Functional Characteristics of Optimized Arterial Tree ... · The major task of the arterial portion of the circulatory system is the transport of oxygen-rich blood and nutrients from

Research Paper

J Vasc Res 2000;37:250–264

Functional Characteristics of OptimizedArterial Tree Models Perfusing Volumesof Different Thickness and Shape

Rudolf Karcha Friederike Neumanna Martin Neumannb

Wolfgang Schreinera

aDepartment of Medical Computer Sciences, and bInstitute of Experimental Physics, University of Vienna,

Vienna, Austria

Received: June 28, 1999

Accepted after revision: January 17, 2000

Dr. Rudolf KarchInstitut für Medizinische ComputerwissenschaftenSpitalgasse 23A–1090 Wien (Austria)Tel. +43 1 40400 6676, Fax +43 1 40400 6677, E-Mail [email protected]

ABCFax + 41 61 306 12 34E-Mail [email protected]

© 2000 S. Karger AG, Basel1018–1172/00/0374–0250$17.50/0

Accessible online at:www.karger.com/journals/jvr

Key WordsComputer simulation W Arterial branching system

model W Arterial network optimization W Blood transport

AbstractThe relationship between the ‘shape of an organ’ and the

‘cost of blood transport’ to perfuse its tissue was evaluat-

ed on the basis of optimized arterial model trees simu-

lated to perfuse square-based 100-cm3 volumes of differ-

ent shape (‘flat’ versus ‘thick’ as defined by the ratio of

thickness to side-length h/s ^1). Specifically, the effects

of ‘shape’ on tree structure, blood transport, and on

hemodynamic characteristics were investigated. Branch-

ing models of arterial trees were generated by con-

strained constructive optimization (CCO), based on an

identical set of model parameters. All model trees were

geometrically and topologically optimized for intravas-

cular volume. Tree structures achieved tremendous sav-

ings of blood (transport medium) in comparison to a sys-

tem of separate tubes. Thickening the perfusion volume

(increasing h/s) resulted in a significant decrease of

mean transport length, deposition time, and intravascu-

lar total volume in the tree. ‘Thick’ perfusion volumes

induced CCO trees to branch more symmetrically into a

number of equivalent subtrees repetitiously splitting into

smaller ones; ‘flat’ structures were dominated through-

out by a few asymmetrically branching major vessels. In

summary, we conclude from systematic variation of

shape that thicker perfusion volumes (h/s 10.1) facilitate

efficient delivery of blood in comparison to large

amounts of ‘dead volume’ to be carried over long dis-

tances in very thin pieces of tissue.Copyright © 2000 S. Karger AG, Basel

Introduction

The major task of the arterial portion of the circulatorysystem is the transport of oxygen-rich blood and nutrientsfrom the heart to all other parts of the body (including theheart itself) and to deliver substances to all perfusion sitesof the respective tissue. Due to the vital importance ofuninterrupted operation of the blood circulation, and dueto the fact that maintenance of the system continuouslyconsumes energy, it is reasonable to assume that somekind of optimization of the arterial network has takenplace during phylogenesis (e.g. [1, 2]). The branching sys-tems are certainly anatomically adapted to the shape ofthe tissue (organ, muscle), and their morphology may beassumed to satisfy physiologic, i.e. functional, require-ments of blood supply. In order to shed some light onoptimization and functional properties of arterial sys-tems, the method of computer simulation can be helpfullyand efficiently applied. If one succeeds to provide a suit-able computer model, i.e. an algorithm to model branch-

Shape-Dependent Perfusion of ArterialTrees

J Vasc Res 2000;37:250–264 251

Fig. 1. Three-dimensional perfusion vol-umes of different shape at constant volume(Vperf = 100 cm3). h and s denote height andside-length of each square slab, and the num-bers for s ! h are given in centimeters.Except for the flattest volume (s = 44.72 cm),all slabs are drawn to the scale.

ing systems comparable to those of real organisms, struc-ture-dependent properties of arteries beyond experimen-tal access can be systematically analyzed and quantified.

The method of constrained constructive optimization(CCO), as introduced in the literature [3, 4], has originallybeen developed as an algorithm to generate model trees ofthe vasculature of the coronary circulation. Without theinput of anatomical information or morphometric data,the growth process of CCO trees is based on principles ofoptimization. The resulting structures, obtained by suc-cessive adding of new segments and optimization of thegrowing tree at each step of development, are geometrical-ly and topologically optimized branching networks whichresemble real vascular trees with regard to visual appear-ance and morphometric parameters [4–7]. However, thealgorithm could only be used for vessel trees in a two-dimensional perfusion area, representing an infinitesi-mally thin layer of tissue. Recently, the procedure hasbeen generalized so as to grow tree models within anythree-dimensional convex shape [8, 9]. Provided withthese new potentials of the algorithm, a series of CCOtrees has been produced for the present study. Using iden-tical constraints, optimization principles and boundaryconditions for pressures and flows, different trees weregrown to perfuse geometric volumes representing piecesof tissue of variable ‘thickness’ and shape.

There were two major points of interest to be investi-gated: first, with CCO available for two- and three-dimen-sional perfusion volumes, it was self-evident to compare‘optimum’ two- and three-dimensional tree structures fortheoretical reasons. A priori, it is by no means clearwhether dimensionality effects a qualitative differencebetween trees or a two-dimensional tree can simply beconsidered as the limiting case of three-dimensional trees

when thickness is reduced to zero. Second, systematicvariation of thickness and shape of the perfusion volumesuggested a comparison of transport properties of therespective trees. In particular, one would expect that theshape of the perfusion volume is directly related to theaverage transport distance and deposition time of bloodcells, as well as to the total intravascular blood volumewhich is required to maintain the flow conditions in thetree. Finally, the visual representation of different treesshowed that bifurcation symmetry played a major role inthe structure of the vessel system and needed to be ana-lyzed in detail.

