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5.2 Steady-State Coronary Blood Flow 325
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Fig. 5.11 Pressure distribution in two views (lateral left and posterolateral oblique left) in the 3D
model of the entire coronary arterial trees consisting of the epicardial, transmural, and perfusion
sub-networks: (a) lateral-left-view pressure, and (b) posterolateral-oblique-left-view pressure.
Reproduced from Huo, Choy, et al. (2009) with permission
The 3D model predicted higher pressures at the epicardial as compared to the
subendocardial surfaces for each respective order (Huo, Kaimovitz, Lanir, Hoffman,
& Kassab, 2009), which is consistent with experimental observations (Chilian,
1991).
5.2.3.2 Flow Heterogeneity with Fractal Nature
The 3D coronary arterial model provides a platform to analyze the distribution and
heterogeneity in small neighboring regions of the myocardium. To be consistent
with experimental measurements, all numerical results are calculated for
cardioplegic solution with the viscosity of 1.1 cp (Huo, Choy, et al., 2009). From
the 3D model for cardioplegic solution, myocardial flows are calculated as 2.04,
2.84, and 2.29 mL/min/g in subepicardium, midwall, and subendocardium, respectively, which is similar to the experimental measurements of 1.95, 2.32, and
2.24 mL/min/g, respectively. Myocardial flows averaged over the entire myocardial
wall are 2.39 and 2.17 mL/min/g in the 3D model and microsphere measurements,
respectively, which correspond to 359 and 326 mL/min in a heart of 150 g.
The relative dispersion (RD) or heterogeneity in the 3D model (Fig. 5.12a)is
smaller than the experi mental measurements. Interestingly, the within layer
(subepicardium, midwall, and subendocardium) RD showed better agreement
between theory and experiment. Both numerical and experimental results show
that the spatial fractal dimension D in three layers (Fig. 5.12b) is larger than in the
entire LV and septum. This implies that the within layer randomness is stronger than
that of the entire thickness whi ch illustrates the role of correlation of radial flow. The
RD in the arrested hearts is larger than that in the normal physiological condition
(Bassingthwaighte et al., 1989), but smaller than that in beating hearts without
Subendocardium
Midwall
Subepicardium
326 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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1
3-D model
Experiment
RD, Relative Dispersion
0.1
0.1 1 10
Mass (grams)
(a)
1
RD, Relative Dispersion
0.1
Fig. 5.12 (a) Fractal regression for spatial flow in define left ventricle (LV) and septum myocar-
dium. (b) The fractal regression for spatial flow in three layers of LV and septum (subepicardium,
midwall, and subendocardium) obtained from experiments (Mean 1 SD) and 3D model. The
solid, dashed, and dotted lines represent the results of 3D model for the subendocardium, midwall,
and subepicardium, respectively. Reproduced from Huo, Kaimovitz, et al. (2009) with permission
vascular tone (Austin et al., 1994). This suggests that vascular tone has a significant
effect on the heterogeneity of myocardial flows as will be discussed later in the
chapter. The vascular tone is determined by various vasoreactive mechanisms which
affect the flows and pressures distal to the site of resistance (Intaglietta, 1981)to
ensure more uniform regional blood flow. The heterogeneity in beating hearts
without vascular tone is affected mainly by mecha nical factors, such as workload,
myocardial contractility, perfusion pressure, and left ventricular pressure. The most
important effect may be the heterogeneity of local work and local ATP hydrolysis for
0.1 1 10
Mass (grams)
(b)
5.2 Steady-State Coronary Blood Flow 327
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contraction (Bassingthwaighte et al., 2001). Prinze n et al. (1992) and Prinzen,
Augustijn, Arts, Allessie, and Reneman (1990) found that early activation of myocardium leads to only small shortening strain during ejection, and late activation
leads to large ejection phase shortening. Rapid shortening against low resistance
reduces ATP hydrolysis and hence flow demand (Landesberg & Sideman, 1994a;
Landesberg & Sideman, 1994b). These mechanical factors during the isovolumetric
phase of systole may increase the heterogeneity in beating hearts without vascular
tone in comparison with an arrested heart.
