A mathematical model of necrotizing enterocolitis

Jared BarberDepartment of Mathematics, University of Pittsburgh

Work with Ivan Yotov and Mark TronzoMay 10, 2011

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Outline

• Background• Necrotizing Enterocolitis Model• Results• Summary and future directions

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Necrotizing Enterocolitis

• Disease characterized by wounds in intestinal wall of premature infants

• Mortality: 2% of premature infants die• Initiation of wounds

– Ischemia/trauma• Worsening of wounds

– Bacterial invasion• Communal bacteria: 103-1012 per mL

– Elevated immune response• What’s causing the damage/how can we make things

better?3/24

Normal Intestine

• Lumen• Epithelial Cells

• Tissue• Blood

Available: http://www.pearsonsuccessnet.com/snpapp/iText/products/0-13-115075-8/text/chapter29/concept29.2.html

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Simplified System-Flattened intestine

0

0.5

1

21

0

21

0

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Computational Domain-

21

0

21

0

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Distribution of macrophages

Available: http://www.foodmuseum.com/exgutparts.html

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Simplified System-Four Layers

• Lumen– Holds commensal

bacteria

• Epithelial layer– Barrier-keeps out bacteria

• Tissue– Intestinal

support/structure• Blood/Mesentery

– Transport to/from rest of body

0

0.5

1

21

0

21

0

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Computational Domain-

21

0

21

0

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Distribution of macrophages

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Simplified System-Players in the disease

• Epithelial cells• ZO1 gap/tight junction

protein• Bacteria• Macrophages

• Activated macrophages• Damage• Activated Neutrophil• Cytokines• Nitric Oxide

0

0.5

1

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Computational Domain-

21

0

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Distribution of macrophages

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Inflammatory Cascade

N a

d

m a

NO

b

c a

c

ZO 1

e c

In flam m ato ryR esp o n se

E p ith e lia l B a rrie rP e rm eab ility

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Partial Differential Equations

• Epithelial Layer

– Diffusion enhanced by• More epithelial cells• Fewer bacteria

– Proliferation– Death enhanced by more activated neutrophils,

pro-inflammatory cytokines, and bacteria

caacccpccc ebcnkeeekebet

e),,()/1()()( max,

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Partial Differential Equations

• Bacteria

• Diffusion• Proliferation• Death by:

– “Passive” local innate immune response– Activated killing by immune cells

)(/1

1max

bNkbmkb

bk

b

bkbD

t

babNaab

bbgb a

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Partial Differential Equations

• Activated Macrophages

• Diffusion• Chemotaxis up gradients of bacteria and

cytokines• Natural death• Recruitment via bacteria, cytokine, and

damage interactions with macrophages

)()( dmkcmkbmkmkcmbmmDt

mdmcmbmamaacabam

aa

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Vertical Diffusion-Epithelial layer permeability

• ZO1 levels (epithelial barrier effectiveness)– Are low when number of epithelial cells are low– Decrease when epithelial cells die– Decrease in the presence of nitric oxide

• Vertical diffusion coefficient modified:

5.1max

5.1max

5.1max

,0,,, 1)11(

)11()(2

ZOZOZO

ZOZODDDD baselinezzbaselinezeffectivez

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Numerics

• Use a cell-centered finite difference approach• Periodic boundary conditions in x and y• Top and bottom, no flux boundary conditions• Initial conditions, zero except

– Constant source of neutrophils from blood– Macrophages nonzero except in lumen and blood– Epithelial layer held at 1 everywhere except in the

wound as is ZO1

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Partial Circular Wound

• Epithelial Cell Integrity in Epithelial Layer 0

0.5

1

21

0

21

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Computational Domain-

21

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Distribution of macrophages

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Circular Hole: Partial Wound

• Dynamics in epithelial layer for healthy case

0

6

x 10-7

d Epith

0.664

0.666

m Epith

0 100 200 300

0

20x 10

-18

na Epith

0

8

x 10-3

b Epith

0

4x 10

-3

m

a Epith

0 100 200 300

0.9

1

ec Epith

0

6

x 10-4

c Epith

0

8

x 10-4

NO Epith

0 100 200 300

0.9

1

ZO1 Epith

Time in hours15/24

Circular Hole: Complete Wound

• Increasing wound severity should lead to more unhealthy outcomes

Epithelial Cell Integrity in Epithelial Layer

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Circular Hole: Complete Wound

0

0.2

d Epith

0.6

0.65

m Epith

0 200 400 600 800

0

3x 10

-3

na Epith

0

0.4

b Epith

0

0.1

ma Epith

0 200 400 600 800

0.8

1

e

c Epith

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20x 10

-3

c Epith

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4

x 10-3

NO Epith

0 200 400 600 800

0.8

1

ZO1 Epith

• Dynamics in epithelial layer for healthy case

Time in hours17/24

Circular Hole: Complete Hole with Breast-feeding

• Add in anti-inflammatory cytokines and anti-microbial peptides

0

15

x 10-5

d Epith

0.66

0.665

m Epith

0

2

x 10-11

na Epith

0 200 400 600 800

0

0.1

ca Epith

0

10

x 10-3

b Epith

0

10x 10

-3

ma Epith

0 200 400 600 8000.8

1

ec Epith

0

15x 10

-4

c Epith

0

10x 10

-4

NO Epith

0 200 400 600 8000.8

1

ZO1 Epith

Time in hours

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Irregular shape (formula fed)

• Same area as large circular wound with same ICs inside and outside wound

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Formula Fed Irregular (dashed) vs Circular Wound (dotted)

• Irregular hole is more healthy

0

0.2

d Epith

0.6

0.65

m Epith

0 200 400 600 800

0

3x 10

-3

na Epith

0

0.4

b Epith

0

0.1

ma Epith

0 200 400 600 800

0.8

1

e

c Epith

0

20x 10

-3

c Epith

0

4

x 10-3

NO Epith

0 200 400 600 800

0.8

1

ZO1 Epith

Time in hours 20/24

Irregular vs Circular Hole:Considering different wound severities

• Initial epithelial integrity: 12.2%

• Note: Reason for reopening—damage-activated macrophage positive feedback loop in tissue 21/24

Irregular vs Circular Hole:Considering different wound severities

• Healthy outcomes– For circular wounds if epithelial IC > 12%– For irregular wounds if epithelial IC > 11%

• In some cases (<1%) irregular wounds may be healthy when circular wounds do not

• PDE model can capture some dynamics that ODE models cannot

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Summary

• Model produces physiologically consistent results– Hole severity– Breast feeding

• Model predicts small but significant dependence of outcomes on shape of wound

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Future Directions

• Finer scale features– Villi– Vasculature

• Model calibration—Parameter estimation• Additional species/layers

– Different types of epithelial cells– Mucous layer– TLR4 receptor activity

• Investigate effects of these changes on the pathogenesis of NEC

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Thank you!

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Physiological consistent results

• Both healthy and unhealthy results possible• Healthiness of outcome increases when

– Wound severity decreases– Breast feeding is used instead of formula feeding

• Do results depend on shape of wound?

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Irregular vs Circular Wounds:Reason for differences

• Why are irregular wound simulations healthier?– Wound closure governed by diffusion– Less distance to travel, more quickly the wound

closes, and the more healthy the situation• Qualitatively, outcomes are the same.• Are there any situations where the outcome

of the simulation (healthy vs unhealthy) depends on the shape of the wound?

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