Modeling of Tumor Induced Angiogenesis II Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis & Nathaniel Whitaker

Bio Recap Angiogenesis: The process of formation of capillary sprouts in response to external chemical stimuli which leads to the formation of blood vessels. Angiogenesis: The process of formation of capillary sprouts in response to external chemical stimuli which leads to the formation of blood vessels. Tumor Angiogenic Factors (TAFs): Stimuli secreted by Tumors Tumor Angiogenic Factors (TAFs): Stimuli secreted by Tumors Extra Cellular Matrix (ECM): The area in which cells interact with the Fibronectin(F). Extra Cellular Matrix (ECM): The area in which cells interact with the Fibronectin(F). Proteases (P): Secreted by tumor to attract cells and destroy Inhibitors. Promotes Angiogenesis. Proteases (P): Secreted by tumor to attract cells and destroy Inhibitors. Promotes Angiogenesis. Inhibitors: Prevent Cells from getting to tumor. Generated by fibronectin cells in the ECM to inactivate proteases. Inhibitors: Prevent Cells from getting to tumor. Generated by fibronectin cells in the ECM to inactivate proteases.

5 “Species” Dynamical Evolution Model (1 Dimension) (1) C t = D c ΔC – ∂/∂x(f F * ∂F/∂x) (1) C t = D c ΔC – ∂/∂x(f F * ∂F/∂x) - ∂/∂x(f T * ∂T/∂x) + ∂/∂x(f I * ∂I/∂x) + k 1 C(1-C) (2) T = e (-(x-L) ² /ε) (2) T = e (-(x-L) ² /ε) (3) F t = -k 2 PF (3) F t = -k 2 PF (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (5) I t = -k 3 PI (5) I t = -k 3 PI f T term represents chemotactic attraction of cells to tumor f F term represents haptotactic response to the Fibronectin f I term represents the “repulsive” effect of inhibitor gradients D c = Diffusion Coefficient f F = a 1 C f T = a 2 C/(1 + a 3 T) f I = a 4 C

After Discretization We Get… C (n, k+1) = P r C (n-1, k) + P s C (n,k) + P l C (n+1, k) C (n, k+1) = P r C (n-1, k) + P s C (n,k) + P l C (n+1, k) F (n, k+1) = F (n,k) *(1 – Δt k 2 P (n,k) ) F (n, k+1) = F (n,k) *(1 – Δt k 2 P (n,k) ) P (n, k+1) = P (n, k) (1 – Δt k 6 – Δt k 3 I (n,k) P (n, k+1) = P (n, k) (1 – Δt k 6 – Δt k 3 I (n,k) + T (n,k) (Δt k 4 C (n,k) + Δt k 5 ) I (n, k+1) = I (n,k) (1 – Δt k 3 P (n,k) ) I (n, k+1) = I (n,k) (1 – Δt k 3 P (n,k) ) T = e -(x – L)²/ε (constant) T = e -(x – L)²/ε (constant)

1 - D results Near Tumor Far from Tumor No inhibitor

Adding an Inhibitor Near tumorFar from tumor weak inhibitor

Another Inhibitor Near tumorFar from tumor Strong Inhibitor

Replenished Inhibitor Examples Near tumor Far from tumor Weak Inhibitor

Replenished cont… Near Tumor Far from tumor Strong Inhibitor

5 Species Dynamic Evolution 2 Dimensional Model (1) C t = D c ΔC – (f F * F) - (f T * T) (1) C t = D c ΔC – (f F * F) - (f T * T) + (f I * I) + k 1 C(1-C) (2) T = e (-(x-L) ² /ε) (2) T = e (-(x-L) ² /ε) (3) F t = -k 2 PF (3) F t = -k 2 PF (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (4) P t = -k 3 PI + k 4 TC + k 5 T – k 6 P (5) I t = -k 3 PI (5) I t = -k 3 PI

After Discretization (2 Dimensions)… C (n, m, k+1) = P r C (n-1, m, k) + P l C (n+1, m, k) C (n, m, k+1) = P r C (n-1, m, k) + P l C (n+1, m, k) + P s C (n, m, k) + P u C (n, m-1, k) + P d C (n, m+1, k) F (n, m, k+1) = F (n, m, k) *(1 – Δt k 2 P (n, m, k) ) F (n, m, k+1) = F (n, m, k) *(1 – Δt k 2 P (n, m, k) ) P (n, m, k+1) = P (n, m, k) (1 – Δt k 6 – Δt k 3 I (n, m, k) P (n, m, k+1) = P (n, m, k) (1 – Δt k 6 – Δt k 3 I (n, m, k) + T (n, m, k) (Δt k 4 C (n, m, k) + Δt k 5 ) I (n, m, k+1) = I (n, m, k) (1 – Δt k 3 P (n, m, k) ) I (n, m, k+1) = I (n, m, k) (1 – Δt k 3 P (n, m, k) ) T = e -[(x – L)² + (y-L) ²]/ε (constant) T = e -[(x – L)² + (y-L) ²]/ε (constant)

2 – D Results Near Tumor – No Inhibitor

Far from Tumor – No Inhibitor

Near Tumor – Weak inhibitor

Far from Tumor – Weak Inhibitor

Angiogenesis in the Cornea ∂C/∂t = DΔC - k C – u L C ∂C/∂t = DΔC - k C – u L C D = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) D = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) k = rate constant of inactivation u = rate constant of uptake k = rate constant of inactivation u = rate constant of uptake L = total vessel length per unit area ΔC = ∂²C/∂x² + ∂²C/∂y² L = total vessel length per unit area ΔC = ∂²C/∂x² + ∂²C/∂y² f(C) = f(C) = C t = Threshold Concentration α = constant that controls shape of the curve C t = Threshold Concentration α = constant that controls shape of the curve n = S max f(C) Δl Δt n = S max f(C) Δl Δt (probability for the formation of 1 sprout from a vessel segment) (probability for the formation of 1 sprout from a vessel segment) S max = rate constant that determines max probability of sprout formation S max = rate constant that determines max probability of sprout formation 0, 0 ≤ C ≤ C t 1 – e -α(C – C t ), Ct ≤ C

Sprout Growth = P + (1-P) E = direction of growth in previous time step E = direction of growth in previous time step G = Direction of concentration gradient of TAF G = Direction of concentration gradient of TAF P = Persistance ratio P = Persistance ratio Δl = V max f(C) Δt(Length increase of sprouts) Δl = V max f(C) Δt(Length increase of sprouts) V max = maximum rate of length increase V max = maximum rate of length increase E x T E xo T G xo T cos θ sin θ E y E yo G yo -sin θ cos θ

Cornea Graphs

Progress & Goals 1-Dimensional Model with “random walker cells” 2-Dimensional Model of Angiogenesis Modeling Angiogenesis in the Cornea (ignoring inhibitors) – In Progress Angiogenesis in the Cornea with Inhibitors and perhaps other factors