Bassingthwaighte and coll eagues (Bassingthwaighte et al., 1987;
Bassingthwaighte et al., 1989; Bassingthwaighte et al., 1990; Bassingthwaighte
et al., 2001; Bassingthwaighte & Beyer, 1991) showed that fractal phenomena
describe regional flows within the heart, skeletal muscle, and other organs. In their
studies, they showed that a simple fractal relationship provides precise descriptions
of the heterogeneity of regional and myocardial blood flows over a wide range of
piece sizes. The fractal form can be represented as RD mðÞ¼RD m
ðÞ
ref
hi
1D
m
m
ref
where m is the mass of the pieces of tissue in grams, D is the spatial fractal
dimension, and the reference level of dispersion, RD(m
be the RD found using pieces of mass m
, which is chosen to be 1 g.
ref
), is taken arbitrarily to
ref
The 3D coronary arterial tree model is used to simulate the spatial flow distribution in the LV and septum of porcine hearts and investigated the fractal nature of
regional myocardial blood flow heterogeneity in diastole in the absence of vessel
tone (Huo, Kaimovitz, et al., 2009 ). Two methods for fractal analysis are
implemented: one (the fractal regression) relates to the effect of plug size on flow
dispersion and the other (correlation versus interval relationship) relates to the effect
of distance between plugs on the correlation between their flows. Figure 5.12a shows
the fractal regression for spatial flow in the LV and septum. A power curve fit reveals
a fractal dimension, D, with value of 1.25 (R
2
¼ 0.98) and 1.27 (R2¼ 0.99) for the
computational model and experimental data (microsphere measurement in six porcine hearts), respectively. The computed fractal dimension is slightly larger than the
value of 1.23 reported by Bassingthwaighte et al. (1989). Furthermore, Fig. 5.12b
shows the fractal regression for spatial flow in three layers of the LV and septum
where a power-law fit for fractal regression in the 3D model shows exponents of 1.45
2
(R
¼ 0.99), 1.46 (R2¼ 0.98), and 1.48 (R2¼ 0.98) for subepicardium, midwall, and
subendocardium, respectively. These agree well with the experimental values (six
hearts) of 1.47, 1.45, and 1.51 for subepicardium, midwall, and subendocardium,
respectively.
Figure 5.13 shows the relationship between the correlation coefficient, r
, and
n
number of inter-myocardial plug intervals, n, at three different levels of myocardial
mass size. Since the fractal dimension, D, can be represented by the equation
D ¼ 2 H (Hurst coefficient), it has the same value (1.25, 1.20) as the prediction
by the 3D model when H is equal to 0.75 and 0.80, respectively. This implies that the
spatial fractal dimension, D, is in the range of 1.20–1.25. In agreement with reports
by Glenny (1992) and Beard and Bassingthwaighte (2000), the correlation coefficient, r
, is found to fall below the curve and may possibly be negative.
5
,
328 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Correlation Coefficient
0.2
0.1
0
012345
Piece size 1.0 g (3-D model)
Piece size 1.0 g (experiment)
Piece size 0.25 g (3-D model)
Piece size 0.125 g (3-D model)
H=0.75
H=0.80
Number of Intervals
Fig. 5.13 The effect of inter-plug distance on the correlation between plugs flows, expressed as the
relationship between correlation coefficient, r
, and number of inter-plug intervals, n, at three
n
different levels of spatial resolution (three from model and one from microsphere measurement).
The correlation coefficients between pieces centered n units apart, r
hi
1
¼
n 1ðÞ2H 2n2Hþ n þ 1ðÞ
r
n
2
2H
, where H is the Hurst coefficient (Bassingthwaighte &
, can be calculated as:
n
Beyer, 1991; Van Beek, Roger, & Bassingthwaighte, 1989). The relationship between Hurst
coefficient and spatial fractal dimension can be represented by the equation H ¼ 2 D, where
D is the fractal dimension. Here, the solid and lines represent the Hurst coefficient of 0.75 and 0.8,
respectively
5.2.4 Role of Vascular Compliance
5.2.4.1 Pressure–Flow Relation in Single Coronary Artery
Kassab (2001) has previously shown that a linear pressure–diameter (P–D)relationship and a small compliance lead to a second-order pressure–flow (P–Q)relationship
in a vessel segment (see Appendix 2). The second-order relationship is a modification
of Poiseuille’s law, which accounts for the distensibility of the blood vessel under the
conditions of Newtonian, steady-state, laminar flow. Distensibility data in the literature
suggest that the smaller coronary arteries and arterioles also obey a linear P–D
relationship in the physiological pressure range with relatively small compliance
(see review in Cornelissen, Dankelman, VanBavel, and Spaan, 2002). Hence, the
second-order P–Q relationship may hold in the various segments of the coronary
arterial tree. It should be noted, however, that in addition to compliance, the curvature
of the pressure–flow relationship depends on number of other factors such as blood
rheology (section below, vessel–myocardium interaction, vascular tone (Chap. 6)). It
is interesting to note that in a diastolic, vasodilated coronary vasculature the P–Q
relationship reveals a second-order relationship (Hoffman & Spaan, 1990).
