2014 by the Japanese Society for the Study of Xenobiotics (JSSX)

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1 Received; January 28, 214 Published online; July 8, 214 Accepted; June 26, 214 doi; /dmpk.DMPK-14-RG-13 A proposal of a pharmacokinetic/pharmacodynamic (PK/PD) index map for selecting an optimal PK/PD index from conventional indices (AUC/MIC, Cmax/MIC, and TAM) for antibiotics Yoshiaki Kitamura, Kenta Yoshida, Makiko Kusama, and Yuichi Sugiyama Discovery Research Laboratories, Kyorin Pharmaceutical Co., Ltd., Tochigi, Japan (Y.K.) Graduate School of Pharmaceutical Sciences, The University of Tokyo, Tokyo, Japan (K.Y. and M.K.) Sugiyama Laboratory, RIKEN Innovation Center, RIKEN Research Cluster for Innovation, Yokohama, Japan (Y.S.) 1 Copyright C 214 by the Japanese Society for the Study of Xenobiotics (JSSX)

2 Running title: PK/PD index map for selecting the best predictor Corresponding author: Yoshiaki Kitamura, Ph.D. Discovery Research Laboratories, Kyorin Pharmaceutical Co., Ltd , Nogi, Nogi-machi, Shimotsuga-gun, Tochigi, , Japan Phone: Fax: Number of text pages: 39 Number of tables: 4 Number of figures: 5 Number of words in the summary: 197 Number of words of text including references: 5 2

3 Summary A pharmacokinetic/pharmacodynamic (PK/PD) analysis is important in antibiotic chemotherapy. Basically, the in vivo efficacy of antibiotics that exert concentration-dependent effects can be predicted using conventional PK/PD indices such as the ratio of the area under the curve to the minimum inhibitory concentration (AUC/MIC) and/or the ratio of the maximum plasma concentration to MIC (Cmax/MIC), whereas that of antibiotics with time-dependent effects can be determined using the period of time for which the drug concentration exceeds the MIC (time above MIC [TAM]). However, an optimal PK/PD index remains to be established for some antibiotics. Thus, a PK/PD model, which describes the PK profile and effect of an antibiotic, was developed, and the results obtained from this model were interpreted to form a PK/PD index map to assess the optimal PK/PD index for the antibiotic. The findings from the map were generally consistent with clinical outcomes even for the antibiotics which became the exception by the conventional classification. For example, AUC/MIC was an optimal index for azithromycin despite its time-dependent bactericidal activity, and Cmax/MIC was a poor index for arbekacin despite its concentration-dependent profile. Thus, the map would be useful for selecting the appropriate PK/PD index for an antibiotic. 3

4 Keywords pharmacokinetics and pharmacodynamics; antibiotics; AUC/MIC; Cmax/MIC, time above MIC 4

5 Introduction For decades, the importance of pharmacokinetic/pharmacodynamic (PK/PD) analysis has been increasing in antibiotic chemotherapy. Understanding the PK/PD relationship, an estimate of in vivo efficacy on the basis of plasma concentration, is essential for determining the dosing regimen in clinical use. Historically, the minimum inhibitory concentration (MIC) was considered the principal PD parameter to determine in vivo efficacy. However, since antibiotics with the same MIC value can have different bactericidal characteristics, 1) MIC alone cannot predict in vivo efficacy. Antibiotics are generally classified into two groups according to the shape of the time-kill curve. Concentration-dependent drugs exhibit a linear relationship between drug concentration and the killing rate after the administration of a clinically approved dose. Time-dependent drugs have a low maximum killing rate, and the bactericidal activity is independent of drug concentration above the MIC; rather, the exposure time predominantly determines the bactericidal activity. These classifications evolved into the following three PK/PD indices: the in vivo effect of concentration-dependent drugs is associated with the ratio of the area under the curve (AUC) to MIC (AUC/MIC) and/or the ratio of the maximum plasma concentration (Cmax) to MIC (Cmax/MIC), whereas the in vivo effect of time-dependent drugs is associated with the time above MIC (TAM). 2) Antibiotics have been conventionally classified into 5

6 these three categories usually according to their class. However, for some antibiotics, the correlation between the identified PK/PD index and the actual clinical efficacy is controversial. For example, for arbekacin, a concentration-dependent aminoglycoside, AUC/MIC data were considered a good predictor, 2 4) whereas Cmax/MIC was concluded as the best index in some studies. 5,6) For vancomycin, a time-dependent drug, both concentration and the TAM were regarded as predictive indices; 7) conversely, Moise-Broder et al. showed that AUC/MIC was the most predictive index. 8) These discrepancies cannot be resolved easily because they necessitate extensive clinical studies. To describe and predict the in vivo antibacterial activity more accurately than the conventional approach, the concept of model-based PK/PD analysis has been proposed, in which the time course of the plasma concentration is considered. 3,4) According to this approach, in addition to MIC, in vitro parameters that are necessary to distinguish between concentration-dependent and time-dependent drugs are incorporated to describe the bactericidal characteristics more precisely. Although the predictability of this model is assumed superior to that of the conventional approach, this model requires modeling and simulation capabilities that may preclude it from general use. Thus, the goal of this study was to develop a method for this model-based analysis without the use of advanced PK/PD modeling and computer simulation. Here, the PK/PD index map was presented to assess the predictability of 6

