Assessing the Combined Antibacterial Effect of Isoniazid and Rifampin on Four Mycobacterium tuberculosis Strains Using In Vitro Experiments and Response-Surface Modeling

ABSTRACT

While isoniazid and rifampin have been the cornerstone of tuberculosis therapy caused by drug-susceptible Mycobacterium tuberculosis for more than 40 years, their combined action has never been thoroughly assessed by modern quantitative pharmacology approaches. The aims of this work were to perform in vitro experiments and mathematical modeling of the antibacterial effect of isoniazid and rifampin alone and in combination against various strains of Mycobacterium tuberculosis. After MIC determination of H37Rv and three strains belonging to the Beijing, Euro-American, and Indo-Oceanic lineages, the antibacterial effects of isoniazid and rifampin alone and in combination were studied in static time-kill experiments. A sigmoidal maximum effect model (Hill equation) and a response-surface model were used to describe the effect of the drugs alone and in combination, respectively. The killing effect of isoniazid and rifampin alone were well described by the Hill equation. Rifampin displayed a more concentration-dependent effect than isoniazid around the MIC. The pharmacodynamics parameters of each drug (maximal effect, median effect concentration, and coefficient of sigmoidicity) were quite similar between the four strains. The response-surface model from Minto et al. fit data of combined effect very well with low bias and imprecision (C. F. Minto, T. W. Schnider, T. G. Short, K. M. Gregg, A. Gentilini, Anesthesiology 92:1603–1616, 2000, https://doi.org/10.1097/00000542-200006000-00017). Response-surface modeling showed that the combined action of isoniazid and rifampin was synergistic for the H37Rv, Beijing, and Euro-American strains but only additive for the Indo-Oceanic strain. This study can serve as a motivating example for preclinical evaluation of combined action of antituberculous drugs.

INTRODUCTION

Tuberculosis (TB) caused by Mycobacterium tuberculosis complex remains one of the most prevalent and deadly infectious diseases worldwide, with an estimated 10.4 million cases and 1.4 million deaths, and an additional 0.4 million deaths resulting from TB disease among people living with HIV, in 2015 (1). The standard short-course treatment of TB for a patient without any history of TB is a 4-drug combination (isoniazid, rifampin, pyrazinamide, and ethambutol) for 2 months in intensive phase, followed by a 2-drug regimen (isoniazid and rifampin) during the 4-month continuous phase. The dosage design of current anti-TB drug regimens has been based on clinical data, as well as cost considerations, from the 1960s to 1970s (2, 3). Since then, many studies have called for optimized dosing regimens of TB drugs, especially higher doses of rifampin (4–7). However, critical determinants of treatment success that would allow updating TB standard of care, such as antimycobacterial effect of drug combinations or determinants of drug resistance, have been poorly documented. In addition, virtually all preclinical studies have focused on the antibacterial effect of single agents. In recent years, WHO officials have emphasized the need for novel approaches in TB research to accelerate the progress from fundamental and translational research into clinical applications, especially to test optimal drug combinations (8). Quantitative pharmacology and pharmacokinetic/pharmacodynamic modeling may be a powerful approach to optimize the design of new anti-TB drug combinations (9).

Moreover, there is both experimental and clinical evidence that M. tuberculosis strains from different lineages vary in their capacity to cause disease and acquire drug resistance. Specifically, M. tuberculosis strains belonging to the East Asian lineage (which includes the Beijing family) have been associated with increased risk of drug resistance compared with strains from the Euro-American lineage (10). A recent study confirmed that M. tuberculosis strains belonging to the East Asian lineage display a higher rate of acquisition of drug resistance and an overall higher mutation rate than that of strains from the Euro-American lineage, even when exposed to the anti-TB agent rifampin (11). The killing effect of drugs may also depend on the bacterial lineage. There is a gap of knowledge about the influence of the phylogeographic characteristics of M. tuberculosis on the response to multidrug therapy. Altogether, these data raise questions of whether drug combinations and dosages currently used are adequate in all clinical situations regardless of the phylogeographic lineages at diagnosis.

