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(5). https://doi.org/10.1115/1.
Chapter 5
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Network Analysis of Coronary Circulation:
I. Steady-State Flow
5.1 Introduction
Coronary heart disease is one of the most prevalent health problems affecting
humans around the world. Mechanically, the main problem in coronary heart disease
is that there is insufficient blood supply to the heart muscle to fulfill the cardiac
metabolic needs. Therefore, the heart fails as an adequate blood pump. Despite the
magnitude of this health problem, the coronary circulation system remains poorly
understood, as clinical work has largely focused on the diseased vessels (atherosclerosis, hypertension, hypercholesteremia, diabetes, etc.) rather than on the dynamics
of the healthy coronary blood flow.
The study of coronary circulation requires a bioengineering understanding of
both the coronary system (complex anatomy consisting of millions of vessels
spanning three orders of magnitude in dimensions, Chap. 2; nonlinear and
non-isotropic passive and active mechanical properties, Chap. 3; nonlinear blood
rheology including non-Newtonian properties in microcirculation, etc.) and the
interactions with the surrounding myocardium. Despite progress in these areas (see
Chaps. 2–4), many issues remain unresolved. Advancements in computer modeling
and numerical methods now make it possible to develop anatomically based computational (distributive) models of the entire coronary vascular system rather than the
“lumped” models used in the past in which the anatomical details of the coronary
vascular system are ignored. Computer simulation and modeling are important tools
in studying the integrated coronary circulation because experimental avenues are
highly limited, particularly in the circulation of the deeper layer of the heart which is
not amenable to direct visualization.
Three kinds of hemodynamic problems are relevant to coronary flow and myocardial perfusion in cardiology. First, the longitudinal distribution of pressure and
blood flow in the coronary blood vessels (i.e., profile of pressure and flow in
successive orders of arteries and veins and capillaries) dictates the level of blood
flow in the capillaries for oxygen and nutrient delivery to adjacent myocytes.
© Springer Science+Business Media, LLC, part of Springer Nature 2019
G. S. Kassab, Coronary Circulation, https://doi.org/10.1007/978-3-030-14819-5_5
309
310 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Morphological
Data
Blood Rheology
Data
Autoregulaon
Data
Boundary
Condions
Fig. 5.1 Block diagram showing interaction between input data (morphological, rheological,
hemodynamics, etc.), theoretical predictions and experimental validations. Reproduced from
Kassab (2001) with permission
Theory
Compliance
Data
Theorecal
Predicon
Experimental
Validaon
Agreement
End
DiagnosisDiscrepancy
Second, the spatial distribution of perfusion in the myocardium is relevant because
too large of heterogeneity in perfusion may pose the risk of local ischemia to some
regions of myocardium. Lastly, the local details of the flow field (e.g., wall shear
stress and respective spatial and temporal gradients) influence the endothelium,
cholesterol transport, atherogenesis, atherosclerosis, stenosis, and dilatation. These
hemodynamic properties interact, e.g., the spatial distribution of myocardial perfusion has an important influence on myocardial infarction, and changes in spatial
distribution are due to changes in longitudinal distribution of blood pressure and
vascular volume. Also, the interaction of blood pressure, vessel elasticity, and
smooth muscle tone in the myocardium may give rise to interesting coronary
phenomena including phasic arterial inflow and venous outflow, significant spatial
and temporal flow heterogeneity, zero-flow pressure, and autoregulation, to mention
just a few (see review by Hoffman & Spaan, 1990).
To understand coronary hemodynamic problems, it is necessary to analyze each
component of system before synthesizing the whole. Accordingly, we need to use
(1) models of vascular geometry and branching pattern, mechanical properties
(passive and active) of the coronary vessels (arteries, capillaries and veins), and
rheology of blood in the coronary vasculature (Chaps. 2–4), (2) apply the basic laws
of physics to write down the governing equations (Chap. 1), and (3) Specify the
appropriate boundary conditions to solve the boundary value problems of coronary
circulation (Fig. 5.1), Chaps. 5–8. When the theoretical and experimental results are
compared as shown in Fig. 5.1, one may find that they do agree. In that case, the
process ends successfully and one gains confidence in the theory which can then be
used to predict the behavior of the physiological system. On the other hand, one may
find that they disagree, and it becomes necessary to examine the cause of the
5.2 Steady-State Coronary Blood Flow 311
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discrepancy. When the theory-experiment discrepancy is “diagnosed” one may wish
to improve the experiment, or the theory, or both. The process is repeated until there
is agreement between theory and experiment. This approach illustrates the use of
physical principles, with the help of anatomy and mechanical properties, to explain
and predict the physiology of the coronary circulation in quantitative terms.
