326

ANTIMICROBIAL AGENTS AND CHEMOTHERAPY, Feb. 1990, p. 326-331 0066-4804/90/020326-06$02.00/0 Copyright © 1990, American Society for Microbiology

Vol. 34, No. 2

Application of Mathematical Model to Experimental Chemotherapy of Fatal Murine Pneumonia TAMOTSU HISHIKAWA,1* TADASHI KUSUNOKI,2 KANJI TSUCHIYA,3 YOSHIO UZUKA,4 TASUKU SAKAMOTO,5 TSUYOSHI NAGATAKE,5 AND KEIZO MATSUMOTO5 New Product Planning and Development Division,' Management and Planning Division,2 and Central Research Division,3 Takeda Chemical Industries, Ltd., Osaka, Third Department of Internal Medicine, Teikyo University, Ichihara,4 and Department of Internal Medicine, Institute of Tropical Medicine, Nagasaki University, Nagasaki,5 Japan Received 7 April 1989/Accepted 6 November 1989

Bacterial reduction is an essential measure in evaluating the effect of antimicrobial therapy in patients and in determining the optimal dosage regimen for antibiotics. However, bacterial c6unts are not always ineasurable in clinical studies, which have other experimental and ethical limitations. To avoid i'uch difulties we established a fatal murine pneumonia model. It was used in the present study on experimental therapy with two beta-lactam antibiotics, cefazolin and cefmenoxime. Since antibiotic therapy cannot be successful without a sufficient concentration of a drug in the infectious focus, many pharmacokinetic studies have been carried out; e.g., studies on antibiotic penetration into tissues by Grasso et al. (5), Ryan et al. (14), and Shibl et al. (15) and protein binding by Bergan et al. (1), Dudley et al. (3), and Wise et al. (19). Furthermore, Craig and Ebert (2) studied the correlation of in vivo bacterial reduction with the area under the timeconcentration curve (AUJC), peak level, and other pharmacokinetic parameters to identify the major factors dominating efficacy. Grasso et al. (5) obtained in vitro bacterial killing curves by using an experimental apparatus that reproduced the antibiotic concentration time course in human serum; the study showed that the regrowth phase took place when the drug concentration dropped below the MIC. These studies are useful for better understanding of antimicrobial efficacy and to construct better clinical treatments. In the present paper we propose a mathematical model to obtain more quantitative informnation about the relationship between bacterial killing and antibiotic pharmacokinetics. Various pharmacokinetic and pharmacodynamic parameters are included in the model to explain the bacterial count time course under the experimental conditions. Of the few studies applying mathematical models, Garrett and Won (4), Hamano et al. (6), Mattie and van der Voet (10), and Tsuji et al. (16) dealt with in vitro killing curves, but only the model of Kono and Asai (8), which is the origin of the model proposed here, dealt with in vivo killing curves *

with the antibiotic concentration time course in serum. The purpose of this study is to evaluate the goodness of fit of the model and to quantify the contribution of the parameters referring to the above factors.

MATERIALS AND METHODS Animal. A total of 100 3-week old male DDY mice were infected with aerosolized Klebsiella pneumoniae B-54 by using the exposure chamber of Matsumoto et al. (9), based on the method of Nishi and Tsuchiya (13). Bacteria. K. pneumoniae B-54 was isolated from a patient with very severe pneumonia in 1978. The 50% lethal dose of this strain was 9.7 CFU per mouse in this experimental pneumonia model. Antibiotics. Cefazolin (Fujisawa Pharmaceutical Co., Japan) and cefmenoxime (Takeda Chemical Industries Ltd., Japan) were used. MICs of cefazolin and cefmenoxime against K. pneumoniae B-54 were measured by the agar well method. The MIC of cefazolin was 1.56 jig/ml and the MIC of cefmenoxime was 0.013 p.g/ml. Doses and dosage regimens,. The doses and the schedules of the four treatments, two dosage regimens each for both drugs (Table 1) were determined to approximate the concentration time course in hutnan serum produced by usual clinical treatment: drip infusion over 1 h with doses of 20 to 40 mg/kg of body weight for both drugs. Since drip infusion was impossible in mice and the time course was much faster in mice than in humans, frequent subcutaneous injections were necessary. Twenty-four hours after infection, the mice were divided into four groups, and each group of mice received one of four treatments (A, B, C, and D; Table 1). Antibiotic assay. Cefazolin and cefmenoxime in plasma samples were assayed by the agar well method with Escherichia coli NIHJ as the test organism (7). The sensitivity was about 0.1 ,ug/ml, and the variation coefficient of replicate analyses was ±7.0%. Measurement of bacterial count. Immediately before and 2, 4, 8, 12, 16, 20, and 24 h after the initial dosing, three mice