The objectives of the present study were to perform in vitro experiments and mathematical modeling of the antibacterial effect of isoniazid and rifampin alone and in combination against various strains of M. tuberculosis.

RESULTS

Isoniazid and rifampin MIC values.

The isoniazid (INH) MIC values were 0.03 mg/liter for the Beijing and Euro-American strains and 0.06 mg/liter for the Indo-Oceanic and reference H37Rv strains. For rifampin (RIF), the MIC values were 0.25 mg/liter for the Euro-American strain and 0.12 mg/liter for the other strains.

Single-drug pharmacodynamics.

Figures 1 and 2 show the raw time-kill data for INH and RIF, respectively, for each strain. No significant regrowth was observed for either drug.

FIG 1.

Effect of isoniazid (INH) on the total colony counts of M. tuberculosis H37Rv strain, Beijing strain, Euro-American strain, and Indo-Oceanic strain. The symbols represent the means of duplicate values; the bars represent standard deviations.

FIG 2.

Effect of rifampin (RIF) on the total colony counts of M. tuberculosis H37Rv strain, Beijing strain, Euro-American strain, and Indo-Oceanic strain. The symbols represent the means of duplicate values; the bars represent standard deviations.

The fit of the Hill pharmacodynamic equation to time-kill data is shown in Fig. 3 and 4 for INH and RIF, respectively. While the maximal effect increased with incubation time, the shapes of the concentration-effect curves at the four time points were similar for both drugs. The model parameters for the effect measured after 7 days are shown in Table 1. The model fit the data very well, with R² values of >0.85 under all conditions. INH and RIF displayed quite different patterns of concentration-effect relationships. For INH, typical values of median effect concentration (EC₅₀) and the Hill coefficient of sigmoidicity (H) were <1 and >1, respectively. Compared to those of INH, RIF exhibited greater maximum effect (E_max) values, lower H values (<1), and lower potency, with an EC₅₀ of >1, except for the H37Rv strain. The effect of INH was remarkably similar between the four strains, as indicated by overlaps between 95% confidence intervals of model parameter values. For RIF, the H37Rv strain was characterized by a significantly lower EC₅₀, which suggests a greater potency. The Indo-Oceanic strain showed the lowest E_max for both drugs.

FIG 3.

Fit of the Hill pharmacodynamic model to isoniazid time-kill data. The y axis represents the difference in log₁₀ CFU/ml relative to the control (concentration equal to zero). The solid lines are the best-fit lines. Each symbol represents the mean value, and the bar represents the range of values from a duplicate experiment. No bar means that duplicates had the same or very similar values.

FIG 4.

Fit of the Hill pharmacodynamic model to rifampin time-kill data. The y axis represents the difference in log₁₀ CFU/ml relative to the control (concentration equal to zero). The solid lines are the best-fit lines. Each symbol represents the mean value, and the bar represents the range of values from a duplicate experiment. No bar means that duplicates had the same or very similar values.

TABLE 1.

Model parameters and fit for single-drug effect measured after 7 days

Drug and parameter	Value(s) for M. tuberculosis straina:
	H37Rv	Beijing	Euro-American	Indo-Oceanic
INH
Emax (decline in log10 CFU/ml)	3.9 (3.8–4.1)	4.5 (4.1–4.9)	4.5 (4.3–4.8)	3.8 (3.4–4.1)
EC50 (× MIC)	0.33 (0.28–0.40)	0.56 (0.41–0.76)	0.50 (0.40–0.64)	0.27 (0.19–0.37)
H	2.2 (1.5–2.9)	1.6 (0.9–2.2)	1.6 (1.1–2.2)	2.1 (0.80–3.4)
Model fit (R2)	0.96	0.93	0.96	0.86
RIF
Emax (decline in log10 CFU/ml)	6.4 (6.1–6.7)	7.5 (6.9–8.1)	7.3 (6.8–7.8)	5.7 (5.2–6.2)
EC50 (× MIC)	0.67 (0.53–0.86)	2.40 (1.6–3.6)	1.40 (1.1–1.9)	2.40 (1.7–3.3)
H	0.68 (0.58–0.78)	0.5 (0.4–0.6)	0.92 (0.72–1.1)	0.9 (0.72–1.1)
Model fit (R2)	0.98	0.99	0.97	0.97

^aParameter values are given as point estimates (95% confidence intervals).