This process is analogous to taking a radio apart into its components (e.g.,
antenna, printed circuit board, resistors, capacitors, coils and transformers, transistors, integrated circuits, speaker) and reassembling it. If we are successful at this
reduction/integration exercise, we gain confidence in our understanding of how the
radio works. What is more interesting is when we are not succes sful. The latter
challenges our understanding of the system and forces us to look deeper (i.e.,
examine our assumptions, logic, function of each component). Greater lessons are
learned from failure than success, and eventual success leads to illumination of our
understanding of the system.
In this chapter, we will use the laws of mechanics (Chap. 1) as a foundation to
integrate or assemble the morphometric data of the coronary vasculature (Chap. 2)
with the material properties of the coronary vasculature (Chaps. 3 and 4), to solve
some boundary value problems of c oronary circulation under steady-state flow. The
solutions will reveal the spatial distributions of transit times, pressure, shear stress,
flow and perfusion; and interesting structure–function relations. This approach
demonstrates the extent to which a bioengineering approach can yield precise
information about coronary circulation leading to myocardial perfusion. The theory
connects physical, morphometric, and rheological variables. Without such an
approach, it would be very difficult to correlate these variables by conventional,
empirical methods. Bel ow, we will solve boundary value problems focused on the
first two kinds of hemodynamic problems noted at above (i.e., longitudinal distribution of pressure and flow and spatial heterogeneity of perfusion) while Chap. 6
will consider the pulsatile flow including myocardial vessel interactions. Finally,
Chap. 8 will address the local details of flow field and local stress distributions in the
vessel wall.
5.2 Steady-State Coronary Blood Flow
To study a system as complex as the complete coronary circulation, it is important to
start at the simplest level, before adding additional levels of complexity. The
simplest case is the steady-state (i.e., no time-dependence) blood flow in the diastolic, maximally vasodilated state of the coronary vasculature, which will be
considered here first.
Blood flow depends on two dimensionless numbers: the Reynolds number, N
(NR¼ ρUD/μ, where U is the mean flow velocity, D is the lumen dimension, ρ is the
density of blood, and μ is the viscosity of blood. Reynolds number affects flow
instability, turbulence, separation, entry and exit disturbances, and resistance of
blood flow. The second important number is the Womersley number, N
R
W
312 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.2 Relation between
Reynolds and Womersley
numbers and order number
of the elements for the
arterial branches of the left
common coronary artery.
Data are fitted by
exponential functions.
Reproduced from Kassab,
Berkley, and Fung (1997)
with permission
(NW¼ D/2(ρω/μ)
1/2
where ω is the circular frequency of pulsatile flow). The
3
10
10
10
10
Nondimensional
10
Hemodynamic Parameters
10
Reynolds number
Womersley number
2
1
0
-1
-2
13579
11
Vessel Order Number
Womersley number affects transient response of blood flow, phase shift, dynamic
amplification, resonance, damping and reflection, and transmission at branching
nodes. These flow phenomena are larger if the N
and NWnumbers are much larger
R
than 1, and smaller if these numbers are much smaller than 1. In majority of coronary
arteries, the Womersley numbers are smaller than 1, tending to 0.01 at the capillary
inlets. Thus, it is reasonable to treat the coronary blood flow as quasi-steady. N
R
approximately 100 in largest coronary arteries (order 11) and decreases exponentially with smaller order number to 0.1 in order 1 arterioles (Fig. 5.2; see Chap. 2 for
definition of diameter-defined Strahler ordering system). Hence, in most coronary
arteries, the flow can be approximated by Poiseuille flow (i.e., laminar flow); only in
the larger coronary arteries is a correction to the Poiseuillean resistance needed.