Corresponding author. 326

Downloaded from aac.asm.org by on April 24, 2009

Two beta-lactam antibiotics, cefazolin and cefmenoxime, were administered in an experimental model of murine pneumonia caused by Kkebsiella pneumonuae in a way which enabled us to approximate the serum antibiotic concentration time course in humans. Bacterial counts during the experiments were subjected to nonlinear least-squares analyses by using a mathematical model that explained the bacterial killing by the antibiotic concentration time course and other factors associated with antimicrobial potency and bacterial growth. Cefazolin gave a killing curve that changed synchronously with the drug levels in serum; in contrast, cefmenoxime gave a curve that was prolonged as compared with the change in the drug levels in serum. Multiple correlation coefficients were about 0.9, and the model worked well for bacteriail count data. Parameters relating to antimicrobial potency of the drugs, bacterial growth rate, and drug distribution into the tissue were estimated numerically.

VOL. 34, 1990

TABLE 1. Doses and dosing regimens for micea Doses (mg/kg) administered at time after initial dosing

Treatment

Total dose (mg/kg) 0

Drug

20 40 1 1.5 2 2.5 3 4 min min min h h h h h h

A B

Cefazolin Cefazolin

126 252

16 20 24 16 16 10 10 8 6 32 40 48 32 32 20 20 16 12

C D

Cefmenoxime Cefmenoxime

84 168

20 20 24 10 40 40 48 20

a Both drugs were

5 10

3 2 6 4

subcutaneously injected in infected mice.

n')

DIP'Atexp(-B1tIE)IE.

(3) = When E 1, equation 3 is the same as equation 2. Since E values of >1.0 prolong the time course of an) and bacterial count, more or less lag time can exist before the onset of killing, and bacterial growth can be suppressed continuously even after the drug concentration in serum drops below the effective level. E has thus a meaning of "delay factor" or "continuity parameter." The generalized equation 3 is applicable to cases such as that noted by Ryan et al. (14), in which there was a marked delay of beta-lactam =

concentration time course in the tissue determined by the surface/volume ratio of the infectious tissue. If Cn) is directly measurable in tissues, pharmacokinetic parameters for the tissue antibiotic concentration time course should be used in equation 3; however, it is not the case in this study. Integration of equation 1 over time t yields p

ln (CbICbo) = Kbt - KDn >.{A.IBj

[1 - exp(-Bit/E)]}

(4)

C

C 100-

Cef azolin

C= 34.2e AUC

60

Cefmenoxi me

100

C = 38.0(1-e

80

co

n attached to D has a specific value corresponding to the tissue. Ai and Bi (i = 1, ..., p) are the pharmacokinetic parameters representing serum concentration time courses and p is the number of exponential terms. In equation 2, Cn") at time t is equal to serum level if n is equal to 1 and less than the serum level if n is less than 1. Since Cf") changes synchronously with the level in serum, the change in bacterial count should also be synchronous with the change in serum drug level and C(n). Some gap in time can exist, however, between the serum and tissue antibiotic concentration time courses, and between the bacterial reduction and the antibiotic concentration time course in serum. To allow the possible gaps, we modified the model by introducing an additional parameter, E. Replacing the exponential term in the equation 2, Atexp(-B,t), with Aiexp(-B1t/E)/E, and assuming that the AUC in the tissue does not change from equation 2, cjn) can be expressed as

=

207

) + 168.5

+55.5e

(1_e-0 40t) ,t'

=

80

t-10

60

C=39.4( 1-e

0

Man

A'.