Response-surface modeling of INH and RIF combined effect.

A total of 328 measures of the effect of INH and RIF alone and in combination were available for the analysis. Experimental data and response-surface models of the combined action of INH and RIF for each of the four M. tuberculosis strains are shown in Fig. 5. Figure 6 illustrates the goodness of fit of the response-surface model. Table 2 provides the parameter estimates and fit criteria of the response-surface model for each M. tuberculosis strain. The Minto model described the combined antibacterial effect very well, with R² values of ≥0.88 and low bias and imprecision values.

FIG 5.

Experimental data and response-surface models of the combined antibacterial effect of isoniazid and rifampin against the four Mycobacterium tuberculosis strains. (Upper left) H37Rv strain; (upper right) Beijing strain; (lower left) Euro-American strain; (lower right) Indo-Oceanic strain. The surfaces represent the model-based predicted effect. The color bars indicate the magnitude of the predicted effect. The sticks represent the antibacterial effect measured after 7 days of exposure (n = 328). For ease of graphical display, only observations above the predicted surfaces are visible.

FIG 6.

Observed combined effect of isoniazid and rifampin versus model-based predictions for four Mycobacterium tuberculosis strains. The solid line is the regression line.

TABLE 2.

Parameter values and goodness of fit of the response-surface model describing the combined action of isoniazid and rifampin

Parametera	Value for M. tuberculosis strainb:
	H37Rv	Beijing	Euro-American	Indo-Oceanic
Emax,INH (decline in log10 CFU/ml)	4.93 (4.70, 5.16)	5.11 (4.90, 5.33)	5.08 (4.90, 5.25)	5.31 (5.04, 5.58)
Emax,RIF (decline in log10 CFU/ml)	7.06 (6.68, 7.44)	6.33 (5.99, 6.67)	6.85 (6.56, 7.14)	5.67 (5.34, 6.00)
βEmax (no unit)	−2.99 (−4.76, −1.22)	−0.86 (−2.26, 0.54)	−1.56 (−3.14, 0.03)	−12.74 (−16.43, −9.05)
HINH (no unit)	3.55 (2.70, 4.39	3.54 (2.64, 4.43)	2.55 (2.16, 2.94)	1.36 (1.15, 1.58)
HRIF (no unit)	0.80 (0.66, 0.94)	0.79 (0.65, 0.93)	0.86 (0.75, 0.96)	0.82 (0.68, 0.95)
βH (no unit)	3.33 (1.65, 5.01)	1.03 (−1.04, 3.09)	2.72 (1.77, 3.67)	1.55 (0.80, 2.31)
EC50,INH (× MIC)	0.40 (0.36, 0.45)	0.53 (0.48–0.58)	0.48 (0.43, 0.52)	0.41 (0.34, 0.48)
EC50,RIF (× MIC)	0.30 (0.23, 0.37)	0.39 (0.30–0.47)	0.50 (0.42, 0.59)	0.64 (0.49, 0.79)
Interaction parameter βU50 (no unit)	1.14 (0.52, 1.76)	1.43 (0.94–1.91)	1.49 (0.96, 2.01)	−1.84 (−4.14, 0.47)
Goodness of fit
Regression equation (obs vs pred)	y = 1.003 + 0.016	y = 0.997x + 0.014	y = 1.008 − 0.040	y = 1.005 − 0.024
R2	0.88	0.90	0.93	0.90
Mean prediction error (SD)	0.0025 (0.42)	−0.0027 (0.63)	0.0073 (0.50)	0.0044 (0.62)
Median absolute prediction error (%)	12.1	12.2	6.8	10.4

^aModel parameters are described in the text (equations 2 and 5 to 8). obs., observed combined antibacterial effect; pred., model-based prediction of the combined antibacterial effect.

