Population distributions of minimum inhibitory concentration increasing accuracy and utility

Transcription

1 Journal of Applied Microbiology ISSN ORIGINAL ARTICLE Population distributions of minimum inhibitory concentration increasing accuracy and utility R 2 -Scientific, Sharnbrook, Bedfordshire, UK Keywords antibiotic resistance, BSAC, EUCAST, minimum inhibitory concentration, MYSTIC, surveillance. Correspondence Ronald Lambert, R 2 -Scientific, 5 Station Road, Sharnbrook, Bedfordshire MK44 PT, UK. rjwlambert@aol.com 25/27: received March 25, revised 3 October 25 and accepted 25 October 25 doi:./j x Abstract Aims: To generate continuous minimum inhibitory concentration (MIC) data that describes the discrete nature of experimentally derived population MIC data. Methods and Results: A logistic model was fitted to experimentally derived MIC population cumulative distributions from clinical isolates of Haemophilus influenzae, Moraxella catarrhalis, Streptococcus pneumoniae and Staphylococcus aureus (European Committee on Antimicrobial Susceptibility Testing, BSAC and MYSTIC population susceptibility databases). From the model continuous distributions of population susceptibility were generated. The experimentally observed population distributions based on discrete MIC could be reproduced from this underlying continuous distribution. Monte Carlo (MC) simulation was used to confirm findings. Where the discrete experimental data contained few or no isolates with MIC greater or less than the antimicrobial concentration range tested, the true mean MIC was a factor of Æ77 times that normally reported and may be of little clinical significance. Where data contained isolates beyond the range of concentration used, the true MIC was dependent on the SD and the number of isolates and could be clinically significant. Subpopulations of differing susceptibilities could be modelled successfully using a modified logistic equation: this allows a more accurate examination of the data from these databases. Conclusions: The mean MIC and SD of population data currently reported are incorrect as the method of obtaining such parameters relies on normally distributed data which current MIC population data are not. Significance and Impact of the Study: Obtaining the distribution parameters from the underlying continuous distribution of MIC can be carried out using a simple logistic equation. MC simulation using these values allows easy visualization of the discrete data. The analyses of subpopulations within the data should increase the usefulness of horizontal studies. Introduction The minimum inhibitory concentration (MIC) is a primary factor in deciding on the clinical possibilities of an effective antimicrobial treatment and has been termed a gold standard in susceptibility testing (Andrews 2). Population data on MIC susceptibility in concert with pharmacokinetics is used to determine whether an organism will respond to antimicrobial therapy (MacGowan and Wise 2; Mouton 22). Routine susceptibility testing is also seen as having public health significance: highlighting evidence of increased resistance through longitudinal studies (Morris and Masterton 22). These surveillance efforts, whether MYSTIC, Alexander project, EUCAST, etc., are truly great scientific endeavours: furnishing data that can lead to initiatives to control the development of antimicrobial resistance (Morris and Masterton 22) and Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26)

2 MIC accuracy which can be used to facilitate optimal therapeutic guidelines (Felmingham 22). Although there are still various ways to obtain an MIC value, attempts to standardize the methodology have been largely successful and large volumes of data have been produced. Extracting useful information from this voluminous data are, however, still problematic (Lewis 22). The presentation of the findings of population susceptibility data usually take the form of either specific values, e.g. the MIC 5 which is the MIC at which 5% of the population is susceptible or in the form of continuous data either tables of the numbers of susceptible isolates at particular MIC values or in the form of simple population frequency plots. MacGowan and Wise (2) examined the establishment of MIC breakpoints and figures showing normal distributions of population MIC were given. However, these figures gave a false impression of experimental fact. Although, population pharmacokinetic parameters are normally distributed, MICs, obtained by broth dilution, are log normal (Mouton 22), but a glance at any graph of population log MIC data does not show a true log-normal distribution the data occur in discrete units based on the dilution series used. Yet, it is this discrete data, which is used to calculate population mean MIC values and SD (e.g. Butler et al. 999). These values are then used in the assignment of clinical breakpoints, e.g. the BSAC formula given by MacGowan and Wise (2) or in therapeutic decision making. The problem is that if the underlying MIC distribution is continuous (and there is absolutely no suggestion that MIC are