7 conventional PK/PD indices by using the model-based analysis. The PK/PD index map enables the selection of the optimal PK/PD index, requiring only a few parameters, such as the elimination rate constant (k e ) and in vitro bactericidal characteristics (ε, γ, and λ). The dosing schedule map was also developed to determine whether once daily dosing is more effective than divided dosing. These maps were developed with regard to efficacy only; the occurrence of adverse events and the emergence of resistant strains were not considered. 7

8 Methods PK/PD modeling The PK/PD model consists of 2 units (Figure 1). The pharmacokinetic unit is a typical 1-compartment model with a gut compartment. The bacterial unit the time profile of the number of bacteria at the infection site is essentially identical to the enhanced-death constant-replication model described by Czock et al. 3) These units are related to each other under the assumption that the free plasma concentration at a certain point in time determines the kill rate of bacteria at the same time. where k a is the absorption rate constant from the gut compartment to the central compartment, k e is the elimination rate constant from the central compartment, X 1 is the amount of drug in the gut compartment, X 2 is the amount of drug in the central compartment, V d is the distribution volume, f p is the free fraction in plasma, C free is the plasma concentration of free drug, N is the number of bacteria, λ is the growth rate 8

9 of bacteria without drug, ε is the maximum kill rate constant, γ is the Hill coefficient, EC 5 is the concentration of drug at which 5% of the maximum effect is obtained, and MIC is the minimum inhibitory concentration, which is equal to C free when dn/dt =. PK/PD simulation under steady state after repeated dosing The PK profile of an antibiotic in vivo and the number of bacteria were calculated at steady state from to 24 h in 4 different dosing schedules once daily dosing at time, twice daily dosing at times and 12 h, 4 times daily dosing every 6 h, and 8 times daily dosing every 3 h by numerical integration using the ode23s function of MATLAB (The MathWorks Inc., Natick, MA), which is based on a modified Rosenbrock formula of order 2. 9) Initial conditions for the numerical integration were given as follows (Supplemental material 1): where F is the oral bioavailability, D is the oral dose, τ is the dosing interval, and N is the number of bacteria at h. X 1 () and X 2 () represent steady-state trough concentrations in the gut compartment 9

10 and the central compartment, respectively. In most cases, X 2 () was calculated using equation (7-1). However, when k a is equal to k e, the denominator of equation (7-1) becomes, which means that X 2 () cannot be calculated. In such cases, equation (7-2) was used instead. PK/PD index mapping The dose at which the number of bacteria at 24 h is equal to that at h was determined using the fminbnd function of MATLAB, which is based on golden section search and parabolic interpolation. This dose was defined as the static dose (Dose static,n ), where n is the daily dosing frequency. For each static dose (Dose static,1, Dose static,2, Dose static,4, and Dose static,8 ), AUC/MIC was calculated using fixed parameters F, V d, f p, k a, k e, ε, γ, λ, and EC 5. Then, AUC/MIC values corresponding to each of the four different dosing schedules were obtained. The ratio of the maximum to minimum among the four values was defined as the index ratio (AUC/MIC) for the parameters F, V d, f p, k a, k e, ε, γ, λ, and EC 5. The same calculation was used to determine the index ratios for C max /MIC and TAM. 1

11 When the index ratio of a PK/PD index approximates 1, the PK/PD index can be regarded as robust, regardless of the dosing schedule. The PK/PD index map was developed by determining the index ratios after varying two selected parameters as described below. (1) Effect of varying k a and k e PK/PD index maps were generated by varying k a (from.1 to 6 h -1 ) and k e (from.5 to 1 h -1 ) at 4 different ε γ pairs (ε = 3 h -1, γ = 1; ε = 3 h -1, γ = 3; ε = 1 h -1, γ = 1; and ε = 1 h -1, γ = 3). Other parameters were fixed at the following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, and EC 5 = 1 μg/ml. (2) Effect of varying ε and γ PK/PD index maps were generated by varying ε (from 1.5 to 15 h -1 ) and γ (from.5 to 1) at 4 different k e values (.1,.2,.5, and 1 h -1 ). Other parameters were fixed at the following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, k a = 1 h -1, and EC 5 = 1 μg/ml. Dosing schedule mapping For the static doses of once daily (Dose static,1 ) and 4 times daily dosing (Dose static,4 ), the total daily dose amounts required to achieve the same antibacterial outcome were calculated, and their ratio (1 11

