A Mathematical Model For Interaction Macrophages, T Lymphocytes and Cytokines at Infection of Mycobacterium tuberculosis with Age Influence

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1 A Mathematical Model For nteraction Macrophages, T Lymphocytes and Cytokines at nfection of Mycobacterium tuberculosis with Age nfluence Usman Pagalay 1, Marjono, Kusworini Handono 3 1 Faculty of Science and Technology, slamic State University of Malang ndonesia Faculty of Science, Brawijaya University of Malang ndonesia 3 Faculty of Medicine, Brawijaya University of Malang ndonesia Malang, East Java, ndonesia 1 usmanpagalay@yahoo.co.id, marjono@ub.ac.id, 3 dr.kusworini@gmail.com Abstract. Tuberculosis (TB) is still a health problem in the world, because of the increasing prevalence and treatment outcomes are less satisfactory. This is presumably due to a complete lack of understanding of the role of the immune system to infection Mycobacterium tuberculosis (Mtb). Currently it is known that the immune response plays a role in controlling the development of the germ cells are macrophages, T lymphocytes and cytokines. This study has made a mathematical model of the interaction between macrophages, T lymphocytes and cytokines with Mtb infection in the lungs. Effect of age was observed through disparate data, young (3 months) and old (18 months). Runge Kutta method order-4 is used to solve the system of non-linear differential equations of the first order. Growth of bacteria (extracellular and intracellular) tends to increase up to 3 months, either in old mice and young mice. Behavior of T cells dropped drastically. Concentration of L- and L-4 is likely to increase at the beginning of infection, TNF-, L-1, FN- and L-1 increased. Resting macrophages tend to fall, infected macrophages tended to rise and activated macrophages fluctuate in the first 3 months. Key-Words: mathematical models, Mycobacterium tuberculosis, macrophages, T lymphocytes, cytokines. 1. ntroduction Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis (Mtb) and infected almost a third of the world population. An approximated 8.9 million people with TB with 3 million people die per year. ndonesia is the third country in the world in order of the number of TB patients after ndia (3%) and China (15%) with a percentage of 1% of the total TB patients in the world. A cell-mediated immune response is essential for control of Mtb in the lungs particularly macrophages and T lymphocytes. Alveolar macrophages that serve as a haven for bacteria as well as to eliminate the bacteria. Eliminates bacteria depends on cytokines FN-, TNF-, L-, L-1 secretion by T cell. T lymphocytes have two important roles in TB infection. First, produce a variety of cytokines in the development of cell-mediated immune response. Second, eliminate infected macrophages through the process of apoptosis. There are four mechanisms of CD8+ T cells contribute to the control of Mtb: (1) the release of cytokines, () cytotoxicity via the granule-dependent exocytosis, (3) mediated cytotoxicity in Fas / Fas ligand interaction, and (4) Directly active as microbicidal. Activity cytotoxic of CD8+ T cells in the process of apoptosis through Fas - Fas ligand pathway and killing through perforin and granulysin. n humans, CD8+ T cells can kill bacteria through the release of intracellular antimicrobial peptide granulysin [3]. CD4+ T cell contributes to TB infection activates macrophages through the production of various cytokines, such as L-1 and FN- for the lysis of infected macrophages. ncreased activation of macrophages in fagositosis can eliminate extracellular bacteria, while lysis process serves chronically infected macrophages to release Mtb []. nterleukin-1 (L-1) is a cytokine that is immunoregulator. These cytokines play