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Phys. Med. Biol. 44 (1999) 2929–2945. Printed in the UK PII: S0031-9155(99)04730-2
On the design of the coronary arterial tree: a generalization of
Murray’s law
Yifang Zhou†, Ghassan S Kassab‡ and Sabee Molloi†§
† Department of Radiological Sciences, University of California, Irvine, CA 92697, USA
‡ Department of Bioengineering, University of California, San Diego, La Jolla, CA 92093, USA
E-mail: symolloi@uci.edu
Received 1 June 1999
Abstract. Murray’s law has been generalized to provide morphometric relationships among
various subtrees as well as between a feeding segment and the subtree it perfuses. The equivalent
resistance of each subtree is empirically determined to be proportional to the cube of a subtree’s
cumulative arterial length (L) and inversely proportional to a subtree’s arterial volume (V) raised to
a power of approximately 2.6. This relationship, along with a minimization of a cost function, and a
linearity assumption between flow andcumulative arterial length, provides a power lawrelationship
between Vand L. These results, in conjunction with conservation of energy, yield relationships
between the diameter of a segment and the length of its distal subtree. The relationships were
tested based on a complete set of anatomical data of the coronary arterial trees using two models.
The first model, called the truncated tree model, is an actual reconstruction of the coronary arterial
tree down to 500 µm in diameter. The second model, called the symmetric tree model, satisfies
all mean anatomical data down to the capillary vessels. Our results show very good agreement
between the theoretical formulation and the measured anatomical data, which may provide insight
into the design of the coronary arterial tree. Furthermore, the established relationships between the
various morphometric parameters of the truncated tree model may provide a basis for assessing the
extent of diffuse coronary artery disease.
1. Introduction
There have been numerous theoretical attempts to explain the design of vascular trees based on
the principles of minimum work (Murray 1926a,b, Oka 1974, 1984), optimal design (Rosen
1967), minimum blood volume (Kamiya and Togawa 1972) and minimum total shear force on
the vessel wall (Zamir 1976, 1977). These attempts have resulted in relationships between the
diameter of the mother and daughter vessels at a bifurcation and the branching angles of the
bifurcations. It was Murray, however, who first derived a relationship between the diameter of
the mother and daughter vessels at a bifurcation. He proposed a cost function that is the sum of
the friction power loss and the metabolic power dissipation proportional to blood volume. By
minimizing the cost function, Murray derived an optimal condition for a vascular bifurcation,
referred to as Murray’s law, which states that the cube of the radius of a parent vessel equals
the sum of the cubes of the radii of the daughter vessels. Murray’s law has been applied to a
number of vascular trees in various organs (Mayrovitz and Roy 1983, Zamir et al 1979, Zamir
§ Corresponding author: Sabee Molloi, PhD, Department of Radiological Sciences, Medical Sciences I, B-140,
University of California, Irvine, CA 92697, USA.
0031-9155/99/122929+17$30.00 © 1999 IOP Publishing Ltd 2929
2930 Y Zhou et al
and Medeiros 1982, Zamir and Brown 1982, Zamir et al 1984, Zamir and Chee 1986, Kassab
and Fung 1995).
Although Murray’s law can explain the design of a bifurcation fairly well (see the review
in Kassab and Fung, 1995), it is local in nature. It would be very useful to formulate an
analysis that is more global and applies to an entire tree or its subtrees. Such analysis would,
for example, relate the diameter or cross-sectional area of a mother vessel to the total length
or volume of its distal tree. Based on angiographic data, there has been important progress
in finding relationships between various morphometric parameters for the epicardial coronary
arterial trees (Koiwa et al 1986, Seiler et al 1992, 1993, Wahle et al 1993, 1994, 1995).
Seiler et al (1992, 1993) reported a power law relationship between the cross-sectional area
of the feeding vessel and the cumulative arterial length of the tree perfusing a myocardial
region. The power law relationship was shown to be comparable in both canine and human
studies. In a more recent human study, a correlation was made between the feeding vessel
and its dependent perfusion bed’s morphometric parameters based on the weighted sum of
15 correlation coefficients (Wahle et al 1993, 1994, 1995). The morphometric parameters
included cross-sectional area, volume and length. Despite this progress, however, there is a
need for a better formulation of the basic design principles underlying the morphology of the
coronary arterial tree.