^bParameter values are given as point estimates (95% confidence intervals).

There were some differences in parameter estimates of single-drug effects (E_max, EC₅₀, and H) compared with the estimation obtained based on single-drug data only (Table 1). For INH, E_max and H values were greater (except for the H value for the Indo-Oceanic strain). For RIF, the main discrepancy was related to EC₅₀, with values estimated by the Minto model being significantly lower than the ones estimated with the Hill equation. All EC₅₀s estimated with the response-surface model were lower than 1 and remarkably similar to EC₅₀s of INH for all strains. Those differences can be attributed to the data used to estimate the parameters. For the Hill equation, parameter estimates were based on 24 experimental points (duplicates of 12 points) for each strain. In contrast, a much larger data set was used to estimate parameters of the response-surface model (n = 328). As a result, estimates from this model may be considered more reliable.

There was no major difference in parameter values of single-drug action between the four bacterial strains, except for the E_max of RIF, which was lower for the Indo-Oceanic than for the other strains. There were more important differences in parameters describing the combined drug action. For example, the β_Emax estimate for the Indo-Oceanic strain was significantly lower than estimates for the three other strains. The most interesting difference was for the interaction parameter β_U50. For the H37Rv, Beijing, and Euro-American strains, β_U50 point estimates were positive and the confidence interval did not include 0, which means that the combined effect was synergistic. In contrast, the point estimate was negative for the Indo-Oceanic strain, which suggests antagonism. However, as the confidence interval included 0, departure from additivity was not statistically significant.

DISCUSSION

Research on TB drugs has been characterized by the following paradox: while multidrug therapy is the standard of care for TB, virtually all preclinical studies have focused on the antibacterial effect of single agents (12). Since anti-TB drugs are always used in combination, it is most important to identify the most effective combinations of drugs based on pharmacodynamic endpoints. In order to accelerate progress in TB research and improve drug therapy, this should be done as early as possible, i.e., during preclinical stages (13).

In this work, we aimed at developing an experimental and modeling framework to assess the combined action of anti-TB drugs in vitro. We selected the INH-RIF combination to illustrate the approach, as this has been the reference two-drug combination in TB therapy for more than 40 years. Our study provided several important results.

First, the in vitro killing effect of INH and RIF alone and in combination was well described by models based on the Hill equation. This model has been successfully used to describe M. tuberculosis time-kill data under other experimental conditions (14–16). Parameter values of the Hill equation applied to single-drug effect, as well as the shape of the corresponding curves as shown in Fig. 3, 4, and 5, confirmed some characteristics of INH and RIF effects. The concentration-effect curve of INH was characterized by a relatively high slope (H coefficient), while that of RIF displayed a lower slope and a greater E_max. For INH, this means that the effect increases very rapidly with increasing drug concentrations and comes close to its maximum for drug concentration between 1 and 10 times the MIC for all strains. Our results are in agreement with results from Jayaram et al. (15), as well as with early bactericidal activity data showing that the antibacterial effect of INH alone appears to plateau for INH doses greater than 150 mg/day in adults (17, 18). It is noteworthy that EC₅₀s were lower than the MIC. We believe that this apparent discrepancy is due to the different experimental conditions of MIC determination and killing experiments. The incubation time was longer for MIC determination, so initial killing followed by regrowth may occur at the MIC. In addition, in killing experiments we consider the effect as the difference between the bacterial count measured with no drug (C = 0) and the count for a given drug concentration. Thus, even a concentration associated with a stable count from time zero is associated with a positive antibacterial effect compared with no drug. It is noteworthy that the in vitro experiments with INH published by Jayaram et al. showed a similar pattern, with EC₅₀ estimated at 0.4× MIC (15).

