Transactions on the Built Environment vol 13, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
Constrained constructive optimization of binary
branching arterial tree models
F. Neumann, W. Schreiner, M. Neumann
Department of Surgery and Institute of Experimental Physics,
University of Vienna, A-1090 Wien, Austria
Abstract
Following realistic principles of coronary blood supply to a given myocardial perfusion area, arterial vascular trees are represented in a computer
simulation model as two-dimensional structures of cylindrical tubes, grown
in accordance with given boundary conditions, a set of constraints, and an
optimization target. In the present work, arterial trees generated by constrained constructive optimization feeding perfusion areas of different shape,
were investigated and compared with respect to their global characteristics.
Perfusion areas allowing for multiple symmetrical branching of arteries were
found to be most efficient in terms of costs of blood transport and delivery.
1 Introduction
Until recently, in computer simulation studies of the coronary vascular system, the highly complex structure of the microcirculation has been reduced
to only a few lumped compartments, each of which represents an entire class
of vessels (e.g. arteries, capillaries, or veins). In order to more adequately
reproduce the branching structure of coronary vessels, in our present model
the arterial tree is simulated in full detail from the feeding artery down
to the level of individual terminal segments. Based on realistic considerations of blood supply and nutrition of myocardial tissue, the model trees
are grown according to given boundary conditions, a set of constraints, and
an optimization target (Schreiner[l], Schreiner et a/.[2]).
In real arterial trees, the branching patterns of the main arteries are
known to exhibit considerable variations (e.g., right or left coronary dominancy, indifferent type), and the shape and size of the tissue areas perfused
by them may differ to a similar degree (McAlpine[3j). Because of its capabil-
Transactions on the Built Environment vol 13, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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ity of reproducing realistic tree structures in full detail, our model was used
to quantitatively describe some properties of arterial trees feeding perfusion
areas of different shape. Nine simulations of model trees, corresponding
to two-dimensional perfusion areas of the same size but widely different
shapes were performed, and the variability between the trees was described
in terms of bifurcation levels, tree dimensions (such as total length, surface
or volume), and symmetry. In the following, after briefly describing our
simulation model, we will show how these global characteristics of arterial
trees depend on the shape of their perfusion area.
2 The model
In our model, anatomical and physiological principles are taken into account by the specific choice of boundary conditions, constraints, and target
function for geometric and structural optimization. Assuming that a given
piece of tissue is to be perfused as homogeneously as possible by a dichotomously branching arterial tree, the terminal segments of the binary model
tree are required to be uniformly distributed within a given perfusion area.
In the present implementation of the model, any two-dimensional perfusion
area bounded by a polygon of arbitrary shape can be chosen to support the
simulated tree structure.
In order to perfuse a piece of tissue under physiological conditions, which
corresponds roughly to the myocardial region supplied by the left coronary
artery in humans, perfusion pressure, terminal pressure, and total perfusion flow were set to the values given in Table 1. In addition, all terminal
segments were required to deliver equal flows (Jterm — Qperf/^term to their
respective perfusion sites. The size of the perfusion area was taken to be
that of a circle of 5 cm radius and was constant in all simulations. The resolution of the model trees is determined by the number of terminal segments.
Because of the rigorously dichotomous branching pattern of the trees, the
total number of segments is given by TVtot = 27Vterm — 1? regardless of the
particular structure of a tree.
Table 1: Global model parameters
parameter
^perf
Pperf
Pterm
Qperf
Wterm
size of perfusion area
perfusion pressure
terminal pressure
perfusion flow
number of terminals
preset value
78.54 cnf
100 mmHg
60 mm Hg
500 ml/min
4000
Transactions on the Built Environment vol 13, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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183
The radii of parent and daughter segments at each bifurcation are assumed to obey a power law
' parent
(1)
In the present study, the exponent was set to 7 = 3 in all simulations, a
condition for uniform shear stress throughout the whole tree (Rodbard[4j).
The model tree is grown by successive addition of terminal segments: A
new terminal site is selected at random and connected to one of the segments of the tree constructed so far, thereby creating a new bifurcation. At
this stage, the segment radii can always be rescaled in such a way that all
terminal segments deliver equal flows (Qterm) at equal pressures (pterm), the
bifurcation law (eqn 1) is obeyed at each bifurcation, and the resistance of
the whole tree is such that the applied pressure drop, Ap = Pperf — Pterm? induces the required total perfusion flow (Qperf) ("constrained optimization").
