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Computational Fluid Dynamics Blood Flow Simulation by Patient-
Specific Modeling of Aortic and Coronary Outlet
Ji-Song Jong., Ji-Song Thak., Sung-Ri Kim*., Hyon-Chol Choe., Kuk-Chol Song
Faculty of Mechanical Science and Technology, Kim Chaek University of Technology, Kyogu-dong
*Corresponding Author
ABSTRACT
We develop a cardiovascular flow simulation system that performs a flow simulation analysis in patient-specific
cardiovascular system and a bypass grafts suggested by a surgeon. In this simulation system, the vessels that are
not reflected in the three-dimensional (3D) model of the vessel are modeled by an electrical circuit analogous
model, and the simulation is carried out by combining the computational fluid dynamics (CFD) analysis of the
vessel 3D model and analysis of the electrical analogous circuit. In order to evaluate the accuracy of the results
by this simulation system we analyzed blood flow in cardiovascular system by ANSYS Fluent 19.2. In this paper,
we study a method of setting boundary conditions for analyzing blood flow by ANSYS Fluent 19.2. We extract
the patient-specific blood pressure (BP) waveforms at the aortic and coronary outlets from the electrical circuit
analogous models of blood vessels such as Windkessel model and lumped parameter model. The parameter
values of the analogous models are optimized to approximate the measured patient-specific systolic and diastolic
pressure. By setting these blood pressure waveforms as outlet boundary conditions, we perform a simulation
analysis of three-dimensional vascular model by ANSYS Fluent 19.2 and compare the results such as blood flow
rates and blood pressures with clinical measurements. The comparison showed that the relative error in systolic
pressure was -3.94~4.42%, diastolic pressure was -2.71~4.43%, and cardiac output was -4.76~4.8%, which
demonstrated the accuracy of our boundary modeling and simulation analysis results.
Keyword: blood flow, cardiovascular, boundary modeling, CFD analysis, analogous model, electrical analogous
circuit
INTRODUCTION
Cardiovascular disease is a leading cause of death worldwide and a lot of researches for improvement of medical
treatment methodology for it are progressed so actively that many successes are achieved with the development
of medical science and technology. Especially a coronary artery bypass graft surgery forms bypass grafts from
aorta to post-stenosed region of coronary artery in order to increase a flow rate of blood to the region. In the
process of applying to clinical treatment, this surgery is admitted as the most effective surgical treatment
methodology for patients with cardiovascular disease. But this treatment methodology is costly and the disease
often recurs. Hence, it is very important to estimate the effect of bypass graft surgery and correctly predict
circulation state of blood in cardiovascular system after surgery, for which computational fluid dynamics blood
flow simulation method is widely used.
The important aspect of cardiovascular flow simulation is the modeling of blood flow and the modeling of
boundary conditions. Accurate assessment and modeling of the cardiovascular circulation has a great impact on
the accuracy of blood flow simulations, and many studies have been carried out by researchers. Blood flow in
cardiovascular system can be considered as laminar flow because of the relatively low Reynolds number flow,
compressible or incompressible flow, Newtonian or non-Newtonian fluid flow depending on characteristics such
as blood density and viscosity. Politis et al. (2007) assumed that blood flow was three-dimensional, steady,
laminar flow and Newtonian flow, and the vessel wall was nonelastic and impermeable, and analyzed
haemodynamic parameters such as velocity, pressure loss, wall shear stress and flow rates at specific locations
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such as anastomosis. Ko et al. (2007) assumed blood flow to be three-dimensional, steady, laminar,
incompressible flow and analyzed it using the continuity equation and Navier-Stokes equation. Blood is assumed
to be Newtonian fluid and its density as 1 000kg/m3 and dynamics viscosity as 0.003 5kg/m•s, respectively.
Bertolotti et al. (1999), Zhu et al. (2015), and Kandail et al. (2018) considered that vessel walls were rigid and
non-porous and blood was viscous, incompressible, Newtonian and homogeneous. Basing on these assumptions,
the conservation equations for three-dimensional, steady, laminar flow were solved by using finite volume based
CFD solver.
However, some studies considered the blood flow as turbulent, considering the pulsatility of the flow and the
swirl at the bifurcation point. Jayendiran et al. (2018) assumed blood to be a Newtonian, incompressible medium
and blood flow to be pulsatile, turbulent and fully developed. Owida et al. (2012) assumed blood to be a
homogeneous, non-deformable, chemically inert Newtonian fluid, and its density to be independent of pressure
changes but only dependent on temperature changes.
