Sylvain Goutelle
email: sylvain.goutelle@chu-lyon.fr

Laurent Bourguignon

Roger W Jelliffe

John E Conte

Pascal Maire

Mathematical modeling of pulmonary tuberculosis therapy: insights from a prototype model with rifampin

There is a critical need for improved and shorter tuberculosis (TB) treatment. Current in vitro models of TB, while valuable, are poor predictors of the antibacterial effect of drugs in vivo. Mathematical models may be useful to overcome the limitations of traditional approaches in TB research. The objective of this study was to set up a prototype mathematical model of TB treatment by rifampin, based on pharmacokinetic, pharmacodynamic and disease submodels. The full mathematical model can simulate the time-course of tuberculous disease from the first day of infection to the last day of therapy. Therapeutic simulations were performed with the full model to study the antibacterial effect of various dosage regimens of rifampin in lungs. The model reproduced some qualitative and quantitative properties of the bactericidal activity of rifampin observed in clinical data. The kill curves simulated with the model showed a typical biphasic decline in the number of extracellular bacteria consistent with observations in TB patients. Simulations performed with more simple pharmacokinetic/pharmacodynamic models indicated a possible role of a protected intracellular bacterial compartment in such a biphasic decline.

This modelling effort strongly suggests that current dosage regimens of RIF may be further optimized. In addition, it suggests a new hypothesis for bacterial persistence during TB treatment.

Introduction

Tuberculosis (TB) remains one of the leading causes of death by infectious disease. In 2007, TB was responsible for approximately 1.75 million deaths, including 450 000 HIV co-infected people (World Health Organization, 2009). In addition, it is estimated that one third of the world population is latently infected by Mycobacterium tuberculosis.

Despite the clinical effectiveness of well-conducted short-course chemotherapy [START_REF] Mitchison | The diagnosis and therapy of tuberculosis during the past 100 years[END_REF], there are several issues associated with current TB treatment. The emergence of multidrug and extensive resistance is a major concern since it might lead to the multiplication of incurable tuberculosis cases [START_REF] Centers | Emergence of Mycobacterium tuberculosis with extensive resistance to second-line drugs[END_REF][START_REF] Gandhi | Extensively drug-resistant tuberculosis as a cause of death in patients co-infected with tuberculosis and HIV in a rural area of South Africa[END_REF].

Another major problem of current tuberculosis treatment is its duration, which is a minimum of 6 months. Shortening the duration of effective TB therapy should have important benefits, including better patients' compliance and lower rates of default, relapse, and drug resistance. Assuming such potential benefits, a simulation study by Salomon and colleagues showed that a shorter 2 month-treatment could greatly reduce TB mortality and incidence of new cases [START_REF] Salomon | Prospects for advancing tuberculosis control efforts through novel therapies[END_REF].

Traditional approaches in pre-clinical tuberculosis research are based on in vitro and animal models. Animal models are valuable but expensive and cannot fully emulate the human disease [START_REF] Gupta | Animal models of tuberculosis[END_REF]. In vitro models provide information on drug potency but they are poorly predictive of the duration and magnitude of drug effect in patients [START_REF] Burman | The value of in vitro drug activity and pharmacokinetics in predicting the effectiveness of antimycobacterial therapy : a critical review[END_REF][START_REF] Nuermberger | Pharmacokinetic and pharmacodynamic issues in the treatment of mycobacterial infections[END_REF].

Mathematical models may be helpful to represent and study current problems associated with TB treatment, and to suggest innovative approaches [START_REF] Young | Highlights > We present a mathematical model of pulmonary tuberculosis therapy by rifampin. >The model is based on pharmacokinetic, pharmacodynamic and disease submodels. >The model simulates the bacterial dynamics of TB from the first day of infection to the last day of therapy. > The model reproduces some properties of the bactericidal activity of rifampin observed in clinical studies. > The model suggests a new hypothesis for bacterial persistence during TB treatment. 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4[END_REF]. In this report, we present a prototype mathematical model which describes the time-course of both tuberculous infection and its treatment by rifampin in the human lung. The full model and simpler pharmacokinetic/pharmacodynamic models were used to simulate the antibacterial effect of various rifampin dosage regimens.

Model description

The full model was based on three submodels: a pharmacokinetic (PK) model, a pharmacodynamic model (PD), and a disease model (or pathophysiological model).

Pharmacokinetic model

A four-compartment, nine-parameter model was used as the PK model. In a previously published population PK study, this model adequately described plasma, epithelial lining fluid (ELF), and alveolar cell (AC) concentrations from 34 non-infected subjects [START_REF] Goutelle | Population modeling and Monte Carlo simulation study of the pharmacokinetics and antituberculosis pharmacodynamics of rifampin in lungs[END_REF]. The PK model had the following system of ordinary differential equations (ODE): dX A /dt = -K A .X A dX 1 /dt = K A .X A -K E .X 1 -K 12 .X 1 + K 21 .X 2 dX 2 /dt = K 12 .X 1 -K 21 .X 2 -K 23 .X 2 + K 32 .X 3 dX 3 /dt = K 23 .X 2 -K 32 .X 3

(1)

where X A , X 1 , X 2 , X 3 are the amounts of drug in the absorptive (oral depot) compartment, the central (plasma concentration) compartment, the pulmonary epithelial lining fluid (ELF) compartment, and the pulmonary alveolar cell (AC) compartment, respectively (in milligrams). K A (h -1 ) is the oral absorptive rate constant. K E (h -1 ) is the elimination rate constant from the central compartment, and K 12 , K 21 , K 23 , K 32 are the intercompartmental transfer rate constants (all in h -1 ).

In addition, three output equations are associated with the above drug amounts, as follows:

C 1 = X 1 /V C C ELF = X 2 /V ELF C CELL = X 3 /V CELL (2)
Where C 1 , C ELF and C CELL are rifampin concentrations in the central (plasma) compartment, the ELF compartment, and the AC compartment, respectively (in mg/L). The symbols V C , V ELF , and V CELL represent the apparent volumes of distribution of the central, ELF and AC compartments, respectively (all in liters).

Pharmacodynamic model

The PD model links rifampin concentration at the effect site with its antibacterial effect.

