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Am J Physiol Heart Circ Physiol 289: H439 –H446, 2005.
First published March 25, 2005; doi:10.1152/ajpheart.00730.2004.
Analysis of blood flow in the entire coronary arterial tree
N. Mittal,1 Y. Zhou,2 C. Linares,1 S. Ung,1 B. Kaimovitz,1 S. Molloi,3 and G. S. Kassab1
1
Department of Biomedical Engineering, University of California, Irvine; 2Department of Environmental Health, Cedars-Sinai
Medical Center, Los Angeles; and 3Department of Radiological Sciences, University of California, Irvine, California
Submitted 21 July 2004; accepted in final form 17 February 2005
vascular reconstruction; coronary morphometry; flow simulation; flow
resistance; transit time
THE CORONARY VASCULAR SYSTEM constitutes the specialized
channels that conduct oxygenated blood throughout the myocardium. The function of this network is to continuously
supply blood to meet the requirements of the beating heart.
Numerous attempts have been made at simulation of blood
flow through these specialized channels to understand the
spatial and temporal distribution of blood flow. Much of the
modeling of the coronary circulation, however, has centered
around lumped-parameter models in which the coronary vasculature or subgroup of vessels are treated as single entities
whose whole behavior is characterized by a limited number of
parameters. Despite the usefulness of such models, they are
generally limited to global aspects of coronary blood flow (10,
21). For example, lumped models cannot be used to predict the
significant spatial distribution of coronary blood flow.
Over a decade ago, a program was initiated to provide the
necessary details of the coronary vascular anatomy (vascular
geometry and branching pattern) to enable anatomically based
modeling of coronary circulation. In this approach, the vascular
system comprised of millions of distensible vessel branches,
strategically distributed and mostly embedded within the myo-
Address for reprint requests and other correspondence: G. S. Kassab, Dept.
of Biomedical Engineering, Univ. of California, Irvine, 204 Rockwell Engineering Center, Irvine, CA 92697-2715 (E-mail: [email protected]).
http://www.ajpheart.org
cardium, must be modeled in as much detail as possible rather
than “lumped.” Although we are still several years away from
accomplishing this goal, some important strides have been
made. As a first step, Kassab and colleagues (13, 15–18)
reconstructed the entire vascular anatomy of the porcine heart
in the framework of a mathematical model of a tree structure,
yielding data on the diameter, lengths, numbers, and connectivity of coronary arteries, capillaries, and veins for every order
or generation number. Almost simultaneously, VanBavel and
Spaan (32) provided morphometric data on the coronary arterial tree. The anatomic programs produced detailed anatomic
data on the vascular geometry and branching pattern in a
statistical framework. The sheer number of vessels involved
and the hemodynamic computations require automation of
reconstruction and analysis. In response, we recently developed (23) a computer program to automate the reconstruction
of the full coronary arterial tree based on our previous morphometric measurements.
The objective of the present study was to use the reconstruction platform to carry out an analysis of blood flow in the entire
coronary arterial tree under steady-flow conditions. This is
only the first step in a program aimed toward understanding the
spatial and temporal distribution of blood flow throughout the
cardiac cycle. The three-dimensional branching pattern embedded in a realistic dynamic model of the beating heart that
integrates the blood vessel elasticity and tissue mechanics can
be built in the future, adding to the degree of sophistication and
realism of the present model. The model’s limitations and
physiological implications are discussed here.
METHODS
Anatomical Model
A flow simulation was carried out in each of the coronary arterial
trees recently reconstructed by Mittal et al. (23) based on measured
morphometric data of Kassab et al. (16). A two-step approach was
used in the reconstruction of the entire coronary arterial tree down to
the capillary level (ⱕ8 ␮m in diameter). Briefly, portions of the
arterial tree [right coronary artery (RCA), left anterior descending
coronary artery (LAD), or left circumflex artery (LCx)] missing from
the cast data were computationally reconstructed from anatomic data.
Missing components of the tree, from broken vessel segments down to
vessels of diameter of 40 ␮m, were reconstructed from the intact cast
data. Portions of the tree made up of vessels with diameter of ⬍40 ␮m
were reconstructed based on histological data. Reconstructed networks were terminated at segments of diameter ⱕ8 ␮m. Any
terminal vessels in the cast data with diameter ⬎8 ␮m was treated
as a broken vessel, and the reconstruction algorithm was applied to
generate a subtree that branched down to the terminal arterioles of
diameter ⱕ8 ␮m.
