Antimicrobial breakpoint estimation accounting for variability in pharmacokinetics
 Goue DenisGohore Bi^{1},
 Jun LI^{2, 3} and
 Fahima Nekka^{1, 2, 4}Email author
DOI: 10.1186/17424682610
© Bi et al; licensee BioMed Central Ltd. 2009
Received: 13 January 2009
Accepted: 26 June 2009
Published: 26 June 2009
Abstract
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/PDbased methods reflect in vivo conditions and are more predictive of efficacy. Unfortunately, they entail the use of one PKderived 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 timedependent 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.
Background
Antimicrobial efficiency and resistance have become a global public health issue and a real challenge for microbiologists, pharmaceutical companies, physicians and other members of the health community. Inadequate use of antibiotics promotes the selection of bacteria with decreased susceptibility. The search for new drugs to treat infectious diseases, the traditional approach to overcoming antibiotic resistance, is growing more challenging because multipleresistance is becoming more prevalent among bacteria, and new targets for antimicrobial antibacterial action remain to be discovered [1–3]. The development of new antimicrobial antibiotics is a long, costly process, which takes a poor second place to the development of more lucrative drugs for an aging population. Therefore, improving the current use of antibiotics is central to preserving their longterm effectiveness in humans and animals. For public health officials, susceptibility testing data are crucial for the surveillance and control of emerging resistance. To collect these data, several susceptibility testing methods including dilution, disk diffusion and automated instrument system methods are currently in routine laboratory use [1–3]. To interpret the susceptibility test results, the breakpoint, a discriminating concentration, has been used to define isolates as susceptible, intermediate or resistant [4–6]. For obvious reasons of drug efficacy and antibiotic resistance problems, estimation of breakpoints has become a necessary step in modern microbiology laboratory practice. Breakpoints are estimated in a variety of ways, the most widely used being the minimal inhibitory concentration (MIC), which is the lowest concentration that completely inhibits microbial growth [1, 3, 7]. Although the MIC is considered the gold standard for breakpoint assessment, its main drawback lies in its in vitro basis, with no drug disposition information being included. In fact, MIC is a threshold value while antibacterial efficacy is a complex consequence of dynamic concentration and timedependent processes. In recent decades, these limitations have led professional groups to make intensive efforts to review pharmacokinetic and clinical data and establish suitable drug breakpoints under in vivo conditions. One of latest tendencies is to integrate PK/PD indices in order to understand the relevance of drug dose and schedule to efficacy [4, 8–18]. The breakpoints obtained, generally called pharmacokinetic/pharmacodynamic (PK/PD) breakpoints, refer to the antibacterial concentrations calculated from the knowledge of a PD parameter and the dimension of that parameter for predicting efficacy in vivo [19]. The specific PK/PD indices correlating with bacteriological efficacy mostly depend on the nature of drug action in bacterial killing, which may be either concentrationdependent or timedependent [20]. There has been a great increase in interest in the use of PK/PD studies to estimate drug efficacy since the foundation of the International Society for Antiinfective Pharmacology (ISAP) in 1991 [20]. Whilst these methods are more realistic as they are adapted to in vivo conditions, they still are empirically based, lacking a theoretical or mechanistic basis. Most importantly, the role of variability between individuals and from other potential sources cannot be explained in a definite way. This situation has clearly restricted the further development of these approaches. Because of this experimental limitation and the complexity of the problem, there is a need to develop new methodologies for drug evaluation. In this work, we provide a theoretical basis for characterizing the "efficiency" of a PK profile under in vivo conditions, which will then be supported by in silico approaches adopted for the two classes of concentration and timedependent antibiotic drugs. Using this approach, breakpoints can be explained and estimated within the context of standard PK/PD analysis.
Report on the antibacterial agents for different activity patterns and methods*
Year  

