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Venous
Arterial
5.3 Structure– Function Relation 345
https://t.me/med1917
4.0
3.0
)
2
x 10
2
2.0
Wall Shear Stress
(dynes/cm
1.0
0.0
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12
Vessel Order Number
Fig. 5.26 The relation between WSS and order number for the coronary arterial (orders 1–11) and
venous (1to12) trees. Reproduced in part from Huo and Kassab (2009) and Wu et al. (2017)
with permission
5.3.1 Transition from “Distributing” to “Delivering” Vessels
A network analysis of coronary arterial blood flow (Appendix 1) is carried out based
on a full set of anatomical data (Chap. 2; Kassab 2005). Figure 5.27a shows the
variation of CSA along the length of the mai n trunk (solid line) and the primary
branches of the RCA tree. The main trunk begins at the root (most proximal
segment) and is defined by the path corresponding to the largest vessel at each
bifurcation down to the first capillary segment. Similarly, the trunk of each primary
branch begins at the segment that arises directly from the main trunk and descends to
the capillaries along the path of the larger diameter at each bifurcation. Since the path
length for the main trunk and the primary branches vary, the axial position of each
path is normalized with respect to the length of the main trunk (normalized cumulative length, NCL). The variation of the CSA shows a “knee” or an abrupt change in
trend (Fig. 5.27a). The knee occurs at diameter (mean SD for the ten simulat ions)
of 404 114 μm (order 8 vessels) where proximal to the knee are EPCA and distal
to the knee are IMCA (orders 11, 10, and 9 are epicardial while orders <8 are
intramural, (Kassab, Rider, et al., 1993)). The main trunk (solid line) has a relatively
constant CSA or diameter in comparison to other EPCA since the trunk of the RCA
maintains a uniform CSA proximal to the posterior descending artery. Hence, the
transition demarcates EPCA from IMCA.
Similar findings are made for the LAD and LCx coronary arterial trees (Kassab,
2005). The “knee” is also clearly seen similar to the RCA which occurs at
503 40.3 μm (order 8 vessels) and 434 102 μm (order 8 vessels) for the LAD
and LCx, respectively. Hence, the first three largest orders are EPCA while the
remaining orders are IMCA (Kassab, 2005). Unlike the RCA, the LAD and LCx
346 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.27 The relationship
between the (a) segment
cross-sectional area and (b)
segment flow and the
normalized cumulative
length of the segment from
the root of the trunk and the
primary branches of RCA
tree. The solid line denotes
the main trunk. (c)An
iso-density plot showing
five layers of frequency
between the velocity in a
vessel segment and the
corresponding diameter of
the vessel for the RCA tree.
The total number of data
points shown are 754 (a, b)
and 1,716,705 (c).
Reproduced from Kassab
(2005) with permission
0
10
A
)
2
–1
10
–2
10
–3
10
–4
10
–5
10
–6
10
Segment Cross–Sectional Area (cm
–7
10
0.0 0.2 0.4
Normalized Cumulative Length from Root
0
10
B
–1
10
–2
10
–3
10
–4
10
–5
10
Segment Flow (ml / s)
–6
10
–7
10
–8
10
0.0 0.2 0.4
Normalized Cumulative Length from Root
2
10
C
0.6 0.8 1.0
0.6 0.8 1.0
1
10
0
10
–1
10
–2
10
Segment Velocity (cm / s)
–3
10
–4
10
0
10
1
10
2
10
Segment Diameter (µm)
Frequency
-6
10
-5
10
-4
10
-3
10
-2
10
3
10
4
10
5.3 Structure– Function Relation 347
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arterial trees lack a distinct main trunk. This is expected since the main trunks of the
LAD and LCx arteries taper gradually unlike the RCA that remains relatively
uniform as it crowns the base of the heart.
