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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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6.2 Pulsatile Flow in Passive Hearts 365
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Fig. 6.1 (a) Schematic of apparatus for pulsatile flow measurements. (b) Comparison of the profile
of pressure produced by the piston pump with measured in vivo aortic pressure. Reproduced from
Huo and Kassab (2006) with permission
366 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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6.2.1 Womersley-Type Model
Huo and Kassab (2006) performed a Womersley-type numerical analysis of pulse
wave transmission in the entire coronary arterial tree as outlined in Appendix 1. The
constitutive equation that relates pressure to cross-sectional area for the several
largest orders is based on experimental data (Kassab and Molloi (2001); Chap. 3).
The Womersley model is modified to account for the measured constitutive relation
of coronary arteries. Previous measurements of coronary wall thickness (Chap. 3)
are also incorporated into the numerical model. The predictions of the mathematical
model are compared with the experimental measurements on diastolic arrested,
vasodilated porcine hearts as outlined below.
6.2.1.1 Low-Frequency Flow Model Compared with Steady-State Flow
It is desirable to verify the analysis for the values of the pressure and flow against
available analytical or numerical solutions. The low-frequency (ω!0) pressure and
flow are calculated by using the current mathematical model and compared to a
previous steady-state model (Huo & Kassab, 2006). The two models are
implemented with the same mean inlet pressure of 100 mmHg. The outlet pressures
in the steady-state computation are set at 16 mmHg and the outlet impedances in the
pulsatile computation are evaluated as
vessel segments (order 0). The wave frequency ( f ) of the pulsatile blood flow is
assumed to be 0.001 Hz. The entire asymmetric LAD, LCx, and RCA trees are
considered in the comparison of the two models. The direct relationship between
mean segment flow |Q|
pressure |P|
(average of 0.5 (|P(0, ω)| + |P (L, ω)|) at every order in the
mean
(average of flow rates at every order) and mean segment
mean
coronary arteries is compared (Huo & Kassab, 2006). The pulsatile flow results at
low frequency agree very well with previous steady-state model both numerically
and in theory (Appendix 1, Section 3).
128 μ
L
capillary
capillary
4
πD
ðÞ
capillary
(g s/cm
4
) at the first capillary
6.2.1.2 Experimental Validation of Womersley Model
The piston pump is used to impose the pressure and flow wave at the inlet of the
LCCA and RCA. The Fourier transform is applied to discretize the real measured
pressure wave into the complex periodic Fourier series. The mathematical methods, as
described in Appendix 1, are implemented to calculate the flow rate at the inlet of the
LCCA and RCA trees based on the detailed morphometric data. The viscosity (μ)of
the perfusion solution is selected as 1.1 cp to mimic our cardioplegic solution
containing Albumin. Figure 6.2 depict the comparison of flow rates between model
predictions and experimental measurement (mean SD; average and standard deviation of the flow rates in six hearts) at the inlet of LCCA and RCA. It is found that the
theoretical predictions are within one standard deviation of the experimental data.
6.2 Pulsatile Flow in Passive Hearts 367
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250A
200
150
100
Mean Flow Rate (ml/min)
50
0
0.2 0.4 0.6 0.8
0
140
B
120
100
80
LCCA
Numerical results
Experimental results
1 1.2 1.4 1.6 1.8 2
Time (second)
RCA
60
40
Mean Flow Rate (ml/min)
20
0
Fig. 6.2 Comparison of flow rates between theoretical predictions and experimental measurements
(means SD; average and SD of the flow rates in 6 hearts) at the inlet of LCCA (a) and RCA (b).
Reproduced from Huo and Kassab (2006) with permission
The mathematical model (as described in Appendix 1) correctly simulates the
pulsatile flow in diastole in the absence of vessel tone in the complex anatomical tree
which has more than one million vessels. It should be noted that the experimental
preparation contains both the arterial and venous trees while the Womersley model
only considers the arterial tree. It appears that the arterial tree dominates the wave
Numerical results
Experimental results
0 0.5 1 1.5 2
Time (second)
368 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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form which may be due to the small diameter of the capillary network that strongly
retards the propagation of the flow wave. Furthermore, the resistance of the arterial
tree is significantly larger than that of the venous counterpart.
