Download Patient-Specific Modeling of Blood Flow and Pressure in Human

Annals of Biomedical Engineering, Vol. 38, No. 10, October 2010 ( 2010) pp. 3195–3209
DOI: 10.1007/s10439-010-0083-6
Patient-Specific Modeling of Blood Flow and Pressure in Human
Coronary Arteries
H. J. KIM,1 I. E. VIGNON-CLEMENTEL,2 J. S. COOGAN,3 C. A. FIGUEROA,3 K. E. JANSEN,1
and C. A. TAYLOR3,4
1
Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309, USA; 2INRIA, Paris-Rocquencourt
BP 105, 78153 Le Chesnay Cedex, France; 3Department of Bioengineering, Stanford University, Stanford, CA 94305, USA; and
4
Department of Surgery, Stanford University, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305, USA
(Received 29 September 2009; accepted 20 May 2010; published online 18 June 2010)
Associate Editor Peter E. McHugh oversaw the review of this article.
and the prediction of the outcomes of interventions.16,23 As the computing capacity and numerical
methods for simulation of blood flow advance, further
applications are anticipated.
However, computational simulations have been
rarely used to predict pulsatile flow and pressure fields
of three-dimensional coronary vascular beds, in part
because the flow rate and pressure in the coronary
vascular beds are highly related to the interactions
between the heart and the arterial system. Unlike flow
in other parts of the arterial system, coronary flow
decreases when the ventricles contract and increase the
intramyocardial pressure, which exerts an extravascular compressive force on the coronary vessels.17 Coronary flow increases when the ventricles relax, thereby,
decreasing the intramyocardial pressure and extravascular compressive force. Therefore, to model coronary
flow and pressure realistically, it is necessary to have a
model of the heart and a model of the arterial system
with consideration of the interactions between the two
models. Because of this complexity in modeling coronary flow and pressure, most three-dimensional computational studies have been conducted with only the
coronary arteries, ignoring the interactions between the
heart and the arterial system and prescribing, not predicting, coronary flow. Further, these studies have not
modeled realistic pressures and generally used tractionfree outlet boundary conditions whether the lateral
surfaces of the coronary arteries are considered as rigid
walls1,6 or compliant walls18,30 or the cardiac motions
due to the contraction and relaxation of the heart are
considered19,21 or not. Migliavacca and coworkers12,16
computed three-dimensional pulsatile coronary flow
and pressure in a single coronary artery, but this study
was performed with an idealized model and low mesh
Abstract—Coronary flow is different from the flow in other
parts of the arterial system because it is influenced by the
contraction and relaxation of the heart. To model coronary
flow realistically, the compressive force of the heart acting on
the coronary vessels needs to be included. In this study, we
developed a method that predicts coronary flow and pressure
of three-dimensional epicardial coronary arteries by considering models of the heart and arterial system and the
interactions between the two models. For each coronary
outlet, a lumped parameter coronary vascular bed model was
assigned to represent the impedance of the downstream
coronary vascular networks absent in the computational
domain. The intramyocardial pressure was represented with
either the left or right ventricular pressure depending on the
location of the coronary arteries. The left and right ventricular pressure were solved from the lumped parameter heart
models coupled to a closed loop system comprising a threedimensional model of the aorta, three-element Windkessel
models of the rest of the systemic circulation and the
pulmonary circulation, and lumped parameter models for the
left and right sides of the heart. The computed coronary flow
and pressure and the aortic flow and pressure waveforms
were realistic as compared to literature data.
Keywords—Blood flow, Coronary flow, Coronary pressure,
Coupled multidomain method.
INTRODUCTION
Computational simulations have been proven useful
in studying blood flow in the cardiovascular system24:
assessing hemodynamics of healthy and diseased blood
vessels,4,6 helping in the design and evaluation of vascular medical devices,1,14 planning of vascular surgeries,
Address correspondence to C. A. Taylor, Department of Surgery,
Stanford University, E350 Clark Center, 318 Campus Drive,
Stanford, CA 94305, USA. Electronic mail: [email protected]
3195
0090-6964/10/1000-3195/0
2010 Biomedical Engineering Society
3196
KIM et al.
resolution. The analytic models used as boundary
conditions were coupled explicitly, necessitating either
subiterations within the same time step or a small time
step size bounded by the stability of an explicit time
integration scheme. To predict physiologically realistic
flow rate and pressure in the coronary arterial trees of a
patient, computational simulations should be robust
and stable enough to handle complex flow characteristics and the coupling with different scales of computer
models should be efficient and versatile.10
In previous studies, Vignon-Clementel et al.27,28
developed boundary conditions for three-dimensional
models of blood flow that can represent physiological
conditions of each subject. For outlet boundaries of
the arterial system excluding the coronary vascular
beds, simple analytic models such as resistance,
impedance, and the three-element Windkessel model
were assigned to model the arterial system absent in
the computational domain. This method is fully
implicit and has proven to be versatile and robust. In a
related study, we implemented a lumped parameter
heart model as an inflow boundary condition of a
three-dimensional finite-element model of the aorta to
consider the interactions between the ventricle and the
arterial system.11 Again, a fully implicit method was
used to couple the heart model to the aortic model. We
showed physiologically realistic flow and pressure
fields for a subject-specific three-dimensional model of
the aorta with these boundary conditions.11 In addition, constraints on the shape of the velocity profile at
the aortic inlet and select outlet boundaries were
enforced using an augmented Lagrangian method to
resolve numerical instabilities that result from reverse
and complex flow structures at boundaries.10
In this paper, we describe a method to calculate the
flow and pressure of three-dimensional coronary vascular beds by considering the models of the left and
right sides of the heart, arterial system, and the interactions between them. For each coronary outlet, a
lumped parameter coronary vascular bed model was
assigned to represent the impedance of downstream
coronary vascular networks absent in the computational domain. We solved for coronary flow and
pressure as well as aortic flow and pressure in patientspecific models for rest and light and moderate exercise
conditions. Additionally, to demonstrate that these
methods can be utilized in clinical investigation, we
studied blood flow and pressure with different degrees
of stenosis. We created a stenosis in the left anterior
descending coronary artery by reducing the local
diameter of the geometric model by 40%, 50%, 60%,
and 75% and simulated a baseline (resting) condition
and a moderate exercise condition, where in the latter
case we assumed that the downstream coronary vascular beds were maximally dilated.
METHODS
Three-Dimensional Finite Element Model of Blood
Flow and Vessel Wall Dynamics
Blood flow in the large vessels of the cardiovascular
system can be approximated by the flow of a Newtonian fluid. Blood flow is modeled using the incompressible Navier–Stokes equations, and the motion of
the vessel wall using the elastodynamics equations. In
these equations, fluid–solid interface conditions, as
well as the initial and boundary conditions, are
required to solve for the blood flow and the vessel wall
dynamics of the fluid and solid domains, respectively.