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0363-6135/05 $8.00 Copyright © 2005 the American Physiological Society
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Mittal, N., Y. Zhou, C. Linares, S. Ung, B. Kaimovitz, S. Molloi,
and G. S. Kassab. Analysis of blood flow in the entire coronary arterial
tree. Am J Physiol Heart Circ Physiol 289: H439 –H446, 2005. First
published March 25, 2005; doi:10.1152/ajpheart.00730.2004.—A hemodynamic analysis of coronary blood flow must be based on the
measured branching pattern and vascular geometry of the coronary
vasculature. We recently developed a computer reconstruction of the
entire coronary arterial tree of the porcine heart based on previously
measured morphometric data. In the present study, we carried out an
analysis of blood flow distribution through a network of millions of
vessels that includes the entire coronary arterial tree down to the first
capillary branch. The pressure and flow are computed throughout the
coronary arterial tree based on conservation of mass and momentum
and appropriate pressure boundary conditions. We found a power law
relationship between the diameter and flow of each vessel branch. The
exponent is ⬃2.2, which deviates from Murray’s prediction of 3.0.
Furthermore, we found the total arterial equivalent resistance to be
0.93, 0.77, and 1.28 mmHg 䡠 ml⫺1 䡠 s⫺1 䡠 g⫺1 for the right coronary
artery, left anterior descending coronary artery, and left circumflex
artery, respectively. The significance of the present study is that it
yields a predictive model that incorporates some of the factors
controlling coronary blood flow. The model of normal hearts will
serve as a physiological reference state. Pathological states can then
be studied in relation to changes in model parameters that alter
coronary perfusion.
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ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
Flow Simulation
After the branching pattern and vascular geometry of the full
arterial network were generated, a network analysis was performed
similar to that of Kassab et al. (19). If we assume that the flow through
a blood vessel is laminar, steady, and free from end effects, then the
volumetric flow Qij in a vessel between any two nodes, represented by
i and j, is given in terms of the pressure differential ⌬Pij and vessel
conductance Gij by
Q ij ⫽
␲
⌬PijGij
128
(1)
RESULTS
␮ vivo ⫽ 1 ⫹ 共6 䡠 e⫺0.085D ⫹ 3.2 ⫺ 2.44e⫺0.06D
0.645
冉
⫺ 1兲
冊册 冉
D
䡠
D ⫺ 1.1
2
冊
D
䡠
D ⫺ 1.1
2
(2)
where D is the vessel diameter. This relationship was used throughout
the coronary vasculature for a given vessel diameter D and hematocrit
of 0.45.
There are two or more vessels that emanate from the jth node
anywhere in the tree, with the number of vessels converging at the jth
node being mj. By conservation of mass we must have
mj
兺Q
ij
⫽0
(3)
i⫽1
where the volumetric flow into a node is considered positive and flow
out of a node is negative for any branch. From Eqs. 1–3 we obtain a
set of linear algebraic equations in pressure for M nodes in the
network, namely
Because the reconstruction process is stochastic, we can
generate many trees that are consistent with the statistical
morphometric data. Mittal et al. (23) recently showed that the
coefficient of variance (CV ⫽ SD/mean) of various morphometric parameters for a given order number was larger within
a single reconstruction or run than that obtained from the mean
of several reconstructions. Similarly, we found that the variability in the hemodynamic parameters of interest within a
simulation run was significantly larger than the variability of
the mean from run to run. The CV in the hemodynamic
parameters—flow and pressure— obtained over five runs for
LCx was found to be ⬍5% for all orders. Hence, all results
shown below correspond to a single reconstruction.
Figure 1 shows the relationship between normalized flow
through a vessel segment and normalized segment diameter for
the LAD arterial tree, excluding the capillaries. This is an
isodensity plot showing five layers of frequency. As expected,
the majority of vessels are the smaller-diameter arterioles. The
diameter and flow are normalized with respect to the inlet,
most proximal segment. The relationship obeys a power law
relation as suggested by Murray’s law. However, the value of
exponent is not 3, as predicted by Murray’s law. As determined
by least-squares fits of the data, the exponent has values of 2.2
(R2 ⫽ 0.993), 2.1 (R2 ⫽ 0.995), and 2.1 (R2 ⫽ 0.994) for RCA,
LAD, and LCx, respectively.
mj
兺 关P ⫺ P 兴G ⫽ 0
i
j
ij
(4)
i⫽1
The set of equations represented by Eq. 4 reduce to a set of simultaneous linear algebraic terms for the nodal pressures once the conductances are evaluated from the geometry and suitable boundary conditions are specified. In matrix form this set of equations is
GP ⫽ GBPB
(5)
where G is the n ⫻ n matrix of conductances, P is a 1 ⫻ n column
vector of the unknown nodal pressures, and GBPB is the column
vector of the conductances times the boundary pressures of their
attached vessels, respectively. Boundary conditions were prescribed
by assigning an inlet pressure of 100 mmHg and a pressure of 26
mmHg at the outlet of the first capillary segment. Because matrix G
is a very sparse matrix, it was represented in a reduced form for
optimal memory utilization. This system of equations was then
solved, using a general mean residual algorithm (27) to determine the
pressure values at all internal nodes of the arterial tree. The pressure
drops as well as the corresponding flows were then computed.
