Inlet and Outlet Boundary Conditions and Uncertainty Quantification in Volumetric Lattice Boltzmann Method for Image-Based Computational Hemodynamics

Abstract

Inlet and outlet boundary conditions (BCs) play an important role in newly emerged image-based computational hemodynamics for blood flows in human arteries anatomically extracted from medical images. We developed physiological inlet and outlet BCs based on patients’ medical data and integrated them into the volumetric lattice Boltzmann method. The inlet BC is a pulsatile paraboloidal velocity profile, which fits the real arterial shape, constructed from the Doppler velocity waveform. The BC of each outlet is a pulsatile pressure calculated from the three-element Windkessel model, in which three physiological parameters are tuned by the corresponding Doppler velocity waveform. Both velocity and pressure BCs are introduced into the lattice Boltzmann equations through Guo’s non-equilibrium extrapolation scheme. Meanwhile, we performed uncertainty quantification for the impact of uncertainties on the computation results. An application study was conducted for six human aortorenal arterial systems. The computed pressure waveforms have good agreement with the medical measurement data. A systematic uncertainty quantification analysis demonstrates the reliability of the computed pressure with associated uncertainties in the Windkessel model. With the developed physiological BCs, the image-based computation hemodynamics is expected to provide a computation potential for the noninvasive evaluation of hemodynamic abnormalities in diseased human vessels.

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Citation: Yu, H.; Khan, M.; Wu, H.;
Zhang, C.; Du, X.; Chen, R.; Fang, X.;
Long, J.; Sawchuk, A.P. Inlet and
Outlet Boundary Conditions and
Uncertainty Quantification in
Volumetric Lattice Boltzmann
Method for Image-Based
Computational Hemodynamics.
Fluids 2022,7, 30. https://doi.org/
10.3390/fluids7010030
Academic Editors:
Mehrdad Massoudi and Goodarz
Ahmadi
Received: 3 November 2021
Accepted: 30 December 2021
Published: 10 January 2022
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fluids
Article
Inlet and Outlet Boundary Conditions and Uncertainty
Quantification in Volumetric Lattice Boltzmann Method for
Image-Based Computational Hemodynamics
Huidan Yu 1,2,* , Monsurul Khan 1, †, Hao Wu 1, Chunze Zhang 3, Xiaoping Du 1, Rou Chen 1,‡ , Xin Fang 4,
Jianyun Long 4and Alan P. Sawchuk 2
1Department of Mechanical and Energy Engineering, Indiana University-Purdue University Indianapolis,
Indianapolis, IN 46202, USA; khan212@purdue.edu (M.K.); hw51@iu.edu (H.W.); duxi@iu.edu (X.D.);
rouchen@cjlu.edu.cn (R.C.)
2Department of Surgery, Division of Vascular Surgery, Indiana University School of Medicine,
Indianapolis, IN 46202, USA; asawchuk@iupui.edu
3Southwest Institute of Water Transport Engineering, Chongqing Jiaotong University,
Chongqing 400074, China; zhangchunze@whu.edu.cn
4
Department of Vascular Surgery, The Affiliated Hangzhou First People’s Hospital, Zhejiang University School
of Medicine, Hangzhou 310006, China; fangxin324@hotmail.com (X.F.); longjianyun1208@126.com (J.L.)
*Correspondence: whyu@iupui.edu
† Current address: School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.
‡ Current address: College of Metrology and Measurement Engineering, China Jiliang University,
Hangzhou 310004, China.
Abstract:
Inlet and outlet boundary conditions (BCs) play an important role in newly emerged image-
based computational hemodynamics for blood flows in human arteries anatomically extracted from
medical images. We developed physiological inlet and outlet BCs based on patients’ medical data and
integrated them into the volumetric lattice Boltzmann method. The inlet BC is a pulsatile paraboloidal
velocity profile, which fits the real arterial shape, constructed from the Doppler velocity waveform.
The BC of each outlet is a pulsatile pressure calculated from the three-element Windkessel model, in
which three physiological parameters are tuned by the corresponding Doppler velocity waveform.
Both velocity and pressure BCs are introduced into the lattice Boltzmann equations through Guo’s
non-equilibrium extrapolation scheme. Meanwhile, we performed uncertainty quantification for
the impact of uncertainties on the computation results. An application study was conducted for six
human aortorenal arterial systems. The computed pressure waveforms have good agreement with
the medical measurement data. A systematic uncertainty quantification analysis demonstrates the
reliability of the computed pressure with associated uncertainties in the Windkessel model. With the
developed physiological BCs, the image-based computation hemodynamics is expected to provide a
computation potential for the noninvasive evaluation of hemodynamic abnormalities in diseased
human vessels.
Keywords:
volumetric lattice Boltzmann method; image-based computational hemodynamics;
three-element Windkessel model; boundary conditions; uncertainty quantification
1. Introduction
With the recent advances in medical imaging, computational power, and mathematical
algorithms, image-based computational hemodynamics (ICHD) has emerged [
1
–
7
] as a new
capability giving rise to the potential for computation-aided diagnostics and therapeutics in
a patient-specific environment for cardiovascular diseases. Based on radiological imaging
data, such as computed tomography angiography (CTA) images and Doppler ultrasound
(DUS) velocity waveforms, ICHD enables noninvasive and patient-specific quantification of
pulsatile hemodynamics in human vessels. Such data, including velocity vector, pressure,
Fluids 2022,7, 30. https://doi.org/10.3390/fluids7010030 https://www.mdpi.com/journal/fluids

