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5.2 Steady-State Coronary Blood Flow 315
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Fig. 5.5 (a) Relation
between blood flow per
vessel element and the order
number for the arterial
branches of the symmetric
and asymmetric models of
the left common coronary
artery (LCCA). Data are
fitted by exponential
functions with use of the
least squares method. (b)
Fractal relation between
myocardial mass and
relative dispersion of blood
flow. Data are fitted by a
power-law function with use
of the least squares method.
Reproduced from Kassab
et al. (1997) with permission
1
10
10
10
10
10
10
Mean Blood Flow
per Element (ml/s)
10
10
1000
Relative Dispersion (RD) of
Blood Flow per Element (%)
Symmetric Model
0
Asymmetric Model
-1
-2
-3
-4
-6
13
Vessel Order Number
100
10
1
0.01 0.1 1 10 100
Myocardial Mass (gm)
5
7911
(a)
order number ratios, respectively) since the values of the viscosity and length ratios
are close to unity for the first several orders. Thus, if the diameter ratio to the fourth
power is decreasing faster than the increase in the number ratio, then a large pressure
drop will occur. This is precisely the case at order 4 vessels, as can be seen in
Fig. 5.4. These observations are similar to the epicardial pressure measurements
reported by Chilian, Eastham, and Marcus (1986), Chilian, Layne, Klausner,
Eastham, and Marcus (1989), Kanatsuka, Lamping, Eastham, Marcus, and
Dellsperger (1991), and Tillmanns, Steinhausen, Leinberger, Thederan, and Kubler
(1981). Direct comparison with these studies, however, warrants caution because
measurements are made in different species, with a different degree of vasodilation,
and with the additional effect of cardiac contraction. Moreover, pressure measurements are made only in the epicardial vessels, unlike the pressure calculated and
presented here which considers the entire vascular tree.
Figure 5.5a shows the coronary blood flow per vessel in the symmetric and
asymmetric left common coronary artery (LCCA) models, respectively. Note that
because only the pressures at the inlet and outlets are specified for the circuits, the
(b)
316 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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total flow of the asymmetric and symmetric model circuits may be unequal, as is
shown at order 11 in Fig. 5.5a. The asymmetric tree carries more flow for the same
pressure drop than the symmetric tree. Only the asymmetric tree, however, can
produce dispersions of blood flow which have been well established in the literature.
Bassingthwaighte, King, and Roger (1989) have shown with the microsphere deposition technique that local blood flow in the myocardium is very non-uniform and
that the measured non-uniformity varies with the volume of the tissue sample. They
found that the relative dispersion of flow and the tissue mass obey a fractal relationship. Based on the relative dispersions in flow and volume in the present model, the
relation between the myocardial mass and the relative dispersion of blood flow per
element is estimated as shown in Fig. 5.5b. The relation has a fractal character with a
fractal dimension of 1.27. This is similar to the results of Bassingthwaighte et al.
(1989) study that reported fractal dimensions of 1.20, 1.16, and 1.22 for
autoregulated baboon, sheep, and rabbit hearts, respectively. Furthermore, the flow
dispersion in subtrees that perfuse 1-g tissue pieces is found to be 16% (Fig. 5.5b),
which is well within the range of 7–43% reported by Bassingthwaighte et al. (1989).
VanBavel and Spaan (1992) modeled the porcine coronary arterial branching
pattern to estimate flow heterogen eity. The quantitative basis for their dichotomous
tree model is provided by defining and measuring the relation between diameters of
parent and daughter segments at arterial nodes, as well as the relation between the
diameter and length of vessel segments. These relations are used to generate
computer models of the coronar y arterial trees for vessels <500 μm in diameter
and subsequently analyzed with Strahler’s ordering scheme. They reconstructed
trees segment for segment and then calculated the flow and pressure drop in each
segment. The flow in their simulated networks is very heterogeneous. They found
that the relationship between the level of flow heterogeneity and the perfused
volume, as expressed by the number of terminal segments in a subtree, obeyed a
fractal relation with a fractal dimension of 1.20.
