A Time-to-Event Pharmacodynamic Model Describing Treatment Response in Patients with Pulmonary Tuberculosis Using Days to Positivity in Automated Liquid Mycobacterial Culture

  Published Ahead of Print 26 November 2012. 10.1128/AAC.01876-12.

2013, 57(2):789. DOI:Antimicrob. Agents Chemother. McIlleron and Mats O. KarlssonVisser, Gary Maartens, Carl M. J. Kirkpatrick, Helen Emmanuel Chigutsa, Kashyap Patel, Paolo Denti, Marianne Mycobacterial CulturePositivity in Automated Liquid

towith Pulmonary Tuberculosis Using Days PatientsDescribing Treatment Response in

A Time-to-Event Pharmacodynamic Model

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A Time-to-Event Pharmacodynamic Model Describing TreatmentResponse in Patients with Pulmonary Tuberculosis Using Days toPositivity in Automated Liquid Mycobacterial Culture

Emmanuel Chigutsa,a Kashyap Patel,b Paolo Denti,a Marianne Visser,c Gary Maartens,a Carl M. J. Kirkpatrick,b Helen McIlleron,a

Mats O. Karlssond

Division of Clinical Pharmacology, Department of Medicine, University of Cape Town, Cape Town, South Africaa; Centre for Medicine Use and Safety, Faculty of Pharmacyand Pharmaceutical Sciences, Monash University, Melbourne, Australiab; School of Public Health, University of the Western Cape, Cape Town, South Africac; Departmentof Pharmaceutical Biosciences, Uppsala University, Uppsala, Swedend

Days to positivity in automated liquid mycobacterial culture have been shown to correlate with mycobacterial load and havebeen proposed as a useful biomarker for treatment responses in tuberculosis. However, there is currently no quantitativemethod or model to analyze the change in days to positivity with time on treatment. The objectives of this study were to describethe decline in numbers of mycobacteria in sputum collected once weekly for 8 weeks from patients on treatment for tuberculosisusing days to positivity in liquid culture. One hundred forty-four patients with smear-positive pulmonary tuberculosis were re-cruited from a tuberculosis clinic in Cape Town, South Africa. A nonlinear mixed-effects repeated-time-to-event modeling ap-proach was used to analyze the time-to-positivity data. A biexponential model described the decline in the estimated number ofbacteria in patients’ sputum samples, while a logistic model with a lag time described the growth of the bacteria in liquid culture.At baseline, the estimated number of rapidly killed bacteria is typically 41 times higher than that of those that are killed slowly.The time to kill half of the rapidly killed bacteria was about 1.8 days, while it was 39 days for slowly killed bacteria. Patients withlung cavitation had higher bacterial loads than patients without lung cavitation. The model successfully described the increase indays to positivity as treatment progressed, differentiating between bacteria that are killed rapidly and those that are killedslowly. Our model can be used to analyze similar data from studies testing new drug regimens.

Serial sputum CFU counts on solid media are an establishedmethod to investigate bactericidal activities of antitubercular

drugs (1–5). However, CFU counting is expensive, labor-inten-sive, and technically challenging, with problems of bacterialclumping. In addition, this procedure is difficult to standardizeacross sites in multicenter studies. Another problem with CFUcounting is that it does not take into account the metabolic activityof the different types of bacteria growing on solid media, i.e., ac-tively replicating and slowly replicating bacteria (persisters). Thisdistinction may be important when investigating antimycobacte-rial activities of drugs, since they act on different replicative statesof the mycobacteria (6). Liquid culture systems may better repre-sent the overall population of bacteria (7), since there are somemycobacterial populations that do not grow on solid media butgrow in liquid media (8). Mycobacterial Growth Indicator Tube(MGIT; Becton Dickinson, Sparks, MD) systems have improvedthe detection of Mycobacterium tuberculosis in clinical samplesand also shortened the time to obtain a positive result (9–11). TheMGIT detection system is based on a silicon rubber disk impreg-nated with ruthenium pentahydrate, a fluorescent indicatorwhose natural fluorescence is quenched in the presence of oxygen.Bacterial growth utilizes oxygen in liquid medium, and the indi-cator then fluoresces, giving a positive result (10). It is possiblethat the lower the number of replicating bacteria inoculated intothe MGIT, the longer it will take for the oxygen tension to fallbelow the threshold that would give a positive result. Indeed, daysto a positive MGIT result have been found to be correlated withCFU counts (12), although one study found only a weak correla-tion between the two (13). Days to culture positivity at baselinehave also been shown to predict sputum culture conversion at 8

weeks (14, 15), which is a widely accepted surrogate marker oftuberculosis cure (16).