This difference in design of EPCA and IMCA vessels is interesting. The EPCA
tend to maintain a relatively uniform CSA and serve to spread the larger branches
over the surface of the heart to reach the entire surface area of the ventricles. These
vessels are coined as “distributing” by Zamir and Silver ( 1985). Furthermore, the
EPCA seem to maintain relatively uniform flow (Fig. 3.8b) so that the various
regions of the heart can receive a similar source of blood supply. The observation
that the EPCA and IMCA have a different branching pattern in terms of the decrease
in segment CSA with increasing distance from the proximal artery has been reported
by Zamir (1998) in humans, by Tanaka et al. (1999) in dogs and by Beighley et al.
(1996, 2004) in the rat.
Figure 5.27b shows the flow corresponding to each of the vessels whose CSA is
shown in Fig. 5.27a. The flow remains relatively constant for the EPCA and
suddenly drops at the IMCA. The shape is nearly identical to the CSA curve with
the location of the knee occurring at approximately the same NCL which corresponds to the same diameter (order 8 vessels). Hence, both CSA and flow remain
relatively constant throughout the larger EPCA but decrease drastically in the
smaller IMCA. The main trunk flow (solid line) can be easily recognized in
comparison with other EPCA similar to the CSA. Similarly, the flow in the epicardial
primary branches can be distinguished from the intramyocardial branches. The
observation that the flow pattern is mirrors to the CSA pattern with the same
EPCA-IMCA transition had not been previously reported. This novel finding illustrates a direct connection between structure (CSA) and function (flow).
5.3.2 Transition from “Conduction” to “Transport”
The relation between the velocity in each vessel segment and the diameter for all
vessels >8 μm is shown in Fig. 5.27c (nearly two million vessels for the RCA tree;
Chap. 2). Due to the enormity of vessels, the figure is represented as an iso-density
plot showing five layers of frequency. The velocity appears relatively uniform for the
larger vessels and abruptly decreases distal to approximately 12.7 0.38 μm
(mean SD for the ten simulations). The characteristic decrease in velocity is
also found for the LAD (at 13.3 0.34 μm vessels) and LCx (at 13.1 0.69 μm
vessels) arterial trees, respectively (Kassab, 2005).
The transition corresponds to the diameter of order 2 vessels (approximately
13 μm in diameter). This is a novel observation that demarcates the functional
significances of various intramyocardial or “delivering” vessels. This may mark
the transition from conductive to transportive flow. The conduction vessel s (orders
3–8) are nearly area conserving (i.e., the exponent of flow–diameter relation is 2).
Their function is to conduct blood without reduction in velocity. In the smaller
arterioles and capillaries (orders 2), however, the velocity must be reduced to
348 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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ensure adequate transit time in the capillaries and consequently sufficient transport of
oxygen and nutrients. Hence, the exponent must become >2; the larger the exponent, the greater is the reduction in velocity. In the entire coronary arterial tree, the
exponent is only 2.2 but this is adequate to reduce the flow velocity by nearly a factor
of 1000 at the pre-capillary arter iole (Fig. 5.27c).
The relative uniformity of velocity in the large epicardial circulation has been
documented experimentally. Stepp et al. (1999) measured the velocity of canine
epicardial coronary arteries in the range of 50–450 μm in diam eter. They showed that
the velocity varies from approximately 10 mm/s (450 μm vessels) to 6 mm/s (50 μm
vessels). The corresponding variation in flow velocity in the adenosine dilated state
are 20 to 12 mm/s, respectively. Unfortunately, the measurements are limited to
vessels >50 μm.
The relatively abrupt transition in velocity demands an explanation. Naturally,
there are two opposing determinants of velocity: (1) the decrease in CSA from larger
to smaller vessels, and (2) the increase in number of vessels. An analytical explanation can be realized if an idealized symmetric circuit is considered such that the flow
at each level is given by
vessels at order n. Conservation of mass implies that the mean velocity U
where A
is the average CSA of each vessel at order n. The change in velocity from a
n
higher to a lower order is given by
Q
in
, where Qinis the inlet flow and Nnis the total number of
N
n
An=A
U
nþ1
U
n
nþ1
¼
, where the numerator and
N
nþ1=Nn
¼
n
Q
AnN
in
denominator are the CSA and branching ratio, respectively. The velocity is
maintained uniform (velocity ratio is one) when the CSA ratio increases in proportion to the branching ratio, i.e., the decrease in CSA from larger to smaller vessels
occurs in proportion to the increase in number of vessels. This seems to be the case
for orders 3. When the increase in number of vessels is more rapid than the
decrease in CSA, however, the velocity ratio becomes <1 and hence the velocity
decreases towards the smaller vessels (orders 2). It should be noted that distal to
the order 1 and 2 arterioles are capillary vessels (order 0a) that undergo further
branching into smaller capillaries (order 00 vessels) which further branch and
anastomose (Kassab & Fung, 1994), which leads to further reduction in blood
velocity.