Specifically, the following questions still remain. Is it possible to provide the underlying
physical principles governing the global design of the arterial tree, from which the already
found morphometric relationships can be predicted? If so, what additional morphometric
relationships can be derived? Motivated by these questions, we generalized Murray’s law
to provide morphometric (diameter or cross-sectional area, length and volume) relationships
among the various subtrees as well as between a feeding segment and the subtree it perfuses.
Some of the hypotheses that underlie the analysis are confirmed using haemodynamic circuit
analysis based on the detailed anatomical data of the porcine coronary arterial trees (Kassab
et al 1993). Two vascular circuits are specified for a haemodynamic analysis of coronary
blood flow. One, called the truncated tree model, is an actual reconstruction of the coronary
arterial tree down proximal to 500 µm in diameter. The other, called the symmetric tree
model, satisfies all the mean anatomical statistical data, but not the standard deviations and
connectivity matrix (Kassab et al 1997). The former model is useful for angiographic studies
whose spatial resolution is approximately 500 µm while the latter model is an idealization of
an entire tree down to the capillary level.
2. Methods
2.1. Theoretical formulation
2.1.1. Optimaldesignofa crown—ageneralizationofMurray’slaw. Inan efforttogeneralize
Murray’s law to the entire coronary arterial tree or subtree, we define the following quantities
similarly to Wahle et al (1994, 1993/94, 1995): a stem is considered as a segment of vessel
between any two successive bifurcations, and a crown is considered as a subtree distal to
a stem. In the foregoing analysis, we shall assume that the terminal vessels of all crowns
have a uniform pressure distribution. Therefore, for each crown, one can define an equivalent
resistance as
Rc=1p
Q(1)
where 1p is the pressure drop along the crown, and Qis the volumetric flow through the
crown’s stem.
A generalization of Murray’s law 2931
We shall now generalize Murray’s law to apply to a crown. For each crown, a power
dissipation function, expressed in terms of total arterial volume (V) and the total arterial
length (L), is proposed as the cost function
F(L,V) =Q
2(L)Rc(L, V ) +kmV(2)
where Qis a function of Lfor a given pressure drop between the coronary ostia to the crown’s
terminal vessels, Rcis a function of Land Vand kmis a metabolic constant of blood. The
first term in equation (2) is the power dissipation due to the viscosity of blood, and the second
term is the metabolic requirement of the blood volume, which is assumed to be proportional
to the arterial volume (Murray 1926a,b, Rosen 1967, McDonald 1960).
For a single vessel, assuming that blood flow obeys Poiseuille’s law, the resistance can be
written as
Rs=8πµl3
v2(3)
where µis the viscosity coefficient, land vare the vessel’s length and volume, respectively.
We shall next propose an empirical form for the equivalent resistance of a crown to be
Rc(L, V ) =kR
L3
V2V
Vref ε(4)
where kRis assumed to be a constant (viscosity coefficient is absorbed in kR). The term
kRL3/V2is analogous to Poiseuille’s resistance with Land Vcorresponding to the cumulative
arterial length and volume of a crown, respectively. The term (V/ Vref)εis a scaling factor
for crown volume with εas an empirical constant to be determined from a least squares fit
of the anatomical data. The scaling factor is motivated by the self-similar, fractal nature of
the arterial tree (Bassingthwaighte et al 1990, Kassab et al 1997), where Vref is taken as a
reference volume. The validity of the equation (4) is tested using the measured anatomical
data, assuming the resistance in each segment obeys equation (3).
Next, we shall nondimensionalize the cost function (equation (2)) with respect to the
metabolic power requirements of the entire tree of interest, kmVmax, to obtain
f≡F(L,V)
k
mV
max
=kR
kmVmaxVε
ref Q2(L) L3
V2−ε+V
Vmax (5)
where Vmax is defined as the arterial volume of the entire tree of interest. By introducing the
following normalized (nondimensional) arterial volume, arterial length and flow,
vN=V
Vmax lN=L
Lmax qN=Q
Qmax (6)
where Lmax and Qmax are the cumulative arterial length of the entire tree of interest and the
entranceflowthrough the most proximal segment,respectively,thedimensionless costfunction
can be written as
f=kRQ2
maxL3
max
kmV3−ε
max Vε
ref q2
Nl3
N
v2−ε
N
+vN.(7)
We shall next hypothesize that the flow through any stem is proportional to the total distal
length of the corresponding crown as given by
Q(L) =kQL(8a)
or in a normalized form as
qN=lN(8b)
2932 Y Zhou et al
where kQis a proportionality constant that depends on the pressure drop between the
coronary ostia and the crown’s terminal vessels. This assumption will be evaluated using
flow simulations in the next section.