To the best of our knowledge, there is no method or modelavailable that quantitatively describes the change in days to posi-tivity in automated liquid culture over time in patients receivingantituberculosis treatment. These kinds of data are becoming in-creasingly available as MGIT tests are becoming common in rou-tine clinical settings because of the advantages described above.The aim of this work is to describe the decline in numbers of viablemycobacteria in the sputum during the 8-week intensive phase ofstandard short-course chemotherapy in patients with pulmonarytuberculosis using the quantitative measure of days to positivity inliquid culture by developing a nonlinear mixed-effects repeated-time-to-event modeling approach. A model that maximizes theuse of the quantitative nature of MGIT days-to-positivity resultsto describe the mycobacterial response to treatment over time insputum samples from patients, combined with the growth kineticsof the mycobacteria, could be useful for quantitatively analyzingsimilar data from studies investigating new drug combinations ortesting the effects of various covariates on disease regression inpatients.

Received 11 September 2012 Returned for modification 16 October 2012Accepted 17 November 2012

Published ahead of print 26 November 2012

Address correspondence to Helen McIlleron, [email protected].

Copyright © 2013, American Society for Microbiology. All Rights Reserved.

doi:10.1128/AAC.01876-12

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MATERIALS AND METHODSThe study received ethical approval from the University of Cape Townresearch ethics committee and was carried out in accordance with theHelsinki Declaration of 1975 (revised in 2008).

Study participants. One hundred fifty-four patients participated in arandomized controlled trial of a micronutrient intervention (vitamin Aand zinc) in Cape Town, South Africa. The micronutrient interventionhad no effect on clinical or microbiological outcomes and was reportedpreviously (17). The study participants had sputum smear-positive pul-monary tuberculosis and were all prescribed the same fixed-dose combi-nation tablets (Rifafour; Aventis Pharma, Johannesburg, South Africa),each containing 150 mg rifampin, 75 mg isoniazid, 400 mg pyrazinamide,and 275 mg ethambutol. The administration of each dose was directlyobserved. Daily doses were administered 5 days a week according to bodyweight, in compliance with standard South African tuberculosis treat-ment guidelines (18): patients weighing 38 to 54 kg received 3 tablets,those weighing 55 to 70 kg received 4 tablets, and those weighing over 70kg received 5 tablets. Patients were monitored up to 8 weeks after treat-ment only, which is the intensive phase of treatment for tuberculosis.

The extent and size of lung cavities at baseline were assessed indepen-dently by two pulmonologists experienced in the use of the Chest Radio-graph Reading and Recording System (CRRS) (19). Disagreements onradiographic readings were resolved by consensus.

Bacteriology. A sputum specimen was collected in the early morningfrom each patient at baseline and then once weekly thereafter for 8 weeks.The specimens were processed for culture on liquid medium using theBactec MGIT 960 system (Becton Dickinson, Sparks, MD). The time toobtain a positive culture result was recorded in days. A lack of growth after42 days of culture was recorded as a negative result.

Data analysis. A nonlinear mixed-effects repeated-time-to-eventmodel was implemented with NONMEM 7.2.0 software (Icon Inc., Ve-rona, PA). An Intel Fortran compiler was used, and the runs were exe-cuted by using Perl-Speaks-NONMEM 3.4.10 (http://psn.sourceforge.net/). Population parameter estimates and variability were obtained byusing the Laplacian estimation method, which has been shown to havelow parameter bias and imprecision when a high proportion of individu-als in the data set have events (20). The objective function value (OFV),visual predictive checks (postprocessed and plotted using the statisticalprogramming language R, version 2.8.1 [21]), and parameter precisionwere used for model building and evaluation. When comparing two hier-archical models, the OFV (�2 � log likelihood) was used. A decrease inthe OFV of at least 3.84 points after the inclusion of one model parameterwas regarded as statistically significant. A bootstrap of 500 replicates wascarried out on the final model.