A close inspection of previously published data (Kassab, Rider, et al., 1993)on
the diameter and hence CSA indeed reveals a significant difference in the diameter
ratio at order 2 (mean diameter of 12 μm). The average CSA ratio (square of diameter
ratio) is 3.58 (for orders 3–11) and 1.97 (orders 0a to 2) for the RCA. There is a
similar significant difference in the area ratio for the LAD (3.61 and 1.98) and LCx
(4.07 and 1.98) arterial trees. No such difference is found in the branching or number
ratio throughout the range of orders for the RCA, LAD, and LCx arterial trees. In
conclusion, the diameters decrease more slowly distal to order 3 (lower CSA ratio)
and hence the total cross-sectional area (product of CSA and total number of vessels)
increases as reported previously by Kassab, Rider, et al. (1993). Consequently, the
velocity decreases more rapidly in those vessels.
The relatively flat velocity distribution followed by an abrupt decrease in velocity
of small arterioles may be unique to the heart. In the bat wing, Mayrovitz, Tuma, and
,
n
5.3 Structure– Function Relation 349
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Wiedeman (1977) reported a linear drop in flow velocity from 80 μm arterioles to the
capillary vessels. In the Pial arteries of the cat, Kobari et al. (1984) reported a
roughly linear decrease from 200 to 50 μm arterioles. Similar conclusions are
made in the human retinal vessels where the velocity varied nearly linearly with
diameters in the range of 40–140 μm (Riva, Grunwald, Sinclair, & Petrig, 1985). In
the cat and human pulmonary vasculature, the increase in CSA towards the capillary
vessels is exponential (Huang, Yen, McLaurine, & Bledsoe, 1996; Singhal,
Cumming, Horsfiled, & Harding, 1973; Yen et al., 1984). The exponential increase
in CSA implies that the velocity will decrease exponentially towards the capillary
vessels. The heart may indeed be different in that the flow velocity is decreased
significantly during each cardiac cycle due to the vessel/muscle interaction
(discussed in later section of this chapter) such that the anatomical increase in
CSA towards the capillaries does not need to be as large as in other organs. This
speculation remains to be validated.
5.3.3 Possible Mechanisms for Functional Hierarchy
Metzger and Kasnow (1999) proposed a common genetic mechanism for all
branching structures, including blood vessels independent of organ region. Furthermore, since EPCA and IMCA are formed from the same extra-cardiac source of
endothelial cells, it is unlikely that the differences are embryological (Baldwin,
1996). A possible explanation for the differences in EPCA and IMCA is dictated
by local demand. In the inner layers of the heart, angiogenesis is stimulated as the
local tissue oxygen gradient increases during postnatal growth of myocardium. The
EPCA, on the other hand, only grow in diameter or CSA in response to increased
flow or wall shear stress (Kassab et al., 2002). There is evidence that capillary
density and the number of small arterioles increase during the postnatal period
(Mattfeldt & Mall, 1987 ; Rakusan & Turek, 1985). The larger vessels, however,
only increase in CSA and segment length (Tomanek, 1996). Similar observations
(increase in diameter and length of larger vessels and increase in number of smaller
vessels) are made in a swine model of flow-overload induced remodeling of the right
ventricular branches in right ventricular hypertrophy (Kassab, Imoto, et al., 1993).