Since this can be achieved for arbitrary locations of the new bifurcation
within the existing tree, it is necessary to introduce a decision rule for the
"optimum" placement of that bifurcation. The criterion of optimality is
defined in terms of the functional structure of the tree. Global properties
of the tree, such as total surface or volume, are suitable cost functions for
structural optimization. In the present work, the cost (or target) function
was chosen to be of the general type
(2)
where , l(i) and r(i) are length and radius of segment i, and TVtot is the total
number of segments in the tree. As suggested by theoretical considerations
(e.g. Kamiya et al. [5]), the exponent A was set to 2. Apart from a multiplicative factor, this is equivalent to minimizing the total intravasal volume
of the tree.
The target function is evaluated for connection of the new bifurcation
to all nearby segments and all possible locations along these segments, and
the bifurcation isfinallylocated in such a way as to achieve the lowest value
of the target function (i.e. total volume). Therefore, minimizing the target
function not only optimizes the geometrical location of a single bifurcation
on a given segment, but also determines the global topological structure of
the tree (i.e., which segment the new terminal site is connected to).
Also, a new terminal site is accepted only when its location is in the
interior of the bounding polygon defining the perfusion area. In the case of
essentially convex shapes, this implies that all segments and bifurcations lie
within the contour, and that the whole tree as well as the individual course
of the large vessels automatically adapts to the given shape.
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3 Global properties of model trees
In contrast to casts of real arterial trees, our simulated model trees are
readily accessible to detailed morphometric analysis without painstaking
measurements. From vessel dimensions of single segments and from their
topological arrangement within the perfusion area, a number of global properties can be determined which characterize the structure of the whole tree.
These properties are related to the functional abilities and efficiency of the
tree, with respect to its task of blood transport and delivery.
The length of the main vessel - analogous to the main river of a drainage
basin was calculated as the cumulative length of segments along the path
starting from the root and always following the larger daughter segment at
each bifurcation. The total (cumulative) length of a model tree, as well as
its total surface and volume are also easily obtained from the individual
segment radii and lengths. Since, in the present work, the total volume was
used as cost function for optimization, while the required total flow and the
size of the perfusion area were kept constant, the volumes calculated for
individual trees indicate the degree of optimal it y which could be achieved
when the shape of the perfusion area was changed in typical ways (see
below).
In order to classify an arterial tree as "conveying" (i.e. transporting
blood over large distances) or "delivering'' (i.e. distributing it into numerous
small branches), we calculated the number of bifurcation levels Z (Zamir[6j)
and the symmetry index £bif (Schreiner ci «/.[7]). Z is defined as the number
of bifurcations proximal to a given segment plus one. The index of symmetry
of a tree with A',erm terminal segments is given by
6>lf =
In a perfectly symmetric binary branching tree, which may also be considered as a perfect deliverer, after log^f A^rm) bifurcations distal of the root
segment , all segments are terminal and £bif — 1. With decreasing symmetry,
£bif < 1, an increasing number of bifurcation levels is necessary to supply
a given number A^rm of terminal segments. A completely asymmetric tree
with AVerm bifurcation levels has £bif = (log^A^erm) -f 1 )/AVerm4 Shape of perfusion area and global tree properties
The shape of the perfusion area of a feeding artery within a particular
organ is determined by the outer form of the organ and by the anatomical
arrangement of surrounding tissues, such as bone, muscle, connecting tissue,
etc. Consequently, the network of blood vessels is forced to adjust itself
topologically as well as spatially to these boundary conditions.
In the present paper, we have selected nine different shapes of perfusion
areas to show, first, that arterial trees perfusing areas of arbitrary shape
Transactions on the Built Environment vol 13, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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185
can actually be generated by constrained constructive optimization; and,
second, that there is a typical variation of the global properties of these
different trees. These shapes (except the perfect circle) and the respective
positions of the inlet site (i.e. the proximal end of the root segment) are
depicted in Fig. 1. Some are purely geometrical forms (circle, ellipse, triangle, hexagon), while others have been designed to resemble more closely
the form of real arterial vessel trees. All shapes were scaled to the same
size, corresponding to the area of a circle of 5 cm radius. In Table 2, the
nine different perfusion areas are listed by increasing intravasal tree volume,
which was used as target function for optimization.
Figure 1: Contours of perfusion areas used for growing arterial tree structures of different shape. All contours are scaled to enclose an area of the
same size. The position of the inlet site is indicated by a diamond. The
labels correspond to the ranking by total intravasal tree volume in Table 2.
Choosing total tree volume as the target function for optimization means
that the task of delivering a given total flow, (Jperf, to a perfusion area of
given (constant) size has to be accomplished with a tree structurefilledwith
the minimum quantity of blood. This minimum volume in turn depends in
a characteristic way on the shape of the perfusion area.
