ABSTRACT
REFERENCES
This work presents a new mathematical model
for tumor-induced angiogenesis. We construct
a continuous mathematical model based on a
system of partial differential equations that
describe the influences that a tumor, as well as
an inhibitor, have on the growth of blood
vessels in the cornea. This is a twodimensional model that incorporates the
diffusion and uptake of tumor angiogenic
factors (the chemical stimuli secreted by
tumors to attract cells), randomness in the rate
and distance of cell growth, anastomosis (the
termination of vessel formation upon
intersection with a pre-existing vessel), and
the presence of an inhibitor which influences
cell growth in the opposite direction of the
tumor. The particular novelty of this model
hinges on the inclusion of the inhibitor effects
and on the systematic delineation of their
importance. We have developed a simulation
of this biological process using MATLAB,
which validates our findings qualitatively by
means of comparison with previous
experimental work. We have performed a
sensitivity analysis on several parameters,
including inhibitor strengths and diffusion
rates. We first establish baseline values for all
the parameters, in accordance with available
experimental data, and subsequently vary the
relevant values motivated by experimental
assays.
Baseline Values:
TAF
concentration
Inhibitor
concentration
DC
DI
Kon
k
Ct
1M
Smax
10 M
Vmax
25×10-4 µm1 -1
h
20 µmh-1
5×10-6 cm2s-1
10-6 cm2s-1
207 M-1h-1
2.89×10-2 h-1
.001 M
∆t
P
u
α
It
2.78×10-4 h
0.8
2000 µmh-1
10
.001 M
Good, Deborah J. et al., “A tumor suppressordependent inhibitor of angiogenesis is
immunologically and functionally
indistinguishable from a fragment of
thrombospondin,” Proc. Natl. Acad. Sci.
USA 87, 5524 (1990).
Kevrekidis, Panayotis G., Nathaniel Whitaker and
Deborah J. Good, “A Minimal Model
for Tumor Angiogenesis,” submitted
to Phys. Rev. E (2004).
Kevrekidis, Panayotis G., Nathaniel Whitaker and
Deborah J. Good, “Towards A
Reduced Model For Angiogenesis: A
Hybrid Approach,” Math. Comp.
Model, in press (2004).
Mathematical
Biology:
Modeling
Tumor-Induced
Angiogenesis
in the Cornea
Tong, Sheng and Fan Yuan. “Numerical
Simulations of Angiogenesis in the
Cornea,”Microvascular Research
61.1(2001):14-27.
CONTACT INFORMATION
Department of Mathematics & Statistics
Lederle Graduate Research Tower
Box 34515
University of Massachusetts Amherst
Amherst, MA 01002-9305, USA
Math Department Phone: (413) 545- 2762
Heather Harrington
[email protected]
Marc Maier
[email protected]
Lé Santha Naidoo
[email protected]
Faculty Advisors:
Panayotis Kevrekidis
[email protected]
Nathaniel Whitaker
[email protected]
University of Massachusetts
Amherst
Heather Harrington
Marc Maier
Lé Santha Naidoo
Faculty Advisors:
Panayotis G. Kevrekidis
Nathaniel Whitaker
January 7th, 2005
BIOLOGY BACKGROUND
Angiogenesis: The formation of blood vessels
from a pre-existing vasculature in response to
chemical stimuli.
EQUATIONS
RESULTS AND SENSITIVITY
ANALYSIS
TAF Gradient
∂C
= DC ∆C − k C − uLC − kon IC
∂t
Tumor Angiogenic Factors (TAFs):
Chemicals secreted by the tumor to attract the
blood vessels (C).
Inhibitor: A chemical that repels cell growth
from the direction of its gradient (I).
Inhibitor Gradient
∂I
= Dt ∆I − k IC
∂t
Dc= Diffusion Coefficient DI = Diffusion Coefficient
u = rate constant of uptake
k = rate constant of inactivation
L = total vessel length per unit area
C = Tumor Angiogenic Factors (TAF)
∂ 2C ∂ 2C
∂ I ∂ I
∆C = 2 + 2
∆I =
+
∂x
∂y
∂x
∂y
kon = rate of Inhibitor depletion influenced by the TAF
No inhibitor. Vessels grow directly toward
the tumor influenced by the TAF gradient.
2
2
2
2
TAF Threshold Function Inhibitor Threshold Function
0 ≤ I ≤ It
0
0 ≤ C ≤ Ct
0
f (I ) = 
f (C ) = 
−α ( I − I t )
−α ( C − C t )
It ≤ I
Ct ≤ C
1 − e
1 − e
Baseline values with inclusion of an
inhibitor. The inhibitor repels vessels away
from its gradient. The dynamic inhibitor
depletes and is periodically replenished;
therefore, at certain times the vessels can
traverse through the inhibitor.
Ct = Threshold Concentration of TAFs
It = Threshold Concentration of Inhibitor
α = constant that controls shape of the curve
Change in Vessel Length
∆l = Vmax |f(C) – f(I)| ∆t
Vmax = maximum rate of length increase
f(C) = TAF threshold function
∆t = time increment between steps
Limbal Vessel: A blood vessel acting as a
border or edge. In the eye, it is the junction
between the cornea and sclera of the eyeball.
Probability of Branching
n = Smax f(C)∆l ∆t
m= -Smax f(I) ∆l ∆t
Branching: The generation of new blood
vessels from the tip of a pre-existing vessel.
Anastomosis: The termination of vessel
formation upon intersection with a preexisting vessel.
Combined Probability
max (n + m, 0)
Smax = rate constant determines max probability
of sprouting
f(I) = Inhibitor threshold function
∆l = total vessel length
Represents positive effect TAF has and negative effect
inhibitor has on branching
Circumscribing Inhibitor (baseline). The
vessels are attracted to the tumor but avoid
areas covered by the inhibitor gradient.
Direction of Vessel Growth






T
T
T
I
 
T 

G




 x 
Ex    Exo 
o  −1− P / 2∗ f  I   xo    cosθ sinθ 
 + 1 − P / 2∗ f (C) 
 = P 




  I

E y    E y  

   − sinθ
cos
θ
G y 


y



 

o
o

o 



5 Times greater inhibitor strength. The
inhibitor is replenished to a greater strength.
The vessels are repelled to a greater distance
around the inhibitor.

Exo, Eyo = Direction of growth in previous time step
Ex, Ey = Direction of growth in current time step
Gxo, Gyo = Direction of concentration gradient of TAF
P = Persistence ratio
θ= Angle of random movement
Ixo, Iyo = Direction of concentration gradient of TAF