really quantal) then the mean (and SD) extracted from discrete data are in error and may not reflect the true, underlying values. These values may be further compromised when data above or below the maximum or minimum dilution range are used (as is commonly done). So although patient mortality and morbidity can be reduced with the availability of reliable and timely information (Felmingham 22), the current manner in which the data are analysed does not reflect the true distribution but rather the distribution imposed by methodological necessity. We undertook a study to see if it were possible to obtain the underlying continuous population distribution of MIC from the discrete MIC data, from which a more accurate representation of the mean and SDs of antibiotic population MIC data could be calculated and to see if this had clinical significance. Methods Population MIC data were obtained from the European Committee on Antimicrobial Susceptibility Testing (EUCAST) website (Jan 25, Lambert (24), from the databases of the Alexander Project ( and MYSTIC (Academy for Infection Management: org). Modelling The discrete nature of population MIC data hides the underlying continuous distribution of log MIC. The values given at each MIC concentration are the cumulative numbers of isolates inhibited between the dilution range being tested. Thus the distribution can be seen to be Total¼ Z a f ðxþdxþ Z b a f ðxþdxþ Z c b f ðxþdxþ Z n f ðxþdx; ðþ where f(x) is the distribution function. The values of each of the separate integrals are simply the number of isolates with MIC between the limits (where the limits are given by the log of the MIC). The total is the number of isolates in the sample size. The integral also includes those isolates which are defined as having an MIC less than or greater than the dilution range used. For a given set of population data solving eqn () will lead to the mean and SD of the population distribution. Unfortunately this is no trivial exercise: Monte Carlo (MC) analysis can be used to find the underlying mean and standard distribution, but becomes difficult if the distribution is bimodal or more complex. A simpler way is to approximate the cumulative distribution function eqn () with the logistic equation eqn (2). Cumulative dist. ffi No. isolates þ expðp ðp 2 xþþ ; ð2þ where x is the log of the inhibitor concentration, P is a measure of the slope, which has a maximum value at (P no. isolates)/4, P 2 is the approximation of the mean value (l, in log concentration), an approximation to the SD (r) of the underlying distribution is given by r ¼ P ln 84 : 6 In this study the cumulative distribution of population MIC data were graphed against the log of inhibitor concentration. The number of distinct steps occurring in the distribution were examined; for a single (no steps) distribution eqn (2) was applied directly; for a bimodal distribution (one step), eqn (3) was applied with n ¼. Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

3 MIC accuracy Cumulative dist. ffi Xnþ Pop i þ expðp ;i ðp 2;i xþþ ; ð3þ i¼ where n is the number of defined steps in the cumulative population distribution and Pop i is the number of isolates per population. The number of isolates per distribution are given approximately by the position of the step, with the proviso in the fitting of the equation that the total number of isolates equals the actual total. Experimental data were analysed by the appropriate form of the equation using nonlinear regression analysis. This can be easily performed using the excel solver routine (Microsoft, Redmond, WA, USA) or by using a mathematically advanced package such as mathematica (Wolfram Research Inc., Champaign, IL, USA). From the equation obtained the MIC,5,9 values were calculated from a simple rearrangement of eqn (2) or (3), allowing direct comparison between values quoted in the literature or on the databases themselves. Monte Carlo simulations For a given population, the mean value l, and the SD r, were obtained using the procedure given above. Up to distributions of continuous MIC were then produced. For data containing two or more subpopulations, random distributions for each population were produced and then combined. These MIC values were then made discrete using the experimental two-fold dilution series. These regenerated or simulated discrete MIC distributions were then compared with the observed. Purely simulated data were produced using random distributions with a given mean and SD. These were used to generate experimental discrete data. Truncation of the concentration range used allowed changes in the mean and SDs to be calculated and compared. Results Population MIC: specific organisms Haemophilus influenzae Two databases were used: the EUCAST wild type and the BSAC (22/23 susceptibility data). The data were plotted as the cumulative distribution and data which showed no step behaviour were analysed by eqn (2), Tables and 2. In no case did the EUCAST wild-type data show any step behaviour. However, data from the BSAC site for several antibiotics showed a single step in the cumulative