How to model the cardiovascular inlet and outlet boundaries has a significant impact on the results of three-
dimensional simulation of blood flow, and extensive research has been conducted. Modeling of boundary
conditions is essentially to model parts that are not reflected in the 3D model for simulation analysis, i.e., to
reflect the variation of the properties in the parts omitted from the 3D model as boundary conditions. Kamangar
et al. (2007, 2017) applied the velocity inlet boundary and pressure outlet boundary as the simplest modeling of
boundary condition. Ko et al. (2007) set the inlet boundary condition as a uniform velocity distribution and the
outlet as a fully developed condition and applied a non-slip condition for the vessel wall. Owida et al. (2012),
Ballarin et al. (2017), and Zhu et al. (2015) specified blood flow measured by duplex ultrasonography as inlet
boundary condition and a zero static pressure as outlet boundary condition. Bertolotti et al. (1999) set three-
dimensional Poiseuille velocity profiles as the inlet boundary condition, the no-slip condition for the vessel wall
and traction-free surfaces as outlet boundary. McGah et al. (2011) applied a time-dependent Womersley velocity
profile to mimic the patient-specific flow rates. Jayendiran et al. (2018) specified pressure distribution as inlet
boundary condition and pressure value as outlet boundary condition. Kandail et al. (2018) numerically simulated
the Navier-Stokes equation by setting transient blood flow waveforms to the aortic inlet and coronary outlets
and a transient pressure waveform to the aortic outlet. Shaik et al. (2008) and Wang et al. (2014) used the blood
flow waveform with heartbeat as an inlet boundary condition to simulate the propagation of wave by heartbeat.
Fayssal et al. (2018), Segalova et al. (2012), and Fan et al. (2016) set the blood pressure or blood flow waveform
as inlet boundary condition and the flow resistance as outlet boundary condition and analyzed the blood pressure
and flow rate distribution by using the three-dimensional finite volume method. Updegrove et al. (2017),
Schiavazzi et al. (2016), Kim et al. (2009, 2010), Lan et al. (2018), Vignon-Clementel et al (2010), Seo et al.
(2020), Zhang et al. (2014), and Mirramezani et al. (2019) modeled heart and vasculars omitted in the three-
dimensional model of a cardiovascular system by parameters such as resistance (R), inertance (L) and
compliance (C) to form an electrical circuit analogous model and carried out the simulation analysis about the
coupling of electrical circuit analogous modeling equation and Navier-Stokes equation of the three-dimensional
model of vessels. Sankaran et al. (2012) and Zhao et al. (2016) modeled the downstream vascular by using a
lumped parameter model comprised of resistance, compliance, inertance and diode obtained from the patient-
specific clinical data and analyzed flow rate and pressure of the three-dimensional model through the finite
element analysis and reflected these results in lumped parameter model again. In the same way they carried out
the simulation analysis repeatedly until the convergence condition was satisfied. Zambrano et al. (2017)
developed computational models considering the stiffness of pulmonary artery and describing Windkessel type
parameter model for each outlet branch. Aboelkassem et al. (2019) modeled the arterial inlet by Windkessel-
Womersley (WK-W) coupled model and the coronary outlet by 6-element Windkessel (WK6) model to predict
the pulsation of blood flow and described the significance in development of basic theory of cardiovascular
mechanics and clinical practice.
It is also important for simulation of blood flow in cardiovascular system to guarantee accuracy and ensure
rapidity of the simulation analysis. Up to now CFD analysis tools such as Fluent, ANSYS CFX, etc. or developed
private analysis tools were used in simulation analysis. Hazer et al. (2006) carried out the transient simulation
analysis of partial cardiovascular system with an inlet velocity waveform by using Fluent 6.2. Horner et al.
(2016) carried out the steady simulation analysis of patient-specific vascular model with mean flow rate and
mean pressure as boundary conditions. Updegrove et al. (2017) and Lan et al. (2018) developed a fully open-
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source SimVascular software package, which was a private tool for the cardiovascular blood flow simulation
analysis. In this package they carry out the transient analysis by using parameter models as boundary conditions.