The effect of rifampin on sensitive bacteria was described by the following equation: max max max 50 50

(1 )(1 )

g k g g k k g k k g dN N C C K N K N dt N C C C C D D D D D D (3)
The bacterial dynamics is assumed to result from logistic bacterial growth and drug-mediated killing. The drug also inhibits the bacterial growth, so the antibacterial effect of the drug results from both killing and growth inhibition. In equation (3), N is the number of bacteria, K gmax is the maximum growth rate constant of M. tuberculosis (in h -1 ), K kmax is the maximum kill rate (h -1 ), N max is the maximum number of bacteria, C is the rifampin concentration at the effect site (in mg/L), Į g and Į k are the Hill coefficients of sigmoidicity for the effect on growth and killing, respectively (no units), and C 50g and C 50k are the median effect concentrations for the effect on growth and killing, respectively (in mg/L).

This equation was derived from the model used by Gumbo et al. to describe the effect of rifampin and other anti-TB drugs on both drug-sensitive and resistant bacteria in an in vitro hollow-fiber system [START_REF] Gumbo | Selection of a moxifloxacin dose that suppresses drug resistance in Mycobacterium tuberculosis, by use of an in vitro pharmacodynamic infection model and mathematical modeling[END_REF]Gumbo et al., 2007c). The effect of rifampin on resistant subpopulations of M. tuberculosis was not included in the present model.

Tuberculous disease model

The immune response model published by Kirschner and colleagues was used to simulate bacterial dynamics from the first day of TB infection [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF][START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF].

Briefly, the lung and lymph node model Only our modifications done to the Kirschner's model for the building of the full model will be described in the next pages. Further details on the disease model can be found in the original publications from this group [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF][START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF])

The final model

The final full model was built by connecting the PK/PD model of rifampin with the TB disease model from Kirschner and colleagues, resulting in a 21 ODE-system. Actually, only the two equations of the bacterial dynamics were altered in their lung and lymph node model, as shown below. The other 15 equations of this model remained unchanged from the original publication [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF]. The PD equation was incorporated into the dynamics of the extracellular bacteria (B E ) in lungs as follows: (1 )( 1)

/ ( ) / ( ) ( ) ( ) ( ) 2 g k g g k k E E E L F E L F g E E k E E E kE L F g E L F T I A E R E I T I m I E I R E m m I I E dB B C C K B K B dt B C C C C T M k M B k M B k N M T M c B B N k NM k M d B IDC B NM B c D D D D D D (4)
The dynamics of intracellular bacteria in lungs (B I ) was modified as shown below: (1

)(1 ) ( ) ( ) ( ) ( ) ( ) 2 / ( ) / g k g g k k m CELL CELL I I g I I k I I m m I I k C E L L g C E L L m I E I R m m I I E T I I T I C C dB B K B K B dt B NM EC C EC C B B N k NM k M B NM B c T M k N M T M c D D D D D D (5)
In the presence of rifampin, we assume that drug concentration in epithelial lining fluid (C ELF )

and alveolar cells (C CELL ) drive the antibacterial effect of rifampin on extracellular and intracellular M. tuberculosis, respectively. Those concentrations are provided by the PK model.

When no drug is present (C ELF and C CELL are equal to zero), the bacterial dynamics is driven only by the disease model. Both extracellular and intracellular bacterial proliferate (at maximal growth rate K gmax(E) and K gmax(I) , in h -1 ). We assume a logistic growth for B E , while the intracellular growth is limited by the number of infected macrophages (M I ) and the maximal bacterial load (N) of this type of cells (the product N*M I ). Extracellular bacteria are killed by activated (M A ) and resident (M R ) macrophages (at rate k 15 and k 18 (h -1 ), respectively). Extracellular TB bacilli are also captured by immature dendritic cells (IDC), at rate d 12 (h -1 ). Internalization of extracellular bacteria by resident macrophages makes extracellular bacilli become intracellular. It is assumed that this process is saturable, and that a macrophage can carry one-half of its maximal bacterial load (N), and so the maximal rate of internalization is k 2 *(N/2) h -1 . In return, intracellular bacilli become extracellular because of bursting and apoptosis infected macrophages. These are also considered as saturable processes. Bursting is limited by the carrying capacity of infected macrophages (the maximal rate of bursting is k 17 *N*M I , in h -1 ). Macrophage apoptosis is assumed to be driven by the entire T-cell lung population (T T is the sum of Th precursors, Th1, and Th2 cells in lungs, see [START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF]). It is also assumed that only a fraction of the maximal bacterial load of infected macrophages is released in the extracellular compartment during apoptosis (N 1 <N). Additional information about the disease model equations and parameters can be found in the original publications from Kirschner's group [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF][START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF].

Parameter values and simulation settings

All simulations with the final model were performed using Matlab software (version 6.5, The MathWorks, Natick, MA, USA). The 21 ODE-system was solved by use of the ode15s solver implemented in Matlab.

Simulations without any drug

First, simulations without any drug present were performed to reproduce different TB progression patterns. Tuberculosis latency was simulated using parameter values published by [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF] for all the parameters of the disease model, except for the maximal growth rate constant of extracellular bacteria, K gmax(E) , which was fixed at 0.01 h -1 instead of 0.005 h -1 . Initial conditions used for simulations with the disease model are shown in table 1.

Then, we modified the value of the two bacterial growth rate constants in order to simulate the time course of TB active disease. Doubling times reported for extracellular H37Rv M.

tuberculosis in mice lungs ranged from 17h to 56 h [START_REF] Manca | Mycobacterium tuberculosis CDC1551 induces a more vigorous host response in vivo and in vitro, but is not more virulent than other clinical isolates[END_REF][START_REF] North | Mycobacterial virulence. Virulent strains of Mycobacteria tuberculosis have faster in vivo doubling times and are better equipped to resist growth-inhibiting functions of macrophages in the presence and absence of specific immunity[END_REF]. For the same strain, in various intracellular conditions, doubling time ranged from about 24 to 80h, approximately [START_REF] Chanwong | Intracellular growth and drug susceptibility of Mycobacterium tuberculosis in macrophages[END_REF][START_REF] Jayaram | Pharmacokinetics-pharmacodynamics of rifampin in an aerosol infection model of tuberculosis[END_REF][START_REF] Paul | Comparable growth of virulent and avirulent Mycobacterium tuberculosis in human macrophages in vitro[END_REF][START_REF] Silver | Expression of virulence of Mycobacterium tuberculosis within human monocytes: virulence correlates with intracellular growth and induction of tumor necrosis factor alpha but not with evasion of lymphocytedependent monocyte effector functions[END_REF]. Based on those published data, the maximum growth rate constants for extracellular and intracellular bacteria were fixed at 0.03 h -1 (doubling time = 23.1 h), and 0.015 h -1 (doubling time = 46.2 h), respectively.