Fluids 2022,7, 30 2 of 15
vorticity vector, and wall-shear stress (WSS) in the entire artery segment with fine spatial
and temporal resolutions, are not readily available from the current standard clinical mea-
surements. Through further postprocessing of the pulsatile hemodynamic data, either the
assessment of the true hemodynamic abnormality or the prediction of potential therapeu-
tic/surgical outcomes from an interventional treatment may aid in clinical decision-making
for various cardiovascular diseases.
A typical ICHD from medical data to medical insights mainly consists of three steps.
They are (1) image extractions of a three-dimensional anatomical geometry of the diseased
artery from CTA data and one-dimensional velocity waveforms from DUS images at inlet
and outlets, (2) computation of pulsatile hemodynamics employing physical parameters
together with the flow environment, and (3) post-processing of the computed pulsatile
hemodynamics with analysis, visualization, and parametrization to the key insights of
the disease assessment and potential therapeutic outcomes. Since only a segment of the
blood circulation system is being computed, boundary conditions (BCs) are required to be
applied at the inlet(s) and outlet(s) of the vessel segment to represent the remaining vascular
network. In general, the introduction of the inlet BC is relatively straightforward, imposing
parabolic-like flow profiles at the cross-section of the inlet. Usually, an inlet cross-section of
a human vessel is not a perfect circle. Therefore, neither a steady Poiseuille-Hagen nor an
oscillating Womersley velocity profile can be directly constructed based on a DUS-measured
velocity waveform. The choices of the outflow BC in ICHD vary among zero pressure or
zero traction conditions, resistance or impedance conditions, reduced-order models which
can be an open or closed loop, and reduced-order one-dimensional wave propagation
equations [
8
–
10
]. To capture the interaction between the local three-dimensional artery
segment and the one-dimensional global circulation, the three-dimensional flow solver
must be coupled to a reduced-order lumped parameter network model. Among them,
the three-element Windkessel model [
11
–
15
] (WK3) has been commonly used to construct
such a network, in which a Windkessel circuit is adopted to model the distal vasculature
with one capacitor, modeling vessel compliance, and two resistors, modeling proximal and
distal flow resistances, respectively. Evidence has shown that the WK3 can well reproduce
physiological pressure waveforms [16,17] in large vessels.
In this work, we present the physiological inlet and outlet BCs in volumetric lattice
Boltzmann method (VLBM) [
18
] for ICHD together with uncertainty quantification (UQ).
The lattice Boltzmann method (LBM) [
19
,
20
] is a class of computational fluid dynamics
(CFD) methods for solving complex flows. Instead of directly solving a set of nonlinear
partial differential equations, i.e., Navier-stokes (NS) equations, the LBM is a discretized
kinetic model on a regular lattice to solve the dynamics of incompressible fluid flow. Due
to its particulate nature and local dynamics, the LBM has its advantages over the NS-
based CFD methods, especially in dealing with complex boundaries [
18
,
21
], incorporating
microscopic interactions [
22
,
23
] in multiphase flows, and implementing GPU (Graphics
Processing Unit) parallel computing [
21
,
24
]. Nevertheless, the LBM has not been extensively
used for ICHD so far and the majority of attempts have imposed non-physiological BCs.
An example is the zero pressure BC [
25
–
28
] at the outlets. The zero pressure BC, although
easy to be implemented, is well known to lead to unrealistic hemodynamics, in part
because of its inability to capture physiologic levels of pressure [
6
]. Few other studies
have used the fully developed BC [
18
,
29
] at the outlets, which is also inappropriate for
a pulsatile flow in arbitrary flow domains. In this paper, we develop the physiological
velocity BC at the inlet based on the DUS waveform and pressure BC at each outlet via
WK3 model tuned by the corresponding DUS waveform and then integrated them into
the VLBM. We study six aortorenal arterial systems, with given CTA image data and DUS
velocity waveforms of each, for noninvasive quantification of pulsatile hemodynamics.
To demonstrate the accuracy of the computation, we compare the computed pressure
waves with the corresponding invasive pressure measurements during digital subtraction
angiography (DSA) in the clinic. Meanwhile, we perform uncertainty quantification to
demonstrate the reliability of the computation.