an important role in down-regulation, including inhibition of macrophage activation. Besides, FN- is a cytokine that is essential in activating macrophages to eliminate Mtb and involved in the process of Tho into Th1 differentiation. Another important cytokine is L-4 and L-1. L-1 is a key cytokine for Th1 type. This cytokine is produced by activating macrophages and infected macrophages in-clicking antigen stimulation. L-1 as well as the regulator primarily to induce differentiation into Th1 lymphocytes, suppress the production of FN-. Cytokine L-4 is a Th cell prototype. These cytokines are also involved in the process of Tho into Th1 differentiation. The role of L-4 in tuberculosis infection is still controversial. Mathematical models of Mtb infection in humans and mice were developed by Wigginton [4]. The model that has been developed is not directly involve cytokines L- and TNF-as well as CD8 + T lymphocytes. Whereas Subject Category: Medical & Health Sciences 5

2 cytokines TNF- and CD8 + T lyarehocytes in TB infection is very important. On TB infection, cytokines TNF- synergize with FN- in activating macrophages in producing various kinds of substances that can suppress the growth of Mtb. Mathematical models of Mtb infection in humans and mice were developed by Friedman [7] with the involvement of the influence of age. The model does not directly involve the cytokines L-4, TNF-, and resting macrophages. Whereas L-4 inhibits activated macrophages, while macrophages and resting circulating equilibrium disrupted by activated macrophages and infected macrophages such resting macrophages depend on time. This study makes a mathematical model interactions, immune system and Mtb involving populations of CD4+, CD8+ T cell; cytokines include interleukin-1 (L-1), L-4, L-1, L-, TNF- and FN- ; infected macrophages, resting macrophages and activated macrophages, the bacteria comprising the extracellular bacteria and intracellular bacteria involving the influence of age.. Material and Methods.1 Material.1.1 The Mathematical Model Mathematical model of the interaction of macrophages and T lymphocytes forms a system of nonlinear differential equations of the first order. The model consists of 13 equations: three equations macrophages, six equations cytokines, two equations T lymphocytes, two equations bacteria. Explanations and models are given below..1. The model equations Macrophage dynamics. The equation that describes the rate of change of macrophage populations during Mtb infection is given in (1) - (3). dm R () t F () t srm 4 A( M A wm ) sr4 B dt F f81 s4b B () E () t t BT F km R k3m R...(1) BE c 9 f1 4 s 1 BT F c8. M MR R dm () A() t t BT F 1 k3m R k4m A dt f1 4 s 1 BT F c 8 1 c3 c 4 () t k3am MA. M A...() () t a ( ) ( ) ( ) E F () t km R k17. M k14bm E ( ) 9 ( ) (. ( )) ( ) 9 1( ) 4b dm t B t B t dt B t c B t N M t F t f t s 1 () t M ( T8 w3t 4) k4m A k3am k14 an. N fracc 1 c3 c 4 a c4. M ( T8 w3t 4) M...(3) M Subject Category: Medical & Health Sciences 6

3 Explanation: Changes in resting macrophages (1) influenced by the production of alveolar macrophages ( sr ), move from activated macrophages 4 A and infected macrophages 4 A w. ( w 1), with the help of TNF- on the recruitment rate sr 4B, the changes become infected macrophages k, activated macrophages k3 as well as changes to the natural mortality rate MR respectively. Changes activated macrophages () is affected by a change in resting macrophages become activated k 3, activated macrophages undergo deactivation k 4, changes become infected macrophages become activated k 3A, and the rate of natural mortality MA. Changes in infected macrophages (3) consecutive influenced by changes in resting macrophages become infected macrophages k, infected macrophages undergo a process of release k 17, changes in infected macrophages become activated macrophages k 4, infected macrophages become activated macrophages k 3A, infected macrophages undergo apoptosis k 14a as well as the rate of natural mortality. MA Cytokines dynamics. Equation that describes the rate of change in cytokine concentrations given in (4) - (9). d1() t s 6 c 7a 7M A k7avm dt 1 f6 s 6 1 c7a T T....