distribution, Fig.. Equation (3) was fitted to these data with n ¼ and the results are given in Table 3. The results suggest the existence of two populations with respect to the antibiotics used, with the proportion of the more sensitive one to the more resistant ranging from 4:to6:. Direct comparison between the two databases is difficult because only ciprofloxacin and moxifloxacin were common to both for these particular data sets: the results for these two antibiotics were, however, comparable analysis of the MIC x data showed good agreement. The BSAC values are dependent on the dilution range used (ciprofloxacin: MIC ¼ Æ8, MIC 5 ¼ Æ8, MIC 9 ¼ Æ5; moxifloxacin: MIC ¼ Æ5, MIC 5 ¼ Æ3, MIC 9 ¼ Æ6), whereas the MIC x values given here are based on the underlying continuous distribution and therefore should give a more accurate value [note that MIC 5 ¼ mean MIC value (mg l ) ) when fitted to eqn (2)]. What is important is that the data normally reported can be obtained through the interpretation of a simple formula with two parameters. Amoxicillin vs. H. influenzae To explain the methodology further, the data for amoxicillin is explored. Figure 2 shows the fit of the model to the observed discrete data along with the underlying nor- Table EUCAST wild-type data for Haemophilus influenzae fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r No. isolates MIC MIC 9 RMSE* Amikacin 8Æ68 (Æ323) Æ62 (Æ5) 4Æ3 Æ Æ38 7Æ494 Æ4 Ciprofloxacin 9Æ32 (Æ327) )2Æ7 (Æ3) Æ85 Æ Æ5 Æ5 Æ79 Gentamicin 7Æ427 (Æ436) )Æ23 (Æ9) Æ62 Æ Æ3 Æ29 Æ85 Levofloxacin Æ62 (Æ9) )Æ994 (Æ) Æ Æ Æ7 Æ6 Æ2 Moxifloxacin Æ47 (Æ275) )Æ846 (Æ2) Æ43 Æ58 98 Æ9 Æ23 Æ55 Ofloxacin 6Æ (Æ3) )Æ637 (Æ) Æ23 Æ4 359 Æ7 Æ32 Æ2 Tobramycin 9Æ58 (Æ67) Æ35 (Æ7) 2Æ2 Æ Æ56 3Æ53 Æ67 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

4 MIC accuracy Table 2 BSAC data for Haemophilus influenzae (926 isolates 22/23) fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r MIC MIC 9 RMSE* Amox/clav 8Æ54 (Æ8) )Æ467 (Æ4) Æ349 Æ26 Æ82 Æ639 Æ73 Cefaclor 6Æ25 (Æ263) Æ265 (Æ7) Æ844 Æ267 Æ86 4Æ56 Æ4 Cefuroxime 7Æ98 (Æ7) )Æ373 (Æ2) Æ4236 Æ28 Æ225 Æ798 Æ259 Cefotaxime 6Æ727 (Æ229) )2Æ (Æ6) Æ Æ246 Æ5 Æ2 Æ3 Erythromycin 6Æ549 (Æ75) Æ495 (Æ2) 3Æ255 Æ253 Æ443 6Æ768 Æ39 Ciprofloxacin 3Æ33 (Æ4) )2Æ37 (Æ6) Æ73 Æ24 Æ5 Æ Æ3 Moxifloxacin 8Æ6 (Æ32) )Æ77 (Æ5) Æ7 Æ23 Æ9 Æ32 Æ3 Tetracycline 3Æ479 (Æ334) )Æ467 (Æ4) Æ345 Æ23 Æ235 Æ497 Æ7 Cumulative number Drug concentration (mg l ) Figure Cumulative distribution for several antibiotics from the BSAC site against Haemophilus influenzae. Cefotaxime ( ) and erythromycin ( ) give simple curves whereas ampicillin (r), amoxicillin (e) and trimethoprim (h) show single-step behaviour. Frequency Amoxicillin MIC (mg l ) Figure 2 Observed and calculated discrete distributions of the susceptibility of Haemophilus influenzae to amoxicillin (hatched and black bars respectively) and the calculated underlying normal distributions of the two subpopulations using eqn (3) with n ¼ and the data from Table 3. Table 3 BSAC data for Haemophilus influenzae (926 isolates, 22/ 23) fitted to eqn (3) with n ¼ Ampicillin Amoxicillin Trimethoprim P, (SE) 6Æ958 (Æ738) 8Æ485 (Æ249) 6Æ72 (Æ2) P 2, (SE) )Æ869 (Æ8) )Æ496 (Æ5) )Æ (Æ3) P,2 (SE) 2Æ73 (Æ9) 3Æ343 (Æ365) 4Æ82 (Æ426) P 2,2 (SE) Æ779 (Æ82) Æ9733 (Æ46) Æ576 (Æ2) Pop mean MIC* Æ35 Æ323 Æ77 Pop r Æ238 Æ95 Æ269 Pop 2 mean MIC 6Æ2 9Æ44 37Æ6 Pop 2 r Æ63 Æ496 Æ344 MIC Æ7 Æ89 Æ364 MIC 5 Æ58 Æ364 Æ878 MIC 9 5Æ56 7Æ98 25Æ8 % Sensitive 8Æ7 82Æ 85Æ5 RMSE Æ97 Æ58 Æ46 *Values are in mg l ). RMSE, root mean square error of the model divided by the number of isolates. mal distributions of the two subpopulations of H. influenzae. Figure 2 shows that eqn (3) with n ¼ is a good fit to the observed data and is therefore an approximation to the underlying normal distribution given by Z ( ) ð Distribution ¼ Pop pffiffiffiffiffi exp x l Þ 2 2p r 2r 2 dx Z ( ) ð þ Pop 2 pffiffiffiffiffi exp x l 2Þ 2 dx; 2p r2 2r 2 2 where the subscripts refer to either population or 2, and where each integral on the right-hand side can be expanded as shown in eqn (). The nonalignment of the normal distribution of each population, shown in Fig. 2, with the observed discrete data are apparent. That this is, however, a closer model to the true distribution can be shown through MC simulation of the observed data using the values for the mean 2 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