In this study, we used a fixed fluid mesh assuming
small displacements of the vessel wall.5
For a fluid domain X with its boundary C and a
solid domain Xs with its boundary Csg ; the following
equations are solved for velocities ~
v; pressure p, and
wall displacement ~
u .5,27
Given ~
f: X ð0; TÞ ! R3 ; ~
fs : Xs ð0; TÞ ! R3 ;
3
s
s
~
g: Cg ð0; TÞ ! R ; ~
g : Cg ð0; TÞ ! R3 ; ~
v0 : X ! R 3 ;
s
s
3
3
~
u0 : X ! R ; and ~
u0;t : X ! R ; find ~
vð~
x; tÞ; pð~
x; tÞ; and
~
uð~
xs ; tÞ for 8~
x 2 X; 8~
xs 2 Xs ; and " t 2 (0, T), such
that the following conditions are satisfied:
q~
v;t þ q~
v r~
v ¼ rp þ divðs Þ þ ~
f
for ð~
x;tÞ 2 X ð0;TÞ
divð~
vÞ ¼ 0 for ð~
x;tÞ 2 X ð0;TÞ
s
s ~s
x s ;tÞ 2 Xs ð0; TÞ ð1Þ
q~
u;tt ¼ r r þ f for ð~
where s ¼ lðr~
v þ ðr~
vÞ T Þ
1
u þ ðr~
uÞT Þ
and rs ¼ C : ðr~
2
with the Dirichlet boundary conditions,
~
vð~
x; tÞ ¼ ~
gð~
x; tÞ for ð~
x; tÞ 2 Cg ð0; TÞ
~
uð~
x s ; tÞ ¼ ~
g s ð~
x s ; tÞ for ð~
x s ; tÞ 2 Csg ð0; TÞ
ð2Þ
the Neumann boundary conditions,
~ v; p; ~
~
n ¼ hð~
x; tÞ for ~
x 2 Ch
t~n ¼ ½p I þ s ~
ð3Þ
and the initial conditions,
~
xÞ for ~
x2X
vð~
x; 0Þ ¼ ~
v0 ð~
s
s
~
uð~
x ; 0Þ ¼ ~
u0 ð~
x Þ for ~
x s 2 Xs
~
x s ; 0Þ ¼ ~
u0;t ð~
x s Þ for ~
x s 2 Xs
u;t ð~
ð4Þ
where the boundary C of the fluid domain is divided
into a Dirichlet boundary portion Cg and a Neumann
boundary portion Ch. Similarly, the boundary Cs of the
solid domain is divided into a Dirichlet boundary
portion Csg and a Neumann boundary portion Csh. They
Patient-Specific Coronary Flow Modeling
a model developed by Mantero et al.15 (Fig. 1). The
coronary venous microcirculation compliance was
eliminated from the original model in order to simplify
the numerics. The coronary pressure and flow waveforms
retained realistic waveforms without the coronary
venous microcirculation compliance. The coronary vascular bed model consists of coronary arterial resistance
Ra, coronary arterial compliance Ca, coronary arterial
microcirculation resistance Ra-micro, myocardial compliance Cim, coronary venous microcirculation resistance
Rv-micro, coronary venous resistance Rv, and intramyocardial pressure Pim(t). The modeling of the intramyocardial pressure is explained in the next subsection.
For each coronary outlet surface Chcork we defined
the operators M and H as follows by replacing the
coronary outlet pressure P(t) with the ordinary differential equation obtained from the lumped parameter coronary vascular bed model:
Z
~
v;pÞ þ H
¼ R
vðtÞ ~
ndC
M ð~
satisfy ðCh [ Cg Þ ¼ C; ðCsh [ Csg Þ ¼ Cs ; Ch \ Cg ¼ /;
and Csh \ Csg ¼ /:
The density q and the dynamic viscosity l of the
fluid, and the density qs of the vessel walls are assumed
to be constant. The external body force on the fluid
domain is represented by ~
f: Similarly, ~
fs is the external
body force on the solid domain, C is a fourth-order
tensor of material constants, and rs is the vessel wall
stress tensor.
To assign coronary outlet boundary conditions, the
coupled multidomain method27 is utilized with a
lumped parameter coronary vascular bed model. Similarly to the treatment of other outflow boundary conditions, a lumped parameter model of coronary
vascular
~c
beds defines boundary operators M ¼ M ; M
m
Chcor
~c
and H ¼ H ; H
that represent the traction and
Chcor
flow at each coronary outlet surface Chcor . For the other
boundaries, the same method is applied to assign an
impedance using a three-element Windkessel model.28
The lateral surface of the fluid domain coincides with a
membrane model of the vessel wall in the coupled
momentum method for fluid–solid interaction.5 A stabilized semi-discrete finite element method25 is utilized
to solve for Eqs. (1)–(4) in this study.
m
þ
m C
hcor
Z
t
ek1 ðtsÞ Z1
Z
0
þ
k
!
~
vðsÞ ~
ndCds I
Chcor
Z
Chcor
k
t
ek2 ðtsÞ Z2
Z
0
!
k
Chcor
~
njChcor
vðsÞ ~
ndCds ~
n s ~
þ s jChcor Aek1 t Bek2 t I
k
Z t
Z t
k1 ðtsÞ
k2 ðtsÞ
e
Y1 Pim ðsÞds e
Y2 Pim ðsÞds I
0
0
h
i
~c
~c ð~
v;pÞ þ H
¼~
vjChcor
M
The Lumped Parameter Model of Downstream
Coronary Vascular Beds
The lumped parameter model of coronary vascular
beds coupled to each coronary outlet surface is based on
Chcor
B
C D EF
k
k
A: Inlet - coupled to lumped parameter heart model
G
RLA-V L LA-V
ELA
ELA
RLA-V
LLA-V
RLV-art
LLV-art
A
V
a
i
ELV ( t)
k
j
Rpd
Rpp
LRV-art
RRV-art
LRA-V
I ð5Þ
k
k
A
c b
e
h gd f
RRA-V
H
ERA
B-H: Outlets - coupled to
three-element Windkessel
model
Rp
Rd
C
RLV-art L LV-art
ELV (t)
V
A -Inlet
2.1
Elastance function
(mmHg/cm3)
m
3197
0
V
Time (s)
0
0.952
a-k: Coronary outlets - coupled to lumped
parameter coronary vascular model
Ra
Ra-micro
-
Rv-mi
v cro
Cim
Cp
ERV (t )
Ca
Pim
V
V
Rv
V
FIGURE 1. Problem specification of the inlet, upper branch vessels, the descending thoracic aorta, and coronary outlets for
simulations of blood flow in a normal thoracic aorta model with coronary outlets. Note that all the outlets of the three-dimensional
computational model feed back in the lumped model at (V).