Fluids 2022,7, 30 3 of 15
2. Methods and Materials
We have previously developed and validated a VLBM solver [
30
] for solving image-
based pore-scale porous media flows. The solver synergistically employs the traditional
node-based LBM for image segmentation and the cell-based VLBM [
18
] for CFD, enabling
a seamless connection between these two parts and unified GPU parallelization for fast
computation [21,31,32].
The VLBM is formulated on a cell-based mesh. Fluid particles are uniformly dis-
tributed in lattice cells, as opposed to sitting at lattice nodes in conventional LBM. As
schematized in Figure 1, an arbitrary boundary (black line) separates a fluid domain (with-
out dots) from a solid boundary structure (with dots). Three distinct cells are characterized
through a volumetric parameter, i.e., the occupation of solid volume
∆Vs(x)
in the cell with
total volume
∆V(x)
, defined as
P(x)≡∆Vs(x)/∆V(x)
. Thus three different cells, fluid cell
(P=0), solid cell (P=1), and boundary cell (0 <P<1), can be distinguished.
Figure 1.
Three types of lattice cells in VLBM: fluid cell (
P=
0), solid cell (
P=
1), and boundary cell
(0 <P<1). The solid line represents an arbitrary boundary of the flow domain.
On a lattice space with bdirections of discrete molecular velocity, VLBM deals with
the time evolution of the particle population, ni(x,t), corresponding to the ith velocity ei
ni(x+eiδt,t+δt)=ni(x,t)−hni(x,t)−neq
i(x,t)i
τ;i=0, . . . , b(1)
where
neq
i(x,t)
and
τ
are the corresponding equilibrium particle population and relaxation
time, respectively. The resulting density
ρ(x,t)
and velocity
u(x,t)
in the fluid domain are
ρ(x,t)=∑ni(x,t)/[1− P(x,t)] (2)
and
u(x,t)=∑eini(x,t)/∑ni(x,t)(3)
The pressure field p(x,t)is then calculated from:
p(x,t)−p0=c2
s[ρ(x,t)−ρ0](4)
where
p0
and
ρ0
are reference pressure and density, respectively. In this work, we adapted
the VLBM solver to ICHD, named InVascular, based on medical imaging data.
The implementation flow chart of InVascular is shown in Figure 2. It starts with the
image segmentation from CTA image data to extract the anatomical geometry using the
conventional LBM with D3Q7 lattice model [33]. A distance field [34] governed by a level
set equation [
35
] is solved in which the zero-level distance represents the morphological
boundary of the vessel segment. From the distance field,
P(x)
of each cell is calculated and
then, together with the inlet and outlet BCs, fed to VLBM [
18
] with D3Q19 lattice model.
Both image segmentation and computational hemodynamics (dashed part in Figure 2)
are carried out on a unified mesh and connected seamlessly. Thus the state-of-the-art
GPU parallelism can be efficiently utilized. The detailed formulation of LBM for image

Fluids 2022,7, 30 4 of 15
segmentation and VLBM computational hemodynamics, as well as the connection between
them, and the GPU parallelization are referred to in our published papers [
21
,
30
]. In this
paper, we focus on the integration of the physiological inlet and outlet BCs with the VLBM,
as highlighted in Figure 2.
Figure 2.
Flow chart of InVascular: (1) 3-D anatomical extraction of vessel segment from CTA image
data; (2) ICHD with the inputs of
P(x)
and inlet and outlet BCs based on DUS image data as well as
three-element WK model; and (3) post-processing for intepretation and medical insights. The unified
LBM (dashed part) is accelerated by GPU parallelism.
2.1. Physiological Inlet and Outlet Boundary Conditions
In this part, we present the algorithms to construct the physiological inlet and outlet
BCs based on the DUS velocity waveforms. The inlet BC is a velocity profile, and the output
BC is a pressure calculated from WK3. Both velocity and pressure BCs are introduced
into VLBM.
2.1.1. Implementation of Velocity and Pressure BCs in VLBM
In the VLBM, we employed the non-equilibrium extrapolation boundary condition
developed by Guo et al. [36] as follows.
ni(xb,t)−neq
i(xb,t)=nixf,t−neq
ixf,t(5)
for i-th direction where
xb
and
xf
are the boundary cell and its next fluid cell along that
direction, respectively. If the velocity,
u(xb,t)
, is known at the boundary cell, the velocity
BC is:
ni(xb,t)=neq
iρxf,t,u(xb,t)+nixf,t−neq
ixf,t(6)
Whereas if the pressure p(xb,t)is given at the boundary cell, the pressure BC reads:
ni(xb,t)=neq
iρ(xb,t),uxf,t+nixf,t−neq
ixf,t(7)
where
ρ(xb,t)
is calculated from Equation (4). We use the velocity BC and pressure BC at
the inlet and each outlet, respectively.
In InVascular, the inlet and outlet BCs are based on the patient’s DUS data, as shown
by the shaded part in Figure 2. We present the introduction of the inlet and outlet BCs in
the following subsections.
2.1.2. Lumen-Fitted Velocity BC Profile at an Inlet
The DUS measured velocity waveform,
uin (t)
, has been commonly used as the inflow
BC [
6
] in ICHD. For a pipe with its radius of R, the typical way to introduce the pulsatile
velocity to drive the flow into the pipe is to construct a parabolic profile of Poiseuille flow,
u(r,t)=uin (r,t)1−r2/R2
, in which ris the distance to the pipe center. However, real ar-
terial lumens are often not perfectly circular. To use this parabolic velocity profile, one needs
to extend the inlet from noncircular to circular, which may introduce an
unrealistic inflow.