(4) df () t BT 3M 31M A 3T4 dt BT f1 4 f71 s 1 T. F...(5) C F d () t BT M ( ) 1 1 A t sg ut4 yt8 dt BT c1 1 s7 1 M A f41 s4 M....(6) M d1 BE c 81 3M R 81M A 11 dt BE c3 c (7) d k1t4 ( k11t 4 k1t8 ) dt () t c1...(8) d4() t 11T4 44 dt...(9) Explanation: L - 1 (4) is produced mainly by activating macrophages with the rate 7, this process is inhibited by FN- and L -1. L - 1 is also produced by infected macrophages at a rate of k 7av, the rate of CD4 + lymphocytes 16 and CD8 + T cell18 as well as on the rate of degradation at rate of 1. TNF- ( 5 ) is produced by macrophages infected with the rate 3, TNF- is produced by activated macrophages with rate 31, this process is inhibited by L - 4 and L - 1. TNF- is also produced by CD4 + T cell at a rate of 3 and on the rate of CD8 + T cell 33 as well Subject Category: Medical & Health Sciences 7

4 as on the rate of degradation F. FN- (6) is produced by Natural Killer cells (NK) in the rate s g. FN- is also produced by CD4 + T cell and the rate of CD8 + T cell in the rate of FN- produced by infected macrophages at a rate of and on the rate of degradation. L-1 (7) is produced by resting macrophages in response infection at u a rate of 3, also produced by activated macrophages at a rate of 81 and degradation at a rate of 1. L - ( 8 ) is produced by CD4 + T cells and the rate k1 of consumption by CD4 + T cells and CD8 + at different speeds k11 and k1 as well as degradation rate. L-4 (9) only produced by CD4 + T cells and degraded at a rate of 4. T cell dynamics. For T cell dynamics given in Equation (1) (11). dt8 F () t x ( M A wm ) 1 sr k44t8 dt F f8. 1 s4b c1 () t Tc T8 M A T 8T8...(1) () t c. dt4 F () t z M A 1 sr3 B k13t4 dt F f8. 1 s4b c11 () t TT4 M A T 4T4...(11) () t c Explanation : CD8 + T cells (1) arrive at infection at a rate of on the rate k 44, the induction of CD8 + T cell apoptosis in the rate Tc x. Recruitment of CD8 + T cells at the rate s r proliferation of L-. CD4, and the rate of natural mortality T 8 + T cells (11) arrive infections at the rate z. Recruitment of CD8 + T cells at the rate sr 3B, proliferation of L- on the rate k 13, the induction of CD8 + T cell apoptosis in the rate T, and the rate of natural mortality T 4. Bacterial dynamics. For bacterial dynamics given in Equation (1) (13). dbe BE BE k5m A BE n3. km R dt BE () t c9. M ( T8 w3t 4) B () t k14 an. N fracc k17 N. M c4. M ( T8 w3t 4) B N. M F () t k14b. N. M F f s 8 1 4b B...(1) Subject Category: Medical & Health Sciences 8

5 db B BE 19B 1 n 3. km R dt B ( ). ( ) BE () t c t N M t 9 B () t M ( T8 w3t 4) k17 N. M k 14aN B ( ). ( ) 4 ( ) ( 8( ) 3 4( )) t N M t c M t T t w T t F () t k14bn. M B...(13) F f8. 1() t s4b Explanation: Extracellular bacteria growing at the rate. The bacteria are killed by activated macrophages at rate k5 and resting macrophages at a rate k, infected macrophages undergo apoptosis k 14a, infected macrophages undergo apoptosis ( k ), TNF- induced apoptosis of infected macrophages with intracellular bacteria natural mortality rate. 17. Methods..1 Computer simulations Numerical simulations are used in a mathematical model to obtain the behavior of each variable is the Runge Kutta method of order 4 with Program Matlab version Parameter values All the variables and parameters used in equations (1) - (13) are given in the table. Table 1 contains the initial value derived from Friedman, 8 and Sud, 6. Table contains the