For the repeated-time-to-event model, a positive MGIT culture resultwas treated as an event, while a negative culture result was treated as aright-censored observation. Mono-, bi-, and triexponential models wereinvestigated in an attempt to describe the decline in bacillary loads inweekly sputum specimens from patients. The relative amount of bacteriain the bacterial growth compartment in the model (corresponding to theMGIT inoculum) was initialized to the value estimated from these mod-els. We use the term relative amount because the amounts do not relate toactual numbers of bacteria but, upon integration according to the growthmodels described below, are an equivalent related to oxygen consump-tion, which is then directly related to the hazard of a positive test result. Inother words, after the subsequent mathematical calculations and integra-tion, the relative amount must translate to a reasonable hazard (or prob-ability of obtaining a positive test result). Therefore, the relative amountin the MGIT inoculum for a particular week is driven by the hazard forthat week. Therefore, had true amounts been measured (which would behigher by several orders of magnitude), they would need to be divided bya very large factor to translate to a reasonable probability, which wouldnot always be 1. An exponential growth model, the Gompertz model (22),and the logistic model (23, 24) were investigated to describe the subse-quent growth of the mycobacteria in the MGIT culture during the days of

incubation. The differential equations for these models implemented inthe ADVAN6 subroutine of NONMEM are as follows:

N�t � Nt � Kgrowth (exponential growth model) (1)

N�t � Nt � Kgrowth � log(Nmax ⁄ Nt) (Gompertz growth model) (2)

N�t � Nt � Kgrowth � �Nmax � Nt� (logistic growth model) (3)

where Nt is the amount of bacteria in the MGIT culture at time t after thestart of incubation, Nmax is the carrying capacity of the MGIT culture, andKgrowth is a constant relating to the growth rate of the bacteria. Thus, at thebeginning of the MGIT incubation, where t is zero, Nt was initialized asdescribed above. The bacteria would then grow until a positive MGITresult was recorded or a negative result was found after 42 days. Thebacterial growth models were investigated with and without a lag phase ofthe mycobacteria before growth commenced. The lag time in days (L) wasestimated as a parameter in the model which delayed the commencementof bacterial growth by a period of time (L days) after the system startsrunning at time zero, such that bacterial growth commenced at timezero � L days. An attempt was made to model different growth rates anddifferent lag times of actively replicating bacteria and the slowly replicat-ing dormant bacilli in the MGIT culture.

The hazard of a positive MGIT result was directly related to the cumu-lative amount of bacteria in the tube by using the following equation:

h(t)dt � Nt � Pr(t � T � (t � dt)�T � t) (4)

where Pr is the probability of having an event within the very short timeinterval dt, provided that one did not have an event before time t.

The probability of not having a positive sputum MGIT culture eventwas a function of the cumulative hazard of a positive event (integral of thehazard with respect to time) for each day of incubation in the MGITculture by using the following equation:

S(t) � e��tj

tj�1 h(t)dt (5)

The probability density function (pdf), i.e., the likelihood of having anevent at time t when an individual had a positive sputum culture event,was calculated as follows:

pdf � h(t) � S(t) (6)

Thus, when an individual had a positive test result, the pdf was estimated,whereas for negative results, the survival to 42 days was estimated.

Although the primary aim of the work was to develop a method uti-lizing “days-to-positivity” data to describe the treatment response in pa-tients with tuberculosis, we investigated the effect of covariates, including

FIG 1 Box-and-whisker plots of days to positivity for each week of treat-ment.

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the presence of lung cavitation and HIV infection, on disease progression.For the patients who did not have lung cavitation data available, a mixturemodel was used to assign an individual to one group or another. Theproportion of patients who had no cavitation was fixed to that calculatedfrom those who did have cavitation results.

RESULTS

Ten of the 154 patients recruited in the study were omitted fromthe analysis because their baseline MGIT culture was negative.Sixteen of the remaining 144 patients (11%) were HIV infected.Ninety-seven (67%) patients had lung cavities visible on chest Xray, 20 (14%) did not have lung cavitation, and 27 (19%) patientshad no available lung cavitation data. Baseline drug susceptibilitytesting results were available for 118 participants (82%). Of these,6 (5%) patients had isoniazid monoresistance, and 1 (0.1%) hadrifampin monoresistance, while resistance to both rifampin andisoniazid was found for 4 (3%) patients. Therefore, the vast ma-jority of patients had drug-susceptible tuberculosis. Only 53% ofpatients had a negative sputum culture result after 8 weeks oftreatment. As patients were monitored only up to 8 weeks aftertreatment initiation, there are no further data on the final patientoutcomes upon the completion of the additional 4-month contin-uation phase. Figure 1 shows the progression of days to positivitywith weeks on treatment. Table 1 shows the increasing percent-ages of patients with negative MGIT culture results as treatment