6.2.1.3 Effect of Various Parameters (e.g., Wave Frequency, Branching
Asymmetry) on Pulsatile Blood Flow
To study the effect of wave frequency, the pressure and flow waves at the inlet of an
artery are produced by the piston pump at a heart rate of 90, 120, and 180 beats/min.
The flow waveform is calculated by using the measured input pressure and the
Womersley model and compared with the measured flow waveform. Agreement is
found between Womersley model predictions and experimental measurements as
described in Huo and Kassab (2006).
The major advantage of Womersley’s method is that it can predict the significant
longitudinal distribution of coronary blood flow, which has physiological and
clinical significance. Therefore, the blood flow and pressure in vessel segments
along the trunk and the primary branches (branches that arise directly from the
trunk) at different harmonic frequencies are investigated for the LAD arterial tree
(Figs. 6.3 and 6.4). The inlet pressure P
(0, ω) is set at 100 mmHg with no phase
inlet
angle at the inlet of the LAD arterial tree. Figure 6.3a shows a schematic of the trunk
and the primary branches, several of which are identified alphabetically (i.e., A-C).
Figure 6.3b shows the relationship between the flow rate |Q(x,ω)| and the cumulative
length (∑L(n)) of vessel segments from the root (order 11) to the first capillary vessel
segment (order 0) along the trunk (root to AP branch to pre-capillary arterioles) with
harmonic wave frequency of 0.001 and 1.5 Hz. Figure 6.3c shows the corresponding
data for the phase angle (phase angle of zero implies the pressure and flow waves are
in phase).
The significant branching asymmetry of coronary arteries is important to understand the longitudinal heterogeneity in coronary flow. The flow wave along the trunk
and major branches of the coronary arterial tree is analyzed with the harmonic wave
frequencies of 0.001 Hz (~steady-state) and 1.5 Hz (90 beats/min). It is noted in
Figs. 6.3a, b that the blood flow through the trunk shows a gradual drop along the
path to the capil lary vessels while the flow from the trunk to the primary branches
has an abrupt drop because the trunk vessel segments have much larger diameter
than the primary branches. There is no significant difference of the amplitudes |Q(x,
ω)| between the harmonic wave frequencies of 0.001 and 1.5 Hz. The phase angle of
Q(x,ω) at the harmonic wave frequency of 1.5 Hz reaches 36
, however, at the inlet
of the trunk and gradually decreases along the trunk. The phase angle of Q( x, ω)of
the primary branches (i.e., A, B, and C) shows a nearly linear decrease towards the
capillaries.Figure 6.4a shows the relationship between the flow rate |Q(x, ω)| and the
cumulative length of vessel segments through the primary branches (A, B, and C) to
the first capillary vessel segment (order 0) with harmonic wave frequency of 0.001
and 1.5 Hz. Figure 6.4b shows the phase angle data for the respective coronary
subtrees at 1.5 Hz.
6.2 Pulsatile Flow in Passive Hearts 369
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B
1.E+00
1.E-01
(ml/s)
1.E-02
1.E-03
A
ROOT
A (1439)
B (966)
C (721)
AP
Q(x,w)|
1.E-04
1.E-05
1.E-06
Segment Flow |
1.E-07
C
Phase Angle of Q(x,w) (degrees)
|Q0| (0.001 Hz)
|Q1| (1.5 Hz)
0
1234
Cumulative Length from Root (cm)
40
35
30
25
20
15
10
5
0
-5
0123456789
Cumulative Length from Root (cm)
567
Phrase of Q0 (0.001 Hz)
Phrase of Q1 (1.5 Hz)
8 9 10 11
10 11
Fig. 6.3 (a) Schematic of the trunk of the LAD artery and several of the primary branches (A-C),
(b) Relationship between the flow rate |Q(x, ω)| and the cumulative length of vessel segments from
the root (order 11) to the first capillary vessel segment (order 0) along the trunk line (root-APcapillary) with the harmonic wave frequency of 0.001 Hz (~steady-state) and 1.5 Hz (90 beats/min),
(c) Relationship between the phase angle of Q(x, ω) and the cumulative length of vessel segments
from the root (order 11) to the first capillary vessel segment (order 0) along the trunk line (root-APcapillary) with the harmonic wave frequency of 0.001and 1.5 Hz. Reproduced from Huo and
Kassab (2006) with permission
In Fig. 6.5, the pressure and flow waves are calculated along the trunk of LAD
artery from the inlet to the first capillary with simulated blood viscosity (Appendix
3, Chap. 5). Figure 6.5a shows the flow waves sequentially at 2.0 cm intervals along
the trunk starting from the inlet of LAD artery. Figure 6.5b shows the pressure waves
sequentially at different diameters along the trunk of LAD artery.
