kinetics and quantification of antibacterial effects of beta-lactams, macrolides, and quinolones...

Infection 33 · 2005 · Supplement 2 © URBAN & VOGEL 3

Kinetics and Quantification of Antibacterial Effects of Beta-Lactams, Macrolides, and Quinolones against Gram-Positive and

Gram-Negative RTI PathogensK.-J. Schaper, S. Schubert, A. Dalhoff

Abstract Traditionally, the in vitro activity of antibacterial agents is characterized by their minimal inhibitory concentrations. However, these endpoints are, by nature, discrete and do not provide information on time-dependent killing of the bacteria during the incubation period. Nevertheless, the pharmacodynamic characteristics of antibacterial agents are almost always defined by correlating a static endpoint describing the antibacterial activity of an agent with the pharmacokinetics, describing the time-dependent fluctuation of drug concentrations. This approach is basically a contra-diction in itself. Therefore, it would be more logical to correlate pharmacokinetics to in vitro parameters describing the time- and concentration-dependent antibacterial action of an agent. Thus, experimental methods and mathematical models quantifying the decrease in growth rate of a bacterial population due to the action of an antibacterial agent as a function of time and drug concentration have been applied to quantitate their pharmacodynamics. The effect of nine antibacterial agents representing drug classes of penicillins, cephalosporins, penems, macrolides, and fluoroquinolones were mathematically analyzed by using three different but related models. The kill rate, maximal kill, the 50%-effective concentration (EC50), the Hill coefficient, and concentra-tions and times needed to obtain a 1,000-fold decrease of the initial number of viable counts were calculated. Both the phenotypic description of the time-kill curves and these five parameters mirror the bacteriostatic or bactericidal activity of all nine agents studied as a function of time and concentration. Therefore, it would be more logical to correlate a parameter quantifying the kinetics of antibacterial in vitro activity with the pharmacokinetics of the drug, thus, replacing static endpoints like minimal inhibitory concentrations.

Infection 2005; 33 (Suppl 2): 3–14DOI 10.1007/s15010-005-8202-2

IntroductionMany methods have been used to elucidate and to describe the effect of antibacterials on the growth of microorgan-isms. Most commonly, minimum inhibitory concentrations

(MICs) are determined; less frequently, minimum bacte-ricidal concentrations (MBCs) are also measured. MICs and MBCs are assessed by exposing bacteria to a constant antibiotic concentration for approximately 18 h. Such static measures provide information on inhibition or killing at an endpoint of 18 h incubation. However, these endpoint measurements are, by nature, discrete and do not provide information on the time-dependent killing rate during the incubation period.

Furthermore, the determination of the MIC can be regarded as being inadequate for the characterization of a bactericidal agent like a beta-lactam, an aminoglycoside or a quinolone. Likewise, the discrete MBC value can be regarded as being inadequate for the characterization of those bactericidal agents exhibiting a concentration-depen-dent activity.

Experimental studies as well as theoretical, math-ematical models discriminating between time- and con-centration-dependent growth or inhibition describe the kinetics and mechanisms of the antibacterial action more adequately than MIC or MBC values.

One approach is to describe phenotypically the killing and regrowth of bacteria in vitro and to derive parameters like minimum antibiotic concentration (MAC) or optimal bactericidal concentration (OBC) from the time kill curves [1, 2].

A variety of attempts to quantify the rate of killing were made, such as measuring the individual cell length and time to lysis [3], the log kill per generation [4], the kill rate [5–7] or the reduction in viable counts per time inter-val [8, 9], calculation of the area under the killing curve [10, 11] and mathematical modeling [12]. These methods

Infection Supplement Article

K.-J. Schaper Structural Biochemistry, Research Center Borstel, Leibniz Center for Medicine and Biosciences, Borstel, Germany S. Schubert, A. Dalhoff (corresponding author)Institute for Infection Medicine, University Hospital Schleswig-Holstein, Campus Kiel, Brunswiker Str. 4, 24105 Kiel, Germany; e-mail: [email protected]

quantify the time-dependent inhibition of bacterial growth at one given concentration.

On the basis of the pioneering studies of Garret et al. [13–15] and their application by Seydel et al. [16–18] ex-perimental methods and mathematical models have been developed to study the kinetics of bacterial growth or in-hibition of growth and to quantify the decrease in growth rate caused by the antibacterial agent as a function of the drug concentrations.

We have used this approach to quantify the concentra-tion-dependent or concentration-independent effects of nine antibacterial agents representing three different drug classes: first, beta-lactams subgrouped into penicillins, cephalospo-rins, penems; second, macrolides; and third, quinolones.

Materials and Methods Bacterial Strains and Drugs Studied

Two strains (one ATCC strain and one recent clinical isolate) of each of the following species were tested: Streptococcus pneu-moniae (S. pneumoniae ATCC 6303, S. pneumoniae 4241), Mo-raxella catarrhalis (M. catarrhalis ATCC 43617, M. catarrhalis Va 14497/00), Haemophilus influenzae (H. influenzae ATCC 49247, H. influenzae Va 1548/00); and one strain of each of the follow-ing was tested: Staphylococcus aureus (S. aureus 133), Escherichia coli (E. coli ATCC 25922) and Klebsiella pneumoniae (K. pneumo-niae 63).