Now we are developing a new simulation analysis system to simulate cardiovascular blood flow by building a
patient-specific three-dimensional vascular model and modeling boundary conditions by parameters. And we
regard CFD simulation analysis results and clinical measurements as comparative data to verify the accuracy of
simulation analysis results from this simulation system. In order to compare simulation results from our
developing simulation system with CFD simulation analysis results, boundary conditions must be the same. This
simulation system models boundary conditions by parameters such as resistance (R), inertance (L) and
compliance (C). To use these boundary models in simulation analysis by ANSYS Fluent 19.2, we extracted
blood pressure waveforms from these analogue circuit models.
In this paper, we suggest the method to extract the patient-specific blood pressure waveforms from analogue
circuit models of cardiovascular downstream by parameters and analyze blood flow by ANSYS Fluent 19.2 with
these blood pressure waveforms.
MATERIALS AND METHODS
Three-dimensional geometric model
A three-dimensional geometry of the cardiovascular system is shown in Fig. 1.
This anatomy is constructed from computed tomography (CT) images by using our simulation system, is similar
to patient-specific blood vessels and can be also used in analysis by ANSYS Fluent. The model consists of a
segment of aortic arch and right and left coronary arteries. And blood flows into the aorto-coronary model
through the aortic inlet and flows out from the aorto-coronary model through the aortic outlet and 5 right
coronary artery (RCA) outlets and 6 left coronary artery (LCA) outlets. As can be seen in the figure, there are
symmetric stenosis of 99% area-reduction and 5 mm length in middle and distal of the third segment of right
coronary artery, respectively, and non-symmetric stenosis of 50% area-reduction and 4mm length in proximal
of the fifth segment of main left coronary artery and non-symmetric stenosis of 70% area-reduction and 18mm
length in proximal of the eleventh segment of left circumflex artery. We created a tetrahedral mesh of this model,
in particular, implemented adapted mesh of stenoses with narrow diameters to ensure the accuracy of the
simulation analysis.
Boundary conditions
Boundary conditions must be applied to the aortic inlet, outlet and coronary outlet in order to carry out CFD
anaysis of this model. Blood flow waveform obtained by cardiac ultrasonography is used as flow rate boundary
condition at the aortic inlet and blood pressure waveforms obtained from electrical circuit analogous models of
downstream vasculars are utilized as pressure boundary conditions at the aortic outlet and coronary outlets.
Patient-specific aortic outlet boundary modeling
The modeling of aortic outlet should reflect the transiency of blood flow and the elasicity of blood vessels. We
used the electrical circuit analogous model such as Fig. 2(a) to reflect these blood flow properties of aortic outlet
(Vignon-Clementel et al., 2010; Zhang et al., 2014; Mirramezani et al., 2019).
As can be seen, aortic outlet is modeled by serially connecting the proximal resistance Rp to the parallel
connection of compliance C and the distal resistance Rd of blood vessel. The blood pressure and flow rate
balance equations at electrical circuit analogous model prescribed at the aortic outlet shown in Fig. 2(a) are as
follows:
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where Q and P are proximal flow rate and pressure, respectly, QC is flow rate accumulated by compliance of
blood vessel and PC is pressure drop due to the compliance of blood vessel, Qd is distal flow rate.
Assuming that the simulation is started at t=0, proximal pressure P(t) at time t is expressed from the above
systems of ordinary differential equations, Eq. 1 as follows:
Eq. 2 means that blood pressure at time t is related to the flow history between time 0 and the current time t.
2.2.2. Patient-specific coronary outlet boundary modeling
The coronary outlet is modeled by coupling of coronary arterial resistance Ra, coronary arterial compliance Ca,
microcirculatory resistance Ram, myocardial compliance Cim, coronary venous resistance Rv and ventricular
pressure Pim as shown in Fig. 2(b) (Mirramezani et al., 2019).
In this case coronary venous compliance and coronary venous microcirculatory resistance omitted from the
analogous model because they have little effect on coronary pressures and flow waveforms. The blood pressure
and flow rate balance equations at subcircuit of electrical circuit analogous model shown in Fig. 2(b) are as
follows:
PCa(t) is obtained from the systems of ordinary differential equations, Eq. 3 as follows.
where the ventricular pressure Pim(t) is expressed by end-diastolic volume of ventricle Vedv and elastance
function of ventricle E(t) as follows (Kim et al., 2009):
where V0 is calculated by end-systolic volume of ventricle Vesv, systolic pressure Psys and the maximum
elastance of ventricle Emax as follows (Kim et al., 2009):
V0=Vesv-0.9Psys/Emax (6)
And the following equations are satisfied from Fig. 2(b).
