1. Modeling of Tumor Induced Angiogenesis III Heather Harrington, Marc Maier & Lé Santha Naidoo Faculty Advisors: Panayotis Kevrekidis, Nathaniel Whitaker, Deborah Good

2. Angiogenesis in the Cornea Biological Terminology 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 Inhibitors: Prevent vessels from getting to tumor. They are given off by the body and can be injected to prevent capillary growth toward the tumor. Inhibitors: Prevent vessels from getting to tumor. They are given off by the body and can be injected to prevent capillary growth toward the tumor. Anastomosis: The termination of vessel formation upon intersection with a pre-existing vessel. Anastomosis: The termination of vessel formation upon intersection with a pre-existing vessel. Branching: The generation of new capillary sprouts from the tip of a pre- existing vessel. Branching: The generation of new capillary sprouts from the tip of a pre- existing vessel.

3. Angiogenesis in the Cornea Mathematical Model ∂C/∂t = D c ΔC - d C – u L C ∂C/∂t = D c ΔC - d C – u L C D c = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) D c = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) d = rate constant of inactivation u = rate constant of uptake d = 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 ∂I/∂t = D I ΔI - k I I C ∂I/∂t = D I ΔI - k I I C D I = Diffusion Coefficient D I = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) C = Tumor Angiogenic Factors (TAF) ΔI = ∂²I/∂x² + ∂²I/∂y² ΔI = ∂²I/∂x² + ∂²I/∂y² k I = rate constant of Inhibitor depletion influenced by the TAF k I = rate constant of Inhibitor depletion influenced by the TAF f(I) = f(I) = I t = Threshold Concentration α = constant that controls shape of the curve I t = Threshold Concentration α = constant that controls shape of the curve 0, 0 ≤ C ≤ C t 1 – e -α(C – Ct),Ct ≤ C 0, 0 ≤ I ≤ I t 1 – e -α(I – It),I t ≤ I

4. Sprout Growth Direction and Length of growth = P + (1-P)/2 * f(C) - (1-P)/2 * f(I) = P + (1-P)/2 * f(C) - (1-P)/2 * f(I) E xo, E yo = Direction of growth in previous time step E xo, E yo = Direction of growth in previous time step E x, E y = Direction of growth in current time step E x, E y = Direction of growth in current time step G = Direction of concentration gradient of TAF G = Direction of concentration gradient of TAF I = Direction of concentration gradient of the Inhibitor I = Direction of concentration gradient of the Inhibitor Threshold functions give relative weights to TAF and Inhibitor vectors Threshold functions give relative weights to TAF and Inhibitor vectors P = Persistance ratio P = Persistance ratio Δl = V max * |f(C) – f(I)| * Δt(Length increase of sprouts) Δl = V max * |f(C) – f(I)| * 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 I xo T cos θ sin θ E y E yo G yo I yo -sin θ cos θ

5. Probability of Branching n = S max f(C) Δl Δt n = S max f(C) Δl Δt Represents positive effect TAF has on branching. Represents positive effect TAF has on branching. m = - S max f(I) Δl Δt m = - S max f(I) Δl Δt Represents negative effect the Inhibitor has on branching. Represents negative effect the Inhibitor has on branching. S max = rate constant that determines max probability of sprout formation. S max = rate constant that determines max probability of sprout formation. Δl = the total vessel length Δl = the total vessel length Combined Probability: max (n + m, 0) Combined Probability: max (n + m, 0)

6. Cornea Graphs

7. Cornea without Inhibitor

8. Cornea with Circumscribed Inhibitor

9. Cornea with Geometric Inhibitor

10. Progress & Goals 1-Dimensional Model with “random walker cells” 1-Dimensional Model with “random walker cells” 2-Dimensional Model of Angiogenesis 2-Dimensional Model of Angiogenesis Modeling Angiogenesis in the Cornea (absence of and constant inhibitors) Modeling Angiogenesis in the Cornea (absence of and constant inhibitors) Angiogenesis in the Cornea with dynamic Inhibitors – In Progress Angiogenesis in the Cornea with dynamic Inhibitors – In Progress