5 MIC accuracy values and SD [from the fitting of eqn (3), with n ¼ ], as starting values. The data, which have the MIC as a continuous variable, are made discrete by applying the experimental dilution range used. When this is carried out the observed data are reproduced; Fig. 3 shows the results of a MC simulation of distributions of 926 isolates using the mean values and SDs given in Table 3. Moraxella catarrhalis Data for the susceptibility of M. catarrhalis were taken from the EUCAST wild type, the BSAC and MYSTIC databases. The data were plotted as the cumulative distribution and data that showed no step behaviour were fitted by eqn (2); Tables 4 and 5. In no case did the wild-type data show any step behaviour. However, data from the BSAC site for ampicillin and amoxicillin showed a single step in the cumulative distribution, Fig. 4. Equation (3) was fitted to these data for n ¼ and the results are given in Table 6. The results suggest the existence of two populations with respect to these two antibiotics, with the proportion of the more sensitive one to the more resistant being : in both cases. Amoxicillin/clavulanate exhibited the profile for a single population. It is interesting to note that the initial inhibition is coincident with that of amoxicillin. This is confirmation of the existence of two populations and that clavulanate essentially removes the resistance advantage. Although the cumulative distribution of cefotaxime did not show a defined step, a shoulder was present and that the normal distribution was atypical (Fig. 5a) suggested the presence of two populations with slightly differing mean values. Equation (3) with n ¼ was fitted to the data and the mean values and SD were obtained (Table 6). These values were used to construct a MC simulation (Fig. 5b). The results suggest the presence of two distinct populations with regards the susceptibility to cefotaxime. Data from the MYSTIC database for the susceptibility of M. catarhalis against cefotaxime also showed a shoulder in the cumulative distribution. An analysis of the data (only 58 isolates) using eqn (3) with n ¼, gave an excellent fit to the data with l pop ¼ Æ49 mg l ), (BSAC ¼ Æ55); l pop2 ¼ Æ293 mg l ) (BSAC ¼ Æ39) and a population ratio of Æ52 (BSAC ¼ Æ66). This shows good coherence between the databases and suggests further the existence of two populations differing slightly in mean MIC Frequency Figure 3 Monte Carlo simulation of the susceptibility of Haemophilus influenzae to amoxicillin, using the mean values and SD given in Table 3 ( distributions of 926 isolates) Amoxicillin MIC (mg l ) >24 Table 4 EUCAST wild-type data for Moraxella catarrhalis fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r No. isolates MIC MIC 9 RMSE* Ciprofloxacin 8Æ45 (Æ37) )Æ477 (Æ4) Æ333 Æ Æ8 Æ6 Æ8 Levofloxacin 5Æ64 (Æ77) )Æ387 (Æ6) Æ4 Æ6 3 Æ3 Æ57 Æ98 Moxifloxacin Æ4 (Æ) )Æ354 (Æ) Æ443 Æ Æ27 Æ73 Æ3 Ofloxacin 2Æ9 (Æ2) )Æ94 (Æ6) Æ85 Æ Æ64 Æ Æ5 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999 3

6 MIC accuracy Table 5 BSAC data for Moraxella catarrhalis (438 isolates) fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r MIC MIC 9 RMSE* Amox/clav 4Æ28 (Æ286) )Æ53 (Æ9) Æ7 Æ42 Æ2 Æ239 Æ269 Cefaclor 3Æ394 (Æ82) Æ336 (Æ8) 2Æ66 Æ489 Æ488 9Æ68 Æ Cefuroxime 7Æ36 (Æ7) )Æ85 (Æ4) Æ654 Æ225 Æ329 Æ3 Æ76 Cefotaxime 3Æ89 (Æ382) )Æ755 (Æ3) Æ76 Æ434 Æ47 Æ662 Æ435 Erythromycin 4Æ33 (Æ27) )Æ3 (Æ2) Æ5 Æ6 Æ35 Æ7 Æ42 Ciprofloxacin 2Æ2 (Æ6) )Æ664 (Æ4) Æ22 Æ82 Æ7 Æ28 Æ5 Moxifloxacin 36Æ2 (34Æ6) )Æ437 (Æ58) Æ37 Æ46 Æ32 Æ42 Æ2 Tetracycline 27Æ95 (4Æ8) )Æ548 (Æ29) Æ283 Æ59 Æ236 Æ339 Æ5 Trimethoprim 5Æ864 (Æ23) Æ96 (Æ7) 8Æ249 Æ283 3Æ48 9Æ547 Æ34 Cefotaxime exhibits a shoulder on the cumulative distribution and the data were also fitted to eqn (3) with n ¼. Cumulative number Drug concentration (mg l ) Figure 4 Cumulative distribution for several antibiotics from the BSAC site against Moraxella catarrhalis. Amoxicillin/clavulanate ( ) has a simple curve whereas ampicillin (h) and amoxicillin (r) show single-step behaviour, cefotaxime (e) exhibits a shoulder. Streptococcus pneumoniae Data for the susceptibility of Strep. pneumoniae were taken from the EUCAST wild type and the BSAC databases. The wild-type data were plotted as the cumulative distribution and the majority of the data showed no step behaviour and were fitted by eqn (2); Table 7. However, the distribution of susceptibility to tobramycin showed two distinct steps. Fitting eqn (3), with n ¼ 2, did not converge to a solution whereas eqn (3) with n ¼ gave a good fit to the data with a ratio of sensitive to resistant of Æ2 (l ¼ Æ556 mg l ), l 2 ¼ Æ6 mgl ) ; r ¼ Æ96, r 2 ¼ Æ265). The BSAC data were plotted as the cumulative distribution and the majority of the data showed single-step behaviour, only ciprofloxacin and moxifloxacin showed Table 6 BSAC data for Moraxella catarrhalis (438 isolates) fitted to eqn (3) Ampicillin Amoxicillin Cefotaxime P, (SE) 3Æ7 (Æ53) Æ89 (9Æ3) 7Æ826 (Æ238) P 2, (SE) )2Æ9 (Æ9) )Æ987 (Æ7) )Æ233 (Æ5) P,2 (SE) 3Æ76 (Æ59) 3Æ59 (Æ94) 2Æ62 (Æ22) P 2,2 (SE) Æ27 (Æ3) Æ946 (Æ8) )Æ495 (Æ3) Pop mean MIC* Æ6 Æ Æ58 Pop r Æ27 Æ52 Æ22 Pop 2 mean MIC Æ64 2Æ43 Æ32 Pop 2 r Æ447 Æ473 Æ3 MIC Æ65 Æ694 Æ42 MIC 5 9Æ34 Æ93 Æ24 MIC 9 39Æ 49Æ9 Æ429 % Sensitive 9Æ 8Æ9 39Æ9 RMSE Æ59 Æ27 Æ2 *MIC values in mg l ). RMSE, root mean square error of the model divided by the number of isolates. no step behaviour (ciprofloxacin: l ¼ Æ85 mg l ), r ¼ Æ3; moxifloxacin: l ¼ Æ3 mg l ), r ¼ Æ92). Table 8 gives the results of fitting eqn (3) with n ¼ to the data. In all cases the proportion of sensitive isolates is c. 9% of the total (ranging from 86% to 95%). In this case it is easy to suspect a high degree of correlation between the susceptibilities; however, without access to the individual strain data no cross-correlations can be made on a sound basis (Lambert 24). Staphylococcus aureus Data for the susceptibility of Staph. aureus were taken from the EUCAST wild-type and the BSAC databases (oxacillin resistant and oxacillin sensitive). The wild-type data were plotted as the cumulative distribution and none of the data sets showed a step behaviour and were fitted by eqn (2) (Table 9). 4 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

