A. P. MacGowan*, M. Wootton, A. J. Hedges, K. E. Bowker, H. A. Holt and D. S. Reeves

Journal of Antimicrobial Chemotherapy (996) 8, 9-0 A new time-kill method of assessing the relative efficacy of antimicrobial agents alone and in combination developed using a representative /Mactam, aminoglycoside and fluoroquinolone A. P. MacGowan*, M. Wootton, A. J. Hedges, K. E. Bowker, H. A. Holt and D. S. Reeves Bristol Centre for Antimicrobial Research & Evaluation, Southmead Health Services NHS Trust and the University of Bristol, Department of Medical Microbiology, Southmead Hospital, Westbury-on-Trym, Bristol BSW 5NB, UK A time-kill curve employing nine sampling times over 6 h was used to provide data which were then used to develop a theoretical (best-fit) curve. From the theoretical curve parameters describing the rate of kill (a), time from addition of antibiotic to initiation of killing (d) and a function of the degree of killing observed (Ym/Yo) were defined. The area-under-the-curve (AUC) was calculated from the theoretical curve. The variability of each parameter was assessed using a theoretical curve to fit the data from experiments done on three occasions and in triplicate. In terms of the parameters a, d, Ym/Yo and AUC, no synergy was demonstrated with combinations of piperacillin/tazobactam plus ciprofloxacin or gentamicin when compared with single antibiotics. The AUC represents the best summary parameter of a time-kill curve but should be supported by other parameters describing the best-fit curve. Introduction The bactericidal activity of antibiotics can be assessed in vitro by sequential sampling and counting viable bacteria in broth following addition of the test antimicrobial agent. This method is often termed the 'time-kill curve' and, if combinations of antibiotics are used it is a recognised means of detecting in-vitro synergy or antagonism between antimicrobial agents. Its use is most convenient when a small number of antibiotics require testing and bactericidal activity is thought to be important. Combinations of antibiotics may need investigation as many patients receive multiple antimicrobial chemotherapy in order to treat all suspected pathogens, to prevent the emergence of resistance, or in order to help improve clinical efficacy. A reliable in-vitro evaluation of synergy or antagonism is therefore invaluable. The chequerboard technique has been criticised by ourselves and others (Blaser, 99; Rand el al., 99; Wootton et al., 995), and a more reproducible and accurate technique is required. Time-kill curve experiments are considered to be more accurate than chequerboards but still have some problems, the most important being a lack of precise descriptive terms to define their Tel: + 44 7 950 5050; Fax: +44 7 959 54; e-mail: lesassays@ukneqasaa.win-uk.net 9 005-745/96/0809 + $.00/0 996 The British Society for Antimicrobial Chemotherapy Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

94 A. P. MacGowan et aj. shape and the use of arbitrary criteria for interpreting 'synergy' or 'antagonism' (Krogstad & Moellering, 986). There are also problems in deciding which is the most appropriate antibiotic concentration to use, the size of the inoculum, medium and the duration of incubation (Norden & Wentzel, 979). A possible solution to some of these problems was suggested by Guerillot, Carret & Flandrois (\99a,b) who described a flexible mathematical model for describing kill-curve kinetics. We used an approach based upon their model to quantify kill-curve kinetics. We have, furthermore, assessed its variability in replicate experiments and used the model's parameters to assess performance of antibiotic combinations in comparison with single drugs. The study aims to provide an alternative method for assessing the efficacy of antibiotics and their combinations. For this purpose we selected a representative /f-lactam (piperacillin/tazobactam), aminoglycoside (gentamicin) and fluoroquinolone (ciprofloxacin) and studied them individually and in combination. Bacterial strains Materials and methods A single isolate of Acinetobacter haemolyticus, strain 564, was used to assess intra- and inter-experiment variability. A further strain of A. haemolyticus and two strains of Serratia marcescens and Morganella morganii plus a single strain of Citrobacter freundii were used to assess antibacterial interactions. The strains were clinical isolates collected at Southmead Hospital since 987 and were stored at 70 C until use. Antimicrobial agents The following antimicrobial agents were used: piperacillin powder, tazobactam powder (Lederle Laboratories, Kent, UK), ciprofloxacin, (Bayer, Newbury, UK) and gentamicin (David Bull Laboratories, Warwick, UK). Piperacillin plus tazobactam were combined in a fixed ratio of 8: for all the experiments. Fixed, pharmacologically achievable concentrations of antimicrobial agents were used as follows: piperacillin/ tazobactam 5 mg/l, ciprofloxacin 0.5 mg/l and gentamicin mg/l. These concentrations gave sufficiently slow kill rates to allow the antibiotic-bacteria interaction to be monitored over several hours. MICs were determined as previously described (Holt, Bywater & Reeves, 990). Time-kill curve method The time-kill experiments were performed as described elsewhere (Krogstad & Moellering, 986). For the experiments used to assess intra- and inter-experiment variability, four inl aliquots of isosensitest broth (three with a single antibiotic incorporated at the concentrations given above and a single growth control) were used. Broths were incubated at 7 C and inoculated to give a final inoculum of 0 6 cfu/ml. These were then sampled, diluted as necessary and viable counts performed using a Spiral plater (Spiral Systems Inc) at time 0, 0 min, h,.5 h and then every 45 min up to 6 h on nutrient agar plates (Difco). At each time point the broth was sampled three times in order to assess the intra-experiment variability, and inter-experiment variability was measured by repeating the procedures three times. Viable counts were read manually after 8 h incubation at 7 C. The results were unaltered following Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

Time-kill method 95 reincubation up to 48 h. The experiments using combinations of antibiotics were similar but incorporated a growth control, piperacillin/tazobactam, ciprofloxacin, gentamicin, piperacillin/tazobactam plus ciprofloxacin and piperacillin/tazobactam plus gentamicin. Mathematical and statistical analysis The components of variance technique was used to estimate the variance of a parameter determined without replication on a single occasion. Analysis of variance yielded estimates of the residual variance (sf) and of the inter-experimental variance (si). Then, with triplicate replication on three occasions, the variance attributable to a single determination is: Jfn = sf + {si - sj)/( x ). The corresponding coefficient of variation CV m is then CV m = 00.WJC Where x is the mean parameter estimate. Kill curves were drawn by plotting log, 0 N, against time (h), where TV, was the average viable count at time = /. These curves generally conformed to one of three types (Figure). ULLL 4 Time(h) Figure. Parameters used to define the different time-kill curves illustrated on a theoretical plot for (a) piperacillin/tazobactam (type ) ; (b) ciprofloxacin (type ) ; (c) gentamicin (type ) O; (d) piperacillin/tazobactam plus ciprofloxacin (type ); and (e) piperacillin/tazobactam plus gentamicin (type ). Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