Simulation of rifampin therapy

All simulations of rifampin therapy were organized in two successive time periods. In the first period, the model was used to simulate the development of active TB disease, as described above (2.5.1.). In this period, there was no drug administration and so, no drug effect was simulated. Parameter values for the PD equations are shown in table 2. Since all parameters had fixed values, only one trajectory was simulated, as shown in the various relevant figures.

In the second period, rifampin therapy was arbitrarily introduced after 6 months, when a high bacterial load had been achieved in lungs. Various rifampin regimens, in terms of duration and dose, were simulated. In this period, PK variability was introduced in the modeling framework by using the individual PK parameter values (Bayesian posterior estimates) of the 34 subjects from an earlier PK study [START_REF] Goutelle | Population modeling and Monte Carlo simulation study of the pharmacokinetics and antituberculosis pharmacodynamics of rifampin in lungs[END_REF]. A summary of the individual PK parameter values used in the simulations is presented in table 3. As a consequence of the PK variability, 34 individual trajectories for PK and PD (B E and B I ) variables may be displayed in period 2.

Simulations with only the PK/PD model

Simulations with a more simple PK/PD model were also performed. The objective was to examine various hypotheses regarding the shape of the killing effect of rifampin more easily than with the full model. This model only featured the four PK and the two PD equations from the full model, but did not include the equations from the disease model. The PD equations describing the bacterial dynamics were modified as follows:

1 1 max( ) max( ) max 50 1 50 1 (1 )(1 ) g k g g k k E E g E E k E E E k g EI E IE I dB B C C K B K B dt B EC C EC C K B K B D D D D D D (6) 2 2 max( ) max( ) max 50 2 50 2 (1 )(1 ) g k g g k k I I g I I k I I I k g EI E IE I dB B C C K B K B dt B EC C EC C K B K B D D D D D D
In this simpler model, we assumed a logistic growth of intracellular bacteria (BI max = 10 7 bacteria / ml), and first order transfer of bacteria from the extracellular to the intracellular compartment, and vice-versa (at rate K EI and K IE , in h -1 ). As this model can only simulate the rifampin treatment period, we assumed initial conditions of high bacterial load in lungs (B E (0)

= 10 9 bacteria / ml and B I (0) = 10 7 bacteria / ml). The individual Bayesian posterior PK parameter values of the 34 subjects from the earlier PK study [START_REF] Goutelle | Population modeling and Monte Carlo simulation study of the pharmacokinetics and antituberculosis pharmacodynamics of rifampin in lungs[END_REF] were used as described above. Other parameter values were the same as described for the simulations with the full model (see 2.5.2. and table 2), unless otherwise specified below.

We compared the antibacterial effect predicted by equation ( 6) under four parameterizations (6a, 6b, 6c, and 6d, respectively). For each simulation, the same rifampin dosage regimen was simulated (1200 mg per day for 20 days). Those four simulations reflected different assumptions concerning the effect of rifampin on M. tuberculosis extracellular and intracellular populations:

-

k EI = k IE = 0 ; C 1 = C ELF ; C 2 = C CELL (6a)
No exchange between B E and B I , specific PK/PD parameters in each bacterial compartment

-k EI k IE 0 ; C 1 = C ELF ; C 2 = C CELL (6b)
Reciprocal transfer between B E and B I , specific PK/PD in each bacterial compartment

- k EI k IE 0 ; C 1 = C 2 = C ELF (6c)
Reciprocal transfer between B E and B I , same rifampin concentrations in the two compartments, specific PD parameters.

-

k EI k IE 0 ; C 1 = C ELF ; C 2 = C CELL ; K kmax(I) = K kmax(E) (6d)
Reciprocal transfer between B E and B I , specific PK in each bacterial compartment, same PD parameters.

Then, the influence of the transfer rate constants K EI and K IE on the shape of the antibacterial effect of rifampin was examined using model 6b, for a 20-day, 600 mg per day rifampin regimen.

Units

Rifampin concentrations in lungs were measured in milligram per liter (mg/L) [START_REF] Conte | Effect of sex and AIDS status on the plasma and intrapulmonary pharmacokinetics of rifampicin[END_REF]. For the other variables in the lung and lymph node compartments, we assumed that 1 cm 3 = 1 mL. All quantities are expressed per mL of volume.

Analysis of the results

The analysis focused mainly on the bacterial dynamics predicted by the full model and the PK/PD model. Results from simulations of rifampin therapy with the full model were compared with clinical data, the early bactericidal activity (EBA) of rifampin. The EBA is based on the log-count of viable bacilli in sputum samples during the early days of TB treatment with a single drug. It is usually measured over the first two or the first five days of therapy. However, measurements up to 14 days may be performed also; they have been called "extended EBA" (see [START_REF] Donald | The early bactericidal activity of anti-tuberculosis drugs: a literature review[END_REF] for further details about the EBA of anti-TB drugs).

An index similar to the EBA was calculated from simulation results as follows: log 10 B E (t 1 ) -

log 10 B E (t 2 )/[t 2 -t 1 ],
where B E (t 1 ) and B E (t 2 ) are the numbers of extracellular bacteria calculated just before the administration of a rifampin dose, at time t 1 and t 2 , respectively. The predicted antibacterial activity of rifampin was calculated between day 0 and day 2, day 0 and day 5, day 2 and day 5, and between day 2 and day 14, for various rifampin dosage regimens. The bactericidal activities were compared with published values of EBA calculated for the same time interval.

Results

Simulations with no drug

Latent tuberculosis

The dynamics of extracellular and intracellular bacteria during latent tuberculosis simulated by the full model are shown in figure 1. Intracellular bacteria constitute the predominant population during latent tuberculosis, while the multiplication of extracellular bacilli is contained by the immune response. Figure 2 represents the dynamics of the different populations of pulmonary macrophages and dendritic cells. Those profiles show good agreement with those from Kirschner and colleagues [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF]. The bacterial dynamics are somewhat slower, the number of extracellular bacteria reaching its maximum on day 300 approximately, instead of day 200 in their simulation of latency, and the late rebound of extracellular bacteria observed after day 1500 seems to be greater. Overall, the bacterial and cellular populations reach latency levels comparable to those presented in the original publication.

