A pharmacodynamic model for the activity of antibiotics against microorganisms under nonsaturable conditions

Tumor growth inhibition modeling in patients with second line biliary tract cancer and first line non-small cell lung cancer based on bintrafusp alfa trials

This analysis aimed to quantify tumor dynamics in patients receiving either bintrafusp alfa (BA) or pembrolizumab, by population pharmacokinetic (PK)-pharmacodynamic modeling, and investigate clinical and molecular covariates describing the variability in tumor dynamics by pharmacometric and machine-learning (ML) approaches. Data originated from two clinical trials in patients with biliary tract cancer (BTC; NCT03833661) receiving BA and non-small cell lung cancer (NSCLC; NCT03631706) receiving BA or pembrolizumab. Individual drug exposure was estimated from previously developed population PK models. Population tumor dynamics models were developed for each drug-indication combination, and covariate evaluations performed using nonlinear mixed-effects modeling (NLME) and ML (elastic net and random forest models) approaches. The three tumor dynamics' model structures all included linear tumor growth components and exponential tumor shrinkage. The final BTC model included the effect of drug exposure (area under the curve) and several covariates (demographics, disease-related, and genetic mutations). Drug exposure was not significant in either of the NSCLC models, which included two, disease-related, covariates in the BA arm, and none in the pembrolizumab arm. The covariates identified by univariable NLME and ML highly overlapped in BTC but showed less agreement in NSCLC analyses. Hyperprogression could be identified by higher tumor growth and lower tumor kill rates and could not be related to BA exposure. Tumor size over time was quantitatively characterized in two tumor types and under two treatments. Factors potentially related to tumor dynamics were assessed using NLME and ML approaches; however, their net impact on tumor size was considered as not clinically relevant.

Assessing the relative importance of bacterial resistance, persistence and hyper-mutation for antibiotic treatment failure

To curb the rising threat of antimicrobial resistance, we need to understand the routes to antimicrobial treatment failure. Bacteria can survive treatment by using both genetic and phenotypic mechanisms to diminish the effect of antimicrobials. We assemble empirical data showing that, for example, Pseudomonas aeruginosa infections frequently contain persisters, transiently non-growing cells unaffected by antibiotics (AB) and hyper-mutators, mutants with elevated mutation rates, and thus higher probability of genetic resistance emergence. Resistance, persistence and hyper-mutation dynamics are difficult to disentangle experimentally. Hence, we use stochastic population modelling and deterministic fitness calculations to investigate the relative importance of genetic and phenotypic mechanisms for immediate treatment failure and establishment of prolonged, chronic infections. We find that persistence causes 'hidden' treatment failure with very low cell numbers if antimicrobial concentrations prevent growth of genetically resistant cells. Persister cells can regrow after treatment is discontinued and allow for resistance evolution in the absence of AB. This leads to different mutational routes during treatment and relapse of an infection. By contrast, hyper-mutation facilitates resistance evolution during treatment, but rarely contributes to treatment failure. Our findings highlight the time and concentration dependence of different bacterial mechanisms to escape AB killing, which should be considered when designing 'failure-proof' treatments.

Mathematical Modeling for an MTT Assay in Fluorine-Containing Graphene Quantum Dots

The paper reports on a new mathematical model, starting with the original Hill equation which is derived to describe cell viability (V) while testing nanomaterials (NMs). Key information on the sample's morphology, such as mean size (⟨s⟩) and size dispersity (σ) is included in the new model via the lognormal distribution function. The new Hill-inspired equation is successfully used to fit MTT (3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide) data from assays performed with the HepG2 cell line challenged by fluorine-containing graphene quantum dots (F:GQDs) under light (400-700 nm wavelength) and dark conditions. The extracted "biological polydispersity" (light: ⟨sMTT⟩=1.77±0.02 nm and σMTT=0.21±0.02); dark: ⟨sMTT⟩=1.87±0.02 nm and σMTT=0.22±0.01) is compared with the "morphological polydispersity" (⟨sTEM⟩=1.98±0.06 nm and σTEM=0.19±0.03), the latter obtained from TEM (transmission electron microscopy). The fitted data are then used to simulate a series of V responses. Two aspects are emphasized in the simulations: (i) fixing σ, one simulates V versus ⟨s⟩ and (ii) fixing ⟨s⟩, one simulates V versus σ. Trends observed in the simulations are supported by a phenomenological model picture describing the monotonic reduction in V as ⟨s⟩ increases (V~pa/(s)p-a; p and a are fitting parameters) and accounting for two opposite trends of V versus σ: under light (V~σ) and under dark (V~1/σ).

Bacteria primed by antimicrobial peptides develop tolerance and persist

Antimicrobial peptides (AMPs) are key components of innate immune defenses. Because of the antibiotic crisis, AMPs have also come into focus as new drugs. Here, we explore whether prior exposure to sub-lethal doses of AMPs increases bacterial survival and abets the evolution of resistance. We show that Escherichia coli primed by sub-lethal doses of AMPs develop tolerance and increase persistence by producing curli or colanic acid, responses linked to biofilm formation. We develop a population dynamic model that predicts that priming delays the clearance of infections and fuels the evolution of resistance. The effects we describe should apply to many AMPs and other drugs that target the cell surface. The optimal strategy to tackle tolerant or persistent cells requires high concentrations of AMPs and fast and long-lasting expression. Our findings also offer a new understanding of non-inherited drug resistance as an adaptive response and could lead to measures that slow the evolution of resistance.

