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Appendix 5: Elliptical Tube Representation of Coronary Veins (Kassab... 355
https://t.me/med1917
1 H
B ¼ 1 þ 6:98
0:4
¼
X
0
D
mother
X
is the minimal fractional blood flow required to draw erythrocyte into the
0
daughter branch, B is the nonlinearity of the relation between FQ
D
mother
D
ð5:20Þ
ð5:21Þ
and FQB, and
E
A is the difference between the relations derived for the two daughter vessels. For
FQ
< X0, FQEis equal to zero and for FQB> 1 X0, FQEequals to 1. A, B, and X
B
have units of μm1and diameter D has a unit of μm.
Appendix 4: Compliance of Entire Coronary Arterial Tree
(Huo & Kassab, 2009)
Table 5.1 Diameter-compliance and normalized diameter-compliance in different diameter-
defined Strahler orders of pig heart
Orders
diameter range
9–11 500 μm Orders
5–880–500 μm Order 8 8.3 10
0–4 80 μm Order 4 6.2 10
Approximate
Diameter-compliance
(cm/mmHg) at
100 mmHg
6.3 10
10–11
Order 9 2.3 10
Order 7 4.5 10
Order 6 2.4 10
Order 5 1.3 10
Order 3 3.9 10
Order 2 2.5 10
Order 1 1.9 10
Order 0 1.4 10
Normalized
diameter-compliance Reference
4
2.4 103mmHg
4
5
2.0 103mmHg
5
5
5
6
2.1 103mmHg
6
6
6
6
1
1
1
Hamza et al.
(2003)
Giezeman et al.
(1994)
Kassab, Le, and
Fung (1999)
0
Appendix 5: Elliptical Tube Representation of Coronary
Veins (Kassab et al., 1994)
At low normal venous pressures, the cross section of a vein can be approximated by
an ellipse. Relative to a set of rectangular Cartesian coordinates x and y with origin
located at the center, the parametric equations of the ellipse with a semi-major axis
a and a semi-minor axis b are as follows:
356 5 Network Analysis of Coronary Circulation: I. Steady-State Flow
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x ¼ a cos θ; y ¼ b sin θ ð5:22Þ
The cross-sectional area is given by:
Area ¼ 1=2Hxdyþ ydxðÞ
Area ¼ 1=2
Area ¼ πab
An equivalent circle of radius R
Z
2π
ab cos2θ þ ab sin2θ
0
will have the same area if the following holds:
e
2
πR
¼ π ab ð5:24aÞ
e
dθ
ð5:23Þ
or
¼ abðÞ
R
e
1=2
¼ ab=aðÞ
1=2
ð5:24bÞ
Kassab et al. (1994) showed that (a/b) varies from 1.25 to 1.96 for vessels between
orders 1 and 12. Hence R
, varies between 0.714a and 0.895a .
e
The circumferential length of the ellipse is given by:
Circumference ¼Hdx
Z
2π
Circumference ¼
0
2
a2sin2θ þ b2cos2θ
þ dy
1=2
2
1=2
dθ
ð5:25Þ
which is an elliptical integral involving a and b. It may be argued that the crosssectional shape of a vein is sensitive to internal pressure, especially if the transmural
pressure is negative, when the compression of the wall causes elastic instability and
buckling, whereas the circumferential length remains constant in the buckling
process. Hence the circumferential lengt h is a more stable parameter than the
major and minor axes a and b. It is, however, difficult to measure this length, and
its value depends on the smoothness of the wall and any irregularity.
The parameters relevant to the flow can be derived from the Navier–Stokes
equation. For a steady longitudinal flow of a Newtonian viscous fluid in a long
cylindrical tube of elliptical cross section subjected to a constant pressure gradient.
In analogy to the exact solution of flow in a circular cylinder, the velocity profile (u):
hi
u ¼ 2U 1 x=aðÞ
2
y=bðÞ
2
ð5:26Þ
Equation (5.26) satisfies the Navier–Stokes equation and the boundary condition that
u is zero on the elliptical wall described by Eq. (5.22). U is the mean velocity over
the cross section. With Eq. (5.26), the Navier–Stokes equation yiel ds the following:
References 357
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dP=dx ¼4μUa2þ b
2
2b2
= a
ð5:27Þ
where μ is the coefficient of viscosity of the fluid, x is the length along the
longitudinal axis of the tube, and dP/dx is the pressure gradient. Then the volume
rate of flow (Q) is given by:
Q ¼ Area U ¼ π ab U ¼π=4μ a
3b3
= a
2
2
þ b
dP=dx ð5:28Þ
The blood flow conductance is given by the following coefficient:
Conductance ¼ π=4μLa
3b3
= a
2
þ b
2
ð5:29Þ
where L is the length of the tube. The resistance to flow is given by the inverse of
conductance as:
Resistance ¼ 4μL=π a
2
2
= a
3b3
þ b
ð5:30Þ
Equations (5.26)–(5.30) show that a and the ratio b/a are the most important
parameters of venous blood flow in which the Womersley number is <1.
Finally, if the cross section is very narrow, the normal cross section may be better
approximated as a rectangular slit rather than an ellipse. If h represents the thickness
of the slit and the tube is rigid, then dP/dx is related to the mean flow velocity by the
equation
2
dP=dx ¼μUh
F ð5:31Þ
in which F is a number that depends on the structure of the slit, red cell dimensions,
and hematocrit. In this case, the elasticity of the tube becomes very important to
hemodynamics.
