Nonlinear Analysis 63 (2005) e2237 – e2246
www.elsevier.com/locate/na
Nonlinear behaviors of capillary formation in a
deterministic angiogenesis model夡
Shuyu Suna,∗ , Mary F. Wheelera , Mandri Obeyesekereb, c ,
Charles Patrick Jr.b, c
a The Institute for Computational Engineering and Sciences, The University of Texas at Austin,
201 E. 24th Street, Austin, TX 78712, USA
b The University of Texas M. D. Anderson Cancer Center, Houston, TX 77030, USA
c The University of Texas Center for Biomedical Engineering, Houston, TX 77030, USA
Abstract
In this paper, we consider a deterministic approach for modeling angiogenesis. The model equations
form a nonlinear coupled system of partial and ordinary differential equations. We propose an efficient,
accurate and locally conservative numerical method to solve the nonlinear system. Computational
results indicate that the model generates the overall dendritic structure and pattern of the capillary
network morphologically similar to those observed in vivo. The influence of the capillary network
and the growth factor distribution on each other and their interaction are investigated using numerical
simulations.
䉷 2005 Elsevier Ltd. All rights reserved.
Keywords: Angiogenesis; Capillary network; Growth factors
1. Introduction
Angiogenesis, the outgrowth of new vessels from a pre-existing vasculature, plays an
important rule in many mammalian growth processes such as early embryogenesis during
the formation of the placenta [6], controlled blood-vessel formation during tissue repair
[4,8] and excessive blood-vessel formation during tumor growth [5]. A deep understanding
夡
Supported in part by a grant from the University of Texas Center for Biomedical Engineering.
∗ Corresponding author. Tel.: +1 512 232 7764; fax: +1 512 232 2445.
E-mail address: [email protected] (S. Sun).
0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.01.066
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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
of angiogenesis at the capillary level is critical for reparative strategies since the capillary
network dictates tissue survival, hemodynamics and mass transport. The angiogenic system
is strongly nonlinear and extremely complex, possessing multiple, integrated modulators
and feedback loops. The nonlinearity and complexity of the system limits the in vitro and in
vivo experiments that may be designed and the amount of nonconfounding information that
can be gleaned. A mathematical formulation describing the intercellular growth patterns
of capillaries within a tissue is essential for understanding and analyzing these complex
phenomena.
In this paper, we consider a deterministic approach for modeling angiogenesis proposed
in [9]. In this model, the anisotropy of extracellular matrix is reflected by the conductivity
of the extracellular matrix for the extension of capillary sprouts. Furthermore, the capillary
network is sharply captured by a capillary indicator function. The model equations form
a nonlinear coupled system of partial and ordinary differential equations. We propose an
efficient, accurate and locally conservative numerical method to solve the equations. The
influence of the capillary network and the growth factor distribution on each other and their
interaction are investigated using numerical simulations.
The remaining parts of this paper are organized as follows. In Section 2, we state the
governing equations of angiogenesis. An efficient and locally conservative numerical algorithm is established in Section 3. Angiogeneses from a single parent vessel and from
two parent vessels are simulated in Section 4, where dendritic and realistic structures of
capillary networks are observed. Finally, a brief summary is given in Section 5 to conclude
this paper.
2. The mathematical model
We restrict our attention to two spatial dimensions here. The capillary presence is represented by a indicator function n, which is a function of space (x, y) and time t, and has
only the value of either 0 or 1 depending on the presence of capillary. The concentration
of chemotactic growth factors (CGFs) is denoted by c. We track the individual behavior of
each sprout tip in our model. Each individual tip at time t is denoted by (xi (t), yi (t)), where
xi (t) and yi (t) are the x and y components of the position of the tip at time t. The collection
of all sprout tips at time t is denoted by a set S(t), i.e.
S(t) = {(xi (t), yi (t)) for all tips}.
We note that the set S might change with time due to sprout branching, extension, and
anastomosis discussed below. The capillary indicator function n is closely related to the
history of the positions of sprout tips. That is, the value of n is 1 on the trajectories of sprout
tips (the trails passed by all sprout tips) and is 0 elsewhere. Mathematically, it can be written
as
n = F (S),
and
1
F (S)(x, y, t) =
0
if (x, y) ∈
otherwise.
