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Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection

  • * Corresponding author: W.-C. LO

    * Corresponding author: W.-C. LO

W.-C. LO is supported by a CityU StUp Grant (No. 7200437)

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  • Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

    Mathematics Subject Classification: Primary: 92B05, 92C42; Secondary: 92C37.

    Citation:

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  • Figure 1.  Model diagram of bacteria, T cells and macrophages interaction.The Sharp arrow means activation; blocked arrow means inhibition. When bacteria encountering resting macrophages, some macrophages will be activated and become activated macrophages ($M_a$), while some will be infected and become infected macrophages ($M_i$). $M_a$ may be deactivated in response to regulatory T cells (Treg). Th1 cells will induce the activation of $M_a$. $M_a$ and $M_r$ will inhibit the growth of bacteria. Bursting of $M_i$ will increase the concentration of bacteria. $M_i$ may be eliminated by Th1 cells. Th1 cells activation will increase in response to $M_i$ and $M_a$, and decrease by Treg cells' inhibition. Treg cells will be activated by Th1 cells and $M_a$

    Figure 2.  Model diagrams of subsystems for analysis.A) Two-equation model. The interaction between bacteria infection and resting macrophages protection. B) Three-equation model. Bacteria infection, macrophages protection with some resting macrophages are infected and become infected macrophages, which will burst and release bacteria

    Figure 3.  Phase plane portraits for the analysis of the two-equation system (7)-(8).A) Case 1 B) Case 2 C) Case 3. Star symbols indicate the steady states; red dotted lines represent $dM_r/dt =0$; blue solid lines represent $dB/dt=0$

    Figure 4.  The dynamics of $B$, $M_r$ and $M_i$ with different values of $\beta_{M_r}$ in the three-equation model (13)-(15).The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=0$. We set $\gamma_i=0.01 \ \text{day}^{-1}$ and the others parameters from Table 3. As the rate of $M_r$ recruitment by $M_i$ and $M_a$ increasing from 0.1 to 10, the number of $M_i$ and $B$ decreases, while the number of $M_r$ is relatively stable

    Figure 5.  Model 1: System (1)-(6) without involving Th1 cell and Treg cell.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Resting macrophages are infected by bacteria and become infected macrophages, leading to a sharp increase in bacteria

    Figure 6.  Model 2: System (1)-(6) with Th1 cell but without Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Without regulation from Treg cells, the population of bacteria, macrophages, and Th1 cells oscillates continuously

    Figure 7.  The effect of Treg cell regulation in Model 3.A) The change in the average of the OI with respect to the increases of $\alpha_{T_r}$, the $T_r$ activation rate by $M_a$ and $\beta_{T_r}$, the $T_r$ activation rate by $T_1$; B) The change in the average of the maximum concentration of bacteria with respect to increases of $\alpha_{T_r}$ and $\beta_{T_r}$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. As $\alpha_{T_r}$ and $\beta_{T_r}$ increase, the oscillation of bacteria becomes smaller, while the maximum value of bacteria decreases first and then increases

    Figure 8.  Model 3: System (1)-(6) with Th1 cell control and low level of Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0.1 \ \text{day}^{-1}$ and the other parameters from Table 3. With Treg cell regulation, the number of bacteria remains stable after a few cycles of oscillation

    Figure 9.  Model 4: System (1)-(6) with Th1 cell control and strong level of Treg cell regulation.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$, $\beta_{M_r}=0.01 \ \text{day}^{-1}$ and the other parameters from Table 3. Strong Treg cell regulation causes sharp reductions of macrophages and Th1 cells, thus, the number of bacteria rapidly increases

    Figure 10.  The effect of enhaced macrophage recruitment in Model 4.A) The change of the average of the OI with respect to the increase of $\beta_{M_r}$; B) The change of the average of the maximum concentration of bacteria with respect to increase of $\beta_{M_r}$, the $M_r$ recruitment rate by $M_a$ and $M_i$; the insert shows the change of the average of the maximum concentration of bacteria when $\beta_{M_r}>0.2$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, We set $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$ and the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. C-D) The simulation results of $B$ with two different values of $\beta_{M_r}$. We set the initial conditions as in (A-B), $\alpha_{T_r}=\beta_{T_r}= 1 \ \text{day}^{-1}$ and the other parameters are listed in Table 3