5.2.2 Coronary Arterial Tree Model: Node-to-Node
Connectivity
The statistical connectivity model presented above does not account for the node-tonode connectivity of the large number of coronary artery vessels. The model
considered some of the parallel vessels as equivalent elements and hence reduced
the number of vessels significantly to decrease the computational cost alth ough it
sacrificed the realism of vascular network. This section provides a hemodynamic
analysis of the full coronary arterial tree based on the entire coronary arterial tree
node-to-node connectivity model presented in Chap. 2 (Mittal, Zhou, Ung, et al.,
2005). Once the full connectivity is adopted for the model with the reconstructed
diameters and lengths of various vessels throughout the entire coronary arterial tree,
the flow analysis is similar to that in Appendix 1 except the outlet capillary pressure
5.2 Steady-State Coronary Blood Flow 317
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are varied as a Gaussian distribution with a mean of 26 mmHg and variable standard
deviation (SD). The total number of vessels for the right coronary artery (RCA), left
anterior descending (LAD), and left circumflex (LCx) arterial trees depicted are
approximately 1.7, 1.9, and 1.1 million, inclu ding the first segment of capillaries.
Figure 5.6 illustrates the relationship between the vascular connectivity to blood
flow and pressure. The coronary 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. 5.6 for the LAD arterial tree. Figure 5.6a shows a schematic of the
trunk and the primary branches, several of which are identified alphabetically
(i.e., A, B, C,..., H). The flow and pressure along the trunk and primary branches
are shown in Fig. 5.6b and c, respectively. The trunk is denoted by a bold line while
several primary branches are denoted alphabetically in Fig. 5.6b, c in correspondence to Fig. 5.6a. The capillary outlet flow and pressure conditions are connected
by a dotted line in Fig. 5.6b and c, respectively. The outlet capillary pressure is
constant for this simulation as imposed by the boundary condition (26 0 mmHg)
while the computed outlet flow is variable as seen in Fig. 5.6b, c, respectively. The
number of circles along each curve in Fig. 5.6b, c represent the number of bifurcations along a 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. 5.6b).
The major pressure drop along the main path (trunk) occurs at length of 11.1, 10.7,
and 7.4 cm from the inlet of the RCA, LAD, and LCx artery, respectively.
It can be noted that the blood flow through the trunk and primary branches
(Fig. 5.6b) shows either abrupt or 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 while the longer paths with more branches show a
more gradual drop of blood flow. The results suggest that a tracer used for experimental purposes or drug used clinically 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 as shown in Fig. 5.6b. 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 Fig. 5.6b, c are
combined into Fig. 5.7.
Figure 5.7 shows the direct relationship between the segment fl
segment pressure (
Inlet þ Outlet
2
) as Fig. 5.6b and c are combined for the trunk (solid
ow and the mean
thick line) and primary branches. 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 various curves corresponding to the various primary branches cluster in
a narrow range and tend to take on similar shape.
In Fig. 5.8, the relationship between the pressures at the outlet of a vessel segment
and the vessel diameter is shown for the entire LAD coronary arterial tree (nearly
two million vessels). There is a gradual drop in pressure in the proximal vessels
followed by a steeper drop in the microvessels. Figure 5.8a, b and c correspond to
three different outlet boundary conditions: 26 0, 26 2, and 26 6 mmHg,
respectively.
g
)
318 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.6 (a) Schematic of
the trunk of the left anterior
descending (LAD) artery
and some of the primary
branches. Relationship
between flow in a vessel
segment (b) and pressure at
the outlet section of a vessel
segment (c), and cumulative
length of the segment from
the root of the trunk for the
primary branches.
Reproduced from Mittal,
Zhou, Linares, Molloi, and
Kassab (2005) with
permission
(A)
(B)
0
10
–1
10
–2
10
–3
10
–4
10
–5
10
–6
10
Segment Flow (ml/s)
–7
10
–8
10
0
ROOT
A (1196)
B (237)
C
F (616)
G (350)
A
C (380)
D
E
G
E (966)
H
F
810
D (1439)
H (721)
B
246
Cumulative Length from Root (cm)
12
(C)
100
80
60
40
20
Segment Exit Pressure (mm Hg)
0
BCAGDE F H
0
246
Cumulative Len
th from Root (cm
81012
g
5.2 Steady-State Coronary Blood Flow 319
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Fig. 5.7 The relation
between flow and mean
pressure (
trunk (bold line) and
primary branches as
depicted in Fig. 5.6.