Therefore, equation (7) can be expressed, in terms of lNand vN,as
f(l
N,v
N)=k
Rk
2
QL
5
max
kmV3−ε
max Vε
ref l5
N
v2−ε
N
+vN.(9)
We shall now minimize the dimensionless cost function given by equation (9) with respect to
the dimensionless arterial volume as
∂f (l N,v
N)
∂vN
=0 (10)
which yields a solution for an optimized arterial volume expressed as
vN=(2−ε)kRk2
QL5
max
kmV3−ε
max Vε
ref 1
3−ε
l
5
3−ε
N.(11)
By definition, vN=1 when lN=1, which leads to the following constraint:
Vmax =(2−ε)kRk2
Q
kmVε
ref 1
3−ε
L
5
3−ε
max (12)
that can be used to simplify equation (11) to
vN=l
5
3−ε
N.(13)
It can be verified that the second derivative of the cost function with respect to vNis
positive, thus equation (11) or (13) does represent a local minimum of the power dissipation.
The minimized dimensionless power can be obtained, by combining equations (9), (12) and
(13), as
fmin(lN)=3−ε
2−εl
5
3−ε
N.(14)
Therefore, equations (12), (13) and (14) represent an optimization hypothesis for the crown’s
structure. Equations (12) and (13) indicate that the optimal morphology obeys a power law
relationship where the exponent depends on the crown’s equivalent resistance parameter ε.
2.1.2. Morphometric relationships between a stem and its crown: conservation of energy. In
order to obtain a stem–crown relationship, we shall apply the law of conservation of energy,
in conjunction with the results obtained above.
A crown can betreated as an entity whoseequivalent resistance, volume–length and power
dissipation obey equations (4), (13) and (14), respectively. The inlet of the crown is its stem
whose mean diameter and mean cross-sectional area are denoted as Dsand As, respectively.
The outlet is the crown’s terminal vessels that have the same outlet pressure. If the heat input
from the stem to the crown, the work done through thevessel wall, the kinetic energy across the
vessel wall and the gravitational potential energy are assumed to be negligible, the equation of
energy conservation can be written (Fung 1997), in terms of work rate and power dissipation,
as
ZAs
pu dA−ZAo
pu dA=ZAo
1
2ρq2udA−ZAs
1
2ρq2udA
+ZV
∂
∂t 1
2ρq2dv+Pvis +Pmeta (15)
A generalization of Murray’s law 2933
where pis the pressure, uthe longitudinal flow velocity, Aothe equivalent cross-sectional area
of the crown’s outlet, Vthe arterial volume of the crown, ρthe blood density in the coronary
arteries and qthe flow speed. The power dissipation due to viscosity and metabolism are
denoted as Pvis and Pmeta, respectively, as expressed in equation (2).
We shall next simplify equation (15) in order to derive a stem–crown morphometric
relationship. The first two terms on the right side of equation (15) represent the rate of
change of kinetic energy flowing in and out of the crown. The third term on the right side is
the rate of change of kinetic energy in the crown volume, which equals zero for a steady-state
flow. As a further simplification, we assume that the rate of change of kinetic energy flowing
into the crown is much larger than that flowing out of the crown. Given these assumptions,
equation (15) can be simplified to
ZAs
pu dA−ZAo
pu dA=−
ZA
s
1
2ρq2udA+Pvis +Pmeta.(16)
Using the condition that the outlet pressure of the crown is uniform, it can be shown that
the left side of equation (16) is equal to Pvis (refer to equation (2)):
ZAs
pu dA−ZAo
pu dA=(ps−po)Q =(RcQ)Q =Pvis (17)
where psis the mean pressure at the inlet of the crown, and pothe mean pressure at the outlet
of the crown. Therefore, equation (16) can be written as
ZAs
1
2ρq2udA=Pmeta.(18)
For an incompressible laminar flow, the left side of equation (18) can be expressed in
terms of the volumetric flow rate and the stem’s cross-sectional area as
ZAs
1
2ρq2udA=aρ Q3
A2
s
(19)
where ais a dimensionless constant which is equal to one for a parabolic velocity profile.
