Modeling of Tumor Induced Angiogenesis III

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Modeling of Tumor Induced AngiogenesisIII

• Heather Harrington, Marc Maier
• Lé Santha Naidoo
• Faculty Advisors Panayotis Kevrekidis, Nathaniel
  Whitaker, Deborah Good

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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.

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Angiogenesis in the CorneaMathematical Model

• ?C/?t Dc?C - 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 ?C
  ?²C/?x² ?²C/?y²
• f(C)
• Ct Threshold Concentration a constant
  that controls shape of the curve
• ?I/?t DI?I - kI I C
• DI Diffusion Coefficient
• C Tumor Angiogenic Factors (TAF)
• ?I ?²I/?x² ?²I/?y²
• kI rate constant of Inhibitor depletion
  influenced by the TAF
• f(I)
• It Threshold Concentration a constant
  that controls shape of the curve

0, 0 C Ct 1 e-a(C Ct), Ct C
0, 0 I It 1 e-a(I It), It I
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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
• ?l Vmax f(C) f(I) ?t (Length increase
  of sprouts)
• Vmax maximum rate of length increase

Ex T Exo T
Gxo T Ixo T cos ?
sin ? Ey Eyo Gyo
Iyo -sin ? cos ?
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Probability of Branching

• n Smax f(C) ?l ?t
• Represents positive effect TAF has on branching.
• m - Smax f(I) ?l ?t
• Represents negative effect the Inhibitor has on
  branching.
• Smax rate constant that determines max
  probability of sprout formation.
• ?l the total vessel length
• Combined Probability max (n m, 0)

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Cornea Graphs
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Cornea without Inhibitor
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Cornea with Circumscribed Inhibitor
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Cornea with Geometric Inhibitor
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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