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blood ¬‚ow ensues. Cellular and biomechanical factors appear to be involved in
determining vascular identity (i.e., whether the vessels are capillaries, arteries, or veins).
Bottom panel: microenvironmental cues (ECM, cell contacts, and organ-associated
growth factors) regulate the organotypic differentiation of a newly formed vascular tree
with distinct types of endothelia. These are not pictured, but include: “continuous”
endothelia in which cells form an unbroken single layer, seen in most capillaries and
larger blood vessels; “discontinuous” endothelia in which adjacent cells have gaps
between them, found in the liver, for example, and “fenestrated” endothelia in which the
membranes of the endothelial cells are perforated, found in the kidney and endocrine
organs. (After Augustin, 2001.)




Cord hollowing

Cell hollowing

Fig. 8.4 Morphological processes of tube formation. Wrapping: a portion of an
epithelial sheet invaginates and curls until the edges of the invaginating region meet and
seal, forming a tube that runs parallel to the plane of the sheet. Budding (also referred to
as sprouting): a group of cells in an existing epithelial tube or sheet migrates out and
forms a new tube as the bud extends. The new tube is a branch of the original tube.
Cavitation: the central cells of a solid cylindrical mass of cells are eliminated, converting
the cylinder into a tube. Cord hollowing: a lumen is created de novo by the rearrangement
of cells, without cell loss, into a thin cylindrical cord. Cell hollowing: a lumen, spanning the
length of a single cell, forms by rearrangement of the cell™s membrane and cytoplasm.
(Reprinted from Lubarsky and Krasnow, 2003, with permission from Elsevier.)

as ˜˜budding” by Lubarsky and Krasnow (2003) and as ˜˜sprouting” by
other investigators, is the main basis of angiogenesis, the formation of
new blood vessels from preexisting ones (Patan, 2000).
Angiogenesis is responsible for creating the anastomosing net-
work of capillaries, which nourishes all tissues in the healthy body,
as well as the ˜˜neovascularization” (abnormal new capillary forma-
tion) of pathological tissues such as tumors (Verheul et al., 2004) and
the diabetic retina (Campochiaro, 2000). Angiogenesis can occur by

˜˜intussusceptive” microvascular growth (IMG), whereby vessels split
by lumenal insertions of tissue-dividing walls, or septa, or by the lon-
gitudinal fold-like splitting of a preexisting vessel (Augustin, 2001).
However, budding or sprouting angiogenesis is the most common
mechanism of capillary network formation (Fig. 8.3). Capillaries in
the embryo and during neovascularization are induced to sprout by
exposure to members of the VEGF (vascular endothelial growth fac-
tor) family of diffusible glycoproteins secreted by target tissues or
by the capillaries themselves. The VEGFs act as chemoattractants to
angioblasts (Folkman, 2003).
Both initial vasculogenesis and remodeling angiogenesis appear
to be self-organizing processes that construct ef¬cient transport net-
works of fractional dimensionality, i.e., these networks are ˜˜fractals.”
It will be helpful to explore the nature of fractals as a prelude to con-
sidering the physical processes that may underlie vascular network