Variable Boundary Conditions
The previous results were implemented for an inlet pressure of 100
mmHg and outlet pressure (at the outlet of the first capillary segment)
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Fig. 1. An isodensity plot showing 5 layers of frequency between normalized
stem flow and normalized diameter of the stem for the left anterior descending
coronary artery (LAD) arterial tree. Diameter and flow are normalized with
respect to the values for the most proximal vessel segment in each respective
tree. Solid line corresponds to least-square fits of the data according to a power
law relation with a correlation coefficient of 0.999. The total number of data
points shown is 936,014, which excludes the capillaries.
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where ⌬Pij ⫽ Pi ⫺ Pj, Gij ⫽ D4ij/␮ijLij, and Dij, Lij, and ␮ij are the
diameter, length, and viscosity, respectively, between nodes i and j.
Data on the variation of viscosity with vessel diameter and hematocrit
given by Pries et al. (26) was used in our model. They proposed a
modified viscosity relationship based on a compilation of literature
data on relative blood viscosity in tube flow in vitro and in vivo
experimental measurements, which reflects the Fahraeus-Lindqvist
effect as given by
冋
of 26 mmHg. To examine the variation of mean transit time with flow,
we varied the inlet boundary condition to 30, 60, 100, 120, 140, 160,
and 180 mmHg. We also examined the effect of changing the outlet
boundary conditions. We maintained the inlet pressure at 100 mmHg
and varied the outlet capillary pressure as a Gaussian distribution with
a mean of 26 mmHg and a SD of 0, 2, 4, 6, or 8 mmHg. We observed
flow reversal at the terminal capillary vessels for SD ⬎ 0. The
frequency of flow reversal was quantified as the fraction of vessels
with negative flow divided by the total number of vessels in the
respective circuit.
ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
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The blood flow and pressure in vessel segments along the
trunk and the primary branches (branches that arise directly
from the trunk) are shown in Fig. 2 for the LAD arterial tree.
Figure 2A shows a schematic of the trunk and the primary
Fig. 2. A: schematic of the trunk of the LAD and some of the primary branches
(A–H). The numbers in parentheses correspond to diameter of primary branches in
␮m. B: relationship between the flow in a vessel segment and the cumulative
length of the segment from the root of the trunk for the primary branches. C:
relationship between the pressure at the outlet section of a vessel segment and the
cumulative length of the segment from the root of the trunk and the primary branches.
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branches, several of which are identified alphabetically (A–H).
The flow and pressure along the trunk and primary branches
are shown in Fig. 2, B and C, respectively. The trunk is denoted
by a bold line, whereas several primary branches are denoted
alphabetically in Fig. 2, B and C, in correspondence to Fig. 2A.
The capillary outlet flow and pressure conditions are connected
by dotted lines in Fig. 2, B and C, respectively. The outlet
capillary pressure is constant for this particular simulation as
imposed by the boundary condition (26 ⫾ 0 mmHg) whereas
the computed outlet flow is variable, as seen in Fig. 2, C and B,
respectively. The number of circles along each curve in Fig. 2,
B and C, represents the number of bifurcations along that
particular path. For example, subtree B has far fewer branches
or bifurcations down to the capillaries than the adjacent subtree
C. It appears that the more bifurcations along the pathway, the
more gradual the decrease in flow (Fig. 2B). The major pressure drop along the main path (trunk) occurs at a length of 11.1,
10.7, and 7.4 cm from the inlet of RCA, LAD, and LCx,
respectively. Figure 3 shows the direct relationship between
the segment flow and the mean segment pressure [(inlet ⫹
outlet)/2] as we combine Fig. 2B and C for the trunk (solid
thick line) and primary branches. It can be seen that the mean
pressure is rather uniform in the large flow regime and drops
rapidly in the lower pressure range. It is also interesting to note
that the curves corresponding to the various primary branches
cluster in a narrow range and tend to take on similar shape.
Figure 4 shows the relationship between the pressures at the
outlet of a vessel segment and its diameter for the entire LAD
coronary arterial tree (nearly 2 million vessels). There is a
gradual drop in pressure in the proximal vessels followed by a
steeper drop in the microvessels. Figures 4, A, B, and C,
correspond to three different outlet boundary conditions: 26 ⫾
0, 26 ⫾ 2, and 26 ⫾ 6 mmHg, respectively.