Equations (18) and (19) can be combined to yield the following relation
aρ Q3
A2
s
=kmV. (20)
Next, we shall write equation (20) in a dimensionless form and combine it with equations (8)
and (13) to yield
aN=saρk 3
QL3
max
kmVmaxA2
smax l
4−3ε
2(3−ε)
N(21)
where aNis the normalized cross-sectional area of the stem with respect to the cross-sectional
area of the most proximal segment (Asmax), which is defined as
aN=As
Asmax .(22)
According to equation (22), we have aN=1 when lN=1, hence equation (21) is simplified
to
aN=l
4−3ε
2(3−ε)
N(23)
which yields the following constraint between Asmax and Lmax:
Asmax =
v
u
u
u
t
aρk
7−3ε
3−ε
QVε
ref
k
2−ε
3−ε
m[(2−ε)kR]
L
4−3ε
2(3−ε)
max .(24)
2934 Y Zhou et al
The relationship between the stem’s normalized diameter (dN) and the crown’s normalized
total arterial length (lN) readily follows as
dN=l
4−3ε
4(3−ε)
N(25)
where dNis defined with respect to the diameter of the most proximal segment (Dsmax) similar
to equation (22). Again, the stem–crown morphometric relationship obeys a power law, whose
exponent is determined by the crown’s equivalent resistance parameter ε.
Another interesting result is the relationship between the normalized stem diameter and
its normalized flow, which can be obtained by combining equations (25) and (8b) to yield
qN=d
4(3−ε)
4−3ε
N.(26)
2.1.3. Relationships between the power exponents. If the power exponents of equations (13)
(vN–lNrelationship), (25) (dN–lNrelationship) and (26) (qN–dNrelationship) are denoted as
β,γand δ, respectively, the following relationships between the exponents are implied:
β+4γ=3 (27)
γδ =1.(28)
Equation (27) is a constraint on the morphometric correlation exponents, and equation (28)
is a constraint on the morphometric–haemodynamic correlation exponents. These constraints
do not depend on the crown’s equivalent resistance parameter ε.
2.2. Available anatomical data
2.2.1. Truncated tree model. Kassab et al 1993 have previously reconstructed the branching
pattern and vascular geometry (diameters and lengths) of the pig’s coronary arterial trees
(body and total heart weights were approximately 30 kg and 150 g, respectively). For the
larger arteries, the detailed vascular connectivity has been recorded along with the segmental
diameters and lengths. To simulate coronary artery angiographic data with a spatial resolution
of approximately 500 µm, the anatomical data considered in the truncated tree model are
proximal to this diameter. Hence, a complete set of morphometric data, down to 500 µm, on
the RCA, LAD and LCX arteries in the normal pig heart was used to test our formulations.
In order to validate the assumptions in equations (4) and (8), and to test the deduced results
of equation (26), a flow simulation was constructed based on the measured anatomical data of
thetruncatedtree model,the conservation of massand momentum andthe appropriate boundary
conditions. The analysis assumes that the flow through each rigid segment is fully developed
and in steady state. By applying conservation of mass at each branching node and conservation
of momentum at each vessel segment, we obtained a set of linear equations. The flow in each
equation was expressed as a product of pressure difference and segmental conductance using
Poiseuille’s law. The boundary conditions were specified at the inlet (100 mm Hg) and outlet
(97 mm Hg) of the entire arterial tree truncated at diameters of 500 µm (Kassab et al 1997).
The viscosity of blood was assumed to be 0.04 g s−1cm−1(poise). In order to set up and solve
the equations automatically, a dedicated software package was developed. The automation
was carried out by mapping the tree nodes to the equation matrix indices. The software can
also handle trifurcations by modelling each trifurcation with two bifurcations connected with
a zero-length segment. After the nodal pressures were solved, the flow through each segment
was computed.
Although a uniform boundary pressure distribution is assumed in the derivations in
section 2.1, we also examined the effect of small variation of the boundary pressures to test
A generalization of Murray’s law 2935
the robustness of the relationships in equations (4), (8) and (26) to this assumption. For this
purpose, the flow simulation was also carried out fornonuniform boundary conditions (random
variation of 10% about a mean of 97 mm Hg). Given this small variation in outlet boundary
pressures, it is necessary to redefine the equivalent resistance of a crown. This can be done by
using the mean outlet pressure as the terminal pressure of a crown and applying equation (1)
to compute the equivalent resistance.
