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Modeling of Tumor Induced Angiogenesis III

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

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Modeling of Tumor Induced Angiogenesis III

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

  2. Angiogenesis in the CorneaBiological Terminology • 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 • 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. • Branching: The generation of new capillary sprouts from the tip of a pre-existing vessel.

  3. Angiogenesis in the CorneaMathematical Model • ∂C/∂t = DcΔC - d C – u L C • Dc = Diffusion Coefficient C = Tumor Angiogenic Factors (TAF) • d = rate constant of inactivation u = rate constant of uptake • L = total vessel length per unit area ΔC = ∂²C/∂x² + ∂²C/∂y² • f(C) = • Ct = Threshold Concentration α = constant that controls shape of the curve • ∂I/∂t = DIΔI - kI I C • DI = Diffusion Coefficient • C = Tumor Angiogenic Factors (TAF) • ΔI = ∂²I/∂x² + ∂²I/∂y² • kI = rate constant of Inhibitor depletion influenced by the TAF • f(I) = • It = Threshold Concentration α = constant that controls shape of the curve 0, 0 ≤ C ≤ Ct 1 – e-α(C – Ct), Ct ≤ C 0, 0 ≤ I ≤ It 1 – e-α(I – It), It ≤ I

  4. Sprout Growth Direction and Length of growth • = P + (1-P)/2 * f(C) - (1-P)/2 * f(I) • Exo, Eyo = Direction of growth in previous time step • Ex, Ey = Direction of growth in current time step • G = Direction of concentration gradient of TAF • I = Direction of concentration gradient of the Inhibitor • Threshold functions give relative weights to TAF and Inhibitor vectors • P = Persistance ratio • Δl = Vmax * |f(C) – f(I)| * Δt (Length increase of sprouts) • Vmax = maximum rate of length increase ExT ExoT Gxo T IxoT cos θ sin θ Ey Eyo Gyo Iyo -sin θ cos θ

  5. Probability of Branching • n = Smax f(C) Δl Δt • Represents positive effect TAF has on branching. • m = - Smax f(I) Δl Δt • Represents negative effect the Inhibitor has on branching. • Smax = rate constant that determines max probability of sprout formation. • Δl = the total vessel length • 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” • 2-Dimensional Model of Angiogenesis • Modeling Angiogenesis in the Cornea (absence of and constant inhibitors) • Angiogenesis in the Cornea with dynamic Inhibitors – In Progress

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