TABLE 1 Percentages of patients (n � 144) with negative sputumculture results and with missing culture results for each week oftreatment for tuberculosis

Wk of treatment % negative results % missing data

0 0 41 7 52 6 33 13 54 15 65 19 96 29 117 33 138 48 10

FIG 2 Visual predictive check from 100 simulations using the final model stratified into each week of treatment. The continuous line is a Kaplan-Meier plot forthe real data. A positive sputum result was regarded as an event and will result in a step on the staircase plot. The shaded area is a 90% prediction interval basedon the simulated data from the model.

Time-to-Event Model for Treatment of Tuberculosis

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progressed as well as the percentages of patients with no MGITculture results at each week, mainly due to contamination of theculture. We sought to develop a quantitative method to analyzethe days-to-positivity data and test its potential use by investigat-ing HIV infection and the presence of lung cavitation as covari-ates.

Model for decline in bacillary load in sputum specimensfrom patients. A biexponential model best described the declinein the number of bacteria in sputum samples over time on treat-ment. The model was of the form described in the following equa-tion:

Amount of bacteria � A .e�.t � B .e�.t (7)

where A is the baseline amount of bacteria that are killed rapidly,with the rate of kill being equal to �, and B is the baseline amountof bacteria that are killed more slowly, with their rate of kill beingequal to �.

A visual predictive check from 100 simulations is shown in Fig.2, which shows that the model described the data well. Table 2shows the final parameter estimates from the model. Derivingfrom alpha, the typical time to kill half the amount of bacteriakilled rapidly is 1.8 days. This means that after about 1 week, mostof these bacteria are no longer present in the patients’ sputum.Similarly, for beta, the time to kill half the amount of bacteriakilled slowly is 39 days, and it can take more than 5 months to killmost of these bacteria.

Figure 3 shows the decline in bacillary burden with time basedon the parameter estimates from the final model for a patient withno lung cavitation. At baseline, the model predicts that the morerapidly killed mycobacteria are typically 41 times more prevalentin the MGIT inoculum, as calculated from the ratio of the baselineamount of rapidly killed bacteria to that of those that are killedslowly. Patients with lung cavitation present were found to have1.73 times the bacterial load of patients with no lung cavitation.The ratio of rapidly killed bacteria to slowly killed bacteria at base-

line in such patients was 23. HIV infection was not found to haveany influence on the decline in the bacillary burden in our data set.

Model for growth of mycobacteria in MGIT culture. Theamount of bacteria from the patients’ sputum was then includedin the next part of the model as a baseline amount of bacteria forinoculation into the MGIT culture. A lag time of 4 days was esti-mated before bacterial growth in the MGIT culture commenced.The logistic growth model best described the growth of the myco-bacteria in the MGIT culture. We also found that the maximumpossible rate of growth decreased from a baseline rate (Kstart) of8.31 day�1 as time on treatment progressed, according to the fol-lowing equation:

Kgrowth(t) � Kstart � e��.t (8)

where Kgrowth(t) is the maximum possible rate of growth at week tand � is the sum of a fractional contribution from bacteria that arekilled rapidly and those that are killed slowly, as follows:

� � 0.863 � Fast Fraction � 0.283 � Slow Fraction (9)

From the equation shown above, the rate of decrease in growth forthe bacteria that were killed rapidly was 3 times higher than thatfor the bacteria that were killed slowly. Population variability in �was estimated by using a log-normal distribution. Figure 4 showsthe change of the maximum MGIT growth rate constant with timeon treatment.

Survival model. An exponential distribution of the time toobtain a positive MGIT culture result was used in the time-to-event model, with a Weibull distribution failing to improve themodel. The hazard was directly related to the cumulative amountof bacteria in the MGIT culture. Figure 5 shows the probability ofobtaining a positive MGIT culture result for up to 42 days ofculture for each week of treatment.

The final model parameter estimates and 95% confidence in-tervals (CIs) from a bootstrap of the final model are reported inTable 2. A schematic diagram of the final model summarizing thedata described above is presented in Fig. 6.