parameter differences between young mice and old mice. Table 3 contains all of the parameters used in equations (1)-(13). When no data is available, was estimated, average, Adjustment or proportional data. Numerical simulation started on day 7 after Mtb infection. The units used for the cell is the number of cells per milliliter of bronchoalveolar lavage (BAL), whereas the concentration of cytokines was picograms per milliliter of BAL. The rate of growth and decay ( 19,, 4, 1 1, g ) is given by using the following standard formula dn() t ln r. N to get r dt Lifetime 3. Results Based on the mathematical model equations (1) - (13), used Runge Kutta order 4, Table 1, and 3 as well as using Matlab.7. software (see figure 1 and ). Subject Category: Medical & Health Sciences 9

6 M(t) (cell/mililiter) TB(t) (cell/mililiter) MR(t) (cell/mililiter) MA(t) (cell/mililiter) M(t)(cell/mililiter) TB(t) (cell/mililiter) MR(t) (sel/mililiter) MA(t) (cell/mililiter) 6 x 15 Figure MR(t) 4 MR(t) muda young MR(t) tua old Figure MA(t) MA(t) young MA(t) old (a) time (day) x 15 Figure M(t) (b) time (day) 1 x 17 Figure TB(t) M(t) young M(t) old (c) time (day) 5 TBA(t) young TB(t) old (d) time (day) Figure 1 Simulation result of total bacteria( TB), Resting macrophages (MR), infected macrophages (M) and activated macrophages (MA) without TNF- 6 x 15 Figure MR(t) Figure MA(t) 4 MR(t) young MR(t) old (a) time (day)` x 15 Figure M(t) MA(t) young MA(t) old (b) time (day)` 1 x 17 Figure TB(t) M(t) young M(t) old (c) time (day)` TB(t) young TB(t) old (d) time (day)` Figure Simulation result of total bacteria (TB), Resting macrophages (MR), infected macrophages (M) and activated macrophages (MA) with TNF-. Figure 1 behavior of macrophages and total bacteria. Figure 1 and consist of a). Resting macrophages for young and old mice. b). Activated macrophages for the young and old mice. c) nfected macrophages are higher in young mice in the first month. d) The behavior of higher total bacterial old mice after a month-3. Figure is the behavior of macrophages and total bacteria involved. Bacterial growth is likely to increase up to 3 months, both young mice and old mice and old mice to bacterial counts higher than the number of bacteria in a mouse model. Total higher when bacteria when compared to without involving. Drawings for lymphocytes T CD4 + and CD8 + as well as cytokine not displayed. Subject Category: Medical & Health Sciences 1

7 4. Conclusion 1) Mathematical model of the interaction of macrophages with T cells in response to infection of Mtb in human form a system of ordinary differential equations, nonlinear involving 13 variables, namely macrophages (resting macrophages, activated macrophages and infected macrophages), T lymphocytes (CD4 + T cells and CD8 + T cells) and cytokines (L-, L-4, L-1, L-1, FN-, TNF- ) and bacterial extracellular and intracellular bacteria. ) Bacterial growth is likely to increase up to 3 months, both young mice and old mice and old mice to bacterial counts higher than the number of bacteria in young mice. 3) Decreased number of T cells ( CD4 + T cells and CD8 + T cells ) when the case without the case with TNF - and TNF - and L - 4 mice both young and old rats. For the case without TNF - and L - 4, increa(g the number of T cells ( CD4 ) cells and CD8 + T cells ) mice both young and old mice. 4) Cytokine L - 1, TNF - and FN-,e likely to increase, both young mice and old mice, while L - 4 and L - tended to decrease. 