Active disease

The evolution of the bacterial populations during active tuberculosis is shown in figure 3.

Compared with bacterial profiles of latent tuberculosis (figure 1), a much higher bacterial load is achieved, and extracellular bacilli represent the predominant population. The dynamics of macrophages and dendritic cells are shown in figure 4A and 4B, respectively. Compared with latent TB, active disease is characterized by higher levels of infected macrophages (MI) and mature dendritic cells (MDC). Of note, those profiles of active tuberculosis do not represent a late reactivation of TB, but rather primary TB without an initial phase of latency.

Again, these profiles show good agreement with the original results from Kirschner's group.

The bacterial populations (B E and B I ) reach their maximum density after about 180 days, which is earlier than in published results with the lung and lymph node model (400 days) [START_REF] Marino | The human immune response to Mycobacterium tuberculosis in lung and lymph node[END_REF], but close to results from the original lung model (150 days) [START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF].

Therapeutic simulations

Effect of rifampin therapy

The evolution of the number of bacteria in lungs during active tuberculosis followed by a 2- In addition, the model was used to study the effect of the rifampin dose size upon the bacterial dynamics. Three rifampin doses were simulated: a 300 mg, a standard 600 mg, and a 1200 mg dose, all administered as a once daily regimen for 2 months. For this simulation, only the median value of the individual pharmacokinetic parameters was used (see table 3). The results are shown in figure 7. With this set of PK parameters, the 300 mg dose was associated with very little reduction in bacterial load. The effect of the 600 mg dose was greater, as expected, but the model predicted that, after an initial phase of decline, the intracellular and extracellular bacterial levels remained stable at a high level after 2 months of therapy. In contrast, the 1200 mg dose produced rapid and complete elimination of intracellular bacilli, and a continuous, large decrease in the number of extracellular bacteria over the 2 months of rifampin treatment.

Exploration of the antibacterial effect of rifampin on extracellular bacteria

The qualitative and quantitative properties of the antibacterial effect of rifampin on the extracellular population of M. tuberculosis over the first two weeks of therapy were studied with the full model, for the three rifampin daily doses that have been used in clinical studies: 300mg, 600 mg, and 1200 mg.

For the first two weeks of therapy, the individual kinetic profiles calculated with the 34 subject dataset are presented in figure 8. For the standard 600 mg dose, a biphasic shape was observed for some profiles corresponding to the highest bactericidal effects. An initial phase of fast killing was observed, followed by a second phase of slower kill. The second phase started somewhere between day 3 and day 10. This biphasic shape was much less apparent with the lower dose of 300 mg, except for the highest antibacterial effect profiles. For the 1200 mg dose, the biphasic killing effect was clearly observed in most profiles, including the median for which the killing slowed down on days 4-5.

The bactericidal activities of rifampin on extracellular bacteria simulated by the full model over the early days of therapy were compared with published data of EBA. Simulated data were calculated for the 34 subjects (PK data set) and for the three dosage regimens of rifampin. The results are presented in table 4. Overall results from the simulations indicated a decline of the initial bactericidal activity over the first 14 days of rifampin therapy. The bactericidal activities calculated between day 0 and day 2 were greater than the activities calculated between day 2 and day 14. This result is in accordance with data from EBA studies. For the 300 mg dose, bactericidal activities from the full model were similar to EBA data for all periods over the first two weeks. For the standard 600 mg dose, the activity calculated between day 0 and 2 (0.277 ± 0.229) was in the range of published data of EBA 0-2 , which has been the most extensively studied measure. Activities predicted for the other timeperiods were greater than the published results. For the 1200 mg dose, the predicted results were significantly greater than published EBA data. However, very few studies have evaluated such a dose in actual clinical practice.

Finally, the study of the antibacterial effect of rifampin on extracellular bacteria simulated with the full model provided three main results. First, the bactericidal effect showed biphasic kinetics. Second, this biphasic behavior was dose-dependent. The greater the dose, the greater the bend in the response. Those two results are in agreement with clinical data of rifampin effect, as illustrated by figure 9. Third, while the model provided realistic values for the antibacterial effect of low and standard rifampin doses, it seemed to overestimate the effect of large rifampin doses.

Simulations with the PK/PD model

The Of note, the kinetics of B E simulated with the four variants of equation 6 were identical during the early 5-6 day long phase.

Then, model 6b was used to assess the influence of the transfer rate constants between the extracellular and the intracellular bacterial populations on the early antibacterial effect. Figure 11 shows the effect of the K EI value on the dynamics of B E and B I over the first 20 days of rifampin therapy (600 mg / day). The biphasic decline of extracellular bacteria was more pronounced when K EI value was greater. For the intracellular bacterial population, an initial increase in the number of bacteria was observed; the higher K EI value, the greater it was. This early rise was followed by a slow decline of BI, with no specific shape.

The effect of K IE value on the bacterial dynamics is depicted in figure 12. As expected, increasing K IE values resulted in a faster decline of the number of intracellular bacteria. For the extracellular bacterial population, a biphasic decline was observed for intermediate values of K IE (0.005 and 0.0005 h -1 ), but it was not seen for the lowest (0.00005 h -1 ) and the highest (0.05 h -1 ) values in the 20-day therapy simulation. However, when the simulation was performed for a longer therapy, a slower killing phase was observed with K IE = 0.00005 h -1 after 20 days, but not with the highest value (data not shown).

In those two simulations, the typical biphasic kinetics of the antibacterial effect of rifampin on extracellular bacteria was observed for K EI /K IE ratio values ranging from 0.2 and 200. It was not observed for K EI /K IE = 0.02 and KIE = 0.05 h -1 , which reflect very fast transfer of bacteria from the intracellular to the extracellular compartment.

In the therapeutic simulations performed with variants of the PK/PD model (equation 6), we identified three conditions associated with the observation of a biphasic antibacterial effect of rifampin on M. tuberculosis:

-reciprocal transfer between intracellular and extracellular bacterial populations -slow transfer of bacteria from intracellular to the extracellular compartment -drug less effective in the intracellular than in the extracellular compartment.