Reproduced from Mittal,
Zhou, Linares et al. (2005)
with permission
Inlet þ Outlet
2
) for the
100
80
60
40
Mean Segment Pressure (mm Hg)
20
10–810–710–610–510–410–310–210–110
ment Flow (ml/s)
Se
When the pressure values at the outlet sections of various segments are considered (Fig. 5.8), the profile showed a flat region followed by a large drop in pressure
for vessels <100 μm in diameter. This agrees with experimental epicardial and
subendocardial pressure measurements (Chilian, 1991; Kanatsuka et al., 1991;
Tillmanns et al., 1981). Furthermore, a very steep drop in pressures is observed at
the smallest arteriolar diameters. It is verified that the steep decline is not due to the
fixed capillary pressure (Fig. 5.8a), 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. 5.8b, c. Instead, the steep drop in pressure is due to the large
asymmetry in subtrees. If flow if considered through a vessel segment, the bifurcation will supply two subtrees. If the subtrees are very asymmetric (i.e., very 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 smaller total
number of vessels will have a very abrupt pressure drop as compared to a more
gradual pressure drop for a subtree with many more vessels. Although the “wall”
appearance is quite pronounced in Fig. 5.8, the total numbers of vessels that give rise
to this appearance are only 5% of the total number of vessels. Interestingly, previous
flow simulation models have not reported the steep drop (Bassingthwaighte, Beard,
Li, & Yipintsoi, 1998; VanBavel & Spaan, 1992); likely, because the required
degree of anatomical detail is not present in the previous studies.
For an inlet pressure of 100 mmHg and outlet pressure of 26 mmHg, respectively;
the inlet flow in the most proximal vessel segment is 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 because the vessels are assumed to be rigid in these simulations. The data are
fitted with a Q ¼ ΔPG
coronary arterial tree, Q is inlet flow, and G
least squares fit of the data revealed G
2
(R
> 0.9999), and 0.0052 mL/s/mmHg (R2¼ 0.999) for the RCA, LAD, and LCx
model where ΔP is the pressure difference along the entire
art
is total arterial conductance. A linear
art
values o f 0.0072 (R2> 0.9999), 0.0086
art
–0
320 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.8 Iso-density plot
showing five layers of
frequency between pressure
at the outlet section of a
vessel segment and the
corresponding diameter of
the vessel for the entire LAD
arterial tree for three
different outlet boundary
conditions: (a)26 0, (b)
26 2, and (c)
26 6 mmHg. Reproduced
from Mittal, Zhou, Linares,
et al. (2005) with permission
(A)
100
80
60
40
20
Segment Exit Pressure (mmHg)
0
10
0
10
1
Segment Diameter (µm)
(B)
100
80
60
40
20
Segment Exit Pressure (mmHg)
0
10
0
10
1
Segment Diameter (µm)
(C)
100
10
10
Frequency
–6
10
–5
10
–4
10
–3
10
–2
10
2
2
10
10
3
3
10
10
4
4
80
60
40
20
Segment Exit Pressure (mmHg)
0
10
0
10
1
Segment Diameter (µm)
10
2
10
3
10
4
5.2 Steady-State Coronary Blood Flow 321
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arterial tree, respectively. It is well known that the majority of flow resistance resides
in the arterial tree; particularly, in small arterioles (Jones, Kuo, Davis, & Chilian,
1993). Hence, the arterial tree constitutes majority of coronary circulation resistance.
The computed linear pressure difference–flow relationship, whose slope is the flow
conductance or inverse of flow resistance, yields values of total equivalent resistance
of 139, 116, and 192 mmHg/mL/s for the RCA, LAD, and LCx arterial trees ,
respectively. If these values are normalized by the total weight of the heart
(150 g), we obtain 0.93 (RCA), 0.77 (LAD), and 1.28 mmHg/mL/s/g (LCx). The
linearity arises from the rigid vessel assumption and linear rheology (no shear ratedependent viscosity, no flow-dependent distribution of red cells and plasma to flow
pathways, etc.). It is well known, however, that the coronary vessels are distensible
and blood rheology is nonlinear which give rise to the nonlinear pressure–flow
relation (Hoffman & Spaan, 1990). The nonlinearity is second order as can be
predicted primarily from the distensibility of the coronary blood vessels as shown
in subsequent section of this chapte r (Kassab, 2001).
