistance is minimal at low antimicrobial challenge and rapidly increases over a range of drug concentration unless a sufficiently high drug concentration kills both the susceptible and resistant populations. These results prompted recommendations for increasing doses, shorter treatment period, and combining several antimicrobials with different mechanisms of action to counter the emergence of drug-resistant bacteria (Mouton et al. 2011) .

8.2.6 Combination Therapies

The resistance to treatment has also rapidly increased hospital mortality due to opportunistic infections (De Kraker et al. 2011). Administering two antibiotic drugs with different mechanisms of action can potentially restore the utilities of these agents. This approach is called combination therapy . The action of many antimicrobial agents has generally been classified based on the PKPD indices, previously described. These indices, however, are relevant primarily to monotherapy. When evaluating combination therapy that includes multiple antibiotics, the pattern may no longer be relevant. This renders the classification of combination therapy and the determination of optimal dosing strategies nontrivial.

Drugs of different mechanisms of action may act synergistically, resulting in greater than fourfold decrease in the MIC of each drug in the same pathogen in vitro (Paul et al. 2004). The use of aminoglycoside/β-lactam combination was practiced in the past (Piccart et al. 1984; Hoepelman et al. 1988a, b; Mondorf et al. 1989). However, the benefits from the combination were later questioned based on a meta-analysis study (Bliziotis et al. 2005). The likely reason could be that the patients who received combination therapy had a higher propensity for mortality since combination antibiotics are more commonly prescribed for the critically ill patients than the single-agent antibiotics. The one subgroup of Gram-negative pathogens, for which the question of combination therapy is currently being investigated in more and more studies, is P. aeruginosa (Louie et al. 2013). This bacterial species is also more common in patients who are severely ill, including the late stage of morbidity in cystic fibrosis (CF) patients (Breen and Aswani 2012). In fact, the Cystic Fibrosis Foundation guidelines recommend that an antipseudomonal β-lactam with an aminoglycoside be used in the treatment of acute pulmonary exacerbations of CF (Flume et al. 2009). CF patients were thought to have higher clearance and larger volumes of distribution, which makes dosing more challenging due to lower exposure (Spino 1991). In a matched control study, no difference was found in aztreonam volume of distribution between CF patients and matched healthy subjects but total body clearance was 30 % higher in CF patients due to enhanced renal clearance as CF patients had 20 % higher free fraction of the drug (Vinks et al. 2007).

The synergy of activities from combination of β-lactam and aminoglycoside would be particularly beneficial in these difficult-to-treat populations. The combination of an aminoglycoside and a β-lactam seems to be the most frequently used combination against P. aeruginosa. Louie et al. (2013) showed that tobramycin in combination with meropenem suppressed resistance amplification in P. aeruginosa at all combination regimens that were tested in the murine pneumonia model.

Safdar et al. (2004) performed a meta-analysis on combination antimicrobial therapy for bacteremia due to Gram-negative bacilli. Their overall results, combining all types of bacteria that were found in their literature search, indicated that combination therapy does not reduce mortality in patients with Gram-negative bacteremia. One limitation of their study was that the literature that they used did not stratify the outcome by the severity of illness. The patients with multiple comorbidities were also more likely to die due to their underlying conditions. In a stratified analysis, they found a significant survival benefit with combination therapy in P. aeruginosa bacteremia, translating to an approximately 50 % mortality reduction (CI: 32–79 %). This specific result provided the rationale for the hypothesis that the combination of aminoglycoside and β-lactam may provide synergism in vivo in a setting where the suspected infection is predominantly P. aeruginosa or other multiresistant Gram-negative bacilli where more than one drug would assure susceptibility to at least one of the antimicrobial agents.

The exact mechanism of action of aminoglycosides is not fully known. It was suggested that aminoglycosides could be either bacteriostatic and/or bactericidal (Bakker 1992). The bacteriostatic effect stops the growing of bacteria by inhibiting protein synthesis as the aminoglycoside binds to the 16S rRNA. The mechanism for its bactericidal effect is by disrupting the integrity of bacterial cell membrane (Shakil et al. 2008). In contrast, the mechanism of action of β-lactam antibiotics is completely known. β-Lactam antibiotics are bactericidal and act by an irreversible inhibition of the penicillin-binding proteins, which normally catalyze the cross-linking of bacterial cell walls. The drug binding to the penicillin-binding proteins kills the bacteria due to the disruption of the cell wall synthesis (Fisher et al. 2005).

The PKPD indices for combination therapy have not been explored yet. It is likely more challenging to develop since the evaluation would require a much larger set of dose fractionation studies especially combining two drugs. The in vitro time-course-based approach may provide a simpler methodology to evaluate dosing regimens for combination therapies than the summary PKPD variables that are MIC based.

8.3 In Vitro Time-Course-Based Approaches

8.3.1 Time-Kill Kinetic Studies

The advancement of PKPD modeling approach came with more defined in vitro methodologies. The in vitro time course of drug–bacterial response characterized by the kill-curve assays has been used as the basis for developing PKPD models to describe bacterial population dynamics, drug effects, and the emergence of resistance. Depending on the objective of the study, the drug concentration in these in vitro time-kill experiments can be relatively constant in the static situation (Garrett et al. 1966; Mielck and Garrett 1969; Garrett and Nolte 1972) or dynamically changing to mimic the in vivo half-life of the drug in humans (Sanfilippo and Morvillo 1968; Sanfilippo and Schioppacassi 1973; Grasso et al. 1978). The data from static time-kill experiments are often used to develop a mathematical model that links the free drug concentrations to the bacterial response whereas dynamic time-kill data are used to validate the model and to predict the outcome in the clinic. The dynamic kill-curve provides an alternative for evaluating PKPD relationships; it simulates the time course of the unbound drug concentrations at the site of action based on a preset half-life. Using multiple pumps, the hollow fiber infection model is used to simulate concentration-time profiles of free drug concentration that mimics the in vivo profiles (Crandon et al. 2012). The effects of different dosing regimens, drug half-lives and even starting inocula can be simulated to study their effects on the bacterial population dynamics over a time period, for example 24 or 48 h.