    Table 1.  The definitions of the parameters used in the equations (1)-(6)

    Parameter Definition
    $\alpha_{M_r}$ $M_r$ source
    $\alpha_B$ Bacteria growth rate
    $\alpha_{T_1}$ $T_1$ activation rate by $M_i$ and $M_a$
    $\alpha_{T_r}$ $T_r$ activation rate by $M_a$
    $\beta_{M_r}$ $M_r$ recruitment by $M_i$ and $M_a$
    $\beta_{T_r}$ $T_r$ activation rate by $T_1$
    $k_{MB}$ Infection rate
    $k_{MT}$ $T_1$ immunity rate
    $k_{BM}$ Death rate of bacteria by $M_r$ and $M_a$
    $\gamma_r$ Deactivation rate of macrophage
    $\gamma_a$ Activation rate of macrophage
    $\gamma_i$ Bursting rate of $M_i$
    $\mu_{M_r}$ Death rate of $M_r$
    $\mu_{M_i}$ Death rate of $M_i$
    $\mu_{M_a}$ Death rate of $M_a$
    $\mu_{T_1}$ Death rate of $T_1$
    $\mu_{T_r}$ Death rate of $T_r$
    $\omega_1$ Ratio in $M_r$ recruitment
    $\omega_2$ Ratio in bacteria killing
    $\omega_3$ Ratio in $T_1$ activation
    $K_B$ Half-saturation constant for $B$ in infection
    $Q_B$ Half-saturation constant for $B$ in macrophage activation
    $K_1$ Half-saturation constant for $T_1$ in immunity
    $K_r$ Half-saturation constant for $T_r$ in inhibition
    $Q_r$ Half-saturation constant for $T_r$ in macrophage deactivation
    $Q_1$ Half-saturation constant for $T_1$ in macrophage activation
    $N$ Estimated number of bacteria per macrophage
    $\bar{N}$ Number of bacteria releasing from macrophage by Th1 cell immunity
     | Show Table
    DownLoad: CSV

    Table 2.  Steady state analysis for the three-equation system (13)-(15). For the conditions, $L_M < L_B$ ($L_M>L_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is low; $R_M < R_B$ ($R_M>R_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is high

    Steady states Status
    $L_M < L_B$
    $R_M < R_B$
    If $R_M < 0, $ Two steady states
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Stable or unstable steady state $(B^*, M_r^*, M_i^*)$
    Latent disease
    (Stable $(B^*, M_r^*, M_i^*)$)
    Active disease
    (Unstable $(B^*, M_r^*, M_i^*)$)
    If $R_M>0, $ One steady state
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Active disease
    $L_M < L_B$
    $R_M>R_B$
    Two steady states
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Stable or unstable steady state $(B^*, M_r^*, M_i^*)$
    Latent disease
    (Stable $(B^*, M_r^*, M_i^*)$)
    Active disease
    (Unstable $(B^*, M_r^*, M_i^*)$)
    $L_M>L_B$
    $R_M < R_B$
    If $R_M>0$ and $L_B>0$, Two steady states
    Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Unstable steady state $(B^*, M_r^*, M_i^*)$
    Active disease or recovery
    (depends on initial values)
    If $R_M>0$ and $L_B < 0$,
    or $R_M < 0$, $L_B < 0$ and $R_B/L_B < R_M/L_M$,
    One steady state
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Active disease
    If $R_M < 0$, $L_B < 0$ and $R_B/L_B>R_M/L_M$,
    Two steady states
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Stable steady state $(B^*, M_r^*, M_i^*)$
    Latent disease
    If $R_M < 0$ and $L_B>0$, One steady state
    Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Recovery
    $L_M>L_B$
    $R_M>R_B$
    If $L_B < 0$, Two steady states
    Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Stable steady state $(B^*, M_r^*, M_i^*)$
    Latent disease
    If $L_B>0$, One steady state
    Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
    Recovery
     | Show Table
    DownLoad: CSV