Discussion

Tuberculosis infection is characterized by a dynamic equilibrium between the development of the pathogen and the host response. After primary infection by Mycobacterium tuberculosis, the human immune response is able to contain the multiplication of the bacteria in most patients, and the infection may remain clinically silent for decades.

Systems biology is a promising tool to study the complex interactions that exist between the host and the pathogen in persistent infections [START_REF] Kirschner | Tuberculosis: global approaches to a global disease[END_REF][START_REF] Young | Highlights > We present a mathematical model of pulmonary tuberculosis therapy by rifampin. >The model is based on pharmacokinetic, pharmacodynamic and disease submodels. >The model simulates the bacterial dynamics of TB from the first day of infection to the last day of therapy. > The model reproduces some properties of the bactericidal activity of rifampin observed in clinical studies. > The model suggests a new hypothesis for bacterial persistence during TB treatment. 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4. Figure 4[END_REF]. In the last 15 years, mathematical models have provided major insights in the knowledge of tuberculosis pathogenesis and human immune response to TB [START_REF] Fang | A systems biology framework for modeling metabolic enzyme inhibition of Mycobacterium tuberculosis[END_REF][START_REF] Marino | TNF and IL-10 are major factors in modulation of the phagocytic cell environment in lung and lymph node in tuberculosis: a next-generation two-compartmental model[END_REF][START_REF] Segovia-Juarez | Identifying control mechanisms of granuloma formation during M. tuberculosis infection using an agent-based model[END_REF][START_REF] Wigginton | A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis[END_REF]). In the meantime, progress has been made in the quantitative description of both pharmacokinetics and in vitro pharmacodynamics of antituberculosis drugs [START_REF] Gumbo | New susceptibility breakpoints for first-line antituberculosis drugs based on antimicrobial pharmacokinetic/pharmacodynamic science and population pharmacokinetic variability[END_REF]Gumbo et al., 2007a;Gumbo et al., 2007c;[START_REF] Jayaram | Pharmacokinetics-pharmacodynamics of rifampin in an aerosol infection model of tuberculosis[END_REF][START_REF] Jayaram | Isoniazid pharmacokinetics-pharmacodynamics in an aerosol infection model of tuberculosis[END_REF][START_REF] Peloquin | Population pharmacokinetic modeling of isoniazid, rifampin, and pyrazinamide[END_REF].

The objective of the present work was to build and study a prototype mathematical model of TB treatment. Because of the major role of the immune response in the bacterial dynamics and the global clearance of M. tuberculosis, our modeling approach was based on quantitative relationships from both PK/PD and systems biology of TB. This construction was possible because current PK/PD models and immune response models at the cell population level both have ODE-based structures.

The full model was able to simulate the bacterial dynamics from the first day of infection to the last day of the treatment by rifampin. Therapeutic simulations showed a considerable contribution of PK variability to the antibacterial effect of rifampin. For the same rifampin dose, the model predicted fast eradication of M. tuberculosis in some subjects, and almost no effect on the bacterial load in others. This is in agreement with clinical observations, as large variations of the EBA of rifampin have been reported [START_REF] Donald | The early bactericidal activity of anti-tuberculosis drugs: a literature review[END_REF].

Our simulations also showed that rifampin doses higher than the current standard 600 mg dose would result in a significantly greater antibacterial effect (see figure 7). Previous works have also suggested that the standard rifampin dose is probably suboptimal [START_REF] Diacon | Early bactericidal activity of high-dose rifampin in patients with pulmonary tuberculosis evidenced by positive sputum smears[END_REF][START_REF] Goutelle | Population modeling and Monte Carlo simulation study of the pharmacokinetics and antituberculosis pharmacodynamics of rifampin in lungs[END_REF]Gumbo et al., 2007c;[START_REF] Jayaram | Pharmacokinetics-pharmacodynamics of rifampin in an aerosol infection model of tuberculosis[END_REF][START_REF] Peloquin | What is the 'right' dose of rifampin?[END_REF]. This means that many patients treated with the standard 600 mg/day rifampin dose may be undertreated. As the standard TB chemotherapy based on a combination of four drugs is effective in most patients, one may assume that the effect of the other drugs compensates for rifampin underdosing. However, optimal dosing of rifampin may be beneficial for many TB patients, with more rapid sputum conversion and shorter TB therapy. Further research is necessary to confirm this hypothesis.

The model was used to analyse in detail the qualitative and quantitative properties of the antibacterial effect of rifampin on extracellular bacteria over the first days of therapy.

Interestingly, the model reproduced the biphasic decline in the number of bacteria that has been observed in clinical studies [START_REF] Davies | Early clinical development of anti-tuberculosis drugs: science, statistics and sterilizing activity[END_REF]). An initial phase of rapid killing in the first 2-5 days was seen, followed by a slower killing phase over approximately the next ten days.

This phenomenon appeared to be dose-dependant in our simulations. This is also in accordance with published results [START_REF] Mitchison | Modern methods for assessing the drugs used in the chemotherapy of mycobacterial disease[END_REF][START_REF] Mitchison | The diagnosis and therapy of tuberculosis during the past 100 years[END_REF]. From a quantitative point of view, the model also provided some realistic results. The bactericidal activities calculated for the first days of rifampin therapy were comparable to early bactericidal activities reported in the litterature for the 300 mg and the 600 mg dose. However, the model seemed to overestimate the effect of higher rifampin doses (1200 mg). This may be due to several limitations of this modelling approach. First, we assumed linear pharmacokinetics of rifampin over the dose range studied. As several studies which compared different rifampin doses reported nonlinear pharmacokinetics in plasma, this might be oversimplistic [START_REF] Diacon | Early bactericidal activity of high-dose rifampin in patients with pulmonary tuberculosis evidenced by positive sputum smears[END_REF][START_REF] Pargal | Non-linear pharmacokinetics of rifampicin in healthy Asian Indian volunteers[END_REF][START_REF] Ruslami | Pharmacokinetics and tolerability of a higher rifampin dose versus the standard dose in pulmonary tuberculosis patients[END_REF]. The pulmonary pharmacokinetic submodel was based on data from subjects who received only 600 mg per day of rifampin. A study on the pulmonary pharmacokinetics of high-dose rifampin would be necessary to clarify this point. Then, this overestimation of the effect of the 1200 mg dose may be due to the nature of the EBA data. The model simulates the number of bacteria in lungs, while EBA is based on the counts of viable bacteria in sputum. Those measurements are likely to underestimate the total bacterial load in the lungs. This may explain such difference between model predictions and clinical data. Alternatively, the larger effect predicted by the model may be questioned. In the Hill equation-based PK/PD model, for the same parameter values, a higher dose (1200 mg) results in a higher effect, closer to the maximum effect. It is possible that the maximum killing effect (K kmax(E) ) value used in the model, which is based on in vitro experiments [START_REF] Jayaram | Pharmacokinetics-pharmacodynamics of rifampin in an aerosol infection model of tuberculosis[END_REF], may overestimate the maximal effect that can be reached in the organism.