8.3.2 PKPD Models of In Vitro Time-Kill Kinetics

The Logistic Growth Models

The PKPD models currently used to describe the in vitro bacterial population dynamics came from models used to study human population dynamics. In 1838, Pierre-François Verhulst described the logistic growth model that many of the modern antimicrobial PKPD models were based on:

$$ \frac{dN}{dt}=r\left(1-\frac{N}{K} \right)N $$

(8.1)

where N is the population number, r is the growth rate, and K is the carrying capacity or the maximum number of individuals that is supported by the environment (Gershenfeld 1999). The analytical solution to Eq. (8.1) is:

$$ N(t)=\frac{K{{N}_{0}}{{e}^{rt}}}{K+{{N}_{0}}({{e}^{rt}}-1)} $$

(8.2)

where N ₀ is the initial population number at time t = 0. The important property of this model is that the limit of this function as time goes to infinity is the carrying capacity: $\underset{t\to \infty }{\mathop{\lim }}\,N(t)=K$. In an in vitro time-kill curves of both static and dynamic systems, the bacterial CFU is restricted from growing indefinitely and usually reaches a plateau, where the net growth is zero. For this reason, the logistic growth model suitably describes this behavior.

The Compartmental Models

The second type of antimicrobial PKPD model can be described in simplistic terms consisting of the natural self-replication and death of bacteria (Eq. 8.3):

$$ \frac{dN}{dt}=({{k}_{\text{growth}}}-{{k}_{\text{death}}})N $$

(8.3)

where N is the bacterial population with the initial count of N ₀, k _growth is the first-order rate constant for bacterial synthesis, and k _death is the first-order rate constant for bacterial death. This common structure to describe bacterial growth is also used in other disease areas such as tumor dynamic models, where a first-order self-replication rate is implemented (Jusko 1971). This model assumes that the bacteria are from a homogeneous population with the same growth and death rate constants, which may not reflect the true population of microbes, which is known to select for resistant strain in the presence of an antimicrobial challenge. The variations based on the compartmental model have improved on this limitation and will be described more thoroughly in later sections of this chapter.

The Mechanistic Models

The third type of antimicrobial models considers the bacterial growth cycle, states of bacterial susceptibility, drug–receptor interaction, and the mechanisms of drug action. This type of models utilized many concepts of mathematical modeling in biology, including the two modeling approaches discussed above. Each of the mechanistic models will be discussed separately as there is no common mathematical approach across these models that can be summarized briefly.

8.3.3 Modifications on the Logistic Growth Model

To incorporate drug action to the capacity limited growth model, Eq. (8.1) can be modified to include a function to describe the drug effect:

$$ \frac{dN}{dt}={{k}_{\text{growth}}}N\left(1-\frac{N}{{{N}_{\max{}}}}\right)-{{f}_{\text{death}}}(\text{drug}) $$

(8.4)

where the added f _death (drug) describes the effect of an antimicrobial agent (Nolting et al. 1996; Mouton et al. 1997; Yano et al. 1998; Mouton and Vinks 2005). In this equation, as N approaches N _max, the growth term approaches a plateau or stationary condition, where there is no net change in the bacterial population. The drug effect is often represented by an E _max or a sigmoidal E _max model such that,

$$ {{f}_{\text{death}}}(\text{drug})=\frac{{{E}_{\max{}}}{{C}^{\gamma }}}{EC_{50}^{\gamma }+{{C}^{\gamma }}}N $$

(8.5)

where C is the drug concentration at any specific time, E _max is the maximum drug effect, and EC _max is the concentration at which the half-maximum effect is achieved. The shape parameter γ is 1 in the E _max model and is a parameter in the sigmoidal E _max model.

During the initial growth phase where $N\ll {{N}_{\max{}}}$ and the growth is linear, Eq. (8.4) can be simplified to the following equation (Nolting et al. 1996):

$$ \frac{dN}{dt}=\left[{{k}_{\text{growth}}}-\frac{{{E}_{\max{}}}{{C}^{\gamma }}}{EC_{50}^{\gamma }+{{C}^{\gamma }}}\right]N $$

(8.6)

By solving for the analytical solution to Eq. (8.6), one can determine the number of bacteria at time (t) through the following equation:

$$ N(t)={{N}_{0}}{{e}^{\left({{k}_{\text{growth}}}-\frac{{{E}_{\max{}}}{{C}^{\gamma }}}{EC_{50}^{\gamma }+{{C}^{\gamma }}}\right)t}}$$

(8.7)

Mouton and Vinks proposed that the stationary concentration (SC), which is defined as the concentration at which the growth rate equals the kill rate and is also the point at which no net change in the number of bacteria is observed, can be derived from Eq. (8.7) (Mouton and Vinks 2005). By taking the natural log of the ratio $N(t)/{{N}_{0}}$ divided by time, which is equivalent to ${{k}_{\text{growth}}}-\frac{{{E}_{\max{}}}{{C}^{\gamma }}}{EC_{50}^{\gamma }+{{C}^{\gamma }}}$, one can obtain the equation for C:

$$ C=E{{C}_{50}}{{\left[\frac{{{k}_{\text{growth}}}-\frac{1}{t}\ \ln \ \frac{N(t)}{{{N}_{0}}}}{{{E}_{\max{}}}-\left({{k}_{\text{growth}}}-\frac{1}{t}\ \ln \ \frac{N(t)}{{{N}_{0}}}\right)} \right]}^{1/\gamma }}$$

(8.8)