Fluids 2022,7, 30 5 of 15
We present an algorithm, as illustrated in Figure 3, for an irregular paraboloid-like
velocity profile that fits the real inlet cross-section. The velocity waveform of
uin (t)
is digi-
tized from the patient’s DUS shown in Figure 3a. It should be noted that, for a blood flow,
the inflow velocity is pulsatile thus the irregular velocity profile needs to be constructed
at every time point and the time resolution should be fine enough, determined through a
temporal convergence check. To refine the temporal resolution, we use linear interpolation.
Figure 3.
Illustration of inlet boundary condition from DUS image data for an irregular artery plane.
(
a
) A generic DUS image recording velocity magnitude waveform
uin (t)
. (
b
) An example of indexing
to construct an irregular paraboloidal velocity profile on the inlet plane. (
c
) Normalized velocity
distribution on inlet plane varying from uin at the center to zero at the edge.
Assume the inlet plane is perpendicular to the z-direction, i.e., the direction of the
bloodstream, and it is located at
z=z0
. On the plane, each cell has known
P(i,j,z0)
with
i=
1,
. . .
,
Nx
and
j=
1,
. . .
,
Ny
. The algorithm to generate an irregular paraboloidal
velocity profile at time tincludes the following steps, schematized in Figure 3b.
(1)
Declare a matrix
Nx×Ny
, i.e.,
Lij
(
i=
1,
. . .
,
Nx
,
j=
1,
. . .
,
Ny
) and initialize it as
Lij =0.
(2)
Loop ifrom 1 to Nxand jfrom 1 to Ny, if
a.
a cell’s
P
is neither 0 nor 1 (i.e., a boundary cell), assign
Lij =
0 for this cell and
define its velocity magnitude 0,
b.
a cell’s
P
is 0 (i.e., a fluid cell) and the
Lij
value of any neighboring cell is 0,
assign Lij =1 for this cell,
c.
a cell’s
P
is 0 and the
Lij
value of any neighboring cell is 1, assign
Lij =
2 for
this cell,
d.
continue until all the fluid cells are assigned. The last index of the cell labeling
is Lij =M.
(3)
Loop ifrom 1 to
Nx
and jfrom 1 to
Ny
and define velocity magnitude as
uij (t)=Li j ×uin (t)/M.
The largest velocity
uin (t)
is recognized at the cell labeled as
Lij =M
. The velocity
reduces radially from
uin (t)
at label M (center) to zero at label
Lij =
0 (wall). Figure 3c
shows two views of paraboloid-like velocity distribution on an irregular inlet plane at a
time point. The inlet velocity profile is introduced in VLBM through Equation (6).
2.1.3. WK3-Based Pressure BC at an Outlet
For the outlet BC, we use the popular WK3 model in an open vessel loop to calculate
the pressure,
p(t)
, on the outlet plane. It has been well-known that WK3 is the best outlet
BC model among other physiologically relevant zero-dimenensional outflow models to
simulate the peripheral vasculature [
37
] and has been popularly used when significant
compliance is located in the modeled distal vasculature [
12
]. As shown in Figure 4, WK3 is
an analogy to an electrical circuit, which models the distal vasculature with one capacitor,
modeling vessel compliance and two resistors, modeling proximal and distal resistance. The