0 <t S(),
S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
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We
remark that the union of tip sets at all the time immediately before current time t,
0 <t S(), represents all the sprouts existing at time t because the sprouts are modeled
as the trajectory of tips here.
We assume that CGFs are released in the extracellular matrix in a constant rate . The
diffusivity of CGFs is denoted by D, a constant diagonal tensor, i.e.
D 0
D=
.
0 D
The consumption (binding) of CGFs by endothelial cells occurs only in the place where
n = 1 and its rate is assumed to be proportional to the concentration of CGFs. Thus the
consumption rate is nc, where is the consumption parameter. Proteins possess a natural
half-life time in the extracellular matrix and in the capillary network due to degradation by
proteases and other entities. We model the natural decay of CGFs by a rate of ∗ c. The mass
balance of CGFs gives the following equation:
jc
= ∇ · (D∇c) + (1 − n) − nc − ∗ c.
jt
(1)
We now focus our attention on the behaviors of tips caused by sprout branching, anastomosis and sprout extension. In our model, the generation of new sprouts from existing
sprout tip (branching) occurs if and only if the following two conditions are simultaneously
satisfied:
• The current sprout has an age greater than a certain threshold branching age a , that is,
new sprouts must mature for a length of time at least equal to a before they are able to
branch.
• The variation of the normalized velocity vector un (un will be defined below) on the
transverse direction of the sprout orientation is greater than a certain threshold value u .
After branching, one sprout tip becomes two tips with a distance of a single capillary
diameter dc . If the existing tip is s ∈ S, we denote the two new tips formed from the
existing tip by b1 (s) and b2 (s). Mathematically, the branching mechanism increases the
number of elements in the tip set S and can be written as
S(t + ) = B(S(t − )),
where the branching operator B(·) is defined as
B(S(t − )) = {b1 (s), b2 (s) : s ∈ B0 (S(t − ))} ∪ S(t − ) \ B0 (S(t − )),
and
B0 (S(t − )) = {s ∈ S(t − ) : age(s)a and |∇un (ps (t), t) · un (ps (t), t)⊥ | u }.
Here, S(t − ) is the set of sprout tips at time t − ( is an arbitrarily small positive number),
B0 (S(t − )) is the set of sprout tips at time t − that will undergo branching at time t, and
S(t + )=B(S(t − )) is the set of sprout tips at time t + immediately after the sprout branching
at time t has performed.
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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
Anastomosis, the fusion of capillary sprouts, is assumed to occur when a sprout tip joins
another sprout tip physically (tip-to-tip anastomosis) or a sprout tip meets another sprout
physically (tip-to-sprout anastomosis). After a tip-to-sprout anastomosis, the tip cell forms
a part of the loop, and will no longer undergo sprout extension, i.e. the tip no longer exists.
A tip-to-tip anastomosis might lead to disappearance of both tips or the nonexistence of one
tip only, depending upon the situation involved. In a “head-on-head” anastomosis, both tips
become inactive, whereas in a “shoulder-on-shoulder” anastomosis, only one of the two tips
become inactive. We can choose either one of the two tips arbitrarily without a loss in the
deterministic nature of the model because these two tips are in the same position and should
have exactly the same behavior in our model. Mathematically, anastomosis mechanism
decreases the number of elements in the tip set S and can be written as
S(t + ) = A(S(t − )),
where the anastomosis operator A(·) is defined as
A(S(t − )) = S(t − )\{s : s ∈ S(t − ) and n(xs , ys , t) = 1}
s1 , s2 : s1 ∈ S(t − ), s2 ∈ S(t − ), s1 = s2 , ps1 = ps2
dps1 dps2
·
<0
and
dt
dt
s1 : s1 ∈ S(t − ), s2 ∈ S(t − ), s1 = s2 , ps1 = ps2
dps1 dps2
and
·
0 .
dt
dt
Here ps denotes the tip position vector (xs , ys ) for tip s. In the above definition of the
anastomosis operator, the second term in the right-hand side of the equation represents the
tip-to-sprout anastomosis. The third and the fourth terms describe the “head-on-head” and
the “shoulder-on-shoulder” tip-to-tip anastomoses, respectively.
Sprout extensions play an important role in angiogenesis. We model sprout extension
using a trajectory tracking method. Let us consider a tip p = (xi , yi ) from the tip set S.