Fractals and their biological signi¬cance
In our discussion of network formation in Chapter 6 we saw that
at the percolation transition drastic changes can take place in a sys-
tem™s behavior. In the example of a telephone communication system,
if the interconnected network of cables ceased to percolate, commu-
nication between distant cities was disrupted (see Fig. 6.6). Let us now
imagine the line segments in Fig. 6.6 to be pipes instead of cables.
If we impose a pressure difference between LA and NY then, once a
percolating network of pipes forms, ¬‚ow between distant locations
will be established. We can now begin to see how the percolation for-
malism, ¬rst introduced to account for the properties of extracellular
matrices, might also be applicable to blood vessel formation.
Another physical example of network formation via percolation is
shown in Fig. 8.5, which is a schematic illustration of an amorphous
material with metallic fragments randomly dispersed in an insulating
matrix. If a voltage difference is set up between the two sides of this
material then below the percolation transition no current can ¬‚ow
across the system because no communication exists between the two
sides through the metallic islands (denoted by identical ¬lled squares
in Fig. 8.5). However, at a critical concentration of the metallic compo-
nent a percolation transition takes place and the metallic islands link
up into an interconnected network through which charges can move
from one side of the system to the other. The tenuous network that
emerges at the percolation transition is a fractal and various biologi-
cal structures have been interpreted in terms of such networks (e.g.,
the vasculature structure and the network of airways in the lung).
To understand the concept of fractals we recall that mass = density
times volume. Thus, a cubic box (having a negligible mass of its own)
containing some material of density ρ, comprises a mass M = ρ L 3 ,
where L is the side of the box. A box with side 2L will hold eight
times as much material, and in general a box with side »L will have

Fig. 8.5 Schematic representation of an amorphous metal. The ¬lled squares denote
the metallic component. The left-hand panel shows the situation below the percolation
threshold. No continuous metallic connection exists between the two contacts on the
left and right, represented by the thick bars, thus no current can ¬‚ow and the bulb is not
lit. The right-hand panel shows the situation above the percolation threshold.
Continuous paths (two in the ¬gure) for the charges exist, the electric circuit is closed,
and the bulb is lit.

a mass M (»L ) = »3 M (L ). The power 3 appears here because the box
is three-dimensional. A similar analysis with ˜˜two-dimensional” boxes
(i.e., squares) would yield a power 2 and, in general, in a d-dimensional
space the analogous relationship is

M (»L ) = »d M (L ). (8.1)

Equation 8.1 can be solved for the function M (L ) on setting » = 1/L ,
to yield as expected M (L ) ∝ L d . (Note that the factor on the left-hand
side of Eq. 8.1 becomes M (1), which is clearly independent of L and
thus can be identi¬ed with the scale-independent density, ρ.)
The above results are well known even without resorting to equa-
tions such as Eq. 8.1. However, it is exactly the relationship in this
equation that is needed to describe fractal objects, for which the Eu-
clidian dimension d in Eq. 8.1 is replaced by the fractal dimension
df . Since df < d (see below), dimensional analysis indicates that the
density of fractal objects, in contrast with ordinary materials, must
be length dependent:

ρ ∝ L df ’d . (8.2)

Thus, upon increase in size, a fractal becomes less dense, which
is illustrated by the Sierpinski carpet (Mandelbrot, 1983), shown in
Fig. 8.6 and discussed in Box 8.1.
It is not surprising that those biological structures whose main
physiological function is material exchange with their surroundings

Fig. 8.6 The construction of the
Sierpinski carpet. For details, see
Box 8.1.

have a fractal character. The vascular system supplies organs with
oxygen and disposes of waste, the respiratory airway network ex-
changes oxygen with carbon dioxide. The ef¬ciency of such exchanges
requires a large surface area: it is precisely here that fractal networks
have an unparalleled advantage when compared to other geometries.
This is quite obvious in the Sierpinski carpet, for example, in which
the perimeter of the holes (i.e., the ˜˜surface area” between the per-
colating ¬lled region and the surrounding material) grows without
limit. We will now see that such an arrangement naturally arises in
a plausible model for blood vessel development.