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From the second and third expressions of Eq. 7, we have:
Sustituting Eq. 4 into Eq. 8, we obtain:
Differentiating both sides of Eq. 9 with respect to t, it is expressed as follows:
Substituting Eq. 9 into Eq. 10, we obtain:
Eq. 11 is a nonhomogeneous second order ordinary differential equation, which can be expressed as follows:
where
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Obtaining Q1 from Eq. 12 and considering Eq. 4, the pressure P0(t) at the coronary outlet is obtained by the first
equation of Eq. 7 as follows:
P0(t)=RaQ0(t)+PCa(t) (13)
where Q0(t) is flow rate at the coronary outlet.
Optimization of parameter values
In order to extract the aortic and coronary outlet pressure waveforms from Eq. 2 and Eq. 13, we must know the
compliance C, distal resistance Rd and proximal resistance Rp of aortic blood vessel and the coronary arterial
resistance Ra, coronary arterial compliance Ca, coronary arterial microcirculatory resistance Ram, myocardial
compliance Cim and coronary venous resistance Rv.
Considering the properties of blood vessels, we assume that these parameters are in the following range.
104Pa•s/m3<Rd, Rp, Ra, Ram, Rv<1011Pa•s/m3
10-10m3/Pa<C, Ca, Cim<10-6m3/Pa
And max-systolic pressure Pmax and the time of the peak systolic tmax, min-diastolic pressure Pmin and the
time of the minimum diastolic tmin can be obtained by clinical examination.
Hence, we can set up the following optimization problems:
Thus, the parameters are optimized such that the maximum systolic pressure at the time of the peak systolic and
the minimum diastolic pressure at the time of the minimum diastolic obtained by clinical measurements are equal
to the values obtained by Eq. 2 and Eq. 13.
The optimization problem such as Eq. 14 and Eq. 15 can be solved using particle swarm optimization (PSO)
algorithm. The PSO algorithm is a stochastic global search approach that selects the optimal solution candidates
of the search space as “particle swarms” and that each particle in the swarm continues to update its position
probabilistically using its optimal information and the optimal information of the swarm at each iteration step,
and then approaches the optimal solution. The PSO algorithm is very simple programmable and very efficient
because it has a simple arithmetic operation.
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The general flowchart of the PSO algorithm is shown in Fig. 3.
The PSO algorithm is compared and evaluated by applying other algorithms such as genetic algorithms to neural
network training problems and standard benchmark functions, and it is verified that the computational effort is
small and convergence is good. Using such particle swarm optimization (PSO) algorithm, the optimization
problems are solved to determine the parameters for the aortic and coronary outlets and thus the blood pressure
waveforms at the aortic and coronary outlets can be obtained by Eq. 2 and Eq. 13.
The parameter settings for the detailed example are given below.
Simulation details
By constructing a cardiovascular geometry of a heart disease patient and by setting the aortic inlet blood flow
waveform and the coronary outlet blood pressure waveforms by the indices obtained from clinical measurement
and the above parameters as boundary conditions, the blood flow simulation analysis is performed by ANSYS
Fluent 19.2.
The characteristic indices of blood flow in cardiovascular system are obtained from clinical examination.
The aortic inlet blood flow waveform of patient with heart rate of 75beats/min, cardiac output of 4.32L/min,
diastolic pressure of 80mmHg and systolic pressure of 110mmHg, respectively, is like as Fig. 4.
And the aortic and coronary outlet parameters obtained by clinical measurements and the optimization problems
Eq. 14 and Eq.15 are listed in Table 1 and Table 2.
According to the above parameters, the blood pressure waveforms at the aortic and coronary outlets obtained
from Eq. 2 and Eq. 13 are shown in Fig. 5.
Fig. 5 shows that the maximum systolic pressure and the minimum diastolic pressure are 110mmHg and
80mmHg at the aortic outlet, respectively, which is consistent with the clinical measurements. And the maximum
systolic pressure is 110mmHg and the minimum diastolic pressure is about 10mmHg in the coronary outlet,
which shows the properties of coronary outlets well.
And blood was approximated as an incompressible Newtonian fluid with density of 1 058 kg/m3 and viscosity
of 0.004 1 Pa•s. These values were measured by blood tests. The vascular walls were considered as rigid, inelastic
and impermeable.
