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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3787_Библиотеки_им_академика_М_И_Перельмана
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5.2 Steady-State Coronary Blood Flow 335
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(a)
60
40
20
0
012345678910
-20
-40
[(High Hct-Control)/Control ×100]
Difference in Flow and Viscosity (%)
-60
(b)
60
40
20
0
012345678910
-20
-40
[(Low Hct-Control)/Control ×100]
Difference in Flow and Viscosity (%)
-60
Hct = 0.60
Viscosity
Flow
Hct = 0.30
Viscosity
Flow
Fig. 5.18 (a) Relative difference (mean SD, averaged in all segments of each order) in flow rate
and effective blood viscosity [expressed as a percentage (value with Hct of 0.60 value with Hct of
0.45)/value with Hct of 0.45 100] in each order. (b) Relative difference (mean SD) in fl ow rate
and effective blood viscosity [expressed as a percentage (value with Hct of 0.30 value with Hct of
0.45)/value with Hct of 0.45 100] in each order. (a, b) Correspond to Fig. 5.17a, b, respectively.
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
volumetric blood flow rate and D and μ are the diameter and coefficient of viscosity,
respectively), it is determined by the product of Q and μ when the diameter is
constant. Figure 5.18 shows relative uniformity of flow and viscosity in order 0–6
and a larger change in orders 7 for different feed Hct. Therefore, WSS is
redistributed, increasing in the larger vessels and remaining unchanged in the smaller
vessels, as shown in Fig. 5.17a.
336 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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In summary, this section presented the first full anatomical model of the entire
coronary arterial tree that evaluates the roles of vessel distensibility and blood
rheology on the pressure–flow relation and wall shear stress. Although both the
distensibility of the blood vessels and the blood rheology may, in principle, contribute to the nonlinearity of the pressure –flow relation, the model established the
distensibility of the blood vessels as the major determinant. The blood rheology,
on the other hand, is a major determinant of flow resistance (smaller vessels) and
wall shear stress (larger vessels). Although this model does not account for the
vasoactivity of vessels (this will be addressed below), it does serve as a platform to
incorporate these and other biological and physiological phenomena. As such, the
model will allow eventually allow for the understanding of such phenomena as flowoverload and iron-overload (with multiple red blood cell transfusions) which are
general risks of intravenous infusion and transfusion (Millam, 1988; Spahn, Leone,
Reves, & Pasch, 1994). Other applications to chronic high altitude (increased
hematocrit) or anemia (decreased hematocrit) will also be possible when additional
realism is added to a fully integrated model of the coronary circulation.
5.2.5 Capillary Network Flow Analysis
The spatial distribution of blood flow into the coronary capillaries has obvious
physiological significance because the nutrition of the heart muscle depends on the
blood flow in the capillaries where oxygen and nutritive transport primarily occurs.
In Chap. 2 , it is emphasized that the topological structure of the coronary arteries and
intramyocardial veins are tree-like but that the coronary capillary blood vessels have
a non-tree-like topology. The capillaries not only branch but also cross-connect
along their lengths. The presence of cross-connections in the myocardial capillaries
may make the pressure and flow distributions in the capillary bed more uniform.
A network simulation of the coronary capillary blood flow has been previously
done by Wieringa, Spaan, Stassen, and Laird (1982) in a model of hexagonally
stacked parallel capillaries with random ly distributed interconnections based on the
experimental data of Bassingthwaighte, Yipintsoi, and Harvey (1974). Wieringa
et al. (1982) assumed that all capillaries have uniform diameter and obey a linear
pressure–flow relationship that did not consider the capillary distensibility and the
non-Newtonian blood rheology.
It is necessary to advance the previous analysis of the coronary capillary blood
flow in four ways:
1. Use the topology and vascular dimensions of the coronary capillary network as
determined by morphometry
2. Use the compliance or distensibility of the coronary capillaries in the network
analysis
3. Incorporate the non-Newtonian blood rheology in the analysis
5.2 Steady-State Coronary Blood Flow 337
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4. Embed the capillary networks between coronary arterial and venous trees realis-
tically according to morphometric data
With regard to 1 and 4 above, we refer to our previously measured morphometric
data (Chap. 2). With regard to 3 above, we refer to the work of Lingard (1979),
Lipowsky, Kovalacheck, and Zweifach (1978), and Pries et al. (1994). For 2 above,
we use the distensibility of epicardial capillaries of the pig left ventricle at the
diastolic state (Chap. 3). With these data, we can use the laws of physics (conservation of mass and momentum) and the appropriate boundary conditions to formulate well-posed boundary value problems for the hemodynamics of the coronary
capillary network. In the context of the analysis below, our main goals are to
determine the effect of capillary cross-connections and the elasticity of vessels on
the pressure and flow distributions in the coronary capillary network.