In addition, we studied the antibacterial effect of rifampin predicted by much simpler PK/PD models, without the equations describing the immune response. For the same rifampin dose, simulations with PK/PD models that did not include a reciprocal transfer between extracellular and intracellular bacteria showed an antibacterial effect on BE faster than with the full model, and with no biphasic shape. The characteristic biphasic decline of BE was observed with equations that included an exchange between BE and BI populations.

To date, the interpretation of the biphasic kinetic clearance of tuberculosis bacilli during TB treatment remains unclear. This has been the subject of an interesting debate in the specialized literature after the publication of a recent work by Gumbo and colleagues on isoniazid (Gumbo et al., 2007b;[START_REF] Mitchison | Isoniazid activity is terminated by bacterial persistence[END_REF][START_REF] Wallis | Persistence, not resistance, is the cause of loss of isoniazid effect[END_REF]. Results from in vitro experiments performed with isoniazid, and also previously with moxifloxacin and ciprofloxacin [START_REF] Gumbo | Pharmacodynamic evidence that ciprofloxacin failure against tuberculosis is not due to poor microbial kill but to rapid emergence of resistance[END_REF][START_REF] Gumbo | Selection of a moxifloxacin dose that suppresses drug resistance in Mycobacterium tuberculosis, by use of an in vitro pharmacodynamic infection model and mathematical modeling[END_REF], suggest that the cessation of the bactericidal activities of those drugs after a few days of therapy may be due to the emergence of a resistant subpopulation of M. tuberculosis. Other scientists have argued that those results, obtained in in vitro studies using a single drug, cannot explain the loss of activity of anti-TB drugs observed in patients treated with multiple drugs for which the bacterial sensitivity was controlled by laboratory tests [START_REF] Wallis | Persistence, not resistance, is the cause of loss of isoniazid effect[END_REF]. Those authors support the classical "special populations" hypothesis to explain clinical data. This hypothesis is depicted in figure 13. It is thought that different populations of bacilli might exist in TB lesions, including both multiplying bacteria and persistent/dormant bacteria characterized by a slower rate of metabolism and growth and occasional spurts of metabolism. The first phase observed in EBA data would reflect the killing of multiplicative bacilli, and the second phase the killing of persistent bacilli [START_REF] Mitchison | Basic mechanisms of chemotherapy[END_REF][START_REF] Mitchison | The diagnosis and therapy of tuberculosis during the past 100 years[END_REF]. The existence of subpopulations of bacilli has been the most popular hypothesis to explain both the dynamics of M. tuberculosis during latent TB and its persistence during anti-TB treatment. Experiments performed on Escherichia coli have provided evidence of the existence of bacterial persisters during antibiotic therapy [START_REF] Balaban | Bacterial persistence as a phenotypic switch[END_REF][START_REF] Connolly | Why is long-term therapy required to cure tuberculosis?[END_REF][START_REF] Keren | Specialized persister cells and the mechanism of multidrug tolerance in Escherichia coli[END_REF]. However, little is known about the real nature, location, and biology of possible persistent bacilli in tuberculosis infection [START_REF] Ehlers | Lazy, dynamic or minimally recrudescent? On the elusive nature and location of the mycobacterium responsible for latent tuberculosis[END_REF].

To summarize, two major hypotheses have been proposed so far to explain the loss of bactericidal activity of anti-TB drugs in the early days of treatment: genetic resistance and phenotypic persistence. Results from our modeling approach eventually suggest a new hypothesis. A biphasic decline of extracellular bacteria during rifampin treatment was observed in the simulations performed with the full model and the variants of the PK/PD model without any assumption of a resistant or metabolically persistent subpopulation. Only two populations were considered in our models: the extracellular and the intracellular bacteria. Based on our simulations with the PK/PD model variants, we hypothesize that this dual location of M. tuberculosis may be principal major cause of this typical shape of the antibacterial effect of anti-TB drugs, and that the intracellular population may constitute a bacterial reservoir. The extracellular bacterial dynamics simulated with the PK/PD models, as presented in figure 10, is illustrative. The initial phase, which is similar with the four models, may reflect "pure" extracellular killing. The bactericidal effect upon the extracellular bacteria does not slow when there is no exchange between the two populations of bacilli, or when rifampin has the same efficacy in both compartments. Otherwise, the intracellular bacterial load appears to be the limiting factor for further killing of extracellular bacteria in the second phase.

Our hypothesis is supported by data from other intracellular pathogens, such as viruses. The kinetic clearance of hepatitis B or C virus during antiviral therapy also shows a typical biphasic shape. It is thought that the initial rapid decline may reflect the clearance of plasma virions, while the slower second phase may represent the rate-limiting process of the clearance of infected cells [START_REF] Bonhoeffer | Virus dynamics and drug therapy[END_REF][START_REF] Lewin | Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed[END_REF][START_REF] Neumann | Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferonalpha therapy[END_REF][START_REF] Tsiang | Biphasic clearance kinetics of hepatitis B virus from patients during adefovir dipivoxil therapy[END_REF].

In addition, our simulations have shown that exchange between intracellular and extracellular bacilli, slow transfer of bacteria from intracellular to the extracellular compartment, and lower ability to kill intracellular bacilli may well be key elements in the existence and maintenance of an intracellular reservoir of bacteria.