5) Resting macrophages decreased dramatically, is caused by resting macrophages and migrate to infected macrophages and activated macrophages, infected macrophages tended to increase at the beginning of infection, activated macrophages is fluctuate, both young mice and old mice. Acknowledgement We are grateful to Prof. Dr. Agus suryanto, M.Sc in help construction mathematical model. We thank Prof. Toto Nusantara, M.Si and Prof. dr. Aris Widodo, MS, SpFK, Ph.D for helpful discussions. We also thank Prof. Dr. r. Budimawan, DEA and Prof. Drs. Sutiman B. Sumitro, SU., D. Sc for early support of this project. References [1] Marino, S. K. D., 4, The human immune response to Mycobacterium tuberculosis in lung and lymph node, J Theor Biol, 7, pp [] Jose L., 4, dentifying control mechanisms of granuloma formation during M. Tuberculosis infection using an agent-based model, J Theor Biol, 31, pp [3] Sud D, Bigbee C, Flynn JL, Kirschner DE, 6, Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, J mmunol, 176, pp [4] Wigginton, J. E., and D. Kirschner, 1, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. mmunol, 166, pp [5] David Gammack, 5, Understanding the mmune Response in Tuberculosis Using Different Mathematical Models and Biological Scales. [6] World Health Organization, 7, Geneva: World Health Organization, pp [7] Friedman A, Turner J, Szomolay B, 8, Amodel on the influence of age on immunity to infection with Mycobacterium tuberculosis, Exp Gerontol, 43, pp [8] Judy, Day, 9, Modeling the immune rheostat of macrophages in the lung in response to infection, J mmunol, pp [9] Kirschner D., 1999, Dynamics of coinfection with M. Tuberculosis and HV-1, Theor. Popul. Biol, pp [1] Magombedze G., 6, Modeling the human immune response mechanisms to Mycobacterium Tuberculosis infection in the lung, [11] Flynn J, Vanja Lazarevic,, CD8+ T Cells in Tuberculosis. [1] Evelyn Guirado, Larry S. Schlesinger, 13, Modeling the Mycobacterium tuberculosis granuloma the critical battle field in host immunity and disease. [13] WHO, 11, Global Tuberculosis Control, Geneva: WHO Press, World Health Organization. [14] Marino S, Sud D, Plessner H, Ling Lin P, Chan J, JoAnne L. Flynn, Denise E. Kirschner, 7, Differences in Reactivation of Tuberculosis nduced from Anti-TNF Treatments Are Based on Bioavailability in Granulomatous Tissue. Subject Category: Medical & Health Sciences 11

8 Table. Variables and initial values [3,7] Symbol Description Young Old Unit M R Density of resting macrophages Cell/ml M A Density of activated macrophages 18 Cell/ml M density of infected macrophages 18 cell/ml concentration of nterleukin- 1 5 pg/ml 4 concentration of nterleukin pg/ml 1 concentration of nterleukin-1 5 pg/ml 1 concentration of nterleukin pg/ml TNF concentration of TNF 5 5 pg/ml FN- concentration of FN-γ 5 5 pg/ml T 4 density of CD4+ T cells cell/ml T 8 density of CD8+ T cells cell/ml B E density of extracellular bacteria 1 1 cell/ml B density of intracellular bacteria 36 4 cell/ml Table. Different parameters between young and old [7] Symbol Description Young Old rate of Major Histocompatibility Complex (MHC) x activation,566 ml/pg/day,854 ml/pg/ day K 3 activation rate of infected macrophages,3415/ day,544/day K 4 deactivation rate of activated macrophages,8876/ day,6177/day K 6 rate of activation of resting macrophages,7768/ day,13539/ day K 7 L-1 production rate by infected macrophages,561/ day,5544/ day K 8 L-1 production rate by activated macrophages,853/ day,5316/ day K 9 L-1 production rate by resting macrophages pg/cell/ day.1 pg/ cell day K pg/cell/ pg/ cell L- production rate by CD4+ T cells day day K 11 loss of L- due to proliferation of CD4+ T cells pg/sel/ day pg/ cell day K 1 loss of L- due to proliferation of CD8+ T cells pg/cell/ day pg/ cell day K 13 rate of proliferation of CD4+ T cells by L pg/cell/ day pg/ cell day K 14 rate of proliferation of CD8+ T cells by L pg/cell/ day.1413 pg/ cell day Table 3. Parameter values [3,7] Symbol Description Range (Value) α 8 L-1 production by MA 8x1-5 pg/ MA/day α 3 TNF- production by M 1-3 x1 - (3x1-3 ) pg/ml M day α 5c FN- production by M.-.6 (.3) pg/ml M α 4A TNF-independent recruitment of MR 5x1 - / day α 3 L-1 production by MR.75x1-5.75x1-4 (x1-4 ) pg/ml MR.3x1 α 31 TNF- production by MA 1.5x1-4 (4x1-3 ) pg/ml MA day α 3 TNF- production by CD4+ T cells 8.16x1-4 pg/ml CD4/ day α 33 TNF- production by CD8+ T cells.6x x1-4 (.5x1-4 ) pg/ml CD8/ day Sr 3B TNF-dependent recruitment of CD4+ T cells 1 3 / day Subject Category: Medical & Health Sciences 1