Even though the model approach is only a simplifiedrepresentation of reality neglecting quite a number ofwell-known effects and dependencies, its advantages ofarbitrary choice of parameters and quantitative availabil-ity of structural characteristics may still be utilized to sys-tematically study properties of the trees which result fromisolated variation of individual parameters.

Methods

Perfusion VolumesIn order to study the functional differences of arterial trees per-

fusing pieces of tissue of different thickness, we defined a series ofnine square slabs of different height, h (with a cube as the limitingcase). Since – in agreement with our previous work on CCO trees –we used a constant perfusion volume of Vperf = 100 cm3, h was chosento be 0.05, 0.1, 0.3, 0.6, 1.0, 1.5, 2.0, 3.0, or 4.64 cm, and the side-length, s, of each slab had to be chosen accordingly such that s2 h =Vperf (44.72 cm 6 s 6 4.64 cm) (fig. 1). For comparisons betweentwo- and three-dimensional CCO structures, we additionally definedtwo flat perfusion ‘volumes’, i.e. genuinely 2D squares, with h = Vperf= 0 and side-lengths corresponding to the two largest values of s in theseries of 3D slabs (44.72 and 31.62 cm). The ratio of height and side-

Vperf

252 J Vasc Res 2000;37:250–264 Karch/Neumann/Neumann/Schreiner

length, characterizing the shape of the perfusion volume, was 0 ̂ h/s^1. Each volume, representing the piece of tissue to be supplied withblood, was to be perfused by means of N = 8,000 terminal segmentsof a CCO-generated arterial tree (see below) whose origin was posi-tioned at a corner point of the square or slab.

The Method of CCOThe CCO algorithm has been described in detail elsewhere [4–7,

9] and is only briefly presented in the following. The arterial networkwithin the perfusion volume is modeled as a binary branching tree ofnonintersecting, rigid cylindrical tubes (‘segments’), perfused atsteady-state and laminar flow conditions (i.e. disregarding turbu-lence or entrance flow) and subject to a set of boundary conditionsand constraints: Nterm terminal segments, yielding

Ntot = 2Nterm – 1 (1)

segments in total, are randomly distributed within the perfusion vol-ume. The resulting structure of the CCO tree induces a spatial hetero-geneity of terminal flows [4] which is comparable to experimentalmeasurements [10].

The feeding artery of the tree (‘root segment’) carries a given totalperfusion flow Qperf at a given pressure pperf. Terminal flows aredefined in order to obtain a stable solution of the hemodynamicequations [11] and set to

Qterm = Qperf/Nterm (2)

for all terminal segments; terminal pressures are constant at a givenvalue pterm. While the relative flow distribution within the CCO treeyields relative radii (i.e. the ratio of radii of any two segments), abso-lute radii are determined by the absolute overall pressure gradient,¢p = pperf – pterm, together with total flow. From the absolute radii,pressure can be calculated for each segment according to Poisseuille’slaw.

At each bifurcation, the radii of parent and daughter segments areassumed to obey a power law of the form

rÁparent = rÁ

l + rÁs (3)

with a constant bifurcation exponent Á 1 0, where r1 and rs are theradii of the ‘larger’ and ‘smaller’ daughter segments, respectively (i.e.r1 6 rs). Such a relationship has also been postulated on theoreticalgrounds. For instance, reflection of pulse waves at bifurcations isminimized with Á = 2.55 [12], whereas Á = 3 allows for uniform shearstress throughout the whole tree [13]. The latter value is also obtainedfrom Murray’s law [14], a necessary condition for minimum energyconsumption in a vascular system.

The computational method of CCO incorporates two tightly cou-pled concepts: growth (‘construction’), and optimization under a setof constraints. Starting from an inlet for the feeding artery at theboundary of the perfusion volume, the tree is grown in a stepwisefashion by subsequently adding new terminal segments while opti-mizing the geometric location and topological site of each new bifur-cation according to a prescribed optimization target function T. Eventhough optimization criteria are – due to the complexity of biologicalsystems – still unknown and a validation of optimality principles islacking, in the model the total intravascular volume of the tree maybe considered as the optimization target to be minimized (e.g. [15,16]), i.e.

T = Ntot

™i = 1

li r 2i → min (4)

Table 1. Global model parameters

Parameter Meaning Value

perfusion volume 100 cm3°pperf perfusion pressure 100 mm Hgpterm terminal pressure 80 mm HgQperf perfusion flow 500 ml min–1†

Nterm number of terminals 8,000Qterm terminal flow 0.0625 ml min–1

Ë viscosity of blood 3.6 cP+

Á bifurcation exponent 3

°For three-dimensional perfusion volumes.†This value was chosen according to the literature for cardiac

arrest [17–19].+Since the effect of shear-rate-dependent viscosity on shear stress

of individual segments could be shown to be of the order of only a fewpercent (4.7% on average [20]), Ë was assumed to be constant.

where li and ri denote length and radius of segment i. The global mod-el parameters used in the present study for the generation of CCOtrees (table 1) have previously been applied to modeling of coronaryarterial trees, for which comparisons to experimental measurementsshowed good agreement with respect to some structural and hemody-namic characteristics (e.g. [4, 5, 9]). It should be mentioned thatapart from the (square) shape of the perfusion area, trees with h =0 cm now represent the two-dimensional CCO models of earlier stud-ies, and simulation results represent ‘cardiac’ vessels, where collater-als are absent.

Reference ValuesIn the following, topological and quantitative properties of CCO-

generated arterial trees are represented by global characteristics, suchas total intravascular volume, and by the distribution of transportroutes, bifurcation symmetry, and segment orientation within thetree. In order to understand the effects of changing shape on theseproperties, and to separate effects originating from changes in treestructure from the purely geometric effect of dimensionality (i.e. 2Dversus 3D), we defined some reference values for idealized, simplearrangements of terminal sites and blood supply.

Transport Properties. To compare the transport properties of treestructures with an arrangement of nonbranching transport paths, weconnected single straight tubes from a fixed inlet location to random-ly distributed terminal sites within the perfusion volume (fig. 2).Geometries and boundary conditions were identical to those of CCOtrees.