5.3.4 Significance of Functional Hierarchy
The above simulations provide direct evidence of the structure–function relation in
the coronary circulation. The transition from EPCA to IMCA is evident in the CSA
(structure) and flow (function) curves. The structure of the EPCA vessels is suited for
distribution of blood flow to various regions of myocardium without significant
diminishment of blood flow. Furthermore, the transition from conductive to
transportive flow further demarcates the functional hierarchy of the IMCA. The
350 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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proximal portion of the IMCA, whose role is flow delivery, maintains a constant
velocity of conduction while the distal vessels significantly reduce the velocity to
ensure ample transit time for transport of oxygen and nutrients. Collectively, these
observations lead to a functional hierarchal model of the coronary arterial tree in
normal hearts. This functional model can serve as a reference state for understanding
coronary disease conditions. Coronary artery disease affects both the CSA and flow
patterns and hence cardiac function. Hence, the CSA, flow, and velocity profiles
presented here may represent the signatures of normal coronary circulation and
deviations from these patterns may indication perfusion abnormalities.
Appendix 1: Asymmetric Coronary Tree Model (Kassab
et al., 1997)
The asymmetric model (Fig. 5.3) simulates the morphometric data of connectivity
matrix (Tables 2.9 and 2.10, Appendix 2 in Chap. 2). Each element shown in Fig. 5.3
may represent one or more elements in parallel. The number of possible pathways
for each element of Fig. 5.2 increases exponentially towards the capillary vessels
(Kassab et al., 1997). A realistic analysis consistent with the morphometric data must
also incorporate the dispersion of diameters and lengths of various orders. In a real
flow, the inflow from the aortic sinus is non-uniformly distributed to the parallel
vessels of order 10 because all order 10 elements do not have the same diameter and
length and offsprings. An idealization is made such that vessels at a given mean
element connectivity (Fig. 5.2 ) are considered parallel and hence have the same
diameter and lengths consistent with the mean morphometric data.
Despite the sophistication of the asymmetric model, it still does not satisfy all of
the statistical data measured previously (Kassab, Rider, et al., 1993). For example,
only one tree topology, corresponding to the mean connectivity matrix, is considered
which ignores the standard deviations of the connectivity matrix. Furthermore, the
connectivity matrix shows a small number of vessels of order n branching from
vessels of order n that the symmetric model does not consider. Finally, the asymmetric circuit is not a bifurcating tree model and cannot satisfy the statistics of the
segment-to-element ratios (S/E) reported in Kassab, Rider, et al. (1993). The asym-
metric model includes the assumption that the S/E ¼ 1 for all orders of vessels that is
not corroborated by experimental measurements (Fig. 2.9, Chap. 2). The analysis
also includes the assumption that certain elements are grouped in parallel definition
of the equivalent conductance G
Once the branching pattern and vascular geometry of the full arterial network are
generated, a steady-state network analysis can be performed (Kassab et al., 1997;
Mittal, Zhou, Linares, et al., 2005). Briefly, if the cylindrical vessel is considered
rigid, long and slender, under laminar and steady flow, the Poiseuille’s law for a
Newtonian fluid can be stated as:
, which simplifies the problem considerably.
eq
Appendix 1: Asymmetric Coronary Tree Model (Kassab et al., 1997) 351
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π
128
ΔP
ijGij
ð5:1Þ
Qij¼
where Q
i and j. ΔP
conductance, G
is the volumetric flow, in a vessel between any two nodes, represented by
ij
is the pressure differential given by ΔPij¼ Pi Pj, and vessel
ij
, is given by Gij¼
ij
4
D
ij
where Dij, L
μijL
ij
and μijare the diameter,
ij,
length, and viscosity, respectively, between nodes i and j. The variation of viscosity
with vessel diameter is given by Pries et al. (1994) as:
"#
μ ¼ 1 þ 6 e
D
D 1:1
0:085D
2
þ 3:2 2:44e
0:06D
0:645
1
D 1:1
2
D
ð5:2Þ
where D is the vessel diameter.