In all simulations, unless otherwise stated, the inlet pressure
was set at 100 mmHg and the pressure at the outlet of the first
arterial capillary segment was 26 mmHg. The inlet flow in the
most proximal vessel segment was found to be 0.53, 0.63, and
0.32 ml/s for RCA, LAD, and LCx, respectively. The pressure
difference (inlet minus capillary outlet)-flow relation is linear
for the RCA, LAD, and LCx arterial trees. The data are fitted
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Fig. 3. The relation between flow and mean pressure [(inlet ⫹ outlet)/2] for
the trunk (bold line) and primary branches as depicted in Fig. 2.
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ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
Fig. 4. An isodensity plot showing 5 layers of frequency between pressure at
the outlet section of a vessel segment and the corresponding diameter of the
vessel for the entire right coronary artery (RCA; A), LAD (B), and left
circumflex artery (LCx; C) arterial trees. The total number of vessels for the
RCA, LAD, and LCx arterial trees depicted are ⬃1.7, 1.9, and 1.1 million,
including the first segment of capillaries.
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Fig. 5. Probability frequency for transit times (t) through the LAD arterial tree.
The dashed vertical line represents the mean t, and the solid line is a nonlinear
least-squares fit of the mean t decay as given by h(t) ⫽ 3.7t⫺3.2 (R2 ⫽ 0.888).
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with a Q ⫽ ⌬PGart model where ⌬P, Q, and Gart are the
pressure difference along the entire coronary arterial tree, inlet
flow, and total arterial conductance, respectively. A linear
least-squares fit of the data revealed Gart values of 0.0072
(R2 ⫽ 1.000), 0.0086 (R2 ⫽ 1.000), and 0.0052 (R2 ⫽ 0.999)
ml 䡠 s⫺1 䡠 mmHg⫺1 for the RCA, LAD, and LCx arterial trees,
respectively.
The mean ⫾ SD arterial transit times were found to be 2.3 ⫾
0.87 (RCA), 1.5 ⫾ 0.56 (LAD), and 1.9 ⫾ 0.67 (LCx) s at inlet
pressure of 100 mmHg; the respective maximum transit times
were 8.8, 7.8, and 4.5 s. The transit times were calculated along
all possible pathways in the arterial tree by adding the transit
times through each individual segment. There were a total of
858,353, 936,014, and 572,632 pathways (equal to the number
of first segment of capillaries) for the RCA, LAD, and LCx
arterial trees, respectively. The probability density function for
the transit times of the LAD arterial tree is shown in Fig. 5. The
vertical dashed line represents the mean value. The decay of
transit time frequency from the mean, h(t), was fitted by the
form h(t) ⫽ ␣t⫺␤, where t represents the transit time. The
empirical constants ␣ and ␤ were determined by using a
nonlinear least-squares fit. We found the values of the exponent ␤ to be 3.4 (R2 ⫽ 0.937), 3.2 (R2 ⫽ 0.888), and 3.2 (R2 ⫽
0.782) for the RCA, LAD, and LCx arterial trees, respectively.
The least-squares fit curve for the LAD arterial tree is shown in
Fig. 5.
The relationship between the inlet flow and the mean transit
time (t៮) was simulated by varying the inlet pressure at 30, 60,
100, 120, 140, 160, and 180 mmHg. The results can be
summarized as hyperbolic relations between t៮ and inlet flow
rate (Qin) as t៮ ⫽ Vt/Qin, where Vt represents the total arterial
volume. The computed value of Vt equals to 1.3, 1.0, and 0.61
ml for the RCA, LAD, and LCx arterial trees, respectively.
Finally, we considered the effect of varying the SD of outlet
boundary conditions at the first capillary segment. Figure 6A
shows the probability density function for the flow at the
capillary outlet for an SD of 6 mmHg. The relation between the
SD of outlet Gaussian pressure distribution (with a mean of 26
mmHg) and the relative dispersion (or CV) of capillary outlet
ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
flow is summarized in Fig. 6B. It is clear that the CV of blood
flow is smallest when the pressure boundary condition has no
variability and increases with an increase in heterogeneity of
capillary pressure. The change in CV is relatively modest over
a significant change in SD of pressure. It was observed that the
variability in capillary pressure can give rise to local flow
reversal within the capillary bifurcation (Fig. 6A). We quantified this reversal as a percentage of capillary vessels with
negative flow normalized with respect to the total number of
capillary vessels. We found that the greatest frequency of
negative capillary flow occurs for the case of SD ⫽ 10 mmHg
(⬃10%).
DISCUSSION
Analysis of Coronary Arterial Flow
Murray’s law. Nearly 80 years ago, Murray deduced a cubed
relationship between flow rate and diameter for a vessel segment based on the minimization of power cost function (24).