Animals and plants defend themselves with ancient molecules called antimicrobial peptides (AMPs) against pathogens. As more and more bacterial diseases have become drug resistant, these AMPs are considered as promising alternatives. In natural situation such as on the skin, bacteria are often exposed to low concentrations of AMPs that do no kill. Here we show that the bacterium Escherichia coli when exposed to such low concentrations becomes recalcitrant to killing concentrations of the same AMPs. We report the ways in which the bacteria alter their surface to do so. We then use a mathematical model to show that these effects caused by low concentrations can drive the evolution of resistance. From the perspective of an organism using AMPs in self-defense, the best option is to deploy high concentrations of AMPs for long. Our findings also offer a new understanding of similar drug resistance mechanisms.

Definitions and guidelines for research on antibiotic persistence

Increasing concerns about the rising rates of antibiotic therapy failure and advances in single-cell analyses have inspired a surge of research into antibiotic persistence. Bacterial persister cells represent a subpopulation of cells that can survive intensive antibiotic treatment without being resistant. Several approaches have emerged to define and measure persistence, and it is now time to agree on the basic definition of persistence and its relation to the other mechanisms by which bacteria survive exposure to bactericidal antibiotic treatments, such as antibiotic resistance, heteroresistance or tolerance. In this Consensus Statement, we provide definitions of persistence phenomena, distinguish between triggered and spontaneous persistence and provide a guide to measuring persistence. Antibiotic persistence is not only an interesting example of non-genetic single-cell heterogeneity, it may also have a role in the failure of antibiotic treatments. Therefore, it is our hope that the guidelines outlined in this article will pave the way for better characterization of antibiotic persistence and for understanding its relevance to clinical outcomes.

Antibiotic persistence contributes to the survival of bacteria during antibiotic treatment. In this Consensus Statement, scientists working on the response of bacteria to antibiotics define antibiotic persistence and provide practical guidance on how to study bacterial persisters.

Antimicrobial combinations: Bliss independence and Loewe additivity derived from mechanistic multi-hit models

Antimicrobial peptides (AMPs) and antibiotics reduce the net growth rate of bacterial populations they target. It is relevant to understand if effects of multiple antimicrobials are synergistic or antagonistic, in particular for AMP responses, because naturally occurring responses involve multiple AMPs. There are several competing proposals describing how multiple types of antimicrobials add up when applied in combination, such as Loewe additivity or Bliss independence. These additivity terms are defined ad hoc from abstract principles explaining the supposed interaction between the antimicrobials. Here, we link these ad hoc combination terms to a mathematical model that represents the dynamics of antimicrobial molecules hitting targets on bacterial cells. In this multi-hit model, bacteria are killed when a certain number of targets are hit by antimicrobials. Using this bottom-up approach reveals that Bliss independence should be the model of choice if no interaction between antimicrobial molecules is expected. Loewe additivity, on the other hand, describes scenarios in which antimicrobials affect the same components of the cell, i.e. are not acting independently. While our approach idealizes the dynamics of antimicrobials, it provides a conceptual underpinning of the additivity terms. The choice of the additivity term is essential to determine synergy or antagonism of antimicrobials.This article is part of the themed issue 'Evolutionary ecology of arthropod antimicrobial peptides'.

Combination Effects of Antimicrobial Peptides

Antimicrobial peptides (AMPs) are ancient and conserved across the tree of life. Their efficacy over evolutionary time has been largely attributed to their mechanisms of killing. Yet, the understanding of their pharmacodynamics both in vivo and in vitro is very limited. This is, however, crucial for applications of AMPs as drugs and also informs the understanding of the action of AMPs in natural immune systems. Here, we selected six different AMPs from different organisms to test their individual and combined effects in vitro. We analyzed their pharmacodynamics based on the Hill function and evaluated the interaction of combinations of two and three AMPs. Interactions of AMPs in our study were mostly synergistic, and three-AMP combinations displayed stronger synergism than two-AMP combinations. This suggests synergism to be a common phenomenon in AMP interaction. Additionally, AMPs displayed a sharp increase in killing within a narrow dose range, contrasting with those of antibiotics. We suggest that our results could lead a way toward better evaluation of AMP application in practice and shed some light on the evolutionary consequences of antimicrobial peptide interactions within the immune system of organisms.

A mathematical model for antibiotic control of bacteria in peritoneal dialysis associated peritonitis

A study of the process of pharmacokinetics-pharmacodynamics (PKPD) of antibiotics and their interaction with bacteria during peritoneal dialysis associated peritonitis (PDAP) is presented. We propose a mathematical model describing the evolution of bacteria population in the presence of antibiotics for different peritoneal dialysis regimens. Using the model along with experimental data, clinical parameters, and physiological values, we compute variations in PD fluid distributions, drug concentrations, and number of bacteria in peritoneal and extra-peritoneal cavities. Scheduling algorithms for the PD exchanges that minimize bacteria count are investigated.

Pharmacokinetic-pharmacodynamic modeling of antibacterial drugs

Pharmacokinetic-pharmacodynamic (PKPD) modeling and simulation has evolved as an important tool for rational drug development and drug use, where developed models characterize both the typical trends in the data and quantify the variability in relationships between dose, concentration, and desired effects and side effects. In parallel, rapid emergence of antibiotic-resistant bacteria imposes new challenges on modern health care. Models that can characterize bacterial growth, bacterial killing by antibiotics and immune system, and selection of resistance can provide valuable information on the interactions between antibiotics, bacteria, and host. Simulations from developed models allow for outcome predictions of untested scenarios, improved study designs, and optimized dosing regimens. Today, much quantitative information on antibiotic PKPD is thrown away by summarizing data into variables with limited possibilities for extrapolation to different dosing regimens and study populations. In vitro studies allow for flexible study designs and valuable information on time courses of antibiotic drug action. Such experiments have formed the basis for development of a variety of PKPD models that primarily differ in how antibiotic drug exposure induces amplification of resistant bacteria. The models have shown promise for efficacy predictions in patients, but few PKPD models describe time courses of antibiotic drug effects in animals and patients. We promote more extensive use of modeling and simulation to speed up development of new antibiotics and promising antibiotic drug combinations. This review summarizes the value of PKPD modeling and provides an overview of the characteristics of available PKPD models of antibiotics based on in vitro, animal, and patient data.

