nathaniel whitaker modeling tumor induced angiogenesis university of massachusetts amherst

Nathaniel WhitakerNathaniel Whitaker

Modeling Tumor Induced Angiogenesis

University of Massachusetts

Amherst

Modeling of Tumor Induced Modeling of Tumor Induced AngiogenesisAngiogenesis

Collaborators: P. Kevrekidis, D.Good, G.Herring, H.Harrington, M. Maier, L. Naidoo , Rong Shao

5 Species Diagram5 Species Diagram

Good et alGood et alPNASPNAS

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

After Discretization We Get…After Discretization We Get…

C(n, k+1) = PrC(n-1, k) + PsC(n,k) + PlC(n+1, k)

F(n, k+1) = F(n,k)*(1 – Δt k2 P(n,k) )

P(n, k+1) = P(n, k) (1 – Δt k6 – Δt k3 I(n,k)

+ T(n,k) (Δt k4 C (n,k) + Δt k5)

I(n, k+1) = I(n,k) (1 – Δt k3 P(n,k) )

T = e-(x – L)²/ε (constant)

Parameters of TermsParameters of Terms

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

1-D without Inhibitor1-D without Inhibitor

1-D with Inhibitor1-D with Inhibitor

After Discretization (2 Dimensions)…

C(n, m, k+1) = Pr C(n-1, m, k) + Pl C(n+1, m, k) + Ps C(n, m, k) + Pu C(n, m-1, k) + Pd C(n, m+1, k)

F(n, m, k+1) = F(n, m, k)*(1 – Δt k2 P(n, m, k) )

P(n, m, k+1) = P(n, m, k) (1 – Δt k6 – Δt k3 I(n, m, k) + T(n, m, k) (Δt k4 C (n, m, k) + Δt k5)

I(n, m, k+1) = I(n, m, k) (1 – Δt k3 P(n, m, k) )

T = e-[(x – L)² + (y-L) ²]/ε (constant)

2-D without inhibitor2-D without inhibitor

2-D Statistics without Inhibitor2-D Statistics without Inhibitor

25 Runs using same initial conditionsAverage arrival time of 6.6Standard deviation 1.6186Average length 3.25 mm(Grid size 10 m m)

2-D with Inhibitor in between2-D with Inhibitor in between

2-D Statistics with Spot 2-D Statistics with Spot InhibitorInhibitor

10 Runs using same initial conditionsAverage arrival time of 33.08Standard deviation 3.2291Average length 9.38 mm(Grid size 10 m m) Fits Arc of circle of radius .49

centered at (.5, .25).

2-D Inhibitor Ring2-D Inhibitor Ring

2-D Statistics with Inhibitor 2-D Statistics with Inhibitor RingRing

10 Runs using same initial conditionsAverage arrival time of 35.08Standard deviation .8550Average length 8.84 mm(Grid size 10 m m)Inhibitor= sech(100*(r-.1)) where r

is the distance from tumor.

Time =0Time =0

Time=10Time=10

Time=20Time=20

Time=35Time=35

Angiogenesis in the CorneaAngiogenesis in the CorneaBiological TerminologyBiological 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.

Angiogenesis: Cornea(Tong Angiogenesis: Cornea(Tong &Yuan)&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) =

– Ct = Threshold Concentration α = constant that controls shape of the curve

n = Smax f(C) Δl Δt– (probability for the formation of 1 sprout from a vessel segment)

– Smax = rate constant that determines max probability of sprout formation

0, 0 ≤ C ≤ Ct

1 – e-α(C – Ct), Ct ≤ C

Angiogenesis in the CorneaAngiogenesis in the CorneaMathematical ModelMathematical 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

Probability of BranchingProbability 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)

Cornea without InhibitorCornea without Inhibitor

Cornea with Geometric Cornea with Geometric InhibitorInhibitor

Initial Inhibitor in CorneaInitial Inhibitor in Cornea

Initial Inhibitor around tumorInitial Inhibitor around tumor

ShaoShaoet et alal

Present and Future WorkPresent and Future Work

Presented 2 models for Angiogenesis withInhibitors. Difficult to find coefficients.Dr Shao, at Baystate medical center, breast

cancer research(experimentalist).Approximate coefficients for models

experimentally, 2 species at a timeExperiment HMVEC and VEGF, Human

Microvascular endothelial cells and Vascular endothelial cell growth factor.

Conclusions and Future WorkConclusions and Future Work

First model for angiogenesis incorporating inhibitors.

PDE is interpreted as biased random walkSecond model with cell motion derived at the

particle level.Experiments to validate the model and

determine correct coefficients.