2.2.2. Symmetric tree model. This is an analytical model that simulates the mean statistical
data of the coronary arterial trees presented in Kassab et al (1993). Physically, it is equivalent
to assuming that all the vessel elements in any order are of equal diameter and length and are
arranged in parallel, and the blood pressures at all of the junctions between specific orders of
vessels are equal (Kassab et al 1997). In this simplified circuit, the flow in each element of
order nis Qmax/Nnwhere Qmax is the total flow into the coronary arterial tree and Nnis the
total number of vessels at order n. The Qmax is determined from the pressure drop between
the most proximal element and the outlets of the pre-capillary vessels (68 mm Hg, see Kassab
et al 1997). It is assumed that the coefficient of viscosity is 0.04 poise in vessels of orders
11 to 5, and decreases linearly with order number to 0.02 in the pre-capillary arterioles (order
1 vessels).
3. Results
3.1. Anatomical data—equivalent resistance of crown
The validity of equation (4) was tested by computing the equivalent resistance of the various
crowns directly from the morphometric data of the arterial trees. Similar to electric circuit
analysis, a crown’s resistance was computed from its subcrown, based on the parallel and serial
arrangement of its segmental resistors, whose resistance was computed from equation (3). The
total arterial length and volume for each crown were also calculated. If equation (4) is valid,
then a plot of Rc/L3versus Von a log–log plot should give a linear relation, with a slope of
ε−2.
The results of the three arterial trees (LAD, LCX and RCA) are shown in figures 1(a) and
1(b) for the truncated and symmetric tree models, respectively. A least squares fit was used to
determine the values of εand the correlation coefficients for the truncated and symmetric tree
models of the LAD, LCX and RCA as listed in table 1. The equivalent resistance assumption
was also verified from the flow simulation, as will be discussed in the next section.
Table 1. Validation of equivalent crown resistance as expressed by equation (4) in the text.
Truncated tree model Symmetric tree model
Arterial tree εr
2εr
2
LAD −0.65 0.988 −0.65 0.998
LCX −0.56 0.990 −0.62 0.999
RCA −0.45 0.980 −0.71 0.999
3.2. Flow simulation
3.2.1. Crown resistance. For each crown, under the uniform boundary pressure condition,
the equivalent resistance was computed from the pressure drop and the entrance flow, as
2936 Y Zhou et al
(a)
(b)
Figure 1. Relationship between the crown’s equivalent resistance (Rc) divided by the length of the
crown (L) cubed and the volume of the crown (V) for the RCA, LAD and LCX artery, respectively,
in (a) truncated tree model, and (b) the symmetric tree model. The data are curve fitted with power
law expressions using least squares fits. The curve fitting parameters are given in table 1.
specified in equation (1). The computed crown resistance was correlated to the crown’s
arterial volume and length. As one might expect, the results exhibit the same relations as
in section 3.1. Therefore, the flow simulation is in agreement with the relationship postulated
in equation (4).
A generalization of Murray’s law 2937
By allowing a small variation of the outlet boundary pressures, the equivalent resistance of
each crown was computed. It was found that the relationships are similar to those determined
under the uniform boundary conditions. Therefore, equation (4) is invariant under a 10%
variation in the outlet boundary pressures.
3.2.2. Stem flow–crown length relationship. Figure 2 supports the hypothesized linear
relationship between stem blood flow and crown length (equation (8a)) for the truncated and
symmetric tree models, respectively, under the assumption of the uniform boundary pressure
condition. A least squares fit results in significant correlations between stem flow and crown
length as summarized in table 2.
Table 2. Validation of the relationship between stem flow and the cumulative arterial length of
crown as given by equation (8a) in the text.
Truncated tree model Symmetric tree model
Arterial tree kQ(ml s−1cm−1)r2kQ(ml s−1cm−1)r2
LAD 0.033 0.988 6 ×10−51.000
LCX 0.029 0.972 5 ×10−51.000
RCA 0.031 0.925 6 ×10−51.000
The stem flow–crown length relationship was also evaluated for the case of non-uniform
boundary conditions in the truncated tree model. The fitting results are identical to those found
using the uniform boundary condition hypothesis.
3.2.3. Stemflow–diameter relationship. Weshallnowexamine the validityof therelationship
between normalized stem diameter and flow as given by equation (26). Figures 3(a) and (b)
show the simulation results of the diameter–flow relationship for the LAD, LCX and RCA
vessels for the truncated and symmetric tree models, respectively. The exponent of a power
law relationship is determined from a least squares fit. Furthermore, the power indices were
predicted from equation (26) based on a computed ε(see section 3.1). Both the simulated
and predicted power indices (δ) are listed in table 3 for comparison. As shown in table 3, the
predictions are in good agreement with the simulated results.