Novel in vitro and mathematical models for the prediction of chemical toxicity

The focus of much scientific and medical research is directed towards understanding the disease process and defining therapeutic intervention strategies. The scientific basis of drug safety is very complex and currently remains poorly understood, despite the fact that adverse drug reactions (ADRs) are a major health concern and a serious impediment to development of new medicines. Toxicity issues account for ∼21% drug attrition during drug development and safety testing strategies require considerable animal use. Mechanistic relationships between drug plasma levels and molecular/cellular events that culminate in whole organ toxicity underpins development of novel safety assessment strategies. Current in vitro test systems are poorly predictive of toxicity of chemicals entering the systemic circulation, particularly to the liver. Such systems fall short because of (1) the physiological gap between cells currently used and human hepatocytes existing in their native state, (2) the lack of physiological integration with other cells/systems within organs, required to amplify the initial toxicological lesion into overt toxicity, (3) the inability to assess how low level cell damage induced by chemicals may develop into overt organ toxicity in a minority of patients, (4) lack of consideration of systemic effects. Reproduction of centrilobular and periportal hepatocyte phenotypes in in vitro culture is crucial for sensitive detection of cellular stress. Hepatocyte metabolism/phenotype is dependent on cell position along the liver lobule, with corresponding differences in exposure to substrate, oxygen and hormone gradients. Application of bioartificial liver (BAL) technology can encompass in vitro predictive toxicity testing with enhanced sensitivity and improved mechanistic understanding. Combining this technology with mechanistic mathematical models describing intracellular metabolism, fluid-flow, substrate, hormone and nutrient distribution provides the opportunity to design the BAL specifically to mimic the in vivo scenario. Such mathematical models enable theoretical hypothesis testing, will inform the design of in vitro experiments, and will enable both refinement and reduction of in vivo animal trials. In this way, development of novel mathematical modelling tools will help to focus and direct in vitro and in vivo research, and can be used as a framework for other areas of drug safety science.

Pharmacokinetic/pharmacodynamic (PK/PD) indices of antibiotics predicted by a semimechanistic PKPD model: a step toward model-based dose optimization

A pharmacokinetic-pharmacodynamic (PKPD) model that characterizes the full time course of in vitro time-kill curve experiments of antibacterial drugs was here evaluated in its capacity to predict the previously determined PK/PD indices. Six drugs (benzylpenicillin, cefuroxime, erythromycin, gentamicin, moxifloxacin, and vancomycin), representing a broad selection of mechanisms of action and PK and PD characteristics, were investigated. For each drug, a dose fractionation study was simulated, using a wide range of total daily doses given as intermittent doses (dosing intervals of 4, 8, 12, or 24 h) or as a constant drug exposure. The time course of the drug concentration (PK model) as well as the bacterial response to drug exposure (in vitro PKPD model) was predicted. Nonlinear least-squares regression analyses determined the PK/PD index (the maximal unbound drug concentration [fC(max)]/MIC, the area under the unbound drug concentration-time curve [fAUC]/MIC, or the percentage of a 24-h time period that the unbound drug concentration exceeds the MIC [fT(>MIC)]) that was most predictive of the effect. The in silico predictions based on the in vitro PKPD model identified the previously determined PK/PD indices, with fT(>MIC) being the best predictor of the effect for β-lactams and fAUC/MIC being the best predictor for the four remaining evaluated drugs. The selection and magnitude of the PK/PD index were, however, shown to be sensitive to differences in PK in subpopulations, uncertainty in MICs, and investigated dosing intervals. In comparison with the use of the PK/PD indices, a model-based approach, where the full time course of effect can be predicted, has a lower sensitivity to study design and allows for PK differences in subpopulations to be considered directly. This study supports the use of PKPD models built from in vitro time-kill curves in the development of optimal dosing regimens for antibacterial drugs.

In vitro pharmacodynamic models to determine the effect of antibacterial drugs

In vitro pharmacodynamic (PD) models are used to obtain useful quantitative information on the effect of either single drugs or drug combinations against bacteria. This review provides an overview of in vitro PD models and their experimental implementation. Models are categorized on the basis of whether the drug concentration remains constant or changes and whether there is a loss of bacteria from the system. Further subdifferentiation is based on whether bacterial loss involves dilution of the medium or is associated with dialysis or diffusion. For comprehension of the underlying principles, experimental settings are simplified and schematically illustrated, including the simulations of various in vivo routes of administration. The different model types are categorized and their (dis)advantages discussed. The application of in vitro models to special organs, infections and pathogens is comprehensively presented. Finally, the relevance and perspectives of in vitro investigations in drug discovery and clinical research are elucidated and discussed.

A simple in vitro PK/PD model system to determine time-kill curves of drugs against Mycobacteria

In vivo tuberculosis is exposed to continually changing drug concentrations for which static minimum inhibitory concentration (MIC) testing may be a poor surrogate. While in vitro approaches to determine time-kill curves for antibiotics have been widely applied in assessing antimicrobial activity against fast growing microorganisms, their availability and application for slow-growing microorganisms including Mycobacterium tuberculosis has so far been scarce. Thus, we developed a novel simple in vitro pharmacokinetic/pharmacodynamic (PK/PD) model for establishing time-kill curves and applied it for evaluating the antimicrobial activity of different dosing regimens of isoniazid (INH) against Mycobacterium bovis BCG as a surrogate for virulent M. tuberculosis. In the in vitro model M. bovis BCG was exposed to INH concentration-time profiles as usually encountered during multiple dose therapy with 25, 100 and 300mg/day in humans who are fast or slow INH metabolizers. Bacterial killing was followed over time by determining viable counts and the resulting time-kill data was analyzed using a semi-mechanistic PK/PD model with an adaptive IC(50) function to describe the emergence of insensitive populations of bacteria over the course of treatment. In agreement with previous studies, the time-kill data suggest that AUC(0-24)/MIC is the PK/PD index that is the most explanatory of the antimicrobial effect of INH. The presented in vitro PK/PD model and associated modeling approach were able to characterize the time-kill kinetics of INH in M. bovis BCG, and may in general serve as a potentially valuable, low cost tool for the assessment of antibacterial activity in slow-growing organisms in drug development and applied pharmacotherapy.