When there is no net change in the number of bacteria, the term $\tfrac{1}{t}\ln \frac{N(t)}{{{N}_{0}}}$ approaches 0 and the SC is defined as:

$$ \text{SC}=E{{C}_{50}}{{\left[\frac{{{k}_{\text{growth}}}}{{{E}_{\max{}}}-{{k}_{\text{growth}}}}\right]}^{1/\gamma }}$$

(8.9)

The SC is not to be confused with the MIC, as SC refers to the concentration where no net bacterial growth occurs. It is often assumed that bacterial growth occurs when the drug concentration is below the MIC. Mouton and Vinks had shown that a correction factor to the SC equation might be required to estimate the MIC (Mouton and Vinks 2005):

$$ \text{MIC}=E{{C}_{50}}{{\left[\frac{{{k}_{\text{growth}}}-0.29}{{{E}_{\max{}}}-({{k}_{\text{growth}}}-0.29)} \right]}^{1/\gamma }}$$

(8.10)

The value 0.29 is obtained from the kill curves such that N(t) reached 10⁸ CFU/mL at 18 h, assuming an initial inoculum of 5 × 10⁵ CFU/mL. This correction factor is therefore dependent on the specific system that is being tested. The relationship between MIC and SC is described in greater detail in Mouton and Vinks (2005).

Tam et al. (2008) modified the logistic growth model to study the effects of gentamicin and amikacin on the in vitro time-kill kinetics of P. aeruginosa ATCC 27853 and Acinetobacter baumannii ATCC BAA 747, respectively, by introducing an adaptation factor to the EC ₅₀ parameter:

$$ {{f}_{\text{death}}}(\text{drug} )=\frac{{{E}_{\max{}}}{{C}^{\gamma }}}{{{(\propto E{{C}_{50}})}^{\gamma }}+{{C}^{\gamma }}}N $$

(8.11)

α is defined as:

$$ \propto =1+\beta (1-{{e}^{-C\tau t}}) $$

(8.12)

where τ is the exponent of the adaptation factor and β is the maximal adaptation. The range of values for the function $1-{{e}^{-C\tau t}}$ is between 0 and 1. The adaptation function α starts from a baseline EC ₅₀ and increases over time to a maximal value of β, if τ is positive.

Delay functions were applied to both the growth rate and the drug effect function to describe the population dynamics of Streptococcus pneumoniae, Haemophilus influenzae, and Moraxella catarrhalis in the presence of azithromycin (Treyaprasert et al. 2007). The delay function has the following form:

$$ \frac{dN}{dt}=\left[{{k}_{\text{growth}}}\left(1-\frac{N}{{{N}_{\max{}}}}\right)(1-{{e}^{-xt}})-\left(\frac{{{k}_{\max{}}}C}{E{{C}_{50}}+C} \right)(1-{{e}^{-yt}}) \right]N $$

(8.13)

One can see that the delay function has a similar form to the adaptation function discussed above. The two equations, $1-{{e}^{-xt}}$ and $1-{{e}^{-yt}}$ (Mouton et al. 1997), behave like a cumulative density function starting from 0 at t = 0 to a maximum value of 1 as $t\to \infty$. The delay function acts as a modulator to allow the curves to conform to the S-shaped pattern of bacterial growth which is often observed during the first couple of hours of the time-kill kinetic experiments in the presence of low antimicrobial agent concentrations. The two functions also shape the transition to plateau after a decrease then increase in bacterial population at the antimicrobial concentrations that allow for bacterial regrowth to occur. An example of model using the delay function is shown in Fig. 8.3. Another modification introduced a second compartment for the persistent bacterial population to differentiate from the first compartment of susceptible bacteria; this alteration was used to model the effect of oxazolidinone on Staphylococcus aureus (Schmidt et al. 2009).

Fig. 8.3

Time-kill kinetics and model prediction of aztreonam-avibactam effect against K. pneumoniae

Bulitta et al. (2009) linked the bacterial population dynamic to cell wall synthesis, and drug effect of ceftazidime on cell wall synthesis to describe the lag time in bactericidal effect of β-lactams. The study examines the inoculum effect of ceftazidime against P. aeruginosa. The natural first-order death rate was dependent on the number of existing CFU in the system and the logistic growth part of the model was dependent on both the cell wall synthesis and CFU. The drug effect acts on the compartment representing the cell wall dynamics, since the primary mode of action of ceftazidime is to inhibit cell wall synthesis:

$$ \frac{dCW}{dt}=\left[\left(1-\frac{{{C}_{B}}}{I{{C}_{50,CW}}+{{C}_{B}}}\right)-CW \right]\cdot {{k}_{\text{out},CW}}$$

(8.14)

$$ \frac{dN}{dt}=\left[{{k}_{\text{growth}}}\left(1-\frac{N}{{{N}_{\max{}}}}\right)CW-{{k}_{\text{death}}}\right]N $$

(8.15)

where CW represents a hypothetical cell wall measurement, whose synthesis is expressed as a fraction of the baseline value. The $I{{C}_{50,CW}}$ is the concentration of ceftazidime in the broth that inhibits 50 % of cell wall synthesis and ${{k}_{\text{out},CW}}$ is the first-order rate constant for the cell wall turnover. The investigators claimed that this model accounts for the slow onset due to the delay between ceftazidime binding to the penicillin-binding proteins and the depletion of cell wall components (Bulitta et al. 2009).