AJP-Heart Circ Physiol • VOL
Numerous papers have been published in the past 80 years on
Murray’s law and on the validation of the exponent. The
studies show support, but with significant scatter (27). The
results for the coronary arterial tree show exponents of 2.1–2.2
for all three arterial trees (Fig. 1). The deviation of the
exponent from 3 was previously explained based on a modification of the Murray formulation that viewed the coronary
arterial tree as an integrated system rather than individual
vessel segments (34). It was found that the exponent is not a
universal constant as required by Murray’s law but depends on
a scaling parameter that is dictated by the equivalent resistance
of the vascular system of interest. The exponent was determined on the basis of flow analysis of anatomic circuits (34)
and in vivo experiments based on angiography (33). In the
former, two anatomic models were considered: 1) a truncated
model representing the complete anatomic data for vessels
proximal to 0.5 mm in diameter and 2) a full symmetric model
of the coronary arterial tree based on mean morphometric data.
It was found that the exponent had values in the ranges of
2.2–2.5 and 2.1–2.2 for the truncated and symmetric models,
respectively, for the coronary arterial trees. In the in vivo
studies based on angiography, the exponent was found to have
a value of 2.7 for the LAD tree proximal to 0.5 mm in
diameter. The present study is the first to determine the exponent of the flow-diameter relation based on the entire asymmetric model of the coronary arterial tree.
Distribution of flow. It can be noted that the blood flow
through the trunk and primary branches (Fig. 2B) shows either
an abrupt or a gradual drop along the path to the capillary blood
vessels. The shorter paths (from trunk to capillary vessels) with
fewer branches show an abrupt drop, whereas the longer paths
with more branches show a more gradual drop of blood flow.
These flow profiles are novel and have not been described
previously because of a lack of a detailed anatomic model. The
results suggest that a tracer will experience very different flow
depending on the path. The flow at the capillary segment of the
various branches is connected by a dotted line in Fig. 2B. The
flow dispersion into the capillary bed is obvious. The pressure
and flow curves for the trunk and various primary branches
reduce to a set of characteristic curves when we combine Fig.
2, B and C, into Fig. 3.
Longitudinal pressure distribution. When the pressure values at the outlet sections of various segments were considered
(Fig. 4), the profile showed a flat region followed by a large
drop in pressure for vessels ⬍100 ␮m in diameter. This is in
agreement with experimental epicardial and subendocardial
pressure measurements (5, 11, 31). Furthermore, a very steep
drop in pressures was observed at the smallest arteriolar diameters. We have verified that the steep decline is not due to the
fixed capillary pressure (Fig. 4A); i.e., the same steep decline is
observed when the capillary pressure is varied according to a
Gaussian distribution with various SDs as shown in Fig. 4, B
and C. Instead, the steep drop in pressure is due to the large
asymmetry in subtrees. If we consider the flow through a vessel
segment, the bifurcation will supply two subtrees. If the subtrees are very asymmetric, i.e., different segment diameters,
different total number of vessels in each subtree, and hence
very different equivalent resistance, it is expected that the flow
and pressure distribution will be quite different. Indeed, we
would expect that the subtree with a smaller total number of
vessels will have a very abrupt pressure drop compared with a
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Fig. 6. A: probability frequency for the capillary flow corresponding to inlet
boundary condition of 100 mmHg and capillary outlet pressure of 26 ⫾ 6
mmHg. B: relationship between the coefficient of variation (CV ⫽ SD/mean ⫻
100) in capillary outlet flow for a Gaussian outlet pressure distribution with
mean of 26 mmHg and various SDs.
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ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
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decreased from 60 to 30 mmHg. The t៮ for the entire coronary
arterial tree was ⬃1–2 s at physiological pressure (100 mmHg)
under steady-flow conditions in rigid vessels. These t៮ values of
the arterial tree are approximately one-half those reported for
the entire coronary circulation (4). It is apparent that, at a given
flow rate, the t៮ for the three vessels is RCA ⬎ LAD ⬎ LCx.
This relates in part to the path length, which is largest for the
RCA, but also to the velocity distribution.
Comparison with Other Models
Numerous attempts have been made to simulate blood flow
in various organs based on detailed anatomic data (13). To our
knowledge, our analysis is the most comprehensive in the
sense that it relies on the most extensive set of anatomic data
to date (13). VanBavel and Spaan (32) presented a flow
simulation based on a partial porcine coronary arterial tree
model in the diameter range of 10 –500 ␮m. More recently, the
same group developed a mathematical model of coronary
arterial blood flow based on VanBavel and Spaan’s data. The
model considered the myogenic response with an idealized
branching structure (6).