In contrast, the killing effect of RIF increased more slowly with increasing RIF concentration and was not maximal at 10 times the MIC for all strains. This is also in agreement with experimental and clinical data and confirms the potential value of increasing RIF exposure to optimize its antibacterial effect (14, 19). Actually, doses of rifampin of ≥30 mg/kg would be necessary to achieve a plasma unbound concentration of rifampin equal to the highest concentration evaluated in this study (256× MIC, i.e., 32 mg/liter) based on the data from Boeree et al. (19). However, it should be noted that concentrations much lower than 256× MIC would achieve a high antibacterial effect as well. Using the model parameters from the Beijing strain experiment (Table 2), we calculated the effect associated with various rifampin concentrations. Concentrations of rifampin (without isoniazid) of 8×, 16×, 64×, 128×, and 256× MIC are associated with effect values (in percentages of E_max) of 92%, 95%, 98%, 99%, and 99.4%, respectively.

The main novelty of our study was the application and result of a response-surface modeling approach. To our knowledge, this is the first study using such an approach for describing and quantifying the combined effect of INH and RIF against clinical strains of M. tuberculosis. A large set of data (n = 328 observations) was produced in vitro. Data analysis showed that the response-surface model proposed by Minto et al. (20) adequately described the combined effect of INH and RIF. Most importantly, the model provided a simple interaction parameter linked to the traditional isobologram approach to drug combination, which allows interpreting the results in terms of synergy/antagonism. Interestingly, while synergy was observed for H37Rv, Beijing, and Euro-American strains, the Indo-Oceanic strain displays only additive combined effect at most. Overall, our experimental and modeling approach has provided evidence of the synergistic potential of this combination that has been used for decades in patients. However, these results also suggest that the combined antibacterial effects of anti-TB drugs vary between strains. Further research is necessary to confirm this finding.

Pioneer works from Dickinson and colleagues on the in vitro bactericidal effect of combinations of two or three major anti-TB drugs indicated complex patterns of response (21). Interestingly, in this study, the combination of INH and RIF was found to be nonadditive (indifference). In mice, INH exhibited antagonism of RIF plus pyrazinamide activity (22, 23). Drusano et al. studied the in vitro effect of the combination of RIF plus moxifloxacin against M. tuberculosis (24). In log-phase cultures of susceptible organisms, the combination did not exhibit a killing effect faster than monotherapies with RIF or moxifloxacin alone. In nonreplicating bacteria, the combination was antagonistic for the bactericidal activity. In another study, this group used a response-surface model, the Greco model (25), to quantify the combined action of RIF and linezolid against M. tuberculosis H37Rv in the hollow-fiber infection model (9). The experimental data indicated that the combined action tended to be antagonistic, although the departure from additivity was not significant. Overall, those published data suggest that there is no simple and universal pattern of combined action of TB drugs and support the need for more quantitative studies in this area. We believe that preclinical response-surface modeling of combined effects of anti-TB drugs may be of great interest in the identification and design of new drug regimens for treating both drug-susceptible and -resistant TB. This approach can provide a rationale for identifying promising synergistic combinations to be tested in clinical studies and discard combinations showing antagonistic properties.

This study has several limitations. First, we performed static in vitro experiments, with constant drug concentration in culture, and we used the effect observed at a single time point (day 7) for response-surface modeling. It is uncertain how results from day 7 would hold true for other time points. Experiments with bacterial counts at several time points would be necessary to clarify this. It has been suggested that dynamic in vitro experiments, such as the hollow-fiber system, can provide more clinically relevant results, as they can reproduce the variation of drug concentration observed in patients and have been shown to be predictive of clinical events (26). We acknowledged that our results need to be confirmed under such experimental conditions.