3.3. Morphometric relationships
3.3.1. Arterial volume–length relationship of crown. Equation (13) suggests a normalized
power law relationship between the arterial volume of a crown and its corresponding length.
Given the crown resistance fitting parameter εin section 3.1, the predicted power indices were
computed. These predicted values can be directly compared with the experimental results
based on the available morphometric data as shown in figures 4(a) and (b). A least squares fit
of the data shows the results for the truncated and symmetric tree models. The predicted as
well as the experimental power indices are listed in table 3. As demonstrated in this table, the
predictions are in good agreement with the measured anatomical data.
3.3.2. Stem diameter–crown length relationship. According to equation (25), a power law
relationshipexistsbetween thenormalized diameterof astem andthe normalizedarterial length
of its crown. From the known values of ε(see section 3.1) and equation (25), the predicted
power indices were calculated. As a comparison, the corresponding results obtained from
2938 Y Zhou et al
(a)
(b)
Figure 2. Relationship between stem flow and length of crown for the RCA, LAD and LCX artery,
respectively, in (a) the truncated tree model, where the inlet and outlet pressure conditions are 100
and 97 mm Hg, respectively, and the viscosity was assumed to be 0.04 g s−1cm−1(poise), and
(b) the symmetric tree model, where the numbers of data points are 10, 9 and 10 for LAD, LCX
and RCA, respectively. The data are curve fitted with power law expressions using least squares
fits. The curve fitting parameters are given in table 2.
the anatomical data are shown in figures 5(a) and 5(b) for the truncated and symmetric tree
models,respectively. Thepowerlawexponentsand thecorrelation coefficientsaresummarized
A generalization of Murray’s law 2939
(a)
(b)
Figure 3. Relationship between normalized stem flow and stem diameter for the RCA, LAD
and LCX artery, respectively, in (a) the truncated tree model, where the inlet and outlet pressure
conditionsare100and97mmHg, respectively,andtheviscositywasassumedtobe0.04g s−1cm−1
(poise), and (b) the symmetric tree model. The flow and diameter are normalized with respect to
their values at the most proximal segment. The data are curve fitted with power law relationships
using least squares fits. The curve fitting parameters are given in table 3.
in table 3 for the truncated and symmetric tree models, respectively. Again, the predictions
are in good agreement with the experimental measurements.
2940 Y Zhou et al
(a)
(b)
Figure 4. Relationship between normalized arterial volume and arterial length of crown for the
RCA, LAD and LCX artery, respectively, in (a) the truncated tree model and (b) the symmetric tree
model. The volume and length are normalized with respect to the total volume and length of the
respective arterial tree. The data are curve fitted with power law expressions using least squares
fits. The curve fitting parameters are given in table 3.
3.4. Relationships between exponents
Equations (27) and (28) provide constraints for the morphometric and haemodynamic
exponents, which do not depend on the crown’s equivalent resistance parameter. Using the
A generalization of Murray’s law 2941
(a)
(b)
Figure 5. Relationship between normalized stem diameter and arterial length of crown for the
RCA, LAD and LCX artery, respectively, in (a) the truncated tree model, and (b) the symmetric
tree model. The stem diameter is normalized with respect to the diameter at the most proximal
segment while crown length is normalized with respect to the total length of the arterial tree. The
dataarecurvefittedwith powerlawexpressionsusingleast squares fits. The curve fitting parameters
are given in table 3.
measured morphometric data and the simulated results as described earlier, we can test these
deduced constraints. The results are listed in table 4, which shows that the theoretical
2942 Y Zhou et al
Table 3. Comparison between the predicted and anatomical/haemodynamic simulation exponents
for the relationships qN=dδ
N(equation (26)), vN=lβ
N(equation (13)) and dN=lγ
N
(equation (25)) (subscripts pand frepresent the predicted values and the fitted values from
anatomical/haemodynamic simulation, respectively).