Thus, Poiseuille law can be used as a first approximation, and corrections for higher
Reynolds number and Womersley number can be made later.
is
5.2.1 Longitudinal Pressure and Flow Distributions
5.2.1.1 Coronary Arterial Tree Model: Statistical Connectivity
To develop a realistic model of the entire coronary vascular circu it for numerical
analysis, the model must have connectivity of the vessels that agree with the
measured anatomical connectivity matrix (Chap. 2). Two vascular circuits have
been specified for the hemodynamic analysis of coronar y flow distribution (Appendix 1). The asymmetric model (Fig. 5.3) satisfies all the statistical anatomical data,
including the means and standard deviations of the diameters and lengths of vessels,
and the mean connectivity matrix. Alternately the symmetric model satisfies all mean
anatomical statistical data, but not the standard deviations and connectivity matrix.
In the symmetric model, the measured connectivity matrix is replaced with an
5.2 Steady-State Coronary Blood Flow 313
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11
...9
...8
...7
...6
*
...5
...4
...7
...6
...7
...4
*
...5
...4
...2
...3
...2
...3
8...
10
6...
*
5...
9
*
5...
8
4...
4...
3...
7
3...
2...
6
3...
5
4
2...
3
1...
0a
0a
1
0a
2
0a
0a
1
0a 0a
1
0a
*
5
0a
123
0a
0a
4
2
0a
0a0a
3
...1
0a
0a
2
0a
1
0a
0a
0a
1
0a
2
0a
1
0a0a
Fig. 5.3 Schematic Asymmetric Model of Coronary Circulation: (Left) Schematic diagram of the
trunk of the left common coronary artery (LCCA) and its immediate branches as specified by its
connectivity matrix. The “...” denotes that each branch arising from the trunk gives rise to further
branches whose branching pattern is also consistent with the connectivity matrix and so on down to
the arterial capillaries (order 0a). (Right) Schematic diagram of a fifth-order arteriole demonstrating
all possible pathways to the capillaries as prescribed by the connectivity matrix. Asterisk shows the
origin of the order 5 arteriole along the trunk of the LCCA (left). Reproduced from Kassab et al.
(1997) with permission
idealized diagonal one. When both the models are subjected to given inlet and outlet
pressures, the longitudinal mean pressure profiles (nodal pressure plotted as a
function of the order number of the blood vessels) are likely to be the same. The
relative dispersion (rat io of the standard deviation of a variable divided by its mean
value) of the pressure and flow, however, will be very different in these two models.
314 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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Fig. 5.4 Relationship
between (a) blood pressure
at the outlet of a blood
vessel and (b) pressure drop
per vessel element; and the
order number of arterial
branches of the symmetric
and asymmetric models of
the left common coronary
artery (LCCA). Curves are
of fifth-order polynomials.
Reproduced from Kassab
et al. (1997) with permission
100
90
80
70
60
50
40
30
at Outlet of Element
20
10
Mean Blood Pressure (mmHg)
0
135
Vessel Order Number
20
15
10
5
Mean Pressure Drop
per Element (mmHg)
Symmetric Model
Asymmetric Model
79 11
(a)
Symmetric Model
Asymmetric Model
Figure 5.4a and b show the mean values of longitudinal pressure distribution and
the pressure drop per vessel element, respectively. Pressure boundary conditions at
the first capillary bifurcations are assumed to be a constant with a value of 26 mmHg.
It is interesting that the symmetric model yields similar mean longitudinal blood
pressure profile and pressure drop per element to that of the asymmetric model. The
major difference between the asymmetric and symmetric models is that the former
produces dispersions in flow, pressure, and volume, while the latter has zero
dispersions. In both models, most of the pressure drop occurs across the first four
orders of vessels (Fig. 5.4). This suggests that the vascular tree topology may not
play as important a role as the vascular geometry in dictating the shape of the
pressure profile.
Based on the Poiseuille equation (Appendix 1), the pressure drop is dictated by
the ratio R
4
nðÞ=RNnðÞ(where n is order numbe r; RDand RNare the diameter and
D
0
13579 11
Vessel Order Number
(b)
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