As described in Chap. 2, all capillaries can be identified as blood vessels of order
number zero and further designated the capillaries as those fed directly by arterioles
(C
), those drained directly into venules (C0v), and those connected to C0aand C
0a
(C00). The capillaries branch in patterns identified as Y, T, H, or HP (hairpin) based
on their geometric shape and anastomose through capillary cross-connections (C
The C
vessels may connect adjacent capillaries or capillaries originating from
cc
cc
different arterioles. A branching pattern of the coronary capillary bed is constructed
on the basis of these patterns (the frequency of Y, T, H, and HP patterns simulated
the measurements in Kassab and Fung (1994)), and the vascular dimensions (diameters and lengths) are prescribed by the morphometric data of (Kassab & Fung,
1994). An idealized case is shown in Fig. 5.19.
The vascular geometry and the flow condition justify the assumptions that the
Reynolds and Womersley numbers of the flow are very small (0.002 and 0.006, i.e.,
viscous and steady flow, respectively) and that the length-to-diameter ratio of each
capillary vessel is large. Under these assumptions, the classic Poiseuille’s law can be
used to describe the local pressure–flow relationship in a circular cylindrical capillary tube as described in Appendix 2. At the capillary dimension, the particulate
nature of the blood cells becomes important and the blood properties become
non-Newtonian. The viscosity in Poiseuille’s law is no longer constant and should
be considered as an apparent viscosity (μ
). In general, the apparent viscosity is a
app
function of vessel diameter, hematocrit, and shear strain rate as described in Appendix 3. The hydrodynamic law can be combined with the elasticity and rheology of
blood (Eq. (5.17), Appendix 3) to yield the desired relationships (Kassab et al.,
1999). A set of equations for each of the capillary segments of the network in
Fig. 5.19 can be reduced to a set of simultaneous algebraic equations that are solved
iteratively for the nodal pressures once the conductances are evaluated from the
geometry and suitable boundary conditions are specified. The boundary conditions
are simulated by a random uniform distribution with the range P(at inlet of
C
) ¼ 32–42 mmHg and P(at outlet of C0v) ¼ 26–28 mmHg. These boundary
0a
conditions represent a mean pressure drop of 10 mmHg, which is in agreement with
the micropres sure measurements of Klassen, Armour, and Garner (1987 ) on the
epicardial surface of the dog left ventricle. The arterioles and venules are chosen
0v
).
338 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.19 A schematic of an idealized capillary network. Asterisk indicates capillaries that connect
to capillary vessels above and below the capillary plane. Reproduced from Kassab et al. (1999) with
permission
randomly, always maintaining a ratio of four arterioles to seven venules, consistent
with our previous morphometric measurements (Kassab, Rider, et al., 1993; Kassab,
Lin, & Fung, 1994).
The pressure drops as well as the corresponding flows are computed (Kassab
et al., 1999). Figure 5.20,a–d, shows the log-transformed, median-normalized flow
distribution in capillaries of orders C
, C00, C0v, and Ccc, respectively. These
0a
distributions correspond to data obtained over 100 runs of the model. The median
flows obtained from 100 runs are within 1.5% of the median flows obtained from
1000 runs. Hence, all the results shown here correspond to 100 runs of the numerical
algorithm.
Figure 5.21a and c show the effect of C
on the median flows and pressure drops
cc
per capillary segment, respectively, for all orders of capillaries (i.e., crossconnections are removed in some simulations by setting their conductance equal to
zero; for comparison). Figure 5.21b and d show the effect of C
on the coefficient of
cc
variance (CV; SD/mean, %) of the flows and pressure drops, respectively. Our
computational results show that the median flow and pressure drop increased in
the presence of C
bed due to the presence of cross-connections. Furthermore, C
. This is due to a decrease in the overall resistance of the capillary
cc
also homogenized
cc
the flow and pressure distributions. Figure 5.21b and d show that the relative
dispersion (or CV) of flow and pressure drop is reduced in the presence of C
cc
which helps homogenize the blood to adjacent myocytes. Although these dispersions
are reduced in the presence of C
, the dispersions at the venous capillaries are still
cc
greater than those at the arterial capillaries, that is, the capillary outlet flow and
pressure are more heterogeneou s than the inlet flow and pressure with or without C
cc
,
.