Kassab et al. (19) presented a model of the entire coronary
arterial tree based on statistical data on diameters, lengths, and
connectivity of vessels. The previous model, however, considered some of the parallel vessels as equivalent elements and
hence reduced the number of vessels significantly to decrease
the computational cost. The vascular network was reduced to a
total of ⬃100,000 vessels, and the nodal pressures were solved
for such a matrix. In the present model, no such assumptions
were made and the nodal pressures were determined for every
arterial segment, which equaled 858,353, 936,014, and 572,632
segments for the RCA, LAD, and LCx arterial trees, respectively.
The models described above lack a three-dimensional
branching structure. Bassingthwaighte et al. (2) developed an
avoidance algorithm to distribute the morphometric measurements of Kassab et al. (15–18) in a cylindrical model of the
heart. The coronary arterial system in the passive cylindrical
model of the heart gave a remarkably complete description of
the distribution of the flows in the heart. Smith et al. (30), also
using Kassab’s morphometric data, developed a similar algorithm to map the distribution of vascular elements in the
myocardium as a nonlinear optimization of individual branch
angles based on the minimum shear stress hypothesis. With
this algorithm, a finite-element model of the largest six generations of arterial coronary tree has been generated. The vascular network was geometrically embedded in the finite-element
ventricular model of Nielsen et al. (25). In a more recent study,
the same group carried out a pulsatile analysis of blood flow in
the partial tree (29).
Model Limitations
Although our analysis is based on detailed measured morphometric data of coronary blood vessels, there are still a
number of assumptions made. For example, the vessels are
assumed to be rigid, which is untrue in reality. This issue
becomes particularly important when considering the flow in a
contracting myocardium because the elasticity of vessels gives
rise to the vessel-muscle interaction. Furthermore, all vessels in
the present model are assumed to be in a vasodilated state
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more gradual pressure drop for a subtree with many more
vessels. Although the “wall” appearance is quite pronounced in
Fig. 4, the total number of vessels that give rise to this
appearance is only 5% of the total number of vessels. Interestingly, previous flow simulation models have not reported the
steep drop (2, 32). We believe that the degree of anatomic
detail in the present model was not present in the previous
studies.
Arterial pressure-flow relation. It is well known that the
majority of flow resistance resides in the arterial tree, particularly in small arterioles (9). Hence, the arterial tree constitutes
the majority of coronary circulation resistance. The computed
linear pressure difference-flow relationship, whose slope is
the flow conductance or inverse of flow resistance, yielded
values of total equivalent resistance of 139, 116, and 192
mmHg 䡠 ml⫺1 䡠 s⫺1 for the RCA, LAD, and LCx arterial trees,
respectively. If we normalize these values by the total
weight of the heart (150 g), we obtain 0.93 (RCA), 0.77
(LAD), and 1.28 (LCx) mmHg 䡠 ml⫺1 䡠 s⫺1 䡠 g⫺1. The linearity
arises from the rigid vessel assumption and linear rheology
(no shear rate-dependent viscosity, no flow-dependent distribution of red blood cells and plasma to flow pathways,
etc.). It is well known, however, that the coronary vessels
are distensible and blood rheology is nonlinear, which gives
rise to the nonlinear pressure-flow relation (7). It turns out
that the nonlinearity is second order, as can be predicted
primarily from the distensibility of the coronary blood
vessels as shown by Kassab (12).
Transit times. The relation between t៮ and inlet flow rate
obeys the classic Stewart-Hamilton relationship, which states
that the t៮ of a fluid through a confined compartment is equal to
the total volume of the compartment divided by the flow rate
into the compartment (35). The theoretical basis for this relation was provided by Meier and Zierler (22). For the coronary
arterial trees, each transit time-flow relation was constructed
from seven different inlet pressures. Each inlet pressure
yielded a different inlet flow rate depending on the equivalent
resistance of the respective coronary arterial tree. The StewartHamilton relationship suggests that the total volume of the
RCA, LAD, and LCx are 1.3, 1.0, and 0.61 ml, respectively.
These values are in good agreement with previous cast measurements of arterial volumes (16). These results provide some
validity for our calculations of t៮ in the coronary arterial tree.
The probability density function of transit times shown in
Fig. 5 is equivalent to the normalized outflow concentrationtime curve under certain conditions. Bassingthwaighte and
Beard (1) showed the downslope of the outflow curves of
tracer-labeled water from the rabbit myocardium to be power
law functions of the form t⫺␤, with ␤ equal to ⬃3. The same
authors later showed that the t⫺3 form is a general property of
a heterogeneous vascular network (3). More recently, Beard
and Bassingthwaighte (4) modeled the left coronary arterial
tree based on the data of Kassab et al. (16), with the capillary
and venous systems lumped, to show that the tails of washout
of intravascular tracer have the form t⫺3.1. This is comparable
to the present finding of t⫺3.2 for the LAD arterial tree. Hence,
the arterial tree seems to be the major determinant of the
washout characteristic.