.a

40

40

A

0 se Q

)+28.9(1-e

2.58 t'

C = 36.3e U C

+14.4

-0. 69 t' e

0

)

t'= t-1 .0

=68.3

AIJC= 6 9.3

20.

co

0

1

2

3

0

1

3

4

5

6

Time after administration(h) Time after administration (h) FIG. 1. Antibiotic concentration time courses in serum with the dose of 20 mg/kg. Multiple subcutaneous injections of treatments A and C (Table 1) were used for mice. Single-drip infusions over 1 h were used for humans. The doses of the drugs were both 20 mg/kg of body weight.

Downloaded from aac.asm.org by on April 24, 2009

of each treatment group were sacrificed, and the lungs of each mouse were aseptically removed. Each lung was homogenized, and the number of viable bacteria was measured by the quantitative culture method. The detection limit was about 10 CFU per lung. Mathematical model. The basic idea is to express the competition between bacterial growth and antimicrobial kill as rate of change in bacterial count (growth/kill) = growth rate - kill rate. Considering the logarithimic growth phase of the bacteria, and assuming that the growth rate is proportional to the bacterial count at the moment and that the kill rate is proportional to the product of the bacterial count and the drug concentration, Kono and Asai (8) have indicated this relationship in the form of a simple differential equation dCJdt = KbCb- KCn )Cb (1) where t denotes the time after administration, Cb and C(n) are the bacterial count and the drug concentration in the infectious tissue at time t, and Kb and K are the growth rate and kill rate, respectively. Garrett (4) used equation 1 a few years later for in vitro experiments. Kono and Asai have assumed that on) is proportional to the level in serum, which is a function of time t expressed by the usual multicompartment model, and described C(n) as C(nl - DnlpA exp(-Bit), (2) where D is the dose of the drug administered and the power

._ _E

327

MATHEMATICAL MODEL FOR EXPERIMENTAL CHEMOTHERAPY

328

HISHIKAWA ET AL.

ANTIMICROB. AGENTS CHEMOTHER.

11

tional assumption (common CbO, and separate Kb, K, n, and E were estimated for each drug), and model 3 is the general model with common Cbo and Kb (common Cbo and Kb, and separate K, n, and E were estimated for each drug). Akaike's information criterion was computed for each model and compared to select the optimal model with the smallest value. RESULTS Pharmacokinetic parameters and simulation of antibiotic concentration time courses. The pharmacokinetic parameters and simulated serum antibiotic concentrations for both drugs in human and mouse serum are shown in Fig. 1. Since the time course of serum drug concentration in mice is similar to that in humans, the design of Table 1 is regarded as an approximation of the time course in humans. By using the differences between the mice and human AUCs, the total dose in each treatment for mice was converted to the corresponding dose for humans: to 17.0 and 34.0 mg of cefazolin per kg in treatments A and B and to 20.3 and 40.6 mg of cefmenoxime per kg in treatments C and D. In the following analyses, these doses and pharmacokinetic parameters in human subjects were used to approximate the antibiotic concentration time course in the infected mice, and the experimental therapy in mice is regarded as a simulation of clinical therapy of pneumonia in humans.

0

Cefmenoxime

Cefazoli n 0

o

0

0

0

Treatment C 0

Q

-I'

0

'1."\

D

0

0

8

A and B 0

0

a.

5 4)

---

C.)

Model I Model 2 *Treatment A oTteatment B

CD

0

8

12

16

100

20

Model I - Model 2 * Treatment C O Treatment D

----

5

24

0 1

4

0

12

16

24

00 -

Cefazolin (Treatment A) 80

8

Cefmenoxime (Treatment C ) 80

-

._ 06 a) ce

4) 0

60 Model 1 Model 2

0 c;

40

60 40 -

Model 1 Model 2 .

.