In such a ‘single-tube arrangement’, the mean transport lengthKLPL for a fixed number of terminal locations, Nterm, with coordinates(xi, yi, zi ) is given by

KLPL = 1

Nterm

Nterm

™i = 1

li = 1

Nterm

Nterm

™i = 1

(x2i + y2

i + z2i )1/2 (5)

where li is the length of the straight line segment connecting terminalsite i to an inlet located at the origin. Likewise, we obtain the total


J Vasc Res 2000;37:250–264 253

Fig. 2. Schematic of two different systems ofblood transport to four identically locatedterminal sites. a Binary branching tree struc-ture with only one segment (‘root segment’,iroot ) connected to the inlet. Thick solid line= path of blood transport (P) from the inletto the site of delivery at segment i; Êi = angleof segment orientation of segment i; dashedline = respective ‘bee-line’. b Single-tube ar-rangement with each tube originating at theinlet location.

volume of the single-tube arrangement perfusing Nterm terminal loca-tions as

Vtot = Nterm

™i = 1

li r2i = !8ËQterm

¢p

Nterm

™i = 1

li3/2 =

!8Ë

¢pQterm

Nterm

™i = 1

(x2i + y2

i + z2i )3/4 (6)

where ¢p = pperf – pterm, and where the radius ri of each tube is calcu-lated by means of Poiseuille’s law:

¢p = S8Ë

D li

r4i

Qterm (7)

Segment Orientation. Let Êi represent the angle between thedirection of blood transport through segment i and a straight line(‘bee-line’) connecting the root segment’s proximal end with the dis-tal end of i:

Êi = arccos (bi Wsi), 0 ^ Ê ^ 180 (8)

where bi and si are the unit vectors pointing in the direction of thebee-line and the symmetry axis of segment i (fig. 2a).

For reference values in a three-dimensional unit volume (cube,sphere), we consider the probability density p(Ê) of the angle Ê D[0, ], which is defined by the line segment joining an arbitrary pointto the fixed location (0,0,1) and a uniformly distributed direction.Since both directions are statistically independent, p(Ê ) is given bythe probability p(Ê ) dÊ of finding the angle Ê between the line seg-ment joining the origin of the unit sphere to a randomly selectedpoint on its surface, and a fixed direction (e.g., the z-axis) in the inter-val [Ê,Ê + dÊ]:

p(Ê) dÊ = 14

dø = 14

2

#0

sin(Ê) dÊ dA = 12

sin(Ê) dÊ (9)

where dø is the corresponding solid angle element.The same arguments hold for the two-dimensional case (e.g. unit

square and fixed location at (0,1)) and one obtains a uniform proba-bility distribution for Ê,

p(Ê) dÊ = 1

dÊ (10)

Tree TopologyFor the topological description of CCO structures, we used the

classification of segments according to their bifurcation level (§bif ),i.e. the segments’ generation number within the tree [21]. Startingwith §bif = 0 at the root segment and following a particular bloodvessel (‘path’) from the root to a terminal segment, §bif is increasedby 1 whenever a bifurcation is met. The ‘main vessel’ of the CCOtree was a unique path obtained by choosing the larger daughter seg-ment at each bifurcation. §bif at the terminal segment of the mainvessel normally represents the maximum bifurcation level of thetree.

Statistical AnalysisAll trees with h 1 0 and one two-dimensional tree (s = 44.72 cm)

were subjected to statistical analysis. Frequency distributions of allvariables were nonnormal and, therefore, represented by the valuesof median, first and third quartile for each tree. Graphically, thesedata were presented as probability density curves. Differences be-tween trees in each variable were tested for significance by applyingKruskal-Wallis analysis of variance to the ranked data and using theDuncan procedure for pair-wise comparisons [22].

Results

Figure 3 shows some examples of the visual appear-ance of the tree structures. Since the pseudo-randomnumber sequence used during tree generation essentiallydetermines the ‘anatomy’ of the major vessels [6], all treeswere generated on the basis of the same sequence and arelatively stable anatomic pattern could be maintained atleast for heights of 0.0 cm ^ h ^ 2.0 cm. In each CCOtree, the topology and geometry of all segments is exactlyknown, and a large variety of characteristics and proper-ties of the tree structure can be statistically analyzed. Inthe following, we will first look at topological differences

a b


Fig. 3. Arterial trees generated by CCO toperfuse volumes of different shape. Side-length s = 44.72, 10.00, 8.16, 7.07, 5.77, and4.64 cm, and height h = 0, 1, 1.5, 2, 3, and4.64 cm (from top left to bottom right).Except for the two-dimensional tree with s =44.72 cm, perfusion volume is constant atVperf = 100 cm3, and figures are drawn to thescale.

Fig. 4. Probability density (p) for the bifur-cation level (§bif) in CCO trees perfusing vol-umes of different shape. Values for s ! hdenote side-length and height of the (square)slabs. Perfusion volume was constant at Vperf= 100 cm3, except for h = 0 cm. The totalnumber of segments was 15,999 in all trees.

between trees when perfusion volumes are varied in thick-ness and shape, and subsequently study a number ofparameters which have been chosen to quantitativelyshow differences between the trees.