8.3.4 Examples of the Compartmental Model

During the linear growth phase just before reaching the plateauing phase, the population dynamics can be described by simple first-order growth and death rates that are dependent on the bacterial burden present at the specific time. In addition to the logistic growth model, other strategies had been utilized to describe the decrease in the net growth rate as the system approaches the plateau wherein the net bacterial growth is zero. One approach is to implement a phenotypic switch between susceptible and persistent population such that the persisters have a markedly reduced growth rate (Balaban et al. 2004). The overall change in the total number of bacteria would then be the sum of those in susceptible (S) and in persistent resting (R) states, such that:

$$ {{A}_{\text{total}}}=S+R $$

(8.16)

The transition between the two states is defined by their respective rate constants. Nielsen et al. presented an example of compartmental model wherein a two-compartment model was used to describe the in vitro effect of a number of antibiotics, including moxifloxacin, vancomycin, benzylpenicillin, cefuroxime, and erythromycin against Streptococcus pyogenes (Nielsen et al. 2007). The delay in the effect of drugs was modeled using an effect compartment model for the drug. The following assumptions were made: (1) The drug effect is to increase the death rate of the susceptible state and (2) the antimicrobials have no effect on the persistent population. The differential equations for the two bacterial populations are shown in the following equations:

$$ \frac{dS}{dt}={{k}_{\text{growth}}}S-{{k}_{\text{death}}}S-{{k}_{SR}}S+{{k}_{RS}}R $$

(8.17)

$$ \frac{dR}{dt}={{k}_{SR}}S-{{k}_{RS}}R-{{k}_{\text{death}}}R $$

(8.18)

As the persistent population is unlikely to return to the susceptible state for the duration of the experiment, the transfer rate for the return to the susceptible state was assumed to be negligible and ${{k}_{RS}}$ was fixed to 0. The transfer rate constant that indicates the rate of change from the susceptible to the persistent states, ${{k}_{SR}}$, dictates the growth-limited capacity using the following equation:

$$ {{k}_{SR}}=\frac{{{k}_{\text{growth}}}-{{k}_{\text{death}}}}{{{B}_{\max{}}}}(S+R) $$

(8.19)

where B _max is the maximum number of bacteria supported by the system.

The investigators evaluated whether the drug decreases the growth rate or increases the rate of death. The later scenario, increase in death rate, was examined as either an additive or a proportional effect. The equations to describe the three different scenarios are shown below:

$$ \frac{dS}{dt}={{k}_{\text{growth}}}(1-f(\text{drug}))S-{{k}_{\text{death}}}S-{{k}_{SR}}S+{{k}_{RS}}R $$

(8.20)

$$ \frac{dS}{dt}={{k}_{\text{growth}}}S-({{k}_{\text{death}}}+f(\text{drug}))S-{{k}_{SR}}S+{{k}_{RS}}R $$

(8.21)

$$ \frac{dS}{dt}={{k}_{\text{growth}}}S-{{k}_{\text{death}}}(1+f(\text{drug}))S-{{k}_{SR}}S+{{k}_{RS}}R $$

(8.22)

where $f\text{(drug})$ is a sigmoidal E _max model to account for the effect of various drug concentrations.

In a follow-up study, the same group described a mechanism for adaptive resistance in E. coli due to gentamicin by introducing two additional compartments that regulate resistance development (Mohamed et al. 2012):

$$ \frac{dA{{R}_{\text{off}}}}{dt}={{k}_{\text{off}}}A{{R}_{\text{on}}}-{{k}_{\text{on}}}A{{R}_{\text{off}}}C $$

(8.23)

$$ \frac{dA{{R}_{\text{on}}}}{dt}={{k}_{\text{on}}}A{{R}_{\text{off}}}C-{{k}_{\text{off}}}A{{R}_{\text{on}}}$$

(8.24)

where AR _off represents the adaptive resistance in dormant stage and AR _on is for the active state; the transfer between states is represented by k _off and k _on; and C refers to gentamicin concentration. A greater flexibility was achieved by the two additional compartments as can be seen in how the model adapted to the data trend. The investigators noted that the model is suitable for gentamicin in the context of compartmental models.

8.3.5 Examples of Mechanistic Models

A similar approach to the compartment model involving three-state susceptibility was used to study colistin effect in P. aeruginosa (Bulitta et al. 2010). In this model, the states of susceptibility included susceptible (S), intermediate (I), and resistant (R). A fourth compartment or state $(CF{{U}_{S,lag}})$ was introduced to account for the difference between the initial total bacterial burden, $CF{{U}_{\text{ALL}}}$, and the initial conditions of the susceptible, intermediate and the resistant populations.

$$ CF{{U}_{\text{ALL}}}=CF{{U}_{S,lag}}+CF{{U}_{S}}+CF{{U}_{I}}+CF{{U}_{R}}$$

(8.25)

The intermediate and resistant populations were assumed to be fractions of the initial total bacterial burden, $CF{{U}_{0}}$:

$$ \frac{dCF{{U}_{S,\text{lag}}}}{dt}=(-{{k}_{\text{lag}}}-IN{{H}_{\text{Kill}}}\cdot {{k}_{2S}}\cdot {{C}_{\text{Colistin,eff}}})\cdot CF{{U}_{S,\text{lag}}}$$

(8.26)

$$ \frac{dCF{{U}_{S}}}{dt}=\left(\frac{IN{{H}_{\text{Rep}}}\cdot V{{G}_{\max{},S}}}{CF{{U}_{m}}+CF{{U}_{\text{ALL}}}}-{{k}_{d}}-IN{{H}_{\text{Kill}}}\cdot {{k}_{2S}}\cdot {{C}_{\text{Colistin,eff}}}\right)\cdot CF{{U}_{S}}+{{k}_{\text{lag}}}$$

(8.27)

$$ \frac{dCF{{U}_{I}}}{dt}=\left(\frac{IN{{H}_{\text{Rep}}}\cdot V{{G}_{\max{},I}}}{CF{{U}_{m}}+CF{{U}_{\text{ALL}}}}-{{k}_{d}}-IN{{H}_{\text{Kill}}}\cdot {{k}_{2I}}\cdot {{C}_{\text{Colistin,eff}}}\right)\cdot CF{{U}_{I}}$$