.I| co c;

Eq-

20

20 -

0

0

16 20 24 0 8 12 4 8 12 16 20 Time after administration (h) Time after administration(h) FIG. 2. Time courses of bacterial counts and tissue drug concentrations. 4

24

Downloaded from aac.asm.org by on April 24, 2009

where CbO is the initial condition, bacterial count at time zero. Estimation of pharmacokinetic parameters. Pharmacokinetic parameters were calculated by applying a two-compartment open model. Simulation curves for mice under frequent dosings were provided, assuming that the concentration time course produced by the succeeding injections could be superimposed linearly, and were compared with those for humans under single-drip infusion. Estimation of Cbo, Kb, K, n and E. Given the values of pharmacokinetic parameters and D, the data of bacterial count in the mice lung were subjected to nonlinear leastsquares analyses to estimate the other unknown parameters, and the overall goodness-of-fit of the model was examined by multiple correlation coefficient (R). The program NONLIN (11) was used, taking Wi as equal to 1 and 1/Cb as the weighting factor, to estimate the unknown parameters. Since both antibiotics have two treatments with different doses, separate estimation of K and n is possible; however, it is impossible if only one dose is used, and the estimation of a compound parameter, KD', becomes inevitable. The above procedures were applied to three models based on equations 3 and 4: model 1 is the Kono model where E = 1.0 (common Cbo, and separate Kb, K, and n were estimated for each drug), model 2 is the general model without addi-

VOL. 34, 1990

MATHEMATICAL MODEL FOR EXPERIMENTAL CHEMOTHERAPY

329

TABLE 2. Estimates of parametersa Model

Drug

ln(Cbo)

Kb

K

E

n

Multiple correlation coefficient

Residual sum of

AIC

squares

1

Cefazolin Cefmenoxime

15.95 ± 0.41

0.43 ± 0.04 0.37 ± 0.03

0.35 ± 0.08 0.27 ± 0.23

1.0 1.0

0.000 ± 0.000 0.382 ± 0.219

0.886

191.8

250.6

2

Cefazolin Cefmenoxime

15.81 ± 0.43

0.42 ± 0.05 0.50 ± 0.06

0.29 ± 0.08 0.79 ± 0.41

0.73 ± 0.49 3.52 ± 0.71

0.003 ± 0.004 0.240 ± 0.125

0.909

154.8

244.9

Cefazolin

15.97 ± 0.39

0.47 ± 0.04

0.40 + 0.08 0.66 ± 0.33

1.08 ± 0.46 3.09 ± 0.64

0.007 ± 0.007 0.271 ± 0.130

0.906

159.7

244.3

3

Cefmenoxime

ln(Cbo),

a Values of Kb, K, E, and n are given as estimates ± standard errors. Model 1 is the Kono model, in which E is 1.0. Model 2 is the general model in equation 4 without additional assumptions. In model 3, Kb is assumed to be common to cefazolin and cefmenoxime. AIC, Akaike's information criterion.

expected little effect of increasing the dose of cefazolin on the in vivo antimicrobial activity in the tissue. The two killing curves with higher and lower doses of cefazolin overlapped because of the small n (Fig. 3). The prolonged simulation curves for cefmenoxime shifted to the right as compared with the curves for cefazolin. The minimum bacterial counts on the curve are arranged in the order of D < C - B - A. The higher dose of cefmenoxime was the most effective. When the minimum bacterial count is reached at time t, just before regrowth takes place, bacterial growth and antimicrobial killing stop, and a relation dCbldt = 0 holds. If Co(') denotes the tissue antibiotic concentration at this moment, Co(n) is derived from equation 1 as Co(n) = Kb/K; using the estimates of Kb and K in Table 2, Co0") is 1.18 ,ug/ml for cefazolin and 0.71 ,ug/ml for cefmenoxime. The length of time until dCbldt = 0 is estimated at 2.0 h for cefazolin and 5.0 h for cefmenoxime. Killing activity of cefmenoxime continues longer than that of cefazolin.

2.

