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J Pharmacokinet Pharmacodyn (2007) 34:727751DOI 10.1007/s10928-007-9069-x

Mechanism-based pharmacokineticpharmacodynamicmodeling of antimicrobial drug effects

David Czock Frieder Keller

Received: 22 March 2007 / Accepted: 17 July 2007 / Published online: 29 September 2007 Springer Science+Business Media, LLC 2007

Abstract Mathematical modeling of drug effects maximizes the information gainedfrom an experiment, provides further insight into the mechanisms of drug effects, andallows for simulations in order to design studies or even to derive clinical treatmentstrategies. We reviewed modeling of antimicrobial drug effects and show that mostof the published mathematical models can be derived from one common mechanism-based PKPD model premised on cell growth and cell killing processes. The generalsigmoid Emax model applies to cell killing and the various parameters can be related tocommon pharmacodynamics, which enabled us to synthesize and compare the differ-ent parameter estimates for a total of 24 antimicrobial drugs from published literature.Furthermore, the common model allows the parameters of these models to be relatedto the MIC and to a common set of PKPD indices. Theoretically, a high Hill coef-ficient and a low maximum kill rate indicate so-called time-dependent antimicrobialeffects, whereas a low Hill coefficient and a high maximum kill rate indicate so-calledconcentration-dependent effects, as illustrated in the garenoxacin and meropenemexamples. Finally, a new equation predicting the time to microorganism eradicationafter repeated drug doses was derived that is based on the area under the kill-rate curve.

Keywords Pharmacokinetics Pharmacodynamics PKPD modeling Antimicrobials Antibiotics Resistance Simulation

D. Czock (B) F. KellerDivision of Nephrology, Medical Department, University Hospital Ulm,Robert-Koch-Str. 8, 89081 Ulm Germanye-mail: david.czock@uniklinik-ulm.de

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Glossary

Abbreviation Name (Unit)

AUC Area under the concentration-time curve (h g/ml)AUETC Area under the effect-time curve (None)C Drug concentration (g/ml)CFU Colony forming unit (None)Cmax Maximum concentration (g/ml)Cstat Stationary concentration (g/ml)EC50 Concentration where the half-maximum effect is present (g/ml)Ed(C) Fractional increase in the death rate depending on concentration

(None)Emax Maximal stimulation of the death rate (None)Er(C) Fractional decrease in the replication rate depending on concentra-

tion (None)H Hill coefficient (None)IC50 Concentration where the half-maximum effect is present (g/ml)Imax Maximum inhibition of the replication rate (None)kdeath0 Death-rate constant (without drug) (h1)kdeath(C) Death rate depending on concentration (h1)ke Elimination rate constant (h1)kgrowth0 Growth-rate constant (without drug) (h1)kgrowth(N ) Growth-rate depending on microorganism number (h1)kkill(C) Kill rate depending on concentration (h1)kkill(t) Kill rate depending on time (h1)kkill max Maximum kill rate (h1)kreplic0 Replication-rate constant (without drug) (h1)kreplic(N ) Replication-rate depending on microorganism number (h1)kreplic max Maximum replication rate (h1)MIC Minimum inhibitory concentration (g/ml)N Number of microorganisms (CFU/ml)N0 Initial number of microorganisms (CFU/ml)Nt Number of microorganisms at time t (CFU/ml)N Number of microorganisms after one dosing interval (CFU/ml)N50 Number of microorganisms at which the replication rate is half

maximal (CFU/ml)Nmax Maximum number of microorganisms (CFU/ml)t Time (h)T>MIC Time above MIC (h)TE Time to microorganism eradication (h) Dosing interval (h)VGmax Maximum velocity of bacterial replication (h1 CFU/ml)

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Introduction

The minimum inhibitory concentration (MIC) of an antimicrobial drug is often usedas a clinical marker of its antimicrobial effect. However, the MIC is an arbitrarymeasurement since microorganism growth is determined at only one point in timeafter an 18- to 24-h exposure to a constant antimicrobial concentration. In contrast,microorganism kill curves after in vitro exposure to fluctuating antimicrobial drugconcentrations more closely reflect the in vivo situation [1,2].

Mathematical models of drug effects provide further insight into drug effects andenable simulations [25]. Generally, a mathematical model should be mechanism-based in order to provide accurate predictions of drug effects [6]. Mechanism-basedmodels of antimicrobials have to include at least a submodel of microorganism repli-cation and a submodel of antimicrobial drug effects. Such models were used to analyzeantimicrobial effects on bacteria [7,32], viruses [33,34], and parasites [33,35]. Modelsthat relate administered doses directly to an outcome measurement are not consideredin this manuscript.