Fluids 2022,7, 30 6 of 15
flow rate (Q) and the mean pressure (
p
) over the outlet plane are related by the following
ordinary differential equation [12]
dp
dt +1
RC p=rdQ
dt +1
RC (r+R)Q(8)
where rand Rrepresent the proximal and distal resistances, and Cis the compliance of the
distal vasculature. Specifically, ris used to absorb the incoming waves and reduce artificial
wave reflections. Equation (8) has an analytical solution.
p(t)=e−t/(RC)Zt
0es/(RC)rdQ(s)/ds +r+RQ(s)
RC ds +pt=0(9)
where pt=0is the initial pressure at the outlet.
Figure 4.
WK3 model consists of one capacitor (C), modeling vessel compliance, and two resistors (r
and R), modeling proximal and distal resistance, respectively.
In Equation (9), the three elements, r,C, and R, specified at each outlet, must be tuned
to obtain the physiological values for the total outflow rate
Qt
and target systolic (
psys
)
and diastolic (
pdia
) pressure, with the mean arterial pressure,
pm=psys +2pdia /
3, based
on patient’s clinical data. For an aortorenal system, see Figure 5below, we use brachial
pressure for a pressure target and DUS velocity waveform for the target flow rate (Q) with
the understanding that the capacitor and resistors have independent functionalities in the
WK3 circuit: a capacitor reflects the pulsatility of blood flow whereas a resistor determines
the flow rate [15]
Figure 5.
Integration of InVascular with velocity BC from DUS at inlet and pressure BCs through the
WK3 model at outlets for quantification of TSPG (
≡pa−pr)
in an aortorenal system extracted from
patient’s CTA.
The integration of the WK3 model [15] and VLBM is described as follows
(1)
Determine the total resistance in the arterial segment
a. Assume the total system compliance Ct=0.1 cm5/dynes.
b. Calculate the total resistance Rt(=r+R)=pm/Qt.

Fluids 2022,7, 30 7 of 15
(2)
Determine rand Rat each outlet based on the published works: the proximal resis-
tance rweights 28% [
38
,
39
] and 5.6% [
40
] out of the total resistance in the renal artery
and abdominal aorta, respectively.
(3)
Tune rand Rbased on DUS flow rate at each outlet.
a.
Integrate the pressure BC from the WK3, Equation (9), into VLBM, Equation (7),
and run InVascular. In one pulsation, r,R, and Cremain the same but Q (t) at
each outlet is obtained from the simulation.
b.
Once a simulation is done, check if the flow rate at each outlet matches that
calculated from DUS imaging data. If yes, rand Rare determined; If not, adjust
Rtand repeat (1) b, (2), and (3).
(4)
Determine the compliance C at each outlet.
a. Distribute Ctto each outlet proportional to the corresponding mean flow rate.
b.
Check if the mean arterial pressure
pCH D
in
matches
pin
at the inlet. If not, adjust
Ctin (1) a. and repeat (1) and (2).
The outlet BC at each outlet is introduced in VLBM through Equations (4) and (7) after
the pressure is obtained from Equation (9) at each time step.
2.2. Uncertainty Quantification
There is no doubt that uncertainty always exists in any modeling and simulation
process [
41
]. In the process shown in the flowchart of InVascular in Figure 2, uncertainties
come from noises introduced during image scanning and the extraction of the arterial
segmentation, the use of empirical blood properties, parameters involved in the boundary
conditions, and so on. The uncertainties in the input variables will affect the output
(hemodynamics) of InVascular. Following the common practice in uncertainty quantification
(UQ), we treat the parameters with uncertainty as random variables and quantify their
effects on the output variables. In this study, we use the common UQ method: the First
Order Second Moment (FOSM) [42] method.
Denote output variables by
Y=(Y1,Y2, . . . , Ym)T
and input variables by
X=(X1,X2, . . . , Xn)T
, where mand
n
are the numbers of output and input variables,
respectively. If the elements of
X
are non-normally distributed and dependent, Rosenblatt
transformation [43] can be used to transform Xinto independent and normal variables.
Suppose the black-box models of InVascular are given by:
Yj=gj(X),j=1, 2, . . . , m(10)
Linearizing a model at the mean values, µ=(µ1,µ2, . . . , µn), of X, yields
Yj≈gj(µ)+∇T(X−µ),j=1, 2, . . . , m(11)
where
∇=∂g
∂X1,∂g
∂X2, . . . , ∂g
∂XnT
is the gradient of
gj(X)
at
µ
. Since
Yj
is approximated as a
linear combination of
x
, it is also normally distributed, denoted by
NµYj,σ2
Yj
with
µYj
and
σYj
, the mean and standard deviation of
Yj
, respectively. The two parameters are given
as follows.
µYj=gj(X),j=1, 2, . . . , m(12)
σ2
Yj=
n
∑
i=1∂gj
∂Xi2
σ2
i,j=1, 2, . . . , m(13)
where
σi
is the standard deviation of
Xi
. The covariance between output variables
Yj
and
Ykis calculated by
Cjk =
n
∑
i=1∂gj
∂Xi∂gk
∂Xiσ2
i(14)