96 A. P. MacGowan et at. Smooth curves were fitted to the experimental results by non-linear least squares (proc NLIN, SAS/STAT statistical package, version 6, The SAS System, SAS Institute Inc., NC, USA). Calculations were done on an IBM 090 mainframe computer. Type curves (piperacillin) were characterised by an initial plateau followed by an exponential reduction of the log numbers, (Figure) which usually resolved onto a final plateau. The fitted curve was the logistic regression function described by Guerillot et al. (99a): Y,= Y o : ift <. d Y,= Y 0 +Y m - (u/v) : ift>d and w = Yo x exp (a(t d)) v = P + exp (<x(? - d)) where: Y, = log, 0 N,,fi = (Y o - Y m )/Y m, d = duration (h) of initial plateau at level = Y o, Y m = level of final plateau. The Gauss-Newton approximation procedure yielded estimates of the parameters Y o, Y m, d, and a (the rate constant) and their confidence intervals. Type curves (ciprofloxacin) were similar to type curves without the initial plateaux. They were fitted by the same procedure, Figure, but with d = 0. Estimated parameters were, therefore, Y o, Y m, and a. Type curves (gentamicin) usually had a rapid linear decrease of the log counts variably preceded by an initial plateau, Figure. The fitted equations were: Y,= Y o - a(t-d) : t> d. The Gauss method of approximation in conjunction with the DUD (secant) method were used to estimate the parameters Y o, d, and a. Areas under the fitted curves (AUCs) were obtained by integration with respect to / of the logistic functions for types and and by simple, geometrical considerations for type curves. Thus, for type curves, where / was taken as 6 h. For type curves, the same expression applies with d = 0 The rate of kill for type curves was often rapid so the convention was adopted that if the log count fell to zero before I= 6, the area was evaluated up to the time of reaching Y x = 0. Thus, AUC = dy 0 + Yl/a. : if a(t - d) Y o AUC = tyt- ct(t - df\ : otherwise where t = 6 h. For types and, d was taken to be zero, when the estimated value of d < 0. Statistical examination of the AUCs and other parameters showed that their distribution was approximately normal (proc UNIVARIATE). Comparisons were, therefore, undertaken by means of analysis of variance (proc ANOVA, proc GLM) and by Student's / test, where appropriate. Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

Time-kill method 97 The model Results The method successfully fitted all of the experimental data to the appropriately chosen theoretical curve as judged by the internal convergence criterion. The parameters we chose to describe the time-kill curves are shown in the Figure. The slope parameter of the curve in the middle time period of sampling is denoted by <z(logio cfu ml - 'h "'); d(h) is the time taken from addition of the antibiotic for the viable count of bacteria to start falling (that is, the initial plateau); Ym/Yo is the log of the viable count of the plateau after killing occurred (that is, the final plateau) divided by the logio of the initial bacterial count, and AUC is the area under the fitted time-kill curve (log cfu h. ml"'). Not all of these parameters were useful in describing the kill curves of each individual antibiotic or combination. Typical kill curve patterns for piperacillin/tazobactam, ciprofloxacin, gentamicin and combinations of piperacillin/ tazobactam, ciprofloxacin or gentamicin are shown in the Figure. For kill curves with piperacillin/tazobactam alone a, Ym/Yo, dand AUC were used to describe the curves; for ciprofloxacin and piperacillin/tazobactam plus ciprofloxacin a, Ym/Yo and AUC were of value, d being zero in all experiments. For gentamicin and piperacillin/tazobactam plus gentamicin a, d and AUC were used because, as no final plateau occurred, Ym/Yo was meaningless. Table I summarises the results of an experiment designed to test reproducibility of the various parameter estimates produced by the method. A single strain of A. haemolyticus was tested with piperacillin/tazobactam, ciprofloxacin or gentamicin alone. The time-kill curves were carried out on three separate occasions with triplicate replication. The