The t៮ decreased only slightly, in a nearly linear fashion,
when the inlet pressure was increased from 120 to 180 mmHg.
However, it increased rapidly when the inlet pressure was
ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
Significance of Model
The present study yields a predictive model that incorporates
some of the detailed anatomic features that influence coronary
blood flow. The accuracy of the model will be determined
through experimental validation. The model of normal hearts
will serve as a physiological reference state. Pathological states
can then be studied in relation to changes in model parameters
that alter coronary perfusion. The proposed study makes use of
physical principles, with the help of anatomy, to explain and
predict the physiology of the coronary arterial circulation in
quantitative terms. The present model will serve as a foundation for future more sophisticated models that incorporate
additional realism and that will serve to quantitatively test
various hypotheses in the coronary circulation.
ACKNOWLEDGMENTS
This research was supported in part by the National Heart, Lung, and Blood
Institute Grants 2-R01-HL-055554 – 06 (G. S. Kassab) and R01-HL-67159 – 03
(S. Molloi) and American Heart Association (AHA) Grant 0315029Y (G. S.
Kassab). G. S. Kassab is an AHA Established Investigator, and N. Mittal is an
AHA Predoctoral Fellow.
REFERENCES
1. Bassingthwaighte JB and Beard DA. Fractal 15O-labeled water washout
from the heart. Circ Res 77: 1212–1221, 1995.
2. Bassingthwaighte JB, Beard DA, Li Z, and Yipintsoi T. Is the fractal
nature of intraorgan spatial flow distributions based on vascular network
growth or on local metabolic needs? In: Vascular Morphogenesis: In Vivo,
In Vitro, In Mente, edited by Little CD, Mironov V, and Sage EH. Boston,
MA: Birkhauser, 1998.
3. Beard DA and Bassingthwaighte JB. Power-law kinetics of tracer
washout from physiological kinetics of tracer washout from physiological
systems. Ann Biomed Eng 26: 775–779, 1998.
4. Beard DA and Bassingthwaighte JB. The fractal nature of myocardial
blood flow emerges from a whole organ model of arterial network. J Vasc
Res 37: 282–296, 2000.
5. Chilian WM. Microvascular pressures and resistances in the left ventricular subepicardium and subendocardium. Circ Res 69: 561–570, 1991.
AJP-Heart Circ Physiol • VOL
6. Cornelissen AJ, Dankelman J, VanBavel E, and Spaan JA. Balance
between myogenic, flow-dependent, and metabolic flow control in coronary arterial tree: a model study. Am J Physiol Heart Circ Physiol 282:
H2224 –H2237, 2002.
7. Hoffman JI and Spaan JA. Pressure-flow relations in the coronary
circulation. Physiol Rev 70: 331–390, 1990.
8. Hoffman JI. Maximal coronary flow and the concept of coronary vascular
reserve. Circulation 70: 153–159, 1984.
9. Jones CJ, Kuo L, Davis MJ, and Chilian WM. Distribution and control
of coronary microvascular resistance. Adv Exp Med Biol 346: 181–188,
1993.
10. Judd RM, Redberg DA, and Mates RE. Diastolic coronary resistance
and capacitance are independent of duration of diastole. Am J Physiol
Heart Circ Physiol 260: H943–H952, 1991.
11. Kanatsuka H, Lamping KG, Eastham CL, Marcus ML, and Dellsperger KC. Coronary microvascular resistance in hypertensive cats. Circ
Res 68: 726 –733, 1991.
12. Kassab GS. Analysis of coronary circulation: a bioengineering approach.
In: Introduction to Bioengineering, edited by Fung YC. Singapore: World
Scientific, 2001.
13. Kassab GS. The coronary vasculature and its reconstruction. Ann Biomed
Eng 28: 903–915, 2000.
14. Kassab GS and Fung YC. The pattern of coronary arteriolar bifurcations
and the uniform shear hypothesis. Ann Biomed Eng 23: 13–20, 1995.
15. Kassab GS and Fung YC. Topology and dimensions of the pig coronary
capillary network. Am J Physiol Heart Circ Physiol 267: H319 –H325,
1994.
16. Kassab GS, Rider CA, Tang NJ, and Fung YC. Morphometry of pig
coronary arterial trees. Am J Physiol Heart Circ Physiol 265: H350 –H365,
1993.
17. Kassab GS, Lin D, and Fung YC. Morphometry of the pig coronary
venous system. Am J Physiol Heart Circ Physiol 267: H2100 –H2113,
1994.
18. Kassab GS, Pallencaoe E, Schatz A, and Fung YC. Longitudinal
position matrix of the pig coronary vasculature and its hemodynamic
implications. Am J Physiol Heart Circ Physiol 273: H2832–H2842, 1997.