Topology of Tree StructuresFigure 4 shows the distribution of bifurcation levels

within CCO trees, plotted as probability densities for therespective trees. Since the maximum number of bifur-cation levels is highly sensitive to changes of shape of

44.72 ! 0.00


J Vasc Res 2000;37:250–264 255

Table 2. Statistical description of different CCO trees perfusing square slabs of side-length s and height h

CCO trees ! hcm ! cm

Segmentradius rmm

Segmentlength lmm

Nearestneighbordistance dn, mm

Area-expansionratio Aexp

Segmentorientation Êdeg

Bifurcationsymmetry Írad

Depositiontime Tds

Bifurcationlevel §bif

0.13 (0.09–0.21) 1.89 (0.11–3.18) 2.35 (1.54–3.31) 1.18 (1.10–1.23) 69.6 (33.2–113.7) 0.56 (0.36–0.74) 1.16 (0.76–1.61) 103 (73–129)44.72 ! 0.05 0.13 (0.08–0.21) 1.88 (1.06–3.18) 2.35 (1.54–3.30) 1.18 (1.10–1.23) 68.3 (33.7–111.6) 0.57 (0.36–0.74) 1.15 (0.76–1.60) 102 (72–125)31.62 ! 0.10 0.12 (0.09–0.19) 1.35 (0.76–2.25) 1.69 (1.14–2.35) 1.18 (1.10–1.24) 69.5 (34.9–112.1) 0.58 (0.37–0.76) 0.68 (0.45–0.96) 91 (64–115)18.26 ! 0.30 0.10 (0.08–0.16) 0.97 (0.60–1.49) 1.36 (1.00–1.75) 1.20 (1.11–1.25) 72.2 (42.6–107.8) 0.63 (0.40–0.81) 0.31 (0.21–0.43) 68 (47–85)12.91 ! 0.60 0.10 (0.08–0.14) 0.89 (0.55–1.40) 1.30 (0.95–1.65) 1.22 (1.15–1.25) 75.8 (49.2–105.4) 0.68 (0.48–0.84) 0.19 (0.13–0.26) 54 (38–68)10.00 ! 1.00 0.09 (0.07–0.13) 0.88 (0.53–1.38) 1.30 (0.95–1.65) 1.22 (1.15–1.25) 75.4 (49.6–103.4) 0.68 (0.50–0.83) 0.15 (0.10–0.19) 45 (32–58)8.16 ! 1.50 0.09 (0.07–0.13) 0.88 (0.54–1.37) 1.28 (0.94–1.63) 1.22 (1.16–1.25) 74.7 (49.2–101.4) 0.68 (0.50–0.84) 0.12 (0.09–0.16) 41 (29–51)7.07 ! 2.00 0.09 (0.07–0.13) 0.88 (0.52–1.37) 1.28 (0.97–1.62) 1.22 (1.15–1.25) 75.5 (49.5–104.5) 0.68 (0.49–0.83) 0.11 (0.08–0.15) 32 (23–41)5.77 ! 3.00 0.09 (0.07–0.13) 0.87 (0.52–1.37) 1.29 (0.96–1.64) 1.22 (1.15–1.25) 74.6 (48.3–103.6) 0.68 (0.50–0.83) 0.10 (0.08–0.13) 30 (22–36)4.64 ! 4.64 0.09 (0.07–0.13) 0.88 (0.53–1.36) 1.30 (0.96–1.63) 1.22 (1.15–1.25) 72.6 (47.1–101.6) 0.68 (0.49–0.83) 0.10 (0.08–0.13) 28 (21–35)

R2 0.079 0.143 0.178 0.032 0.002 0.032 0.714 0.532

Except for the two-dimensional tree (h = 0), the perfused volume was 100 cm3.Values are median, first and third quartile (in brackets). R2 values are taken from analyses of variance of ranked data. Due to the large number of segments in

each tree, the overall effect of s ! h on each variable was statistically significant (at the level of significance of · = 0.0001), but only at larger values of R2 aconsiderable number of pair-wise comparisons between trees were significant.

the perfusion volume, the curves are distinctly different,and – except for the two flattest trees – all pair-wise com-parisons of (ranked) means were statistically significant(table 2). Increasing the side-length of the perfusion vol-ume (4.64 cm ̂ s ̂ 44.72 cm), we observed for the mainvessel a roughly eight-fold increase of length (from 8.87cm in the cube to 68.17 cm in the 2D tree). At the sametime, however, figure 4 shows that the maximum numberof bifurcation levels increased only three-fold (from 57 to180). In other words, the number of bifurcation levels perunit length, or ‘bifurcation density’, is drastically reduced(from 6.43 to 2.64) when the perfusion volume flattensfrom cubic to square.

Quantitative Analysis of TreesPath Analysis: Transport Route and Deposition Time.

Considering quantitative properties of CCO trees, ourmajor concern was to characterize how pathways of bloodtransport depend on thickness and shape of the volume tobe perfused. The length of a path P, which connects theroot segment with one of 8,000 terminal segments anddelivers blood to the respective perfusion site, is definedby

LP = ™ li, i D path P (11)

i.e., LP is the sum of lengths of all segments i along P(fig. 2a). It should be mentioned that blood cells to betransported to different terminal locations may use com-mon pathways even over large distances before separating

and finally arriving at individual terminal destinations.Figure 5 shows the mean transport length, 7LP 8, i.e. thedistance blood cells have to cover on average. In CCOtrees (denoted by symbols), the mean transport lengthincreased from 5.88 to 39.6 cm when the side-lengthbecame longer (from 4.6 to 44.72 cm) and the structuremore and more flat. Two-dimensional trees were almostidentical with a very thin (0.05 or 0.1 cm) tree of equalside-length. The solid line in figure 5 represents 7LP 8 forthe single-tube arrangement, i.e. the reference value ofmean transport length at a given ratio h/s of the perfusionvolume if blood delivery to 8,000 randomly distributedterminal sites were performed by individual straight cy-lindrical tubes (Eq. 5). Obviously, due to directionalchanges of segments and curvature of vessels, blood trans-port by tree structures is less straightforward in compari-son to a system of individual, noninterconnected tubes.Quantitatively, 7LP 8 of trees exceeded that of straighttubes by 22% (2D) to 32% (cube).