computer output produced best estimates of the parameters a, d Yo/Ym together with their standard errors and approximate 95% confidence intervals (data not shown). The residual variances, a measure of how good a fit was obtained, and the AUC are also shown in Table I. Table II shows the variation of the mean ± S.E. for a, Ym/Yo, d and AUC among replicates on each occasion. Analyses of variance (ANOVAs) were conducted for all parameters and these showed that the variance of estimates made on different occasions was roughly ten-fold greater than that among replicates on a single occasion. The components of variance technique was used to estimate the CV,,, for a single determination made on a single occasion. This allowed a fair comparison of the various estimates and the results are shown on Table III. Tables II and III show that in general the AUC gave the most reproducible results. The only other parameter, a, that could be calculated for all antibiotics showed a greater variability both within and between occasions. It is also clear that gentamicin yielded the most variable results for the kill-curve determinations. This is largely due to its rapidly bactericidal action at the concentration chosen which resulted in fewer points being available for fitting. Antimicrobial combinations Six strains, 5. marcescens 570 and 5690, M. morganii 5496 and 579, A. haemolyticus 5644 and C. freundii 50, were tested in time-kill curves with each agent on its own and with combinations of piperacillin/tazobactam plus gentamicin and piperacillin/ tazobactam plus ciprofloxacin. As before, the curves were described by a, Ym/Yo, dand AUC. Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

98 A. P. MacGowan et aj. Table I. Intra- and inter-experiment variations of parameters a, Yo/Ym, d, AUC using a single isolate A. haemolyticus, strain 564 Experiment Replicate a Ym/Yo Parameter d AUC Residual variance Piperacillin/ tazobactam (MIC = 0.5 mg/l) 0.406 0.68 0.477 0.494 0.45 0.499 0.457 0.588 0.496 4.4.7 4.50....67.08.06..6.7.78.79.66.50.6.48 6. 5.7 6.0 7.6 7. 7.9 5. 4.94 5.05 0.0075 0.050 0.05 0.058 0.07 0.069 0.04 0.088 0.050 Ciprofloxacin (MIC = 0.06 mg/l) 0.5787 0.5 0.57 0.4890 0.589 0.5008 0.5504 0.5574 0.554.09..68.54.46.56.0.47.55 8.7 8.06 7.76 0.6 0.65 0.58 8.7 7.98 8.07 0.045 0.08 0.0857 0.0549 0.056 0.050 0.0895 0.0 0.09 Gentamicin (MIC = mg/l).00.979.80 4.4.60 4.044.608.6009.97 _ 0.67 0.69 0.66 0.70 0.6 0.67 0.0 0.5 0 6.89 6.77 6.79 6.4 6.46 6.5 8.85 9.04 0.4 0.4 0.555 0.480 0.5 0.45 0.45 0.9 0.44 0.94 For piperacillin/tazobactam plus ciprofloxacin the values of the parameters used a, Ym/Yo and AUC were similar to ciprofloxacin alone. There were no statistically significant differences among the AUCs for any of the strains with the combination compared with ciprofloxacin alone (Table IV). Similarly, for gentamicin plus TaWe II. Inter-experiment variation of a, Ym/Yo, d and AUC for A. haemolyticus (strain 564) Piperacillin/ tazobactam Ciprofloxacin Gentamicin Experiment a 0.406 (0.068) 0.45 (0.008) 0.494 (0.06) 0.558 (0.0) 0.50(0.05) 0.55 (0.004).5 (0.045).9 (0.8).50(0.8) Parameter Ym/Yo 4.0(0.6).(0.0).7 (0.5).9 (0.).5 (0.0).5 (0.8) mean ± S.E. d. (0.08).74 (0.07).5(0.07) 0.67 (0.0) 0.66 (0.04) 0.08 (0.07) AUC 5.99 (0.5) 7.9 (0.09) 5. (0.0) 8.8(0.49) 0.6 (0.04) 8.5(0.40) 6.8 (0.06) 6.8 (0.07) 9.44 (0.85) Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