7 MIC accuracy (a) Frequency (b) Frequency Data taken from the BSAC database for Staph. aureus (oxacillin resistant MIC >4 mg l ) ) for five of the antibiotics gave simple (one population) distributions, which were fitted by eqn (2) (Table ). Gentamicin had 2 4 Cefotaxime MIC (mg l ) Cefotaxime MIC (mg l ) Figure 5 (a) Observed and calculated discrete distributions of the susceptibility of Moraxella catarrhalis to cefotaxime (hatched and black bars respectively). (b) Monte Carlo simulation of the susceptibility of M. catarrhalis to cefotaxime, using the mean values and SD given in Table 6 ( distributions of 438 isolates) a single step present and was modelled using eqn (3) with n ¼ (l ¼ Æ34, r ¼ Æ69; l 2 ¼ 35Æ5, r 2 ¼ Æ48, ratio of S : R ¼ 6Æ5 : ), whereas trimethoprim appears to show two steps and was fitted to eqn (3) with n ¼ 2 (l ¼ Æ43, r ¼ Æ98; l 2 ¼ 5Æ9, r 2 ¼ Æ35; l 3 ¼ 8Æ, r 3 ¼ Æ46, ratio of S : M : R ¼ 9Æ7 :Æ : ). For the other antibiotics a complicating factor is the truncation of the dilution series: for oxacillin, clindamycin and erythromycin there are at least two populations present, but they have 8%, 82% and 25% of the isolates noted as being 256 mg l ) for their respective antibiotics. Therefore this information allows us only to state that for oxacillin 4% of isolates have a mean of 7Æ66 mg l ) (r ¼ Æ4), for erythromycin 6% of isolates have a mean value of Æ28 (r ¼ Æ23) and for clindamycin 74% of isolates have a mean of Æ mg l ) (r ¼ Æ58). The other isolates have a mean value or mean values greater than the highest concentration tested. For ciprofloxacin there are at least three populations present; however, 25% of the isolates are labelled as 256 mg l ). Figure 6a shows the distribution of the MIC data for ciprofloxacin against Staph. aureus along with the modelled fit. Figure 6b displays the results of the MC simulation ( distributions of isolates shown). Although an extremely good fit is found, the data are misleading: the distribution parameters for the two most susceptible populations may have real validity but the parameters for the most resistant have not. Effect of truncation of the dilution range Simulated population MIC data Simulated normal distributions ( isolates) of log MIC were generated with l ¼ oræ33 (Æ mg ) or 2 mg l ) respectively) and r ranging from Æ to 3. The discrete natures of the experimental MIC values were then produced from these normal distributions using the two-fold dilution range Æ2 to >24. The average mean and SD of the generated MIC plots were obtained. Table 7 EUCAST wild-type data for Streptococcus pneumoniae fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r No. isolates MIC MIC 9 RMSE* Ciprofloxacin 8Æ739 (Æ57) )Æ43 (Æ3) Æ722 Æ Æ44 Æ285 Æ59 Levofloxacin 2Æ98 (Æ3) )Æ28 (Æ2) Æ646 Æ Æ49 Æ893 Æ4 Linezolid 9Æ985 (Æ74) )Æ225 (Æ2) Æ595 Æ Æ358 Æ988 Æ44 Moxifloxacin Æ64 (Æ4) )Æ46 (Æ) Æ9 Æ Æ58 Æ39 Æ2 Netilmicin 4Æ58 (Æ22) Æ949 (Æ2) 8Æ8953 Æ Æ93 27Æ257 Æ94 Norfloxacin 7Æ793 (Æ477) Æ993 (Æ9) 9Æ848 Æ23 5 5Æ45 8Æ85 Æ9 Ofloxacin 4Æ87 (Æ2) Æ97 (Æ2) Æ2499 Æ 488 Æ89 Æ756 Æ27 Teicoplanin Æ4 (Æ7) )Æ483 (Æ) Æ329 Æ63 26 Æ2 Æ54 Æ32 Vancomycin 5Æ3 (Æ78) )Æ65 (Æ) Æ2424 Æ 9 38 Æ74 Æ339 Æ36 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999 5

8 MIC accuracy Table 8 BSAC data for Streptococcus pneumoniae (772 isolates) fitted to eqn (3) Penicillin Amoxicillin Cefaclor Cefuroxime Cefotaxime Erythromycin Clindamycin Tetracycline Trimethoprim P, (SE) Æ36 (Æ3) 2Æ85 (Æ7) Æ (Æ22) 5Æ8 (Æ92) 3Æ34 (Æ62) 2Æ2 (2Æ) 8Æ96 (Æ3) 5Æ54 (Æ69) Æ83 (Æ8) P 2, (SE) )2Æ82 (Æ3) )2Æ3 (Æ2) )Æ795 (Æ3) )Æ964 (Æ8) )2Æ73 (Æ4) )Æ27 (Æ7) )Æ6 (Æ) )Æ878 (Æ) Æ522 (Æ) P,2 (SE) 4Æ854 (Æ928) 5Æ73 (Æ8) 3Æ846 (Æ64) 2Æ239 (Æ54) 3Æ554 (Æ636) 5Æ32 (Æ833) 29Æ8 (2Æ2) 4Æ74 (4Æ84) 34Æ67 (35Æ6) P 2,2 (SE) )Æ388 (Æ44) )Æ266 (Æ4) Æ349 (Æ5) )Æ386 (Æ73) )Æ82 (Æ65) Æ35 (Æ35) Æ389 (Æ76) Æ257 (Æ23) Æ57 (Æ265) Pop mean MIC* Æ7 Æ8 Æ6 Æ Æ7 Æ75 Æ87 Æ32 3Æ327 Pop r Æ46 Æ29 Æ5 Æ Æ24 Æ82 Æ87 Æ7 Æ4 Pop2 mean MIC Æ49 Æ542 22Æ34 Æ4 Æ5 3Æ65 2Æ449 8Æ7 37Æ24 Pop 2 r Æ342 Æ29 Æ43 Æ74 Æ467 Æ32 Æ56 Æ2 Æ48 MIC Æ4 Æ5 Æ3 Æ8 Æ5 Æ59 Æ67 Æ97 2Æ28 MIC 5 Æ7 Æ8 Æ66 Æ Æ7 Æ76 Æ88 Æ35 3Æ475 MIC 9 Æ9 Æ6 Æ348 Æ68 Æ64 Æ48 Æ22 Æ2 6Æ48 % Sensitive 9Æ4 9Æ4 92Æ2 86Æ8% 87Æ3% 9Æ2% 95Æ4% 94% 9% RMSE Æ59 Æ53 Æ47 Æ5 Æ7 Æ53 Æ3 Æ27 Æ3 *MIC values in mg l ). RMSE, root mean square error of the model divided by the number of isolates. The dilution range was truncated in a series of tests and the mean and SD recalculated, the results are given in Tables 3. Discrete distributions which had no isolates beyond the dilution range used (i.e. no isolates labelled as Æ2 or >24) had mean values that were Æ4 times higher than the underlying continuous data. This factor is simply the p root ffiffi of the dilution series used (for two-fold