In addition, the experiments and analysis considered only a single population of M. tuberculosis in the exponential phase of growth. We did not explore more complex phenomena, such as stationary growth phase, anaerobic growth conditions, regrowth, and drug resistance, that have been observed in previous works (16, 24). We considered this work a pilot study aiming to show the feasibility of quantitative preclinical evaluation of anti-TB drug combinations. We designed a simple experiment of short-term in vitro combined antibacterial effect as a proof of concept of the approach. In future works, it would be most interesting to apply this approach to dormant, persistent, or resistant subpopulations of M. tuberculosis. Finally, we chose the Minto model to describe the combined drug action and did not evaluate other candidate models. The Greco model probably has been the most used model to describe the combined action of anti-infective agents so far (25, 27). Although this model has been a benchmark in pharmacology, it has some limitations. It is not flexible in the description of the drug combined effect. The interaction parameter and the patterns of combined action are assumed to be invariant over the whole response surface. In contrast, the Minto model allows each drug ratio to have its own interaction behavior and set of parameters. While the Minto model described the data very well, a thorough comparative evaluation of several response-surface models would be most interesting to perform in further research. In addition, the model should be evaluated with other data sets, including a combination with known antagonism for further confirmation of its value.

Conclusions.

This is the first study quantifying and modeling the combined in vitro effect of INH and RIF against clinical strains of M. tuberculosis. The response-surface approach has established the synergy of the INH-RIF combination that has been the cornerstone of TB therapy for more than 4 decades. Those results may help to design optimal dosage regimens taking into account both bacterial and treatment-related factors. Furthermore, our experimental and modeling approaches may serve as a motivating example for future preclinical evaluation of new TB drug combination regimens.

MATERIALS AND METHODS

Antimicrobials.

Isoniazid (INH) and rifampin (RIF) were purchased from Sigma (Saint Louis, MO, USA). Stock solutions were prepared by dissolving the antibiotics in 100% dimethyl sulfoxide, and appropriate dilutions were made in Middlebrook 7H9 medium (Becton Dickinson, Sparks, MD) supplemented with 10% OADC (oleic acid, albumin, dextrose, catalase; Becton Dickinson, Sparks, MD) immediately before use.

Bacterial cultures.

Three clinical M. tuberculosis-susceptible strains, fully characterized by mycobacterial interspersed repetitive-unit typing and belonging to three major M. tuberculosis lineages (Beijing family, Indo-Oceanic, and Euro-American), were selected from the laboratory strain collection. A reference strain H37Rv (ATCC 27294) was also tested as a control. Bacteria were grown in Middlebrook 7H9 medium supplemented with 10% OADC. Cell densities were estimated by plating on Middlebrook 7H10 agar (Becton Dickinson) supplemented with 10% OADC and incubated for 3 to 4 weeks for viable count analysis.

MIC determination.

MICs of INH and RIF were determined for each strain by a standard microdilution method. The assay was performed in a 96-well microtiter plate. Serial 2-fold dilutions of INH and RIF were put in the wells, with the concentrations ranging from 16 to 0.0015 mg/liter. Each well was inoculated with a final inoculum of approximately 5 × 10⁵ CFU/ml. The plates were incubated at 37°C for 12 to 18 days, and the wells were assessed for visible turbidity. The lowest concentration at which there was no visible turbidity was defined as the MIC. Concentrations of INH and RIF were normalized to MIC values and are expressed as multiples of the MIC throughout.

Killing kinetics of single drugs.

For each strain, an inoculum of approximately 5 × 10⁵ CFU/ml was prepared in 7H9 Middlebrook medium. The inoculum was incubated without any antibiotic (growth control) and with INH or RIF at concentrations of 1/32×, 1/16×, 1/8×, 1/4×, 1/2×, 1×, 2×, 4×, 16×, 64×, and 256× MIC. Incubation was performed in 96-deep-well plates (1-ml working volume) at 37°C. Each well was homogenized by pipetting and vortexing, serially diluted, and plated on 7H10 agar. The numbers of CFU per milliliter were determined on alternate days after 3, 5, 7, and 10 days of drug exposure. Each experiment was performed twice.

Killing kinetics of INH-RIF combination.