Time-kill method 99 Table III. Co-efficient of variation for a single determination in one experiment for a, Ym/Yo, d and AUC for A. haemolyticus strain 564 Parameter a Ym/Yo d AUC Piperacillin/ tazobactam.5. 8. 4. Ciprofloxacin 6.4 6.7 7.5 Gentamicin 4.6 79.9.7 piperacillin/tazobactam the combination produced similar AUC values to gentamicin alone for all the strains except M. morganii strain 5496. For this strain the AUC was significantly smaller (P < 0.05) for the combination than for gentamicin alone. The other parameters were similar to the AUC, showing occasional significant differences with some strains. For ciprofloxacin, a was not statistically different from a for ciprofloxacin plus piperacillin/tazobactam for all strains, and a for gentamicin was the same as a for gentamicin plus piperacillin/tazobactam for all strains except S. marcescens, strain 570, when a was significantly larger for the combination (P < 0.05) Ym/Yo was not significantly different for ciprofloxacin plus piperacillin/tazobactam when compared with ciprofloxacin alone for any of the strains, and d was the same in all strains when gentamicin plus piperacillin/tazobactam and gentamicin were compared. M. morganii, strain 5496 and S. marcescens, strain 570, had a significant difference in only one parameter, that is for strain 5496 when tested with gentamicin plus piperacillin/tazobactam a and d were the same but AUC was not, while for strain 570 and gentamicin plus piperacillin/tazobactam d and AUC were the same but a was not. Convergence of the method to a best fit The non-linear least squares procedure (SAS: Proc NLIN) demands (a) initial guesses of the parameters' values (b) a choice of approximation algorithm. In this study the GAUSS, NEWTON and DUD (secant) methods of approximation were chosen. There are three possible outcomes: () Failure to converge to an acceptable fit; () Convergence to a unique set of parameters; and () Convergence to one of several sets of parameters that are acceptable to the convergence criterion. Table V shows the results obtained from attempts to fit two sets of data under variations of (a) and (b). Four acceptable solutions were found for piperacillin/tazobactam alone using A. haemolyticus strain 564 on one occasion, and five for the same strain on another. Often combinations of (a) and (b) failed to converge. Residual variances are shown on Table V for each of the attempts and measures of the variability of the various estimates are also shown. Again it is clear that the AUCs have the highest precision of estimation. Discussion Owing to the difficulties of assessing the effect of combinations of antibiotics in clinical practice, it is likely that we will continue to depend on laboratory evaluations to give at least an initial impression of any likely interactions. Hence, reliable methodology Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

CO Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07 Table IV. a, Ym/Yo, d and AUC values for piperacillin/tazobactam, ciprofloxacin, gentamicin, combinations of piperacillin/tazobactam plus ciprofloxacin and piperacillin/tazobactam plus gentamicin against S. marcescens (n = ), M. morganii (n = ) A. haemolyticus and C. freundii S. marcescens (strain 570) piperacillin/tazabactam (MIC* = 4) ciprofloxacin (MIC = 0.) gentamicin (MIC = ) piperacillin/tazabactam + gentamicin S. marcescens (strain 5690) piperacillin/tazobactam (MIC = 4) ciprofloxacin (MIC = 0.) gentamicin (MIC = ) piperacillin/tazobactam + gentamicin M. morganii (strain 5496) piperacillin/tazobactam (MIC = 0.5) ciprofloxacin (MIC = 0.0) gentamicin (MIC = 0.5) piperacillin/tazobactam + gentamicin M. morganii (strain 579) piperacillin/tazobactam (MIC = 8) ciprofloxacin (MIC = 0.) gentamicin (MIC = 0.5) piperacillin/tazobactam + gentamicin A. haemolyticus (strain 5644) piperacillin/tazobactam (MIC = 0.5) ciprofloxacin (MIC = 0.0) gentamicin (MIC = I) piperacillin/tazobactam + gentamicin C. fruendii (strain 50) piperacillin/tazobactam (MIC = 64) ciprofloxacin (MIC = 0.0) gentamicin (MIC = ) piperacillin/tazobactam + gentamicin a 0.8 (0.0) 0.676 (0.0) 0.0 (0.04) 0.76 (0.00) 0.468 (0.040) 0.48 (0.5) 0.80 (0.4).7 (0.500) 0.84 (0.50).5 (0.404) 0.94 (0.) 