Blood flow in a cardiovascular system may be regarded as laminar flow because Reynolds number Re is
relatively small, but it is more accurate to consider turbulent flow because the vortex is involved in the separation
of the cardiovascular system. Hence, the flow model for the blood flow analysis was regarded as the Realizable
k-ε turbulent model.
We also used the Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) scheme as the
pressure-velocity coupling scheme, the Standard scheme for pressure and the Second Order Upwind schemes
for momentum, turbulent kinetic energy and turbulent dissipation rate as spatial discretization schemes, and
Crank-Nicholson approximation scheme as the time-stepping scheme.
For the initial condition the whole model is initializes using the inlet flow rate at time t=0s.
RESULTS AND DISCUSSION
Setting the above aortic inlet blood flow waveform, aortic outlet blood pressure waveform and coronary outlet
blood pressure waveforms as inlet and outlet boundary conditions, we carried out analysis of vascular model as
like Fig. 1 by analysis program, ANSYS Fluent 19.2 during two cardiac cycles.
To analyze the effect of mesh number on the simulation results, the analysis was performed by varying mesh
number to calculate the mean blood pressure at the aortic inlet and the results are shown in Fig. 6.
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As shown in the figure, with the increase of mesh number, the variation of mean aortic pressure gradually
decreased, and it was very small, less than 1% since the mesh number was about 500 000. From this mesh
independence analysis, the mesh model was composed of 496 617 meshes and 65 225 nodes, and the time step
was set to 10-4 s, and transient analysis was performed to obtain blood pressure distribution and blood flow
waveforms.
Blood pressure distributions
To verify the accuracy of the simulation, the blood pressure distribution in the vessel wall is analyzed and
compared with the results obtained by clinical measurements.
Fig. 7 shows the blood pressure contours of cardiovascular system at the time of peak systole and minimum
diastole.
This figure shows the blood pressure drop from the aorta to the coronary artery branches visually, especially in
the constrictive branches, a significant blood pressure drop was observed.
The results of the analysis shows that the aortic blood pressure at the time of peak systole and at the time of
minimum diastole was 112.77mmHg and 79.12mmHg, respectively, which are in close agreement with the
systolic pressure of 110mmHg and diastolic pressure of 80mmHg obtained by clinical measurements. The
relative error is 2.55% at the peak systolic pressure and -1.12% at the minimum diastolic pressure, which shows
the accuracy of the simulation analysis.
Blood flow waveforms
Blood flow at the vascular outlet is very important to meet the oxygen demand of the cardiac circulation, and
therefore, insufficient blood flow is a major cause of the development of blood deficiency. Thus, blood flow
rates at each vascular outlet were calculated.
Fig. 8 shows the aortic and coronary outlet blood flow waveforms.
From the figure, the blood flow rate at the coronary arteries is large in the diastole and small in the systole
because the intramyocardial pressure caused by the ventricular pressure is high in the systole.
Table 3 shows the mean flow rate to every branch of coronaty artery.
From the table, the flow rates to RCA_1 and RCA_5 are much lower than to the others because the vascular
diameter at their proximal part became narrow by stenosis. And the flow rate to the right coronary artery is 2.606
mL/s, the flow rate to the left coronary artery is 2.152 mL/s, which yields a total coronary flow of 4.758 mL/s.
A typical 3/7 flow split between RCA and LCA is enforced (Sankaran et al., 2012) but the above result shows
that the flow rate to the left coronary artery is rather less than that to the right coronary artery, because there is a
stenosis with 50% area-reduction in proximal of the main left coronary artery. If there is no stenosis in proximal
of the main left coronary artery, the flow rate will be markedly incresed to the left coronary artery and the typical
flow split between RCA and LCA will be satisfied.
And from the simulation results, the flow rate to the aortic outlet is 50.1 mL/s, and consequently cardiac output
computed from simulation analysis is 4.11L/min, which nearly coincides with 4.32L/min as the cardiac output
obteined by clinical examination
Finally, it shows that the boundary modeling method and the simulartion analysis method are well-suited to the
patient-specific characteristics.
Setting the boundary conditions by above inlet and outlet boundary modeling method for cardiovascular models
of 30 patients with cardiovascular disease, we analyzed blood flow in cardiovascular system by ANSYS Fluent
19.2 and compared with clinical measurements. The comparison results are shown in Table 4.
Comparing the simulation analysis results and the clinical measurements, Table 4 shows that the relative error
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of systolic pressure, diastolic pressure and cardiac output are -3.94~4.42%, -2.71~4.43% and -4.76~4.8%
respectively, which verifys the accuracy of boundary modeling and simulation analysis method.