Based on these results, one might reason why transportof nutrients for the tissue is usually not accomplished byindividual straight tubes originating in the source of bloodsupply, and why vessel systems rather emerge as tree-shaped structures during the development of the organ-ism. Apart from the problem of spatial arrangement, suchsingle-tube perfusion is obviously not compatible withnature’s optimization principles because of costly re-sources other than path length. Figure 6 impressivelyshows the tremendous cost of total blood volume (i.e.,


Fig. 5. Mean length of blood transportroutes (7LP8) in CCO trees perfusing (square-based) volumes of different side-length s andheight h, represented by their ratio h/s. Ab-solute values of side-length and height areindicated by s ! h for each tree (symbols).Except for h = 0 cm, perfusion volume wasconstant at Vperf = 100 cm3. The number ofblood-delivering terminal segments was8.000 in all trees. Solid line = mean transportlength for 8,000 randomly distributed termi-nals perfused by single-tube arrangement (cf.Methods).

Fig. 6. Total intravascular volume (Vtot) ofCCO trees perfusing (square-based) volumesof different side-length s and height h, repre-sented by their ratio h/s. Absolute values ofside-length and height are indicated by s ! hfor each tree (symbols). Except for h = 0 cm,perfusion volume was constant at Vperf = 100cm3. The number of blood-delivering termi-nal segments was 8,000 in all trees. Solid line= total ‘intravascular volume’ of a single-tube arrangement perfusing 8,000 randomlydistributed terminals.

intravascular volume required to simply fill the vessel sys-tem and serving as transport medium of nutrients) incomparison to tree structures at equal in- and outflowconditions. Vessel trees (symbols in fig. 6) carry only 4%(2D) to 9% (cube) of the blood volume of single-tube sys-tems (solid line, again representing a theoretical curve forsingle-tube perfusion; see also Eq. 6).

Combining these two quantitative results, we couldshow for square slabs that at relatively low additional

costs of transport length, a tremendous saving of blood (i.e.of transport medium) is attained by using tree structures toperfuse tissue of any thickness and shape. Comparing totalintravascular volume between CCO trees, changes of shapefrom cubic to flat resulted in an increase of one order ofmagnitude. This finding corresponds to results from a pre-vious study in two-dimensional CCO trees, where signifi-cant departures of shape from circular resulted in a con-comitant increase of intravascular volume [23].


J Vasc Res 2000;37:250–264 257

Fig. 7. Probability density (p) for the lengthof blood transport routes (LP). Values for s !h denote side-length and height of the(square-based) perfusion volume (Vperf = 100cm3 ). The number of blood-delivering ter-minal segments was 8,000 in all trees. Thecurve for h = 0 cm was almost identical tothat for h = 0.05 cm and has been omitted.

Fig. 8. Probability density (p) for the deposi-tion time of blood cells (Td ). Values for s ! hdenote side-length and height of the (square-based) perfusion volume (Vperf = 100 cm3).The number of blood-delivering terminalsegments was 8,000 in all trees. The curve forh = 0 cm was almost identical to that for h =0.05 cm and has been omitted.

Figure 7 gives a more detailed view of how path lengthof blood transport is distributed within the trees. In corre-spondence to mean transport length (see above), weobserve a shift of the probability density curves to theright when shape changes from cubic to flat, accompaniedby a change of skewness of the curves.

Of more practical significance, we can consider thetime (‘deposition time’, Td) it takes a blood cell to travelalong its path from the root segment to the perfusion site.

The distribution of deposition times within a tree is simi-lar, but not identical, to that of transport lengths (fig. 8).Assuming that flow velocity, v, is constant and uniformwithin each segment, then

¢ti = li/vi (12)

is the deposition time of blood transport through segmenti. Since (due to bifurcations) all vi’s along a whole trans-port route P (from root to terminal) are different, deposi-


tion time and transport length along a given path are notsimply proportional. But qualitatively, figures 7 and 8 aresimilar, and we observe a distinct increase of transportlength and time when the perfusion volume becomes flat.Statistically, all pair-wise comparisons of (ranked) meanswere significant, except for h ^ 0.1 cm. Deposition timeof the CCO-simulated vessel trees cannot be directly relat-ed to measurements of transit time after bolus adminis-tration in studies on perfusion or pharmacokinetics aslong as the model does not comprise the microcirculationand the venous portions of the circulatory system.

Analysis of Segments and Bifurcations. Properties oftransport paths as presented so far could have been intui-tively expected to depend on the side-length (and basis) ofthe perfusion volume, or, in other words, on the maxi-mum elongation of the tree structure (fig. 3). Clear depen-dencies were confirmed by high R2 values of explainedvariation (table 2, columns 7 and 8) and significant differ-ences between trees for almost all pair-wise comparisons.However, when looking at individual segments and bifur-cations, shape-dependent trends are no longer evidentand less clear (R2 values were low and Duncan groupingshowed only subsets of similar trees differing significantlyfrom each other).

Segment Radii, Lengths and Spacing. On average, bothsegment radii and lengths gradually decreased as long as hwas increased from 0 to 0.6 cm, but remained stable inthicker perfusion volumes (table 2, columns 1 and 2). Thisresult corresponds closely to the values of total volumegiven in figure 6. Similarly, the average spacing within thetree, as roughly characterized by the nearest-neighbor dis-tance (dn) of terminal sites, decreased drastically (45%)when h was increased from 0 to 0.6 cm and remainedalmost constant in thicker perfusion volumes (table 2, col-umn 3).

Area-Expansion Ratio. The growth of the cross-sec-tional area at bifurcations due to the radii of the parent (r)and its daughter segments (r1 and rs) can be characterizedby the area-expansion ratio [24]:

Aexp = r1

2 + rs2

r2, Aexp 6 1 for Á 6 2 (13)

Table 2, column 4, shows an average area-expansion ratioof approximately 1.22, except for slightly smaller valuesin very thin perfusion volumes. Within each tree, Aexp wasrelatively stable over a wide range of bifurcation levels;smaller values were observed for proximal portions of thetree (at low values of §bif ), large values for distal portionswith high values of §bif.