DISCUSSION

Ours is the first semimechanistic model describing the progres-sion of days to positivity of MGIT culture using a repeated-time-to-event modeling approach and taking into account (i) biexpo-nential decline in bacillary load in patients, (ii) differingreductions in the ability of the mycobacterial subpopulations to

TABLE 2 Population parameter estimates

Parameter Estimated value (95% CI)

Baseline amt of rapidly killed bacteria 0.0573 (0.00387, 0.119)Rate of kill for rapidly killed bacteria (wk�1) 2.68 (1.07, 4.32)Baseline amt of slowly killed bacteria 0.00141 (0.0003, 0.00248)Fractional increase in baseline amt of slowly

killed bacteria for presence of lungcavitation

0.728 (0.0782, 4.04)

Rate of kill for slowly killed bacteria (wk�1) 0.124 (0.0126, 0.173)Bacterial lag time before growth in MGIT

culture commences (days)4.00 (2.96, 4.00)

Baseline maximum growth rate of bacteriain MGIT culture (day�1)

8.31 (4.94, 12.9)

Rate of decrease of maximum growth rate inMGIT culture of rapidly killed bacteria(wk�1)

0.863 (0.582, 1.83)

Rate of decrease of maximum growth rate inMGIT culture of slowly killed bacteria(wk�1)

0.283 (0.223, 0.370)

Maximum carrying capacity of MGIT 0.158 (0.133, 0.212)Population variability in rate of decrease in

maximum growth rate of bacteria inMGIT (%)

56.6 (34.1, 71.6)

FIG 3 Typical relative amount of Mycobacterium tuberculosis bacteria pro-cessed from patients’ weekly sputum samples and inoculated into the MGIT.

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grow in the MGIT tube during treatment, (iii) saturable growth inthe MGIT system, and (iv) the numerical nature of the MGIT data,i.e., the time-to-event-type data. Although the time to positivity inMGIT data was highly variable, a general upward trend with timeon treatment can be seen in Fig. 1, with the boxes plateauing lateon treatment because of the censoring of the data at 42 days. Thevisual predictive check using Kaplan-Meier plots shows that ourmodel can predict the observed data quite well.

To test the potential use of our model, we investigated theeffect of 2 covariates on some model parameters. Our finding thatpatients with lung cavitation have larger amounts of bacteria thatare killed slowly is in accord with previous studies that showedlower 2-month sputum conversion rates (15, 25) and a shortertime to detection (or days to positivity) (26, 27) for patients withlung cavitation. The small number of patients with HIV infectionmay explain our failure to identify HIV infection as a significantcovariate. However, other studies have also shown that HIV infec-tion does not appear to have an effect on sputum conversion rates(28–30).

Depending on the type of bacteria in culture, the model bestdescribing their growth can differ (24), with the Gompertz model

performing better than the logistic model in some cases (31), orvice versa (23). In our case, the logistic model described the databetter. It is probable that a positive MGIT culture result will havebeen recorded by the time the carrying capacity of the MGIT sys-tem is reached. Indeed, our model predicts that the carrying ca-pacity could be reached only with the baseline sputum sample(data not shown); none of the subsequent samples had enoughbacteria to reach the carrying capacity, even when cultured for upto 42 days. Nonetheless, the correct growth model must be used.The lag time that was estimated by the model is probably due tothe bacteria having to recover from harsh treatment with sodiumhydroxide and other antibiotics used during the decontaminationprocess for the sputum before the inoculum can start growing inthe MGIT culture system. Another possible reason for a lag timewould be postantibiotic effects of the drugs administered to thepatients during treatment. This presents a particularly importantissue in that different drug classes may have differing postantibi-otic effects, which could confound the association between time topositivity and the number of viable bacteria in the inoculum andnegatively impact the ability of time to positivity to serve as asurrogate for the bacterial load when comparing the efficacies ofdifferent drug regimens. Care should be taken to address this pos-sibility by testing the drug regimens as covariates on the estimatedlag times. A drug regimen with greater postantibiotic effects wouldtherefore have a longer lag time.