The antibacterials moxifloxacin (MFX), amoxicillin (AMX), cefixime (CFI), cefuroxime (CFX), imipenem (IMI), meropenem (MER), faropenem (FAR), and ertapenem (ERT) as well as clar-ithromycin (CLA) studied were commercially obtained or kindly provided by the manufacturers. The MICs of the nine agents stud-

K.-J. Schaper et al. Kinetics and Quantification of Antibacterial Effects

4 Infection 33 · 2005 · Supplement 2 © URBAN & VOGEL

MIC (µg/ml)Strain MXF AMX CFI CFX IMI MER FAR ERT CLA

S. pneumoniae 0.125 0.015 0.5 0.03 0.007 0.015 0.007 0.06 0.03ATCC 6303

S. pneumoniae 4241 0.25 0.015 0.5 0.03 0.03 0.015 0.015 0.06 0.03

S. aureus 133 0.03 0.125 32a 2 0.015 0.06 0.125 0.5 0.125

H. influenzae ATCC 49247 0.06 16a 0.5 32a 1 1 4 0.5 16a

H. influenzae Va 15481/00 0.03 0.5 0.06 2 1 0.06 0.5 0.125 16a

M. catarrhalis ATCC 43617 0.03 2 0.5 1 0.015 0.003 0.5 0.007 0.125

M. catarrhalis Va 14497/00 0.06 0.007 0.06 0.5 0.015 0.003 0.06 0.015 0.03

M. catarrhalisEngland 11 0.03 0.25 0.03 1.0 0.06 0.003 0.06 0.015 0.06

E. coliATCC 25922 0.007 8a 1 nd 0.5 0.015 0.5 0.015 nd

K. pneumoniae 63 0.06 64a 0.25 nd 2 0.03 0.5 0.03 nd

a these strain/drug associations were not studied because MIC values were too high; nd: not done

Table 1Minimum inhibitory concentrations (MIC, µg/ml) of the agents studied against the test strains.

ied against the test strains are shown in table 1. MICs were deter-mined by microbroth dilution method according to DIN 58940. Bacteria were cultivated in brain-heart infusion (BHI) broth with-out any supplementation; all strains tested grew well under these experimental conditions.

Killing KineticsTime-kill studies were performed under batch culture conditions. At zero time, a static phase inoculum was added to the BHI broth to a final density of approximately 105–107 colony forming units (CFU)/ml. Simultaneously multiples of the individual MICs of the various test drugs were added to the cultures, drug concentrations ranging from one to 32 times the MIC to ensure that bioequivalent concentrations were used. Samples were taken prior to the addi-tion of the drugs and at 30 min, 1, 2, 4, 6, and 8 h and subcultured quantitatively on drug-free BHI agar. The antibacterial activity of the various agents against the test strains is expressed by the change in viable counts with time.

Mathematical Models Used for the Calculation of Kill Kinetics

The bactericidal effect (time-kill kinetics and its concentration de-pendence) was analyzed using three different but related methods. The basis of all of them is the assumption of a first-order exponen-tial decrease with time of the number N of viable cells (CFU/ml) treated by a drug at a certain concentration C (using multiples x of the MIC: C=x X MIC with x=1, 4, 8, 16, and 32) according to:

N = N0 X e –kt (1)

where N is the number of bacteria at any given time point, N0 is the number of bacteria in the initial inoculum, and k is the first-order kill rate constant.



Furthermore, it was assumed that the apparent kill rate con-stant k increases with increasing bactericidal drug concentration from k = 0 at C = 0 to a constant maximum value kmax at high drug concentration. This nonlinear concentration-response relationship generally can be quantitatively described by the Hill equation:

k = kmax X Cb / (EC50b + Cb) (2)

where EC50 is the concentration resulting in k = kmax /2, and b is the Hill coefficient characterizing the steepness and sigmoidicity of the observed concentration response profile.

Method 1 (M1) of the above-mentioned three methods con-sists in a separate fit of Eq. (1) to the time-dependent number of CFUs at a given concentration resulting (for each concentration) in an estimate of the kill rate constant k and of No. In a subsequent step Eq. (2) is fitted to the previously obtained k values. For each drug this analysis results in an estimate for kmax and EC50, which are quantitative measures of the drug’s bactericidal activity.

Method 2 (M2) combines the two steps of M1 and performs a simultaneous analysis of the time-dependent AND concentration-dependent decrease in the number of CFUs. This is possible if Eq. (2) is inserted into Eq. (1). Whereas M1 overall performs an esti-mate of 13 values (N0 and k from kill kinetics data at five different concentrations, and kmax, EC50, b from kill dynamics data) M2 requires only the estimation of eight values (five N0 values, corre-sponding to each of the five concentrations, and kmax, EC50, b, the latter three values being valid for the whole data set). Therefore, obviously for M2 the number of degrees of freedom (i.e. the dif-ference between the number of analyzed N values and the number of estimated regression coefficients) is higher than for M1. This may result in a better significance of the estimated data.

The development of method 3 (M3) was stimulated by the observation of CFU data non-monotonically decreasing with time. In these cases (and also other cases) M3 was used in the hope of obtaining more detailed information about changes of the kill rate constants with time in an experiment with a given drug concen-tration. Using a transformation of Eq. (1) for each observed time point a single-point kill rate constant can be obtained by:

k = – [ln (N / No)]/t (3)

If the change in the number of CFUs (at a given concentra-tion of drug) is well described by the model Eq. (1) then all k values obtained by Eq. (3) should be similar. In most cases this expectation was fulfilled. Generally, for each experimental con-dition the highest and the lowest k values were deleted and the mean value calculated from the remaining single-point kill con-stants. This mean value was used in the subsequent analysis of the concentration response profile as described for M1.

