© 2002 American Thoracic Society
A Reiterative Method for Calculating the Early Bactericidal Activity of Antituberculosis DrugsAcademic Department of Medical Microbiology, Royal Free and University College Medical School, London, United Kingdom Correspondence and requests for reprints should be addressed to Stephen H. Gillespie, Academic Department of Medical Microbiology, Royal Free Campus, Royal Free and University College Medical School, Rowland Hill Street, London NW3 2PF, UK. E-mail stepheng{at}rfc.ucl.ac.uk
Studies of early bactericidal activity (EBA) are important in the rapid evaluation of new antituberculosis drugs. Historically, these have concentrated on the log fall in the viable count in sputum during the first 48 hours of therapy. In this paper, we provide a mathematical model that suggests that the viable count in sputum follows an exponential decay curve with the equation V = S + Me-kt (where V is the viable count, M the population of bacteria susceptible to the test drug, S the population susceptible only to sterilizing agents, t the day of sputum collection as related to start of therapy, k the rate constant for the bacteria killed each day, and e the Napierian constant). We demonstrate that data from clinical trials fits the exponential decay model. We propose that future EBA studies should be performed by measuring daily quantitative counts for at least 5 days. We also propose that the comparison of the early bactericidal activity of antituberculosis drugs should be evaluated using the time taken to reduce the viable count by 50% (vt50). A further reiterative refinement following a rule set based on statistically the best fit to the exponential decay model is described that will allow investigators to identify anomalous results and thus enhance the accuracy in measuring early bactericidal activity.
Key Words: tuberculosis, pulmonary • clinical trials • therapy • bacterial count
Tuberculosis has been recognized once again as a serious threat to human health (1). The first modern campaign to control it was complimented by a series of trials performed by the British Medical Research Council and the U.S. public health service, so therapeutic regimens have always been built on a strong evidence base (2–5). More recently, international initiatives have been launched to develop new drugs that will shorten treatment duration and improve overall recovery rates (6). If these ambitious plans are to be realized, the research community must rapidly develop accurate methods to evaluate the potential activity of candidate antituberculosis agents (7). Historically, early bactericidal activity (EBA) has been a keystone measurement when comparing the specific activity of different antituberculosis drugs to overcome the difficulty of determining the activity of an individual drug when given in a combination. This is ethically justified, as theoretical and practical evidence from the initial and subsequent studies show that there is no risk of resistance developing during the short period of monotherapy (8). In a study in the late 1970s, Jindani and coworkers developed a method to investigate the effect of antituberculosis agents given singly or in combination (8). They determined the sequential viable count of Mycobacterium tuberculosis found in 12-hour overnight sputa collected on two pretreatment days (Days -1 and 0) and during treatment on Days 2, 4, 6, 8, 10, 12, and 14. The viable counts were transformed into log10 values and plotted against time and the mean log-fall being calculated, a separate graph being made for each drug under investigation. From these graphs, three separate regression coefficients were calculated for 0- to 2-, 2- to 14-, and 0- to 14-day periods for each drug regimen. The authors found that the differences in the mean log-fall during Days 0–2 were highly significant between regimens, whereas those for Days 2–14 did not attain statistical significance. It was decided therefore that EBA would be defined as the rate of fall of colony forming units (cfu) during the first 2 days of treatment and should be expressed as log10cfu/day. The ability of a drug to reduce the viable count quickly is an important consideration because this rapidly renders the patient noninfectious. In addition, by reducing the bacterial load, patient well-being is quickly improved, and recovery from the immunologic derangement associated with tuberculosis may lead to improved treatment outcomes (8–10). Since this original work, many studies have examined EBA using a similar methodology. The definition of measuring the viable count within the first 48 hours of treatment became standard practice (11–14). This paper questions the validity and appropriateness of this definition and suggests an improved method of evaluating this kind of data. One of the main characteristics associated with EBA studies is the substantial inter- and intra-patient variation with respect to their colony counts (12). There have been several approaches to dealing with this anomaly. One study excluded patients whose sputum colony counts drifted away from the regression line (15). However, as most EBA studies are numerically small (group numbers are usually between three and eight subjects), removal of a single patient can introduce considerable statistical error. To reduce the problems associated with the variability reflected by the amount of saliva present in specimens, some studies calculated the EBA by also measuring the total count by microscopy and applying the formula: EBA = log10 (v1/t1) – log10 (v3/t3) (where v is the viable count as measured by the cfu and t the total count as measured by microscopy). Although this approach has been validated in an international multicenter trial (12), a group working in the United States has systematically studied protocols for EBA studies and concluded that microscopy does not reduce variation significantly (13). They also demonstrated that 10- to 12-hour sputum collections are preferable to 2-hour or spot sputum collections and that measurement of EBA over 5 days reduced the inter-patient variation. Thus, there is a need to explore the underlying reasons for such variability of results and to devise robust experimental protocols that reduce inter-patient variation or allow it to be readily identified. In this paper, we provide evidence that the current 48-hour definition of early bactericidal activity results in the loss of important data that reduces the accuracy of the studies. We also present an alternative method for evaluating these data that enables easier comparison between agents and allows anomalous results and true biologic variation to be identified.