The patterns of waves in the LAD, LCx, and RCA trees are similar from inlet of
coronary artery to the first capillary. As seen in Fig. 6.5a, decrease in amplitude of
370 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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Fig. 6.4 (a) Relationship between the flow rate |Q(x, ω)| and the cumulative length of vessel
segments from the primary branches (A, B, and C) to the first capillary vessel segment (order 0)
with the harmonic wave frequency of 0.001 and 1.5 Hz, (b) Relationship between the phase angle of
Q(x, ω) and the cumulative length of vessel segments from the primary branches (A, B, and C) to
the first capillary vessel segment (order 0) with the harmonic wave frequency of 0.001 and 1.5 Hz.
Reproduced from Huo and Kassab (2006) with permission
6.2 Pulsatile Flow in Passive Hearts 371
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Fig. 6.5 (a) Flow waves sequentially at 2.0 cm intervals along the trunk starting from the inlet of
LAD artery, (b) Pressure waves sequentially at different diameters along the trunk of LAD artery.
Reproduced from Huo and Kassab (2006) with permission
372 6 Network Analysis of Coronary Circulation: II. Pulsatile Flow
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the flow wave is apparent at 2.0 cm intervals along the trunk because the primary
branches shunt the flow away from the trunk. A small phase angle shift of the flow
waves is gradually developed along the trunk from the inlet to the first capillary.
Figure 6.5b depicts the relation between pressure waves and diameter along the trunk
of LAD artery. It is noted that the attenuation of the amplitude of the pressure wave is
rather small for larger vessels and becomes significant for vessel <100 μm(<order
6). The phase angle shift of the pressure wave shows a similar tenden cy to that of the
flow wave.
6.2.2 Hybrid One-Dimensional/Womersley Model
The frequency-domain Womersley-type approach described above represents a
linearization of the fluid dynamic approach as it ignores higher order nonlinear
effects such as convective losses and the diffusion term in the Navier–Stokes
equation. Although this approximation may be reasonable for smaller vessels, the
nonlinear affects must be considered for larger vessels where the Reynolds and
Womersley numbers are higher (Kassab et al., 1993). Hence, to accurately predict
the distribution of transient pressure and flow at any position in the main trunk and
primary branches, the vessels should be discretized and the time-domain fluid
dynamics method must be used.
There are two major time-domain approaches to simulate the pulsatile blood flow
and pressure waves in the vascular system: the Hugh’ s “1D” model (Hughes &
Lubliner, 1973) and Parker’s “wave-intensity analysis” model (Parker & Jones,
1990; Sherwin, Franke, Peiró, & Parker, 2003; Sun, Anderson, Parker, & Tyberg,
2000; Wang & Parker, 2004). Since the large computational cost becomes prohib-
itive when the entire coronary arterial tree is considered, the models require appropriate outflow boundary conditions to simulate a truncated tree. Different outflow
boundary conditions have been developed to satisfy the time-domain equations in
lumped distal models, such as the pure resistive load boundary conditions (Stettler
et al., 1981), the Windkessel model (Judd et al., 1991; Lee et al., 1984), the
nonreflecting outlet boundary conditions (Formaggia et al., 2001), or the impedance
boundary conditions (Olufsen, 1999, 2000). Despite the utility of lumped models,
however, they provide limited insight into regional issues (Hoffman & Chilian,
2000). Hence, it is necessary to develop a hybrid 1D/Womersley model which not
only incorporates the nonlinear effects in the larger epicardial vessels but also
integrates the smaller vessels down to the capillaries.