If the analyzed data sets are well described by model Eqs. (1) and (2) then further characteristics of the bactericidal efficiency of drugs can be calculated. A well-known equation for the calcula-tion of the half-life of a first-order exponential decay process (as assumed above) is:

t1/2 = (ln 2) / k (4)

For clinical microbiologists the more relevant time to obtain a decrease in the initial number of CFUs from N0 to N0/1000 may be calculated by the more general equation:

t1/f = (ln f) / k (5)

with f = 1000 (or any other factor). Given the case that the drug-specific constants kmax and EC50 of Eq. (2) are known then Eq. (5) combined with Eq. (2), on the one hand, provides the possibility to calculate the time to obtain a reduction of N0 to N0/f in a medium containing a given drug concentration C by:

t1/f = [(EC50b + Cb)/Cb] X (ln f)/kmax (6)

On the other hand, the concentration needed to obtain the same effect (i.e. No ➝ No/f) at a desired time t (which in this case is t1/f) may be calculated by an equation obtained by transforma-tion of Eq. (6): C = EC50 /{[t1/f X kmax/(ln f) – 1]1/b} (7)

Eq. (7) requires that t1/f X kmax > (ln f). Obviously with decreasing t1/f increasing concentrations C are needed to obtain No ➝ No/f. Therefore, at some desired (too short) time t1/f we obtain:

t1/f X kmax = (ln f) and t1/f X kmax/(ln f) – 1 = 0

and thus the unrealizable condition: C = EC50 / 0 = ∞.Furthermore, if the product (t1/f X kmax) approaches the value of (ln f) the statistical uncertainty in estimated values of kmax and EC50 leads to a very poor prediction of C by Eq. (7).

Observed raw data (CFU = F(t, C)) were analyzed using at least one of the three methods presented above (M1, M2 or M3). After comparing results (i.e. EC50, kmax, b, t1/1000 at C = 4 X MIC, and C to obtain N0/1000 after 6 h) obtained from M1 and/or M2 and/or M3, those values with the best statistical significance were collected and are listed in table 2.

Results Effects on Viable Counts

In general, moxifloxacin and the four penems tested (imi-penem, meropenem, ertapenem, and faropenem) exhib-ited the most pronounced effects against all the bacterial species studied. Among the gram-negative bacteria, moxi-floxacin eliminated the inocula of H. influenzae ATCC 49247, H. influenzae Va 15481/00 (at 8-times the MIC), E. coli and K. pneumoniae at 4-times their MICs within 4–8 h. Likewise, the penems were rapidly bactericidal against the gram-negative bacteria and eliminated the inocula of S. pneumoniae 4241 as well as S. aureus 133 at 4–8 times their MICs within 4–8 h. Among the penams and cephems, ce-fixime exhibited the most pronounced bactericidal effects; amoxicillin and in particular cefuroxime tended to be less bactericidally active than cefixime.

Application of Mathematical ModelsAs an example, some details of the analysis of two data sets generated for (i) amoxicillin against S. pneumoniae 4241 (Figures 1 and 2), and (ii) moxifloxacin against M. catarrhalis England 11 (Figures 3 and 4) are presented in the following:

Amoxicillin/S. pneumoniae 4241. Table 3 shows the ob-served raw data of the bactericidal effect of amoxicillin. Method M1: The fit of Eq. (1) to the number of CFUs/ml


Tabl

e 2

Bact

eric

idal

eff

ect c

hara

cter

istic

s obt

aine

d by

non

linea

r reg

ress

ion

anal

ysis

of t

he ti

me

depe

nden

t red

uctio

n of

CFU

s obs

erve

d at

diff

eren

t dru

g co

ncen

trat

ions

(C =

x X

MIC

with

x =

0,

1, 4

, 16,

32)

. Cod

es M

1, M

2, M

3 in

dica

te th

e ap

plie

d th

ree

met

hods

of d

ata

anal

ysis

.

Bact

eria

l str

ain

St. pn

eum

onia

eAT

CC 6

303

M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) c

onc.

n

eede

d fo

r 3

log

kill

in 6

h t(

1‰ 4

MIC

)(h)

t

for

3 lo

g ki

ll at

4 M

ICS.

pne

umon

iae

Bay

4241

M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

S. a

ureu

sBa

y 13

3 M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

K. p

neum

onia

eBa

y 63

M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

E. c

oli

ATCC

259

22 M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

Faro

pene

m

0.00

71.

7

M

20.

052

1 0.12

12 0.01

52.

4

M

20.

012

1 0.01

3.4

0.12

50.

72

M

10.

186

1 ∞b

13 0.5

4.2

M2

1.53

1 0.58

2.9

0.5

1.5

M1

1.9

1 6.7

9.2

Erta

pene

m

0.06

~ 0.

9

M

3<

0.06

∞b

0.06

1.4

M

10.

073

1 0.4

6.5

0.5

1

M

30.

64a,

c

1 ∞b

8a,c

0.03

1.7

M3

0.01

31 0.

03

4 0.01

5 1.

8

M

30.

0012

a,c

1 0.00

22

3.9a,

c

Mer

open

em

0.01

50.

6 M

30.

011

1 ∞ b

13 0.15

2.4

M

30.

031 0.

03

4.4

0.06

3.5c

M

30.

6c

1 0.3

7c 0.3

1.8

M

30.

05c

1 0.08

5

5c 0.01

55.

0 M

30.

090

1 0.02

7 3.

4

Imip

enem

0.00

7~

1.6

M

3<

0.02

8

0.03

2.6

M

30.

041 0.

03

3.5

0.01

53.

8 M

30.

053 0.

04

3 2 4.5

M

31.

8a,c

1 0.6a,

c

1.9a,

c

0.5

4.7

M

35.

6 1 1.

83 5.

6

Cefi

xim

e

0.5

1.1

M

30.

171 ∞

b

6.9

0.5

~ 0.

7 M

1<

0.5

∞b

0.25

2.4

M

30.

186

1 0.17

3.4

1 2.1

M

30.

621 0.

74

3.8

Cefu

roxi

me

0.03

~ 0.

4 M

1<

0.03

∞b

0.03

1.4

M

20.

051 0.

23

6.9

2 ~ 0.

5a M

1<

2a

∞b

Amox

icill

in

0.01

50.

7 M

30.