For each strain, a bacterial inoculum was prepared as described above. Twelve concentrations of INH (0, 1/32, 1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8, 16, and 64 multiples of the MIC) were combined with 12 possible concentrations of RIF (same multiples of MIC as those for INH) in an incomplete checkerboard design. For each INH concentration, the combination experiments were conducted in quadruplicates, but for each replicate, the exact number of RIF concentrations may vary from 2 to 12. For example, for the highest concentration of INH (64 mg/liter), only two concentrations of RIF (0 and 2 mg/liter) were combined, because this situation is virtually a monotherapy with isoniazid and the expected effect is close to the maximum whatever the RIF concentration is. The experiments were done this way to get better coverage of the response surface and avoid uninformative data. The incubation was performed as described for single-drug experiments. The numbers of CFU per milliliter were determined after day 7 of drug exposure only. A total number of 328 measures were available for each strain.

Mathematical modeling of single-drug effect.

The relationship between the antibacterial effect of each drug alone and its concentration was described using a Hill equation (28), whose general form is

$$E=\frac{E_{\max }{}_{\cdot }{C}^H}{{\text{EC}}_{50}^H+C^H}=\frac{E_{\max }}{1+{\left(\frac{\mathrm{E}{\mathrm{C}}_{50}}{C}\right)}^H}$$ (1)

where E is the drug effect, C is the drug concentration, E_max is the maximal effect (a measure of efficacy), EC₅₀ is the median effect concentration (a measure of potency), and H is the Hill coefficient of sigmoidicity, a measure of the steepness of the concentration-effect curve.

This model was fit to log-transformed concentration data using GraphPad Prism for Windows, version 5.02 (GraphPad Software, La Jolla, CA, USA). In the software, the equation was rearranged as

$$E=\frac{E_{\max }}{1+{10}^{^}{\log }_{10}\left[{\left(\frac{{\text{EC}}_{50}}{C}\right)}^H\right]}=\frac{E_{\max }}{1+{10}^{^}(H\times ({\log }_{10}({\text{EC}}_{50})-\log C))}$$ (2)

The antibacterial effect (E) considered was the net difference between the bacterial count measured with no drug (C = 0) and the count for a given drug concentration at the same time point. Thus, this effect was positive, ranging from 0 when C = 0 to E_max, the maximal antibacterial effect. The parameters EC₅₀ and H are as defined above. For each strain, INH and RIF concentrations were normalized to the MIC, so C is expressed as the ratio of the actual concentration of drug in culture divided by the MIC.

For each strain and each time point (days 3, 5, 7, and 10), a nonlinear least-square fit of all data from duplicate experiments was performed. GraphPad Prism provided point estimates of model parameters along with confidence intervals and goodness-of-fit results, including R² values.

Response-surface modeling of INH and RIF combined effect.

We used the Minto model (20) to describe the combined antimicrobial effect of INH and RIF based on data measured after 7 days of therapy.

The Minto model is based on the Hill pharmacodynamic model. In the Minto model, the concentration of each drug has to be normalized to the drug potency, as quantified by the EC₅₀ of the single drug. In our example, these normalized concentrations, denoted X_INH and X_RIF, are

$$X_{\text{INH}}=\frac{C_{\text{INH}}}{{\text{EC}}_{50,\text{INH}}}\text{\hspace{0.17em}and\hspace{0.17em}}{X}_{\text{RIF}}=\frac{C_{\text{RIF}}}{{\text{EC}}_{50,\text{RIF}}}$$ (3)

where C_INH and C_RIF are INH and RIF concentrations in the drug combination, respectively, and EC_50,INH and EC_50,RIF are the median effect concentrations of INH and RIF, as defined in equations 1 and 2.

Another variable in the model is U, which quantifies the ratio of each drug in the combination:

$$U=\frac{X_{\text{RIF}}}{X_{\text{INH}}+X_{\text{RIF}}}$$ (4)

As a result, U may range from 0 (INH only) to 1 (RIF only).

