1. Nathaniel Whitaker Modeling Tumor Induced Angiogenesis University of Massachusetts Amherst

2. Modeling of Tumor Induced Angiogenesis Collaborators: P. Kevrekidis, D.Good, G.Herring, H.Harrington, M. Maier, L. Naidoo, Rong Shao

4. 5 Species Diagram

5. Good et al PNAS

6. Anderson and Chaplain(1998) Kevrekidis, Whitaker, Good(2004) Kevrekidis, Whitaker, Good, Herring(2006) Stokes and Lauffenburger(1991) Levine et al (2002) Anderson (2005)

8. 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) 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) + 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) ) T = e -(x – L)²/ε (constant)

9. Parameters of Terms DC=.00035, DP=1, k2=k3=k4=.1 K5=.2, fF=a1*c, fT=a4*c, fI=a2*c/(1+a3*T) Epsilon=.45 Relevant length scale=2mm Relevant time scale 1.5 days

10. 1-D without Inhibitor

11. 1-D with Inhibitor

14. 2-D without inhibitor

15. 2-D Statistics without Inhibitor 25 Runs using same initial conditions Average arrival time of 6.6 Standard deviation 1.6186 Average length 3.25 mm(Grid size 10 m m)

16. 2-D with Inhibitor in between

17. 2-D Statistics with Spot Inhibitor 10 Runs using same initial conditions Average arrival time of 33.08 Standard deviation 3.2291 Average length 9.38 mm(Grid size 10 m m) Fits Arc of circle of radius.49 centered at (.5,.25).

18. 2-D Inhibitor Ring

19. 2-D Statistics with Inhibitor Ring 10 Runs using same initial conditions Average arrival time of 35.08 Standard deviation.8550 Average length 8.84 mm(Grid size 10 m m) Inhibitor= sech(100*(r-.1)) where r is the distance from tumor.

20. Time =0

21. Time=10

22. Time=20

23. Time=35

24. 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. 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.

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

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

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

30. Cornea without Inhibitor

31. Cornea with Geometric Inhibitor

32. Initial Inhibitor in Cornea

33. Initial Inhibitor around tumor

34. Shao et al

46. Present and Future Work Presented 2 models for Angiogenesis with Inhibitors. Difficult to find coefficients. Dr Shao, at Baystate medical center, breast cancer research(experimentalist). Approximate coefficients for models experimentally, 2 species at a time Experiment HMVEC and VEGF, Human Microvascular endothelial cells and Vascular endothelial cell growth factor.

47. Conclusions and Future Work First model for angiogenesis incorporating inhibitors. PDE is interpreted as biased random walk Second model with cell motion derived at the particle level. Experiments to validate the model and determine correct coefficients.