031 ∞

b

14 0.01

51.

4 M

30.

041 0.

22

8.4

0.12

51

M

30.

21 ∞

b

10

Mox

iflo

xaci

n

0.12

50.

55 M

20.

221 ∞

b

18 0.25

1.4

M

30.

61 2.

2

7.5

0.03

3.0

M

30.

41 0.

25

9.9

0.06

5.1

M

20.

226

3.4

0.16

2.5

Clar

ithr

omyc

in

0.03

~ 0.

5a M

1<

0.03

a

∞b

0.03

1a M

30.

007a

1a ∞b

7.6

0.12

50.

86 M

30.

51 ∞

b

16



Tabl

e 2

(con

tinue

d)

M. ca

tarrha

lisAT

CC 4

3617

M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

M. ca

tarrha

lisKi

el V

a 14

497/

00 M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

M. ca

tarrha

lisEn

glan

d 11

M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

H. in

fluen

zae

ATCC

492

47 M

IC (

µg/m

l) k(

kill)

max

. (h

-1)

EC

50 (

µg/m

l) H

ill c

oeff

. b

C(

N 0/1

000

6h)

(µ

g/m

l) t(

1‰ 4

MIC

)(h)

H. in

fluen

zae

Kiel

Va

1548

1/00

M

IC [

µg/m

l] k(

kill)

max

. [h

-1]

EC

50 [

µg/m

l] H

ill c

oeff

. b

C(

N 0/1

000

6h)

[µ

g/m

l] t(

1‰ 4

MIC

)[h]

0.5

1.3

M3

0.12

a,c

1 1.1

5.75

a,c

0.06

5.

31

M

20.

33

1 0.09

3.1

0.06

2.

95

M

20.

265

1 0.17

4.9

4

0.53

M1

6.3

1 ∞ b

18 0.5

1.3

M

11.

71 12 9.

7

0.00

7 0.

69

M

10.

0006

4 1 ∞

b

11.6

0.01

5 5.

8

M2

0.12

8

1 0.03

2

4 n.d.

0.5

1.

6

M3

3.93

51 11 13

.1

0.12

51.

5

M1

0.17

51 0.

65

8

0.00

3

0.00

3

n.d.

1

2.3

M

22.

5 1 2.

60 5 0.

061.

7 M

20.

094

1 0.2

5.6

0.01

5

0.01

56.

4c M

30.

111c

1 0.02

5c

3c n.d.

1

3.4

M

21.

40

1 0.72

2.7

1 8.3

M

15.

41 0.

9

2

0.5

1.3

M

10.

371 2.

3

6 0.06

3.

3 M

20.

091

1 0.04

8 3 n.

d.

0.5

1.

1 M

21.

13

1 ∞b

9.7

0.06

0.7

a,c

M

30.

04 a,

c

1 ∞b

11 a,

c

1 7.6

M

22 1 0.

36

1.4

0.5

7.5

M

20.

25

1 0.04

4

1 1.0

7.6

M

1 2.

51 0.

44

1.5

2 0.5a

M

20.

9a,c

1 ∞b

11.4

a,c

2 2.5

M

33.

81 3.

2

4 0.00

7 1.

5 M

10.

046

1 0.15

12 0.25

2.3

M

32.

451 2.

40

10.3

0.5

~ 0.

9 M

3<

0.5a

∞b

0.03

3.2

M

30.

151 0.

09

5 0.06

5.2

M

30.

726

1 0.20

5.3

0.03

14.6

M

10.

252.

60.

098

3.8

0.06

12.8

M

20.

331.

60.

08

1.5

0.03

14.1

M

30.

781 0.

07

3.7

0.12

50.

7 M

30.

7c1 ∞

b

25c

0.03

1.1

M

30.

031 ∞

b

7.9

0.06

1.2

M

30.

022

1 0.3

6.1

Bact

eria

l str

ain

Fa

rope

nem

Ert

apen

em

Mer

open

em

Im

ipen

em

Cef

ixim

e

Cef

urox

ime

Am

oxic

illin

Mox

iflo

xaci

n

Cl

arit

hrom

ycin

a at

C=

~1 X

MIC

the

obs

erve

d ki

ll ra

te is

~k(

kill)

max

; th

eref

ore,

EC 5

0(ca

lc.)

is <

1 X

MIC

and

is n

ot s

igni

fican

tly

diff

eren

t fr

om E

C 50 =

0; b

cal

cula

ted

k(ki

ll)m

ax is

too

low

to

obta

in 9

9.9%

kill

at

6 h

wit

h C→

∞;

c un

cert

ain

resu

lt w

ith

low

sta

tist

ical

sig

nific

ance



following exposure to amoxicillin at C = x X MIC (with x = 1, 4, 8, 16, and 32) results in five different N0 values (not shown) and in the corresponding kill rate constants which are listed in the left part of table 4.

Observed CFUs and decay curves calculated using these kill rates are shown in figure 1. A plot of the kill rates against the concentration of amoxicillin (multiples of its

MIC!) is shown in figure 2. Obviously the kill rate increases steeply at low concentrations (up to C ~ 4 X MIC) and at about C = 16 X MIC approaches a constant maximum. Such profiles with a kmax of about 1.1 h-1 are typical for the penicillins and cephalosporins tested.