Transactions on the Built Environment vol 13, © 1995 WIT Press, www.witpress.com, ISSN 1743-3509
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Table 2: Global characteristics of arterial trees feeding perfusion areas of
different shape, listed by increasing intravasal volume. The numbers in
column 1 (except no. 6, which denotes a circle) correspond to the labels of
Fig. 1. £bif is the symmetry index and Z the number of bifurcation levels,
as defined in the text.
main
shape' vessel
no.
length
(cm)
1
9.59
9.62
2
9.60
3
11.41
4
14.60
5
11.22
6
12.25
7
14.39
8
16.52
9
total
total
surface
total
volume
fc,
%
(cm)
701.1
701.7
702.5
701.8
702.6
700.3
701.5
699.9
704.4
(cm?)
62.01
62.34
63.81
64.21
64.37
64.75
65.64
66.56
69.22
0.8163
0.8296
0.8920
0.9220
0.9297
0.9453
0.9897
1.0658
1.1900
0,.1271
0,.1189
0..1168
0,.1108
0..0894
0,.1147
0..1099
0..0998
0..0795
102
109
111
117
145
113
118
130
163
The results in Table 2 show that areas perfused by long main vessels
with a large number of bifurcation levels need approximately up to 40 %
more intravasal volume than areas supplied by short main vessels. The relationship between main vessel length and volume is approximately linear
with a correlation coefficient of r = 0.88 (R* — 0.78). Obviously, long main
vessels carry a large "dead" volume of blood which has to be transported
over long distances before being delivered at the respective perfusion sites.
This tree property of blood conveyance can be characterized numerically by
the symmetry index £bif (eqn 3): Long main vessels show a large number
of asymmetrical bifurcations with large differences between the major and
the minor daughter segments (the former forming the continuation of the
main vessel, the latter being only small side branches). This requires automatically a large number of bifurcation levels Z and yields a small £bif for
the whole tree, which may thus be classified of "conveying" type. On the
other hand, in trees with short main vessels, splitting mostly into symmetrical bifurcations of daughter segments of approximately equal caliber, £bif
will be relatively high, and trees of this type may be classified as "delivering" . Quantifying the relationship between the length of the main vessel
and the symmetry index £bif, an almost perfect linear correlation could be
established (r = 0.96 with R* = 0.93).
Figure 2 shows four complete trees grown within different perfusion areas, and gives a visual impression of the resulting structural differences
when the vascular network is forced into a particular shape. The deformed
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187
cardioid (top left; shape no. 1 of Fig. 1 and Table 2) is characterized by the
shortest main vessel length, the smallest total length, surface and volume.
It also features the lowest number of bifurcation levels and, thus, the highest
symmetry index. In terms of costs of the transport medium, the deformed
cardioid turns out to be the most efficient perfusion area, i.e. splitting symmetrically and carrying the least amount of blood over short distances. By
contrast, the triangle with its root segment at the upper vertex is the most
"expensive" shape to be perfused, showing the longest main vessel, greatest
total length, surface and volume, the greatest number of bifurcation levels
and low symmetry. The ellipse with its root at the minor vertex, and the
circle have intermediate values of all global characteristics, with a slight
advantage of the ellipse in terms of blood-transport costs.
Figure 2: Dependence of tree structure on the shape of the perfusion
area. The main vessel length drastically increases by more than 60 % between the deformed cardioid (top left) and the triangle, with the flat ellipse
and the circle taking on intermediate values. All trees were optimized for
intravasal volume and scaled to the size (area) of the circle.
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In this paper we have demonstrated that by using our method of constrained constructive optimization, binary branching model trees can be
generated for perfusion areas of more or less arbitrary shape. An investigation of selected global properties of these trees showed that the way in which
the task of blood transport and delivery is achieved (i.e. more or less costly)
depends crucially, through the topological and geometrical characteristics
of the vascular network that evolves, on the given shape of the perfusion
area.
References
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Journal of Biomedical Engineering, 1993, 15, 148-150.
2. Schreiner, W. & Buxbaum, P. Computer-optimization of vascular
trees, IEEE Transactions of Biomedical Engineering, 1993, 40, 482491.
3. McAlpine, W.A. Heart and Coronary Arteries, Springer- Verlag Berlin
and New York, 1975.
4. Rodbard, S. Vascular caliber, Cardiology, 1975, 60, 4-49.
5. Kamiya, A. & Togawa, T. Optimal branching structure of the vascular
tree, Bulletin of Mathematical Biophysics, 1972, 34, 431-438.
6. Zamir, M. Distributing and delivering vessels of the human heart,
o/ GeneroZ P&z/aW^z/, 1988, 91, 725-735.
7. Schreiner, W., Neumann, F., Neumann, M., End, A. & Miiller, M.R.
Structural quantification and bifurcation symmetry in arterial tree
models generated by constrained constructive optimization, Microvascular Research, 1995, submitted.