5.2 Steady-State Coronary Blood Flow 339
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Fig. 5.20 Log-transformed, median-normalized flow distributions in (a) capillaries fed directly by
arterioles (C
venules (C
permission. Q
); (b) capillaries connected to C0aand C0v(C00); (c) capillaries drained directly into
0a
); and (d) capillary cross-connections (Ccc). Reproduced from Kassab et al. (1999) with
0v
, Q00, Q0v, and Qcc, flow in C0a, C00, C0v, and Ccc, respectively; Q median,
0a
median flow
340 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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AB
6.00E-08
5.00E-08
4.00E-08
3.00E-08
2.00E-08
1.00E-08
Median Flow per Capillary Segment (ml/s)
0.00E+00
0a 00
Capillary Order Number
With Ccc
Without Ccc
0v 0a 00 0vCcc
180
160
140
120
100
CV of Flow per Capillary Segment (%)
With Ccc
Without Ccc
80
60
40
20
0
Capillary Order Number
CD
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Median ΔP per Capillary Segment (mmHg)
0
0a 00
Capillary Order Number
With Ccc
Without Ccc
0v
Ccc
80
60
40
20
00
80
60
40
20
CV of Pressure Drop per Capillary Segment (%)
0
With Ccc
Without Ccc
0a 00
Capillary Order Number
0v
Fig. 5.21 (a) Relationship between median flow per capillary segment and capillary order number,
with and without C
flow and capillary order number, with and without C
drop (ΔP) per capillary segment and capillary order number, with and without C
between CV of pressure drop and capillary order number, with and without C
Kassab et al. (1999) with permission
This result may stem from the fact that the number of venous capillaries is greater
than the number of arterial capillaries (Kassab et al., 1994). Hence, an increase in the
number of possible pathways may lead to an increase in the variability of the
hemodynamic parameters.
The effect of C
Fig. 5.22.QT
in
.(b) Relationship between coefficient of variance (CV; SD/mean, %) of blood
cc
on the total flow into the capillary network (QTin) is shown in
cc
.(c) Relationship between median pressure
cc
.(d) Relationship
cc
. Reproduced from
cc
is normalized with respect to the total flow into the network in the
5.2 Steady-State Coronary Blood Flow 341
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Fig. 5.22 Relationship between median total flow into capillary network (QTin) and number of Ccc.
Flow is normalized with respect to QT
(1999) with permission
with presence of all Ccc. Reproduced from Kassab et al.
in
presence of all Ccc. The total flow into the capillary network increased in the
presence of C
. The increase in total flow occurred for the same pressure drop
cc
across the capillary network. Hence, the resistance to flow is decreased by the
presence of C
.
cc
Finally, the effect of cross-connections on the median velocity of various capillary orders is shown in Fig. 5.23. The median blood velocity is computed from the
flows and cross-sectional areas of the capillary vessels. The median velocities range
from 300 μm/s in capillaries of order C
to 1400 μm/s in capillaries of order C0a.
00
Tillmanns et al. (1974) found average diastolic capillary red cell velocities of
909 and 1428 μm/s on the epicardial surface of turtle and dog left ventricles,
respectively. Direct comparison canno t be made, however, because the order of
capillary vessels is unspecified in the measurements of Tillmanns et al. (1974).
Wieringa et al. (1982) found a mean value of 1428 μm/s in their capillary network
simulation.
The pressure drop in Eq. (5.14) (Appendix 2) can be plotted as a function of the
compliance α for the various orders of capillary vessels as shown in Fig. 5.24. When
the compliance is zero (rigid vessel), the pressure drop corresponds to that given by
Poiseuille’s equation. When the compliance is non-zer o, however, the pressure drop
is smaller than that given by Poiseuille’s equation and varies for the different orders
of capillary vessels. For the measured values of compliance (Kassab et al., 1999), the
pressure drop predicted by Eq. (5.14) is nearly equal to Poiseuille’s pressure drop as
shown in Fig. 5.24. Hence, it can be concluded that the effect of epicardial surface
capillary compliance on the hemodynamics of the capillary network is negligible in
the physiological range of pressures. These are only the epicardial surface vessels,
however, and it remains unknown whether the intramyocardial vessels are equally
stiff.
342 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.23 Relationship between median flow velocity per capillary segment and capillary order
number, with and without C
Fig. 5.24 Relationship between pressure drop per vessel segment (ΔP) and logarithm of compli-
ance constant for various orders of capillary vessels. Pressure drop is normalized with respect to
Poiseuille’s pressure drop (ΔP
. Reproduced from Kassab et al. (1999) with permission
cc
p
). Reproduced from Kassab et al. (1999) with permission
In summary, this network analysis has shown that the capillary cross-connections
tend to homogenize the pressure and flow distributions and reduce the flow resistance of the capillary network (Kassab et al., 1999). Hence, the analysis clarifies an
important hemodynamic function of the capillary cross-connections, showing that
the cross-connections play an important role in the structure–function relationship.