By fitting the Hill equation (Eq. (2), using b = 1) to the amoxicillin rate constants of table 4 (together with k = 0 at

CFU

/ml

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0Time (h)

0

200,000

400,000

600,000

800,000

1000,000

1200,000

1400,000

1600,000

200,000

400,000

600,000

800,000

1000,000

1200,000

1400,000

1600,000

Moxifloxacin / M. catarrhalis England 11

Figure 3. Bactericidal effect of moxifloxacin against M. catarrhalis En-gland 11 as a function of time and drug concentration C (C shown in multiples of the MIC; MIC = 0.03 µg/ml). Time-dependent numbers of CFUs have been observed at concentration C=x X MIC with x=1, 4, 8, 16, and 32 (symbols ö, ∆, ˙, õ, ó, respectively). Calculated curves have been obtained using the kill rate constants of table 4. For the sake of clarity experimental CFUs at t=0 have been vertically shifted to 1.5*106, 1.45*106, 1.4*106, 1.35*106, and 1.3*106; all other CF Us as well as calculated curves have been shifted by corresponding factors.

k(ki

ll) (

h)-1

0 4 8 12 16 20 24 28 32 36 40

0 4 8 12 16 20 24 28 32 36 40

x (C = x X MIC)

0.0

2

4

6

8

10

12

14

0.0

2

4

6

8

10

12

14

Figure 4. Bactericidal effect of moxifloxacin against M. catarrhalis England 11 as a function of the drug concentration C (C shown in multiples of the MIC; MIC = 0.03 µg/ml). Plotted experimental kill rate constants (ó) of table 4 have been obtained by fitting Eq. (1) to the time-dependent number of CFUs counted at C=x X MIC with x=0, 1, 4, 8, 16, 32. The concentration response curve has been calcu-lated by Eq. (2) using values of kmax, EC50 and Hill coefficient b shown in the figure.

CFU

/ml

50000

0 1 2 3 4

0 1 2 3 4

40000

30000

20000

10000

Time (h)

0

50000

40000

30000

20000

10000

0

Amoxicillin / S. pneumoniae Bay 4241

Figure 1. Bactericidal effect of amoxicillin against S. pneumoniae 4241 as a function of time and drug concentration C (C is shown in multi-ples of the MIC; MIC = 0.015 µg/ml). Time-dependent numbers of CFUs have been observed at concentration C = x X MIC with x = 1, 4, 8, 16, and 32 (symbols ö, ∆, ˙, õ, ó, respectively). Calculated curves have been obtained using the kill rate constants of table 4. For the sake of clarity experimental CFUs at t = 0 have been vertically shifted to 50,000, 45,000, 40,000, 35,000, and 30,000; all other CFUs as well as calculated curves have been shifted by corresponding factors.

k(ki

ll) (

h)-1

0 5 10 15 20 25 30 35

0 5 10 15 20 25 30 35

x (C = x X MIC)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 2. Bactericidal effect of amoxicillin against S. pneumoniae 4241 as a function of the drug concentration C (C shown in multiples of the MIC; MIC = 0.015 µg/ml). Plotted experimental kill rate con-stants (ó) of table 4 have been obtained by fitting Eq. (1) to the time-dependent number of CFUs counted at C=x X MIC with x=0, 1, 4, 8, 16, 32. The concentration response curve has been calculated by Eq. (2) using values of kmax and EC50 shown in the figure.



C = 0) the following characteristics of the above combina-tion of drug (i.e. amoxicillin) and bacterial strain (i.e. S. pneumoniae 4241) are obtained (Eq. [8]):

Like in most other cases (see Table 2) this drug/bug association (AMX/S. pneumoniae) leads to the best solu-

tion with Hill coefficient b = 1. As the calculation has been performed with the concentration expressed by mul-tiples of the MIC the concentration needed to obtain 50%

of the maximal kill rate is EC50 = 2.13 X MIC (MIC = 0.015 µg/ml; table 2 lists the EC50 values in units of µg/ml). The concentra-

tion response profile calculated by Eqs. (2) and (8) is shown by figure 2.

Time (h) Control C = 1 X MIC C = 4 X MIC C = 8 X MIC C = 16 X MIC C = 32 X MIC

0 53500 39000 36000 49000 42000 34000 0.5 54500 40500 33000 32500 27000 26000 1 45500 29000 21500 20500 9550 11000 2 43500 15000 4850 2550 2920 1800 4 720000 (17500) 500 405 300 235 6 22500000 (58500) 110 70 35 40 8 33500000 (1100000) 20 10 1 1

Table 3Bactericidal effect of amoxicillin (MIC = 0.015 µg/ml) against S. pneumoniae 4241. CFUs/ml determined at different times and different concentrations C of drug in BHI broth without serum.

Amoxicillin, AMX; S. pneumoniae 4241 Moxifloxacin, MXF; M. catarrhalisC = x X MIC N = N0 e-kt N = N0 e-kt

x = kill constant k (h)-1 ± 95% CI kill constant k (h)-1 ± 95% CI

1 0.422 0.428 (90%) 0.051 0.1 (90%) 4 0.703 0.362 1.516 0.951 8 0.990 0.254 7.154 0.101 16 1.232 0.350 11.941 0.404 32 1.027 0.417 14.342 0.505

Table 4First order kill rate constants k and their 95% confidence intervals obtained by nonlinear regression analysis of the corresponding data in columns 3–7 of table 3 (for AMX) and of table 6 (for MXF).

Time (h) C = 1*MIC C = 4*MIC C = 8*MIC C = 16*MIC C = 32*MIC

0.5 – (0.174) (0.821) (0.884) (0.537)1 0.296 0.515 0.871 (1.481) 1.1282 (0.478) 1.002 (1.478) 1.333 (1.469)4 (0.200) (1.069) 1.199 1.235 1.2446 – 0.965 1.092 1.182 1.1248 – 0.937 1.062 – –

k(mean) 0.296 n = 1 0.855 n = 4 1.056 n = 4 1.250 n = 3 1.165 n = 3

Table 5Single-point kill rate constants k for amoxicillin and S. pneumoniae 4241 (and their mean values, see text) calculated by Eq. (3) using data of columns 3–7 of table 3.

