A mathematical and computational model of necrotizing enterocolitis

Transcription

1 A mathematical and computational model of necrotizing enterocolitis Ivan Yotov Department of Mathematics, University of Pittsburgh McGowan Institute Scientific Retreat March 10-12, 2008 Acknowledgment: Gilles Clermont, Joshua Sullivan, and Yoram Vodovotz, University of Pittsburgh; Jeffrey Upperman, University of Southern California. Department of Mathematics, University of Pittsburgh 1

2 Problem Description NEC: lethal gastrointestinal disease in premature infants Risk factors: prematurity, hypoxia, formula feeding, bacterial infection Mathematical and computational model: predict the effect of various risk factors on the outcome of the disease Dynamic effects: inflammatory reactions Spatial effects: 4 compartments: lumen, gut epithelial layer, body tissue, blood epithelial cell migration, diffusion, chemotaxis Department of Mathematics, University of Pittsburgh 2

3 Department of Mathematics, University of Pittsburgh 3

4 System components e c - epithelial cells: barrier to infection b - bacteria: pathogen m - macrophage, N - neutrophil: stationary dormant defense system m a - activated macrophage, N a - activated neutrophil: immune defense c - pro-inflammatory cytokine: response promoter NO - nitric oxide: waste chemical that damages wall integrity ZO1 - tight junction protein: restricts passage through the wall d - damage: measure of tissue damage c a - anti-inflammatory cytokine: response inhibitor Department of Mathematics, University of Pittsburgh 4

5 Inflammatory reactions m + b m a + b m a c + m c + N N a + b N a e c e c NO + ZO1 d + m d + N k bm m a + b macrophage activation k ab m a + c macrophage bacteria destruction k m ac m a + c macrophage cytokine release k cm m a + NO macrophage reaction promotion k cn N a + N neutrophil reaction promotion k N ab N a + c neutrophil bacteria destruction k N ac N a + c neutrophil cytokine release k p 2e c proliferation of epi cells k a apoptosis of epi cells k nz N O tight junction destruction k dm m a macrophage activation from damage k dn N a neutrophil activation from damage Department of Mathematics, University of Pittsburgh 5

6 Spatial modeling x2 x1 Concentration - number of cells per unit volume For each spatial point x, b(x) is the concentration of bacteria at the point x. Diffusion Flux across: D b (b(left) b(right)) Conservation of mass: b(t + dt) b(t) dt flux(x + dx) flux(x) = dx x x + dx Department of Mathematics, University of Pittsburgh 6

7 Partial differential equations e c t + (β(e c)u(e c, b)) = k p e c (1 e c /e max c ) k a (b, c)e c, e 2 c β(e c ) = e 2 c + (e max c e c ) 2, u(e c, b) = α(b) e c b t D b b = k bg b(1 b/b max ) k b b/(1 + b/ɛ) R(c a )(k ab m a b + k Na bn a b) m t = k m0m 0 (1 m/m max ) k m m R(c a )(k bm bm + k cm cm + k dm dm) m a (D ma m a γ 0 m a c γ 1 m a b) t = k m m a + R(c a )(k bm bm + k cm cm + k dm dm) c t D c c = k c c + k ma cm a + k Na cn a +R(c a )(k ab m a b + k Na bn a b k cn cn k cm cm) c a t D Q c a c a = k ca c a + s c + k cnn 1 + Q Department of Mathematics, University of Pittsburgh 7

8 Partial differential equations, cont. NO = k no NO + k cm cm t ZO1 = k zo1 h(e c, e max c, 1/4)ZO1 0 (1 ZO1/ZO1 max ) k nz NO ZO1 t N a (D Na N a γ 2 N a c) = k Na N a + R(c a )(k cn cn + k dn dn) t d t D T q d d = k d (c a )d + k dn x q dn + T q γ 1 >> γ 0, T = R(c a ) = m a + N a 1 + k nc (c a / c a ) 2, Q = m a + N a + k cnd d 1 + k nc (c/ c a ) k nc (c a / c a ) 2, h(a, b, q) = a q a q + (b a) q Department of Mathematics, University of Pittsburgh 8

9 Model domain and initial conditions Lumen b = b 0 Epithelial layer m = m max m a = 0 N a = 0 e c = e c,0 b = 0 c = 0 NO = 0 ZO1 = ZO1 0 Tissue b = 0 c = 0 m = m max m a = 0 N a = 0 NO = 0 Blood b = 0 c = 0 m = m max m a = 0 N = N 0 N a = 0 NO = 0 Department of Mathematics, University of Pittsburgh 9

10 Some features of the model Diffusion coefficients are specified for each components and each layer. Vertical and horizontal diffusion may differ. The epithelial layer permeability depends on the amount of the tight junction protein ZO1. This is modeled by multiplying the vertical diffusion on the interface between the epithelial layer and the tissue by a permeability function, e.g., Db z Db z h(zo1 max ZO1, ZO1 max, 2). Blood/tissue barrier is generally very restrictive unless it is damaged: D z b D z b h(d d max, d max, 2). Epithelial cell migration is modeled via a conservation law with nonlinear flux β(e c )u(e c, b), u(e c, b) = α(b) e c. It is affected by the presence of LPS - endotoxin present in the lumen bacteria: α(b) = γ 3 h(b max b, b max, 1/2) Department of Mathematics, University of Pittsburgh 10

11 Computer simulator Matlab implementation Space-time finite difference discretization on 3D rectangular grids Rates, initial conditions, and other parameters are easy to manipulate Run time graphical user interface Department of Mathematics, University of Pittsburgh 11

12 Graphical user interface Department of Mathematics, University of Pittsburgh 12

13 Simulations: risk factors hypoxia and formula feed Hypoxia is modeled via an initial damage to the intestinal wall Initial condition: small wall damage Initial condition: large wall damage Breast milk effect is modeled via presence of anti-inflammatory cytokines Department of Mathematics, University of Pittsburgh 13

14 Small wall damage, breast milk feed: healthy outcome Department of Mathematics, University of Pittsburgh 14

15 Small wall damage, formula feed: sustained inflammation Department of Mathematics, University of Pittsburgh 15

16 Large wall damage, breast milk feed: approaches recovery Department of Mathematics, University of Pittsburgh 16

17 Large wall damage, formula feed: unhealthy outcome Department of Mathematics, University of Pittsburgh 17

18 Large damage, breast milk with reduced antibacterial effect Department of Mathematics, University of Pittsburgh 18