3198
KIM et al.
where P(t) and Q(t) are the pressure and flow at the
coronary outlet surface and I is an identity tensor. The
coefficients k1, k2, A, B, R, Z1, Y1, Z2, and Y2 are
defined using the lumped parameter model of downstream coronary vascular beds as follows:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p1 þ p21 4p0 p2
k1 ¼
2p2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p1 p21 4p0 p2
k2 ¼
2p2
"
1
dQ
A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq2 k1 þ q1 ÞQð0Þ þ q2 ð0Þ
dt
p21 4p0 p2
#
dP
þ b1 Pim ð0Þ þ p2 k2 Pð0Þ ð0Þ
dt
"
1
dQ
B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq2 k2 þ q1 ÞQð0Þ þ q2 ð0Þ
dt
p21 4p0 p2
#
dP
þ b1 Pim ð0Þ þ p2 k1 Pð0Þ ð0Þ
dt
q2
R¼
p2
Z1 ¼
q2 k21 þ q1 k1 þ q0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p21 4p0 p2
b1 k1 þ b0
Y1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p21 4p0 p2
Z2 ¼
q2 k22 þ q1 k2 þ q0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p21 4p0 p2
b1 k2 þ b0
Y2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p21 4p0 p2
with the following coefficients:
p0 ¼ 1
p1 ¼ Ra-micro Ca þ ðRv þ Rv-micro ÞðCa þ Cim Þ
p2 ¼ Ca Cim Ra-micro ðRv þ Rv-micro Þ
q0 ¼ Ra þ Ra-micro þ Rv þ Rv-micro
q1 ¼ Ra Ca ðRa-micro þ Rv þ Rv-micro Þ
þ Cim ðRa þ Ra-micro ÞðRv þ Rv-micro Þ
q2 ¼ Ca Cim Ra Ra-micro ðRv þ Rv-micro Þ
b0 ¼ 0
b1 ¼ Cim ðRv þ Rv-micro Þ
Using these operators defined above, we couple the
flow and pressure at each coronary outlet surface
between the upstream three-dimensional finite element
model and the downstream lumped parameter model.
The velocity and pressure fields in the three-dimensional
domain and its boundaries were solved implicitly.
Computation of the Intramyocardial Pressure
Depending on the location of the coronary vascular
beds, the coronary beds experience different intramyocardial pressure Pim. Coronary flow in the left coronary vascular beds is decreased significantly in systole
because the left ventricle operates in a high pressure
range. However, the right coronary flow does not
decrease in systole because the right ventricle operates
in a lower pressure range than the left ventricle.17 Even
the maximum right ventricular pressure is one-fifth of
the average aortic pressure.17 To accommodate the
change in the intramyorcardial pressure depending on
the location of the coronary vascular beds, we implemented a closed loop system and computed the left and
right ventricular pressure. Previously, we had developed a boundary condition coupling a lumped
parameter model representing the left side of the heart
to the inlet of the thoracic aorta to compute the left
ventricular pressure.11 In this study, another lumped
parameter right heart model was coupled to the pulmonary circulation to compute the right ventricular
pressure. Depending on the location of the coronary
outlet boundaries, we used the left ventricular pressure
to represent the intramyocardial pressure of the left
coronary arteries and the right ventricular pressure to
represent the intramyocardial pressure of the right
coronary arteries. The closed loop system implemented
in these simulations is plotted in Fig. 1.
Choice of the Parameter Values for the Coronary Model
The flow distribution to each primary branch of the
coronary arteries was calculated on the basis of morphology data and data from the literature.31 We
assumed that the mean coronary flow is 4.0% of the
cardiac output.17 For each coronary outlet surface,
coronary venous resistance was calculated on the basis
of the mean flow and assigned venous pressure
according to literature data.17 We then computed
coronary arterial resistance and coronary arterial
microcirculation resistance on the basis of mean flow,
mean arterial pressure, and the coronary impedance
spectrum using literature data.3,26 The capacitance
values were adjusted to give physiologically realistic
coronary flow and pressure waveforms using literature
data.3
During simulated exercise, the mean flow to the
coronary vascular bed was increased to maintain the
mean flow at 4.0% of the cardiac output. However, we
set a threshold value below which the resistance cannot
decrease. The coronary parameter values for each
Patient-Specific Coronary Flow Modeling
coronary outlet surface were modified by decreasing
the total resistances and increasing the capacitances
and the ratio of the coronary arterial resistance to the
total coronary resistance based on literature data.3
The parameter values of the lumped parameter
heart model and three-element Windkessel models
were determined to match subject-specific cardiac
output and pulse pressure.11
Exercise Simulations
In this study, we simulated two different levels of
exercise conditions. First, for the simulation with
normal coronary arteries, we simulated rest and light
exercise conditions. The light exercise condition is
similar to a brisk walk. We doubled the cardiac output
and heart rate, and the coronary flow was doubled to
meet the increased metabolic demands accordingly.
However, the downstream coronary vascular networks
were not dilated maximally. Second, for the simulations with different degrees of stenosis, we simulated
moderate exercise to approximate maximal dilation of
downstream coronary vascular networks. This way, we
can study how different degrees of stenosis reduce the
coronary flow reserve affecting the coronary flow and
pressure waveforms. To simulate moderate exercise,
we tripled the cardiac output and heart rate and the
mean coronary flow was increased by 350%.
Simulation Details
Blood was approximated as incompressible Newtonian fluid with a density of 1.06 g/cm3 and a
dynamic viscosity of 0.04 dynes/cm2 s for all the
simulations. The blood vessel walls were modeled with
a linearly elastic material. Poisson’s ratio was 0.5 with
a wall density of 1.0 g/cm3 and a uniform wall thickness of 0.1 cm. The inlet and the outlet rings were fixed
in space and time.5 The mesh generated was anisotropic with local refinement on exterior surfaces and
coronary vascular beds with five semi-structured layers
within the boundary layer.20 The solutions were run
until the relative pressure fields at the inlet and outlets
did not change more than 1.0% compared to those in
the solutions at the same phase in the previous cardiac
cycle.
We constructed the geometrical model used in these
simulations from cardiac-gated computer tomography
data of a 36-year-old healthy male subject. The model
started from the root of the aorta, ended above the
diaphragm, and included major coronary arteries (left
anterior descending, left circumflex, and right coronary
arteries) and major branches of them and main upper
branch vessels (right subclavian, left subclavian, right
vertebral, left vertebral, right carotid, and left carotid
3199
arteries). For the inlet, we coupled the lumped
parameter heart model.11 For the coronary outlets, we
assigned lumped parameter coronary vascular models.15 For the upper branch vessels and the descending
thoracic aorta, we assigned three-element Windkessel
models (Fig. 1). The flow distribution used to evaluate
the resistance/impedance of each coronary, upper
branch, and descending thoracic aortic outlet was set
based on literature data.29 Boundary conditions were
adjusted to match both the flow distribution and the
measured brachial artery pulse pressure.13,22 Additionally, we adjusted the Young’s modulus of the
blood vessel walls to match the measured wall deformations of the aorta on the basis of the cardiac-gated
computer tomography data. In this study, we modeled
the wall deformability with the uniform Young’s
modulus of 6.26 9 106 dynes/cm2. We also found
tethering areas using this patient’s cardiac-gated computer tomography data and fixed those areas in space
and time in the computer model. For the coronary
vessels, we identified thin strips approximating the
intersection between the coronary arteries and epicardial surface of the heart and fixed the surface of the
coronary arteries along the strips.