(8.28)

$$ \frac{dCF{{U}_{R}}}{dt}=\left(\frac{IN{{H}_{\text{Rep}}}\cdot V{{G}_{\max{},R}}}{CF{{U}_{m}}+CF{{U}_{\text{ALL}}}}-{{k}_{d}}-IN{{H}_{\text{Kill}}}\cdot {{k}_{2R}}\cdot {{C}_{\text{Colistin,eff}}}\right)\cdot CF{{U}_{R}}$$

(8.29)

where $V{{G}_{\max{}}}$ refers to the maximal rate of bacterial growth in the unit of CFU/(mL.h), $CF{{U}_{m}}$ is the bacterial density that produces 50 % of the maximal growth rate, ${{k}_{\text{d}}}$ is the natural death rate, ${{k}_{\text{lag}}}$ represents the first-order growth rate constant associated with the slower initial growth phase of the susceptible population in the lag compartment, $IN{{H}_{\text{Kill}}}$ and $IN{{H}_{\text{Rep}}}$ are represented by the following equations:

$$ IN{{H}_{\text{Kill}}}=1-{{I}_{\text{max,Kill}}}\cdot \frac{{{C}_{\text{Signal}}}}{I{{C}_{50}}+{{C}_{\text{Signal}}}}$$

(8.30)

$$ IN{{H}_{\text{Rep}}}=1-{{I}_{\text{max,Rep}}}\cdot \frac{{{C}_{\text{Signal}}}}{I{{C}_{50}}+{{C}_{\text{Signal}}}}$$

(8.31)

$IN{{H}_{\text{Kill}}}$ and $IN{{H}_{\text{Rep}}}$ are inhibition of killing and of replication by signal molecules, ${{C}_{\text{signal}}}$. The synthesis of freely diffusible signal molecules ${{C}_{\text{Signal}}}$ by the bacteria was assumed to inhibit or to slow down the killing effects of colistin. The ${{I}_{\text{max,Kill}}}$ and ${{I}_{\text{max,Rep}}}$ were the maximum inhibition of killing and of replication; the concentration of signal molecules to achieve 50 % of the maximum inhibition is the $I{{C}_{50}}$. The kinetic behavior of the hypothetical signal molecule is described by the following differential equation:

$$ \frac{d{{C}_{\text{Signal}}}}{dt}=\left(CF{{U}_{\text{ALL}}}\frac{ml}{CFU}-{{C}_{\text{Signal}}}\right){{k}_{\text{deg}}}$$

(8.32)

where ${{k}_{\text{deg}}}$ is the degradation rate constant.

The assumption was made that the effect of colistin is to displace competitively both Mg²⁺ and Ca²⁺ from the binding sites in the outer membrane and the resulting displacement is responsible for colistin killing effect. The model utilizes receptor occupancy model to describe the competitive inhibition of colistin with Mg²⁺ and Ca²⁺ for the membrane binding sites. The fractional occupancy of these cations as a function of the molar sum of Mg²⁺ and Ca²⁺ concentrations as well as colistin concentration in mg/L is defined such that:

$$ F{{r}_{\text{Cations}}}=\frac{{{C}_{\text{Cations}}}}{{{K}_{d,\text{Cations}}}+{{C}_{\text{Cations}}}+\frac{{{K}_{d,\text{Cations}}}}{{{K}_{d,\text{Colistin}}}}\cdot \frac{{{C}_{\text{Colistin}}}}{{{M}_{m}}}}$$

(8.33)

where ${{K}_{d,Cations}}$ and ${{K}_{d,Colistin}}$ are the dissociation constants for the two cations and colistin, respectively; ${{M}_{m}}$ is the average molecular mass of colistin A and B, which are the two primary components of colistin; and $C$ refers to the concentration of the respective components in the broth. The fraction of the receptors that were not bound to Mg²⁺ or Ca²⁺ was used to compute the effective colistin concentration at the target site. The effective colistin concentration, ${{C}_{\text{Colistin,eff}}}$, was a Hill function dependent on the $F{{r}_{\text{Cations}}}$ and colistin concentration in broth, ${{C}_{\text{Colistin}}}$:

$$ {{C}_{\text{Colistin,eff}}}=\frac{{{(1-F{{r}_{\text{Cations}}})}^{\gamma }}}{EC_{50}^{\gamma }+{{(1-F{{r}_{\text{Cations}}})}^{\gamma }}}$$

(8.34)

An important difference between this model and the logistic growth or the compartment models is that it assumes that the drug has an effect on all types of bacteria from the susceptible to the intermediate to the resistant ones. The limitation of such a complex model is that the data from an in vitro time-kill study will not be sufficient to characterize the model parameters and that many of the parameter values will have to be derived from the literature.

Another compartmental-type model incorporated mechanisms involved in the life cycle of bacterial replication, autolysis, and ceftazidime effect against P. aeruginosa (Bulitta et al. 2009). The model utilizes a two-compartment model for the bacteria population and a turnover model to describe the time-kill behavior in the presence of antimicrobial agents. The bacterial life-cycle model utilizes two states for the susceptible population, wherein the first state, ${{S}_{1}}$, describes the cycle immediately after cell replication whereas the second state, ${{S}_{2}}$, occurs just before replication:

$$ \frac{d{{S}_{1}}}{dt}=2\left[1-\frac{CF{{U}_{\text{total}}}}{CF{{U}_{\text{total}}}+CF{{U}_{\max{}}}}\right][1-ALy{{s}_{S}}]\cdot {{k}_{21}}\cdot {{S}_{2}}-In{{h}_{k12}}\cdot {{k}_{12}}\cdot {{S}_{1}}$$

(8.35)

$$ \frac{d{{S}_{2}}}{dt}=-{{k}_{21}}\cdot {{S}_{2}}+In{{h}_{k12}}\cdot {{k}_{12}}\cdot {{S}_{1}}$$