No appreciable gap was noted between the cefazolin concentration time course with models 1 and 2 and between the corresponding time courses of bacterial counts (Fig. 2). In contrast, the simulation for cefmenoxime, based on model 2 with E = 3.52, showed a better fit to the observed bacterial counts versus time. The cefmenoxime concentration time course with E = 3.52 showed a marked delay compared with that based on model 1 with E = 1.0. These results suggest that the E of cefazolin does not differ from 1.0 and that the E of cefmenoxime is larger than 1.0. Optimal model. Models 2 and 3 had almost the same optimality in terms of Akaike information criterion (244.9 and 244.3) and residual sum of squares (154.8 and 159.7) (Table 2). However, model 3 is considered to be optimal, since it meets the requirement of parsimony with the minimum number of parameters to be estimated. The assumption of model 3 that Kb is common regardless of the drugs is also preferred. Figure 3 shows the simulation based on model 3 and the results of Table 2. The continuity parameter E was estimated in model 3 to be 1.08 and 3.09 for cefazolin and cefmenoxime, respectively. Comparison between two drugs in parameter estimates from model 3. The kill rate constant K was estimated at 0.66 for cefmenoxime. It was about 1.5 times larger than the K of 0.40 for cefazolin. These facts show that cefmenoxime has a stronger in vivo potency than that of cefazolin against K. pneumoniae B-54. The estimate of n was less than 1.0 for both drugs. It was especially small for cefazolin (n = 0.007), which was almost 1/40 of that for cefmenoxime (n = 0.271). Consequently, we

DISCUSSION The shape of the killing curves can be divided into depth and length of killing phase. In the model proposed here the 1 0-

// /l

9-

le

8-

7-77

I

7,I -a,-

6Treatments A and B (Cefazolin )

5

---

TTreatmenit C' (Cefmenoxime)

.Treat meidt

0

4

8

1

12

D (Ce fmenoxime)

16

20

24

Time af ter administration (h) FIG. 3. Simulated time courses of bacterial counts with model 3.

Downloaded from aac.asm.org by on April 24, 2009

Bacterial count time courses. The log1o CFU of the three mice at time t and the simulated time courses with models 1 and 2 and the tissue antibiotic concentration time courses in treatments A and C are shown in Fig. 2. The initial mean log1o CFU was 6.94 for all treatments. The lowest mean log1o CFU and the times they were observed were 6.26 (8 h), 6.12 (4 h), 6.35 (8 h), and 5.84 (8 h) in treatments A, B, C, and D, respectively. The higher dose of cefmenoxime (40.6 mg/kg) gave the lowest mean CFU among the four treatments. Comparison between model 1 (Kono model) and model 2 (general model). The estimates of the parameters and other results are summarized in Table 2. Since the choice of weighting factors (Wi = 1 or JICb) did not affect the results, those from Wi = 1 were adopted here. Models 1 and 2 showed high multiple correlation coefficients of 0.886 and 0.909. The residual sum of squares decreased from 191.8 in model 1 to 154.8 in model 2, and the Akaike information criterion decreased from 250.6 in model 1 to 244.9 in model

330

ANTIMICROB. AGENTS CHEMOTHER.

HISHIKAWA ET AL.

(A)

160-

-,

3 120-

0

80-

co

E= 3.0

-_-- E= 1.0

It.4

II

I'

ru 40 -

C) 0

v

0

8

4

12

16

20

24

Time after administration (h)

2.0

-

CB)

1.5-

,

-. 1.0-

ol

A* I

I

AUC

0.5- II

I

Q

O-oo

I

0

¢

I0 4

0

4

I

8

12

16

2

24

20

24

0 _-l

a

-

0C 0C; Q

6 5-

Q._

~>

A

a c; C. 0

:s

.t

24 16 20 8 12 4 0 FIG. 4. Relation between parameter E and antibiotic concentration time courses, AUC, and bacterial counts.

crobial kill (-KD'AUC) to form the overall bacterial counts time course as Fig. 4D which was represented in equation 4. Since this model worked well to explain the observed bacterial counts by pharmacokinetic data with an appreciable goodness of fit, it is a possible approach to quantitation of the in vivo relationship between antimicrobial effects, realized as bacterial killing curves, and factors inherent in antibiotic-microorganism-animal tissue combinations. The results obtained here will be useful in considering optimal clinical treatment with cefazolin and cefmenoxime. ACKNOWLEDGMENTS

