Computational pharmacokinetics/pharmacodynamics of rifampin in a mouse tuberculosis infection model

Abstract

One critical approach to preclinical evaluation of anti-tuberculosis (anti-TB) drugs is the study of correlations between drug exposure and efficacy in animal TB infection models. While such pharmacokinetic/pharmacodynamic (PK/PD) studies are useful for the identification of optimal clinical dosing regimens, they are resource intensive and are not routinely performed. A mathematical model capable of simulating the PK/PD properties of drug therapy for experimental TB offers a way to mitigate some of the practical obstacles to determining the PK/PD index that best correlates with efficacy. Here, we present a preliminary physiologically based PK/PD model of rifampin therapy in a mouse TB infection model. The computational framework integrates whole-body rifampin PKs, cell population dynamics for the host immune response to Mycobacterium tuberculosis infection, drug-bacteria interactions, and a Bayesian method for parameter estimation. As an initial application, we calibrated the model to a set of available rifampin PK/PD data and simulated a separate dose fractionation experiment for bacterial killing kinetics in the lungs of TB-infected mice. The simulation results qualitatively agreed with the experimentally observed PK/PD correlations, including the identification of area under the concentration-time curve as best correlating with efficacy. This single-drug framework is aimed toward extension to multiple anti-TB drugs in order to facilitate development of optimal combination regimens.

Appendix

The PK/PD model used in the present work is described by Eqs. (3–23) and the corresponding compartmental structure shown in Fig. 5. The equations are those from Friedman et al. [24] and Lyons et al. [23] together with the modifications described in the "Materials and methods" section.

Fig. 5

PK/PD model structure. a Intracellular (In) and extracellular (Ex) bacterial populations and killing kinetics parameters. $B_{E}$: extracellular bacteria, $B_{A}$: bacteria residing in activated macrophages, $B_{I}$: bacteria residing in infected macrophages, $k_{max}$: maximum kill rate, $EC_{50}$: half-maximum effect drug concentration, and $\gamma$: Hill coefficient. b Lung compartment with free rifampin drug concentration ($f_{LU}C_{LU}$) together with cell and cytokine populations (adapted from Friedman et al. [24]). $M_{R}, M_{A}, \text {and}\; M_{I}$: resting, activated, and infected macrophages. $I_{10}, I_{12}, I_{2}, \text {and}\; I_{\gamma }$: IL-10, IL-12, IL-2, and IFN-$\gamma$. $T_{4}$ and $T_{8}$: CD4$^{+}$ and CD8$^{+}$ T cells. c Rifampin PBPK model compartments and blood flow (adapted from Lyons et al. [23])

Pharmacodynamics: TB model

The TB model and drug-bacteria interactions are described by Eqs. (3–13). The variable names, descriptions, and initial conditions used in the the present work are shown in Table 4. The parameter values that were not updated in the Bayesian calibration are shown in Table 5, while the updated parameters are the posterior mean values shown in Table 2.

$$\begin{aligned} \frac{dB_{I}}{dt}= \,&\alpha _{I}B_{I} \left( 1 - \frac{B_{I}^2}{B_{I}^{2} + (N M_{I})^{2}} \right) + k_{1} n_{3} M_{R} \frac{B_{E}}{B_{E} + c_{1}} - k_{2} N M_{I} \frac{B_{I}^{2}}{B_{I}^{2} + (N M_{I})^{2}} \nonumber \\&- n_{1} k_{3} B_{I} \frac{I_{\gamma }}{I_{\gamma } + c_{2}} + n_{2} k_{4} B_{A} \frac{I_{10}}{I_{10} + c_{3} I_{\gamma } + c_{4}} \nonumber \\&- B_{I} \frac{k_{max}^{In}(f_{LU} C_{LU})^{\gamma ^{In}}}{(EC_{50}^{In})^{\gamma ^{In}} + (f_{LU} C_{LU})^{\gamma ^{In}}} \end{aligned}$$

(3)

$$\begin{aligned} \frac{dB_{A}}{dt}= \,&\alpha _{A}B_{A} - n_{2} k_{4} B_{A} \frac{I_{10}}{I_{10} + c_{3} I_{\gamma } + c_{4}} + n_{1} k_{3} B_{I} \frac{I_{\gamma }}{I_{\gamma } + c_{2}} - n_{2} \mu _{MA}B_{A} \nonumber \\ {}&- B_{A} \frac{k_{max}^{In}(f_{LU} C_{LU})^{\gamma ^{In}}}{(EC_{50}^{In})^{\gamma ^{In}} + (f_{LU} C_{LU})^{\gamma ^{In}}}\end{aligned}$$