Huo and Kassab (2007) used Hughes’s 1D model (Hughes & Lubliner, 1973) for
the study of pulsatile blood flow in the main trunk and primary branches along with
the impedance boundary conditions (Appendix 2). The impedance at each junction
between the large and small vessels is calculated through Womersley-type numerical
computations based on the morphometric data of the full coronary arterial tree. The
implicit finite difference solver is developed to solve the 1D hybrid model as
outlined in Appendix 2. The predictions of the mathematical model are compared
6.2 Pulsatile Flow in Passive Hearts 373
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below with the Womersley-type numerical analysis and the experimental measurements on vasodilated, potassium-arrested porcine hearts as described above.
The computational model described in Appendix 2 is used to predict the transient
pressure and flow waves in the main trunk and primary branches. Extensive numerical simulations are carried out for various parameters and computational domains
(Huo & Kassab, 2007). A selection of computed results is given below, which are
compared with the above experimental and Womersley results. The computational
domains in 1D hybrid model are shown in Appendix 2 Figs. 6.14a, b for the RCA,
and LAD and LCx arteries, respectively. The 1D hybrid model is restricted to the
trunk of the major arteries (proximal artery to vessel approximately 2 mm in
diameter).
6.2.2.1 Pressure Boundary Conditions at the Inlet of LAD and LCx
Arteries
It is desirable to verify the time-domain computer code for the values of the pressure
and flow with available experimental or numerical solutions. The calculated flow
waves using the hybrid 1D/Womersley model with pressure and impedance boundary conditions at the inlet and outlet of RCA, LAD, and LCx arteries, respectively,
has been compared with the experimental measurements and the full Womersley
model as described above (Huo & Kassab, 2007). Figures 6.6a, b show the comparison of flow wave at the inlet of RCA, LAD, and LCx arteries, as shown in
Figs. 6.14a, b or Appendix 2, respectively. The hybrid model results are slightly
better than the Womersley model in comparison with the experimental results. Both
models are within 1 SD of the experimental data, as shown in Fig. 6.6 which
provides confidence in the models.
The 1D hybrid model is also simulated without the second-order viscous term in
the Navier–Stokes equation (Chap. 1). The resulting flow waves (data not shown) are
indistinguishable from those in Fig. 6.6 that include the viscous term. The computed
time average ratios of pressure-to-velocity head and convective-to-pr essure term at
the inlet of vessel segments are listed in Table 6.1 from Appendix 2 for various
branch angles, respectively. It is found that the pressure term is much larger than the
convective term (Chap. 1). Furthermore, the primary branches have smaller ratios of
convective-to-pressure term than the trunk vessels.The agreement between the
hybrid and Womersley models implies that the nonlinear convective term plays a
minor role even in the larger epicardial vessels in an arrested heart. Table 6.1
(Appendix 2) shows the pressure term is at least 100 times larger than the convective
term in the Navier–Stokes equation and the pressure head has the dominant effect
over the velocity head in the Bernoulli equation. It is also found that the viscous term
is small because the solution of the 1D model with and without the second-order
viscous term yielded very similar results.
The hybrid model can predict the transient blood flow accurately in the main
trunk and primary branches. The flow waves are calculated along the main trunk of
LAD artery. Figure 6.7a shows the flow waves sequentially at different spatial
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Fig. 6.6 Pulsatile flow in hybrid one-dimensional (1D)/Womersley model and full Womersley-
type model and experimental results at the inlet of RCA (a) and LAD and LCx (b). Both models are
implemented in the entire coronary arterial tree. Reproduced from Huo and Kassab (2007) with
permission
positions along the main trunk starting from the inlet of LAD artery. The decrease in
the amplitude is apparent, but the flow wave form remains relatively unchanged
except for a very small phase angle shift in the potassium-arrested heart. Figure 6.7b
shows the flow waves at the inlet of various primary branches. When the data is
normalized relative to the mean of each waveform, the patterns look remarkably
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