Mathematical Modeling An assumption was made that there are at least two populations of M. tuberculosis coexisting in every patient with active tuberculosis (16). One of these populations consists of rapidly multiplying bacteria that behave in a way similar to organisms growing continuously in vitro; the other are semidormant bacteria (17, 18). The rapidly multiplying population is susceptible to those antibiotics that kill actively dividing bacteria, such as isoniazid and quinolones (16). The second population was assumed to be only minimally susceptible to antibiotics with rapid bactericidal activity and can only be killed by the action of drugs with sterilizing activity, such as rifampicin and pyrazinamide (16). Based on this assumption a mathematical model was constructed to describe the steep decline in sputum viable count that has been previously documented in clinical trials (8, 10). These trials suggested that, as the effect of sterilizing drugs is slow and as studies of EBA take place at the start of treatment, the effect of sterilizing activity could be safely discounted. Evidence from in vitro studies suggested that drugs kill a fixed proportion of mycobacteria per unit time and that this produces an exponential fall in the viable count (19). Thus, the viable count (V) can be calculated as the total number of rapidly multiplying bacteria susceptible to drugs with bactericidal drugs (M) added to the number of persisters susceptible only to sterilizing agents (S) and where t is the day of sputum collection as related to the start of therapy, k the rate constant for the bacteria killed each day, and e the Napierian constant.
Applying this formula, models were performed with k values of between 0.1 and 0.8 over the 7 days of therapy. Values were entered into Microsoft Excel (Microsoft Corp., Redmond, WA).
Re-evaluation of Published and Prospective Evaluation of Clinical Data
When the new formula is used to describe the outcome of a modeled EBA study, there is a clear relationship between the bactericidal activity represented by the value of k and the steepness of the fall of viable count. Thus, a drug with modest bactericidal activity causes a slow decline to a plateau, whereas a drug with higher bactericidal activity will cause a very rapid decline and a shorter time before a plateau is reached. As anticipated, the slope of all of the graph sets was similar: the point at which the plateau is reached differed. The result of one of the mathematical simulations is illustrated in Figure 1 . From this model we may conclude that if clinical studies are ended after 48 hours, modest bactericidal activity may not be detected.
Clinical Data Clinical data obtained from the previously published study of five patients given isoniazid and five given ciprofloxacin was re-evaluated (10). In addition, patients with open pulmonary tuberculosis recruited using similar inclusion/exclusion criteria at the same location (the Kibong'oto National Tuberculosis Hospital, Sanya Juu, Tanzania) given isoniazid 300 mg for 5 days (four patients) and rifampicin 600 mg for 5 days (two patients) were evaluated. The sputum viable count values were entered into the curve-fitting program and showed that the fall in sputum viable count followed an exponential decay curve with an r2 > 0.95 in seven patients treated with isoniazid, two patients treated with ciprofloxacin, and two of the patients treated with rifampicin alone or in combination with isoniazid. This curve was described using the new formula proposed above: V = S + Me-kt. (Equation 1). The data were also fitted to a two-phase equation:
From this, it was possible to calculate the time (in days) to reduce the viable count by 50% (vt50) using the formula
Example: Patient I1 value of k = 1.277, therefore
A Reiterative Refinement It has always been recognized that a particular source of variation is seen in patients that shed organisms irregularly, for which there are many different reasons (12, 20). Variation from irregular shedding or sampling errors would produce results that deviate from the exponential decay curve described above and reduce the goodness of fit to the model. Another source of variation is found with patients with resistant organisms or those with a poor immune response. Data from these patients would fit the model although the values of vt50 would be greater than for patients infected with susceptible organisms or who are immunocompetent. But variation caused by resistance or poor clinical response is part of the biologic system, unlike anomalous results that confuse estimations of drug efficacy. It was essential, therefore, to refine the mathematical model in a way that would differentiate between valid biologic variability and anomalous results due to irregular excretion of organisms and deal with them in such a way as to improve the goodness of fit to the exponential decay model. We therefore proposed and evaluated the following set of rules that constitute the reiterative refinement using r2, calculated by Prism Graphpad, as a measure of goodness of fit where
Patient data could be accepted if the r2 value calculated from at least four points was greater than 0.95. If this was not the case, then a single point should be removed sequentially, and taking the remaining points a value of r2 recalculated. All possible combinations of data should be calculated and the best fit compared. If only one change brought the r2 to above 0.95 the result could be accepted, but if more than one of the changes brought a value of r2 > 0.95 the data set with the lowest variance was accepted as the best estimate. If removal of a single point was insufficient then additional points could be removed, provided that at least four points were available and the results treated as above. It was deemed essential that a rigid rule set be proposed and followed so that the final value was statistically the "best fit" to the exponential decay model (maximum r2 with the least variance) and not chosen by the investigator. Applying this reiterative refinement to the results of Patient I2, the data point for Day 0 was removed from the calculations. It is notable that the viable count at the beginning of treatment was lower than that on subsequent days and lower than the average presenting colony count for patients in this study and others reported previously, suggesting that this was an anomalous result (12). This refinement resulted in a recalculated r2 of 1.0; no other changes resulted in an improved fit (this process is illustrated in Figure 2) . Removal of a single data result did not produce an r 2 > 0.95 for patient I4, but removal of both the Day 0 and Day 1 values left a curve of four points with an r2 of 1.0 (see Table 1). This process is illustrated in Figure 3 . The calculations for other reiterated data are not shown.