Although we report a biexponential model, a model with 3exponents had an objective function value (OFV) that was 11points lower than that of the biexponential model. However, dueto the small difference in the OFV upon the addition of 2 param-eters (intercept and slope) as well as wide confidence intervalsfrom a bootstrap, the biexponential model was chosen, since it wasmore stable. The marginally lower OFV for a triexponential modelfor bacillary decline in patients suggests the existence of at least athird bacterial population. Richer data than our once-weeklyMGIT results may be able to properly define the third population.The difference between the OFV for the biexponential model andthat for the monoexponential model was 29 points; hence, the

FIG 4 Maximum growth rate of mycobacteria in MGIT culture for the logisticgrowth model as weeks on treatment progress.

FIG 5 Probability of obtaining a positive MGIT culture result upon incubation for up to 42 days as weeks on treatment progress.

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biexponential model was significantly better and was deemed todescribe data adequately.

A strength of our model is that it can differentiate an increase indays to positivity due to killing of the bacteria in the patient froman increase due to changes in the growth kinetics of the bacteria inMGIT culture. We report a decrease in the proliferative ability ofthe bacteria in the MGIT culture as time on treatment progresses,suggesting a change in the fitness of the bacteria. A possible expla-nation for the decrease in proliferative ability with time is that astreatment progresses, only persister mycobacteria remain in thesputum, and these may take a longer time to grow in the MGITculture. Another possibility is drug pressure resulting in the selec-tion for or acquisition of some mutations at a fitness cost (32–35),although this does not occur indefinitely, since an evolution ofcompensatory mutations (36, 37) is likely to occur. One otherpossible explanation is the development of an adaptive mecha-nism of change in expression of different pathways by the myco-bacteria in response to the altered environment caused by drugpressure and other stresses. We report 57% population variabilityin the rate of change of the bacterial growth rate in the MGITculture. A previous report found genetically similar mycobacteriaobtained from different patients having different fitness (38),showing that host factors also play a role. The higher rate of de-clining fitness of the rapidly killed bacteria might be because oftheir higher rate of multiplication in the human host, such thatthey acquire mutations that impede fitness much faster than thebacteria that are already growing slowly, since mutations can oc-cur during DNA replication (39).

Although our semimechanistic model predicts the data welland is based on biologically plausible mechanisms, in vitro data tosupport our findings are lacking. Further work to determine the invitro trends in growth rates and changes in the fitness of differentsubpopulations of M. tuberculosis during treatment is needed. Todevelop an even more mechanistic model, experiments to identifya calibration curve to relate days to positivity from 0 to 42 withinitial bacterial amounts, followed by identifying the exact (possi-bly nonlinear) relationship between oxygen consumption and thehazard of a positive test result, would be ideal. It is also essential inthis case to know the metabolic activity of the bacteria upon in-troduction to the culture. Identification of the exact threshold inoxygen tension and other factors that trigger a positive resultwould also enable more mechanistic models to be developed. Wemade an attempt to separate the bacterial growth compartmentfrom the oxygen consumption compartment in the model. This

resulted in a worse fit than the model that we present. This, how-ever, means that what is in the bacterial growth compartment withthe logistic function is actually a combination of bacterial growthand oxygen consumption and that Kgrowth is a combination of therate of bacterial growth and oxygen consumption. We also tried toestimate between-subject variability in the amount of bacteria, butthis could not be supported by the data and created model insta-bility. Only one random effect could be estimated (as is typical fortime-to-event-type data), and this was on the change in viability ofbacteria with time on treatment.

In summary, we present a novel method for analyzing the re-sponse of tuberculosis patients to treatment using data comprisedof days to positivity in MGIT cultures of serially collected sputumsamples. This model can be used to investigate covariates such asvarious drug regimens, drug exposure, strains of mycobacteria,patient/disease characteristics, or any other covariates on any ofthe model parameters in a manner similar to what we have shownwith lung cavitation.

ACKNOWLEDGMENTS

This work was supported by grants from Clinical Infectious Diseases Re-search Initiative (CIDRI) Wellcome Trust Fund grant 412164; the Na-tional Research Foundation (NRF) South Africa (grants 2067444 andRCN 180353/S50); the Norwegian Programme for Development, Re-search and Higher Education (grant NUFUPRO-2007/10183); the Re-search Council of Norway (RCN) (grant 183694/S50); the SwedishResearch Council (grant 521-2011-3442); and the South African MedicalResearch Council.

We are indebted to Juergen Bulitta, Cornelia Landersdorfer, GeraintDavies, and Robert Wallis for invaluable input into the modeling process.

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