Pharmacokinetics and pharmacodynamics of antimicrobial drugs

BACKGROUND

Antimicrobial drugs exhibit different characteristics in their correlation between antimicrobial drug concentrations and effects on microorganisms. These correlations have been studied using different approaches including in vitro analyses with constant and fluctuating concentrations and in vivo analyses involving animals and humans. Mathematical analysis includes correlation of pharmacokinetic-pharmacodynamic (PK-PD) indices to an outcome parameter. Further insight can be gained by mechanism-based modelling of antimicrobial drug effects.

METHODS AND RESULTS

This review aims to provide an overview on the various approaches used to analyse antimicrobial pharmacodynamics, to discuss the limitations of these approaches, to indicate recent developments and to summarise the current knowledge on PK-PD target values as derived from human studies.

CONCLUSION

It is expected that PK-PD analysis of antimicrobial drug effects will lead to a more efficient and possibly also less toxic antimicrobial drug therapy.

Antimicrobial breakpoint estimation accounting for variability in pharmacokinetics

Background

Pharmacokinetic and pharmacodynamic (PK/PD) indices are increasingly being used in the microbiological field to assess the efficacy of a dosing regimen. In contrast to methods using MIC, PK/PD-based methods reflect in vivo conditions and are more predictive of efficacy. Unfortunately, they entail the use of one PK-derived value such as AUC or Cmax and may thus lead to biased efficiency information when the variability is large. The aim of the present work was to evaluate the efficacy of a treatment by adjusting classical breakpoint estimation methods to the situation of variable PK profiles.

Methods and results

We propose a logical generalisation of the usual AUC methods by introducing the concept of "efficiency" for a PK profile, which involves the efficacy function as a weight. We formulated these methods for both classes of concentration- and time-dependent antibiotics. Using drug models and in silico approaches, we provide a theoretical basis for characterizing the efficiency of a PK profile under in vivo conditions. We also used the particular case of variable drug intake to assess the effect of the variable PK profiles generated and to analyse the implications for breakpoint estimation.

Conclusion

Compared to traditional methods, our weighted AUC approach gives a more powerful PK/PD link and reveals, through examples, interesting issues about the uniqueness of therapeutic outcome indices and antibiotic resistance problems.

Mechanism-based pharmacokinetic–pharmacodynamic modeling of antimicrobial drug effects

Mathematical modeling of drug effects maximizes the information gained from an experiment, provides further insight into the mechanisms of drug effects, and allows for simulations in order to design studies or even to derive clinical treatment strategies. We reviewed modeling of antimicrobial drug effects and show that most of the published mathematical models can be derived from one common mechanism-based PK-PD model premised on cell growth and cell killing processes. The general sigmoid Emax model applies to cell killing and the various parameters can be related to common pharmacodynamics, which enabled us to synthesize and compare the different parameter estimates for a total of 24 antimicrobial drugs from published literature. Furthermore, the common model allows the parameters of these models to be related to the MIC and to a common set of PK-PD indices. Theoretically, a high Hill coefficient and a low maximum kill rate indicate so-called time-dependent antimicrobial effects, whereas a low Hill coefficient and a high maximum kill rate indicate so-called concentration-dependent effects, as illustrated in the garenoxacin and meropenem examples. Finally, a new equation predicting the time to microorganism eradication after repeated drug doses was derived that is based on the area under the kill-rate curve.

Concentration-effect relationship of ceftazidime explains why the time above the MIC is 40 percent for a static effect in vivo

Growth-kill dynamics were characterized in vitro, and the parameter estimates were used to simulate bacterial growth and kill in vivo using both mouse and human pharmacokinetics. The parameter estimates obtained in vitro predicted a time above the MIC of between 35 and 38% for a static effect in mice after 24 h of treatment.

Pharmacodynamics of non-replicating viruses, bacteriocins and lysins

The pharmacodynamics of antibiotics and many other chemotherapeutic agents is often governed by a 'multi-hit' kinetics, which requires the binding of several molecules of the therapeutic agent for the killing of their targets. In contrast, the pharmacodynamics of novel alternative therapeutic agents, such as phages and bacteriocins against bacterial infections or viruses engineered to target tumour cells, is governed by a 'single-hit' kinetics according to which the agent will kill once it is bound to its target. In addition to requiring only a single molecule for killing, these agents bind irreversibly to their targets. Here, we explore the pharmacodynamics of such 'irreversible, single-hit inhibitors' using mathematical models. We focus on agents that do not replicate, i.e. in the case of phage therapy, we deal only with non-lytic phages and in the case of cancer treatment, we restrict our analysis to replication of incompetent viruses. We study the impact of adsorption on dead cells, heterogeneity in adsorption rates and spatial compartmentalization.