    Table 3.  Parameters used in the equations (1)-(6)

    Parameter Range for
    stability tests
    in Section 4
    Values for the
    simulations in
    Figs. 4-6 and
    Figs. 8-10
    Range for $OI$ tests
    in Fig. 7 and Fig. 10
    Unit Reference
    $\alpha_{M_r}$ $3.3\times10^{2}-1\times 10^3$ $5\times10^{2}$ $3.3\times10^{2}-1\times 10^3$ $\text{ml}^{-1} \text{day}^{-1}$ [2,31]
    $\alpha_B$ $0.01-0.1$ $5\times10^{-2}$ $0.025-0.075$ $\text{day}^{-1}$ [14,31]
    $\alpha_{T_1}$ - $5.2\times 10^{-2}$ $0.026-0.078$ $\text{day}^{-1}$ [14]
    $\alpha_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
    $\beta_{M_r}$ $10^{-2}-10 $ $1\times 10^{-2}$ $10^{-2}-0.1$ $\text{day}^{-1}$ [14,31]
    $\beta_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
    $k_{MB}$ $0.2-0.5$ $5\times 10^{-1}$ $0.2-0.5$ $\text{day}^{-1}$ [14]
    $k_{MT}$ - $1\times 10^{-1} $ $0.08-0.12$ $\text{day}^{-1}$ Estimated
    $k_{BM}$ $1.25\times 10^{-9}$
    $-2\times 10^{-8}$
    $2\times 10^{-8}$ $1.25\times 10^{-9}-2\times 10^{-8}$ $\text{ml}\;\text{day}^{-1}$ [3,14]
    $\gamma_r$ - $2\times 10^{-1}$ $0.1-0.5$ $\text{day}^{-1}$ Estimated
    $\gamma_a$ - $4\times 10^{-1}$ $0.2-0.6$ $\text{day}^{-1}$ Estimated
    $\gamma_i$ $0.01-0.1$ $4\times 10^{-2} $ $0.2-0.4$ $\text{day}^{-1}$ Estimated
    $\mu_{M_r}$ $1\times 10^{-2} $ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
    $\mu_{M_i}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
    $\mu_{M_a}$ - $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
    $\mu_{T_1}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
    $\mu_{T_r}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
    $\omega_1$ - $7.14$ $7.14$ [14]
    $\omega_2$ - $10$ $10$ [14]
    $\omega_3$ - $10$ $10$ Estimated
    $K_B$ $1\times 10^{7} $ $1\times 10^{7} $ $1\times 10^{7}$ $\text{ml}^{-1}$ [31]
    $Q_B$ - $1\times 10^{6}$ $1\times 10^{6}$ $\text{ml}^{-1}$ [31]
    $K_1$ - $1$ 1 Estimated
    $K_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
    $Q_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
    $Q_1$ - $1\times 10^{2}$ $1\times 10^{2}$ $\text{ml}^{-1}$ Estimated
    $N$ $50$ $50$ $50$ [14,32]
    $\bar{N}$ - $10$ $10$ [14]
     | Show Table
    DownLoad: CSV
  •   A. Chaudhry , R. M. Samstein , P. Treuting , Y. Liang , M. C. Pils , J.-M. Heinrich , R. S. Jack , F. T. Wunderlich , J. C. Bruning , W. Muller  and  A. Y. Rudensky , Interleukin-10 signaling in regulatory T cells is required for suppression of Th17 cell-mediated inflammation, Immunity, 34 (2011) , 566-578.  doi: 10.1016/j.immuni.2011.03.018.
      R. Condos , W. N. Rom , Y. M. Liu  and  N. W. Schluger , Local immune responses correlate with presentation and outcome in tuberculosis, American Journal of Respiratory and Critical Care Medicine, 157 (1998) , 729-735.  doi: 10.1164/ajrccm.157.3.9705044.
      I. E. A. Flesch  and  S. H. E. Kaufmann , Activation of tuberculostatic macrophage functions by gamma interferon, interleukin-4, and tumor necrosis factor, Infection and Immunity, 58 (1990) , 2675-2677. 
      J. L. Flynn  and  J. Chan , Immunology of tuberculosis, Annual Review of Immunology, 19 (2001) , 93-129. 