Two or more vessels emanate from the jth node anywhere in the tree with the
number of vessels converging at the jth node being m
. By conservation of mass, the
j
following must hold:
m
j
X
Qij¼ 0 ð5:3Þ
i¼1
where the volumetric flow into a node is considered positive and flow out of a node is
negative for any branch. From Eqs. (5.1), (5.2), and (5.3), a set of linear algebraic
equations in pressure for M nodes in the network is obtained as:
The set of equations represented by Eq. (5.4) 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 ¼ G
(idealized to ~850 for LCCA), P is a 1 n column vector of the unknown nodal
pressures, and G
pressures of their attached vessels, respectively. Boundary conditions are prescribed
by assigning an inlet pressure of 100 mmHg and a uniform pressure of 25 mmHg at
the outlet of the first capillary segment. Since matrix G is a very sparse matrix, it can
be represented in a reduced form for optimal memory utilization. This system of
equations can be solved to determine the pressure values at all internal nodes of the
arterial tree. The pressure drops as well as the corresponding flows can be subsequently calculated.
m
j
X
Pi P
i¼1
where G is the n n matrix of conductances
BPB
is the column vector of the conductances times the boundary
BPB
G
¼ 0 ð5:4Þ
j
ij
352 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Symmetric Model
This is an analytical model that simulates the mean statistical data (assumes all
standard deviations are 0) but replaces the connectivity matrix by a diagonal matrix
whose non-vanishing components in row m and column m + 1 are the branching ratios
of the number of the elements of order m divided by the number of elements of order
m + 1. Physically, it is equivalent to assuming that all the vessel elements in any order
are in parallel, and the blood pressures at all the junctions between specific orders of
vessels are equal. In this simplified circuit, the flow in each element of order n obeys
Poiseuille’s formula as given by Eq. (5.1). There are N, elements of order n in parallel.
If the total flow is Q
yields the pressure drop. The pressure at the Valsalva sinus, P
,thenqn, in each vessel of order n is QT/Nn. Equation (5.1)then
T
at n ¼ 11 being given
11
(e.g., an inlet diastolic coronary artery pressure of 100 mmHg similar to asymmetric
model), P
, P9, ... P0a(0a refers to an arteriolar capillary) can be computed in turn.
10
Using the mean morphometric data, the pressure profile can be obtained under the
assumptions that the pressure at the first bifurcation of the capillary bed, P
,isa
0a
constant with a value of 25 mmHg as noted for the asymmetric model.
Appendix 2: Steady Laminar Flow in an Elastic Tube
(Kassab, 2001)
If the distensibility of the blood vessels is known, the mechanics of the blood vessel
can be coupled to the mechanics of blood flow to yield a pressure–flow relation for
each vessel segment. This can be demonstrated for the cylindrical coronary arteries
as follows: assume that the tube is long and slender, that the flow is laminar and
steady, that the disturbances due to entry and exit are negligible, and that the
deformed tube remains smooth and slender. These assumptions permit the use of
Poiseuille’s law for a Newtonian fluid that can be stated as:
dP=dx ¼ 128μ=πD
4
Q ð5:5Þ
where P is the pressure, x is the axial coordinate, Q is the volume-flow rate and D, L,
and μ are the diameter, length, and viscosity, respectively. In a stationary,
non-permeable tube, Q is a constant throughout the length of the tube. The tube
diameter is a function of x because of the elastic deformation. Pressure–diameter data