Indeed, the Minto model is an extension of the Hill equation (equation 1) to any ratio U of two drugs:

$$E=\frac{E_{\max }\text{(}U\text{)}\cdot {\left(\frac{X_{\text{INH}}+X_{\text{RIF}}}{U_{50}(U)}\right)}^{H(U)}}{1+{\left(\frac{X_{\text{INH}}+X_{\text{RIF}}}{U_{50}(U)}\right)}^{H(U)}}$$ (5)

where E_max(U) is the maximal effect at ratio U, U₅₀(U) is the number of units of X_RIF associated with 50% of the maximal effect at ratio U, and H(U) is the coefficient of sigmoidicity at ratio U. Each of the three parameters of the Hill equation are functions of U, the drug ratio, which implies that each ratio of INH and RIF behaves as a specific drug and has its own sigmoidal concentration-effect relationship and its own set of parameters. The concentration-effect curve associated with each ratio defined the contour of a response surface of the drug combination. The simple Hill equation with one drug (equation 1) is just a specific case and defines the margins of the surface.

As suggested by Minto et al. (20), we used a polynomial function to describe E_max(U), U₅₀(U), and H(U):

$$E_{\max }(U)=E_{\max ,\text{INH}}+(E_{\max ,\text{RIF}}-E_{\max ,\text{INH}}-{\text{β}}_{E\max })\cdot U+{\text{β}}_{E\max }\cdot U^2$$ (6)

where E_max,INH and E_max,RIF are the maximal effects of INH and RIF alone, respectively, and βE_max is the coefficient of the two-order polynomial for E_max(U):

$$H(U)=H_{\text{INH}}+(H_{\text{RIF}}-H_{\text{INH}}-{\text{β}}_H)\cdot U+{\text{β}}_H\cdot U^2$$ (7)

where H_INH and H_RIF are the Hill coefficients of sigmoidicity for INH and RIF alone, respectively, and β_H is the coefficient of the two-order polynomial for H(U), and

$$U_{50}(U)=1-{\text{β}}_{U50\cdot \hspace{0.17em}}U+{\text{β}}_{U50\cdot \hspace{0.17em}}{U}^2$$ (8)

where β_U50 is the coefficient of the two-order polynomial for U₅₀(U). Two-order polynomials were sufficient to adequately describe the data. Four-order polynomials suggested by Minto et al. (20) were also tested but did not improve the fit (data not shown).

The last equation (equation 8) is of importance, as it permits us to interpret the response surface in terms of synergy/antagonism and represents the link between response surface and the traditional isobologram representation of combined drug action (20, 27). If β_U50 is not different from 0, U₅₀ is equal to 1 for all values of U and the interaction is additive. If β_U50 is significantly greater than 0, the curve shows an inward curvature. In this case, U₅₀ is lower than 1 for all values of U between 0 and 1. The normalized drug mixture, i.e., [(X_INH + X_RIF)/U₅₀(U)], is more than additive, which means synergy. Finally, if β_U50 is significantly lower than 0, the curve displays an outward curvature, with U₅₀ being greater than 1 for all values of U between 0 and 1. The normalized drug mixture [(X_INH + X_RIF)/U₅₀(U)] is less than additive, which means antagonism.

The Minto model (equations 3 to 8) was fit to the entire data set, including data from single-drug experiments, by using nonlinear regression (Levenberg-Marquardt algorithm implemented in the nlinfit and nlintool functions) within the Matlab software (version 2011b; MathWorks, Natick, MA, USA). Point estimates of model parameters were obtained along with their confidence intervals. Goodness of fit of the model was assessed by analysis of plots of observed versus predicted antibacterial effects as well as calculation of bias and imprecision. Prediction error was defined as predicted effect minus observed effect. Bias was defined as the mean prediction error. Imprecision was defined as the median percent absolute error of prediction. The percent absolute error of prediction was calculated as |prediction − observation|/observation.

Values and 95% confidence intervals of the β_U50 coefficient were examined to interpret the combined action in terms of synergy/antagonism. Synergy was confirmed when the lower bound of the 95% confidence interval was greater than 0. Antagonism was stated when the upper bound of the 95% confidence interval was lower than 0. When the 95% confidence interval included 0, the interaction was considered additive.