      F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959.
      A. M. Green , J. T. Mattila , C. L. Bigbee , K. S. Bongers , P. L. Lin  and  J. L. Flynn , CD4(+) regulatory T cells in a cynomolgus macaque model of Mycobacterium tuberculosis infection, The Journal of Infectious Diseases, 202 (2010) , 533-541. 
      C. A. Janeway, P. Travers, M. Walport and M. Shlomchik, Immunobiology: The Immune System in Health and Disease, New York: Garland Science, 2001.
      M. Kursar , M. Koch , H.-W. Mittrücker , G. Nouailles , K. Bonhagen , T. Kamradt  and  S. H. E. Kaufmann , Cutting Edge: Regulatory T cells prevent efficient clearance of Mycobacterium tuberculosis, Journal of Immunology, 178 (2007) , 2661-2665.  doi: 10.4049/jimmunol.178.5.2661.
      S. H. Lee , Diagnosis and treatment of latent tuberculosis infection, Tuberculosis and Respiratory Diseases, 78 (2015) , 56-63.  doi: 10.4046/trd.2015.78.2.56.
      W. -C. Lo, V. Arsenescu, R. I. Arsenescu and A. Friedman, Inflammatory bowel disease: How effective is TNF-alpha suppression? PLoS ONE, 12 (2017), e01708g5. doi: 10.1371/journal.pone.0165782.
      W.-C. Lo , R. I. Arsenescu  and  A. Friedman , Mathematical model of the roles of T cells in inflammatory bowel disease, Bulletin of Mathematical Biology, 75 (2013) , 1417-1433.  doi: 10.1007/s11538-013-9853-2.
      K. J. Maloy  and  F. Powrie , Intestinal homeostasis and its breakdown in inflammatory bowel disease, Nature, 474 (2011) , 298-306.  doi: 10.1038/nature10208.
      S. Marino , S. Pawar , C. L. Fuller , T. A. Reinhart , J. L. Flynn  and  D. E. Kirschner , Dendritic cell trafficking and antigen presentation in the human immune response to Mycobacterium tuberculosis, Journal of Immunology, 173 (2004) , 494-506.  doi: 10.4049/jimmunol.173.1.494.
      S. Marino  and  D. E. Kirschner , The human immune response to Mycobacterium tuberculosis in lung and lymph node, Journal of Theoretical Biology, 227 (2004) , 463-486.  doi: 10.1016/j.jtbi.2003.11.023.
      F. O. Martinez and S. Gordon, The M1 and M2 paradigm of macrophage activation: time for reassessment, F1000prime Reports, 6 (2014), p13. doi: 10.12703/P6-13.
      K. A. McDonough , Y. Kress  and  B. R. Bloom , Pathogenesis of tuberculosis: Interaction of Mycobacterium tuberculosis with macrophages, Infection and Immunity, 61 (1993) , 2763-2773. 
      T. Mogues , M. E. Goodrich , L. Ryan , R. LaCourse  and  R. J. North , The relative importance of T cell subsets in immunity and immunopathology of airborne Mycobacterium tuberculosis infection in mice, The Journal of Experimental Medicine, 193 (2001) , 271-280. 
      D. M. Nancy , C. P. Sara , M. V. Viviana , A. R. Carlos , R. Mauricio  and  F. G. Luis , Regulatory T cell frequency and modulation of IFN-gamma and IL-17 in active and latent tuberculosis, Tuberculosis, 90 (2010) , 252-261. 
      G. Pedruzzi , K. V. S. Rao  and  S. Chatterjee , Mathematical model of mycobacterium - host interaction describes physiology of persistence, Journal of Theoretical Biology, 376 (2015) , 105-117.  doi: 10.1016/j.jtbi.2015.03.031.