Semimechanistic pharmacokinetic/pharmacodynamic model for assessment of activity of antibacterial agents from time-kill curve experiments

Dosing of antibacterial agents is generally based on point estimates of the effect, even though bacteria exposed to antibiotics show complex kinetic behaviors. The use of the whole time course of the observed effects would be more advantageous. The aim of the present study was to develop a semimechanistic pharmacokinetic (PK)/pharmacodynamic (PD) model characterizing the events seen in a bacterial system when it is exposed to antibacterial agents with different mechanisms of action. Time-kill curve experiments were performed with a strain of Streptococcus pyogenes exposed to a wide range of concentrations of the following antibiotics: benzylpenicillin, cefuroxime, erythromycin, moxifloxacin, and vancomycin. Bacterial counts were monitored with frequent sampling during the experiment. A simultaneous fit of all data was accomplished. The degradation of the drugs was monitored and corrected for in the model, and a link model was used to account for an effect delay. In the final PK/PD model, the total bacterial population was divided into two subpopulations: one growing drug-susceptible population and one resting insusceptible population. The drug effect was included as an increase of the killing rate of bacteria in the susceptible state, according to a maximum-effect (E(max)) model. An internal model validation showed that the model was robust and had good predictability. In conclusion, for all drugs, the final PK/PD model successfully described bacterial growth and killing kinetics when the bacteria were exposed to different antibiotic concentrations. The semimechanistic model that was developed might, after further refinement, serve as a tool for the development of optimal dosing strategies for antibacterial agents.

Modelling time–kill studies to discern the pharmacodynamics of meropenem

OBJECTIVES

Time-kill studies are commonly used in investigations of new antimicrobial agents. However, they typically provide descriptive information on pharmacodynamics. We developed a mathematical model to capture the relationship between microbial burden and antimicrobial agent concentrations.

METHODS

Time-kill studies were performed with 10(8) cfu/mL of Pseudomonas aeruginosa at baseline. Meropenem at 0, 0.25, 1, 4, 16 and 64 x MIC was used (MIC = 1 mg/L). Serial samples were obtained to quantify bacterial burden over 24 h. The data were analysed by a population analysis using the non-parametric adaptive grid program. The rate of change of bacteria over time was expressed as the difference between linear bacterial growth rate and sigmoidal kill rate. Regrowth was attributed to adaptation, which was explicitly modelled as increase in C(50k) (concentration to achieve 50% maximal kill rate), using a saturable function of selective pressure (both meropenem concentration and time).

RESULTS

The best-fit model consisted of eight parameters and the fit to the data was satisfactory. The r2 of maximum a-posteriori probability Bayesian predictions based on the mean parameter estimates was 0.984. Maximal killing rate at baseline was found to be 4.7 h(-1); C(90k) was achieved with meropenem at 5.0 mg/L. The model was validated by time-kill studies using 2x and 32x MIC of meropenem.

CONCLUSIONS

Our model reasonably described and predicted the time course of P. aeruginosa in time-kill studies, and provided quantitative information on the pharmacodynamics of meropenem. The structural model appeared robust and could be used to provide a realistic expectation of the killing performance of antimicrobial agents.

Principles of pharmacodynamics and their applications in veterinary pharmacology

Pharmacodynamics (PDs) is the science of drug action on the body or on microorganisms and other parasites within or on the body. It may be studied at many organizational levels--sub-molecular, molecular, cellular, tissue/organ and whole body--using in vivo, ex vivo and in vitro methods and utilizing a wide range of techniques. A few drugs owe their PD properties to some physico-chemical property or action and, in such cases, detailed molecular drug structure plays little or no role in the response elicited. For the great majority of drugs, however, action on the body is crucially dependent on chemical structure, so that a very small change, e.g. substitution of a proton by a methyl group, can markedly alter the potency of the drug, even to the point of loss of activity. In the late 19th century and first half of the 20th century recognition of these facts by Langley, Ehrlich, Dale, Clarke and others provided the foundation for the receptor site hypothesis of drug action. According to these early ideas the drug, in order to elicit its effect, had to first combine with a specific 'target molecule' on either the cell surface or an intracellular organelle. It was soon realized that the 'right' chemical structure was required for drug-target site interaction (and the subsequent pharmacological response). In addition, from this requirement, for specificity of chemical structure requirement, developed not only the modern science of pharmacology but also that of toxicology. In relation to drug actions on microbes and parasites, for example, the early work of Ehrlich led to the introduction of molecules selectively toxic for them and relatively safe for the animal host. In the whole animal drugs may act on many target molecules in many tissues. These actions may lead to primary responses which, in turn, may induce secondary responses, that may either enhance or diminish the primary response. Therefore, it is common to investigate drug pharmacodynamics (PDs) in the first instance at molecular, cellular and tissue levels in vitro, so that the primary effects can be better understood without interference from the complexities involved in whole animal studies. When a drug, hormone or neurotransmitter combines with a target molecule, it is described as a ligand. Ligands are classified into two groups, agonists (which initiate a chain of reactions leading, usually via the release or formation of secondary messengers, to the response) and antagonists (which fail to initiate the transduction pathways but nevertheless compete with agonists for occupancy of receptor sites and thereby inhibit their actions). The parameters which characterize drug receptor interaction are affinity, efficacy, potency and sensitivity, each of which can be elucidated quantitatively for a particular drug acting on a particular receptor in a particular tissue. The most fundamental objective of PDs is to use the derived numerical values for these parameters to classify and sub-classify receptors and to compare and classify drugs on the basis of their affinity, efficacy, potency and sensitivity. This review introduces and summarizes the principles of PDs and illustrates them with examples drawn from both basic and veterinary pharmacology. Drugs acting on adrenoceptors and cardiovascular, non-steroidal anti-inflammatory and antimicrobial drugs are considered briefly to provide a foundation for subsequent reviews in this issue which deal with pharmacokinetic (PK)-PD modelling and integration of these drug classes. Drug action on receptors has many features in common with enzyme kinetics and gas adsorption onto surfaces, as defined by Michaelis-Menten and Langmuir absorption equations, respectively. These and other derived equations are outlined in this review. There is, however, no single theory which adequately explains all aspects of drug-receptor interaction. The early 'occupation' and 'rate' theories each explain some, but not all, experimental observations. From these basic theories the operational model and the two-state theory have been developed. For a discussion of more advanced theories see Kenakin (1997).