Eq. (8) 95% CI 90% CI Eq. (9) 95% CI Eq. (10) 95% CI

EC50 (MIC units) 2.13 ± 2.31 ± 1.77 2.07 ± 1.66 2.62 ± 1.80 kmax (h-1) 1.22 ± 0.30 ± 0.23 1.21 ± 0.30 1.36 ± 0.23

n = 6; r = 0.976; s = 0.110 n = 6; r = 0.986; s = 3101 n = 6; r = 0.991; s = 0.077



The confidence intervals (CI) of Eq. (8) show that the fitted EC50 value of amoxicillin is significant only at the 90% level. A statistically more significant result was ob-tained by method M2 (simultaneous analysis of time and concentration dependence) resulting in Eq. (9).

The estimated values of EC50 and kmax in Eqs. (8) and (9) are very similar but a considerably smaller confidence interval for EC50 is obtained in Eq. (9). Note that in Eq. (9) the five N0 values calculated for the five concentration levels are not shown, and that the standard error of the estimate s is given in terms of CFU/ml, whereas in Eq. (8) and Eq. (10) it is expressed by the error in the calculated kill rate constants.

For the first example (AMX/S. pneumoniae) the best result is obtained by method M3 (single-point calculation of kill rates, see section Methods). Using Eq. (3) the mean rate constants listed in table 5 were obtained. A fit of Eq. (2) to these kill rates resulted in Eq. (10). This final result was used to calculate two additional characteristics:(a) the time required to obtain a kill of 3 log N units at the concentration of 4 X MIC, and (b) the concentration required to observe the same effect after 6 h.

Using Eq. (6) with b = 1, EC50 = 2.62, kmax = 1.36 [h]-1, f = 1000, and C = 4 a value of t1/1000 = 8.4 h is obtained. Table 3 shows that this calculated value is quite precise, because starting at C = 4 X MIC with No = 36,000 the number of CFUs/ml is reduced to 20 at t = 8 h. On the other hand, Eq.

(7) with t1/1000 = 6 h results in C = 14.45 units of MIC (with MIC = 0.015 µg/ml ➝ C = 0.22 µg/ml), whereas the observed number of CFUs/ml at t = 6 h and C = 16 X MIC is 35.

As explained before (see Materials and Methods; Mathematical Models) Eq. (7) generally requires that t1/f X kmax > (ln f). This means that with a desired 3 log N kill in 6 h (i.e. f = 1,000 and t1/f= 6) the maximal kill rate has to be kmax > (ln 1000)/6 > ~1.2 [h]-1.

Table 2 shows that in many cases this requirement was not fulfilled (see entries for k(kill)max.) and that even a concentration approaching C ~ ∞ can not lead to a kill of 3 log N units within 6 h.

Moxifloxacin/M. catarrhalis England 11. In a few cases, the kill rate was observed to be almost at zero level at C = 1 X MIC and was increasing very steeply at higher concen-trations. The consequence of this fact is that in these cases the concentration/kill rate profile could not be described by the Hill equation (Eq. [2]) with b = 1 but that a value b > 1 was required (for results see table 2). Interestingly this effect was observed mainly for moxifloxacin (vs S. pneumoniae PP 62, K. pneumoniae Bay 63, M. catarrhalis England 11, H. influenzae ATCC 49247) and only once for imipenem/S. aureus Bay 133.

Table 6 shows the observed raw data of the bacteri-cidal effect of MXF/M. catarrhalis 11. Method M1 leads to the kill rate constants listed in the right part of table 4 and

Time (h) Control C=1 X MIC C=4 X MIC C=8 X MIC C=16 X MIC C=32 X MIC

0 1.87 106 1.55 106 1.30 106 1.56 106 1.51 106 2.67 106

0.5 1.69 106 1.44 106 8.75 105 4.35 104 3.85 103 2.05 103

1 2.01 106 1.51 106 7.05 104 2.85 103 560 372 2 2.38 106 (2.50 106) 5.40 103 1.00 103 98 45 4 3.58 106 (4.80 106) 485 141 45 12 6 4.82 108 (1.63 107) 100 35 5 1 8 1.82 109 (1.95 107) 20 10 1 1

Table 6Bactericidal effect of moxifloxacin (MIC = 0.03 µg/ml) against M. catarrhalis England 11. CFUs/ml determined at different times and different concentrations C of drug in BHI broth without serum.

Time (h) C = 1 X MIC C = 4 X MIC C = 8 X MIC C = 16 X MIC C = 32 X MIC

0.5 0.147 (0.792) (7.159) (11.944) (14.344) 1 0.026 (2.915) 6.305 7.900 8.879 2 – 2.742 3.676 4.821 5.495 4 – 1.973 2.328 2.605 (3.078) 6 – 1.579 1.784 (2.103) – 8 – 1.385 (1.495) – –

k(mean) 0.087 n = 2 1.920 n = 4 3.523 n = 4 5.109 n = 3 7.187 n = 2

Table 7Single point kill rate constants k for moxifloxacin and M. catarrhalis England 11 (and their mean values) calculated by Eq. (3) using data of columns 3–7 of table 6.



the best fit for these concentration-dependent kill rates is presented in Eq. (11):

These values of EC50, kmax, and b are used for the calculation of the profiles of figures 3 and 4. Figure 3 on the one hand clearly shows that at C = 1 X MIC (MIC = 0.03 µg/ml) only a slight bactericidal effect of MXF is ob-served. On the other hand, a very high kmax value is achieved by this drug towards M. catarrhalis (see Eq. (11)) as well as towards most of the other bacterial strains (see table 2; see also resulting low values of t[1‰ 4MIC]). By analysis of the mean single-point kill rates (obtained from the data in table 6 and listed in table 7; method M3) and also by method M2 much less significant results were obtained.