Types or Methods  1970–1980  1981–1990  1991–2000  2001–2008  Drugs or parameters 
Timedependent  15  60  104  228  Betalactams Macrolides 
Concentrationdependent  10  80  225  315  Aminoglycosides Fluoroquinolone 
PK/PDbased methods  0  20  141  401  AUC_{24}/MIC Cmax/MIC C_{BP} T_{>MIC} 
This paper is organized as follows: In the Methods Section, we propose a logical extension of the known efficacy function in order to define the efficiency of a PK profile. In the Application and Results Section, we discuss some useful properties of our new approach and apply it to the particular case of variable drug intake. Finally, we give a general discussion to position our approach and findings within the current status of the field.
Methods
Weighted AUC: a rational parameter for assessing PK/PD efficiency
As mentioned in the background, recently introduced PK/PDbased breakpoint estimation was put forward to overcome drawbacks of threshold criteria, namely MIC, which determines in vitro antimicrobial efficacy. However, these PK/PDbased methods use drug exposure mainly through the AUC value (the amount of drug absorbed), whereas the variability in drug concentration time course is not integrated. This variability turns out to be an important factor in drug efficacy, as widely reported [27]. In bioequivalence studies, for example, it is common to combine AUC and Cmax to compare PK profiles and thus indirectly assess the expected drug efficacy. Therefore, to rely solely on the use of these PK parameters may not be sufficient for drawing reliable conclusions on drug efficacy. To employ these parameters efficiently and optimize their use for specific purposes, we need to adapt them by adding more information on drug PK/PD properties.
AUCbased drug efficacy is generally assessed through statistical methods such as scatter plots. Since PK/PD properties are not fully exploited, the relationship between drug efficacy and PK parameters only partially reflects the pharmacological properties. If additional PK/PD properties can be accounted for, the capacity of the actual empirical PK/PDbased breakpoint estimation is likely to be improved. Ideally, when a PK/PD relationship can be determined in vivo, the power of drug efficacy prediction can be maximized. However, exact doseresponse relationships under in vivo situations are not easily accessible. This is the main restriction that prevents full exploration of drug efficacy prediction. Alternatively, combining the in vitro efficacy function (E) – the PK/PD relationship measurable in the laboratory – with AUC provides a better relationship (being more informationloaded) than that of drug efficacy in terms of AUC. As we will see, this combination can be considered an extension of the definition of AUC, thus relating to specific information on drug response.
where Emax is the maximum effect (normalized to one in this paper), EC_{50} the drug concentration that attains 50% of Emax, and H is the Hill constant [31]. Since this efficacy function carries rich information about the response in terms of concentration, it should and could be translated under in vivo conditions. In fact, the in vivo situation can be considered as a composite of many "local" in vitro cases. "Locally in vitro" here means that once the antibiotic reaches a certain site in the body (a target organ for example), it behaves in a similar way as in vitro. In the following, we will include this efficacy function E in our approach to predicting the drug's therapeutic performance and apply it to the case of concentrationdependent antibiotics.
where again E is a function related to drug efficacy, T is the therapeutic duration used as a reference period and n = 0, 1, ...
Compared to AUC, E here plays the role of a weighting function. We use it to include the information on the PK/PD relationship as an integral part of drug efficiency measurement expressed through Eff. This information can always be updated and integrated for this purpose. For the particular case of E = 1 and n = 1, we obtain the usual AUC definition, thus making our newly introduced efficiency function a direct extension of AUC.
Concentrationdependent antibiotics: weighted AUC method for antimicrobial efficiency
We notice that Eff_{1} contains information related to both AUC and concentration variation levels. In this newly proposed formula, these two elements are well integrated to reflect their contributions to the evaluation of drug performance. Eff_{1} can thus be considered an extension of the classical approach [14]. We refer to Eff_{1} as the weighted AUC and denote it by AUC_{W}.
Timedependent antibiotics: an analytic expression for total antimicrobial efficiency
where E = χ is the indicative function. We recall here that the indicative function χ_{A} of a set A is defined as: χ_{A} (t) = 1 if t belongs to A; 0 otherwise. Hence, expressed in this way, χ will be 1 if C(t) > C_{BP} and will be 0 otherwise. We notice that Eff_{0} is simply the cumulative time during which C(t) remains above the specific concentration value C_{BP}, which turns out to be exactly the same classic definition for evaluating timedependent efficacy. However, expressing it in this way, with explicit reference to the zeroorder general efficiency Eff_{ n }function proposed above, helps us to understand the direct relationship of efficiency for different drug groups.