Pharmacokinetic/Pharmacodynamic Modelling of Antibacterials In Vitro and In Vivo Using Bacterial Growth and Kill Kinetics

BACKGROUND

The minimum inhibitory concentration (MIC) is the in vitro reference value to describe the activity of an antibacterial against micro-organisms. It does not represent the dynamic effect of the antimicrobial at any point in time, but rather the total antimicrobial effect over the incubation period at a fixed concentration.

OBJECTIVE

To explore the concentration-effect relationship of antimicrobial concentrations against micro-organisms in relation to the MIC.

METHODS

Time-kill curves were generated for ceftazidime, meropenem and tobramycin against Pseudomonas aeruginosa. The Hill equation with variable slope was fit to the time-kill data, and mathematical models of growth and kill were explored with reference to the MIC.

RESULTS

With declining concentrations, bacterial killing will decrease until a specific threshold concentration is reached. This concentration, at which bacteria are neither killed nor able to grow, is named the stationary concentration (SC) and is not equal to the MIC. Pharmacokinetic/pharmacodynamic simulations over a range of kill rates, growth rates and slope factors showed that for beta-lactam antibacterials, the SC is close to the MIC value, which may explain why concentrations in vivo need to be above the MIC, while regrowth of bacteria occurs when concentrations decline below the MIC. For concentration-dependent antibacterials, such as aminoglycosides and quinolones, the SC is shown to be markedly different from the MIC and, in general, is much lower.

CONCLUSION

The MIC is not a good pharmacodynamic parameter to characterise the concentration effect relationship of a given antimicrobial. For 'concentration independent' antimicrobials the SC is likely to be close to the MIC, but may be much lower for 'concentration dependent' antimicrobials, and may explain sub-MIC effects.

Pharmacodynamic functions: a multiparameter approach to the design of antibiotic treatment regimens

There is a complex quantitative relationship between the concentrations of antibiotics and the growth and death rates of bacteria. Despite this complexity, in most cases only a single pharmacodynamic parameter, the MIC of the drug, is employed for the rational development of antibiotic treatment regimens. In this report, we use a mathematical model based on a Hill function-which we call the pharmacodynamic function and which is related to previously published E(max) models-to describe the relationship between the bacterial net growth rates and the concentrations of antibiotics of five different classes: ampicillin, ciprofloxacin, tetracycline, streptomycin, and rifampin. Using Escherichia coli O18:K1:H7, we illustrate how precise estimates of the four parameters of the pharmacodynamic function can be obtained from in vitro time-kill data. We show that, in addition to their respective MICs, these antibiotics differ in the values of the other pharmacodynamic parameters. Using a computer simulation of antibiotic treatment in vivo, we demonstrate that, as a consequence of differences in pharmacodynamic parameters, such as the steepness of the Hill function and the minimum bacterial net growth rate attained at high antibiotic concentrations, there can be profound differences in the microbiological efficacy of antibiotics with identical MICs. We discuss the clinical implications and limitations of these results.

Aminoglycoside nephrotoxicity: modeling, simulation, and control

The main constraints on the administration of aminoglycosides are the risks of nephrotoxicity and ototoxicity, which can lead to acute, renal, vestibular, and auditory toxicities. In the present study we focused on nephrotoxicity. No reliable predictor of nephrotoxicity has been found to date. We have developed a deterministic model which describes the pharmacokinetic behavior of aminoglycosides (with a two-compartment model), the kinetics of aminoglycoside accumulation in the renal cortex, the effects of aminoglycosides on renal cells, the resulting effects on renal function by tubuloglomerular feedback, and the resulting effects on serum creatinine concentrations. The pharmacokinetic parameter values were estimated by use of the NPEM program. The estimated pharmacodynamic parameter values were obtained after minimization of the least-squares objective function between the measured and the calculated serum creatinine concentrations. A simulation program assessed the influences of the dosage regimens on the occurrence of nephrotoxicity. We have also demonstrated the relevancy of modeling of the circadian rhythm of the renal function. We have shown the ability of the model to fit with 49 observed serum creatinine concentrations for a group of eight patients treated for endocarditis by comparison with 49 calculated serum creatinine concentrations (r(2) = 0.988; P < 0.001). We have found that for the same daily dose, the nephrotoxicity observed with a thrice-daily administration schedule appears more rapidly, induces a greater decrease in renal function, and is more prolonged than those that occur with less frequent administration schedules (for example, once-daily administration). Moreover, for once-daily administration, we have demonstrated that the time of day of administration can influence the incidence of aminoglycoside nephrotoxicity. The lowest level of nephrotoxicity was observed when aminoglycosides were administered at 1:30 p.m. Clinical application of this model might make it possible to adjust aminoglycoside dosage regimens by taking into account both the efficacies and toxicities of the drugs.

The application of population pharmacokinetic modeling to individualized antibiotic therapy

This paper describes applications of population pharmacokinetic modeling to the optimization of antibiotic dosing. Parametric and nonparametric pharmacokinetic modeling approaches are discussed. Population models can be important extensions of therapeutic drug monitoring (TDM) in infectious disease. The concept of population model-based individualized antimicrobial therapy is described. With the availability of population modeling for obtaining PK parameter estimates, the focus has shifted to quantifying the antimicrobial effect and linking kinetics to drug effects. Examples of integrated pharmacokinetic-pharmacodynamic (PK-PD) models to describe bacterial killing as a function of drug concentration are discussed. Application of PK-PD mathematical models that correlate with microbiological and clinical outcomes will provide us with a better rationale for the proper dose selection of anti-infective therapy in different patient populations.