To demonstrate that these methods can be utilized
in clinical investigation, we studied blood flow and
pressure with different degrees of stenosis. We created
a stenosis in the left anterior descending coronary
artery by reducing the local diameter of the geometric
model by 40%, 50%, 60%, and 75% and simulated a
rest condition and a moderate exercise condition,
where in the latter case we assumed that the downstream coronary vascular beds were maximally dilated.
RESULTS
Simulations of Coronary Flow and Pressure of a Normal
Thoracic Aortic Model with Epicardial Coronary
Arteries at Rest and During Light Exercise
In these simulations, we studied coronary flow and
pressure of normal coronary arteries for rest and light
exercise conditions. Solutions were obtained using a
1,768,953 element (339,664 node) finite element mesh
with a time step size of 0.25 ms to simulate a resting
condition and 0.125 ms to simulate a light exercise
condition. The shape of the velocity profiles of the inlet
and of the outlets of the upper branch vessels, and the
descending thoracic aorta was constrained to an axisymmetric shape with a profile order of nine.10
To simulate light exercise, the resistance value of the
descending thoracic aorta was decreased in order to
increase flow to the lower extremities. The cardiac
cycle was shortened to simulate increased heart rate
until the systolic pressure of the thoracic aorta
3200
KIM et al.
increased by 20% compared to the resting state.2 The
boundary conditions of the upper branch vessels were
unchanged. The parameter values of the closed loop
system are shown in Table 1, and the parameter values
of the Windkessel models are shown in Table 2. The
contractility of this subject was not changed for the
light exercise simulation. Table 3 shows the parameter
values of the lumped parameter coronary vascular
models assigned to each coronary outlet for rest and
light exercise conditions.
Figure 2 shows computed pressure and flow waveforms of the inlet and the outlets for rest and light
exercise conditions of the normal thoracic aorta. The
pressure–volume loops of the left and right ventricles
for both conditions are also shown. The cardiac cycle
decreased from 1.0 to 0.5 s, the cardiac output
increased from 5 to 10.3 L/min, and systolic blood
pressure increased from 124 to 149 mmHg. However,
the stroke volume increased little from 83 to 86 cc
signifying the increase in the cardiac output was mainly
due to the shortening of the cardiac cycle, not due to
the increase of the stroke volume.
Figure 2 also shows that the upper branch vessels
experience retrograde flow in diastole. Retrograde flow
to the upper branch vessels is severe in the light exercise condition, even though the same boundary conditions were assigned to the upper branch vessels,
likely due to the increased flow demand of the
descending thoracic aorta. Figure 2 also shows the
pressure waveforms of the upper branch vessels and
the descending thoracic aorta. The pressure waveform
of the descending thoracic aorta decays faster during
exercise than in the resting condition.
In Fig. 3, coronary flow and pressure waveforms of
the left anterior descending, left circumflex, and right
coronary arteries are plotted for rest and exercise
TABLE 1. Parameter values of the closed loop system at rest (R), during light exercise (E1), and during moderate exercise (E2)
for the simulations of thoracic aorta with coronary arteries.
RLA-V (dynes s/cm5)
LLA-V (dynes s2/cm5)
RLV-art(dynes s/cm5)
LLV-art (dynes s2/cm5)
ELV,max (mmHg/cc)
VLV, 0 (cc)
VLA, 0 (cc)
ELA (mmHg/cc)
tmax (s)
Rpp (dynes s/cm5)
Rpd (dynes s/cm5)
R
E1
E2
5
5
10
0.69
2.0
0
260
270
0.33
16
144
5
5
10
0.69
2.0
0
260
350
0.25
16
144
5
5
10
0.69
2.2
0
260
360
0.15
2
38
RRA-V (dynes s/cm5)
LRA-V (dynes s2/cm5)
RRV-art (dynes s/cm5)
LRV-art (dynes s2/cm5)
ERV,max (mmHg/cc)
VRV, 0 (cc)
VRA, 0 (cc)
ERA (mmHg/cc)
Cardiac cycle (s)
Cp (cm5/dynes)
R
E1
E2
5
1
10
0.55
0.5
0
260
60
1.0
0.022
5
1
10
0.55
0.5
0
260
80
0.5
0.022
5
1
10
0.55
0.5
0
260
80
0.3
0.02
TABLE 2. Parameter values of the three-element Windkessel models at rest (R), during light exercise (E1), and during moderate
exercise (E2) for the simulations of thoracic aorta with coronary arteries.
Rp (103 dynes s/cm5)
C (1026cm5/dynes)
Rd (103 s/cm5)
Rp (103dynes s/cm5)
C (1026 cm5/dynes)
Rd (103 dynes s/cm5)
3
5
Rp (10 dynes s/cm )
C (1026 cm5/dynes)
Rd (103 dynes s/cm5)
B-R
B-El
B-E2
C-R
C-El
C-E2
D-R
D-El
D-E2
1.49
235
15.1
1.49
235
15.1
0.20
31.8
20.0
1.41
248
14.3
1.41
248
14.3
0.19
33.5
19.0
10.7
32.9
108
10.7
32.9
108
1.44
4.45
143
E-R
E-El
E-E2
F-R
F-El
F-E2
G-R
G-El
G-E2
1.75
201
17.6
1.75
201
17.6
0.24
27.1
23.4
7.96
44.0
80.5
7.96
44.0
80.5
1.08
5.95
107
1.80
195
18.2
1.08
195
18.2
0.24
26.3
24.1
H-R
H-El
H-E2
0.227
1540
2.29
0.225
1600
0.676
0.08
1200
0.45
B denotes right subclavian artery, C denotes right carotid artery, D denotes right vertebral artery, E denotes left carotid artery, F denotes left
vertebral artery, G denotes left subclavian artery, and H denotes descending thoracic aorta. Note that the parameter values of the upper
branch vessels are the same for the light exercise and resting conditions and change for the moderate exercise condition.
Patient-Specific Coronary Flow Modeling
3201
TABLE 3. Parameter values of the lumped parameter models of the coronary vascular beds at rest (R), during light exercise (E1),
and during moderate exercise (E2) for the simulations of thoracic aorta with coronary arteries.