12 Dose static,1 /4 Dose static,4 ) was defined as the dose ratio. When the dose ratio is <1, once daily dosing requires less amount of antibiotic than 4 times daily dosing to exert the same antibacterial effect, suggesting that single dose is better than multiple dosing; a ratio >1 suggests that 4 times daily dosing is better. The dosing schedule map was developed by calculating the dose ratios after varying ε (from 1.5 to 15 h -1 ) and γ (from.5 to 1) at 4 different k e values (.1,.2,.5, and 1 h -1 ). Other parameters were fixed at the following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, k a = 1 h -1, and EC 5 = 1 μg/ml. Effect of MIC on the calculation of TAM The static dose of each dosing schedule was calculated at 4 different k e values (.1,.2,.5, and 1 h -1 ) under the following condition: F = 1, V d = 1 L/kg, f p = 1, k a = 1 h -1, ε = 3 h -1, γ = 1, λ = 1 h -1, and EC 5 = 1 μg/ml. MIC was theoretically determined to be.5 μg/ml from the given parameters by using equation (5). To investigate the impact of the accuracy of MIC on the calculation of TAM, TAM values at the static doses were calculated using 2 different MIC values around the theoretical value (.4 and.6 μg/ml). Actual data collection The plasma concentration of 6 antibiotics on the market (arbekacin, cefditoren, levofloxacin, 12

13 tebipenem, vancomycin, and azithromycin) belonging to different classes (aminoglycoside, cephem, fluoroquinolone, carbapenem, glycopeptide, and macrolide, respectively) were obtained from the package inserts (Package insert of Habekacin Injections 7th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Meiact MS Tablets 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Cravit Tablets 7th ed. Tokyo, Japan, Daiichi Sankyo Company, Limited; 211; Package insert of Orapenem Fine Granules 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Vancomycin 12th ed. Osaka, Japan, Shionogi & Co., Ltd.; 29; Package insert of Zithromac Tablets 18th ed. Tokyo, Japan, Pfizer Japan Inc.; 213). The elimination rate constant (k e ) was obtained by fitting the plasma concentration to a 1-compartment model (weight: 1/conc 2 ). Parameter optimization was performed using the computer program WinNonlin (version 6.3; Certara, Saint Louis, MO). In the case of azithromycin, only the data up to 24 h were used because the plasma concentrations up to 72 h did not align well with a 1-compartment model. In vitro antibiotic parameters for arbekacin, cefditoren, levofloxacin, tebipenem, and vancomycin were obtained from the literature. 1) For azithromycin, the killing profile derived by den Hollander et al. 11) was converted to a time-kill curve. The number of bacteria was transformed to its natural logarithm, and the initial slope of the time-kill curve was plotted against the drug concentration (C). The growth rate of bacteria (λ) was determined from the slope in the absence of drug. Other in vitro 13

14 antibiotic parameters (ε, γ, and EC 5 ) were determined by fitting these plots to the following equation. 14

15 Results PK/PD index mapping (1) Effect of varying of k a and k e on the index ratio PK/PD index map is shown in Figure 2. According to the flip-flop phenomenon in pharmacokinetics, the k a and the k e are essentially interchangeable in this pharmacokinetic model analysis. The analysis regarding the case of k a < k e (shown in gray triangles in Figure 2) can be substituted for the equivalent case by alternating k a and k e to one another. For each ε γ pair, k e had a significant effect on the index ratio, whereas k a had a marginal influence on the index ratio when k a > k e. (2) Effect of varying ε and γ on the index ratio The PK/PD index value is dependent on ε/λ, not their absolute values (Supplemental material 2). Small ε/λ corresponds to drugs that demonstrated time-dependent antibacterial activity, where saturation of the killing rate occurred at low multiples of the MIC usually around four to five times the MIC. 12) Concentrations above these values did not kill the organisms any faster. Large ε/λ corresponds to those drugs that demonstrated concentration-dependent characteristics, where bactericidal activity increased with increased concentrations of the antibiotic, across a wide range of concentrations. The PK/PD 15

16 index can be regarded as a good predictor when the index ratio approximates 1 (shown in red), and a poor predictor when its index ratio is large (shown in pale blue) (Figure 3). The map suggests that the best PK/PD index is dependent on both of ε/λ and γ. The map appeared differently with different k e values, showing that the k e is another important factor. This is consistent with the observation by the k a k e maps in Figure 2. For time-dependent drugs (i.e., small ε/λ), TAM was always a good predictor, regardless of γ and k e ; if k e was low, AUC/MIC was also a good predictor. For concentration-dependent drugs (i.e., large ε/λ), the predictability of AUC/MIC and Cmax/MIC was dependent on γ and k e. Generally, AUC/MIC was a good predictor if γ was small; Cmax/MIC was a good predictor if γ was large. If k e was large, both AUC/MIC and Cmax/MIC were poor predictors at γ = 3 4. Dosing schedule mapping The dosing schedule map was developed by plotting the dose ratio. According to the dosing schedule map, the dose regimen was very important for antibiotics with large k e (Figure 4). For time-dependent drugs, divided dosing was strongly recommended. For concentration-dependent drugs, single dosing was preferred if γ was large. In contrast, for antibiotics with small k e, the selection of the dose regimen exerted only a slight influence on the treatment effectiveness. A comparison between the 16