(Chap. 3) show that, in the physiological pressure range, the elastic deformation can
be described approximately by a linear relationship as:
D D
where D is the diameter at a given intravascular pressure P, D
corresponding to a pressure P
*
¼ α P P
and α is the compliance constant of the vessel
ðÞ ð5:6Þ
*
is the diameter
(Kassab et al., 1999). Using Eq. (5.6), differentiation yields:
Appendix 2: Steady Laminar Flow in an Elastic Tube (Kassab, 2001) 353
https://t.me/med1917
dP=dx ¼ dP=dD dD=dx ¼ 1=αdD=dx ð5:7Þ
On substituting Eq. (5.6) into Eq. (5.5) and rearranging terms, we obtain the
following:
4
D
dD ¼ 128μαQ=πðÞdx ð5:8Þ
Since the right-hand side term is a constant independent of x, we obtain the
integrated result:
5
D
xðÞ¼ 640μαQ=πðÞx þ D50ðÞ ð5:9Þ
The integration constant can be determined by the boundary condition at the entry
section of the capillary, that when x ¼ 0, D(x ) ¼ D(0). Putting x ¼ L, at the exit
section of a capillary, in Eq. (5.9) yields
5
D
LðÞD50ðÞ¼640μ
αQL=π ð5:10Þ
app
We now seek an approximate expression of Eq. (5.10) when D(L ) D(0) is small,
i.e., the vessel compliance is small. Letting D(L ) ¼ D(0) + ε, expanding the left-hand
side of Eq. (5.10) in power series of ε, and retaining only terms up to ε
2
, we obtain
the approximation:
DLðÞD 0ðÞ½1 þ 2 DLðÞD 0ðÞ½=D 0ðÞ
fg
¼ 128μ
app
αLQ
= πD40ðÞ
ð5:11Þ
Using Eq. (5.6) first at x ¼ L and then at x ¼ 0 and subtracting, we have:
DLðÞD 0ðÞ¼α PLðÞP 0ðÞ½ ð5:12Þ
Combining Eqs. (5.11) and (5.12), and writing D
ΔP þ 2α=D
ðÞΔP2¼ 128μ
0
for D(0), we obtain:
0
LQ=πD
app
4
0
ð5:13Þ
where ΔP ¼ P(L ) P(0). The solution to Eq. (5.13) takes the form:
where ΔP
hi
ΔP
¼Dnþ D
n
p
is the Poiseuille’s pressure drop as given by the right-hand side of
2
þ 8αnΔPpn=D
n
1=2
=4α
n
n
ð5:14Þ
Eq. (5.13) and applies to each arterial vessel of order n. It is noted that 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 various orders of
vessels.
354 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Appendix 3: Models of Blood Rheology (Huo & Kassab,
2009)
Fahraeus–Lindqvist effect
The relative apparent viscosity in a vessel segment in vivo, previously reported
(Pries et al., 1994 ), can be written as:
μ
vivo
"#
¼ 1 þ μ
0:45
1
1 H
ðÞ
1 0:45ðÞ
C
D
1
C
1
D 1:1
2
D
D 1:1
2
D
ð5:15Þ
where μ
and HDare the viscosity and discharge hematocri t (Hct), respectively.
vivo
The apparent viscosity equals to the product of relative ap parent viscosity and 1.3 cp
(the viscosity of plasma). μ
μ
C ¼ 0:8 þ e
The units for μ
0075D
and D are cP and μm, respectively. Equation (5.15)reflects the
vivo
and exponent C are defined as follows:
0:45
0:45
0:085D
¼ 6 e
1 þ
þ 3:2 2:44 e
1
11
1 þ 10
D
0:06D
þ
12
1 þ 10
0:645
1
11
D
ð5:16Þ
ð5:17Þ
12
Fahraeus–Lindqvist effect.
Phase-separation effect
In order to consider the phase-separation effect, Pries et al. (1989) have studied the
distribution of erythrocyte at microvascular bifurcations. The fraction of the erythrocyte flow and volumetric blood flow from the mother vessel to a daughter vessel is
defined as FQ
q
daughter
¼
E
q
mother
and FQB¼
Q
daughter
, respectively. Here, capital Q and small
Q
mother
q represent the volumetric blood flow and erythrocyte flow, respectively. An empir-
ical relation (Pries et al., 1990) has been developed to describe the distribution of
volumetric blood flow and erythrocyte flow at an individual bifurcation, which can
be written as:
where Logit FQ
ðÞ¼ln
E
ðÞ¼A þ B Logit
Logit FQ
E
FQ
E
. A, B, and X
1FQ
E
6:96 ln
A ¼
FQB X
1 2X
can be written as:
0
D
left daughter
D
right daughter
D
mother
0
0
ð5:18Þ
ð5:19Þ
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