Ra
a: LAD1
b: LAD2
c: LAD3
d: LAD4
e: LCX1
f: LCX2
g: LCX3
h: LCX4
i: RCA1
j: RCA2
k: RCA3
c*: LAD3
c**: LAD3
c***: LAD3
c****: LAD3
d*: LAD4
d**: LAD4
d***: LAD4
d****: LAD4
Ra-micro
Rv + Rv-micro
Ca
cim
R
E1
E2
R
E1
E2
R
E1
E2
R
E1
E2
R
E1
E2
183
131
123
75
49
160
216
170
168
236
266
100
89
80
57
65
54
48
35
177
126
91
55
47
154
208
164
163
229
257
76
52
51
31
20
65
87
68
71
98
110
51
51
51
51
31
31
31
31
299
214
148
90
80
261
353
277
274
385
435
146
146
98
93
89
89
58
57
56
40
39
24
15
49
66
52
51
72
81
24
16
16
10
6.2
20
27
21
22
31
35
16
16
16
16
10
10
10
10
94
67
65
40
25
82
111
87
86
121
136
65
65
65
65
40
40
40
40
44
32
31
19
12
39
53
41
40
56
64
18
13
12
7
5
15
21
16
16
23
25
12
12
12
12
7
7
7
7
0.34
0.48
0.49
0.80
1.28
0.39
0.29
0.37
0.37
0.26
0.23
0.49
0.49
0.49
0.49
0.80
0.80
0.80
0.80
0.75
1.02
1.07
1.74
2.79
0.85
0.63
0.80
0.83
0.59
0.52
0.75
1.02
1.07
1.74
2.79
0.85
0.63
0.80
0.83
0.59
0.52
1.07
1.07
1.07
1.07
1.74
1.74
1.74
1.74
2.89
4.04
4.16
6.82
10.8
3.31
2.45
3.12
3.15
2.24
1.99
4.16
4.16
4.16
4.16
6.82
6.82
6.82
6.82
6.88
9.34
9.74
15.9
25.4
7.78
5.74
7.29
7.60
5.38
4.72
6.88
9.34
9.74
15.9
25.4
7.78
5.74
7.29
7.60
5.38
4.72
9.74
9.74
9.74
9.74
15.9
15.9
15.9
15.9
c* denotes LAD3 with 40% diameter reduction, c** denotes LAD3 with 50% diameter reduction, c*** denotes LAD3 with 60% diameter
reduction, and c**** denotes LAD3 with 75% diameter reduction. Similarly, d* denotes LAD4 with 40% diameter reduction, d** denotes LAD4
with 50% diameter reduction, d*** denotes LAD4 with 60% diameter reduction, and d**** denotes LAD4 with 75% diameter reduction. Note
that the unit of the resistance values are in 103 dynes s/cm5 and the unit of the capacitance values are in 1026 cm5/dynes.
conditions. For the left anterior descending and circumflex coronary arteries, coronary flow is high in
diastole and low in systole because the intramyocardial
pressure approximated by the left ventricular pressure
is elevated in systole. On the contrary, the right coronary artery has high flow in systole and low flow in
diastole because the intramyocardial pressure caused
by the right ventricular pressure operates in a low
pressure range. By coupling the lumped parameter
heart models and the lumped parameter coronary
vascular models to a three-dimensional finite element
model of the aorta, we were able to obtain realistic
coronary flow and pressure waveforms17 because we
considered the effects of the contraction and relaxation
of the left and right ventricles. The coronary pressure
waveforms are similar to the aortic pressure waveform
unlike the coronary flow waveforms, which are
dependent on the location of the myocardium the
coronary arteries feed. During light exercise, the coronary flow doubled to meet the demands of the heart.
For the resting condition, the mean coronary flow to
the left anterior descending coronary artery was 1.3 cc/s,
the mean flow to the left circumflex coronary artery
was 1.5 cc/s, and the mean flow to the right coronary
artery was 0.6 cc/s, to yield a total coronary flow of
3.4 cc/s. During light exercise, the coronary flow
doubled achieving a mean flow of 2.8 cc/s to the left
anterior descending coronary artery, 3.1 cc/s to the left
circumflex coronary artery, and 1.1 cc/s to the right
coronary artery, and a total coronary flow of 7.0 cc/s.
Wall shear stress of the coronary arteries for the
resting condition and the light exercise condition are
also plotted for peak-systole, peak left coronary flow,
and late diastole in Fig. 4. For the light exercise condition, the wall shear stress increased as higher volume
of flow traveled to the coronary arteries from the left
ventricle. The left coronary arteries experience higher
wall shear stress fields in late systole and late diastole
whereas the right coronary arteries experience higher
wall shear stress fields in systole due to the asynchrony
of the maximum flow in the left and right coronary
arteries.
Simulations of Coronary Flow and Pressure
with Different Degrees of Stenosis at the Left
Anterior Descending Coronary Artery
In these simulations, we studied how the degree of a
stenosis in the left anterior descending coronary artery
affects coronary flow and pressure. For the normal
case with no stenosis in the left anterior descending
coronary artery, solutions were obtained using a
1,768,953 element and a 339,664 node mesh with a time
step size of 0.25 ms to simulate a resting condition and
0.1 ms to simulate a moderate exercise condition. The
shape of the velocity profiles at the inlet and at the
3202
KIM et al.
Pressure-volume loops
D-Left vertebral
150
E-Right carotid
138
Pressure (mmHg)
Left ventricle (rest)
Right ventricle (rest)
Left ventricle (exercise)
Right ventricle (exercise)
Pressure (mmHg)
Pressure (mmHg)
138
0
68
Time (s)
1
68
0
D-Left vertebral
Flow rate (cc/s)
C D
E F
G
0
1
39
B
-1.5
Time (s)
E-Right carotid
Rest
Exercise
7.5
0
150
Volume (cc)
Flow rate (cc/s)
0
1
-7 0
Time (s)
1
Time (s)
G-Right subclavian
C-Left carotid
138
Pressure (mmHg)
Pressure (mmHg)
138
68
68
0
Time (s)
0
1
A
C-Left carotid
Time (s)
1
G-Right subclavian
37
Flow rate (cc/s)
Flow rate (cc/s)
31
0
1
-7
Time (s)
H
H-Descending thoracic aorta
138
138
Pressure (mmHg)
Rest
Exercise
Pressure (mmHg)
Pressure (mmHg)
138
68
68
Time (s)
68
0
1
B-Left subclavian
1
Time (s)
0
A-Aortic inlet
30
1
400
Rest
Exercise
Flow rate (cc/s)
Flow rate (cc/s)
1
Time (s)
Time (s)
H-Descending thoracic aorta
580
-5 0
1
Time (s)
A-Aortic inlet
B-Left subclavian
0
0
Flow rate (cc/s)
-6
-20
-40
0
Time (s)
1
0
Time (s)
1
FIGURE 2. Pressure–volume loops of the left and right ventricles and flow and pressure waveforms of the descending thoracic
aorta and upper branch vessels for resting and light exercise cases with normal coronary anatomy.
Patient-Specific Coronary Flow Modeling
Pressure
3203
Flow
Rest
a-d: Left anterior descending coronary
a-d: Left anterior descending coronary
Exercise
4.5
Flow rate (cc/s)
Pressure (mmHg)
138
68
0
0
Time (s)
0
1
Time (s)
1
e-h: Left circumflex coronary
e-h: Left circumflex coronary
138
Flow rate (cc/s)
Pressure (mmHg)
5
a
i
68
0
Time (s)
1
0
0
c
k
i-k: Right coronary
138
b
Time (s)
1
Right coronary
j
h gd
68
f
e
Flow rate (cc/s)
Pressure (mmHg)
2
0
0
Time (s)
1
0
Time (s)
1
FIGURE 3. Flow and pressure waveforms of coronary arteries at rest and during light exercise case.
outlets of the upper branch vessels and the descending
thoracic aorta were constrained to an axisymmetric
shape with a profile order of nine.10 For a stenosis case
with 40% diameter reduction in the left anterior
descending coronary artery, solutions were obtained
using a 1,813,097 element and a 350,123 node mesh
with the same time step size as in the normal case. For
a stenosis case with 50% diameter reduction in the left
anterior descending coronary artery, solutions were
obtained using a 2,576,721 element and a 475,723 node
mesh with the same time step size as in the normal case.