Under certain biological conditions, cells behind the sprout tips undergo mitosis and sprout
extension subsequently occurs. The position movement of an individual sprout tip during
proliferation depends on the direction and the speed of the sprout extension. We make the
usual assumptions [1,7] that the endothelial cells respond chemotactically to gradients of
CGFs. Using the theory of Darcy’s flow in heterogeneous and anisotropic porous media and
the trajectory tracking theory, we have the following governing equation for the position of
each sprout tip:
dp(t)
= up (xi , yi , t)
dt
with p(t) = (xi (t), yi (t)), ∀(xi (t), yi (t)) ∈ S(t),
up (x, y, t) = fp (un (x, y, t)),
(2)
(3)
S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
un (x, y, t) =
u(x, y, t)
,
|u(x, y, t)|
u(x, y, t) = K∇c,
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(4)
(5)
where up is the sprout extension velocity vector, un is the normalized velocity vector, u
is the auxiliary sprout extension velocity vector, fp (u) is the velocity modulator based on
the proliferation rate, and K is the conductivity of the extracellular matrix for the movement/extension of capillary cells. We remark that un is a well-defined quantity for all the
time and space because u will never be zero due to the CGFs consumption. We note that
only the direction information of u is used in the model. We take fp (un ) = kp un , where kp is
the proliferation parameter. We assume kp (c) = lc /tc (c) for c c∗ and kp (c) = 0 for c < c∗ ,
where lc is the endothelial cell length, tc is the cell cycle time, and c∗ is the minimum CGFs
concentration at which endothelial cells will proliferate. The cell cycle time tc is assumed
to be approximated by a function of c given as
tc = (1 + ec̄/c−1 ),
where is the limiting cell proliferation time of endothelial cells and c̄ is the concentration
related to the doubling of .
The conductivity K is an intrinsic property of the extracellular matrix and it is assumed
to be independent of time but varies in space. We consider K being a general second-order
tensor that may be heterogeneous and anisotropic. The heterogeneity of the extracellular
matrix is natural since different types of tissues impose different obstacles to the extension
of capillary sprouts. The anisotropy of the extracellular matrix means that the resistance of
the extracellular matrix for the sprout extension might be strong in some direction and weak
in other directions, which is also natural because tissues often have layered orientation. The
anisotropy of the extracellular matrix implies that the direction of sprout extension does not
necessarily coincide with the direction of the gradient of growth factors.
Combining all the above mechanisms, we have a system of governing equations:
dp
u
= kp (c)
, p = (xi , yi ), ∀(xi , yi ) ∈ S,
dt
|u|
u = K∇c,
n = F (S),
jc
= ∇ · (D∇c) + (1 − n) − nc − ∗ c,
jt
S(t + ) = A(B(S(t − ))).
This is a nonlinearly coupled system of partial differential equations (PDEs) and ordinary
differential equations (ODEs). We note that anastomosis and sprout branching are described
as sudden events here, whereas sprout extension is modeled as a continuous-in-time process.
Since the set S changes with time, the number of the unknowns of the system may vary with
time.
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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
3. Numerical algorithm
We partition the rectangular domain = (0, Lx ) × (0, Ly ) using a possibly nonuniform
x × y rectangular mesh, where x and y are the partitions in x and in y directions,
respectively, and defined below.
x : 0 = x0 < x1 < · · · < xM = Lx ,
y : 0 = y0 < y1 < · · · < yN = Ly .
Now we define
xi+1/2 =
yj +1/2 =
1
2
1
2
(xi + xi+1 ),
(yj + yj +1 )
for i = 0, 1, 2, . . . , M − 1 and j = 0, 1, 2, . . . , N − 1. The simulation time [0, T ] is divided
into K intervals by 0 = t0 < t1 < · · · < tK−1 < tK = T .
We use the forward Euler method to solve the trajectory equation for the position of an
individual sprout tip.
p(tk+1 ) − p(tk )
u(xs , ys , tk )
,
= kp
tk+1 − tk
|u(xs , ys , tk )|
(xs , ys ) ∈ S(tk ),
where p(tk )=(xs , ys ) is given at time tk+1 , kp is evaluated using the computed concentration
in time step tk and u is computed by replacing derivatives by finite differences in Eq. (5).