Segment Orientation. A descriptor much more closelycorrelated with the ‘dimensionality’ of the tissue sample isgiven by the orientation of individual segments, repre-sented by the angle (Ê) between the direction of bloodtransport through the segment and a straight line connect-ing the segment with the inlet of the tree (see Eq. 8 andfig. 2a). Segments characterized by small values of Ê carryblood ‘away from’ the root, while large values of Ê indi-cate the opposite transport direction ‘toward root’. Fig-ure 9 shows the distribution of Ê in five selected trees. Weobserved an approximately straight line for the two-dimensional tree, while the distribution was a unimodalcurve in all three-dimensional trees (with the maximummoving to the right when shape changed from flat to cub-ic). This drastic difference suggested a fundamental effectof the geometry, i.e. the dimensionality, of the perfusionvolume. To elucidate this presumption, we calculated ref-erence densities of Ê for randomly selected ‘perfusionsites’ and ‘flow directions’ within a unit square (2D; Eq.10) and cube (3D; Eq. 9). These reference curves as shownin figure 9 yield a horizontal line (uniform distribution)for 2D, and a sine function for 3D, quantitatively reflect-ing the difference already known from theoretical consid-erations (see Methods). Deviations of the tree data fromthe reference curves can be explained by the ‘orderingeffect’ of a tree structure, as opposed to randomly ar-ranged single segments. In the tree, at least for large seg-ments, there is a general orientation of segments in thedirection of blood flow, even if smaller and terminal seg-ments may run in the opposite direction. Quantitatively,the deviations of the density curves for three-dimensionaltree data from the sine curve can be quantified by theskewness, which is (for theoretical reasons) zero for thesine curve (i.e. the distribution is symmetric about Ê =90°). In CCO trees, skewness was in the range of 0.22–0.36. In two dimensions, fitting a straight line to the da-ta of the tree yielded a non-zero slope of –2.99 ! 10–5

deg–2.Bifurcation Symmetry. The radii of the two daughter

segments of a bifurcation are obtained as a result of theoptimization process. The ratio of these radii can be char-acterized by a symmetry index

Írad = rs/r1, 0 ! Írad ^ 1 (14)

where r1 and rs denote the radii of the larger and smallerdaughter segment. Considering probability densitiesagain (fig. 10), it was evident (and in agreement with sta-tistical comparisons of means) that flat trees with h ^0.3 cm differed from others by a lower portion of symmet-ric bifurcations (i.e. Írad close to 1).


J Vasc Res 2000;37:250–264 259

Fig. 9. Probability density (p) for the angleof segment orientation (Ê). Values for s ! hdenote side-length and height of the (square-based) perfusion volumes. Except for h = 0cm, perfusion volume was constant at Vperf =100 cm3. The total number of segments was15,999 in all trees. Two reference curvesrepresent the distribution of Ê for randomlychosen points in a unit square (2D) and aunit cube (3D; cf. Methods).

Fig. 10. Probability density (p) for the bifur-cation symmetry (Írad). Values for s ! hdenote side-length and height of the (square-based) perfusion volume (Vperf = 100 cm3).The number of bifurcating segments was7,999 in all trees. For reasons of clarity, treeswith h = 0, 1.5, and 3 cm have been omit-ted.

Visual inspection of the basic pattern of the major,noticeable branches shows a great similarity between alltrees (fig. 3) which is due to the same ‘seed’ of the pseudorandom number sequence used in all simulations [6].However, while the major vessels of trees in flatter vol-umes (h ^ 1.5 cm) (fig. 3a–c) show an almost constantpattern, we observe a drastic change in the rest of thetrees: looking at the cube or the thick slab with h = 3 cmgives the impression of the tree being divided into a num-

ber of equivalent subtrees, while the structure of flattertrees seems to be dominated by three major vessels. Theslab with h = 2 cm assumes an intermediate position.Could this effect result from different symmetry proper-ties of the bifurcations in different trees? In other words,is it the branching pattern of the major vessels whichcauses a different distribution of Írad in ‘flat’ and ‘thick’perfusion volumes? To address this question, we focusedon the branching properties of the main vessel (as the


Fig. 11. Average radii of parent segments ofsubtrees branching from the main vessel. Ac-cording to the definition of the main vesselas route following the larger daughter seg-ment at each bifurcation, each subtree par-ent is the smaller daughter (rs ) of a main ves-sel segment. Segments are classified accord-ing to their relative distance from the rootsegment (i.e. the ratio of main vessel length)and represented by the median of rs. Valuesfor s ! h denote side-length and height of the(square-based) slabs. Except for h = 0 cm,perfusion volume was constant at Vperf =100 cm3.

most prominent representative of major vessels), as wellas on the subset of pre-terminal segments which bifurcateinto no more than two terminal segments. This latter setof segments comprises those 13–15% of segments of a treewhich split highly symmetrically into exactly two terminalsegments (with medians of Írad in the range of 0.77–0.83in all trees). In general, all other segments of the tree bifur-cate less symmetrically into subtrees of unequal ‘size’ (i.e.number of segments), accompanied by a decrease of Írad

from pre-terminals to the segments of the major vessels.For the main vessel, we obtained the medians of Írad rang-ing from 0.11 in the two-dimensional tree to 0.41 in thecube, reinforcing the previous assumption that inter-treedifferences of Írad (fig. 10) result from the branching pat-tern of the major vessels.

Furthermore, visual inspection of the trees clearly sug-gests for the two-dimensional tree that most of the sub-trees branching out from major vessels are relatively smalland look similar (fig. 3a). This implies that the radii of theparent (i.e. ‘root’) segments of the branching subtreesshould be similar. On the other hand, subtrees of consid-erably different size appear to branch from major vesselsin thick perfusion volumes (fig. 3e, f). Figure 11 shows theparent radii of subtrees branching out of the main vessel,regionally classified by their relative distance from theinlet. It is evident that flat structures (with h ^ 1 cm)show a much more uniform distribution of branchingradii throughout all sections of the main vessel than thick

slabs. Within 60% distance from the root along the mainvessel, the branching segment radii in the cube are 2–3times larger compared with the two-dimensional tree,while in the distal sections of the main vessel the differ-ences become more and more negligible.