CONCLUSION
In this paper, we suggested a method to extract patient-specific aortic and coronary outlet blood pressure
waveforms from the electrical circuit analogous models by solving the analogous circuit equations, and set those
waveforms as boundary conditions to analyze blood flow in the cardiovascular system by using ANSYS Fluent
19.2. Through the study of the simulation analysis results of several patient-specific cardiovascular models we
confirmed the accuracy of the electrical circuit analogous models for the aortic and coronary outlets, and the
validity of the optimization for choosing parameters and the computational fluid dynamics blood fluid flow
simulation method. Our developing simulation system includes the optimizing process for choosing optimal
parameters, the discretization process of the basic equations, the modeling process of blood flow and the solving
process discretization schemes for simulating and analyzing blood flow, so it is not easy to verify the correctness
of every step in process of developing. Hence, we suggested the comparison with the analysis results by ANSYS
Fluent 19.2 as a solution to verify the accuracy of analysis results by our simulation system. In order to set
boundary conditions for simulation analysis by ANSYS Fluent 19.2, we extracted outlet blood pressure
waveforms from the electrical circuit analogous models modeled by our simulation system. And we carried out
blood flow simulation analysis of patient-specific three-dimensional vascular models made by our simulation
system by using ANSYS Fluent 19.2 and compared that result with clinical measurements. The comparison
between the results from ANSYS Fluent 19.2 and the clinical measurements shows that the relative errors of the
clinical indices such as blood pressure and cardiac output are all less than ±5%. These study results verify the
correctness of blood pressure waveform extraction method and blood flow simulation method by using ANSYS
Fluent 19.2 and demonstrate the validity of analysis results from ANSYS Fluent 19.2 as comparative proposition.
And those made accelerate our simulation system development by verifying the correctness of blood simulation
results from our simulation system. The boundary setting method and simulation method suggested in this paper
can be applied effectively to CFD analysis of blood flow in vascular.
ACKNOWLEDGMENT
The authors are grateful to Dr. Il-Guk Jo for his valuable suggestions during the completion of this paper.
Funding information
None
Authors' contribution
Ji-Song Jong conceived the study concept and design, participated in analysis and interpretation, and drafted and
revised the manuscript.
Ji-Song Thak studied the boundary condition modeling method.
Sung-Ri Kim performed literature study and data analysis.
Hyon-Chol Choe participated in CFD analysis of blood flow.
Kuk-Chol Song participated in clinical data acquisition and analysis.
All authors read and approved the final manuscript.
Conflict of interests
The authors declare that they have no conflict of interest.
Ethical approval
The conducted research is not related to either human or animal use.
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Data availability statement
The data that support the findings of this study are available from [third party name] but restrictions apply to the
availability of these data, which were used under license for the current study, and so are not publicly available.
Data are however available from the authors upon reasonable request and with permission of [third party name].
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Table 1. The aortic outlet parameters.
R
p
(10
7
Pa·s/m
3
)
C(10
-7
m
3
/Pa)
R
d
(10
9
Pa·s/m
3
)
1.626
9.35
3.87
Table 2. The coronary outlet parameters.
Branch
R
a
(10
7
Pa·s/m
3
)
C
a
(10
-9
m
3
/Pa)
R
am
(10
6
Pa·s/m
3
)
C
im
(10
-8
m
3
/Pa)
R
v
(10
5
Pa·s/m
3
)
RCA_1
6.48
7.36
2.73
6.58
1.4
RCA_2
8.23
5.89
4.36
4.75
2.67
RCA_3
8.23
5.89
4.36
4.75
2.67
RCA_4
8.23
5.89
4.36
4.75
2.67
RCA_5
6.48
7.36
2.73
6.58
1.4
LCA_1
4.23
2.98
4.82
1.76
1.9
LCA_2
2.65
4.32
2.87
5.67
3.7
LCA_3
5.46
5.64
3.52
3.98
4.78
LCA_4
5.46
5.64
3.52
3.98
4.78
LCA_5
3.23
6.07
3.45
9.73
6.07
LCA_6
3.23
6.07
3.45
9.73
6.07
Table 3. The coronary outlet blood flow rate.