(8.36)

where ${{k}_{12}}$ and ${{k}_{21}}$ are first-order transition rates between the first and second states. ${{k}_{12}}$ is determined from the mean generation time (MGT) such that ${{k}_{12}}=1/MG{{T}_{12}}$. The MGT is discussed later in this section. $In{{h}_{k12}}$ is identical to Eqs. (8.30) and (8.31). The factor 2 was used to represent bacterial doubling during cell replication. The autolysin activity $ALy{{s}_{S}}$ is stimulated by ceftazidime and is described by a turnover model:

$$ \frac{dALy{{s}_{S}}}{dt}=\left[\frac{{{S}_{\max,S}}\cdot {{C}_{B}}}{S{{C}_{50}}+{{C}_{B}}}-\left(1+\frac{{{S}_{\text{max,loss}}}\cdot {{C}_{Sig1}}}{{{C}_{\text{50,Sig}}}+{{C}_{\text{Sig1}}}}\right)\cdot ALy{{s}_{S}}\right]\cdot {{k}_{\text{out}}}$$

(8.37)

The ${{S}_{\max,S}}$ value limits the maximum value of $ALy{{s}_{S}}$ to 1, indicating that a high drug concentration can completely inhibit replication. $S{{C}_{50}}$ refers to the drug concentration at which autolysin is half maximally stimulated. ${{S}_{\text{max,loss}}}$ represents the maximum extent of the inoculum effect at high signal molecule concentrations. In contrast to the previous model, the current one assumes two signal molecules with the following behavior:

$$ \frac{d{{C}_{\text{Sig1}}}}{dt}=(CF{{U}_{\text{ALL}}}-{{C}_{\text{Sig1}}})\cdot {{k}_{\deg }}-{{k}_{S12}}\cdot {{C}_{\text{Sig1}}}+{{k}_{S21}}\cdot {{C}_{\text{Sig2}}}$$

(8.38)

$$ \frac{d{{C}_{\text{Sig}2}}}{dt}={{k}_{S12}}\cdot {{C}_{\text{Sig1}}}-{{k}_{S21}}\cdot {{C}_{\text{Sig2}}}$$

where the initial conditions for ${{C}_{\text{Sig}1}}$ and ${{C}_{\text{Sig2}}}$ were $CF{{U}_{0}}$ and $CF{{U}_{0}}\cdot {{k}_{S12}}/{{k}_{S21}}$. The role of the first signal molecule ${{C}_{\text{Sig1}}}$ is to slow down the replication rate and ${{C}_{\text{Sig}1}}$ is in equilibrium with ${{C}_{\text{Sig2}}}$.

The life-cycle growth model involving two-stage susceptibility was applied to the study of linezolid against vancomycin-resistant enterococci and S. aureus (Tsuji et al. 2012a, b). Linezolid effect was assumed to inhibit protein synthesis; a turnover-type model was used to describe its effect:

$$ \frac{dP}{dt}={{k}_{\text{Prot}}}\left[\left(1-\frac{{{C}_{\text{drug}}}}{I{{C}_{50}}+{{C}_{\text{drug}}}}\right)-P \right] $$

(8.39)

where $P$ represents the protein pool whose steady-state maximum value is 1, $I{{C}_{50}}$ is the linezolid concentration that produces half-maximum inhibition of protein synthesis and ${{k}_{\text{prot}}}$ is the turnover rate constant of the protein pool. The probability of death due to lack of protein represented by ($Lack=1-P$) was defined by the following function:

$$ Pro{{b}_{\text{death}}}={{I}_{\text{max,Rep}}}Lack $$

(8.40)

The E _max -type model was used to describe the plateau phase of the primary susceptible bacterial population. A two-state susceptibility bacterial population was described for ${{S}_{1}}$ and ${{S}_{2}}$, similar to the model described in Eqs. (8.35) and (8.36), with the function $1-Pro{{b}_{\text{death}}}$ replacing the $1-ALy{{s}_{s}}$ and $In{{h}_{12}}$ was fixed to 1. The number of sensitive alleles (${{N}_{\text{Sen}}}$) was a covariate to determine the $I{{C}_{50}}$ of linezolid in inhibiting protein synthesis:

$$ I{{C}_{50}}=I{{C}_{50\text{Sen}0}}\left(1-\frac{{{I}_{\text{max,Sen}}}\cdot N_{\text{sen}}^{\text{HSen}}}{N_{\text{50,Sen}}^{\text{HSen}}+N_{\text{sen}}^{\text{HSen}}}\right)\cdot {{f}_{HFIM}}$$

(8.41)

where $I{{C}_{\text{50Sen0}}}$ refers to the $I{{C}_{50}}$ for a strain with no sensitive alleles, I _max,Sen refers to the maximum fractional decline for $I{{C}_{50}}$, ${{f}_{HFIM}}$ refers to the ratio of $I{{C}_{50}}$ in the hollow fiber infection model compared with the static time-kill model, and $HSen$ is the hill coefficient.

The MGT is defined as the doubling time required for the bacteria to double in number and is computed from the net growth rate, similar to the computation of half-life (Garrett 1978):

$$ MGT=\frac{\ln (2)}{{{k}_{net}}}$$

(8.42)

where ${{k}_{net}}={{k}_{growth}}-{{k}_{death}}$. In the model, the number of resistant alleles $({{N}_{res}})$ was used as a covariate to compute the MGT ($MG{{T}_{12}}$) wherein ${{k}_{12}}$ was computed as $1/MG{{T}_{12}}$:

$$ MG{{T}_{12}}=MG{{T}_{0}}\cdot \left(1+\frac{{{S}_{max,Res}}\cdot N_{res}^{HRes}}{N_{50,Res}^{HRes}+N_{res}^{HRes}}\right) $$

(8.43)

where ${{N}_{50,Res}}$ refers to the number of resistant alleles that produce 50 % of ${{S}_{max,Res}}$,which is the maximum fractional increase in $MG{{T}_{12}}$ due to resistant alleles, and $HRes$ is the Hill coefficient.