      E. Pienaar  and  M. Lerm , A mathematical model of the initial interaction between Mycobacterium tuberculosis and macrophages, Journal of Theoretical Biology, 342 (2014) , 23-32.  doi: 10.1016/j.jtbi.2013.09.029.
      K. M. Quinn , R. S. McHugh , F. J. Rich , L. M. Goldsack , G. W. De Lisle , B. M. Buddle , B. Delahunt  and  J. R. Kirman , Inactivation of CD4+CD25+ regulatory T cells during early mycobacterial infection increases cytokine production but does not affect pathogen load, Immunology and Cell Biology, 84 (2006) , 467-474.  doi: 10.1111/j.1440-1711.2006.01460.x.
      G. A. Rook , J. Steele , M. Ainsworth  and  B. R. Champion , Activation of macrophages to inhibit proliferation of Mycobacterium tuberculosis: comparison of the effects of recombinant gamma-interferon on human monocytes and murine peritoneal macrophages, Immunology, 59 (1986) , 333-338. 
      P. Salgame , Host innate and Th1 responses and the bacterial factors that control Mycobacterium tuberculosis infection, Current Opinion in Immunology, 17 (2005) , 374-380.  doi: 10.1016/j.coi.2005.06.006.
      S. K. Schwander , M. Torres , E. Sada , C. Carranza , E. Ramos , M. Tary-Lehmann , R. S. Wallis , J. Sierra  and  E. a. Rich , Enhanced responses to Mycobacterium tuberculosis antigens by human alveolar lymphocytes during active pulmonary tuberculosis, The Journal of Infectious Diseases, 178 (1998) , 1434-1445. 
      D. K. Sojka , Y. H. Huang  and  D. J. Fowell , Mechanisms of regulatory t-cell suppression - a diverse arsenal for a moving target, Immunology, 124 (2008) , 13-22.  doi: 10.1111/j.1365-2567.2008.02813.x.
      D. Sud , C. Bigbee , J. L. Flynn  and  D. E. Kirschner , Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, Journal of Immunology, 176 (2006) , 4296-4314. 
      J. Tan , D. Canaday , W. Boom , K. Balaji , S. Schwander  and  E. Rich , Human alveolar T lymphocyte responses to Mycobacterium tuberculosis antigens - Role for CD4(+) and CD8(+) cytotoxic T cells and relative resistance of alveolar macrophages to lysis, Journal of Immunology, 159 (1997) , 290-297. 
      K. Tsukaguchi , B. de Lange  and  W. H. Boom , Differential regulation of IFN-gamma, TNF-alpha, and IL-10 production by CD4(+) alphabetaTCR+ T cells and vdelta2(+) gammadelta T cells in response to monocytes infected with Mycobacterium tuberculosis-H37Ra, Cellular Immunology, 194 (1999) , 12-20. 
      R. Van Furth , M. C. Diesselhoff-den Dulk  and  H. Mattie , Quantitative study on the production and kinetics of mononuclear phagocytes during an acute inflammatory reaction, Journal of Experimental Medicine, 138 (1973) , 1314-1330. 
      I. Wergeland , J. Assmus  and  A. M. Dyrhol-Riise , T regulatory cells and immune activation in Mycobacterium tuberculosis infection and the effect of preventive therapy, Scandinavian Journal of Immunology, 73 (2011) , 234-242. 
      J. E. Wigginton  and  D. Kirschner , A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, Journal of Immunology, 166 (2001) , 1951-1967.  doi: 10.4049/jimmunol.166.3.1951.
      M. Zhang , J. Gong , Y. Lin  and  P. F. Barnes , Growth of virulent and avirulent Mycobacterium tuberculosis strains in human macrophages, Infection and immunity, 66 (1998) , 794-799. 
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