Box 8.1 The Sierpinski carpet

The Sierpinski carpet is a fractal embedded in two-dimensional Euclidean space.
To construct this object one starts with a ¬lled square of side 1, Fig. 8.6, panel 0,
and creates a hole by removing the middle ninth of its area as shown in panel 1.
In the next step the middle ninth of each of the remaining eight small squares is
removed (panel 2) and the process is iterated ad in¬nitum. Clearly, the density (the
brown, ¬lled, fraction of the total area) of the ultimate Sierpinski carpet vanishes
while the total perimeter of the holes is in¬nite. To determine the fractal dimension
of the Sierpinski carpet let us calculate its density as a function of the square™s side
at each iteration. For the original square with side 1, the density is 1; after the
¬rst iteration it is 8/9 (out of the nine squares with side 1/3, eight are ¬lled, and
one is empty; the side of the original square in terms of the side of the smaller
squares is 3). After the second iteration the density is (8/9)2 (out of the 81 squares
with side 1/9, 64 are ¬lled; the side of the original square is now 9). Finally, after
the kth iteration ρ = (8/9)k and the side of the original square in terms of the
smallest squares is L = 3k . Using Eq. 8.2 (which, as a result of the fact that we are
working with dimensionless quantities, is now an equality) the fractal dimension
of the Sierpinski carpet is given by (8/9)k = (3k )df ’d , which yields (with d = 2)
d f = 3 ln 2/ ln 3 ≈ 1.893 < 2.

Early vasculogenesis: the analysis of Bussolino
and coworkers
As discussed above, the early stages of vasculogenesis involve (i) the
chemotactically driven migration of endothelial cells and network for-
mation, (ii) network remodeling, and (iii) differentiation into tubular
structures. The network can best perform its biological functions if it

is a fractal. However, fractal geometry is the end result of a dynamical
process and a priori it is not evident how the interaction and migra-
tion of endothelial cells can bring about this nontrivial structure.
There have been numerous attempts to model vasculogenesis and
angiogenesis in terms of percolating fractal networks. Here we con-
centrate on the model of Bussolino and coworkers (Gamba et al., 2003;
Serini et al., 2003), who considered the early stages of vascular network
formation. These authors performed in vitro experiments with human
endothelial cells plated on Matrigel, an extracellular matrix prepara-
tion. The role of the chemotaxis of endothelial cells in response to a
form of VEGF was analyzed, as well as the dependence of vessel assem-
bly on cell density. To gain insight into the mechanisms driving net-
work formation, the dynamics of the various interacting components
was modeled using biologically motivated assumptions. The model
calculations reproduced the experimental observations and strongly
suggested that endothelial cells have the inherent capacity to self-
organize into percolating networks with fractal geometry.
The experimental images in Fig. 8.7 (upper row) show the time evo-
lution of capillary network formation, when the original plating cell
density ρ0 is above the percolation threshold (ρ0 ≈ 100 cells/mm2 ) and


0 hour 3 hours 6 hours 9 hours


0 hour 3 hours 6 hours

Fig. 8.7 In vitro vasculogenesis and, for comparison, the predictions of the model of
Bussolino and coworkers. (A“D) Human endothelial cells were plated (at 125 cells/mm2 )
on Matrigel, and the time course of network formation was recorded by the time-lapse
videomicroscopy of a 4 mm2 portion of the surface. (E“G) The positions of the cell
centroids obtained by the model of Serini et al. (2003) (i.e., the two-dimensional analogs
of Eqs. 8.3“8.5) using the same number of cells as in panels A“D. In the model, an initially
random distribution of cells was prepared by imposing conditions in the form of a set of
randomly distributed bell-shaped bumps each having a ¬nite width of the order of the
average cell diameter and zero velocity. The dots in panels E“G denote the cell
centroids, the centers of these bumps. Nonzero velocities are built up by the
chemoattractive term, starting from inhomogeneities in the density distribution.
Subsequently, the dynamics ampli¬es the density variations and forms a capillary-like
network. (Reprinted from Serini et al., 2003, with permission.)

the concentration c of VEGF is below its saturation value (c ≈ 20 nM).
The initially randomly distributed cells start to move, interact, and
adhere to their neighbors, eventually forming an interconnected mul-
ticellular network. The migration of cells is directed toward zones of
higher cell concentration. The analogues of the telephone cables in
Fig. 6.6 are now the chords formed by the adhering cells. The net-


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