For a stenosis case with 60% diameter reduction in the
left anterior descending coronary artery, solutions
were obtained using a 3,146,766 element and a 575,122
node mesh with the same time step size as in the normal case. Finally, for a stenosis case with 75% diameter reduction in the left anterior descending coronary
artery, solutions were obtained using a 3,836,663 element and a 722,518 node mesh with the same time step
size as in the normal case. The increase in the mesh size
was due to the additional refinements in the area of
stenosis.
For resting conditions, we decreased the resistance
of the coronary model downstream of the stenosed
artery until we obtained the same mean flow to these
outlets as with the normal case. We recorded the
pressure loss through these stenosed regions in
Table 4. To dilate the downstream coronary vascular
networks to maximum, we simulated moderate exercise by decreasing the resistance value of the descending thoracic aorta and shortened the cardiac cycle to
increase cardiac output threefold compared to the
resting state. In the exercise simulations, we assumed
that the downstream coronary vasculatures were
maximally dilated and assigned the same boundary
conditions with the normal case to all the stenosis
cases. The flow difference downstream of the stenosed
artery was recorded in Table 4. Table 3 shows the
parameter values of the lumped parameter coronary
vascular models assigned to each coronary outlet for
rest and moderate exercise conditions for the normal
case and all the stenosis cases.
At rest, the downstream coronary boundary conditions were modified until the same mean flow was
achieved through the stenosed artery. However, as the
same mean flow traveled through the stenosed artery, a
pressure loss occurred due to the viscous loss through
the stenosis and complex flow structures distal to the
3204
KIM et al.
Inlet flow rate (cc/s)
580
a
Rest
Exercise
A
-20
b c B
C
Time (s)
0
1
Rest
A
B
C
Exercise
a
b
c
Wall shear stress (dynes/cm2 )
0
7.5
15
22.5
30
FIGURE 4. Wall shear stress of coronary arteries for peak systole, peak left coronary flow rate, and mid-diastole at rest and during
light exercise.
TABLE 4. Mean left anterior descending coronary artery flow
and pressure at rest and during moderate exercise for the
simulations of coronary flow and pressure with different
degrees of stenosis in the left anterior descending coronary
artery.
Mean coronary
flow (cc/s)
Mean pressure
(mmHg)
Mean coronary flow and pressure at rest
Normal
1.1
94
40% Diameter reduction
1.1
93
50% Diameter reduction
1.1
92
60% Diameter reduction
1.1
86
75% Diameter reduction
1.1
78
Mean coronary flow and pressure at moderate exercise
Normal
3.7
85
40% Diameter reduction
3.4
77
50% Diameter reduction
3.2
73
60% Diameter reduction
3.0
67
75% Diameter reduction
1.8
36
stenosed region as shown in Table 4. During exercise,
the same coronary boundary conditions were assigned
to the coronary outlets. A noticeable pressure loss and
flow rate decrease occurred in the case of the stenosis
with 75% diameter reduction.
The boundary conditions of the upper branch vessels were changed to obtain the same mean flow rate as
in the resting state. The parameter values of the closed
loop system is shown in Table 1 and the parameter
values of the Windkessel models are shown in Table 2.
The contractility of this subject was increased 10% for
this moderate exercise simulation.2
Figure 5 depicts flow and pressure waveforms of the
left anterior descending coronary artery for rest and
moderate exercise conditions. Note that we plotted
the flow and pressure waveforms of the 50% and
75% diameter reduction cases only. We see that the
Patient-Specific Coronary Flow Modeling
50% diameter reduction
Normal
75% diameter reduction
Left anterior descending coronary flow at rest
Left anterior descending coronary pressure at rest
3
Normal
50% diameter reduction
75% diameter reduction
Pressure (mmHg)
Flow rate (cc/s)
130
Normal
50% diameter reduction
75% diameter reduction
0
3205
60
0
0
1
Time (s)
Left anterior descending coronary flow at exercise
11
Left anterior descending coronary pressure at exercise
130
Normal
50% diameter reduction
75% diameter reduction
Pressure (mmHg)
Normal
50% diameter reduction
75% diameter reduction
Flow rate (cc/s)
1
Time (s)
0
0
0
0.3
Time (s)
0
Time (s)
0.3
FIGURE 5. Effect of stenosis on coronary artery flow and pressure at rest and during moderate exercise. The pressure was
computed downstream of the stenosis. Note that only 50% and 75% diameter reduction cases are plotted.
to the stenosed regions and they propagate longer for
more severe stenosis. Note that we selected 50% and
75% diameter reduction cases only.
Left anterior descending coronary
Flow rate (cc/s)
5
4
3
2
DISCUSSION
1
0
0
20
40
60
80
100
Diameter reduction (%)
Rest
Exercise
Rest (Gould et al)
Hyperemia (Gould et al)
FIGURE 6. Mean flow rate of the left anterior descending
coronary artery vs. degree of stenosis for rest and moderate
exercise.
coronary flow tripled and the pressure pulse also
became larger for the exercise condition. The pressure
loss increased as more flow traveled through the stenosed regions in the cases of 50% and 75% diameter
reduction.
Figure 6 shows the mean flow in the stenosed artery
in both resting and moderate exercise conditions. We
also plotted the experimental results of Gould et al. for
comparison.7 Figure 7 shows volume rendered velocity
magnitudes of the coronary arteries during moderate
exercise. Complex flow structures are observed distal
We have successfully developed and implemented a
coronary boundary condition that couples a lumped
parameter coronary vascular model to each coronary
outlet of a three-dimensional finite element model of the
aorta and epicardial coronary arteries. We also used an
inflow boundary condition coupling a lumped parameter heart model and a closed loop model to represent the
intramyocardial pressure by considering the interactions between the heart and arterial system. Fluid–
structure interaction simulations were performed to
better represent flow and pressure waveforms. Additionally, we were able to obtain robust and stable solutions by constraining the shape of the velocity profiles of
the boundaries that experienced retrograde flow.
Using the lumped parameter coronary vascular
model along with the inflow boundary condition that
couples the lumped parameter heart model and the
closed loop system, we studied how changes in cardiac and arterial properties affect coronary flow and
3206
KIM et al.