Our aim was to review mathematical pharmacokineticpharmacodynamic(PKPD) modeling of antimicrobial drug effects and to show that the various mo-dels can be derived from a common mechanism-based model (Eqs. 24).

Antimicrobial PKPD models

Generally, a mechanism-based antimicrobial PKPD model includes equations descri-bing microorganism growth (i.e., the microorganism submodel), the effect of antimi-crobial drugs (i.e., the antimicrobial submodel), and changing drug concentrations (i.e.,the pharmacokinetic submodel). Firstly, equations of a common model are derived.Secondly, complexities such as capacity-limited microorganism growth and resistantsubpopulations are discussed.

The microorganism submodel

Common model

The most common model of microorganism survival is shown in Eq. 1 where N is thenumber of microorganisms and kreplic0 and kdeath0 are the first-order rate constants des-cribing natural replication and death of the microorganisms without any antimicrobialdrug [31,36].

dN

dt= kreplic0 N kdeath0 N (1)

The starting number of microorganisma, N (0), is the initial condition of this and laterequations. A first-order rate constant for the observed growth (kgrowth0 = kreplic0 kdeath0) is often applied (kapp in previous publications), since it can be difficult toseparate microorganism replication and death by a traditional kill curve analysis. This

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assumes that the number N of microorganisms able to replicate is the same as thenumber of microorganisms subjected to death.

Limited growth

Capacity-limited microorganism growth has been modeled by linking the microor-ganism growth rate to N , either linearly as kgrowth(N ) = kgrowth0 (1 N/Nmax),commonly known as the logistic function, where Nmax is the maximum number ofmicroorganisms [8,13,14,20,23,2628,30,35] or nonlinearly as kreplic(N ) = VGmax(N50 + N ), where VGmax is the maximal velocity of bacterial replication, N50 is thenumber of microorganisms at which the replication rate is half maximum and themaximum replication rate is kreplic max = VGmax/N50 [19,32]. From a mechanisticpoint of view, a modification for capacity-limited microorganism growth should beapplied to the replication rate (kreplic). When such a modification is applied to thegrowth rate (kgrowth), it is assumed that the rate of microorganism death also decreaseswith increasing number of microorganisms.

Another approach to model capacity-limited microorganism growth is to dividemicroorganisms into a growing and a resting subpopulation where transformation fromthe growing to the resting stage is triggered by a high total microorganism level [31].

Replication of resistant subpopulations

It can be necessary to assume different replication rates for susceptible and resis-tant microorganism subpopulations [37] since some resistant subpopulations appearto grow more slowly [8,13,19,26,28,32]. When a modification for capacity-limitedmicroorganism replication is applied, the replication rate of a subpopulation should belinked to the total number of microorganisms and not to that from one subpopulation.

The antimicrobial submodel

Common model

Antimicrobial drugs can either decrease the rate of microorganism replication (kreplic)or increase the rate of microorganism death (kdeath). Thus, the most common model ofantimicrobial effects is shown in Fig. 1 and Eq. 2, where the pharmacodynamic func-tions Er(C) and Ed(C) relate the replication rate and the death rate to the antimicrobialdrug concentration C [31,32].

dN

dt= kreplic0 Er (C) N kdeath0 Ed (C) N (2)

Inhibition of the rate of microorganism replication can be related to C using thesigmoid Emax model, where Imax is the maximum inhibition, IC50 is the concentrationwhere half-maximum inhibition is present, and the Hill coefficient H describes thesigmoidicity (Eq. 3). The sigmoid Emax model can describe the sequence of theconcave, linear, and convex (i.e., saturated) relationship for increasing concentrations.

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Replicationkreplic0(h

-1) Er(C)Death

kdeath0(h-1) Ed(C)

N(CFU / ml)

Fig. 1 Schematic of the common model for microorganism survival and antimicrobial drug effects. Nis the number of microorganisms, kreplic0 is the replication rate constant, and kdeath0 is the death rateconstant. Antimicrobial drugs can act by inhibition of replication or by stimulation of microorganism death.Er(C) and Ed(C) represent the equations that relate drug concentration C to the effect on microorganismreplication and death

Er (C) = 1 Imax CH

ICH50 + C H; with 0 Imax 1 (3)

Similarly, the increase in the rate of microorganism death can also be related toC using the sigmoid Emax model, where Emax is the maximum stimulation of thedeath rate, EC50 is the concentration wh