The mean SD of the arterial transit times are found to be 2.3 0.87 (RCA),
1.5 0.56 (LAD), and 1.9 0.67 s (LCx) at an inlet pressure of 100 mmHg; the
respective maximum transit times are 8.8, 7.8, and 4.5 s. The transit times are
calculated along all possible pathways in the arterial tree by adding the transit times
through each individual segment. There is 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. The relation between mean transit time and inlet flow rate obeys the
classical Stewart–Hamilton relationship which states that the mean transit time of a
fluid through a confined compartment is equal to the total volume of the compartment
divided by the flow rate into the compartment (Zierler, 2000). The theoretical basis for
this relation is provided by Meier and Zierler (1954). For the coronary arterial trees,
each transit time–flow relation is constructed from seven different inlet pressures. Each
inlet pressure yields a different inlet flow rate depending on the equivalent resistance
of the respective coronary arterial tree. The Stewart–Hamilton relation 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 (Kassab, Rider, Tang, & Fung, 1993) and provide some validity of our
calculations of mean transit times in the coronary arterial tree.
The probability density function for the transit times of LAD arterial tree is shown
in Fig. 5.9. The vertical dotted line represents the mean value. The decay of transit
time frequency from the mean, h(t), is fitted by the form h(t) ¼ αt
β
where
t represents the transit time, and α and β are constants. The empirical constants α
and β are determined using a nonlinear least squares fit. The values of the exponents
β are found to be 3.4 (R
2
3.2 (R
¼ 0.782) for the LCx arterial trees. As an example, the least squares fit curve
2
¼ 0.937) for the RCA, 3.2 (R2¼ 0.888) for the LAD, and
for the LAD arterial tree is shown in Fig. 5.9. The probability density function of
transit times show n in this figure is equivalent to the normalized outflow
concentration–time curve under certain conditions. Bassingthwaighte and Beard
(1995) showed that the downslope of the outflow curves of tracer-labeled water
from the rabbit myocardium to be power-law functions of the form t
to approximately 3. The same authors later showed that the t
β
, with β equal
3
form is a general
322 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.9 The probability
frequency for transit times
through the LAD arterial
tree. The dotted vertical line
represents the mean transit
time while the solid line is a
nonlinear least squares fitof
the mean transit time decay
as given by h(t)¼3.7t
(R2¼ 0.888). Reproduced
from Mittal, Zhou, Linares,
et al. (2005) with permission
3.2
0.7
0.6
0.5
0.4
0.3
Frequency
0.2
0.1
0.0
0
123
Transit Time (s)
456
property of a heterogenous vascular network (Beard & Bassingthwaighte, 1998).
Two years later, Beard and Bassingthwaighte (2000) modeled the left coronary
arterial tree based on Kassab et al.’s data (Kassab, Imoto, et al., 1993; Kassab,
Rider, et al., 1993, ), and the capillary and venous system as lumped, to show that the
tails of ishout of intravascular tracer have the form t
finding of t
3.2
for the LAD arterial tree. Hence, the arterial tree seems to be the
3.1
. This is comparable to the
major determinant of is washout pattern or profile.
The relationship between the inlet flow and the mean transit time is 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 mean transit time (t) and inlet
flow rate (Q
value of V
) as:t ¼
in
equals to 1.3, 1.0, and 0.61 mL for the RCA, LAD, and LCx arterial trees,
t
V
t
where Vtrepresents the total arterial volume. The computed
Q
in
respectively. The mean transit time decreased only slightly, in a nearly linear
fashion, when the inlet pressure is increased from 120 to 180 mmHg. It increased,
however, rapidly when the inlet pressure is decreased from 60 to 30 mmHg. The
mean transit time for the entire coronary arterial tree is ~1–2 s at physiological
pressure (100 mmHg) under steady flow conditions in rigid vessels. These mean
transit times of the arterial tree are approximately one half of those reported for the
entire coronary circulation (Beard & Bassingthwaighte, 2000). It is apparent that at a
given flow rate, the mean transit time 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 since transit time is the quotient of length and velocity.
5.2.3 Spatial Heterogeneity of Coronary Flow
It is well established that the distribution of myocardial blood flow is heterogeneous
in small regions of the myocardium (Austin, Aldea, Coggins, Flynn, & Hoffman,
1990; Austin, Smedi ra Jr., Squiers, & Hoffman, 1994; Bassingthwaighte et al., 1987;
5.2 Steady-State Coronary Blood Flow 323
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Bassingthwaighte et al., 1989; Bassingthwaighte et al., 1990; Bassingthwaighte &
Beard, 1995; Bassingthwaighte, Beard, & Li, 2001; Bassingthwaighte & Beyer,
1991; King, Bassingthwaighte, Hales, & Rowell, 1985; Mori et al., 1995; Stapleton,
van Beek, Roger, Baskin, & Bassingthwaighte, 1988). It is difficult to measure the
heterogeneity of myocardial blood flow in vivo. Computer simulation and mathematical models can play an important role in understanding coronary blood flow
since experimental avenues to the problem are highly limited, particularly, deep into
the heart wall. The heterogeneity of myocardial blood flow is affected by many
physical factors such as the architecture of coronary vasculature, duration of systole,
internal and external chamber pressures, regional stresses and strains in the
contracting myocardium, vasoactivity of microvasculature, and so on (Hoffman &
Spaan, 1990). To understand the many factors that impact coronary blood flow, it is
necessary to adapt a staged model approach that gradually considers each factor to
increase the degree of realism. Here a 3D model of the coronary vasculature is
needed to simulate the spatial heterogeneity of coronary blood flow.