Application and results
In the following, we will focus on concentrationdependent antibiotics to illustrate how the newly introduced weighted AUC method can be used.
Efficiency equivalence between in vivo and in vitro
For timedependent drugs, the situation is different. The efficiency is the percentage of time during which the concentration remains above a specific value C_{BP}. As in vitro concentrations can only have binary efficiency, we have to determine the threshold of time percentage for an effective drug regime. The efficiency of a PK profile can be compared with this threshold to measure its efficacy.
Weighted AUC method and irregular drug intake
As an application of the AUC_{W} method, we will consider the case of variability in PK profile generated by irregular drug intake. It is common sense that a deviation between real drug intake and the ideal prescribed dosing regimen is likely to have an impact on the pharmacokinetic profile and eventually the drug response. Noncompliance characteristics can be translated into some derived PK/PD parameters and pharmacological indices.
In a previous study, we investigated the impact of animal feeding behaviour on the pharmacokinetics of chlortetracycline (CTC), a widelyused antibiotic usually given through animal feed [32]. We modeled a widely reported animal feeding behaviour and associated it with the CTC disposition model to obtain a feeding behaviourPK (FBPK) model. Using this model, we revealed the PK variability induced by random drug intake and assessed its main characteristics [32].
In the present paper, we will focus on the estimation of efficiency of CTC in this particular context of irregular drug intake. We have to mention that similar reasoning and analysis can be accomplished using other sources of variability impacting pharmacokinetics. Since CTC is a concentrationdependent antibiotic widely used in collective medical therapy, we will base our analysis on the method we propose for this antibiotic class. For the purpose of illustration, we will use the individual FBPK we previously developed [32]. The case of an animal population can be developed by adding the intervariability to the associated PK parameters. In the following, we will answer the following questions: Can the "efficacy performance" of PK profile be characterized uniquely by its average concentration value? What is the extent of in vitro equivalent concentrations that an average concentration can reach? Since we only have access to the drug concentration in feed, what can we say about the potential efficacy of various drug concentrations in feed compared to that of MIC?
An advanced PK model integrating swine feeding behaviour: an FBPK model
In veterinary medicine, the problem of optimal use can arise for drugs administered through feed, a widelyused practice for therapeutic, metaphylactic or prophylactic treatment of bacterial infections [33]. As a consequence, animal feeding behaviour directly influences systemic exposure to drugs. However, variation in the feeding behaviour of animals medicated through feed has been overlooked for more than 50 years, during which feed antibiotic therapy remained empirical. Using widelyreported descriptions of swine feeding behaviour, we have mathematically formulated and integrated this behaviour model into a PK model (FBPK) in order to analyze its influence on systemic exposure to drugs quantitatively [32]. We include here a brief review of the FBPK model. Complete details about the model and its analysis can be found in [32].
where [T_{s}, T_{e}] is the duration of the peak period, DOSE is the drug concentration mixed in the feed, with units in ppm, is the average ingestion rate, F is the bioavailability, K_{a} and K_{e} are the absorption and elimination rates respectively, and V is the volume of distribution.
Estimation of MIC breakpoints in animal populations
By definition, MIC breakpoints refer to critical drug concentrations that characterize specific antibacterial activities. The values of these MIC breakpoints are highly pertinent to the pharmacokinetic properties as well as to the pharmacodynamic killing capacities of these drugs with respect to particular bacterial strains. In the clinical setting, MIC is considered an important reference index in choosing effective dose regimens. However, because of the evident large variation in concentration time course and the unavoidable pharmacokinetic variability under the in vivo situation, the true PK/PD relationship is generally more complex. Using a single static value of MIC for the decision process is dubious or even misleading. Therefore we have to take account of dynamic in vivo properties when estimating drug efficacy.
In the following, we will use the abovedeveloped feeding behaviourPK model to show how one can obtain breakpoint information, and of what kind, for an in vivo situation.
To do this, we adopt a Monte Carlo approach to generate, for an animal X, possible drug inputs prior to drug disposition. The corresponding concentration time courses are then produced with these drug inputs. To explain our approach, we need to introduce some new concepts and their notations.