Branch
Blood flow rate (mL/s)
Branch
Blood flow rate (mL/s)
RCA_1
0.058
LCA_1
0.411
RCA_2
0.994
LCA_2
0.556
RCA_3
0.572
LCA_3
0.319
RCA_4
0.911
LCA_4
0.594
RCA_5
0.071
LCA_5
0.154
LCA_6
0.118
RCA
2.606
LCA
2.152
Table 4. Comparison between simulation results and clinical measurements.
Model
Systolic pressure (mmHg)
Diastolic pressure (mmHg)
Cardiac output (mL)
Simulation
result
Clinical
data
Relative
error
(%)
Simulation
result
Clinical
data
Relative
error
(%)
Simulation
result
Clinical
data
Relative
error
(%)
1
114.3
110
3.91
72.3
70
3.29
4.24
4.42
-4.1
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2
103.4
100
3.4
61.5
60
2.5
4.37
4.22
3.36
3
115.7
120
-3.58
83.4
80
4.25
3.6
3.48
3.37
4
113.2
110
2.91
68.5
70
-2.14
3.68
3.82
-3.6
5
119.5
115
3.91
69.3
70
-1
3.86
3.94
-2.05
6
138.2
135
2.37
82.6
80
3.25
4.57
4.47
2.46
7
96.4
100
-3.6
61.9
60
3.17
3.72
3.62
2.54
8
106.3
110
-3.36
72.8
70
4
4.1
4.05
1.07
9
159.3
155
2.77
98.3
100
-1.7
4.94
4.75
3.86
10
131.2
135
-2.81
82.7
85
-2.71
4.5
4.39
2.5
11
134.5
130
3.46
88.4
90
-1.78
4.02
4.15
-3.26
12
144.2
140
3
82.1
80
2.63
4.32
4.2
2.92
13
127.8
130
-1.69
83.5
80
4.38
4.22
4.03
4.60
14
118.2
120
-1.5
78.4
80
-2
3.83
3.67
4.19
15
147.2
150
-1.87
88.7
90
-1.44
4.22
4.06
4.10
16
112.8
110
2.55
79.1
80
-1.12
4.11
4.32
-4.76
17
141.2
140
0.86
90.7
90
0.78
4.77
4.87
-2.13
18
149.3
150
-0.47
81.5
80
1.88
4.02
3.86
4.04
19
153.5
150
2.33
88.9
90
-1.22
4.76
4.61
3.39
20
138.4
140
-1.14
89.1
90
-1.0
4.63
4.42
4.8
21
132.1
130
1.62
81.6
80
2
4.32
4.2
2.93
22
164.8
160
3
93.2
90
3.56
3.84
3.67
4.63
23
125.3
120
4.42
79.2
80
-1
3.82
3.7
3.15
24
153.7
160
-3.94
102.4
100
2.4
3.87
3.74
3.42
25
115.3
120
-3.92
73.1
70
4.43
3.95
4.06
-2.60
26
136.7
140
-2.36
93.4
90
3.78
4.7
4.49
4.61
27
143.2
140
2.29
81.3
80
1.6
3.85
3.94
-2.20
28
117.9
120
-1.75
69.2
70
-1.14
4.05
3.89
4.04
29
124.6
120
3.83
79.4
80
-0.75
3.82
3.65
4.69
30
153.8
150
2.53
81.5
80
1.88
3.83
3.94
-2.80
Figure Legends
Figure 1: Patient-specific three-dimensional geometry of the cardiovascular system.
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Figure 2: Electrical circuit analogous model prescribed at the aortic and coronary outlet: (a) aortic outlet (b)
coronary outelt.
Figure 3: The general flowchart of the PSO algorithm.
Figure 4: The aortic inlet blood flow waveform.
Figure 5: The aortic and coronary outlet pressure waveforms: (a) aortic outlet (b) LCA_1; (c) LCA_2; (d) LCA_3
and LCA_4; (e) LCA_5 and LCA_6; (f) RCA_1 and RCA_5; (g) RCA_2, RCA_3 and RCA_4.
Figure 6: Grid independence analysis.
Figure 7: Blood pressure contour of cardiovascular system: (a) at the time point of peak systole; (b) at the time
point of minimum diastole.
Figure 8: The aortic and coronary outlet blood flow waveforms: (a) aortic outlet; (b) LCA_1; (c) LCA_2; (d)
LCA_3; (e) LCA_4; (f) LCA_5; (g) LCA_6; (h) RCA_1; (i) RCA_2; (j) RCA_3; (k) RCA_4; (l) RCA_5.
Figure 2
Figure 3