5.2 Steady-State Coronary Blood Flow 329
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5.2.4.2 Role of Compliance and Blood Rheology on Pressure–Flow
Relation in Entire Coronary Arterial Tree
As shown above, the vessel compliance is an important determinant of the pressure–
flow relation of a vessel segment and, in turn, in an organ. Indeed, the pressure–flow
relation of a circulatory network is expected to be linear if the vessels are rigid and
the fluid is Newtonian according to Poiseuille’s equation (Chap. 1). Since neither is
strictly true, the effect of compliance and blood rheology (hematocrit) on the
nonlinearity of the pressure–flow relation can be evaluated in the entire coronary
arterial tree.
The relation between the apparent viscosity and hematocrit (Hct) for different size
vessels is highly nonlinear as described by the Fahraeus–Lindqvist effect (Fahraeus
& Lindqvist, 1931). The effect of this nonlinearity on the pressure–flow relation as
well as on different size vessels is interesting in the context of a full vascular network
and will be explored below. It is likely that the effect of vessel compliance is a
stronger determinant of the nonlinearity of the pressure–flow relation than the blood
rheology since the change in diameter is amplified by the fourth power as predicted
by Poiseuille’s equation. The blood rheology, on the other hand, plays a larger role
on wall shear stress (WSS ¼ (32μQ)/πD
3
for Poiseuille flow where μ, Q, and D are
blood viscosity, flow rate, and vessel diameter, respectively). Specifically, the
changes in Hct are likely to have a greater effect on WSS of larger than smaller
vessels because of the Fahraeus–Lindqvist effect. Accordingly, the effects of
changes in inlet feed Hct on wall shear stress (WSS) are considered throughout the
entire coronary arterial tree as described below.
The compliance of arteries (see Chap. 3) and in vivo viscosity model (see
Appendix 3) are considered in each vessel order (Table 5.1 in Appendix 4). The
cascading effect of numerous bifurcations on the microvascular Hct (Pries, Ley,
Claassen, & Gaehtgens, 1989; Pries, Secomb, Gaehtgens, & Gross, 1990)is
included in the model. The predictions of the mathematical model have been
compared with the experimentally measured pressure–flow relationship of right
coronary arterial (RCA) tree perfused by the cardioplegic solution (Kassab, Rider,
et al., 1993). Figure 5.14a shows the computed pressure–flow relation with inlet
pressure in the range of 60–100 mmHg. The model of constant viscosity (1.3 cp)
agrees very well with the experimental measurements (Kassab, Imoto, et al., 1993).
As determined by least squares fits of the data, the pressure–flow relation that
accounts for compliance of vessels depicts a second-order polynomial function
2
(R
> 0.9999) in comparison with the linear pressure–flow in a rigid tree model
subject to Poiseuille’s law. Figure 5.14b shows that the total arterial volume
increases linearly with inlet pressure with a slope of 2.3 10
(volume compliance) which is consistent with the previous measurements
(2.6 1.8 10
3
mL/mmHg, see Hamza et al. (2003)).
3
mL/mmHg
Figure 5.15a shows the pressure–flow relations at the inlet of LCx arterial tree
with various feed Hcts ranging from 0.25 to 0.65 as the inlet pressure is varied from
60 to 140 mmHg. Figure 5.15b shows the flow rates as a second-order polynomial
330 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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1.6
(a)
1.4
1.2
1
0.8
0.6
Inlet Flow Rate (ml/s)
0.4
0.2
0
0.8
(b)
0.7
Experimental results
Elastic model with constant viscosity (1.3 cp)
Rigid model with constant viscosity (1.3 cp)
50
60 70 80
RCA Inlet Pressure (mmHg)
90 100 110
0.6
0.5
0.4
0.3
Total Arterial Volume (ml)
0.2
0.1
0
40 60 80 100 120 140
LCx Inlet Pressure (mmHg)
160
Fig. 5.14 (a) Pressure–flow relation at the inlet of the right coronary arterial (RCA) tree as inlet
pressure changes from 60 to 100 mmHg. Experimental results are obtained from Kassab, Rider,
et al. (1993). Elastic and rigid models represent pressure–flow results with elastic and rigid vessel
walls, respectively. Since pressure at each outlet (first capillary segment) is fixed at 26 mmHg,
pressure–flow relations have an intercept of 26 mmHg on the horizontal axis. (b) Relation between
total arterial volume (sum of all vessel volumes of the entire tree) and pressure at the inlet of the left
circumflex arterial (LCx) tree. Least squares fit of data shows a linear relation (R
2
¼ 0.997).