Discussion

In the present paper, we analyzed a set of optimizedarterial tree models to evaluate how topological propertiesand functional characteristics depend on the shape of theperfusion volume. In particular, we focused on routes ofblood transport, symmetry properties of bifurcations, andthe direction of blood flow of individual segments in aseries of CCO-generated arterial trees whose 3D perfusionvolume was constant at Vperf = 100 cm3. All model param-eters were kept constant, while only the shape of the per-fusion volumes varied from a ‘thick’ cube to a ‘flat’ slab,with side-lengths (s) increasing from 4.64 to 44.72 cm.Additionally, two 2D (Vperf = 0) trees were generated toperfuse squares of s = 31.62 cm and s = 44.72 cm.

Blood Transport and HemodynamicsAs expected, a decrease of ‘thickness’ (h) and a con-

comitant increase of side-length, i.e. a decrease of thecharacteristic h/s ratio, were closely related to the averagetransport distance and deposition time of blood cells.


J Vasc Res 2000;37:250–264 261

Reducing h/s from 1 to 0 by more and more stretching theside-length and flattening the perfusion volume results ina seven-fold increase of mean transport length and atwelve-fold increase of deposition time. As a conse-quence, the total intravascular volume of blood, i.e. of thetransport medium, increased as well (up to nine-fold). Asto the topological structure, this variation of shape wasaccompanied by a tripling of bifurcation levels of thetree.

Transport Systems. Comparing a tree structure with asystem of individual straight tubes originating in a com-mon source of blood supply and terminating at randomlyselected ‘perfusion sites’ convincingly showed the superi-ority of tree structures in terms of ‘costs’: at the additionalexpense of approximately 20% length of transport dis-tances, tree structures fulfill their task of blood transportat the cost of only "5% of blood volume of a nonbranch-ing system! This effect of organized tree structure on vol-ume reduction significantly dominates the effect of op-timization: using intravascular volume, Vtot, as the targetfunction for tree generation (which is in agreement withtotal volume being generally accepted as an optimizationprinciple for single bifurcations and tree structures [15,25–27]), minimum total volume is obtained. Tree optimi-zation according to other criteria results in an only moder-ate variation of Vtot (by a factor of 2.5 at constant perfu-sion area, boundary conditions, and constraints [16]).

Pressure and Flow. Until recently, experimental mea-surements and hemodynamic data on individual seg-ments of vascular trees were scarce. However, it couldalready be shown [4, 9] that classifying radii by bifurca-tion levels and comparing means and standard deviationsof CCO models to real trees [21] yields good agreementfor coronary arteries.

As an example of the hemodynamics in the presentCCO models, we consider blood pressures and flows forsegments of different radii in one tree and refer to a nowavailable complete database of microvascular networkstructure and hemodynamics for the rat mesentery [28].Experimental measurements of diameter, length, hemato-crit (7 networks; 3,129 segments), and flow velocity (3 net-works; 1,321 segments) in arterioles, capillaries, and ven-ules were obtained from intravital microscopy. Fromthese measurements, the missing flow velocities were cal-culated from diameters by an empirical linear regression;pressures and volumetric flow rates per segment were esti-mated by means of network flow simulations based on theexperimentally determined topology, geometry, and he-modynamic boundary conditions [29]. Pressure and flowdata for arterioles (150 B 26 segments over 10.2 B 4.4

generations per network) were assigned to classes of seg-ment diameters (in the range of 0.004–0.034 mm) andreported as means and standard errors of the mean for theseven networks. Pressures were normalized to a scaleranging from 1 (pressure in the main input arteriole) to 0(main draining venule).

Accordingly, we classified all non-terminal segments ofthe two-dimensional (flat) model tree by diameters (cov-ering a range of 0.2–4.2 mm), and normalized perfusionpressure and (constant) terminal pressures to 1 and 0,respectively. Figure 12a shows relative experimental andmodel pressures (prel) on different, yet proportional, scalesof radii (top and bottom) in satisfying qualitative agree-ment.

For volumetric flow (Q), power laws were fitted to boththe experimental and model data points, resulting instraight lines on the double-logarithmic plot of figure 12b.We obtained excellent quantitative agreement in terms ofalmost identical slopes of the radius-flow relationship, ascan be seen from Eq. 15 and 16 for experimental andmodel data, respectively:

log10(Q) = –0.347 + 2.789 log10(r) (15)

log10(Q) = –0.206 + 2.753 log10(r) (16)

with Q expressed in cm3 s–1 and r in mm. 1 In other words,downstream reduction of radii by one order of magnitudewithin the arterial tree was accompanied by a flow reduc-tion of "2.75 orders of magnitude in both cases. The pre-diction of the radius-flow relationship for arteriolar radiiin the range of 10–3 to 10–2 mm by extrapolation of themodel regression line roughly coincides with the straightline fitted to the experimental data points. The differenceof flow on the absolute scale can readily be explained as aconsequence of the specific model parameters (see below).It should be kept in mind that the present set of parame-ters has originally been chosen to model the myocardialarteries and might not adequately reproduce the mesen-teric tree. Finally, a similar shift on the absolute scale offlow may as well be found experimentally in a comparisonof the radius-flow relationship between different ranges ofradii of the same organ.

It is obvious from figure 12 that a straight line does notcorrectly represent the experimental relationship betweenradius and flow, and the power law suggested by the mod-

1 Estimated standard errors of intercept and slope are 0.350 cm3 s–1 and0.166 ml min–1 mm–1 for Eq. 15, and 0.047 cm3 s–1 and 0.127 ml min–1 mm–1

for Eq. 16.


Fig. 12. Comparison of results from a two-dimensional CCO model tree with data from experimental measurementsin the rat mesentery [28]: relative mid-segment pressure (prel ) and volumetric flow (Q) as classified by radii (r). Model([) and experimental data (P) are represented as means. a Bars indicate standard deviations of the model data forprel. Axes on the bottom and top refer to model and experiment, respectively. b Straight lines were fitted to the pointsof model (––––) and experimental data (–––) according to Eq. 15 and 16.

el data is not strictly obeyed. At least in part, this could beexplained by the model assumption of constant viscosityof blood throughout the whole tree, which certainly doesnot hold for real trees. And also in real organs, the shape ofthe radius-flow relationship may be different between pre-capillary and much larger (two orders of magnitude) arter-ies.