The diffusion–reaction equation for the concentration of CGFs, c, is solved by the implicit
cell center finite difference (CCFD) method. That is, the space is discretized using CCFDs
and the time is discretized using the backward Euler method. The implicit CCFD method
is used because it is efficient to implement, it possesses element-wise conservation locally,
and it is stable [3,2]. We approximate the concentration c by an element-wise constant
function, or it can be thought that the unknowns are at the center of each cell. We denote the
k
approximated solution of c(xi+1/2 , yj +1/2 , tk ) by Ci+1/2,j
+1/2 . For i = 1, 2, 3, . . . , M − 2
and j = 1, 2, 3, . . . , N − 2, the implicit CCFD method applied to the diffusion–reaction
equation for the CGFs concentration c is as follows:
k+1
k
Ci+1/2,j
+1/2 − Ci+1/2,j +1/2
t k+1 − t k
k+1
k+1
k+1
k+1
Ci+3/2,j
Ci+1/2,j
1
+1/2 −Ci+1/2,j +1/2
+1/2 − Ci−1/2,j +1/2
D
=
−D
xi+1 −xi
xi+3/2 − xi+1/2
xi+1/2 − xi−1/2
k+1
k+1
k+1
k+1
Ci+1/2,j
Ci+1/2,j
1
+3/2 −Ci+1/2,j +1/2
+1/2 −Ci+1/2,j −1/2
×
D
−D
yj +1 −yj
yj +3/2 −yj +1/2
yj +1/2 − yj −1/2
k+1
+ (1 − n(xi+1/2 , yj +1/2 , tk )) − ∗ Ci+1/2,j
+1/2
k+1
− n(xi+1/2 , yj +1/2 , tk )Ci+1/2,j
+1/2 .
For i = 0 or M − 1 or j = 0 or N − 1, the equation has a similar form except the terms
corresponding to boundary conditions. We remark that this scheme is implicit only in terms
of the concentration c, and is explicit in terms of n.
S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
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The evolution equation for sprout tip set S(t + ) = A(B(S(t − ))) is simulated in a point-topoint tracking method, where we check the conditions for sprout branching and anastomosis
at each tk (k = 1, 2, . . . , K) for each sprout tip, and if sprout branching or anastomosis is
triggered, the corresponding mechanism is performed. The capillary indicator function n is
computed in the following way:
1 if distance((x, y), kn=0 S(tn )) < h,
n(tk ) =
0 otherwise,
where distance(a, A) is the distance between a point a and a point set A in the usual Euclidean
sense, and h is the local element size of the mesh.
4. Computational results
We consider two angiogenesis scenarios as shown in Fig. 1. The domain is (0, Lx ) ×
(0, Ly ) with Lx = Ly = 1 mm. In the first case, there is one parent blood vessel on the
left boundary, and initially, there are five active sprout tips in the parent blood vessels at
the positions of y = 0.1, 0.3, 0.5, 0.7, and 0.9 mm. In the second case, we consider the
angiogenesis from two parent vessels separated by the extracellular matrix, and initially,
there are five active sprout tips in each of the parent blood vessels at the positions of y = 0.1,
0.3, 0.5, 0.7, and 0.9 mm.
The conductivity tensor K is generated by the following formula:
2
kcond
−vx vy
vy2
vx
v x vy
K = kcond
+
,
vx vy
vy2
vx2
trans −vx vy
where kcond is taken to be 1.0, trans is generated randomly from 1.0 to 10.0 (uniform
distribution), and (vx , vy ) is a normalized vector generated randomly (uniform distribution
among all possible directions). The vector (vx , vy ) represents the local orientation of the
extracellular matrix and is the direction on which the extracellular matrix possesses lowest
resistance for the sprout extension. It helps to modify the velocity imposed by the gradient
of CGFs concentration according to the extracellular characteristics. Once conductivities
are generated, they are fixed as given data for all angiogenesis simulations. We set the
limiting cell proliferation time to be 0.2 days. All other modeling parameters, the initial
Extracellular Matrix
(ECM)
y=0
Extracellular Matrix
(ECM)
Parent Vessels
Parent Vessels
y=L
Parent Vessels
y=L
y=0
x=0
x=L
x=0
x=L
Fig. 1. The geometries for the single parental vessel case (left) and for the two parent vessels case (right).