Reproduced from Huo and Kassab (2009) with permission
5.2 Steady-State Coronary Blood Flow 331
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function of feed Hct at the inlet of LCx tree when the inlet pressure equals to
100 mmHg. It is noted that the low feed Hct results in high coronary inlet flow.
The slope of the pressure–flow relation is related to the conductance of flow which is
the inverse of flow resistance. Clearly, the total resistance to flow decreases with a
decrease in inlet hematocrit, which is intuitive since red blood cells contribute to the
resistance to flow.
The compliance of coronary vessels is important because it affects the pressure–
flow relation and hence the resistance to flow (Hoffman & Spaan, 1990). The results
(Figs. 5.14 and 5.15) show that vessel compliance mainly affects the nonlinearity of
the pressure–flow relation. The vessel compliance has a very small effect on WSS in
the entire coronary arterial tree, which is consistent with the finding in the epicardial
coronary arterial tree (Huo & Kassab, 2009).
The relation between WSS and vessel diameter in the vessels of LCx arterial tree
with inlet feed Hct of 0.45 is shown in Fig. 5.16. A least squares fit shows a powerlaw relation (WSS¼3716 D
0.957
, R2¼ 0.989) between WSS (dyne cm2) and
vessel diameter, D (μm), in the diameter range of 30–1000 μm. This is consistent
with the experimental measurements in vessels of diameters >50 μm of dog hearts
(Stepp et al., 1999). There is a relatively uniform WSS in arterioles, however, with
diameters between 10 and 30 μm (orders 1–3), which agrees reasonably well with
experimental measurements (Pries, Secomb, & Gaehtgens, 1995) but otherwise
decreases with larger vessels nearly as an inverse relation. The consistency between
computational results and experimental measurements from fluorescence
microangiography (Stepp et al., 1999) depicts an inverse relationship between
WSS and vessel diameter in epicardial and transmural sub-networks. The area
expansion ratio has a value of unity in the corresponding sub-networks (Kaimovitz,
Huo, Lanir, & Kassab, 2008). This implies a uniform flow velocity in the
sub-networks, which has been validated by experimental measurements (Stepp
et al., 1999) and theoretical analysis (Kassab, 2005). There is a relatively uniform
WSS in arterioles, however, with diameters between 10 and 30 μm as shown in
Fig. 5.16. The same trend has been reported in mesenteric arterioles with diameter
<30 μm (see Fig. 2 in Pries et al. 1995). This is due to the increase of area expansion
ratio with the decrease of diameter in the perfusion sub-network (Kaimovitz et al.,
2008). This trend of uniform shear stress, however, does not extend throughout the
coronary microcirculation (see Chap. 7).
Figure 5.17a and b show the Hct-induced relative difference of WSS in each order
with inlet feed Hct of 0.6 and 0.3, respectively, where the inlet pressure is fixed at
100 mmHg. It is found that the change of inlet feed Hct has a much larger effect on
WSS in the epicardial coronary arteries (orders 8–10) than in the transmural and
perfusion arterioles (orders <8). The WSS in orders 8–10 increases by 12–20% with
the increase of feed Hct from 0.45 to 0.6 and decreases by 13–21% with the decrease
of feed Hct from 0.45 to 0.3. The WSS in orders <7 is not significantly affected by
the change of Hct and WSS in order 7 (a transition from larger arteries to smaller
arterioles) changes by about ~8%. There is no significant difference of WSS in the
arterioles (orders <7) as Hct changes.