(4)

$$\begin{aligned} \frac{dB_{E}}{dt}=\, & \alpha _{E}B_{E} - k_{1} n_{3} M_{R} \frac{B_{E}}{B_{E} + c_{1}} + k_{2} N M_{I} \frac{B_{I}^{2}}{B_{I}^{2} + (N M_{I})^{2}} - k_{5} M_{A} B_{E} \nonumber \\ {}&+ n_{2} \mu _{MA} B_{A} - B_{E} \frac{k_{max}^{Ex}(f_{LU} C_{LU})^{\gamma ^{Ex}}}{(EC_{50}^{Ex})^{\gamma ^{Ex}} + (f_{LU} C_{LU})^{\gamma ^{Ex}}} \end{aligned}$$

(5)

$$\begin{aligned} \frac{dM_{I}}{dt}=\, & k_{1} M_{R} \frac{B_{E}}{B_{E} + c_{1}} - k_{2} M_{I} \frac{B_{I}^2}{B_{I}^{2} + (N M_{I})^{2}} - k_{3} M_{I} \frac{I_{\gamma }}{I_{\gamma } + c_{2}} \nonumber \\ {}&+ k_{4} M_{A} \frac{I_{10}}{I_{10} + c_{3} I_{\gamma } + c_{4}} - \mu _{MI} M_{I} \end{aligned}$$

(6)

$$\begin{aligned} \frac{dM_{A}}{dt}= \,& - k_{4} M_{A} \frac{I_{10}}{I_{10} + c_{3} I_{\gamma } + c_{4}} + k_{3} M_{I} \frac{I_{\gamma }}{I_{\gamma } + c_{2}} - \mu _{MA} M_{A}\nonumber \\&+ k_{6} M_{R} \frac{B_{E}}{B_{E} + c_{5}} \frac{I_{\gamma }}{I_{\gamma } + c_{6}} \end{aligned}$$

(7)

$$\begin{aligned} \frac{dI_{10}}{dt}= \,& k_{7} M_{I} \frac{c_{7}}{I_{10} + c_{7}} - \mu _{10}I_{10} \end{aligned}$$

(8)

$$\begin{aligned} \frac{dI_{12}}{dt}=\, & k_{8} M_{A} \frac{c_{8}}{I_{10} + c_{8}} + k_{9} M_{R} \frac{B_{E}}{B_{E} + c_{9}} - \mu _{12} I_{12} \end{aligned}$$

(9)

$$\begin{aligned} \frac{dI_{2}}{dt}= \,& k_{10} T_{4} - \left( k_{11}T_{4} + k_{12} T_{8} \right) \frac{I_{2}}{I_{2} + c_{10}} - \mu _{2}I_{2} \end{aligned}$$

(10)

$$\begin{aligned} \frac{dI_{\gamma }}{dt}=\, & \left( \lambda _{u}(t) T_{4} + \lambda _{y}(t) T_{8}\right) \frac{I_{12}}{I_{12} + c_{11}} - \mu _{\gamma } I_{\gamma } \end{aligned}$$

(11)

$$\begin{aligned} \frac{dT_{4}}{dt}=\, & {} \lambda _{z}(t) M_{A} I_{12} + k_{13} T_{4} \frac{I_{2}}{I_{2} + c_{10}} - \mu _{T4} T_{4} \end{aligned}$$

(12)

$$\begin{aligned} \frac{dT_{8}}{dt}=\, & {} \lambda _{x}(t) \left( M_{A} + M_{I} \right) I_{12} + k_{14}T_{8} \frac{I_{2}}{I_{2} + c_{10}} - \mu _{T8}T_{8} \end{aligned}$$

(13)

The functions $\lambda _{i}(t), \; i = \{x,y,z,u\}$ were implemented here as $\lambda _{i}(t) = \lambda _{i} \cdot \theta (t - t_{delay})$, where $t_{delay} = 14$ days, and $\theta (x) = 1 (0)$ for $x\ge 0(x<0)$ is a step function.