Fluids 2022,7, 30 8 of 15
Then the joint probability density function (PDF) of output
Y=(Y1,Y2, . . . , Ym)T
is de-
termined by the mean vector
µY=µY1,µY2, . . . , µYmT
and covariance matrix
ΣY=Cjkj,k=1,2,...,m
. Since
gj(X)
is a black box, its gradient is evaluated numerically
by the finite difference method. The total computational cost, measured by the number of
function (model) calls, is
n+
1. The efficiency is high since the number of function calls is
linearly proportional to the dimensionality of input variables.
2.3. Materials
We studied six human aortorenal arterial systems. The medical data of each case
included CTA images and DUS waveforms at the inlet and outlets, obtained from the elec-
tronic medical libraries in Indiana University Methodist Hospital in Indianapolis, Indiana,
USA, and Hangzhou First People’s Hospital, Hangzhou, China. The CTA resolutions are
approximately 0.752 ×2.5 mm3(US cases) and 0.652 ×0.6mm3(China cases).
We show one representative case in Figure 5to demonstrate the integration of VLBM
and physiological BCs at inlet and outlets, for InVascular. The aortorenal arterial system,
anatomically extracted from the patient’s CTA data, consists of the aortic artery (AA), left
renal artery (LRA), and right renal artery (RRA). The inlet BC based on the DUS velocity
waveform and outlet BCs of WK3 are imposed at the inlet and three outlets, respectively.
A minor lumen reduction (circled, about 20% lumen reduction) is seen in the LRA. The
DUS images are available at the AA inlet to construct a paraboloidal velocity profile and
outlets of AA, LRA, and RRA to tune the r, R, and C parameters. The physical flow domain
is 63
×
116
×
84 mm
3
. The cardiac cycle is 0.68 s with a time resolution of 6.79 ms. The
density and kinematic viscosity are 1.06
×
10
3
kg/m
3
and 3.3
×
10
−6
m/s
2
, respectively.
The dimensionless relaxation time τin VLBM is 0.5079.
The WK3 parameters r, R, and C at the three outlets are listed in Table 1. The pulsatile
pressure waveforms in AA, LRA, and RRA were invasively measured during a clinical
intervention, which are used to validate the computed pressure below. As seen in Figure 6
below, for a given pressure waveform, the pressure values at the peak and the end of
the waveform are called systolic blood pressure,
psys
, and diastolic blood pressure,
pdia
,
respectively. The mean arterial pressure (MAP) is defined as
MAP =psys −pdi a/
3
+pdia
.
Table 1.
Values of resistance and compliance parameters, r,R, and C, in WK3 model at corresponding
outlets tuned from the DUS data.
Outlet r
(dynes×s/cm5)
R
(dynes×s/cm5)
105C
(cm5/dynes)
AA 88.0 2773.1 1.8
LRA 2982.4 7666.03 0.36
RRA 5972.8 15358.7 0.32
Figure 6.
Comparisons of pressure waveforms in (
a
) AA, (
b
) RRA, and (
c
) LRA between noninvasive
computation (solid line) and invasive measurement (dashed line).

Fluids 2022,7, 30 9 of 15
The spatial and temporal convergence checks are exhibited in Table 2. The relative
errors are the normalized differences of the mean arterial pressure (MAP) and systolic
pressure (
Psys
) between two successive grids and cycles, respectively. To balance the
accuracy and the computation cost, we chose 200
×
368
×
265 as the resolution for the
simulation and run 10 cycles to produce the computational results.
Table 2.
Spatial (left) and temporal (right) convergence check. The spatial reoslution is represented
by the grid number along the flow direction. MAP and
psys
stand for mean arterial presure and
systolic pressure, respectively. The relative error is the normalized difference of the correponding
pressure values between two sucsessive grids and cycles.
Spatial Temporal
Grid MAP(mmHg) Relative Error (%) Cycle Psys (mmHg) Relative Error (%)
170 100 1 150
180 87.5 12.5 3 154 2.7
190 89 1.71 5 152 −1.3
200 90 0.34 10 155 2.0
210 90.15 0.19 15 155 0
220 90.20 0.05 20 155 0
3. Results
In this section, we demonstrate the applicability and reliability of InVascular in
two aspects.
First, we use the representative study case to show the computed pulsatile
pressure, velocity, and vorticity fields in the arterial system. The noninvasively computed
pressure waveforms are compared with the invasively measured ones at three locations.
Second, we perform a systematic UQ analysis for the representative case and for all
six cases
to study how the r, R, and C parameters impact the computed pressure.
3.1. Pulsatile Hemodynamics in an Aortorenal Arterial System
We first demonstrate the accuracy of InVascular for the quantification of the pul-
satile pressure field. Figure 6shows the comparisons of the cyclic pressure waveforms in
(a) AA, (b) RRA, and (c) LRA between noninvasive computation (solid lines) and invasive
measurements (dashed lines).
The computed pressure waveforms agree very well with the medical measured wave-
forms. The pressure contours on (a) the AA-LRA plane, (b)AA-RRA plane, and (c) repre-
sentative cross-sections are plotted in Figure 7. The trans-stenotic pressure gradient (
TSPG
)
in the LRA can be calculated through ether
MAP
or
psys
. The comparison of the TSPGs
between noninvasive computation and invasive measurement is shown in Table 3. Again,
both are in good agreement.
Figure 7.
Systolic pressure contours (
a
) the AA-LRA plane, (
b
) AA-RRA plane, and (
c
) representative
cross-sections.