The goal of anti-infective therapy is to administer an effective dose of drug to a patient with a high probability of achieving therapeutic success while minimizing toxicity. Mechanistic models are believed to better describe the processes and subtleties of nature that may not be apparent in an empirical model (Lo et al. 2011). Whether mechanistic models are better predictors of clinical outcome over the empirical and semi-mechanistic models is yet to be proven.

8.3.6 Models of Combination Therapy

We have recently modified the logistic growth model to study the enhanced potency of aztreonam by avibactam using a shift in EC₅₀ that approximates the reduction in MIC values at increasing avibactam concentrations against K. pneumoniae (Fig. 8.3; Sy et al. 2013) . The reduction in aztreonam EC₅₀ as a function of avibactam concentration was approximated by an empirical bi-exponential decay equation. Avibactam, being a β-lactamase inhibitor, has no antimicrobial activity against K. pneumoniae, but rather restores the potency of aztreonam by inhibiting aztreonam removal and degradation by the β-lactamase enzymes. The advantages of this simple approach are that the model-predicted aztreonam EC₅₀ closely mimicked the MIC value and the generated curves described well the observed bacterial dynamics in response to the combination therapy.

The Loewe additivity was utilized to evaluate the effect of combination therapy consisting of a novel aminoglycoside, vertilmicin, and ceftazidime on P. aeruginosa (Zhuang et al. 2013). The bacterial population model was based on a two-state logistic growth model. In contrast to the aztreonam–avibactam study, both agents have antimicrobial effects with different mechanisms of action. Greco et al. proposed a generalized sigmoidal E _max equation to describe Loewe additivity of the combined effect of two agents (Greco et al. 1995):

$$ 1=\frac{{{C}_{1}}}{E{{C}_{50,1}}{{\left(\frac{{{E}_{\max{}}}}{{{E}_{\max{}}}-{{k}_{\max{}}}}\right)}^{1/{{m}_{1}}}}}\frac{{{C}_{2}}}{E{{C}_{50,2}}{{\left(\frac{{{E}_{\max{}}}}{{{E}_{\max{}}}-{{k}_{\max{}}}}\right)}^{1/{{m}_{2}}}}}+\frac{\gamma \cdot {{C}_{1}}\cdot {{C}_{2}}}{E{{C}_{50,1}}\cdot E{{C}_{50,2}}{{\left(\frac{{{E}_{\max{}}}}{{{E}_{\max{}}}-{{k}_{\max{}}}}\right)}^{(1/2{{m}_{1}}+1/2{{m}_{2}})}}}$$

(8.44)

The additive effect of the two agents is described by the sum of the first two terms and the third term is an interaction term, wherein $\gamma$ is a parameter that indicates synergism–antagonism interaction. The interaction is additive if the 95 % confidence interval of the $\gamma$ estimate overlaps the zero value. If $\gamma>0$ or $\gamma <0$, the interaction is either synergistic or antagonistic, respectively. In the model, ${{m}_{1}}$ and ${{m}_{2}}$ were assumed to be equal. An additional interaction term $\lambda$ was incorporated to the effect of an initial killing rate of both agents (${{k}_{\max{}}}$) such that:

$$ {{k}_{\max{}}}={{k}_{\text{max,1}}}+{{k}_{\text{max,2}}}+\lambda \cdot {{k}_{\text{max,1}}}\cdot {{k}_{\text{max,2}}}$$

(8.45)

The resulting E _max model to evaluate the combination of vertilmicin and ceftazidime was:

$$ {{E}_{\max{}}}=\frac{{{k}_{\max{}}}{{\left(\frac{{{C}_{1}}}{{{\alpha }_{1}}E{{C}_{50,1}}}+\frac{{{C}_{2}}}{{{\alpha }_{2}}E{{C}_{50,2}}}+\frac{\gamma {{C}_{1}}{{C}_{2}}}{{{\alpha }_{1}}{{\alpha }_{2}}\cdot E{{C}_{50,2}}\cdot E{{C}_{50,2}}}\right)}^{k}}}{1+{{\left(\frac{{{C}_{1}}}{{{\alpha }_{1}}E{{C}_{50,1}}}+\frac{{{C}_{2}}}{{{\alpha }_{2}}E{{C}_{50,2}}}+\frac{\gamma {{C}_{1}}{{C}_{2}}}{{{\alpha }_{1}}{{\alpha }_{2}}\cdot E{{C}_{50,2}}\cdot E{{C}_{50,2}}}\right)}^{k}}}$$

(8.46)

where ${{\alpha }_{i}}$ refers to the adaptation factor mentioned in Eq. (8.12). This empirical approach described well the combined effects of an aminoglycoside and a β-lactam against P. aeruginosa .

8.3.7 Models Estimating Resistant Subpopulation

To quantify the resistant bacterial subpopulation in an in vitro time-kill experiment, one can plate the bacteria in agar plates that contained the antimicrobial agent at three times the MIC or greater. This approach ensures that susceptible bacteria are removed by the drug. The choice for at least thrice the MIC is that twice the MIC level is still within the error measurement of susceptibility determination. An alternative method to determine resistance development is to determine the MIC after the 24-h time-kill experiment.