0.950(0.9).64 (0.7).45 (.58).80 (0.669) 0.45 (0.04) 0.5 (0.089).865 (.569) 0.644 (0.67).848 (0.890) 0.98 (0.65).0(0.99).99 (0.777).6 (0.444).04 (0.75) 0.98.56.685.78.49 "MICs in mg/l; [], Negative value;, not significant. Parameter Ym/Yo.60 (0.76).8 (0.).5 (0.04) 4.58 (.84).0 (0.).48 (0.).76 (0.50).98 (0.79).80 (0.5).6 (.9).80 (0.09) 4.8 (0.67).9 (0.50).7 (0.).76 (0.54).7.96.8 d Mean.9 (0.6) [0.5] (0.8) 0.4 (0.08).8 (0.) 0.50 (0.7) 0.8 (0.).6 (0.7) [0.4] (0.9) [0.07] (0.7).56 (0.5) [0.] (0.8) 0. (0.0).9 (0.0) [0.7] (0.5) 0. (0.05) 0.99 0.4 0.06 (S.E.S) AUC.7 (.69) 9.89 (0.9) 0.64 (.0) 9.78 (0.40) 8.05 (0.4).57 (0.9).65 (.05).47 (.58).06 (.7).7 (.7) 5.8 (.) 6.0 (.7). (.9) 6.0(.58) 8.98 (.).8 (.) 4.5 (.8) 6.0 (.0).4 (.99).8 (6.96).58 (0.80) 4.86 (.8) 4.99 (7.84) 4.47 (.90) 9.75 (.58) 0.84 7.0 0. 7.44.7 P < 0.05

Time-kill method 0 Table V. Variation in parameter estimates when different initial methods and conditions were employed (strain 564) Antimicrobial agent Pipcracillin/ tazobactam PiperaciUin/ tazobactam CV SE attempt (s) 4 (s) 4 5 a 0.07.565 0.88 0.9 0.9.687 0.8 0.66 0.9 68.7 0.55 Parameter Ym/Yo.9.4.59.59.7.44.6.5.55 4..04 d.50.6.0.6..66.0 0.96.9.7 0.6 AUC 7.7 7.7 7. 7.40 7.6 7.4 7.8 7.7 7.5 0.4 0. A(r)*.86.6.78 9.7 9..50.60.70 0.4 40.0 6.6 A(c)*.69 4.75 5.54 7.68 8.4 4.84 5.68 4.0 7. 68.0 6.6 Residual variance 0.06 0.0070 0.0079 0.00 0.0 0.008 0.0096 0.048 0.0 *A(r), area of the rectangle defined by t = 0, Y = Ym\ t = 6, Y Ym (see Figure); A(c), area under curve above A(r) (see Figure). which can provide objective measurements is of great importance, otherwise it will not be possible to make the clinical-laboratory correlations necessary to validate laboratory methods for assessing antimicrobial activity or interactions. We believe the methodology described here offers an objective way of describing time-kill curves, and, by fitting them to theoretical functions, enables a superior approach to the determination of synergy or antagonism compared with traditional methods (Krogstad & Moellering, 986). The use of best-fit (theoretical) curves and theoretical functions, including AUC, which is the area under the best-fit (theoretical) curve, enables estimates of the precision of the various parameters to be determined. In addition, variabilities are averaged out by the fitting procedure thereby ensuring a more precise estimation of each parameter. Careful characterisation of the process by repeating time-kill curves on several occasions and on the same occasion in triplicate enabled us to assess the various parameters and their relative precision. In this regard the AUC is the best parameter as a single overall measure. However, overall agreement among all the parameters for a single strain, or for a combination of antimicrobial agents showing significant differences from single antibiotics is important in deciding if synergy or antagonism is present. If this approach is taken with our data then there is no evidence of synergy between piperacillin/tazobactam plus ciprofloxacin or piperacillin/tazobactam plus gentamicin. When using the methods we have described, ideally the initial parameter estimates should be varied, as should the approximation procedures in order to obtain a selection of possible 'fits' and descriptive parameters. When the experimental data approximate closely to one of the theoretical curves it is likely that a unique fit has been obtained. However, in a number of cases a variety of'acceptable' solutions may result, which may reveal large variations in 'acceptable' estimates of some parameters, for example, a may vary five-fold (Table V). If presented with several competing solutions, ideally the choice should be made by selecting the one with the smallest residual variance. In practice, however, an investigator may not have sufficient time to try many competing possibilities, nor access to information technology as sophisticated as that used in this study. In these circumstances the AUC of the theoretical (best fit) curve is robust against Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