17 recommendation by the dosing schedule map and the clinical usage is summarized in Table 4. Most of the clinical dosing regimens were consistent with those suggested by the dosing schedule map, supporting the validity of the map. Effect of MIC on the calculation of PK/PD indices Small differences in MIC strongly affected the calculated TAM values when k e was low and the compound was administered multiple times per day (Figure 5). For example, if k e =.1 h -1, the theoretical TAM value (MIC =.5 μg/ml) at the static dose for a drug administered 4 times daily (Dose static,4 ) was 54%; the calculated TAM values at Dose static,4 was 1% and % with MIC values of.4 and.6 μg/ml, respectively. However, if k e = 1 h -1 and the drug was administered 4 times daily, the theoretical TAM value (MIC =.5 μg/ml) at Dose static,4 was 5%, and the calculated TAM values were 58% and 46% with MIC values of.4 and.6 μg/ml, respectively. Conversely, variability in AUC/MIC and Cmax/MIC was always inversely proportional to the variability in MIC. Comparison between prediction and actual data Actual antibiotic and pharmacokinetic parameters of 6 antibiotics from different classes are summarized in Tables 1 and 2, and plotted on the PK/PD index maps and the dosing schedule maps. 17

18 Because maps are provided only for k e =.1,.2,.5, and 1 h -1, each drug was plotted on the map with the k e that was closest to the actual value (Figures 3 and 4). The PK/PD index map shows that the in vivo effects of cefditoren and tebipenem are associated with TAM, because these compounds lie in the area of good predictor (red color) on the TAM map, and in the area of a poor predictor (blue pale color) on the AUC/MIC and the Cmax/MIC maps (Figure 3). Similarly, the in vivo effects of arbekacin, levofloxacin, and azithromycin are associated with AUC/MIC. According to the dosing schedule map, divided dosing was better than once daily dosing for cefditoren, tebipenem, and vancomycin (Figure 4). For arbekacin, the dosing schedule map predicts that divided dosing might have a slightly better outcome. Conversely, administration frequency has only limited influence on the in vivo effectiveness of levofloxacin and azithromycin. 18

19 Discussion A PK/PD index map was devised to assess the optimal PK/PD index for a given antibiotic with the use of a model-based PK/PD analysis. The map suggests that the elimination rate constant (k e ) is an important in vivo PK parameter for selecting the relevant PK/PD index; the absorption rate constant (k a ) plays a marginal role. Selection of the optimal PK/PD index was also dependent on in vitro characteristics, including the ratio of the maximum kill rate constant to the growth rate of bacteria without drug (ε/λ) and the Hill coefficient (γ) of the concentration kill rate curve. The PK/PD classifications by the map were mostly in agreement with convention; TAM is a good index for the in vivo effects of time-dependent drugs, and Cmax/MIC and AUC/MIC are good indices for concentration-dependent drugs. 2) The selection of Cmax/MIC versus AUC/MIC was predominantly dependent on γ. Our results suggest that indices incorporating MIC must consider the error in the measured value inherent in the method used to derive it, i.e. 2-fold dilution concentration series. 4) Errors in AUC/MIC and Cmax/MIC are always inversely proportional to the error in MIC, irrespective of k e. Conversely, small errors in MIC are amplified in the calculation of TAM when k e is low (Figure 5). For example, the calculated AUC/MIC and Cmax/MIC always decrease by 33% when the MIC changed from.4 to 19

20 .6 μg/ml, irrespective of the dosing frequency and k e. Our simulation demonstrated that the calculated TAM values at Dose static,4 for a drug with small k e (.1 h -1 ) decreased dramatically from 1% to % when the MIC changed from.4 to.6 μg/ml, whereas the reduction was only 12% (from 58% to 46%) for a drug with large k e (1 h -1 ). Thus, our results suggest that TAM should not be used for a drug with k e less than.2 h -1, even if the drug belongs to the time-dependent category. The PK/PD index map suggests that the in vivo effects of cefditoren and tebipenem are related more closely with TAM, whereas those of arbekacin, levofloxacin, and azithromycin are associated with AUC/MIC (Figure 3). These predictions by the map are largely in good agreement with the clinical results, 2 4,7) suggesting that the map is fairly reliable (Table 3). Regarding azithromycin, the k e estimated using WinNonlin (.94 h -1 ) was quite different from that calculated from the half-life on the package insert (.11 h -1 ). This difference can be explained by the different data used to estimate the parameter. The index ratio of AUC/MIC always decreases with a decreasing k e when the values of ε/λ and γ are fixed (Figure 2). According to the PK/PD index map of k e =.1 h -1, AUC/MIC is a good predictor for azithromycin (Figure 3), because the k e of azithromycin is smaller than.1 h -1 in either case. There has been some difference between the clinically observed effect and the conventional classification of the antibacterial effect. For example, aminoglycosides have generally been classified 2