Pharmacokinetic/pharmacodynamic integration in drug development and dosage-regimen optimization for veterinary medicine

Pharmacokinetic (PK)/pharmacodynamic (PD) modeling is a scientific tool to help developers select a rational dosage regimen for confirmatory clinical testing. This article describes some of the limitations associated with traditional dose-titration designs (parallel and crossover designs) for determining an appropriate dosage regimen. It also explains how a PK/PD model integrates the PK model (describing the relationship between dose, systemic drug concentrations, and time) with the PD model (describing the relationship between systemic drug concentration and the effect vs time profile) and a statistical model (particularly, the intra- and interindividual variability of PK and/or PD origin). Of equal importance is the utility of these models for promoting rational drug selection on the basis of effectiveness and selectivity. PK/PD modeling can be executed using various approaches, such as direct versus indirect response models and parametric versus nonparametric models. PK/PD concepts can be applied to individual dose optimization. Examples of the application of PK/PD approaches in veterinary drug development are provided, with particular emphasis given to nonsteroidal anti-inflammatory drugs. The limits of PK/PD approaches include the development of appropriate models, the validity of surrogate endpoints, and the acceptance of these models in a regulatory environment.

Antibiotic efficacy in vivo predicted by in vitro activity

A brief overview of arguments found in the literature is presented to apply the E(max) concept to experimental studies of antibiotics as well as to their clinical application. It may turn out to be more flexible than schedules based on arbitrary parameters that have the disadvantage that they have to be proven in each individual situation.

Parameters of bacterial killing and regrowth kinetics and antimicrobial effect examined in terms of area under the concentration-time curve relationships: action of ciprofloxacin against Escherichia coli in an in vitro dynamic model

Although many parameters have been described to quantitate the killing and regrowth of bacteria, substantial shortcomings are inherent in most of them, such as low sensitivity to pharmacokinetic determinants of the antimicrobial effect, an inability to predict a total effect, insufficient robustness, and uncertain interrelations between the parameters that prevent an ultimate determination of the effect. To examine different parameters, the kinetics of killing and regrowth of Escherichia coli (MIC, 0.013 microg/ml) were studied in vitro by simulating a series of ciprofloxacin monoexponential pharmacokinetic profiles. Initial ciprofloxacin concentrations varied from 0.02 to 19.2 microg/ml, whereas the half-life of 4 h was the same in all experiments. The following parameters were calculated and estimated: the time to reduce the initial inoculum (N0) 10-, 100-, and 1,000-fold (T90%, T99%, and T99.9%, respectively), the rate constant of bacterial elimination (k(elb)), the nadir level (Nmin) in the viable count (N)-versus-time (t) curve, the time to reach Nmin (t(min)), the numbers of bacteria that survived (Ntau) by the end of the observation period (tau), the area under the bacterial killing and regrowth curve (log N(A)-t curve) from the zero point (time zero) to tau (AUBC), the area above this curve (AAC), the area between the control growth curve (log N(C)-t curve) and the bacterial killing and regrowth curve (log N(A)-t curve) from the zero point to tau (ABBC) or to the time point when log N(A) reaches the maximal values observed in the log N(C)-t curve (I(E); intensity of the effect), and the time shift between the control growth and regrowth curves (T(E); duration of the effect). Being highly sensitive to the AUC, I(E), and T(E) showed the most regular AUC relationships: the effect expressed by I(E) or T(E) increased systematically when the AUC or initial concentration of ciprofloxacin rose. Other parameters, especially T90%, T99%, T99.9%, t(min), and log N0 - log Nmin = delta log Nmin, related to the AUC less regularly and were poorly sensitive to the AUC. T(E) proved to be the best predictor and t(min) proved to be the worst predictor of the total antimicrobial effect reflected by I(E). Distinct feedback relationships between the effect determination and the experimental design were demonstrated. It was shown that unjustified shortening of the observation period, i.e., cutting off the log N(A)-t curves, may lead to the degeneration of the AUC-response relationships, as expressed by log N0 - log Ntau = delta log Ntau, AUBC, AAC, or ABBC, to a point where it gives rise to the false idea of an AUC- or concentration-independent effect. Thus, use of I(E) and T(E) provides the most unbiased, robust, and comprehensive means of determining the antimicrobial effect.

Pharmacokinetic-pharmacodynamic modeling of activity of ceftazidime during continuous and intermittent infusion

We developed and applied pharmacokinetic-pharmacodynamic (PK-PD) models to characterize in vitro bacterial rate of killing as a function of ceftazidime concentrations over time. For PK-PD modeling, data obtained during continuous and intermittent infusion of ceftazidime in Pseudomonas aeruginosa killing experiments with an in vitro pharmacokinetic model were used. The basic PK-PD model was a maximum-effect model which described the number of viable bacteria (N) as a function of the growth rate (lambda) and killing rate (epsilon) according to the equation dN/dt = [lambda - epsilon x [Cgamma(EC50gamma + Cgamma)]] N, where gamma is the Hill factor, C is the concentration of antibiotic, and EC50 is the concentration of antibiotic at which 50% of the maximum effect is obtained. Next, four different models with increasing complexity were analyzed by using the EDSIM program (MediWare, Groningen, The Netherlands). These models incorporated either an adaptation rate factor and a maximum number of bacteria (Nmax) factor or combinations of the two parameters. In addition, a two-population model was evaluated. Model discrimination was by Akaike's information criterion. The experimental data were best described by the model which included an Nmax term and a rate term for adaptation for a period up to 36 h. The absolute values for maximal growth rate and killing rate in this model were different from those in the original experiment, but net growth rates were comparable. It is concluded that the derived models can describe bacterial growth and killing in the presence of antibiotic concentrations mimicking human pharmacokinetics. Application of these models will eventually provide us with parameters which can be used for further dosage optimization.