Comparison of the Drug Classes and Agents Studied

In general, it is obvious that for every drug class and agent studied the kill rates kmax of the gram-positive species were lowest, whereas those of the gram-negative species were at least two and up to 20 times higher.

All the beta-lactams studied typically behaved like amoxicillin (Figures 1, 2, and 5). Viable count vs time curves decrease steeply (Figure 1) or vice versa the kill

rates (Figures 2 and 5) increase steeply at low concentra-tions and approach the maximum kill rate slowly up to a concentration of optimal bactericidal effect. Beyond a con-centration mediating almost the optimal bactericidal effect the kill rates approach the maximum kill rate slowly at C ≥ 8 X MIC. Cefuroxime exhibits the most pronounced bacte-ricidal effect amongst the penams and cephems studied.

For the four penems studied, there is a trend towards a more pronounced concentration-dependent bactericidal activity. The kill rates increase continuously for almost all penem and strain associations studied and slowly approach a constant maximum at C = 16 X MIC. Among the penems, imipenem seems to be the most bactericidal agent (see MIC and kmax values in table 2). However, it is interesting to note that faropenem is particular not only among the penems, but in comparison to the other drug classes as well.

Faropenem exhibits a marked bactericidal effect, i.e. a high kill rate at a concentration of C = 1 X MIC (data not shown). This strong bactericidal effect at low concentra-tions is particularly noted for the two M. catarrhalis test strains and at C = 4 X MIC for E. coli and K. pneumoniae as well. Ertapenem shows analogous effects against K. pneumoniae and E. coli as well against S. aureus 133 and S. pneumoniae ATCC 6303.

Among all the drugs tested, moxifloxacin (Figures 3, 4, and 6) exhibits in general the most pronounced bacteri-cidal effects throughout the entire range of concentrations studied. The kill rates increase steeply (Figure 6); for all the gram-negative bacteria studied a significant, although a shallower increase in the kill rate slope was noted at con-centrations of C = 16 X MIC and 32 X MIC. For H. influen-zae 49247, E. coli 25922, and S. pneumoniae 6303 kill rates were estimated by method M1, whereas for the other bac-

Moxifloxacin/M. catarrhalis England 11

Eq. (11) 95% C.I.

EC50 (MIC units) 8.37 ± 1.25 kmax [h-1] 14.6 ± 1.49 Hill coeff. b 2.62 ± 0.87

n = 6 r = 0.999 s = 0.352

k(ki

ll) (

h)-1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

x (C = x X MIC)

0.0

0.5

1.0

1.5

2.0

2.5

0.0

0.5

1.0

1.5

2.0

2.5

Figure 5. Kill rate/concentration profiles of amoxicillin against M. catarrhalis ATCC 43617 (x), M. catarrhalis Va 14497/00 (õ), S. pneu-moniae Bay 4241 (◊), H. influenzae Va 15481/00 (ò), S. aureus Bay 133 (+), and S. pneumoniae ATCC 6303 (�). Single-point kill rates (mean values) have been obtained by method M3.

k(ki

ll) (

h)-1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

x (C = x X MIC)

0.0

2

4

6

8

10

12

0.0

2

4

6

8

10

12

Figure 6. Kill rate/concentration profiles of moxifloxacin against H. influenzae ATCC 49247 (✳), E. coli ATCC 25922 (ó ), H. influenzae Va 15481/00 (ò), K. pneumoniae Bay 63 (∆), M. catarrhalis Va 14497/00 (õ), M. catarrhalis ATCC 43617 (x), S. aureus Bay 133 (+), S. pneumoniae Bay 4241 (◊), and S. pneumoniae ATCC 6303 (�).



teria single-point kill rates (mean values) were obtained by method M3. Because of the linear increase in the kill rate with concentration of MXF no EC50 and kmax values could be calculated for E. coli.

In contrast to all the other agents studied the maximum kill rates for clarithromycin were low, in general ≤ 1.0 h-1.

For several drug/strain associations studied the kill rate at C = 1 X MIC was observed to be already at the level of kmax. In these cases, a statistically significant EC50 value could not be determined. Here, therefore, in table 2 only the kmax value is given, which was obtained by method M1 or M3 (almost constant levels of kill rate [M1] or of mean kill rate [M3] at all concentrations) or by method M2 (re-sulting in a significant kmax but in non-significant, very low EC50). In a few other cases (mainly for MXF) the kill rate at C = 1 X MIC was observed to be almost at zero level. In these cases a value b > 1 was required in the Hill equation. For both cases, i.e. (i) all kill rates ~kmax, and (ii) k ~ 0 at C = 1 X MIC, statistically more significant results would have been obtained by determining the bactericidal effect also at, e.g. C = 0.5 X MIC and C = 2 X MIC.

A clinically relevant parameter may be the reduction of the bacterial burden at the focus of infection; under in vitro conditions, this would correspond to the reduction of the inoculum within a given period of time. Thus, the time to reduce the inoculum by 3 log10 titers at a concen-tration of 4 times the individual MICs [t (1‰ 4MIC)] was calculated. In general, and in agreement with the calcu-lated kill rates, more time was required for any agent to reduce the inoculum of the gram-positive bacteria than the time needed to affect the gram-negative bacteria (Table 2). Although for several drug-bug associations studied no significant results could be obtained (because of the rea-sons mentioned above), there was a general trend that the penems and the quinolones studied reduced the inoculum of all the strains studied most effectively. Likewise, the concentrations required to observe the same effect within 6 h (C[N0/1000 6 h]) were lowest for the penems and the quinolones. In many cases the concentrations of penams and cephems needed to reduce the inoculum could not be calculated because of the above-mentioned reasons.

Discussion Phenotypically this study confirms the well-established pharmacodynamics of various antibacterial drug classes. Among the beta-lactams the bactericidal activity of the penams and cephems increases up to a concentration producing a maximal effect; higher penam or cephem concentrations do not further augment their bactericidal activity, i.e. kill rate, significantly [1, 2, 19].