5.2.3.1 Steady Flow Analysis in a 3D Coronary Arterial Model
A steady flow simulation (Appendix 1) is carried out in the 3D right coronary artery
(RCA), left anterior descending artery (LAD artery), and left circumflex artery (LCx
artery) tree models described in Chap. 2. The prolate left ventricle (LV) model
encompassing the 3D coronary arterial tree is numerically divided into small plugs
that have the same size and spatial distribution (Huo, Choy, Svendsen, Sinha, &
Kassab, 2009). Briefly, the hollow truncated ellipsoid is formed by rotation of two
ellipses about their major axis (z-axis). The detailed dimension of ellipsoid is
presented in Kaimovitz, Lanir, and Kassab (2005) which is first divided into six
rings along its major axis (z-axis), as shown in Fig. 5.10a. Each ring is then divided
into 7, 7, 7, 4, 3, and 2 mm thick plugs (including the subepicardium, midwall,
subendocardium) rotated around its major axis corresponding to the experimental
measurements. Each thick plug is then divided into the subepicardium, midwall,
subendocardium plugs from outer to inner surfaces, as shown in Fig. 5.10b. Each
plug is further divided into eight pieces of 0.125 g myocardium, as shown in
Fig. 5.10c, to analyze the fractal nature of predicted regional myocardial blood
flow heterogeneity. The regional flow is calculated as the sum of the magnitude of
flow through all first segments of capillary vessels in each plug.
Experimental validations of spatial flow heterogeneity are reported in Huo, Choy,
et al. (2009). Regional coronary blood flow is measured with injections of fluorescent microspheres of 15 μm diameter (Molecular Probes; Eugene, OR). A minimum
of 400 microspheres are needed per tissue piece to be 95% confi
measurement is within 10% of the true value (Buckberg et al., 1971). Before
injection, the microspheres are agitated and dispersed through vigorous agitation
and ultrasonic water bath, respectively. The microspheres are injected into the
cannulated coronary arteries with the cardioplegic solution through Tygon tubing.
dent that the flow
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Fig. 5.10 Schematic representation of experimental sectioning of the porcine hearts to: (a)
different rings and (b) small plugs (subepicardium, midwall, subendocardium). (c) Numerical
sectioning of each plug in 3D model corresponding to experimental sections. Reproduced from
Huo, Choy, et al. (2009) with permission
A slow and steady rate of injection is implemented to keep pressure and flow rate
constant.
The flow and pressure are calculated in each of five 3D reconstructions of the
coronary arterial tree. The mean 1 SD (both mean pressure and mean flow over
each order are averaged in five reconstructions) for pressure–flow relationship and
are found to be very similar (small SD). The mean inlet flow and equivalent
resistance for the five simulations of the LCx arterial tree are 0.44 0.02 (mL/s)
and 167 9 (mmHg ∙ s/mL) for blood, respectively, and 1.27 0.05 (mL/s) and
58 4 (mmHg ∙ s/mL) for cardioplegic solution. The experimentally measured flow
at the inlet of LCx artery is 1.16 0.12 (mL/s) for cardioplegic solution, which
agrees well with the numerical result.
Figure 5.11 illustrates the pressure distribution in two views (lateral left and
posterolateral oblique left) of the entire coronary arterial tree model down to the
first capillary segments. It is clear that the pressure distribution is fairly uniform in
larger vessels and changes significantly in smaller vessels (<100 μm), which is in
agreement with experimental measurements (Chilian, 1991; Chilian et al., 1989;
Kanatsuka et al., 1991). The 3D model’s prediction of longitudinal pressure distributions compared very well to the above results from a model that did not consider
the 3D spatial geometry (Mittal, Zhou, Ung, et al., 2005). The 3D model allows the
differentiation of arteriolar pressure distributions at epicardium and endocardium.
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