Previous theoretical works have considered the role of a "protected compartment", or a "refuge", in the long-term persistence of M. tuberculosis and the bacterial dynamics during TB therapy [START_REF] Antia | Models of the within-host dynamics of persistent mycobacterial infections[END_REF][START_REF] Lipsitch | Population dynamics of tuberculosis treatment: mathematical models of the roles of non-compliance and bacterial heterogeneity in the evolution of drug resistance[END_REF]. However, those studies did not suggest a clear identification of such a protected compartment, and they did not consider its importance for the interpretation of clinical data.

Of note, the hypothesis of an intracellular bacterial reservoir does not reject the role of genetic resistance and metabolic persistence in the dynamics of M. tuberculosis. It is possible that the three phenomena co-exist in infected patients. While further research is necessary to confirm these findings, the importance of the intracellular bacterial compartment might have major implications for research concerning new anti-TB drugs. High cellular diffusion and potent intracellular killing effect could be important properties for new drug candidates.

This study has many limitations that should be considered in the interpretation of the results.

The construction of the full model was based on pre-existing submodels. The parameter values of the immune response model, as well as those of the pharmacodynamic model, are based on many heterogeneous experimental data. In addition, the PK model was based on data from non-infected subjects [START_REF] Conte | Effect of sex and AIDS status on the plasma and intrapulmonary pharmacokinetics of rifampicin[END_REF]. The intrapulmonary pharmacokinetics of rifampin may be different in TB infected patients, because of the specific pulmonary lesions.

As a result, extrapolation of our model results to the human disease remains hypothetical.

We only studied the role of pharmacokinetic variability in the antibacterial effect of rifampin.

It would be interesting to study the influence of other sources of variability (for example pharmacodynamic or physiological variability) on the results.

In addition, it is clear that the assumptions embodied in this prototype model represent a considerable simplification of the dynamics of real TB infection and therapy.

The present model only includes one antituberculosis drug, while TB therapy always consists of a combination of several drugs. The incorporation of more than one drug in the model will require additional modeling of the combined action of drugs. Except for a recent study on this subject [START_REF] Drusano | The Combination of Rifampin plus Moxifloxacin Is Synergistic for Suppression of Resistance but Antagonistic for Cell Kill of Mycobacterium tuberculosis as Determined in a Hollow-Fiber Infection Model[END_REF], little information is available to date. There is a need for work in population PK/PD modelling of combination drug therapy in TB patients.

We also assumed that rifampin exerts its action only by killing and inhibiting the growth of M. tuberculosis. This might be overly simplistic. The mathematical model of tuberculosis therapy from Magombedze and colleagues hypothesized a more complex action of rifampin, including a modification of the immune response by the drugs [START_REF] Magombedze | Mathematical modeling of chemotherapy of human TB infection[END_REF].

However, strong evidence of immunomodulation properties of rifampin is still lacking [START_REF] Jaffuel | Rifampicin is not an activator of the glucocorticoid receptor in A549 human alveolar cells[END_REF][START_REF] Tauber | Immunomodulatory properties of antibiotics[END_REF]. Further research is necessary to clarify this question.

We did not incorporate resistant subpopulations of bacilli in the model. Drug resistance is a major concern for TB control, and various studies have shown that drug resistance may play a key role in the bacterial dynamics during TB therapy (Gumbo et al., 2007b;[START_REF] Lipsitch | Population dynamics of tuberculosis treatment: mathematical models of the roles of non-compliance and bacterial heterogeneity in the evolution of drug resistance[END_REF]. Some other important characteristics of the pharmacokinetics and pharmacodynamics of rifampin were not included in this prototype model, such as rifampin auto-induction, or post antibiotic effect. As a long post-antibiotic effect of rifampin has been reported [START_REF] Chan | Postantibiotic effects of antituberculosis agents alone and in combination[END_REF]Gumbo et al., 2007c), it could explain the clinical efficacy of extended-interval rifampin dosage regimens. All these limitations should be considered for future development of this modeling approach.

To conclude, a mathematical model has been built which describes the time-course of TB disease and its treatment, from the first day of infection to the last day of therapy. [START_REF] Jayaram | Pharmacokinetics-pharmacodynamics of rifampin in an aerosol infection model of tuberculosis[END_REF] Table 3. Pharmacokinetic parameter values used for simulations of rifampin therapy [START_REF] Jindani | Bactericidal and sterilizing activities of antituberculosis drugs during the first 14 days[END_REF][START_REF] Jindani | The early bactericidal activity of drugs in patients with pulmonary tuberculosis[END_REF] 1200 mg or 20 mg/kg 0-2 0.659 (0.512) 0.693 0.383 (0.326) 0.44 (0.24) [START_REF] Jindani | Bactericidal and sterilizing activities of antituberculosis drugs during the first 14 days[END_REF][START_REF] Jindani | The early bactericidal activity of drugs in patients with pulmonary tuberculosis[END_REF] [START_REF] Jindani | Bactericidal and sterilizing activities of antituberculosis drugs during the first 14 days[END_REF][START_REF] Jindani | The early bactericidal activity of drugs in patients with pulmonary tuberculosis[END_REF] a Results from 34 simulated subjects; b Data from the Cape town center ; c Data from the Hong Kong center Bactericidal activity predicted by the model; B, EBA data from Jindani [START_REF] Jindani | Bactericidal and sterilizing activities of antituberculosis drugs during the first 14 days[END_REF][START_REF] Jindani | The early bactericidal activity of drugs in patients with pulmonary tuberculosis[END_REF]. [START_REF] Mitchison | Modern methods for assessing the drugs used in the chemotherapy of mycobacterial disease[END_REF] and drawn with data from Davies [START_REF] Davies | Use of nonlinear mixed-effects analysis for improved precision of early pharmacodynamic measures in tuberculosis treatment[END_REF]

K A (h -1 ) K E (h -1 ) K 12 (h -1 ) K 21 (h -1 ) K 23 (h -1 ) K 32 (h -1 ) V 1 (L) V 2 (L) V 3 (L) Median 2

  is a system of 17 ODE which describe the timecourse of the human immune response in lung and lymph node during TB infection. In the lung compartment, the variables included are: resident (M R ), activated (M A ), and infected (M I ) macrophages; interferon gamma (IFNȖ) and interleukins IL 12 , IL 10 , and IL 4 ; Tlymphocyte precursors (Th 0 ), Th 1 , and Th 2 lymphocytes; immature dentritic cells (IDC); and extracellular (B E ) and intracellular (B I ) M. tuberculosis bacilli. For the lymph node compartment, there are four variables: naïve T-cells (T),T-lymphocyte precursors (Th 0ln ), IL 12 (IL 12ln ), and mature dendritic cells (MDC).