0 A. P. MacGowan et al. variation of individual parameters (CV = 0.4%), and we advocate its use as the best simple characteristic to assess these in-vitro processes. Many software packages for personal computers now contain programs for fitting non-linear regressions and, even where a facility for automatic calculation of AUCs is not provided, simple numerical procedures such as Simpson's Rule may be applied. It could be argued that the apparent robustness of the AUC is related to its having a larger mean value than the other parameters, so depressing the CV, and to the dominance of the rectangle beneath the curve (Figure). However, not only is the CV of the AUC small, but so is the standard error in absolute terms. Furthermore, the values of A(r) and A(c) demonstrate that both of these component areas are as variable as the other individual parameters and it is the combined AUC that shows high precision. Finally, it is possible to envisage a situation where two AUCs may be the same, within experimental error, but when examination of the other parameters may reveal distinction between two dynamic systems. Despite this, we believe that the AUC still represents the best single summary parameter for describing time-kill curves and recommend its use to assess antimicrobial combinations. It should, however, be supported by the other parameters we describe. Acknowledgements This study was supported by a grant from Lederle Laboratories, Gosport, Kent. We also thank Dr Mike Allan of Lederle Laboratories UK for his help. References Blaser, J. (99). Interactions of antimicrobial combinations in vitro: the relativity of synergism. Scandanavian Journal of Infectious Diseases 74, Suppl., 7-9. Guerillot F., Carret G. & Flandrois, J. P. (99a). Mathematical model for comparison of time-killing curves. Antimicrobial Agents and Chemotherapy 7, 685-9. Guerillot, F., Carret, G. & Flandrois, J. P. (996). A statistical evaluation of the bacterial effects of ceftibuten in combination with aminoglycosides and ciprofloxacin. Journal of Antimicrobial Chemotherapy, 685-94. Holt, H. A., Bywater, M. J. & Reeves, D. S. (990). In vitro activity of cefpodoxime against 84 isolates from domiciliary infections at 0 UK centres. Journal of Antimicrobial Chemotherapy 6, Suppl. E, 7-. Krogstad, D. J. & Moellering, R. C. (986). Antimicrobial combinations. In Antibiotics in Laboratory Medicine, nd edn (Lorian, V., Ed.), pp. 57-78. Williams & Wilkins, Baltimore. Norden, C. W., Wentzel, H. & Keleti, E. (979). Comparison of techniques for measurement of in vitro synergy. Journal of Infectious Diseases 04, 69-. Rand, K. H., Houck, H. J., Brown, P. & Bennett, D. (99). Reproducability of the microdilution checkerboard method for antibiotic synergy. Antimicrobial Agents and Chemotherapy 7, 6-5. Wootton, M., Hedges, A. J., Bowker, K. E., Holt, H. A., Reeves, D. S. & MacGowan, A. P. (995). A critical assessment of the agar dilution chequerboard technique for studying in-vitro antimicrobial interactions using a representative /f-lactam, aminoglycoside and fluoroquinolone Journal of Antimicrobial Chemotherapy 5, 569-76. {Received 4 March 995; returned August 995; revised 9 February 996; accepted 6 April 996) Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07

Time-kill method 0 Appendix Differential coefficients needed for the approximation procedures For types and curves, when t > d and writing y = u/v : _ dy 0 dy (_L_JL\ - _JL_ \Y m Y.J' ~dy\ ~ Y 0 Y m dy\( Y ) dy dy_ dd ~ a (dy\ cpy (t - d)\da)' PY t - djjda ' <fy Y 0 \(dy u cfy dyodd a ( cpy \ cpy (t - d)\dyodaj' dyjia cpy cpy \ cpy ) For types curves when t < d: dy. dy l u V, Y 0 )[ l dy w The DUD (secant) method determines numerical estimates of the derivatives directly from the data and, therefore, does not require any of the algebra above. Downloaded from https://academic.oup.com/jac/article-abstract/8//9/79479 on 7 November 07