Download Inflammation in the Lung

Jared BarberSeminar, Oct 4, 2011
Joint work with Ivan Yotov and Gilles Clermont


Background on pneumonia and inflammation
ODE Model
◦ Model
◦ Desired behaviors
◦ Results

PDE Model
◦ Model additions
◦ Results


Conclusions
Future Work



A condition where inflammation in the lung
compromises lung function
A leading cause of death in elderly, very
young, chronically ill, and third world
Caused by bacteria, virus, fungi, parasites
◦ Bacteria associated with most severe cases
◦ Flu can cause pneumonia


Associated with coughing, fever, chills, lack
of breath, confusion in elderly.
Treated with fluids, antibiotics, oxygen
therapy, breathing treatments

Players
◦ Pathogen-introduced via air
 Bacteria-b
◦ Immune cells
 Neutrophils-n
 Macrophages-m
m
◦ Cytokines
 Pro-inflammatory-cp
 Anti-inflammatory-ca

Process (to right)
cp
b
n
ca

Parameters chosen so that
◦ Healthy steady state is stable
◦ Neutrophils outkill/outnumber macrophages
◦ At maximal anti-inflammation levels, immune
response is reduced by 75-80%
◦ Pro and anti-inflammatory cytokine levels are of the
same order
◦ Anti-inflammatory cytokines delayed wrt proinflammatory cytokines

There are some rarer desired behaviors that
are not currently reproducible by the model
0.005
0
0
50
-4
2
x 10
100
150
cp
200
1

Bacterial infection is cleared by local immune
response without needing to activate
macrophages and neutrophils
0
0
50
100
150
ca
200
50
100
150
200
150
200
150
200
-5
5
x 10
0
0
-5
b
0.01
4
0.005
2
0
0
50
-4
2
x 10
100
150
cp
200
2
50
100
150
ca
200
50
100
200
-5
x 10
0
0
-5
2
0
50
100
n
x 10
1
0
0
4
x 10
-4
1
5
0
m
x 10
150
t in mhrs
0
0
50
100
t in hrs
0.1
0
0
50
b
2
0.1
1
0.05
0
50
100
150
cp
200
0
0.02
0.1
0.01
0
50
b
100
150
ca
200
0
0.1
0.2
1
0.05
0.1
50
100
150
cp
200
0
0.02
0.1
0.01
0
50
0.05
0.01
100
150
200
0
0
50
100
150
200

0
50
0
0
50
t in hrs
n
0.2
Note: We have all desired behaviors
0.1
50
100
150
m
0.02
200
t in hrs
0.1
0
100
150
ca
50
0
m
0.2
0
0
50
200
0
0
50
100
150
100
150
200
100
150
200
150
200
n
2
0
0
m
0.2
0
0
200
Bacterial infection grows initially and then is
destroyed by activated immune cells which
subsequently decay to zero
0

100
150
ca
200
100
t in hrs
0.5
0
0
50
b
20
0.4
10
0.2
Bacterial infection is initially reduced but
recovers once anti-inflammatory cytokines
kick in
0

0
50
100
150
cp
200
0
0.1
0.5
0.05
0
50
b
100
150
ca
200
0
0.4
0.4
10
0.2
0.2
50
100
150
cp
200
0
0.1
0.5
0.05
0
50
100
150
ca
200
t in hrs
0.4
0.05
100
150
200
0
50
100
150
200
t in hrs
n
0.2
0
50
100
150
m
0.1
0
0.4
0.2
0
50
m
1
0
0
200
0
0
50
100
150
50
100
150
200
0
50
100
150
200
150
200
n
20
0
0
200
m
1
0
0
100
150
ca
200
0
0
50
100
t in hrs
0.2
0
0
50
b

