Transcript Discrete Spatial Model of Granuloma Development

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
Infected macrophage
Replication
T cells
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
ODEs:
 interactions within each
compartment
 movement of cells
between compartments
i
j
Bacteria,
T cells,
macrophages,
etc.
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
T (i,j)
MR (i,j)
MI (i,j)
MA (i,j)
BE (i,j)





d

B
E (i , j ) 

B
E (i , j )   T (i , j ) 
MR (i , j )   k 2 MR (i , j ) 
  k 3 MR ( i , j ) 



 BE ( i , j )  c8   T ( i , j )  s 3 
B
E
(
i
,
j
)

c
9
dt





 
 

d
 BE (B

i, E
j )( i , j) T ( i , j )
  
  ...
MAI((ii,, jj))kk3M
2M
M
R ( iR, (ji), j ) 
  BE ( i , j )  c9 

dt
 ( i , j )  c8   T ( i , j )  s 3 
 BE
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
QuickTime™ and a YUV 420 codec decompressor are needed to see this picture.
Extracellular
bacteria
Activated
Infected
Resting
macrophages macrophages macrophages
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 “microcompartments”
 “Fine” discretization (n large)
 each micro-compartment can
contain a single macrophage
agent and a single T cell
agent
Rules to govern:
 interactions within each microcompartment
 movement of agents between
micro-compartments
i
T
M
j
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
Within micro-compartment (i, j):
Time t
T
Time t+1
T
MI
T cell agent and
macrophage agent in
infected state
M.state = infected
MA
Macrophage agent changes
to activated state
M.state = activated
ABM: preliminary results
Macrophages
Bacteria
Goals
 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…