We are grateful to G. L. Drusano and two reviewers for their constructive comments and suggestions. We also thank Y. Kibune,

Downloaded from aac.asm.org by on April 24, 2009

length is determined by the antibiotic time course in serum and the parameter E. In contrast to the fact that cefmenoxime showed a prolonged killing curve associated with E = 3.09, no appreciable delay was noted in cefazolin with E = 1.08. The Kono model, which is the origin of our model, was applicable to cefazolin. We have already reported similar preliminary results of E = 1 for cefazolin and E = 3 for cefmenoxime with a lag before onset and a delayed end of the killing phase (12). But separate estimation of the parameters K and n was impossible because only one dose level each for both drugs was used in that experiment. Despite this, reproducibility problems will remain if delayed killing curves are attributed solely to the simultaneous delay of antibiotic concentration time course in tissues. The postantibiotic effect proposed by Vogelman and Craig (18) and phenotypic tolerance in stationary phase proposed by Tuomanen et al. (17) should also be considered. The roles of E and other parameters in the model are illustrated in Fig. 4, which shows a simulation of treatment D of cefmenoxime based on model 3 and numerical estimates of the parameters other than E. The solid and dotted lines denote the simulations using E = 3 and E = 1. An E of 3 (>1) caused a delay of antibiotic concentration time course in the tissue and a decrease of peak concentration (Fig. 4A), and this delay produced a little lag time before the onset of kill and a prolonged killing phase (Fig. 4D). Garrett and Won used lag time as another parameter to explain this kind of phenomenon; however, the model in equation 4 is correct if the true factor is the antibiotic time course in the tissue. Additional experiments and careful explanations are necessary to identify the true factor and to select the correct model. The AUC shown in Fig. 4B, dose level (D), kill rate (K), and distribution into the tissue (n) cooperate each other in the fashion of the second term of equation 4, KD'AUC, to determine the depth (Fig. 4C). Larger values of K, D', and AUC produce greater antimicrobial effect and greater depth. Protein binding, studied by Dudley et al. (3), would be a real mechanism affecting n. Emphasis is necessary on the fact that the strength of killing is dominated solely by the magnitude of K when n approaches 0 (n << 1), as noted for cefazolin. K is the direct expression of in vivo antimicrobial potency, having specific values for the respective antibiotics. It might be compared to the MIC, the representative parameter of in vitro antimicrobial potency. A more appropriate parameter for the in vivo situation, Co(n), was constructed from our model. Comparing the empirical COW with the MIC, COt" = 1.18 for cefazolin is comparable to the MIC of 1.56 ,ug/ml, but Co(n) = 0.71 for cefmenoxime was larger than the MIC of 0.013 ,ug/ml. In Fig. 4C the difference between the solid and the dotted lines (E = 1 and E = 3) is noted only during the killing phase, when the AUCs are expanding and two lines are getting together when the AUCs are reaching to the common limit, AUC°. The killing phase is the first 8 h, which include only three points of bacterial count measurements, 2, 4, and 8 h; the other four points are located on the regrowth phase (Fig. 4C and D). This makes the estimate of Kb reliable; however, it is unfavorable for accuracy of parameter estimates relating to killing effects. Despite the above problems in definition, interpretation, and accuracy of various parameters, we combined the competition between the bacterial growth (Kbt) and antimi-

VOL. 34, 1990

MATHEMATICAL MODEL FOR EXPERIMENTAL CHEMOTHERAPY

T. Morikawa, and M. Yoshida for their kind advice and encouragement.

10. 11.

12.

13.

14.

15. 16.

17. 18.