21 in the Cmax/MIC or the AUC/MIC category 2,4) because they exhibit an obvious concentration-dependent antibiotic profile (large ε). However, this is inconsistent with many clinical studies in which in vivo antibacterial effects were similar regardless of the dosing frequency within the same total daily dose, 13 15) suggesting that the Cmax/MIC is a poor index for aminoglycosides. The PK/PD index map (Figure 3) indicates that the Cmax/MIC is a poor index for arbekacin, an aminoglycoside, despite its concentration-dependent characteristic. For azithromycin, a time-dependent drug, convention dictates that the in vivo efficacy is dependent on TAM; however the clinical observation shows that the AUC/MIC is a good index. 16) Generally this inconsistency has been attributed to factors like post antibiotic effect (PAE) or sub-mic effect (SME). 17) However, the PK/PD index map would suggest that AUC/MIC is a good index for a time-dependent drug such as azithromycin, without employing additional factors. These observations suggest that the PK/PD index map can be a better predictor of the antibacterial effect than the conventional classification. Regarding vancomycin, although AUC/MIC and TAM have been identified as predictive indices, the PK/PD index map showed that none of the conventional PK/PD indices seemed to be useful. Since the k e is.16 h -1, the TAM cannot be appropriate for practical use as mentioned above. The Cmax/MIC appeared to be a poor index and the AUC/MIC would be of only limited use. Thus, the PK/PD index map deduced that none of conventional PK/PD indices are predictable and the clinical regimen should be decided by a 21

22 model-based PK/PD analysis on a case-by-case basis. The PK/PD index map shows that the selection of the PK/PD index is dependent on the k e of antibiotics. It is known that pharmacokinetic parameters show inter-individual variability in humans for various reasons. Drug-drug interactions can also change the pharmacokinetics of antibiotics; for example, concomitant probenecid administration decreases the k e of some cephalosporin antibiotics, primarily by reducing their renal clearance. 18) Renal impairment also accounts for the decreased renal clearance of antibiotics such as ciprofloxacin and levofloxacin. 19) Single nucleotide polymorphisms on the genes encoding metabolic enzymes and transporters 2 22) may affect the metabolism and excretion of some antibiotics. Pediatric patients may exhibit different pharmacokinetic properties compared to adults. 23) In such cases, a different PK/PD index might have a better correlation with the therapeutic efficacy due to an individual difference in the k e. Since the k e can also have some species differences between experimental animals and humans, 24) a PK/PD index identified in an animal study would not be always applicable to the clinical prediction. In addition to the PK/PD index map, the dosing schedule map was developed to predict the relative effectiveness of once daily versus divided dosing (Figure 4). According to the dosing schedule map, the selection of the dosing regimen has little effect on the clinical outcome for drugs with low k e, such as levofloxacin and azithromycin. In contrast, for drugs with large k e, the dosing regimen exerts a greater 22

23 influence. The map suggests that divided dosing is better for time-dependent drugs such as cefditoren, tebipenem, and vancomycin. Contrary to convention, divided dosing might be better for a concentration-dependent drug when it has very low γ and high k e. One limitation of this study is that we only considered the in vivo effect of antibiotics. Risks such as the occurrence of adverse events and the evolution of resistant strains were beyond our scope because incorporating the quantitative analysis of these risk factors is overly complicated. However, benefits and risks of a drug are considered simultaneously in deciding the clinical dosage and dose regimen. For example, aminoglycosides are known to cause nephrotoxicity. 25) Because the trough concentration is associated with the frequency of toxicity, an extended dosing interval is often recommended. Another example is QT interval prolongation caused by fluoroquinolones, 26) in which Cmax is associated with the risk of torsade de pointes in clinical use. 27) Antibiotic resistance is also a significant issue posed by antibiotic treatment. The concept of the mutant selection window (MSW) was introduced to optimize the dosing regimens. 28) Since drug resistance is acquired within the MSW, the drug concentration should exceed the MSW for a particular duration of the dosing interval. Another limitation is that these maps are only applicable for antibiotics that induce cell death, because the PK/PD simulation was performed based on the enhanced-death constant-replication model. Strictly speaking, for an antibiotic that inhibits cell replication, further analysis with a model 23

24 incorporating the inhibition of replication is necessary. However, it is not easy to determine whether the drug effect involves replication inhibition. In the PD model of most studies, only the increase in death rate is considered. 29) Furthermore, for drugs with other pharmacological effects, additional analysis based on an appropriate PD model is required. Preparation of a PK/PD index map and a dosing schedule map based on different PD models would be beneficial to expand the concept of these maps for a wide variety of drugs. In summary, a PK/PD index map was proposed to assess the predictability of in vivo efficacy of each PK/PD index. The underlying assumption in deriving the index map was that the bactericidal activity of a drug in vitro is identical to that in vivo. The fact that most of the clinical results show good agreement with the predictions obtained from the map suggests this model analysis is reliable. The map also suggests that the appropriate PK/PD index for each antibiotic is dependent on both in vitro (ε/λ and γ) and in vivo PK (k e ) parameters. Moreover, a dosing schedule map was generated using this model-based analysis to predict whether once daily or divided dosing is more effective. The PK/PD index map and the dosing schedule map are expected to be a practical guide for optimizing antibiotic therapy, by exploiting the advantages of the model-based analysis without the need for advanced PK/PD modeling and computer simulation. 24