Furthermore, the measured compliance of the epicardial surface coronary capillary
vessels is relatively small in the physiological pressure range. The analysis has
shown that the effect of the measured epicardial surface vessel compliance on the
hemodynamics of the coronary capillary is negligible in the diastolic state of the
heart. The compliance of intramyocardial blood vessels remains unknown, however,
5.3 Structure– Function Relation 343
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and the interaction of the muscle contraction and blood vessel elasticity in systole
will be described in subsequent sections of this chapter.
5.2.6 Venous Network Flow Analysis
The venous system has unique anatomy that includes non-circular cross sections
(approximated by ellipse as demonstrated in Appendix 5), interconnections, and
anastomoses; trifurcation and quadrification branching; and so on as outlined in
Chap. 2. A network steady-state flow analysis is performed in the venous system that
accounts for the unique morphology and branching pattern (Wu, Kassab, Tan, &
Huo, 2017). Figure 5.25a shows the pressure–diameter relationship obtained from
network simulations in comparison with experimental literature measurements. The
computed pressures in coronary venules are consistent with the experimental measurements in coronary venules (vessel diameter of 150–500 μm) (Chilian et al.,
1989). Figure 5.25b shows the longitudinal pressure distribution in each order of
entire coronary arterial and sinusal venous trees. There is an abrupt drop of pressure
as the vessel diameter decreases in the arteriolar and venular beds as shown in
Fig. 5.25b. Hence, the vascular flow resistances are mainly in the arteriolar and
venular vessels.
Moreover, the values of wall shear stress (WSS) are computed based on flow and
diameter in each vessel segment. Figure 5.26 shows the WSS as a function of order
number for entire coronary venous and arterial tree (adapted from Fig. 5.16). The
variations of WSS in the arterial and venous system clearly do not mirror each other.
The WSS is generally lower in the venous system and does not obey the same
inverse relation observed for the arterial system (Fig. 5.1 6 ).
5.3 Structure–Function Relation
The question of functional hierarchy of the coronary arterial has been the subject of
numerous investigations (Beighley, Thomas, Jorgensen, & Ritman, 1996; Kassab,
2006; Zamir, 1990; Zamir, 1998; Zamir & Silve r, 1985). Zamir and colleagues have
suggested that the arterial system can be classified into “distributing” vessels that
remain on the surface of distinct zones of the heart and give rise to “delivering”
vessels that penetrate those zones to implement the delivery of blood (Zamir, 1998;
Zamir & Silver, 1985). As a basis for this classification, Zamir and Silver (1985)
showed that the distributing vessels have a lower branching rate than delivering
vessels. The delivering vessels are found to divide more profusely and terminate
more rapidly than the distributing vessels. Additional evidence for the differences in
the branching pattern of epicardial coronary arteries (EPCA) and intramyocardial
coronary arteries (IMCA) came from X-ray studies of Tanaka et al. (1999). They
found that the self-similar branching pattern of coronary arteries is discrete at the
344 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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35
30
25
-1
20
15
Pressure (mm Hg)
10
5
-2
-3
-4
10 100 1000
-6
-7
-8
-5
-9
-10
Diameter (μm)
(A)
120
100
80
60
40
Pressure (mm Hg)
20
0
-1
2 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Vessel Order Number
(B)
Model
Cat Coronary
-11
Venous
Arterial
-12
Fig. 5.25 (a) Pressure–diameter relationship obtained from computational model and experiments
(Chilian, 1991) in orders 1to12 of coronary sinusal venous tree. (b) Integration of data from
Figs. 5.4 and 5.25a to yield the longitudinal pressure distribution throughout the coronary arterial
(orders 1–11) and venous trees (orders 1to12). Reproduced in part from Kassab et al. (1997)
and Wu et al. (2017) with permission
connection between the EPCA and IMCA (Tanaka et al., 1999). Finally, Ritman and
colleagues found a similar discontinuity between EPCA and IMCA vessels using
micro-CT (Beighley et al., 1996; Beighley, Britton, Gerard-Koch, & Ritman, 2004).
Although these past studies suggest the existence of anatomical differences between
EPCA and IMCA, these differences are not connected to the hemodynamics (e.g.,
flow, velocity) of the coronary circulation.
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