Inflow waveform
Flow rate (cc/s)
1000
Velocity magnitude (cm/s)
A
0
15
30
45
60
B
-50
0
0.3
Time (s)
A
B
Normal
50% diameter
reduction
75% diameter
reduction
FIGURE 7. Volume rendered velocity magnitudes of coronary arteries with different degrees of stenosis during moderate exercise. Note that only 50% and 75% diameter reduction cases are plotted.
pressure. We examined how coronary flow and pressure change for resting and light exercise conditions for
normal coronary anatomy. We then created stenoses in
the left anterior descending coronary artery to investigate how the coronary flow and pressure change for
different degrees of stenoses at rest and moderate
exercise condition when the downstream coronary
vascular beds are maximally dilated.
For the simulations at rest and during light exercise,
the computed coronary flow and pressure waveforms
Patient-Specific Coronary Flow Modeling
were realistic and the asynchrony of the left and right
coronary arteries were represented as we approximated
the intramyocardial pressure with the left and right
ventricular pressure depending on the location of the
coronary arteries. For the simulations with different
degrees of stenosis in the left anterior descending coronary artery at rest and moderate exercise, the reduction in the coronary flow was more apparent for the
moderate exercise case because more energy loss
occurs as more coronary flow travels through the stenosed region. We also demonstrated a greater pressure
loss across the stenosed region as the degree of stenosis
increases. Especially, the pressure loss did not increase
linearly resulting from the energy dissipated due to the
viscous loss and the turbulence in the stenosed region.
Figure 6 showed that the relationship between the
mean coronary flow and the degree of stenosis is
comparable to results obtained from experimental
techniques.7 Gould et al.7 created temporary stenosis
in the left circumflex artery of canines and injected
Hypaque to simulate hyperemic conditions in canine
coronary arteries. They observed that the mean flow
did not decrease up to 85% diameter reduction at
rest and 30–45% diameter reduction at hyperemia.
Although our study focused on different physiologic
conditions of moderate exercise rather than hyperemia,
our results with resting and moderate exercise conditions yield similar behaviors because in both maximal
hyperemia and simulated moderate exercise conditions, we assume that the downstream coronary vascular beds are maximally dilated to receive more flow
from the aorta. We also observed that the flow
remained the same up to 75% diameter reduction case
at rest. In Gould et al.’s plot, the mean flow at rest
slightly increased for mild stenosis cases but in our
study, we aimed to have the same mean flow for mild
stenosis cases, thus, we maintained the same mean flow
at rest up to 75% diameter reduction case. The slope
during moderate exercise is comparable to the slope in
hyperemia although there is a difference in the magnitude of the mean flow. In Gould et al.’s paper, the
mean flow at hyperemic state increased 409% compared to the baseline value whereas in our study, we
designed that the mean flow increased 350% at moderate exercise compared to the baseline value.
Our method has three primary limitations. First, we
did not consider the motion of the heart during the
cardiac cycle. We fixed the coronary arteries to the
epicardial surface of the heart and fixed the surface in
space and time. In reality, the heart moves significantly
to contract and relax during the cardiac cycle. This
movement is large and cannot be modeled using the
coupled momentum method,5 which is a linearized
approach using a fixed fluid mesh. A different
approach, such as an arbitrary Lagrangian–Eulerian
3207
formulation, would be needed to represent the movement of the heart over the cardiac cycle. However,
previous studies showed that the effects due to the
movement of the heart were secondary and did not
affect the pressure and flow fields as much as the
geometry and the boundary conditions.18,19,21,30
Second, we assumed that the left coronary arteries
transport blood to the left ventricle and the right coronary arteries transport blood to the right ventricle. In
reality, however, the coronary vascular networks
experience nonuniform intramyocardial pressure
depending on the location of the coronary networks.
To consider nonuniform intramyocardial pressure, a
three-dimensional nonuniform model of the heart as
well as the mapping of the coronary arteries to the
location of the myocardium that the arteries perfuse
should be considered. For example, a three-dimensional model of cardiac ventricular mechanics can be
used to compute nonuniform intramyocardial pressure
acting on the coronary vascular beds.8,9
Third, we assumed a uniform Young’s modulus for
the whole computational model although we know
that the vessel wall properties vary spatially. To consider nonuniform vessel wall properties, noninvasive
methods of estimating wall thickness and elastic (viscoelastic) wall properties should be developed. In this
study, rather than considering nonuniform wall properties, we adjusted the vessel wall properties so that the
wall deformation of the descending thoracic aorta fit
the experimental data reasonably well.
CONCLUSIONS
A coronary boundary condition that couples a
lumped parameter coronary vascular model to each
coronary outlet of a three-dimensional finite element
model of the aorta and epicardial coronary arteries is
developed. An inflow boundary condition coupling a
lumped parameter heart model and a closed loop
model is implemented to represent the intramyocardial pressure by considering the interactions between
the heart and arterial system. We considered fluid–
structure interaction to better represent flow and
pressure waveforms. Using the lumped parameter
coronary vascular model along with the inflow
boundary condition that couples the lumped parameter heart model and the closed loop system, we can
predict coronary flow and pressure realistically using
anatomic data obtained from medical imaging techniques and study how changes in cardiac and arterial
properties affect coronary flow and pressure and vice
versa. Further, we can utilize these methods to
investigate different vascular interventions of cardiovascular disease.
KIM et al.
3208
ACKNOWLEDGMENTS
Hyun Jin Kim was supported by a Stanford Graduate Fellowship. This material is based upon work
supported by the National Science Foundation under
Grant No. 0205741. The authors gratefully acknowledge the assistance of Dr. Nathan M. Wilson for
assistance with software development. The authors
gratefully acknowledge Dr. Farzin Shakib for the use
of his linear algebra package AcuSolveTM (http://www.
acusim.com) and the support of Simmetrix, Inc. for the
use of the MeshSimTM (http://www.simmetrix.com)
mesh generator.
REFERENCES
1
Berry, J. L., A. Santamarina, J. E. Moore,
S. Roychowdhury, and W. D. Routh. Experimental and
computational flow evaluation of coronary stents. Ann.
Biomed. Eng. 28(4):386–398, 2000.
2
Brooks, G. A., T. D. Fahey, T. P. White, and K. M.
Baldwin. Exercise Physiology Human Bioenergetics and Its
Applications. Berkshire, UK: McGraw-Hill Companies,
2004.
3
Burattini, R., P. Sipkema, G. van Huis, and N. Westerhof.
Identification of canine coronary resistance and intramyocardial compliance on the basis of the waterfall model.
Ann. Biomed. Eng. 13(5):385–404, 1985.
4
Cebral, J. R., M. A. Castro, J. E. Burgess, R. S. Pergolizzi,
M. J. Sheridan, and C. M. Putman. Characterization of
cerebral aneurysms for assessing risk of rupture by using
patient-specific computational hemodynamics models. Am.
J. Neuroradiol. 26(10):2550–2559, 2005.
5
Figueroa, C. A., I. E. Vignon-Clementel, K. E. Jansen, T.
J. R. Hughes, and C. A. Taylor. A coupled momentum
method for modeling blood flow in three-dimensional
deformable arteries. Comput. Methods Appl. Mech. Eng.
195(41–43):5685–5706, 2006.