332 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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(a)
1.2
1
0.8
0.6
0.4
Inlet Flow Rate (ml/s)Inlet Flow Rate (ml/s)
0.2
0
30 40 50 60 70 80 90
20
(b)
0.6
0.55
0.5
0.25
0.35
0.45
0.55
0.65
100 110 120 130 140 150
LCx Inlet Pressure (mmHg)
0.45
0.4
0.35
0.3
0.25
0.2
0.2 0.3 0.4 0.5 0.6 0.7
Hematocrit
Fig. 5.15 (a) Pressure–flow relation at the inlet of the left circumflex (LCx) tree with different feed
hematocrit (Hct) values (0.25, 0.35, 0.45, 0.55, and 0.65) as inlet pressure varies from 60 to
140 mmHg. (b) Flow rate as a function of feed Hct at the inlet of the LCx tree when inlet pressure
is 100 mmHg. Solid line, least squares fit of data according to a second-order polynomial function
2
> 0.9999). Reproduced from Huo and Kassab (2009) with permission
(R
Fahraeus and Lindqvist (1931) found a reduction of blood viscosity in small
tubes, which is due to a decrease of Hct as the vessel diameter decreases (Barbee &
Cokelet, 1971). Pries and his colleagues showed that the in vitro (Pries, Neuhaus, &
Gaehtgens, 1992) and in vivo (Pries et al., 1994) relations predict the relative
apparent blood viscosity from vessel diameter and Hct. The in vivo flow resistances
5.2 Steady-State Coronary Blood Flow 333
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1000
)
-2
100
10
Wall Shear Stress (dyne·cm
1
1
Vessel Diameter (mm)
Fig. 5.16 Relation between wall shear stress (WSS) and vessel diameter in orders 1–10 (diameter
>8 μm) of LCx arterial tree with inlet feed Hct of 0.45, where the inlet and outlet pressures are
assumed to be 100 and 26 mmHg, respectively. The dark dots represent the experimental data
obtained by Stepp et al. (1999). A least squares fit shows an approximate power-law relationship
(WSS ¼ 3716 Diameter
from Huo and Kassab (2009) with permission
0.957
, R2¼ 0.989) in the diameter range of 30–1000 μm. Reproduced
Model
Stepp et al, 1999
10000100010010
with diameter <40 μm are markedly higher and show a stronger dependence on
hematocrit than in vitro. From the in vivo viscosity law (Pries et al., 1994), the
viscosity decreases in the epicardial and transmural sub-networks (orders >4) and
increases in the perfusion sub-network (orders 4) as the vessel diameter decreases,
and the minimum occurs at order 4 with vessel diameter of about 30 μm. This occurs
due to the Fahraeus–Lindqvist effect.
The model of phase separation shows a disproportionate distribution of red blood
cells and plasma at arteriolar bifurcations (Prie s et al., 1989, 1990). For a given
fractional blood flow, the smaller branch will receive more red blood cells than the
larger branch. This model leads to the larger heterogeneity of blood flow in the
smaller orders as reflected by the standard deviations shown in Fig. 5.18.
The relative differences of flow rate and effective blood viscosity in each order
are shown in Fig. 5.18a, b in correspondence with Fig. 5.17a, b, respectively. The
relative change of flow rate is uniform in each order of vessels. The mean flow rate
(averaged over all vessel segments in each order) decreases by approximately 22%
and increases by 16% with the change of feed Hct from 0.45 to 0.6 and 0.3,
respectively. The mean viscosity (averaged over all vessel segments in each order)
changes more significantly in the larger arteries while the values in orders 0–6 have
an approximat ely uniform increase of 30% and decrease of 22% with the change of
feed Hct from 0.45 to 0.6 and 0.3, respectively.
334 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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(a)
30
20
10
Perfusion
Sub-network
Hct=0.60
Transmural
Sub-network
Epicardial
Sub-network
****
0
0123
-10
Difference in WSS (%)
-20
[(High Hct-Control)/Control ×100]
-30
30
(b)
20
10
0
0123456789
-10
Difference in WSS (%)
-20
[(Low Hct-Control)/Control ×100]
-30
56789
4
Hct=0.30
10
10
****
Fig. 5.17 (a) Relative difference (mean SD, averaged in all segments of each order) in wall shear
stress (WSS) [expressed as a percentage (WSS with Hct of 0.60 WSS with Hct of 0.45)/WSS with
Hct of 0.45 100] in each order where asterisks represent the statistical significance ( p-value
<0.05) of WSS between Hct values of 0.45 and 0.60. (b) Relative difference (mean SD) in WSS
[expressed as a percentage (WSS with Hct of 0.30 WSS with Hct of 0.45)/WSS with Hct of
0.45 100] in each order where asterisks represent the statistical significance ( p-value <0.05) of
WSS between Hct values of 0.45 and 0.30. Here, the inlet and outlet pressures are assumed to be
100 and 26 mmHg, respectively, and inlet feed Hct of 0.45 is used for control. The error bar in each
order represents the standard deviation calculated over values in all vessel segments in that order.
Reproduced from Huo and Kassab (2009) with permission
Furthermore, the redistribution of WSS induced by Hct is much larger in the
epicardial sub-network than in the transmural and perfusion sub-networks, as shown
in Fig. 5.17. Because of the Fahraeus – Lindqvist effect , the dependence of viscosity
on Hct is weaker in the transmural and perfusion sub-networks than the epicardial
sub-network and thus an increase of feed Hct causes a larger increase in larger-vessel
viscosity than in smaller-vessel viscosity, as shown in Fig. 5.18. Figure 5.18 also
shows a uniform flow rate change in each order. Since WSS ¼
32μQ
πD
(Q is the
3
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