25 References 1) Regoes, R. R., Wiuff, C., Zappala, R. M., Garner, K. N., Baquero, F. and Levin, B. R.: Pharmacodynamic functions: a multiparameter approach to the design of antibiotic treatment regimens. Antimicrob. Agents Chemother., 48: (24). 2) Craig, W. A.: Pharmacokinetic/pharmacodynamic parameters: rationale for antibacterial dosing of mice and men. Clin. Infect. Dis., 26: 1 1; quiz (1998). 3) Czock, D., Markert, C., Hartman, B. and Keller, F.: Pharmacokinetics and pharmacodynamics of antimicrobial drugs. Expert Opin. Drug Metab. Toxicol., 5: (29). 4) Nielsen, E. I., Cars, O. and Friberg, L. E.: Pharmacokinetic/pharmacodynamic (PK/PD) indices of antibiotics predicted by a semimechanistic PKPD model: a step toward model-based dose optimization. Antimicrob. Agents Chemother., 55: (211). 5) Moore, R. D., Lietman, P. S. and Smith, C. R.: Clinical response to aminoglycoside therapy: importance of the ratio of peak concentration to minimal inhibitory concentration. J. Infect. Dis., 155: (1987). 25

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30 27) Redfern, W. S., Carlsson, L., Davis, A. S., Lynch, W. G., MacKenzie, I., Palethorpe, S., Siegl, P. K. S., Strang, I., Sullivan, A. T., et al.: Relationships between preclinical cardiac electrophysiology, clinical QT interval prolongation and torsade de pointes for a broad range of drugs: evidence for a provisional safety margin in drug development. Cardiovasc. Res., 58: (23). 28) Drlica, K. and Zhao, X.: Mutant selection window hypothesis updated. Clin. Infect. Dis., 44: (27). 29) Czock, D. and Keller, F.: Mechanism-based pharmacokinetic-pharmacodynamic modeling of antimicrobial drug effects. J. Pharmacokinet. Pharmacodyn., 34: (27). 3

31 Tables Table 1. In vitro antibiotic parameters Parameters were obtained from the literature 1) or calculated from the concentration-killing curve. 11) λ is the growth rate of bacteria without drug, ε is the maximum kill rate constant, and γ is the Hill coefficient of the concentration-kill rate curve. Class Bacterial species ε/λ γ Arbekacin aminoglycoside MRSA Cefditoren cephem S. pneumoniae 2.2, , 1.4 Levofloxacin fluoroquinolone S. pneumoniae 3.1, , 2.63 Tebipenem carbapenem S. pneumoniae 2.5, , 3.27 Vancomycin glycopeptide MRSA Azithromycin macrolide S. pneumoniae MRSA: methicillin-resistant Staphyloccocus aureus 31

32 Table 2. In vivo pharmacokinetic parameters in human The plasma concentration was obtained from package inserts (Package insert of Habekacin Injections 7th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Meiact MS Tablets 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Cravit Tablets 7th ed. Tokyo, Japan, Daiichi Sankyo Company, Limited; 211; Package insert of Orapenem Fine Granules 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Vancomycin 12th ed. Osaka, Japan, Shionogi & Co., Ltd.; 29; Package insert of Zithromac Tablets 18th ed. Tokyo, Japan, Pfizer Japan Inc.; 213). The elimination rate constant (k e ) was obtained by fitting the plasma concentration to a 1-compartment model using WinNonlin. For azithromycin, plasma concentrations up to 24 h were used for the calculation. Dosing amount and route k e (h -1 ) Arbekacin 2 mg, iv infusion (1 h).3 ±.3 Cefditoren 2 mg, po.56 ±.23 Levofloxacin 5 mg, po.12 ±.2 Tebipenem 25 mg, po.89 ±.1 Vancomycin 5 mg, iv infusion (1 h).16 ±.4 32

33 Azithromycin 5 mg, po.94 ±.33 33

34 Table 3. Comparison of antibiotic selection: recommendation by pharmacokinetic/pharmacodynamic (PK/PD) index map vs. conventional classification Conventional classification was quoted from the literature. 2 4,7) For arbekacin, levofloxacin, and vancomycin, two indices have been identified depending on the source. It is conventionally suggested that the post-antibiotic effect (PAE) is involved in the in vivo effect of azithromycin. PK/PD index map Conventional classification Arbekacin AUC/MIC AUC/MIC, Cmax/MIC Cefditoren TAM TAM Levofloxacin AUC/MIC AUC/MIC, Cmax/MIC Tebipenem TAM TAM Vancomycin no good index # AUC/MIC, TAM Azithromycin AUC/MIC AUC/MIC (PAE) AUC: area under the curve; MIC: minimum inhibitory concentration; TAM: time above MIC #: for vancomycin, model-based PK/PD analysis is necessary on a case-by-case basis 34

