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

2. 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

3. 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?

4. 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

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

6. Human granuloma: cross-section of lung tissue

7. 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

8. 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

9. 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

10. 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

11. 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

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

13. 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

14. 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

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

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

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

18. 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

19. 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))

20. 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

21. ABM: preliminary results
Macrophages
Bacteria

22. 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

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