6
Gijsen, F. J. H., J. J. Wentzel, A. Thury, F. Mastik, J. A.
Schaar, J. C. H. Schuurbiers, C. J. Slager, W. J. van der
Giessen, P. J. de Feyter, A. F. W. van der Steen, and P. W.
Serruys. Strain distribution over plaques in human coronary arteries relates to shear stress. Am. J. Physiol. Heart
Circ. Physiol. 295(4):H1608–1614, 2008.
7
Gould, K. L., K. Lipscomb, and G. W. Hamilton. Physiologic basis for assessing critical coronary stenosis.
Instantaneous flow response and regional distribution
during coronary hyperemia as measures of coronary flow
reserve. Am. J. Cardiol. 33(1):87–94, 1974.
8
Hunter, P. J., A. J. Pullan, and B. H. Smaill. Modeling
total heart function. Annu. Rev. Biomed. Eng. 5(1):147–177,
2003.
9
Kerckhoffs, R. C. P., M. L. Neal, Q. Gu, J. B.
Bassingthwaighte, J. H. Omens, and A. D. McCulloch.
Coupling of a 3D finite element model of cardiac ventricular
mechanics to lumped systems models of the systemic and
pulmonic circulation. Ann. Biomed. Eng. 35(1):1–18, 2007.
10
Kim, H. J., C. A. Figueroa, T. J. R. Hughes, K. E. Jansen,
and C. A. Taylor. Augmented Lagrangian method for
constraining the shape of velocity profiles at outlet
boundaries for three-dimensional finite element simulations
of blood flow. Comput. Methods Appl. Mech. Eng. 198(45–
46):3551–3566, 2009.
11
Kim, H. J., I. E. Vignon-Clementel, C. A. Figueroa, J. F.
LaDisa, K. E. Jansen, J. A. Feinstein, and C. A. Taylor. On
coupling a lumped parameter heart model and a threedimensional finite element aorta model. Ann. Biomed. Eng.
37(11):2153–2169, 2009.
12
Lagana, K., R. Balossino, F. Migliavacca, G. Pennati,
E. L. Bove, M. R. de Leval, and G. Dubini. Multiscale
modeling of the cardiovascular system: application to the
study of pulmonary and coronary perfusions in the univentricular circulation. J. Biomech. 38(5):1129–41, 2005.
13
Laskey, W. K., H. G. Parker, V. A. Ferrari, W. G.
Kussmaul, and A. Noordergraaf. Estimation of total systemic arterial compliance in humans. J. Appl. Physiol.
69(1):112–119, 1990.
14
Li, Z., and C. Kleinstreuer. Blood flow and structure
interactions in a stented abdominal aortic aneurysm model.
Med. Eng. Phys. 27(5):369–382, 2005.
15
Mantero, S., R. Pietrabissa, and R. Fumero. The coronary
bed and its role in the cardiovascular system: a review and
an introductory single-branch model. J. Biomed. Eng.
14:109–115, 1992.
16
Migliavacca, F., R. Balossino, G. Pennati, G. Dubini, T. Y.
Hsia, M. R. de Leval, and E. L. Bove. Multiscale modelling
in biofluidynamics: application to reconstructive paediatric
cardiac surgery. J. Biomech. 39(6):1010–1020, 2006.
17
Opie, L. H. Heart Physiology: From Cell to Circulation.
Philadelphia, PA, USA: Lippincott Williams and Wilkins,
2003.
18
Qiu, Y., and J. M. Tarbell. Numerical simulation of pulsatile flow in a compliant curved tube model of a coronary
artery. J. Biomech. Eng. 122(1):77–85, 2000.
19
Ramaswamy, S. D., S. C. Vigmostad, A. Wahle, Y. G. Lai, M.
E. Olszewski, K. C. Braddy, T. M. H. Brennan, J. D. Rossen,
M. Sonka, and K. B. Chandran. Fluid dynamic analysis in a
human left anterior descending coronary artery with arterial
motion. Ann. Biomed. Eng. 32(12):1628–1641, 2004.
20
Sahni, O., J. Muller, K. E. Jansen, M. S. Shephard, and
C. A. Taylor. Efficient anisotropic adaptive discretization
of the cardiovascular system. Comput. Methods Appl.
Mech. Eng. 195(41–43):5634–5655, 2006.
21
Santamarina, A., E. Weydahl, Jr. J. M. Siegel, and J. E.
Moore, Jr. Computational analysis of flow in a curved tube
model of the coronary arteries: effects of time-varying
curvature. Ann. Biomed. Eng. 26:944–954, 1998.
22
Stergiopulos, N., P. Segers, and N. Westerhof. Use of pulse
pressure method for estimating total arterial compliance in
vivo. Am. J. Physiol. Heart Circ. Physiol. 276(2):H424–
H428, 1999.
23
Taylor, C. A., M. T. Draney, J. P. Ku, D. Parker, B. N.
Steele, K. Wang, and C. K. Zarins. Predictive medicine:
computational techniques in therapeutic decision-making.
Comput. Aided Surg. 4(5):231–247, 1999.
24
Taylor, C. A., and C. A. Figueroa. Patient-specific model
of cardiovascular mechanics. Annu. Rev. Biomed. Eng. 11:
109–134, 2009.
25
Taylor, C. A., T. J. R. Hughes, and C. K. Zarins. Finite
element modeling of blood flow in arteries. Comput.
Methods Appl. Mech. Eng. 158(1–2):155–196, 1998.
26
Van Huis, G. A., P. Sipkema, and N. Westerhof. Coronary
input impedance during cardiac cycle as determined by
impulse response method. Am. J. Physiol. Heart Circ.
Physiol. 253(2):H317–H324, 1987.
Patient-Specific Coronary Flow Modeling
27
Vignon-Clementel, I.E., C.A. Figueroa, K.E. Jansen, and
C.A. Taylor. Outflow boundary conditions for threedimensional finite element modeling of blood flow and
pressure in arteries. Comput. Methods Appl. Mech. Eng.
195(29–32):3776–3796, 2006.
28
Vignon-Clementel, I. E., C. A. Figueroa, K. E. Jansen,
and C. A. Taylor. Outflow boundary conditions for
three-dimensional simulations of non-periodic blood
flow and pressure fields in deformable arteries.
Comput. Methods Biomech. Biomed. Eng., 2008. doi:
10.1080/10255840903413565.
29
3209
Zamir, M., P. Sinclair, and T.H. Wonnacott. Relation
between diameter and flow in major branches of the arch of
the aorta. J. Biomech. 25(11):1303–1310, 1992.
30
Zeng, D., E. Boutsianis, M. Ammann, K. Boomsma,
S. Wildermuth, and D. Poulikakos. A study on the
compliance of a right coronary artery and its impact
on wall shear stress. J. Biomech. Eng. 130(4):041014,
2008.
31
Zhou, Y., G. S. Kassab, and S. Molloi. On the design of the
coronary arterial tree: a generalization of Murray’s law.
Phys. Med. Biol. 44:2929–2945, 1999.