dilutions ¼ 2 ).As the SD of the continuous distribution was increased, the number of isolates beyond the dilution range increased, the calculated mean of the discrete distributions fell, and the calculated SD approached a maximum value. For example, a discrete distribution obtained from a continuous distribution with l ¼ 2mgl ) and r ¼ Æ4, has a calculated mean ¼ 2Æ82 mg l ) and r ¼ Æ4, with no isolates beyond the dilution range; a discrete distribution obtained from a continuous distribution with l ¼ 2mgl ) and r ¼ 2, has a calculated mean ¼ Æ4 mg l ) and r ¼ Æ5, with 5% of isolates beyond the dilution range. In a series of calculations a discrete distribution obtained from a continuous distribution based on l ¼ ( mg l ) ) and r ¼ Æ5, was truncated by either removal of the highest dilution values (lowest concentrations) or by the removal of the lowest dilutions (highest inhibitor concentrations). In the former case the calculation of the summation of the number of isolates equal to or less than a specified MIC was straightforward. In the latter case as the MIC is defined as the lowest concentration showing no visible growth, any isolates which had MIC greater than a specified value had to be censored from the calculation of mean and SD. In the literature the use of terminology such as MIC 256 mg l ) is incorrect given the definition of the method itself. Tables 2 and 3 show that if the population MIC data have a significant percentage of isolates which are labelled as either less than or equal to lowest MIC or greater than highest MIC tested then there is a significant effect on the perceived mean value. It is possible to assign a mean value, which is up to eight times the actual MIC depending on how the data has been analysed. Triclosan vs. Staph. aureus (methicillin-sensitive Staph. aureus and methicillin-resistant Staph. aureus) The MIC from a total of 69 clinical isolates of methicillin-sensitive Staph. aureus (MSSA) and 97 of methicillinresistant Staph. aureus (MRSA) against the antimicrobial biocide triclosan were examined (Lambert 24). When analysed together, the cumulative distribution was fitted to eqn (2), giving l ¼ Æ24 mg l ) and r ¼ Æ62. Of the combined data 86 isolates were labelled as Æ5 mg l ) : MSSA (79%), MRSA (56%). The discrete distribution gave a mean ¼ Æ77 mg l ) with r ¼ 6 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

9 MIC accuracy Table 9 EUCAST wild-type data for Staphylococcus aureus fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r No. isolates MIC MIC 9 RMSE* Amikacin 5Æ97 (Æ2) Æ29 (Æ4) Æ956 Æ28 8 Æ832 4Æ599 Æ69 Ciprofloxacin 6Æ59 (Æ97) )Æ647 (Æ5) Æ226 Æ Æ4 Æ49 Æ98 Gentamicin 6Æ22 (Æ9) )Æ543 (Æ3) Æ287 Æ Æ27 Æ647 Æ59 Levofloxacin 9Æ273 (Æ25) )Æ934 (Æ) Æ6 Æ Æ67 Æ2 Æ3 Linezolid Æ34 (Æ79) Æ2 (Æ) Æ265 Æ Æ8 Æ976 Æ9 Moxifloxacin 6Æ77 (Æ35) )Æ44 (Æ8) Æ36 Æ Æ7 Æ76 Æ49 Norfloxacin 9Æ83 (Æ4) )Æ48 (Æ2) Æ7 Æ8 236 Æ4 Æ234 Æ4 Ofloxacin 9Æ958 (Æ34) )Æ652 (Æ) Æ223 Æ Æ34 Æ37 Æ3 Teicoplanin 8Æ25 (Æ6) )Æ336 (Æ2) Æ46 Æ Æ246 Æ867 Æ36 Tobramycin 6Æ26 (Æ443) )Æ547 (Æ3) Æ284 Æ267 4 Æ26 Æ64 Æ239 Vancomycin 3Æ38 (Æ86) )Æ95 (Æ) Æ639 Æ Æ438 Æ932 Æ5 Table BSAC data for Staphylococcus aureus (Oxa R, isolates) fitted to eqn (2) Antibiotics P (SE) P 2 (SE) Mean MIC (mg l ) ) r MIC MIC 9 RMSE* Teicoplanin 9Æ954 (Æ35) )Æ9 (Æ2) Æ6452 Æ67 Æ388 Æ73 Æ4 Vancomycin 8Æ37 (Æ) )Æ6 (7 )5 ) Æ694 Æ9 Æ525 Æ9 Æ Tetracycline 49Æ7 (78) )Æ538 (Æ337) Æ2896 Æ33 Æ262 Æ32 Æ58 Minocycline 22Æ6 (Æ5) )Æ97 (Æ3) Æ8 Æ75 Æ64 Æ Æ29 Linezolid 3Æ4 (Æ47) Æ8 (Æ) Æ225 Æ55 Æ8 Æ42 Æ28 Æ359. This is consistent with the findings of Table 3, where a large truncation of the data gives rise to an apparently higher mean and lower SD than the continuous distribution. MC simulation of 266 isolates with the parameters from eqn (2), using the two-fold dilution series ( Æ5,, >Æ64) was able to reproduce the observed discrete distribution. When analysed separately as MSSA and MRSA, a distinction between the two was found. From the discrete experimental distribution l ¼ )2Æ336 (Æ7 mg l ) ) and r ¼ Æ38 for MSSA and l ¼ )2Æ86 (Æ8 mg l ) ) and r ¼ Æ35 for MRSA. The observed discrete data for MSSA could be reproduced from a normal distribution of continuous log-mic data where l ¼ )2Æ693 (Æ2 mg l ) ) and r ¼ Æ89. With MRSA the experimental data could be reproduced from a normal distribution of continuous log MIC data where l ¼ )2Æ35 (Æ45 mg l ) ) and r ¼ Æ387. The two distributions are significantly different (P < Æ5). Discussion The careful analysis of population data are essential if the rewards of the large-scale surveillance projects are to be gleaned from the voluminous data generated. Within this study we have attempted to recreate the underlying continuous MIC distribution(s) to obtain a true value for the mean and SD of the population(s). The simple rationale was to take the standard frequency data and calculate the cumulative distribution and use the logistic equation to model the true distribution. From the fitted parameters of this model mean and SD could be obtained; that these values were closer to the true mean and SD of the population could be gauged through MC