DOSE: drug concentration mixed in feed, with units of ppm.

: average over a time duration T of one concentration time course generated by Monte Carlo; it is AUCbased.

: global mean of all average concentrations .

: 95% higher mean concentration where 95% of are below this concentration.

: 95% lower mean concentration where 95% of are above this concentration.

: in vitro equivalent concentration (Eq. 2) of C_{i}(t), where 0 ≤ t ≤ T; it is AUC_{W}based.

: global mean of all in vitro equivalent concentrations .

95% higher in vitro equivalent concentrations , where 95% of are below this concentration

: 95% lower in vitro equivalent concentrations , where 95% of are above this concentration.
[ , ] = [0.397 mg/L, 0.435 mg/L].
We can consider that a DOSE is at least 95% efficiencyequivalent to an in vitro concentration C_{eff} by defining 95% of equivalent in vitro concentrations generated by this DOSE as being above C_{ eff }. In our case, for a given DOSE, we have the relationship C_{ eff }= (Dose) according to this 95% efficiencyequivalence criterion.
However, with each DOSE, we can also associate a 95% confidence interval of average concentrations represented by [ (Dose), (Dose)] as illustrated in the right panel of Figure 4.
To a given target value Ce (in vitro target), there corresponds a DOSE that gives an interval of equivalent concentrations (hence equivalent efficacy) lying above Ce.
Robustness of weighted AUC approach
Discussion
Unlike the ideal in vitro conditions, where major guidelines for drug efficacy are routinely established for stable drug concentrations, it is natural that high variability arises in vivo and thus raises concerns about the applicability of in vitroestablished principles. This in vivo variability may have various origins and forms [34, 35]. One of these sources is structural and is directly linked to drug disposition and elimination processes (generally referred to by ADME: Absorption, Distribution, Metabolism and Elimination), where the drug concentration time course is often described using ordinary differential equations. These ADME scenario components are generally mimicked, separately, under laboratory conditions but hardly synthesized as a whole. The well known PK parameters such as AUC and Cmax are specifically designed to reflect this drug exposure variation in the PK/PD association. Beyond this structural variability, other pharmacokinetic variability is widely recognized and turns out to be an important influence on drug efficacy. Neglecting variability when assessing therapeutic efficacy may lead to wrong conclusions [32, 35–37]. In the current article, we have shown how, instead of relying solely on AUC or other single parameters, the entire (in vitro or in vivo) pharmacodynamic function should be considered in a more integrated way for evaluating and developing antibiotic treatment protocols. Being concerned with this issue, we have directly generalized the classical AUCbased methods and rendered drug evaluation more efficient by including richer information on the PK profile.
As a static efficacythreshold parameter widely used for breakpoint assessment, MIC does not include drug disposition or other potential variability information. In fact, MIC is measured under almost deterministic conditions since variability is likely to be smaller in vitro than in vivo. However, antibacterial efficacy is the result of a complex dynamic process that depends on concentration and time. Hence, relying on such in vitro values may be risky since real in viv o values can spread over a relatively large range. Generally, these in vitro values are used to refer to mean in vivo values. However, we have seen here that using the average concentration as a reference value can lead to ambiguous interpretation of drug efficacy since various PK profiles are likely to share the same average concentration while having different therapeutic performances. Under in vivo conditions, all these parameters should be reconsidered and adapted to reflect this varying situation. In this context, it is thus common sense to have recourse to a probabilistic approach, as we illustrated in the examples above.