9 Sr 4B TNF-dependent recruitment of MR x1 4 MR/ day f 9 Ratio adjustment, TNF- /L1 1-1 (5) f 7 Effect of L-1 on FN- -induced CD4+ T cells 1 f 8 Ratio Adjustment, L-1/TNF- on MR Recruitment 1 1 (1) S 4b Half-sat, TNF- on MR recruitment () pg/ml/ day C 3 Half-sat, BT on L-1 by MR 1 3-5x1 6 (5x1 3 ) BT/ml w Max, percentage contribution of M-produced chemokines to MR recruitment.15 μ Ty FN- induced apoptosis rate of CD4+ T cells (1-4 ) /MA day μ B B turnover to BE due to M death, other mechanisms -.5 (.4)/ day μ TNF decay rate of TNF- 1.11/ day k 14a Fas-FasL-induced apoptosis of M (.1) / day k 14b TNF induced apoptosis of M (.1) / day s 1 Half-sat, FN- on TNF- production by MA 5-1 (8) pg/ml δ 7 L-1 production by MA (.1) pg/ml MA α BE growth rate -.6 (.5) / day α 19 B growth rate (.4) / day α 16 L-1 production by CD4+ T cells (x1 - ) pg/cd4 day S rm MR recruitment rate 6-1 (1) MR/ day f 6 Adjustment, FN- on L (.5) f 4 Adjustment, L-1/L-1 on FN () f 1 Adjustment, L-4/FN () s Half sat., L (5) pg/ml s 6 Half-sat, L-1 self-inhibition in MA 51-6 (6) pg/ml s 4 Half-sat, L-1 on FN- 5-1 (5) pg/ml s 7 Half-sat, L-1 on FN- by NK cells 5 1 (4) pg/ml s 1 Half-sat, FN- on MR to MA 5 11 (7) c 9 Half-sat, BE on MR infection (x1 6 ) BE c 8 Half-sat, BT on MR activation 5x1 4 5x1 5 (1 5 ) B T /ml c 4 Half-sat, T cells /M ratio for M lysis 1 6 (4) T/M c 1 Half-sat, bacteria on FN by NK cells (1 3 ) BT/ml μ MR death rate of MR.33/ day μ M death rate of M.11/ day μ MA death rate of MA.7/ day μ γ decay rate of FN (.16) / day μ 4 decay rate of L-4.77/ day μ 1 decay rate of L (5) / day μ 1 decay rate of L / day k MR infection rate. -.4 (.4) / day k 3 MR activation rate. -.4 (.1) / day k 17 Max. M death due to B. -.8 (.) / day k 4 MA deactivation by L (.8) / day s g FN-γ production by Natural killer (NK) cells 1 (1) pg/ml day N carrying capacity of infected macrophages 1 1 () B/M scaling factor of TNF for MR to MA (1 ) BT/pg scaling factor of BT for TNF- production by MA (1-3 ) rate of FN-γ production by u CD4+ T cells pg/cell day rate of FN-γ production by y CD8+ T cells pg/ cell day rate of MHC activation.566 pg/ cell day x Subject Category: Medical & Health Sciences 13

10 z rate of MHC activation.153 pg/ cell day α 11 L-4 production by CD4+ T cells x1-3 (1-4 ) pg/cd4+ day k 4 deactivation rate of activated macrophages.8876 / day k 3A activated M by NF / day k 7av L-1 production rate by infected macrophages.561 pg/ml cell k 1 L- production rate by CD4+ T cells pg/ cell day k 11 loss of L- due to proliferation of CD4+ T cells pg/ cell day c 3 FN-γ inhibition for deactivation of activated macrophages 3 c 1 saturation for T cell proliferation by L- 5 pg/ml c 7a saturation for L-1 inhibition by L-1 5 pg/ml μ T4 death rate of CD4+ T cells.33 / day μ T8 death rate of CD8+ T cells.33 / day μ g decay rate of FN (.16) / day a saturation for activation of infected macrophage 5 pg/m c 81 saturation for L-1 inhibition by L-1 pg/ml n average number of B in an activated macrophages 5 n3 threshold at which a resting macrophage becomes infected 1 μ decay rate of L / day α 18 L-1 production by CD8+ T cells (x1 - ) pg/cd8 day BE, extracellular bacteria; B, intracellular bacteria; BT, total bacteria; Half-sat, half-saturation; MA, activated macrophages; M, infected macrophages; MR, resting macrophages. Subject Category: Medical & Health Sciences 14