Tree model Arterial tree δpδrr2βpβrr2γpγrr2
LAD 2.45 2.21 0.93 1.37 1.37 0.99 0.41 0.44 0.93
Truncated LCX 2.51 2.51 0.87 1.40 1.42 0.99 0.40 0.38 0.96
RCA 2.58 2.18 0.80 1.45 1.44 0.98 0.39 0.40 0.87
LAD 2.45 2.18 0.99 1.37 1.41 1.00 0.41 0.45 0.99
Symmetric LCX 2.47 2.05 0.99 1.38 1.45 1.00 0.40 0.47 0.99
RCA 2.42 2.17 0.99 1.35 1.38 1.00 0.41 0.45 1.00
Table 4. The anatomical/simulated results on the power exponent relationship. As described in
the text, the predicted relations between the exponents are: β+4γ=3 (equation (27)) and γδ=1
(equation (28)).
Truncated tree model Symmetric tree model
Arterial tree β+4γγδ β+4γγδ
LAD 3.13 0.97 3.32 0.96
LCX 2.94 0.95 3.38 0.94
RCA 3.04 0.87 3.25 0.95
predictions are in good agreement with the anatomical data/haemodynamic simulations for
the truncated and symmetric tree models.
4. Discussion and conclusion
Murray (1926a,b) attempted to explain the design of vascular trees by predicting a design
criterion that constrains the relative diameter of the daughter vessels at each bifurcation. In
this paper, we attempt to describe the design of the coronary arterial tree from the most
proximal stem whose distal crown includes the entire tree down to the most distal stem
whose crown terminates either at the truncated diameter or at the capillary level. This is
done by progressively considering smaller crowns of the coronary arterial tree. The domain
of the largest crown includes the entire coronary arterial tree, and each subcrown represents
a subdomain of the larger crown, and so on down to the capillary vessels. For a given stem
diameter, our formulations predict the correlation between the total length and volume of the
distal crown. This formulation, however, does not impose any criterion on the connectivity
of the individual segments of the crown. It is for this reason that the symmetric tree model,
which ignores the actual connectivity of the vascular tree, yields very similar results to the
truncated tree model, which specifies the actual connectivity of the coronary arterial vessels.
Hence, our predicted design criterion is globalwith respect to thetotal length and volumeof the
arterial vessels without regard to the specific connectivity. Both the present formulations that
lead to the stem–diameter and crown–length and crown–volume relationships and Murray’s
constraints on the ratio of the diameters and bifurcation angles are a result of a minimization of
acost function. The combination ofthese design constraints maylead to a betterunderstanding
of the design of vascular trees.
It has been well established that coronary arterial trees obey self-similarity patterns (see
reviewinBassingthwaighte et al 1990). In the present study, ithas been found thatthe diameter
A generalization of Murray’s law 2943
or cross-sectional area of a stem and the volume of a crown are related to the crown length
through power law relationships (equations (13), (23) and (25)) which is suggestive of a self-
similar pattern over each crown or stem–crown set. The self-similar pattern also holds for
the equivalent resistance relationship for each crown (equation (4)). This haemodynamic self-
similarity serves as a foundation on which the morphometric relationships are derived. The
form of equation (4) was tested using a least squares fit of the anatomical data where it was
found to have a significant correlation coefficient. Moreover, the crown resistances of both the
truncated and symmetric tree models obey a similar relationship.
Therehavebeenpreviousstudies thathavecorrelatedthe variousmorphometricparameters
of the larger coronary arteries. Seiler et al (1992, 1993) have found a power law relationship
between the stem cross-sectional area and the crown cumulative arterial length for both canine
(body weight: 22 kg–32 kg) and human studies. In the present study, a similar relationship
has been derived based on optimality and conservation of energy for a laminar, steady state
flow. The power exponents for the cross-sectional area–length relationship are twice those of
the diameter–length relationship given in table 3 (i.e. 0.814 for LAD, 0.812 for LCX and 0.776
for RCA). Seiler et al (1992, 1993) reported a value of 0.82 for in vivo human studies of left
coronary arteries and similar values for the canine studies. It is interesting that our model fits
the data from various species.
Seiler et al (1992, 1993) have compared their cross-sectional area–length results with a
prediction using Murray’s law. They applied Murray’s law in conjunction with a constant
perfusion value (0.8 ml min−1g−1) and obtained a predicted power exponent of 0.67. This
predicted value is smaller than the experimental finding of Seiler et al. The fact that our
predicted values are closer to the experimental data implies that our model is not equivalent
to Murray’s law but a generalization of it. Whereas Murray’s law is an optimization that
applies to vessel segments, our generalization extends to an entire tree or subtrees. In fact, the
generalization of our model can be further demonstrated by the following analysis. If ε=0is
assumed, equation (21) or (23) will result in a power index of 2/3 (0.67) for the relationship
between a stem cross-sectional area and its crown’s arterial length, which is consistent with
Murray’s law. In addition, ε=0 in equation (26) predicts an exponent of 3 for the relationship
between a stem flow and the stem diameter. Again, this is in agreement with Murray’s law.