  month treatment with rifampin is depicted in figures 5A and 5B. One can visualize the two periods of time in the modeling framework. From day 0 to day 180, there is only one trajectory, since only the disease model drives the dynamics, with all parameter having fixed values. The second period starts when rifampin is introduced on day 180, with a 600 mg oral dose administered every 24 hours, during 60 days. The use of each of the 34 subject's individual PK parameter values in the simulations results in variable drug exposure and drug effect. The individual curves for B E and B I observed after day 180 in figures 5A and 5B show the considerable effect of pharmacokinetic variability upon the bacterial dynamics during rifampin therapy. For the same rifampin dosage regimen, there was almost no antibacterial effect for some subjects, while a sharp decline in the number of B E and B I was observed for others. In most individual profiles of extracellular bacteria, same oscillations are observed.These result from the variation of individual rifampin exposure over the 24-hour dose interval. The simulated PK profiles of the 34 subjects in plasma and lungs over only the first three days of rifampin therapy are shown in figures 6A and 6B, respectively.

  bacterial dynamics simulated with the four variants of the PK/PD model (equation 6) are shown in figure10. The regimen simulated was 1200 mg / day during 20 days. For model 6b, 6c, and 6d, the values of the transfer constants K EI and K IE were fixed at 0.001 h -1 and 0.0005 h -1 , respectively. Profiles obtained with model 6a, which did not include a transfer between extracellular and intracellular bacterial populations, showed a one-phase, steady decline of both B E and B I , with a slower kill for the latter. Results from model 6b, which incorporated reciprocal transfer between B E and B I , were characterized by a biphasic decline of B E , the second phase of slower kill starting about day 5. In addition, the killing of B I was slower than observed with model 6a. Model 6c was similar to equation 6b, except that rifampin concentrations in the two bacterial compartments were set equal to the concentration in the ELF, and so concentrations in the intracellular compartment were lower. Compared with profiles from model 6b, those from equation 6c showed comparable shape of the antibacterial effect of rifampin on B E and B I . However, killing of B I was slower, while killing of B E was essentially similar during the first phase, and slower in the second phase. Model 6d was also a variant of equation 6b, with the same PD parameters in the extracellular and intracellular bacterial compartments. The profiles associated with model 6d were characterized by a monophasic decline of B E (a little bit slower than observed with model 6a), and a fast monophasic decrease of B I , basically parallel to B E .

Figure 1 .

 1 Figure 1. Dynamics of extracellular and intracellular bacteria during latent tuberculosis. BE,

Figure 7 .Figure 8 .

 78 Figure 7. Comparison of the effect of three rifampin dosage regimens on the bacterial

Figure 10 .

 10 Figure 10. Dynamics of extracellular and intracellular bacteria simulated with the PK/PD

Figure 11 .

 11 Figure 11. Influence of the transfer rate constant from the extracellular to the intracellular

Table 1 .

 1 The model reproduces some important characteristics of the antibacterial effect of rifampin observed in patients. Simulations have shown that the intracellular population of M. tuberculosis may play a major role in the early loss of bactericidal activity observed in clinical practice. Further experimental and clinical studies are required to evaluate these results. However, by suggesting new hypotheses and experiments, this work confirms the utility of mathematical modeling in TB research. Initial conditions used for simulations with the full model

	Tables
	Variable a	Initial condition	Unit
	Resident macrophages in lungs (M R )	5*10 5	Cells / ml
	Naive T-cells in lymph node (T)	4*10 3	Cells / m 3 of tissue
	Immature dendritic cells in lymph node (IDC)	5*10 4	Cells / m 3 of tissue
	Extracellular bacteria in lungs (B E )	500 (latency) 100 (active disease)	Bacteria / ml

a Initial conditions for the other variables were set at zero

Table 2 .

 2 Parameter values of the pharmacodynamic model used for simulations of rifampin

	therapy
	Parameter	Value	Unit	Definition	Reference
	K gmax(E)	0,03	h -1	Maximal growth rate of B E	(Manca et al., 1999; North and Izzo, 1993)
	B Emax	10 9	B E /ml	Maximal number of B E	(Allen and Mitchison, 1992)
					(Chanwong et al., 2007;
	K gmax(I)	0,015	h -1	Maximal growth rate of B I	Jayaram et al., 2003; Paul et al., 1996; Silver et al.,
					1998)
	m	2	None	Hill coefficient	(Marino and Kirschner, 2004)
	N	50	B I /M I	Maximal bacterial load of M I	(Marino and Kirschner, 2004)
	EC 50g	1,932	mg/L	Median effective concentration for RIF effect on bacterial growth	(Gumbo et al., 2007c)
	Į g	0,361	None	Hill coefficient for RIF effect on bacterial growth	(Gumbo et al., 2007c)
	k kmax(E)	0,576	h -1	Maximal kill rate of RIF on B E	(Jayaram et al., 2003)
	EC 50k	7,652	mg/L	Median effective concentration for RIF killing	(Gumbo et al., 2007c)
	Į k	1,388	None	Hill coefficient for RIF killing effect	(Gumbo et al., 2007c)
	k kmax(I)	0,0504	h -1	Maximal kill rate of RIF on B I

Table 4 .

 4 Comparison of extracellular bactericidal activities of three rifampin dosage regimens predicted by the full model with published data of early bactericidal activity

		.00	1.18	39.43	19.29	46.01	22.18	5.31	78.0	27.09
	Min	0.25	0.025	1.57	2.53	0.034	1.28	1.55	24.20	8.61
	Max	10.0	5.0	49.9	50.0	50.0	50.0	46.4	200	200

Acknowledgements

We thank Dr Gérard Carret, from Laboratoire de Bactériologie, Centre Hospitalier Lyon Sud, Hospices Civils de Lyon and Université Lyon 1, UMR CNRS 5558 & Faculté de Médecine Lyon Sud for his helpful suggestions during the preparation of the manuscript.

Conflicts of interest statement

The authors have no conflicts of interest.