Another interesting issue arising directly from our method concerns bacterial antibiotic resistance. It is known that underexposure of bacterial strains to antibiotics is the main cause of resistance. When traditional exposure indices such as AUC or Cmax are used to evaluate drug efficacy, the prediction is linearly related to antibiotic exposure. Since these derived indices are proportional to dose, the real mechanism of drug killing is not incorporated as the linear property remains unchanged when either drug exposure or dose is used. In some recent work, a trend in this direction can be noticed [28, 38–40]. Using our efficiency evaluation approach, we observe that for low doses the traditional AUCbased method gives an optimistic efficacy evaluation as the drug killing properties are ignored in its expression form. However, when we account for killing properties through the efficacy curve as we did in our efficiency formula, we clearly see that the drug efficacy evolves more slowly than the corresponding dose. In our example, under a 500 ppm DOSE, the drug efficiency estimated using our method is half that of the AUCbased method. Hence, for lower doses, there is a good chance of being in low efficiency situations where the risk of antibiotic resistance is higher than can be assessed using traditional methods. These results suggest that further investigation in this direction is needed, especially because lower doses are usually related to irregular drug intake, such as drug holidays or cases of antibiotic abuse. We believe that more advanced methods should be developed to address this problem. Our approach is one step towards this end. We propose here a logical way of evaluating drug efficiency on the basis of in vitro efficacy information and the PK profile. This can be relevant to antibiotic development, especially for the estimation of the efficacy dose in phase II. In this work, we have used killing curves to illustrate our methods. This does not prevent extension of our approach to other antibacterial drugs. In fact, we have already suggested a possible approach for timedependent drugs. However, for other complex types of effect, there is a need for further investigation of the mechanism and for embedding the results in the efficiency form that we proposed. In fact, the complex facets of antibacterial activities are not limited to this simple classification. For some antibiotics, such as glycopeptides, a combination of concentration and time of exposure may both be relevant [7, 26]. The coexistence of bactericidal and bacteriostatic properties in these drugs makes them codependent on both concentration and time. Nonetheless, the classification into concentrationdependent and timedependent suggested by the ISAP for antibiotics provides an objective basis for judging antibiotic performance. It is interesting to note that these two patterns fall within the two extreme cases of antibiotic efficacy, as illustrated by the curves of Figure 2[28].
Finally, owing to the complexity of biological systems – the human body here, as well as the mechanisms involved in the bacteria killing capacities of drugs – new methods are being developed every day. Some of them use very sophisticated theories that include every known facet of mechanism; others seek to ignore these complexities and use elementary mathematics with an empirical philosophy. The former struggle with the applicability of their methods, while the latter often lack logistic links to the underlying mechanism. A tradeoff should be found to balance these tendencies. We think that our article may inspire progress on this path.
Conclusion
In this paper, we have proposed a logical generalisation of the classical AUC method by introducing the "efficiency" of a PK profile, which involves the efficacy function as a weight. We have formulated these methods for both classes of concentrationdependent and timedependent antibiotics. We have illustrated the approach developed using the particular case of variable drug intake. We have also shown how the new approach can overcome some limitations of the classical methods for assessing drug efficacy.
Declarations
Acknowledgements
This work has been supported by NSERC through NSERC UFA and NSERC grant (RGPIN227118), hold by F. Nekka. The Centre of Excellence in Mathematics of Information Technology and Complex Systems (MITACS) is also acknowledged for its support as well as the Centre de Recherches Mathématiques (CRM) and Faculté de Pharmacie of Université de Montréal.
Authors’ Affiliations
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