Fluids 2022,7, 30 10 of 15
Table 3. Comparison of TSPG in LRA and RRA based on MAP andpsys.
TSPG
MAP psys
Computed Measured Computed Measured
pa−pr, left 2.5 2.6 4.1 4.0
pa−pr, right 2.0 2.0 4.0 4.0
We found that the fully-developed BC and DUS-based velocity BC, which are com-
monly used in the LBM, cannot capture the physiological pressure waveform. Figure 8
shows the cyclic evolution of the systolic pressure at a representative location in the arterial
system under three different BCs with identical computation environments and conditions.
Neither the fully-developed BC (long dash) nor DUS-based velocity BC (short dash) is
convergent. The pressure (left scale) asymptotically increases with time and exceeds the
human blood pressure after a few cardiac cycles, whereas the WK3-based pressure BC
(solid line) leads to a convergent systolic pressure (
psys ≈
154.8 mmHg ) (right vertical
scale) after 10 cardiac cycles.
Figure 8.
Cyclic evolution of systolic pressure in a representative location using three different BCs:
velocity BC, pressure BC via WK3, and fully-developed BC.
Besides the pressure field, InVascular simultaneously computes the pulsatile velocity
field, from which the vorticity and shear stress fields can be calculated. Figures 9and 10
show the velocity field with magnitude contours and streamlines and vorticity contours,
respectively, at t = (a) 0.1, (b) 0.23, and (c) 0.63 in seconds in one cardiac cycle corresponding
to systole (heart contraction, flow acceleration), diastole (heart relaxation, flow deceleration),
and the end of diastole respectively. In Figure 9, flow in AA is stronger at systole (a) than
at diastole (b) but remains intensive in LRA and RRA at both time points and is better
organized at systole than at diastole. Whereas at the end of diastole, the flow is weak but
chaotic. The vorticity contours shown in Figure 10 are similarly intensive in (a) and (b)
with a large degree of skewness in AA, demonstrating the complexity of the flow in the
real arteries. At the end of diastole, vorticity contours are much smaller and chaotic.

Fluids 2022,7, 30 11 of 15
Figure 9.
Velocity contours and streamlines at t = (
a
) 0.10 (systole), (
b
) 0.23, and (
c
) 0.63 (end of
diastole) in seconds.
Figure 10.
Z-component (vertically up) and x-component (horizontally right) vorticity contours at
t=(a) 0.10, (b) 0.23, and (c) 0.63 in seconds.
3.2. Impact of r, C, and R Parameters on Pressure Quantification
Although the WK3 has been popularly used to model the physiological BC at each
outlet of the artery segment (see Figure 5), its parameters reflecting resistances, r and R,
and compliance, C, are determined empirically [
38
–
40
], which is subjected to uncertainty.
To demonstrate the impact of the uncertainty in r-C-R parameters on the quantification
of proximal and distal pressure, we performed a UQ analysis using FOSM. The input
variables are the r, R, and C parameters in WK3, defined in Table 4. The elements of
X
are independently and normally distributed. The output variables are the pressure values
in AA, LRA, and RRA, defined in Table 5. In this study, we assumed that the standard
deviation of a random input variable is 3% of its mean. We performed UQ for five cases.
The input distributions for the representative case in Section 3.1 are shown in Table 4.
Table 4. Input distributions for the representative case in Section 3.1.
Artery Parameter Variables Mean Standard
Deviation Distribution Type
AA
r(dynes
×
s/cm
5
)
X1108.12 3.24 Normal
AA
R(dynes
×
s/cm
5
)
X23386.38 101.59 Normal
LRA
r(dynes
×
s/cm
5
)
X32879.76 86.39 Normal
LRA
R(dynes
×
s/cm
5
)
X47386.06 221.58 Normal
RRA
r(dynes
×
s/cm
5
)
X53306.39 99.19 Normal
RRA
R(dynes
×
s/cm
5
)
X68505.96 255.18 Normal
AA
C(cm
5
/dynes)
X71.0 ×10−53.0 ×10−7Normal
LRA
C(cm
5
/dynes)
X85.4 ×10−61.62 ×10−7Normal
RRA
C(cm
5
/dynes)
X94.8 ×10−61.43 ×10−7Normal