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Chapter 6
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Network Analysis of Coronary Circulation:
II. Pulsatile Flow
6.1 Introduction
The blood flow in the coronary arteries is very pulsatile (i.e., time-dependent) with
zero or even reversing (negative) flow in systole. The two major determinants of
coronary flow pulsatility are the: (1) Pulsatile aortic pressure due to cyclic cardiac
contraction, and (2) Myocardial/vascular interaction where the heart cyclically
compresses the coronary vasculature within the myocardium. In order to understand
coronary blood flow, each of these two effects must be understood in turn. We will
start with the former in this section and the latter is described in a subsequent section.
The theory of pulsatile blood flow is initiated by Leonhard Euler in 1775 and
Thomas Young in 1808. Later, numerous attempts have been made at the study of
pulsatile blood flow in the vascular system, e.g., Zamir’s Womersley-type analysis
(Duan & Zamir, 1992, 1995; Zamir, 1998, 2000), and Hughes’ one-dimensional
theory (Hughes & Lubliner, 1973). Basically, there are three major approaches to
simulate the pulsatile blood flow and pressure waves in the vascular system:
1. Womersley’s solution of pulsatile blood flow (Avolio, 1980; Duan & Zamir,
1995; Gan & Moodie, 1989; Gan & Yen, 1994; Gao, Huang, & Yen, 2000;
Huang, Tian, Gao, & Yen, 1998; Womersley, 1955, 1957),
2. One-dimensional wave propagation solution (Formaggia, Gerbeau, Nobile, &
Quarteroni, 2001; Hughes & Lubliner, 1973; Olufsen, 1999, 2000; Stergiopulos,
Young, & Rogge, 1992; Vignon & Taylor, 2004);
3. Three-dimensional solution of pulsatile blood flow (Formaggia et al., 2001).
The latter two methods require appropriate outflow boundary conditions to
simulate the pulsatile blood flow wave. Different outflow boundary conditions
have been developed to satisfy the 1D or 3D equations, such as the pure resistive
load boundary conditions (Stettler, Niederer, & Anliker, 1981), the Windkessel
model (Judd, Redberg, & Mates, 1991; Lee, Chambers, Akizuki, & Downey,
1984) the nonreflecting outlet boundary conditions (Formaggia et al., 2001), or the
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impedance boundary conditions (Formaggia et al., 2001; Olufsen, 1999, 2000;
Vignon & Taylor, 2004). All boundary conditions may be classified as lumped
parameter model boundary conditions, meaning that a distributive vascular tree is
idealized as a discrete or “lumped” equivalence that approximates the outlet flow or
pressure outl et conditions. Despite the usefulness of lumped models, however, they
are generally limited to the global aspects of coronary blood flow and fail to reveal
the flow and pressure wave distribution through the vascular system. For example,
lumped models cannot be used to predict the significant spatial distribution of
coronary blood flow, e.g., transmural distribution of blood flow in the myocardium.
Womersley’s solution avoids those shortcomings because it assumes that pressure
and flow are decoupled. Hence, this approximation can be applied throughout a
complex vascular network such as the entire coronary arterial tree.
In this chapter, we will extend the boundary value formulation initiated in Chap. 5
to pulsatile flow analysis that accounts for pulsatile inlet boundary condition (i.e.,
aortic pressure) in diastolic heart as well as cyclic boundary conditions external to
the coronary vessels from the contracting surrounding myocardium (i.e.,
myocardial–vessel interaction). The analysis will be compared to available experimental data to refute or verify the models.
6.2 Pulsatile Flow in Passive Hearts
To validate the pulsatile flow analysis outlined below, experiments are performed on
isolated, arrested hearts (Huo & Kassab, 2006). Briefly, the isolated heart is placed in
a cold (0
C) saline bath as shown in Fig. 6.1a. The major coronary arteries are
cannulated under saline to avoid air bubbles. The coronary arteries are perfused with
a piston pump that provides a continuous sawtooth pulsatile flow. The sawtooth
pulsatile flow produced by the piston pump is confirmed to be similar to that
measured in vivo at a heart rate of 90 beats/min (Fig. 6.1b). Therefore, the piston
pump is used to simulate the in vivo aortic pressure. The wave amplitude of the
pulsatile pressure is in the range of 50–100 mmHg with a mean pressure of around
75 mm Hg. In order to mimic the profile and range of the in vivo pressure wave, tubes
of appropriate resistance and compliance arranged between the piston pump and
pressure transducer are selected. It is found that the phase difference is very small
(<20 ms), which is within the range of relative error of the flow transducers.
The discussion below is divided into three subsections. First, low-frequency flow
(ω!0) is compared with steady-state flow. Second, the pressure produced by the
piston pump is applied as the inlet pressure boundary condition to calculate the
pressure and flow distribution in the entire coronary arterial tree by using the current
mathematical model in conjunction with a Fourier transform. The mathematical
predictions are in turn compared with the experimental results. Finally, the effects
of various parameters (e.g., frequency, viscosity) on pulsatile flow are examined for
the entire coronary arterial tree.
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