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Antimicrobial Agents and Chemotherapy, September 2006, p. 3154-3156, Vol. 50, No. 9
0066-4804/06/$08.00+0     doi:10.1128/AAC.00774-05
Copyright © 2006, American Society for Microbiology. All Rights Reserved.

Use of Nonlinear Mixed-Effects Analysis for Improved Precision of Early Pharmacodynamic Measures in Tuberculosis Treatment

Geraint R. Davies,1* Richard Brindle,2 Saye H. Khoo,3 and Leon J. Aarons4

Wellcome Trust Centre for Clinical Tropical Medicine, University of Liverpool, Liverpool, United Kingdom,1 Department of Clinical Microbiology, Portsmouth Hospitals, Portsmouth, United Kingdom,2 Department of Pharmacology and Therapeutics, University of Liverpool, Liverpool, United Kingdom,3 School of Pharmacy and Pharmaceutical Sciences, University of Manchester, Manchester, United Kingdom4

Received 20 June 2005/ Returned for modification 7 March 2006/ Accepted 12 June 2006


   ABSTRACT
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Nonlinear mixed-effects analysis of serial sputum colony-counting data supports the existence of two bacillary subpopulations in sputum, eliminated at different rates. It distinguishes between combination regimens, removes bias, and greatly improves precision, with significant implications for the analysis of surrogate endpoints of "sterilization" in the development of new antituberculosis regimens.


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Improving tuberculosis treatment depends on enhancing the "sterilizing" activity of drug regimens, and surrogate endpoints of this activity are urgently needed (9). Serial sputum colony counting (SSCC) of Mycobacterium tuberculosis early in therapy has been advocated for this purpose. "Early-bactericidal-activity" studies measure the effects of single agents over the first 2 days (10, 20), probably reflecting activity against actively dividing bacilli within cavities rather than "sterilizing" activity. Extending the sampling period could provide such a measure (11), but prolonged monotherapy raises ethical concerns. The only SSCC study employing a modern combination regimen and methods without decontamination of sputum was performed in Kenya in 1989 and 1990 (2). Reanalysis of this study (1) using summary measures (SM) (13) supports the concept of extended sampling but does not make optimal use of the data. We describe a nonlinear mixed-effects (NLME) approach offering greater efficiency.

The study design and conduct were described previously (2). A total of 122 new pulmonary-tuberculosis patients received streptomycin and isoniazid and either thiacetazone (the SHT regimen) or rifampin and pyrazinamide (the SHRZ regimen). All harbored drug-sensitive strains and consented to human immunodeficiency virus (HIV) serology (29% seropositive). Sputa were sampled for SSCC on days 0, 2, 7, 14, and 28.

The nlme package (18) in R (19) was used for NLME analysis with monoexponential and biexponential models. Only positive results were included, as in the previous analysis (1). A log10 transformation of the response and a variance function were used to account for heteroscedasticity. With an exponentiated parameterization to enforce positivity of the parameters (5), the biexponential model has the following form:

Formula
Fitted models were compared using the Akaike information criterion, the likelihood ratio test, and residual plots. For comparability, parameter estimates were reexpressed as intercepts A and B ({theta}1 and {theta}3 divided by 2.303) and rate constants {alpha} and ß [e.g., Formula] on the log10 scale (1). An analytical approximation, Formula, was used to compute the elimination times of subpopulations. A linear model was used for covariate effects on the intercepts. Inferences relied on Wald tests, but other methods were used to explore and confirm the findings (quantile-quantile plots and the Shapiro-Wilks W test for normality, unpaired Student or Welch t tests, and Wilcoxon and Mann-Whitney rank sum U tests for tests of significance, as appropriate).

One hundred subjects (54 SHT; 46 SHRZ) had more than one data point available, and 48 subjects had five positive data points. For both data sets, model fitting unequivocally selected a biexponential model (Table 1 and Fig. 1), suggesting that at least two subpopulations of bacilli, eliminated at different rates, are present in sputum (referred to henceforth as "fast" and "slow"). Parameter estimates were consistent with previous SM estimates (1) and the effects of these regimens in clinical trials (7), with the "slow" rate constant, ß, larger for SHRZ than for SHT (0.094 versus 0.063 log CFU/ml/day for regimen contrast within the whole data set [P = 0.013]; 0.094 versus 0.076 log CFU/ml/day for fitted data by regimen [P < 0.001]). The addition of random effects significantly improved the fit (P < 0.001) (Table 1), accounting for high variability in parameters between subjects, greater in the intercepts A and B than the rate constants {alpha} and ß.