19. Kassab GS, Berkley J, and Fung YC. Analysis of pig’s coronary arterial
blood flow with detailed anatomical data. Ann Biomed Eng 25: 204 –217,
1997.
20. Kassab GS, Le KN, and Fung YC. A hemodynamic analysis of coronary
capillary blood flow based on detailed anatomic and distensibility data.
Am J Physiol Heart Circ Physiol 277: H2158 –H2166, 1999.
21. Lee J, Chambers DE, Akizuki S, and Downey JM. The role of vascular
capacitance in coronary arteries. Circ Res 55: 751–762, 1984.
22. Meier P and Zierler KL. On the theory of the indicator-dilution method
for measurement of blood flow and volume. J Appl Physiol 6: 731–744,
1954.
23. Mittal N, Zhou Y, Ung S, Linares C, Molloi S, and Kassab GS. A
computer reconstruction of the entire coronary arterial tree based on
detailed morphometric data. Ann Biomed Eng. In press.
24. Murray CD. The physiological principle of minimum work. I. The
vascular system and the cost of blood volume. Proc Natl Acad Sci USA 12:
207–214, 1926.
25. Nielsen PM, Le Grice IJ, Smaill BH, and Hunter PJ. Mathematical
model of geometry and fibrous structure of the heart. Am J Physiol Heart
Circ Physiol 260: H1365–H1378, 1991.
26. Pries AR, Neuhaus D, and Gaehtgens P. Blood viscosity in tube flow:
dependence on diameter and hematocrit. Am J Physiol Heart Circ Physiol
263: H1770 –H1778, 1992.
27. Saad Y and Schulter M. GMRES: a generalized minimum residual
algorithm for solving nonsymmetric linear systems. SIAM J Sci Statist
Comput 7: 856 – 869, 1986.
28. Sherman TF. On connecting large vessels to small: the meaning of
Murray’s law. J Gen Physiol 78: 431– 453, 1981.
29. Smith NP, Pullan AJ, and Hunter PJ. An anatomically based model of
transient coronary blood flow in the heart. SIAM J Appl Math 62:
990 –1018, 2002.
30. Smith NP, Pullan AJ, and Hunter PJ. Generation of an anatomically
based geometric coronary model. Ann Biomed Eng 28: 14 –25, 2000.
31. Tillmanns H, Steinhausen M, Leinberger H, Thederan H, and Kubler
W. Pressure measurements in the terminal vascular bed of the epimyocardium of rats and cats. Circ Res 49: 1202–1211, 1981.
289 • JULY 2005 •
www.ajpheart.org
Downloaded from http://ajpheart.physiology.org/ by 10.220.33.2 on May 5, 2017
where coronary flow reserve is substantially reduced. It is
important to eventually include vasoregulatory mechanisms to
investigate many clinically relevant phenomena that relate to
coronary flow reserve (8).
In the future, the present arterial circuit will be extended to
the entire coronary vasculature; i.e., we will connect the
capillaries to the entire venous system. Hence, we will replace
the ad hoc pressure boundary condition on the capillary vessel
with a measured boundary condition at the outlet of the venous
system. Because the capillary boundary conditions are ad hoc,
we examined the effect of variation of capillary pressure.
Figure 6B shows a modest effect of variation of capillary outlet
pressure on the CV of capillary flow. We previously showed
(20) that the cross-connections may serve to homogenize the
pressure and flow distribution in the capillary bed.
One effect of increase in capillary pressure dispersion is the
creation of local flow reversal as seen in Fig. 6A. The flow
reversal does not extend beyond the segment of a capillary
vessel. This was due to the fact that as the outlet pressure
distribution became broad, it was possible that the pressure
gradient was reversed locally at the capillary segment. It can be
seen that the frequency of such an occurrence is quite small,
which is ensured by the homogenization function of the capillary network (20).
H445
H446
ANALYSIS OF CORONARY ARTERIAL BLOOD FLOW
32. VanBavel F and Spaan JA. Branching patterns in the porcine coronary
arterial tree. Estimation of flow heterogeneity. Circ Res 71: 1200 –1212,
1992.
33. Zhou Y, Kassab GS, and Molloi S. In vivo validation of the design rules
of the coronary arteries and their application in the assessment of diffuse
disease. Phys Med Biol 47: 977–993, 2002.
34. Zhou Y, Kassab GS, and Molloi S. On the design of the coronary arterial
tree: a generalization of Murray’s law. Phys Med Biol 44: 2929 –2945,
1999.
35. Zierler K. Indicator dilution methods for measuring blood flow, volume
and other properties of biological systems: a brief history and memoir. Ann
Biomed Eng 28: 836 – 848, 2000.
Downloaded from http://ajpheart.physiology.org/ by 10.220.33.2 on May 5, 2017
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