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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
Fig. 2. Capillary network (top) and CGFs concentration (bottom: concentration with a unit of ×10−16 mol/m2 )
for the single parental vessel case. The time steps are 100 (left), 150 (middle) and 200 (right).
concentration and boundary conditions are the same as in [9]. The domain is partitioned
by a 200 × 200 uniform mesh and the simulation time is discretized into multiple intervals
with a uniform time step of t = 0.07 days.
4.1. Angiogenesis from one parent vessel into extracellular matrix
The capillary network is represented by the capillary indicator function n. The first row
of Fig. 2 contains plots of the capillary indicator function n, where the area with n = 1
forms the capillary network. The endothelial cells at the capillary sprout tips are migrating from the parent vessels (at x = 0) to the right induced by the gradient of the CGFs
concentration. The simulations produce capillary networks with a very realistic structure
and morphology without resorting to stochastic differential equations or a random discrete
model. The dendritic structure of network is sharply captured. The anastomosis can be seen
clearly in the right top plot in Fig. 2, as at least five large and numerous smaller capillary
loops are formed. The front of capillary network is not uniform because of the anisotropic
extracellular matrix and CGFs concentration-dependent cell cycle time. We observe in the
simulation that several capillary sprouts grow backward while the majority of them follow
the trend. It should be noted that similar behaviors are also observed in the capillary network
in vivo. These “backward angiogenesis” phenomena are reproduced in the model due to the
heterogeneity and the anisotropy of the conductivity for the extracellular matrix.
The CGFs concentrations at different times are shown in the second row of Fig. 2. Large
concentrations are found in the areas without the presence of capillaries whereas small
concentrations are located near the capillary networks, mainly due to the CGFs consumption
by endothelial cells. Because of the nonzero CGFs diffusivity, we see that the concentration
profile is continuous in space. Interestingly, there are several spots near x = 0.1 mm with
high CGFs concentration, and CGFs on these spots remain high concentration even after
S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
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Fig. 3. Capillary network (top) and CGFs concentration (bottom: concentration with a unit of ×10−16 mol/m2 )
in the angiogenesis from two parent vessels separated by the extracellular matrix. The time steps are 80 (left), 100
(middle) and 140 (right).
250 time steps when the concentration on most of the areas is reduced greatly. This is
because sprout tips have migrated way too far to the right of the domain at the later time
and cannot be attracted back by the CGFs spots near x = 0.1 mm.
4.2. Angiogenesis from two parent vessels separated by the extracellular matrix
Fig. 3 shows the capillary networks and concentration profiles at different time steps.
Clearly, the endothelial cells at the capillary sprout tips are migrating from the parent vessels
(at x = 0 and 1 mm) toward the domain’s midline (x = 0.5 mm), and they connect when
they merge in the middle of the domain by the “head-on-head” anastomosis mechanism. We
observe that anastomosis is more pronounced in this case than in the single parent vessel
case. One can see that the number of endothelial cells gradually reaches a steady state after
140 time steps (or 9.8 days). This is because the number of sprout tips is eventually reduced
to zero due to the “head-on-head” anastomosis in the center of the domain. Comparing
the two cases (single versus two parent vessels), one observes that the overall angiogenic
profiles are similar. Hence, branching and extension are not significantly affected by the
geometries tested.
5. Conclusions
A deterministic approach was used to model growth factor-induced angiogenesis in this
paper. A capillary indicator function is employed to describe the capillary network structure.
The capillary indicator function has binary values, and possesses a value of 1 on the space
occupied by capillaries and has a value of 0 otherwise. The replacement of the traditional
endothelial cell density by the capillary indicator function empowers the model to capture
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S. Sun et al. / Nonlinear Analysis 63 (2005) e2237 – e2246
the capillary network precisely at fine scales. The conductivity of the extracellular matrix
for the movement/extension of sprouts is utilized in the model to describe the heterogeneity
and anisotropy of the extracellular matrix. A numerical algorithm based on cell-centered
finite difference method is proposed to solve the nonlinear system of the continuous model.
A case of angiogenesis with a single parent vessel is simulated, where the nonuniform
invading front of capillaries is observed. The simulation of an angiogenesis process involving
two parallel parent vessels demonstrates the important role of anastomosis. Computational
results indicate that the model generates the overall dendritic structure and pattern of the
capillary network morphologically similar to those observed in vivo.
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