Spatial Models of Tuberculosis: Granuloma Formation Suman Ganguli Kirschner Group Dept. of Microbiology & Immunology University of Michigan

Outline Background: M. tuberculosis & granuloma Spatial metapopulation model “Coarsely” discretized spatial domain ODEs Joint work with D. Gammack & D. Kirschner Agent-based model “Finely” discretized spatial domain Discrete rules Joint work with J. Segovia-Juarez & D. Kirschner

Mycobacterium tuberculosis Estimated 1/3 of world’s population infected Leading cause of death by infectious disease approx. 3 million deaths per year 90% of infected individuals achieve and maintain latency 5% progress rapidly to active disease 5% initially latent, but infection reactivates What factors lead to these different outcomes?

Infection & immune response M. tuberculosis ingested by macrophages in the lung Macrophages may be unable to clear bacteria Bacteria replicate inside these macrophages Leads to cell-mediated immune response infected macrophages release chemotactic signals immune effector cells (T cells, macrophages) migrate to site of infection form characteristic spatial pattern: granuloma

Sketch of granuloma formation Replication T cells Infected macrophage Activated macrophages bacteria Active disease Latency (Dannenberg & Rook, “Pathogenesis of Pulmonary Tuberculosis”, 1994)

Human granuloma: cross-section of lung tissue

Granuloma & disease outcomes Latency: Properly functioning granuloma forms activated macrophages and T cells contain infection Active Disease: Poorly formed granuloma bacteria spreads, extensive tissue damage Reactivation: Functioning granuloma breaks down bacteria escapes, active disease develops Develop mathematical models to help understand : the complex spatio-temporal process of granuloma formation it role in disease outcome

Modeling host-pathogen interactions of Mtb. infection Wigginton & Kirschner (J. Immunology, 2001) ODE model temporal dynamics of bacteria, macrophages, T cells, key cytokines 2-compartmental ODE model (Marino) Trafficking between lung and lymph node Spatio-temporal models of granuloma formation PDE model (Gammack, Kirschner & Doering, J. Mathematical Biology, 2003) Metapopulation model Agent-based model

Metapopulation model of granuloma formation Discretize spatial domain (lung tissue): n x n lattice of compartments “Coarse” discretization (n small) subpopulations of each cell type in each compartment i j Bacteria, T cells, macrophages, etc. ODEs: interactions within each compartment movement of cells between compartments

Cell subpopulations For each compartment (i, j): 3 types of macrophages resting (MR (i,j)), activated (MA (i,j)), infected (MI (i,j)) 2 types of bacteria extracellular (BE (i,j)) and intracellular (BI (i,j)) T cells (T(i,j)) chemokine (C(i,j)) molecules that direct cell movement ODE for each subpopulation => system of 7·n2 ODEs

ODE terms: dynamics within each compartment Model the interactions of subpopulations within each compartment Simplified version of Wigginton & Kirschner’s temporal ODE model for each compartment

Example: Resting macrophage dynamics MA (i,j) T (i,j) MR (i,j) MI (i,j) BE (i,j)

ODE terms: movement between compartments Unbiased movement (diffusion): chemokine diffuses equally in all directions Biased movement: T cells, macrophages tend to move up chemokine gradient Continuously update coefficients in diffusion terms as a function of changing chemokine environment

Metapopulation Model: Results 5 x 5 lattice bacteria begins in and is restricted to center compartment study spatial recruitment of immune cells Clearance: bacteria eliminated Latency: bacterial growth contained all populations achieve steady-state Active disease: uncontrolled bacterial growth Bifurcation parameters include those governing recruitment & movement of immune cells

Clearance: spatial distributions Time (days) Extracellular bacteria Resting macrophages Infected macrophages Activated macrophages T cells Chemokine

Clearance: spatial distributions Resting macrophages Infected macrophages Activated macrophages Extracellular bacteria T cells Chemokine

Latency: spatial distributions Time (days) Extracellular bacteria Resting macrophages Infected macrophages Activated macrophages T cells Chemokine

Agent-based model of granuloma formation Discretize spatial domain n x n lattice of “micro-compartments” “Fine” discretization (n large) each micro-compartment can contain a single macrophage agent and a single T cell agent i j T M Rules to govern: interactions within each micro-compartment movement of agents between micro-compartments

ABM: agents & continuous entities 2 types of agents Macrophages (each in resting, infected, chronically infected, or activated state) T cells Continuous entities extracellular bacteria (BE (i,j)) chemokine (C(i,j))

ABM Rules: Example MI MA Within micro-compartment (i, j): Time t T cell agent and macrophage agent in infected state T MA Time t+1 Macrophage agent changes to activated state M.state = infected M.state = activated

ABM: preliminary results Macrophages Bacteria

Goals Mechanisms & bifurcation parameters Disease outcomes in ABM Mechanisms & bifurcation parameters Spatio-temporal organization of immune cells Comparison with metapopulation, PDE models Combine various modeling approaches to model tuberculosis infection at multiple scales

Acknowledgements Denise Kirschner David Gammack Jose Segovia-Juarez Members of the Kirschner lab…