19.

pneumonia due to gram negative bacilli by air-borne infection. Nippon Kyoubushikkan Gakkai Zasshi 16:581-588. (In Japanese.) Mattie, H., and G. B. van der Voet. 1979. Influence of aminopenicillins on bacterial growth kinetics in vitro in comparison with the antibacterial effect in vivo. Infection 7:434-437. Metzler, C. M., G. L. Elfring, and A. J. McEvan. 1974. A users manual for NONLIN and associated programs. The Upjohn Co., Kalamazoo, Mich. Neu, H. C. 1985. Antibacterial concentration-response relationships in vitro, in animals and in patients, p. 2703-2708. In J. Ishigami (ed.), Recent advances in chemotherapy, antimicrobial section 3: proceedings of the 14th International Congress of Chemotherapy, Kyoto, 1985. University of Tokyo Press, Tokyo. Nishi, T., and K. Tsuchiya. 1980. Experimental respiratory tract infection with Klebsiella pneumoniae DT-S in mice: chemotherapy with kanamycin. Antimicrob. Agents Chemother. 17:494505. Ryan, D. M., 0. Cars, and B. Hoffstedt. 1986. The use of antibiotic serum levels to predict concentrations in tissues. Scand. J. Infect. Dis. 18:381-388. Shibi, A. M., C. J. Hackbarth, and M. A. Sande. 1986. Evaluation of pefloxacin in experimental Escherichia coli meningitis. Antimicrob. Agents Chemother. 29:409-411. Tsuji, A., S. Hamano, T. Asano, E. Nakashima, T. Yamana, and S. Mitsuhashi. 1984. Microbial kinetics of P-lactam antibiotics against Escherichia coli. J. Pharm. Sci. 73:1418-1422. Tuomanen, E., D. T. Durack, and A. Tomasz. 1986. Antibiotic tolerance among clinical isolates of bacteria. Antimicrob. Agents Chemother. 30:521-527. Vogelman, B. S., and W. A. Craig. 1985. Postantibiotic effects. J. Antimicrob. Chemother. 15(Suppl. A):37-46. Wise, R., A. P. Gillett, B. Cadge, S. R. Durham, and S. Baker. 1980. The influence of protein binding upon tissue fluid levels of six P-lactam antibiotics. J. Infect. Dis. 142:77-82.

Downloaded from aac.asm.org by on April 24, 2009

LITERATURE CITED 1. Bergan, T., A. Engeset, W. Olszewski, N. 0stby, and R. Solberg. 1986. Extravascular penetration of highly protein-bound flucloxacillin. Antimicrob. Agents Chemother. 30:729-732. 2. Craig, W. A., and S. C. Ebert. 1987. Importance of pharmacokinetics and pharmacodynamics in determining the efficacy of P-lactams, p. 269-284. In U. Hamano (ed.), Frontiers of antibiotic research. Academic Press of Japan, Tokyo. 3. Dudley, M. N., W. C. Shyu, C. H. Nightingale, and R. Quintiliani. 1986. Effect of saturable serum protein binding on the pharmacokinetics of unbound cefonicid in humans. Antimicrob. Agents Chemother. 30:565-569. 4. Garrett, E. R., and C. M. Won. 1973. Kinetics and mechanisms of drug action on microorganisms. XVII. Bactericidal effects of penicillin, kanamycin and rifampin with and without organism pretreatment with bacteriostatic chloramphenicol, tetracycline, and novobiocin. J. Pharm. Sci. 62:1666-1673. 5. Grasso, S., G. Meinardi, I. de Carneri, and V. Tamassia. 1978. New in vitro model to study the effect of antibiotic concentration and rate of elimination on antibacterial activity. Antimicrob. Agents Chemother. 13:570-576. 6. Hamano, S., A. Tsuji, T. Asano, I. Tamai, E. Nakashima, T. Yamana, and S. Mitsuhashi. 1984. Kinetic analysis and characterization of the bacterial regrowth after treatment of Escherichia coli with P-lactam antibiotics. J. Pharm. Sci. 73:14221427. 7. Kita, Y., T. Fugono, and A. Imada. 1986. Comparative pharmacokinetics of carumonam and aztreonam in mice, rats, rabbits, dogs, and cynomolgus monkeys. Antimicrob. Agents Chemother. 29:127-134. 8. Kono, T., and T. Asai. 1969. Kinetics of fermentation processes. Biotechnol. Bioeng. 11:293-321. 9. Matsumoto, K., Y. Uzuka, T. Nagatake, H. Shishido, H. Suzuki, Y. Noguchi, K. Tamaki, S. Rah, and M. Ide. 1978. Experimental

331