Branching SymmetryWith regard to bifurcation symmetry, defined by the

ratio (Írad ) of the radii of the smaller and larger daughtersegments of each bifurcation, perfusion volumes with athickness of at least 0.6 cm had a higher number of sym-metric bifurcations than flat trees. This result encouragedfurther analysis of the distribution of Írad within the tree.Visual inspection of the trees showed that the few, rela-tively long, major vessels of flat trees were flanked allalong their course by a great number of small subtrees ofapproximately equal size. In contrast, in thick perfusionvolumes major vessels seemed to divide into subtrees

repetitiously splitting into smaller and smaller ones. Thisobservation could be verified by statistical analysis of sub-trees branching from the main vessel, in comparison toterminal subtrees (i.e. to pre-terminal segments splittinginto two terminals). While the symmetry of terminal sub-trees is similar in all trees, the average Írad of the mainvessel of the cube was drastically higher compared to thetwo-dimensional tree.

This concept of major vessels branching into large sub-trees is obviously realized in real arterial trees: for in-stance, rather than winding around the anterior and pos-terior sections of the left ventricle as a single large vesseland repeatedly dismissing small side branches, the leftmain coronary artery splits into the subtrees of the leftanterior descending and the circumflex artery and theirrespective main branches (e.g. [30]). In combination withour results on transport length, deposition time and totalintravascular volume, we conclude that branching intolarge subtrees of similar size facilitates blood transportover short distances at small total blood volume, in com-


J Vasc Res 2000;37:250–264 263

parison to one major vessel inefficiently carrying a largevolume of blood (mostly ‘dead volume’) through a systemof multiple length.

DimensionalityAs to the question of dimensionality, we compared a

two-dimensional tree of side-length s = 44.72 cm withthree-dimensional trees of various shapes. Average trans-port length and deposition time, total intravascular vol-ume, and the distribution of the bifurcation symmetryindex within the 2D tree were found to be almost identicalto that of a ‘very thin’ structure (h = 0.05 cm) of equalside-length (s = 44.72 cm). This observation similarlyholds for two-dimensional trees of any side-length, and weconclude that with respect to these properties, two-dimen-sional CCO trees can be considered conceptually as thelimiting case of three-dimensional trees.

In contrast, segment orientation as characterized bythe angle Ê between the segment’s symmetry axis and thedirection to the origin of the tree (as rather ‘away from’ or‘toward’ the root segment) could be shown to be qualita-tively different in two- and three-dimensional CCO trees,and to produce probability distributions of fundamentallydifferent shape. While Ê is uniformly distributed in a two-dimensional area perfused at random terminal sites, theprobability density of Ê is a sine curve for random perfu-sion sites in a three-dimensional volume. Deviations ofCCO trees from these idealized density curves (i.e. astraight line with non-zero slope in the 2D tree, skewedunimodal curves in 3D trees) result from the nonrandomstructure of the tree, i.e. from the general tendency of ves-sels to direct flow ‘away from’ the root. Moreover, theparameter of the optimization target function [16, 31] or,as in the present case, the choice of the thickness/length(h/s) ratio of the perfusion volume determines the partic-ular shape of the curve.

Model Assumptions Affecting ResultsAs mentioned in the beginning, any model approach is

a simplified representation of reality, neglecting quite anumber of well known effects and dependencies of realsystems. While in CCO models the effect of rheologicalproperties and flow conditions (i.e. fine-tuning for exactquantitative reproducibility of flow) has not yet been tak-en into account, a systematic variation of individualparameters shows how the results depend on the specificassumptions of the model. For instance, reducing theoverall pressure gradient (¢p) of the CCO tree wouldresult in a downward shift of the radius-flow relationshipof figure 12b. A more complex effect can be observed

when the value of the exponent Á of the bifurcation law(Eq. 3) assumes different values. Variation of Á within rea-sonable limits (e.g. between 2 and 3) results in a consider-able change of the pressure profile of the tree [4]. Plottingpressures over (logarithmic) radii of the segments of aCCO tree yields an approximately straight line for Á "2.5. For Á ! 2.5, the pressure profile is fairly flat across thelarger branches of the tree and growing steeper towardsmaller radii, whereas the reverse is true for Á 1 2.5.

Conclusion

The CCO method offers the unique possibility of gen-erating vessel trees without the input of anatomical ormorphometric information, and without using evolution-ary laws for the creation of the tree structure. Instead, atgiven boundary conditions and constraints, optimizationprinciples govern the geometry and topology of the result-ing vessel tree. In the present study, a set of arterial CCOtrees, optimized for total intravascular blood volume,were grown to perfuse volumes (‘pieces of tissue’) of dif-ferent shape, ranging from a two-dimensional square to athree-dimensional cube. Effects of shape variation on theresulting structure could be clearly demonstrated forsome, mostly quantitative, characteristics of the trees. Insummary, a thickness-to-side-length ratio of h/s 1 0.1facilitates fast delivery of blood at lower total intravascu-lar volume and shorter transport distances, in comparisonto large amounts of dead volume to be carried a long wayby the vessel system at lower h/s.

One could argue that our optimization principles werenot very realistic at all, or at least that optimizationaccording to any other property (e.g. constant shear stressin all segments) might have produced different results.But even if the results quantitatively depend on the partic-ular choice of optimization principles, this study showedthat (1) CCO trees could be grown as geometricallyarrangeable, visually ‘realistic’ structures within any ofthe shapes, (2) the tree structure definitely changed ifshape was systematically modified, and (3) a great num-ber of ‘morphometric’ properties is available in CCO treesto evaluate the effects of parameter variation by statisticalanalysis.


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