20
0.4
10
0.2
100
150
ca
Introduction of additional bacteria later on
can turn a healthy situation (Simulation 2)
into an unhealthy one
0
0
50
100
150
cp
200
0
0
50
100
200
150
200
150
200
150
200
m
0.4
0.05
0.2
0
0
50
b
100
150
ca
200
0
0.4
0.2
10
0.2
0.1
0
50
100
150
cp
200
0
0
50
100
150
200
150
200
150
200
m
0.4
0.05
0.2
0
0
50
100
150
ca
200
0
0.2
0.2
0.1
0
50
100
150
m
0.05
50
100
n
0.4
0
0
200
0
0
50
100
50
100
n
20
0
0
0
0
50
100

Patients seen are usually Type II or Type III
We want O(Type II) ≈ O(Type III), not the case
2
Type III Region
1.8
bacteria initial condition

1.6
Type II Region
1.4
1.2
1
0.8
Type I Region
0
1
2
3
4
5
6
7
8
relative speed at which immune cells act
9
10

Diffusion
◦ All species
◦ Smaller species (cytokines) diffuse more than larger
species (inflammatory cells)

Chemotaxis
◦ Macrophages migrate towards regions of high
bacterial and cytokine concentration
◦ Neutrophils migrate towards regions of high
cytokine concentration

Lung made up of three components:
◦ Air/Alveolar region (A-90% of the lung)
◦ Blood (B-5% of the lung)
◦ Tissue (T-5% of the lung)


Inflammation indicator function
n( x, y, z )  kc p n c p ( x, y, z )
 ( x, y , z ) 
1  kca ca ( x, y, z )
Local saturation function
1  ST ,ref
ST ( x, y, z )  1 
1  mT  ( x, y, z )

Saturation for other components:
1  ST ,ref
ST ( x, y , z )  1 
1  mT  ( x, y, z )
 1  ST ( x, y , z ) 

S A ( x, y, z )  S A,ref 
 1 S

T
,
ref


 1  ST ( x, y , z ) 

S B ( x, y, z )  S B ,ref 
 1 S

T
,
ref



Effective diffusion/chemotaxis coefficients
depend on air, blood, and tissue saturation:
D( x, y, z )  DA S A ( x, y, z )  DB S B ( x, y, z )  DT ST ( x, y, z )
 ( x, y, z )   A S A ( x, y, z )   B S B ( x, y, z )   T ST ( x, y, z )

For macrophages:
Dm ( x, y, z )  0.00015  S A ( x, y, z )  0  S B ( x, y, z ) 
0.00045  ST ( x, y, z )

Bacterial infection is cleared and immune
system returns to original steady state
Note:
Actual
Comp
Domain
20x20 cm

Time: Each profile 2hrs apart
B
5
0.08
0.3
4
0.06
0.2
3
0.04
2
0
0.1
0.02
1
0
50
100
0
0
50
100
0
0
50
100
ST
0.05
0.6
0.04
1
0.4
0.03
0.02
0.5
0.2
0.01
0
0
50
100
0
0
50
100
0
0
50
t = 104
100
Note:
Actual
Comp
Domain
20x20 cm

Bacterial infection is not cleared and system
proceeds to death
Note:
Actual
Comp
Domain
20x20 cm
Bacterial infection is not cleared and system
proceeds to death
10
9
Type III Region PDEs
8
bacteria initial condition

Note:
Actual
Comp
Domain
20x20 cm
7
6
PDE Divider?
5
4
Type II Region PDEs
3
ODE Divider
2
1
0
1
2
3
4
5
6
7
relative speed at which immune cells act
8
9
10
Type I Region


Both models can produce desired behavior
PDE model allows more Type II simulations
◦ PDE system starts with less bacterial load
PDE
vs
ODE
◦ Diffusion lessens virility of bacterial growth
◦ Chemotaxis allows inflammatory cells to gang up
on the bacteria

PDE model gives much more flexibility

X-rays pick up mostly water

X-ray density = ST + SB

Use Kalman Filter to compare with actual data


Further refine ODE model to obtain more desired
behaviors
Consider including other members
◦ Damage
◦ Adaptive immune response

For chemotaxis and diffusion coefficients,
◦ Maximize number of physiologically realistic simulations
◦ Find conditions to limit pattern formation from occurring
◦ Use smaller initial size of infection

Obtain average values of neutrophils and
cytokines in addition to x-rays to use for
parameter estimation