Thus, at high concentrations beta-lactams are con-sidered to act concentration independently, so that their pharmacodynamics are determined by time for which the serum concentrations significantly exceed the MIC of the causative pathogen [19–26]. In agreement with these find-ings, the results from this study clearly demonstrated that

beyond a concentration of four to eight times their MIC, the bactericidal activity and consequently the kill rates no longer increased significantly.

In contrast to the penams and cephems, the penems tended to act more in a concentration-dependent manner. Penems exhibit a concentration-dependent effect against gram-positive and gram-negative bacteria. Penems like imipenem and meropenem [27–29] or the newer ones like artapenem and faropenem [30, 31] show increasing bacterial killing with increasing concentrations. Conse-quently, there is a significant trend toward shorter time periods with C > MIC with penems [31, 32] as compared to penams or cephems. These findings on the pharmaco-dynamic characteristics validate the theory that carbapen-ems and penems are not typical of the beta-lactam class.

Among the penems, imipenem is characterized by the most pronounced bactericidal effect and highest kill rates for almost all indicator organisms studied. Farope-nem and in part ertapenem, too, exhibited high kill rates at low concentrations.

In contrast to all the other drug classes and agents studied moxifloxacin’s activity against both gram-positive and gram-negative species was clearly concentration de-pendent. These data agree very well with previous results [33–35].

As expected, the bactericidal activity of clarithromy-cin was almost negligible.

Apart from the phenotypic comparative description of the bactericidal activities of different drug classes, the primary aim of this study was to apply mathematical mod-els in order to quantify the time- and concentration-de-pendent antibacterial effects of the agents studied.

Experimental methods and mathematical models have been applied in this study to quantify the antibacte-rial effects of the nine agents studied, representing three different drug classes. The parameters calculated to quan-tify the extent of bacteriostatic or bactericidal effects are 1) kill rate, 2) maximal kill, 3) EC50 values, 4) Hill coeffi-cient, and 5) time to reduce the inoculum by 3 log10 titers. In general, these five parameters mirror an antibiotic’s bactericidal or bacteriostatic activity as a function of time and/or concentration; in contrast, MICs are discrete and static endpoints. Therefore, it would be more conclusive to correlate a parameter quantifying the kinetics of anti-bacterial in vitro activity with the pharmacokinetics of the drug. Currently, the pharmacodynamic characteristics of antibacterial agents are almost always described by one of three surrogate parameters: i) time (t) of exposure of a bacterium to serum concentrations exceeding the MIC (t with C > MIC); ii) the ratio of peak concentration (Cmax) of an agent to the MIC of the drug for the bac-terium (Cmax/MIC); and iii) the ratio of the area under the serum concentration versus time curve (AUC) to the MIC (AUC/MIC). Even if all possible drug exposures for standard dosage regimens and all MIC values likely to be found for the clinical isolates would be integrated into the



calculations, the principle contradiction remains unsolved that pharmacokinetics are correlated to a discrete, static microbiological endpoint, i.e. the MIC value. It would be more logical to correlate pharmacokinetics to in vitro pa-rameters quantifying the time- and concentration-depen-dent antibacterial activity. Such parameters could be the ones calculated in this study.

It has been demonstrated previously that the only in vitro parameter correlating to in vivo efficacy in a rabbit endocarditis model was the in vitro kill rate, but not the MIC or MBC value [36, 37]. The relationship between the in vitro parameters MIC, MBC and kill rate with the in vivo efficacy of a single antibiotic dose in an experimental rabbit model of E. coli endocarditis [36] or Enterobac-ter cloacae and Serratia marcescens endocarditis [37] was evaluated. The drugs tested were tobramycin, netilmicin, gentamicin, amikacin, pefloxacin, ciprofloxacin, ceftriax-one, moxalactam, cefotaxime, ceftazidime, and amoxicil-lin. Neither the MICs nor the MBCs, but the killing rate was the major determinant for in vivo efficacy. Likewise, in an in vitro pharmacodynamic model comparing the ac-tivities of moxifloxacin monotherapy with those of a com-bination/regimen of levofloxacin plus metronidazole in a mixed infection model of E. coli and Bacteroides fragilis, it became obvious that the MICs were not predictive for activity in the pharmacodynamic in vitro model [38]. This lack of correlation between pharmacodynamic activity and in vitro susceptibility testing results is also discussed elsewhere in this issue [39].

Basically, the experimental data needed to calculate these dynamic parameters can easily be generated by, e.g. performing time-kill experiments. In principle, automated test systems constitute another source of information. The effect of antibacterial drugs on the viability of bacteria is continuously monitored in the automated susceptibility test systems as well (e.g. Biomérieux’s Vitek® or the Mi-croscan® from Dade/Behring).

Another option to replace the static endpoints by a parameter describing more adequately the dynamics of antibacterial action of a drug would be to quantify the effect of fluctuating drug concentrations simulating drug levels in serum or at the target site on the bacterial pop-ulation by calculating the area under the bacterial kill curve (AUBKC) [10, 39–41] or by calculating the kill-constants.

Although MICs are, on the one hand, easy to deter-mine and provide a basis for, e.g. comparisons of the anti-bacterial spectra of antibacterial agents or routine suscep-tibility testing, it is, on the other hand, illogical to derive pharmacodynamic surrogates from static endpoints. It would be more conclusive to provide pharmacokinetic/pharmacodynamic (PK/PD) measures which have been calculated on the basis of parameters integrating the ki-netics of the drug in the macroorganism with the kinetics of antibacterial action of the drug in the microorganism.

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kinetics and quantification of antibacterial effects of beta-lactams, macrolides, and quinolones...

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