Simultaneous pharmacodynamic analysis of the lag and bactericidal phases exhibited by beta-lactams against Escherichia coli

Antibiotic-bacterium interactions are complex in nature. In many cases, bacterial killing does not commence immediately after the addition of an antibiotic, and a lag period is observed. Antibiotic permeation and/or the intermediate steps that exist between antibiotic-receptor binding and expression of cell death are two major possible causes for such lag period. This study was primarily designed to determine the relationship, if any, between antibiotic concentrations and the lag periods by a modeling approach. Short-term time-kill studies were conducted for amoxicillin, ampicillin, penicillin-G, oxacillin, and dicloxacillin against Escherichia coli. In conjunction with the use of a saturable rate model to describe the concentration-dependent killing process, a first-order induction (initiation) rate constant was used to characterize the delay in bacterial killing during the lag period. For all of the beta-lactams tested, parameters describing the bactericidal effect suggest that amoxicillin and ampicillin were much more potent than oxacillin and dicloxacillin. The induction rate constant estimates for both ampicillin and amoxicillin were found to relate linearly to concentrations. Nevertheless, these induction rate constant estimates were lower for penicillin-G, oxacillin, and dicloxacillin and increased nonlinearly with concentrations until an apparent plateau was observed. These findings support the hypothesis that the permeation process is potentially a rate-limiting step for the rapid bactericidal beta-lactams such as ampicillin and amoxicillin. However, as suggested by previous observations of the various morphological changes induced by beta-lactams, the contribution of the steps following antibiotic-receptor complex formation to the lag period might be significant for the less bactericidal antibiotics such as oxacillin and dicloxacillin. Findings from the present modeling approach can potentially be used to guide future bench experimentation.

Pharmacokinetic-pharmacodynamic model for spiramycin in staphylococcal mastitis

Simultaneous pharmacokinetic-pharmacodynamic (PK/PD) modelling for spiramycin in staphylococcal infections of the mammary gland of cows was used to predict the efficacy of spiramycin. A differential equation derived from the Zhi model was fitted to an in vitro killing curve and post-antibiotic effect determination. A seven-compartment PK model, in which 4 compartments representing each quarter of the mammary gland which was considered to be the effect compartment, was included. The PD model linked to the PK model was able to describe the in vivo spiramycin effect against Staphylococcus aureus. The parameters calculated from in vitro data predicted a rapid decrease for the first 12-24 h, and regrowth within 72 h following the treatment, whereas in vivo the bacterial effect was much less after 24 h than that predicted by the in vitro data. PK/PD modelling permitted the simulation of various doses to optimize the efficacy of the antibiotic, taking into account such dynamic parameters as bacterial growth rate constant, bacterial killing rate constant and the Michaelis-Menten type saturation constant. An optimal dosage regimen of 20000 IU/kg per day for 3 days was predicted for the treatment of Staphylococcus aureus mastitis.

The fractional maximal effect method: a new way to characterize the effect of antibiotic combinations and other nonlinear pharmacodynamic interactions

The checkerboard technique leading to the fractional inhibitory concentration indexes and the killing curve method are currently the most widely used methods to study antibiotic combinations. For both methods, experimental conditions and interpretation criteria are somewhat arbitrary. The relevance of the fractional inhibitory concentration index computation, in the classic case of additivity [P = d1/(D1)p + d2/(D2)p, where d1 and d2 are the doses of drugs 1 and 2 in combination to produce an effect at a percent level (P) and (D1)p and (D2)p are the doses required for the two respective drugs alone to produce the same effect] relies on the assumption of a linear relationship between the MIC and the concentration of the test antibiotics. In addition, there is no consensus as to the definition of synergy in killing curve interpretation. The fractional maximal effect (FME) method is a new approach which was developed to handle the nonlinear pharmacodynamics exhibited by antibiotics and other drugs. This method relies on the mathematical linearization of the nonlinear concentration-effect scales and eventual construction of an isobologram-type data plot. The FME method was applied to study interactions between several antibiotic combinations: amoxicillin and tetracycline, ciprofloxacin and erythromycin, and ticarcillin and tobramycin. These combinations were selected because the pharmacologic basis for their interactions has been previously described. The FME method correctly identified antagonism for the first two combinations and synergism for the last combination. Conclusions were reproducible across the range of concentrations studied. Besides providing information on the nature of the interaction, the method can rapidly explore the effect of changing concentration ratios of two antimicrobial agents on the degrees of interaction. The FME method may be applied to interactions between drugs or agents with either a linear or nonlinear endpoint measurement. Methods frequently used for drug combination testing are also discussed in the paper.

Microbial pharmacodynamics of piperacillin in neutropenic mice of systematic infection due to Pseudomonas aeruginosa

Mathematical solutions for two possible pharmacodynamic interactions (linear nonsaturable and nonlinear saturable) between antibiotics and microorganisms derived from the incorporation of clinically relevant antibiotic dosage regimens such as single bolus dosing, multiple doses, and constant infusion at steady state have been obtained. It is concluded that the saturable nonlinear interaction model between the tested antibiotic and microorganism appears appropriate. The model and its derived equations are capable of describing in vivo bacterial growth of P. aeruginosa after single bolus dosing and multiple doses of piperacillin as described by a linear one-compartment pharmacokinetic model. The activity of piperacillin against P. aeruginosa in the neutropenic mouse systemic infection model can be described by an equation with three dynamic parameters: the bacterial growth rate constant kapp, 0.02345 min-1, the bacterial killing rate constant k'kill, 0.02623 min-1, and the Michaelis-Menten type saturation constant Km, 0.05467 microgram/ml. The concept and derived equations for the optimal dosing interval and minimum critical concentration are of clinical importance for the proper selection of antibiotic dosage regimens.