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

Biological Background Important terms 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). Haptotaxis: The attraction of cells to ECM. Haptotaxis: The attraction of cells to ECM. 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.

More Bio Cells: Diffuse in the ECM, get generated and die. They are driven to the TAF gradient (chemotaxis), as well as the ECM concentration gradient and are “repelled” from inhibitor gradients Cells: Diffuse in the ECM, get generated and die. They are driven to the TAF gradient (chemotaxis), as well as the ECM concentration gradient and are “repelled” from inhibitor gradients TAFs: We assume TAF gradient is fixed TAFs: We assume TAF gradient is fixed ECM: Secretion of Proteases by the tumor gradually degrades the ECM; proteases involved in the angiogenic process bind to the Fibronectin depleting it allowing the cells to be driven to the tumor Proteases: Neutralized by inhibitors Inhibitor: Mutually Neutralizes with Protease.

5 “Species” Dynamical Evolution Model (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

What’s Next? 1-Dimensional Model with “random walker cells” 1-Dimensional Model with “random walker cells” 2-Dimensional Model of Angiogenesis 2-Dimensional Model of Angiogenesis Modelling Angiogenesis in the Cornea (ignoring inhibitors) Modelling Angiogenesis in the Cornea (ignoring inhibitors) (If Time Permits) Angiogenesis in the Cornea with Inhibitors (If Time Permits) Angiogenesis in the Cornea with Inhibitors