The reiterative refinement was also applied to results obtained from the patients given ciprofloxacin: of these, two fitted the model with r2 > 0.95 (Patients C3 and C5). Of the three patients who initially did not fit the curve, only one produced a value of r2 > 0.95 by removal of a single result (patient C1): this change resulted in an r2 of 0.99. Removal of the Day 0 and Day 3 values produced a good fit for two patients: C2 (r2 = 1) and C4 (r2 = 0.95). The time taken to reduce viable count by 50% (vt50) was then calculated as shown in Table 1.
Comparison of Methods
Using the "standard" EBA method the value for isoniazid was 0.49 (95% confidence interval [CI], 0.17–0.80; variance 0.19) and for ciprofloxacin 0.34 (95% CI, 0.004–0.67; variance 0.073). The corresponding values using the reiterative exponential decay model were 0.40 (95% CI, 0.31–0.48; variance 0.032) and 0.79 (95% CI, -0.24–1.88; variance 0.706), respectively. These data are included in Table 1. It is notable that by using the standard method, 2 of the 17 patients was lost because no Day 2 value was obtained. This is an important consideration, as it means that studies can be completed with fewer patients because they will not be lost due to the absence of a single result. More significantly, some of the values obtained using the conventional measure were irrational as there were two patients treated with isoniazid whose viable counts were higher on Day 2 than on the pretreatment day (Patients I2 and I4; Table 1). This results in a negative value for EBA, but this result was obscured by others where the fall was very high. The data for these patients calculated by the reiterative exponential decay model suggested that these results were anomalous. When the means ± SD, 95% CIs, and variance are calculated for the isoniazid-treated patients by both methods, the reiterative exponential decay model generated no such anomalous EBA values and less variance. The wide variation seen with the "standard" EBA is the inevitable consequence of a calculation method based on only two data points. For the ciprofloxacin data set the 95% CIs and variance were higher for the patient data evaluated by the reiterative exponential decay model. This effect is mainly due to Patient C5; the sputum counts for this patient fit the exponential decay curve very well but the calculated vt50 is five times greater than that of the other four patients treated with this drug. As the isolate was not resistant and the patient went on to a normal treatment regimen with bacteriologic cure, the reason for this result cannot be determined (21). However, the fact that the data fits the model suggests that the result represents real biologic variation. This demonstrates the advantage of the reiterative exponential decay model, as the slow response of this patient could not be distinguished using the conventional method. Also, the conventional method gives no indication as to whether variation is due to methodologic problems or real biologic variation (12, 20). This may prove to be a significant advantage, as it means that once we can be more certain of the values for individual patients, we will be able to more reliably distinguish between different treatment regimens and between different types of patients (e.g., HIV seropositive or HIV seronegative).
Conclusion For a minority of patients, the results of sputum viable counts do not fit the model and these outlying data points contribute to the variability found in previous studies (12, 20). But by using the reiterative refinement described, we have been able to identify and remove these anomalous results, thus reducing intrapatient variability. We have demonstrated that this can be achieved without removing genuine biologic variation. If adopted, this model requires a change in the design of early bactericidal activity studies. Sputum samples should be collected daily over 5–7 days. This is needed to allow for the proportion of samples that cannot be included because the cultures are contaminated or to allow anomalous results to be removed. With the recent renewed international effort to develop new drugs for the treatment of tuberculosis, it is essential to have a simple, rapid, and inexpensive mechanism to evaluate new drugs (7). Our proposal for 5- to 7-day monotherapy trials, and the use of the exponential decay curve model with the reiterative refinement to define vt50, will enable researchers to confirm that candidate drugs are not only active in the human host but provide a measure of their activity in comparison with established agents. The ability to identify and remove anomalous results will do much to clarify true biologic variability and eliminate the confusion that has previously dogged early bactericidal studies.
The authors gratefully acknowledge the critical discussions of Prof. Andrew Nunn, Dr. Amina Jindani, and Dr. Janet Gillespie in this work. R. G. is supported by a research grant from Bayer AG. Received in original form December 26, 2001; accepted in final form March 25, 2002
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