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  		TABLE 1. Point estimates and 95% confidence intervals of the fixed effects and goodness of fit for the two regimens from pooled and mixed-effects fits using the whole data set (SHT n = 54, SHRZ n = 46)

 

Figure 1
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  		FIG. 1. Locations of the monoexponential (dashed lines) and biexponential (solid lines) regression functions fitted to the subsets of complete profiles for the SHT (n = 29) and SHRZ (n = 19) regimens.

 
Differences in the A intercept by HIV status were confounded by the extent of cavitation (P = 0.432 for HIV and 0.018 for the cavitation score), but controlling for this, the B intercept remained reduced in the HIV-positive group (P = 0.008). ß did not differ by HIV status for either regimen (SHT, P = 0.928; SHRZ, P = 0.367). For the SHT and SHRZ regimens, the predicted elimination times were 7.5 (5.6 to 9.5) and 13.2 (10.3 to 16.0) days for the "fast" subpopulation and 74.3 (68.8 to 79.8) and 56.2 (52.9 to 59.5) days for the "slow" subpopulation.

We compared the NLME estimates of ß with different SM estimates, with respect to bias (Table 2) and precision (Table 3). Any SM estimator involving day 2 overestimated ß by 16 to 139%, reflecting the persistence of the "fast" component within the first week, but over later periods agreed to within 5 to 29%. Overall, comparing the SM ß (2 to 28) coefficient with NLME estimates of ß, the percent coefficient of variation was reduced sixfold, from 62% to under 10%.


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  		TABLE 2. Bias: magnitude of SM and NLME estimators of ß and their ratioa

 

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  		TABLE 3. Precision: standard errors and percent coefficients of variation of SM and NLME estimators of ß

 
We conclude that a biexponential model fits the data best, a finding consistent with the presence of multiple bacillary subpopulations. Alternative explanations, such as selection of drug-tolerant bacilli from an initially homogeneous population during therapy, seem less plausible given the spatial compartmentalization of bacilli in pulmonary tuberculosis (4, 6), the pharmacodynamic properties of combination chemotherapy (14, 15), and the conditions for induction of drug tolerance in vitro (16). Some studies have reported only a single population in their data (8, 12, 21) but invariably employed sputum decontamination, presumably selectively eliminating any "fast" subpopulation to various extents (3). Since the only observed population then declines at a constant rate, these data support the hypothesis of multiple baseline subpopulations.

Since the biexponential model better represents the data, it can eliminate bias in SM estimates computed over periods prior to elimination of the "fast" subpopulation. It can in principle discriminate between effects on subpopulations, a critical methodological issue for accurate estimation of "sterilizing" activity. The activities of antituberculosis drugs against different subpopulations vary in vitro (17). Agents active against multiple subpopulations or interacting pharmacodynamically with other components of a regimen in vivo could create unpredictable bias in SM estimates.

The estimated elimination times of the "slow" component raise the question of whether this subpopulation is the one from which relapses subsequently arise and therefore an appropriate model for ultimate sterilization and cure. A third subpopulation at an abundance below 10 to 103 CFU/ml of sputum may not be detected by SSCC (3), and a lack of degrees of freedom restricts the complexity of the models fitted here. Our findings need to be supported and extended by further experimental work and clinical trials. Though SSCC parameters have greater prima facie plausibility than early-bactericidal-activity measures, whether they accurately reflect clinical endpoints of "sterilization," like relapse, requires ongoing evaluation.

NLME analysis can explicitly model complicated error structures, account for interindividual variability, and make efficient use of incomplete data. It may be a useful method of reducing bias and improving precision in assessment of the "sterilizing" activities of antituberculosis regimens.


   ACKNOWLEDGMENTS
 
G. R. Davies is supported by a Wellcome Trust Training Fellowship in Clinical Tropical Medicine (grant no. GR067910MA).

We thank Denis Mitchison for his foundational work in this area and for his comments on earlier drafts of this paper and Kasia Stepniewska for useful discussions regarding the statistical methods.


   FOOTNOTES
 
* Corresponding author. Mailing address: Wellcome Trust-Mahidol University-Oxford Tropical Medicine Research Unit, Faculty of Tropical Medicine, Mahidol University, 420/6 Rajavithi Road, Bangkok 10400, Thailand. Phone: 66 23549172. Fax: 66 23549169. E-mail: gerrydavies{at}doctors.org.uk. Back


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Antimicrobial Agents and Chemotherapy, September 2006, p. 3154-3156, Vol. 50, No. 9
0066-4804/06/$08.00+0     doi:10.1128/AAC.00774-05
Copyright © 2006, American Society for Microbiology. All Rights Reserved.




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