simulation. The MC simulation of the continuous MIC was rendered discrete by the application of experimental dilution series. In all the populations studied this technique was successful in describing the experimentally observed discrete population pattern. As such we consider this technique to be a superior method of analysis than the simple analysis of discrete data. Where evidence of two separate populations occurred, the logistic-based model [eqn (3)] was used to obtain the mean values and SD of the populations and the ratios of sensitive to resistant. MC simulations of these distributions were also successful, further suggesting the applicability of this method to the study of such data. However, as the number of steps increased the degrees of freedom to fit the model decreased. In such cases the model is best used as a guide to the mean and SD values of the possible population types for use in a standard MC simulation. The ability to simulate the observed discrete distributions of MIC from an underlying continuous distribution of MIC must lead to an increase in the accuracy of the Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999 7

10 MIC accuracy (a) 5 Frequency (b) 7 Frequency Ciprofloxacin MIC (mg l ) Ciprofloxacin MIC (mg l ) Figure 6 (a) Observed and modelled Staphylococcus aureus (oxacillin >4 mg l ) ) distribution of ciprofloxacin minimum inhibitory concentration data obtained from the BSAC database (hatched and black bars respectively). Equation (3) was fitted to the data with n ¼ 2: Pop ¼ 6%, l ¼ Æ5, r ¼ Æ4; Pop 2 ¼ 5Æ5%, l 2 ¼ 4Æ, r 2 ¼ Æ6; Pop 3 ¼ 78Æ5%, l 3 ¼ 8Æ4, r 3 ¼ Æ54. (b) Monte Carlo simulation of the susceptibility of S. aureus (oxacillin >4 mg l ) ) to ciprofloxacin, using the mean values and SD given in (a) ( distributions of isolates). Table Effect of SD on the estimation of the mean MIC from discrete data produced from a continuous distribution of MIC with mean ¼ or 2 mg l ), with SD of Æ 3 for isolates True l (log MIC/mg l ) ) True r Observed l for discrete data Observed r for discrete data Æ Æ45 Æ52 Æ Æ2 Æ45 Æ28 Æ Æ4 Æ42 Æ4 Æ Æ6 Æ422 Æ6 Æ Æ8 Æ434 Æ8 Æ Æ48 Æ99 Æ5 Æ2 Æ38 Æ56 Æ8 Æ4 Æ282 Æ297 4Æ3 Æ6 Æ79 Æ45 7Æ5 Æ8 Æ37 Æ488 Æ6 2 Æ97 Æ56 5Æ7 3 Æ479 Æ69 34Æ5 Æ33 Æ2 2Æ825 Æ28 Æ Æ4 2Æ82 Æ49 Æ Æ6 2Æ842 Æ66 Æ Æ8 2Æ84 Æ797 Æ 2Æ796 Æ982 Æ5 Æ2 2Æ624 Æ4 Æ8 Æ4 2Æ326 Æ277 4Æ3 Æ6 2Æ7 Æ375 7Æ6 Æ8 Æ657 Æ459 Æ4 2 Æ44 Æ524 5Æ3 3 Æ646 Æ664 34Æ2 Percentage of isolates labelled as Æ2 or >24 Table 2 Effect on the mean MIC and r as the lower series of the dilution range are truncated Observed l for discrete data Observed r for discrete data Full range Æ428 Æ56 Æ Æ64 Æ43 Æ55 Æ7 Æ25 Æ437 Æ499 3Æ3 Æ25 Æ47 Æ48 Æ4 Æ5 Æ59 Æ433 27Æ Æ92 Æ35 49Æ6 2 2Æ78 Æ248 72Æ3 4 4Æ47 Æ49 88Æ5 Percentage of isolates labelled as MIC Test: true mean ¼ mgl ) and r ¼ Æ5. Full range: MIC: Æ2 MIC < 24. information being provided to the clinician. Furthermore, understanding the true underlying continuous distribution must also lead to more accurate representations of longitudinal analyses. At its most basic, if MIC population data have few or no isolates beyond the dilution range used, and if there is only one population represented, then the mean value of the discrete data are simply Æ4 times higher than the real value. The SD found is largely unchanged. Thus with such data there may be little clinical impact on having a more accurate value. However, if data were present which contain large numbers of isolates beyond the dilution range used, then the assign- 8 Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999

11 MIC accuracy Table 3 Effect on the mean MIC and r as the upper series of the dilution range are truncated Observed l for discrete data Observed r for discrete data Full range Æ42 Æ56 Æ >52 Æ42 Æ56 Æ >256 Æ42 Æ56 Æ >28 Æ42 Æ56 Æ >64 Æ42 Æ55 Æ >32 Æ45 Æ53 Æ >6 Æ386 Æ492 Æ8 >8 Æ29 Æ464 3Æ6 >4 Æ96 Æ46 Æ5 >2 Æ823 Æ355 27Æ5 > Æ552 Æ293 5Æ >Æ5 Æ337 Æ234 72Æ8 >Æ25 Æ95 Æ9 88Æ5 >Æ25 Æ25 Æ296 96Æ4 Percentage of isolates labelled as >MIC Test: true mean ¼ mgl ) and r ¼ Æ5. Full range: MIC: Æ2 MIC < 24. ment of a mean value using the discrete data are in error and may have a clinical significance. Hanberger et al. (999) examined a variety of antibiotics against several Gram-negative species using BSAC, NCCLS and Swedish Reference Group (SRGA) definitions for breakpoints. They stated that the surveillance studies must be precise and accurate and that sufficiently broad ranges of MIC must be used in order to detect small increases in susceptibility levels. Their study suggested that the SRGA breakpoints detected higher frequencies of decreased susceptibility in some populations than the NCCLS for some antibiotics. Yet many of the population distributions had a high percentage of isolates labelled as less than or equal to minimum MIC or greater than or equal to maximum MIC. The simple methods used to examine these discrete distributions took no account of this. Yet the conclusion was that with quantitatively accurate MIC methods the early signs of susceptibility shifts could be detected and remedial strategies introduced. From the evidence of Tables 3, small (and large) decreases in the apparent susceptibility can be caused through having too many isolates at the extremes of the dilution range. Butler et al. (999) examined 5252 Strep. pneumoniae isolates, tested in the Alexander Project. The study focused on whether amoxicillin and co-amoxiclav had lower MICs than penicillin, concluding that indeed they did. In this study, penicillin and amoxicillin both showed the presence of two populations. The most sensitive population to penicillin had a lower MIC than amoxicillin (Æ7 vs Æ8 mg l ) ), but amoxicillin had a higher proportion of isolates with that lower MIC (92% vs 9%). From the cumulative distribution curve there was little difference between the two. The method used by Butler et al. (999) to examine the MIC distribution relied on the use of the discrete values, which returns MIC values in accordance with the dilution series used. In this study the generation of the underlying continuous distribution allows a more accurate evaluation of common values such as MIC 9. The MIC 9 of amoxicillin was lower than that of penicillin. The method outlined herein can reconstruct the observed discrete distribution and can provide parameters for a MC simulation of the data. The data can also be used to provide the starting values for a more in-depth nonlinear modelling process. However, that the methods outlined can all be undertaken within the Microsoft excel package adds much to its merit. When the cumulative distribution is more complex than a single step (suggesting the presence of more than two subpopulations), the model can still be fitted to the data, and in fact will give a perfect fit in some cases the degrees of freedom reduce by 3 per step found. For a finite dilution series this poses a problem of obtaining accurate measures of the mean values and SD. In these cases it has been found better to use the values to prime an MC simulation and to use these to refine the values further. In many of the cases studied from the BSAC database, there would appear to be the suggestion that correlations exist between different antibiotics. Such correlations are, of course, well known (e.g. Lambert 24). From the data given in all these databases, however, there is no way to prove the existence of correlations the data for each strain are absent from the public databases and this reduces the effectiveness of these efforts. There is a need to have access to the individual strain data if cross-resistance patterns are to be firmly established. The standardization of susceptibility testing is required, of that there is little doubt. The problem is that as a standard gains acceptance, new insight and new methods come about and upset the attempts to rigidify the procedure. As Andrews states (Andrews 2), Like all standardized procedures, the method must be adhered to and may not be adapted by the user. Taken at face value this statement occludes any attempts to improve methodology. As improvements in other areas of antimicrobial science continue, MIC testing cannot continue with its early 2th century methods. An ability to embrace new ideas within the strict codification of standard methods must be sought. Acknowledgements The authors would like to thank the Soap and Detergent Association and the Cosmetic, Toiletry and Fragrance Association for funding this study. Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999 9

12 MIC accuracy References Andrews, J.M. (2) Determination of minimum inhibitory concentrations. J Antimicrob Chemother 48(Suppl. S), 5 6. Butler, D.L., Gagnon, R.C., Miller, L.A., Poupard, J.A., Felmingham, D. and Gruneberg, R.N. (999) Differences between the activity of penicillin, amoxycillin, and co-amoxyclav against 5252 Streptococcus pneumoniae isolates tested in the Alexander Project J Antimicrob Chemother 43, Felmingham, D. (22) The need for antimicrobial resistance surveillance. J Antimicrob Chemother 5(Suppl. S), 7. Hanberger, H., Nilsson, L.E., Claesson, B., Kärnell, A., Larsson, P., Rylander, M., Svensson, E., Sörberg, M. et al. (999) New species-related MIC breakpoints for early detection of development of resistance among Gram-negative bacteria in Swedish intensive care units. J Antimicrob Chemother 44, Lambert, R.J.W. (24) Comparative analysis of antibiotic and antimicrobial biocide susceptibility data in clinical isolates of methicillin-sensitive Staphylococcus aureus, methicillinresistant Staphylococcus aureus and Pseudomonas aeruginosa between 989 and 2. J Appl Microbiol 97, Lewis, D. (22) Antimicrobial resistance surveillance: methods will depend on objectives. J Antimicrob Chemother 49, 3 5. MacGowan, A.P. and Wise, R. (2) Establishing MIC breakpoints and the interpretation of in vitro susceptibility tests. J Antimicrob Chemother 48(Suppl. S), Morris, A.K. and Masterton, R.G. (22) Antibiotic resistance surveillance: action for international studies. J Antimicrob Chemother 49, 7. Mouton, J.W. (22) Breakpoints: current practice and future perspectives. Int J Antimicrob Agents 9, Journal compilation ª 26 The Society for Applied Microbiology, Journal of Applied Microbiology (26) 999