Table 4 PD model variables and initial conditions (day 7 post-infection) used in the present work. The variable names, descriptions, and values are from Friedman et al. [24]

Table 5 Baseline PD model parameters that were not updated in the Bayesian calibration procedure. The parameter names, descriptions, and values are from Friedman et al. [24]

Pharmacokinetics: PBPK model

The PBPK model is described by Eqs. (14–23) with initial conditions all set to zero. The subscripts, T, on the tissue volumes, $V_{T}$, drug concentrations, $C_{T}$, blood flow rates, $Q_{T}$, and tissue/blood partition coefficients, $P_{T}$, denote abbreviations for the model compartments as; V: venous blood, LU: lung, A: arterial blood, BR: brain, F: fat, SK: skin, K: kidney, S: spleen, G: gut, GL: gut lumen, L: liver, LA: hepatic artery, CR: carcass. Fractional tissue volumes, $V_{TC}$, and blood flow rates, $Q_{TC}$, were scaled to total values as $V_{T} = V_{TC}\cdot BW$ and $Q_{T} = Q_{CC}\cdot BW^{0.75}$. BW is body weight, and $Q_{CC}$ is an allometric coefficient for cardiac output. Fractional clearance was also scaled as $CL = CL_{C} \cdot BW^{0.75}$. Drug concentration in plasma was determined from concentration in venous blood as $C_{plasma} = C_{V}/BP$. The parameter values that were not updated in the Bayesian calibration are shown in Tables 6 and 7, while the updated parameters are given in Table 2.

$$\begin{aligned} V_{V}\frac{dC_{V}}{dt}=\, & {} \sum _{T} Q_{T} C_{T}/P_{T} - Q_{C}C_{V},\nonumber \\&\qquad T = \{BR,F,H,M,B,SK,K,L,CR\} \end{aligned}$$

(14)

$$\begin{aligned} V_{LU} \frac{dC_{LU}}{dt}= \,& {} Q_{C}\left( C_{V} - C_{LU}/P_{LU}\right) \end{aligned}$$

(15)

$$\begin{aligned} V_{A}\frac{dC_{A}}{dt}=\, & {} Q_{C} \left( C_{LU}/P_{LU} - C_{A} \right) \end{aligned}$$

(16)

$$\begin{aligned} V_{T}\frac{dC_{T}}{dt}=\, & {} Q_{T} \left( C_{V} - C_{T}/P_{T}\right) , \quad T = \{BR,\,F,\,H,\,M,\,B,\,SK,\,S,\,CR\} \end{aligned}$$

(17)

$$\begin{aligned} V_{K}\frac{dC_{K}}{dt}=\, & {} Q_{K}\left( C_{A} - C_{K}/P_{K} \right) - f_{R}\cdot CL \cdot C_{A} \end{aligned}$$

(18)

$$\begin{aligned} V_{L} \frac{dC_{L}}{dt}=\, & {} Q_{LA}C_{A} + Q_{S}C_{S}/P_{S} + C_{G}Q_{G}/P_{G} - Q_{L}C_{L}/P_{L} \end{aligned}$$

(19)

$$\begin{aligned}&- (1-f_{R})\cdot CL \cdot \left( Q_{LA}C_{A} + Q_{S}C_{S}/P_{S} + Q_{G}C_{G}/P_{G}\right) /Q_{L} \end{aligned}$$

(20)

$$\begin{aligned} V_{G}\frac{dC_{G}}{dt}=\,Q_{G}\left( C_{A} - C_{G}/P_{G}\right) + k_{a}A_{OD} + k_{r}A_{GL} \end{aligned}$$

(21)

$$\begin{aligned} \frac{dA_{GL}}{dt}=\,&(1 - f_{R})\cdot CL \cdot \left( Q_{LA}C_{A} + Q_{S}C_{S}/P_{S} + Q_{G}C_{G}/P_{G}\right) /Q_{L} \nonumber \\&- \left( k_{r} + k_{F}\right) A_{GL} \end{aligned}$$

(22)

$$\begin{aligned} \frac{dA_{OD}}{dt}=\, & {} - k_{a}\cdot A_{OD} + F_{a}\cdot D \cdot \sum _{n=0} \delta (t - t_{n}) \end{aligned}$$

(23)

The summation in Eq. (14) is over all tissues draining into the venous blood compartment; for liver, $Q_{L} = \left( Q_{LA}+Q_{S}+Q_{G} \right)$. The symbols $A_{OD}$ and $A_{GL}$ in Eqs. (21–23) are the amounts of drug input to the gut and in the gut lumen. The delta function in Eq. (23) describes pulsed oral bolus dosing (D = dose), with times of administration, $t_{n}, \; n = 0, \ldots , n_{max}-1$, with $n_{max}$ the maximum number of doses in the treatment interval.

Table 6 Baseline anatomical/physiological PBPK model parameters that were not updated in the Bayesian calibration procedure

Table 7 Baseline rifampin-dependent PBPK model parameters that were not updated in the Bayesian calibration procedure