35 Table 4. Comparison of daily dosing regimen: recommendation by dosing schedule map vs. clinical regimen Clinical regimens in Japan and U.S. were quoted from documents (Package insert of Habekacin Injections 7th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Meiact MS Tablets 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Cravit Tablets 7th ed. Tokyo, Japan, Daiichi Sankyo Company, Limited; 211; Package insert of Orapenem Fine Granules 5th ed. Tokyo, Japan, Meiji Seika Pharma, Co., Ltd.; 211; Package insert of Vancomycin 12th ed. Osaka, Japan, Shionogi & Co., Ltd.; 29; Package insert of Zithromac Tablets 18th ed. Tokyo, Japan, Pfizer Japan Inc.; 213; Label of Spectracef (cefditoren pivoxil) Tablets, Cornerstone Therapeutics Inc., Cary, NC, FDA Reference ID: 31793; Full prescribing information for LEVAQUIN (levofloxacin) Tablet, Janssen Pharmaceuticals, Inc., Titusville, NJ, FDA Reference ID: ; Full prescribing information for VANCOCIN (vancomycin hydrochloride, USP) Capsules, ViroPharma Incorporated, Exton, PA, FDA Reference ID: ; Label of ZITHROMAX (azithromycin tablets), Pfizer Labs, NY, NY, FDA Reference ID: ). Dosing schedule map Clinical regimen Clinical regimen in Japan in the U.S. 35

36 Arbekacin divided once once (unapproved) Cefditoren divided >> once 3 times twice Levofloxacin divided = once once once Tebipenem divided >> once twice (unapproved) Vancomycin divided >> once 2 4 times 3 4 times Azithromycin divided = once once once 36

37 Legends for figures Figure 1. Schematic illustration of pharmacokinetic/pharmacodynamic (PK/PD) model The PK/PD model comprises 2 units: the PK profile of an antibiotic in the human body and the time profile of the number of bacteria at the infection site. These units are related under the assumption that the free plasma concentration at a certain point in time determines the kill rate of bacteria at the same time. k a, the absorption rate constant from the gut compartment to the central compartment; k e, the elimination rate constant from the central compartment; λ, the growth rate of bacteria without drug; ε, the maximum kill rate constant; γ, the Hill coefficient; X 1, the amount of a drug in the gut compartment; X 2, the amount of a drug in the central compartment; N, the number of bacteria at the infection site. Figure 2. Pharmacokinetic/pharmacodynamic (PK/PD) index map (k a k e plot) The PK/PD index maps with regard to AUC/MIC, Cmax/MIC, and TAM are depicted, varying k a (from.1 to 6 h -1 ) and k e (from.5 to 1 h -1 ) at 4 different ε γ pairs (ε = 3 h -1, γ = 1; ε = 3 h -1, γ = 3; ε = 1 h -1, γ = 1; and ε = 1 h -1, γ = 3). Other parameters were fixed at the following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, and EC 5 = 1 μg/ml. According to the flip-flop phenomenon of pharmacokinetics, k a and k e are essentially interchangeable in this pharmacokinetic model analysis. The analysis regarding the 37

38 case of k a < k e (shown by the gray triangles) can be substituted for the equivalent case, alternating k a and k e one another. Figure 3. Pharmacokinetic/pharmacodynamic (PK/PD) index map (ε/λ γ plot) The PK/PD index maps with regard to AUC/MIC, Cmax/MIC, and TAM are depicted, varying ε (from 1.5 to 15 h -1 ) and γ (from.5 to 1) at 4 different k e values (.1,.2,.5, and 1 h -1 ). Other parameters were fixed at following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, k a = 1 h -1, and EC 5 = 1 μg/ml. The parameters of actual antibiotic drugs are from Tables 1 and 2. TAM is not suitable for practical use for a drug with low k e (surrounded by the gray rectangle). Figure 4. Dosing schedule map (ε/λ γ plot) Static daily dose was determined for once daily and 4 times daily dosing; the ratio was defined as dose ratio. The dose ratios were plotted, varying ε (from 1.5 to 15 h -1 ) and γ (from.5 to 1) at 4 different k e values (.1,.2,.5, and 1 h -1 ) to generate the dosing schedule map. Other parameters were fixed at following values: F = 1, V d = 1 L/kg, f p = 1, λ = 1 h -1, k a = 1 h -1, and EC 5 = 1 μg/ml. The parameters of actual antibiotic drugs are from Tables 1 and 2. 38

39 Figure 5. Effect of minimum inhibitory concentration (MIC) on the time above MIC (TAM) To investigate the effect of varying MIC, TAM values were calculated assuming 3 different MIC values (.4,.5, and.6 μg/ml). TAM is very sensitive to MIC when k e is small and dosing is frequent. 39

40 Figure 1 pharmacokinetic model unit bacterial system unit dose gut compartment X 1 k a central compartment X 2 k growth (λ) N k e (ε, γ) bactericidal activity 4