Therefore, it is the deviation of εfrom 0 that is responsible for the difference between our
model and that of Murray. The value of εin our model yields exponents that are in better
agreement with the anatomical and simulated results (see table 3, and Seiler et al).
Myocardialperfusion (volumetric flow per mass of tissue) isessential for the heart because
it affects its health and function. The linearity between stem flow and crown length in the
truncated tree model is very interesting because it relates to myocardial perfusion. According
to Seiler et al (1993), in the larger coronary arteries, the total arterial length of a crown (in cm)
was found to be linearly related to the canine regional myocardial mass (in grams) with a slope
of about one. If it is assumed that theporcine myocardium obeysa similar relationship, then the
crown’s total arterial length (in cm) in the truncated tree model can be equated to the regional
myocardial mass (in grams). Accordingly, the ratio of stem flow to the crown’s total arterial
length will correspond to the regional myocardial perfusion (in ml min−1g−1). The simulation
results in figure 3(a) show slopes of 0.033 ml s−1cm−1for LAD, 0.029 ml s−1cm−1for LCX
and 0.031 ml s−1cm−1for RCA. By converting these slopes to myocardial perfusion, the
results are 1.98 ml min−1g−1,1.74 ml min−1g−1and 1.86 ml min−1g−1, for LAD, LCX
and RCA, respectively. Carli et al (1995) used PET to measure perfusion in human hearts
and reported a value of 2.3±0.6 ml min−1g−1for the normal human coronary arteries in
the vasodilated state. Our results are consistent with these data given that the anatomical data
were obtained in the vasodilated state.
2944 Y Zhou et al
Oneimportant application ofthe morphometricrelationships for thetruncated treemodel is
the assessment of coronary artery disease (CAD). CAD is usually quantified by the percentage
stenosis of arterial diameter, which is an important index of coronary lesion (Gould et al 1975,
Gould and Lipscomb 1975, Gould and Kelly 1982, Walinsky et al 1979). Using this index
to evaluate diffuse CAD, however, has been seriously questioned (Marcus et al 1988). The
most obvious problem is that the normal reference segment does not exist in the case of diffuse
disease. Therefore, it is essential to establish morphometric relationships for the coronary
arterial tree in the normal heart. Pathological changes may alter these relationships where the
deviations from normal can then be used to diagnose diffuse CAD.
The relationship between arterial volume and length in the normal heart can quantitatively
define the coronary arterial tree. Interestingly, the derived constraint in equation (27) between
the two exponents does not depend on the phenomenological parameter ε. Therefore, this
constraint may serve as an index of the normal coronary artery tree, where deviations from this
index may quantitatively establish the extent of diffuse CAD using angiographic techniques.
Our analysis can also clarify some ambiguities in collecting angiographic data for the
purpose of finding morphological correlations (Wahle et al 1995). For example, there are two
criteria to define the truncation of arterial length, either according to the generation number
from the root, or according to some prescribed diameter. Our model suggests that a uniform
diameter truncation is an appropriate choice.
As shown in section 3, the stem diameter–crown length and stem flow–diameter data
showed some scatter, especially for RCA data. This scatter is due to the fact that, in some
cases, the daughter vessel may have a slightly larger diameter than its parent vessel. Although
this occurrence is rare, it was observed more frequently in the RCA data than the LAD
and LCX data. The crown’s equivalent resistance is not sensitive to this feature, since it
includes the accumulative arterial length and volume. However, the stem–crown and the stem
flow–diameter data are more sensitive, especially the latter, because the dependent variable is
the stem diameter.
In conclusion, we have established important morphometric and haemodynamic
relationships based on a detailed anatomical database, optimality and conservation principles.
These relationships provide insight into the design of the coronary arterial tree and may be
helpful in the assessment of diffuse coronary artery disease.
Acknowledgments
We wish to thank Drs John Breault and Fuming Wang for the helpful discussion on the
modelling, and Mr George Chiou and Mr Changzheng Huang for the helpful suggestions
on the flow simulation. This work was supported in part by an Established Investigation Grant
from the American Heart Association. Dr Kassab is a recipient of the NIH First Award.
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