The subpopulation of resistant bacteria is a very small fraction of the total bacterial population that is predominantly drug-susceptible wild-type population. The probability of detecting the resistant subpopulation is dependent on the inoculum size or total bacterial burden and the frequency of mutation to the drug-specific resistance (Jumbe and Drusano 2011). For a resistant subpopulation to amplify effectively, both the fitness of the mutant and the selection pressure presented by the antimicrobial agent are important determining factors. Jumbe and Drusano proposed equations that incorporate probability estimates to a general model that governs the natural replication of bacteria and the death due to antimicrobial effect (Jumbe and Drusano 2011):

$$ \frac{dS}{dt}={{\zeta }_{G,s}}\cdot {{E}_{R,s}}[a(t ) ]\cdot (1-P )S(t )+{{\zeta }_{G,r}}\cdot {{E}_{R,r}}[a(t ) ]\cdot P\cdot \Gamma \cdot R(t )-{{\Psi }_{K,s}}\cdot {{E}_{D,s}}[a(t ) ]\cdot S(t) $$

(8.47)

$$ \frac{dR}{dt}={{\zeta }_{G,r}}\cdot {{E}_{R,r}}[a(t ) ]\cdot (1-P )\cdot \Gamma \cdot R(t )+{{\zeta }_{G,s}}\cdot {{E}_{R,s}}[a(t ) ]\cdot P\cdot S(t )-{{\Psi }_{K,r}}\cdot {{E}_{D,r}}[a(t ) ]\cdot R(t) $$

(8.48)

where $S$ and $R$ represent susceptible and resistant bacterial population, ${{\zeta }_{G}}$ and ${{\Psi }_{K}}$ are rates related to the natural replication and bacterial death, and ${{E}_{R}}[a(t ) ]$ and ${{E}_{D}}[a(t ) ]$ refer to antimicrobial effects on replication and death. $P$ is the probability related to mutation occurrence and $\Gamma$ determines the relative fitness of the susceptible to resistant population. This modeling strategy of estimating the proportion of each subpopulation was adopted in the model used to study colistin effect in P. aeruginosa that was previously discussed (Bulitta et al. 2010). Tam et al. (2005, 2007b) applied the PKPD model to describe the dynamics of garenoxacin-sensitive and -resistant subpopulations of P. aeruginosa and S. aureus in response to fluctuating concentrations of quinolone drugs. Their study showed that exposure below a specific breakpoint allowed resistant subpopulation to grow rapidly. Jumbe et al. (2003) showed the predictive value of modeling and simulation in determining the proliferation of resistant population in insufficient antimicrobial therapy.

8.3.8 Models Incorporating Host Defense

Rodent studies provide a good model to evaluate the effect of the host’s immune system on the time course of antimicrobial effect by chemotherapeutic agents, as well as the antimicrobial effect imposed by the immune system. The effect of the immune system can be quantified by comparing the immune-competent mice and the neutropenic mice. To evaluate the contribution of granulocytes on bacterial kill, studies performed in the mouse thigh-infection model and the murine model of pneumonia showed that granulocytes alone are potent in eradicating bacteria at a low inoculum size whereas for bacterial burden of ≥ 10⁷ CFU/g of tissue, a net bacterial growth was observed after 24 h (Drusano et al. 2010, 2011a, b). The model incorporating host defense has similar mathematical form as that of drug effects and combination study (Jumbe and Drusano 2011):

$$ \frac{dN}{dt}={{\zeta }_{G}}\cdot {{E}_{R}}[a(t ) ]\cdot N(t )-{{C}_{K}}\cdot {{E}_{D}}[a(t )+{{E}_{P}}+{{E}_{P}}\cdot {{E}_{I}}+{{E}_{I}}]\cdot N(t) $$

(8.49)

where ${{E}_{P}}$ and ${{E}_{I}}$ refer to the humoral and cellular response of the host and the innate adaptive immunity interaction is characterized by ${{E}_{P}}\cdot {{E}_{I}}$. When the combined effect of the two host’s defense processes is greater than the microorganism natural proliferation rate, ${{C}_{K}}\gg {{\zeta }_{G}}$, the host immunity can remove the infection without therapeutic intervention (Jumbe and Drusano 2011).

8.3.9 Linking In Vitro Models to PKPD Indices

The concentration–effect relationship of ceftazidime established from in vitro time-kill curves was used to explain the PKPD index that % fT > MIC of 40 % is required for a static effect in vivo (Mouton et al. 2007). The logistic-growth model was used to simulate the bacterial kill over time in dosing regimens of 1 mg every 2 h to 256 mg every 8 h. The pharmacokinetic profiles were simulated using parameters for mice and humans. The dosing regimens that resulted in a predicted static effect (i.e., CFU at 24 h ≤ CFU at 0 h) were then evaluated and the corresponding % fT > MIC for the dosing regimen was determined. For a 2 log₁₀ decrease after 24 h, the authors estimated that % fT > MIC of at least 50 % is required. Neilsen et al. (2011) used a semi-mechanistic PKPD model based on the compartment approach to predict the PKPD indices of several antibiotics. They have shown that simulation studies using the information from in vitro studies can be used to predict the PKPD indices of antimicrobial agents but cautioned that the determination of the suitable PKPD index for a particular drug is sensitive to the study conditions including dosing frequency as well as uncertainty in the MIC values.

8.4 Summary

Due to rapid evolution of bacterial resistance to many of the commercially available antimicrobial agents, many investigators have called for drug discovery and development programs that target suppression of resistance selection and eradication of drug-resistant infections (Jumbe and Drusano 2011; Nielsen and Friberg 2013). Pharmacometrics has an important role in developing dosing strategies to effectively achieve these two goals. The search for regimens and drug combinations in anti-infectives has benefitted tremendously from modeling and simulation. Many of the current dosing regimens were based on understanding of the PK–PD relationship between antimicrobial agents and bacterial infection.

In this chapter, we have summarized the pharmacometric models that were used to derive the current state-of-the-art treatment paradigm. More progress can still be made to maximize patients’ benefits through implementing treatment programs based on sound analysis of all available information from in vitro studies, animal models of infections, and clinical data.