41 index ratio TAM k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) Cmax/MIC k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) AUC/MIC k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) k e (h -1 ) Drug Metabolism and Pharmacokinetics (DMPK) Advance Publication by J-STAGE Figure 2 ε=3 h -1, γ=1 ε=3 h -1, γ=3 ε=1 h -1, γ=1 ε=1 h -1, γ= k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) good predictor ; 1~1.2 ; 1.2~1.5 ; 1.5~2 ; 2~5 ; 5~ poor predictor k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) k a (h -1 ) 41

42 Drug Metabolism and Pharmacokinetics (DMPK) Advance Publication by J-STAGE Figure ε (h-1) γ arbekacin ε (h-1) 15 vancomycin levofloxacin ε (h -1) 8 arbekacin ε (h -1) 15 vancomycin 2 azithromycin 5 1 εε/λ (h -1) 1 ε (h-1) 15 arbekacin εε/λ (h -1) ε (h-1) tebipenem 5 1 ε (h -1) ε (h -1) 15 8 cefditoren γ cefditoren 6 γ γγ levofloxacin 1 tebipenem γ azithromycin γ 6 TAM cefditoren 6 8 γγ Cmax/MIC γ azithromycin 1 γ levofloxacin 2 1 index ratio vancomycin 6 4 ke=1 h-1 1 γ 6 ke=.5 h-1 γ 1 good predictor ; ; ; ; 2 5 ; 5 poor predictor ke=.2 h-1 1 γγ AUC/MIC ke=.1 h-1 tebipenem 5 1 εε/λ (h -1) εε/λ (h -1) 15

43 dose ratio γ γ γ γ γ Drug Metabolism and Pharmacokinetics (DMPK) Advance Publication by J-STAGE Figure 4 better for once daily dose ; ~.5 ;.5 ~.8 ;.8 ~1.25 ;1.25 ~2 ;2 ~ better for divided dose k e =.1 h -1 k k e =1 h -1 e =.2 h -1 k e =.5 h -1 levofloxacin azithromycin ε (h -1 ) vancomycin arbekacin ε (h -1 ) cefditoren ε (h -1 ) ε (h -1 ) ε/λ ε/λ ε/λ ε/λ tebipenem 43

44 TAM TAM (% (% of 24 h) h) TAM TAM (% (% of 24 h) h) TAM TAM (% (% of 24 h) h) TAM TAM (% of 24 h) h) Drug Metabolism and Pharmacokinetics (DMPK) Advance Publication by J-STAGE Figure 5 1 k e =.1 h -1 1 k e =.2 h MICobs=.4 MICobs=.5 4 MICobs=.6 once twice 4 times 4 88 times once twice 4 times 4 88 times 2 1 k e =.5 h -1 k e =1 h -1 8 MIC=.4 μg/ml MIC=.5 μg/ml MIC=.6 μg/ml MICobs=.4 MICobs=.5 MICobs= MICobs=.4 MICobs= once twice 4 times times MICobs=.5 4 MICobs=.6 2 once twice 4 times 4 88 times MICobs=.5 MICobs=.6 44

45 Supplemental material 1: Calculation of initial condition of antibiotics at steady state The amounts of antibiotic in the gut (X 1 ) and central compartments (X 2 ) after single administration are expressed in the following equations:... (S1-1)... (S1-2)... (S1-3) Since the antibiotic is administered at the constant interval of τ, the amount of antibiotic at the time of administration (t = in the main text) at steady state can be expressed as follows:... (S1-4)... (S1-5)... (S1-6) 45

46 The equations (S1-4) to (S1-6) are summarized in equations (6), (7-1), and (7-2), respectively, in the main text. 46

47 Supplemental material 2: Interrelated effects of λ and ε values on bacterial kinetics The following is an explanation why static dose is dependent only on the ratio between ε and λ (ε/λ), and not on the absolute values, for the simple bacterial kinetic model shown in equations (1) to (4) in the main text. First, the following ordinary differential equations were prepared to describe the amount of bacteria N after the administration of antibiotics:... (S2-1)... (S2-2) where dot denotes time derivative. In this section, two different bacteria N 1 and N 2, whose parameter values were λ 1 and λ 2 for the growth rates of bacteria without drug, were treated with an antibiotic drug having the same parameter values for the Hill coefficient (γ), the concentration of drug at which 5% of the maximum effect is obtained (EC 5 ), and the concentrations over time (C(t)) against N 1 and N 2. The maximum kill rate constants of the drug were different for the two bacteria: ε 1 and ε 2 for N 1 and N 2, respectively. When kλ 1 = λ 2 and kε 1 = ε 2, equations (S2-3) to (S2-6) can be derived from equations (S2-1) and (S2-2). 47

48 ... (S2-3)... (S2-4)... (S2-5) (S2-6) Let α be the ratio between the amount of bacteria N 1 before and after time τ:... (S2-7) By combining equations (S2-6) and (S2-7), equations (S2-8) to (S2-1) can be derived. (S2-8) (S2-9)... (S2-1) 48

49 If N 1 (τ) equals N 1 () ( α = 1), N 2 (τ) also equals N 2 (). Hence, we only need to consider ε/λ values in our analyses. 49