Fluids 2022,7, 30 12 of 15
Table 5. UQ results.
Artery Output Variable Mean µy1Standard Deviation σy195% Confidence Interval
AA Y1(mmHg) 155.80 1.37 [153.05, 158.55]
LRA Y2(mmHg) 141.72 1.12 [139.49, 143.95]
RRA Y3(mmHg) 144.61 1.12 [142.37, 147.86]
UQ results are given in Table 5. All the model output variables are normally distributed.
For example,
Y1∼NµY1,σY1=N(
155.80 mmHg,1.37 ) . With these results, we know
complete information about the simulation predictions, including the 95% confidence
intervals of the model predictions. The formula for 95% confidence interval is
µYi±
2
σYi
.
For example, the 95% confidence interval of
Y1
is [153.051, 158.55] mmHg unit. This means
that the chance the actual value of
Y1
falling into the interval is 95%, or we have 95%
confidence that the actual value of
Y1
is between 153.05 mmHg and 158.56 mmHg. The
results of mean and standard deviation from five patients are also given in Table 6. The
95% confidence intervals of the model predictions of five patients are in Table 7.
Table 6. Mean and standard deviation of five patient cases.
Case Y1(mmHg) Y2(mmHg) Y3(mmHg)
1N156.80, 1.372N141.72, 1.112N144.61, 1.122
2N163.61, 2.062N154.25, 1.932N56.42, 0.092
3N157.22, 1.542N152.35, 1.452N153.85, 1.472
4N109.71, 0.932N106.49, 0.892N74.50, 0.562
5N123.21, 0.972N117.34, 0.892N102.17, 1.242
Table 7. 95% confidence intervals of model predictions of five patient cases.
Case Y1(mmHg) Y2(mmHg) Y3(mmHg)
1 [153.05, 158.55] [139.49, 143.95] [142.37, 146.86]
2 [159.48, 167.74] [150.38, 158.12] [56.25, 56.59]
3 [154.14, 160.30] [149.45,155.25] [150.92,156.79]
4 [107.84, 111.58] [104.72,108.27] [73.37,75.62]
5 [121.28, 125.15] [115.57, 119.11] [99.69,104.64]
4. Discussion
We have presented the physiological inlet and outlet BCs for ICHD and integrated
them into our in-house computational platform, InVascular. Using the unified LBM mod-
eling for image segmentation and computational hemodynamics, InVascular seamlessly
integrates the anatomical extraction of the interested arterial segment and quantification of
pulsatile hemodynamics and achieves fast computation via GPU parallel computing. The
inlet BC is a pulsatile velocity. A paraboloidal velocity profile is constructed based on the
DUS velocity waveform, which fits the real shape of the arterial lumen (usually noncircular).
Each outlet BC is a pulsatile pressure determined by WK3 during the simulation. The
inlet velocity and outlet pressure BCs are introduced in the VLBM via a non-equilibrium
extrapolation BC scheme. Using InVascular, we performed UQ analysis to quantify the
impact of input variations caused by uncertainties. We applied InVascular into a human
aortorenal arterial system extracted from medical CTA imaging data and demonstrated
the applicability and reliability of InVascular for a real-world flow system. Six cases were
studied. The pressure waveforms in AA, LRA, and RRA computed from InVascular have
excellent agreements with the invasive measurements. The pulsatile velocity and then

Fluids 2022,7, 30 13 of 15
vorticity fields are shown as well. Due to the lack of available data, the validation of the
velocity quantification has not been conducted. A systematic UQ analysis focuses on the
impact of the variation of r, R, and C parameters on the quantification of the pressure field.
Results include joint probability density of the computed pressure, which also provides
the uncertainty or the confidence of the prediction. Due to the suitability of LBM for GPU
parallel computing, InVascular features exceptionally fast computation speed. With a great
potential to further speed up through parallel optimization and/or multiple GPU cards,
the computation time is expected to be around 10 min per patient case. Such a computation
capability is critically important for the clinical use of InVascular, enabling massive numeri-
cal analysis through parametrization to assess the true degree of existing arterial stenosis,
either severe for immediate therapeutics or mild to avoid unnecessary intervention, within
clinic permitted time.
Author Contributions:
Conceptualization, H.Y. and X.D.; methodology, H.Y., X.D., M.K. and C.Z.;
formal analysis, M.K., H.W. and R.C.; investigation, H.Y., X.D., M.K. and H.W.; data curation, M.K.
and H.Y.; resources: A.P.S., X.F. and J.L.; writing—original draft preparation, M.K., H.Y. and X.D.;
writing—review and editing, H.Y. and X.D.; visualization, M.K. and H.W.; supervision, H.Y., X.D.
and A.P.S.; project administration, H.Y.; funding acquisition, H.Y., X.D. and A.P.S. All authors have
read and agreed to the published version of the manuscript.
Funding:
This research was funded by NSF through grant CBET 1803845. This work used the
Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National
Science Foundation Grant No. ACI-1548562. The 1st and corresponding author would like to also
acknowledge the IUPUI University Fellowship and IUPUI MEE Graduate Fellowship.
Institutional Review Board Statement:
The IRB approval (#1309233521R003|N) was obtained for the
patients enrolled at Indiana University. The study (#116-01) was approved by the Ethics Committee
of Hangzhou First People’s Hospital. The investigation conformed to the principles outlined in the
Declaration of Helsinki.
Informed Consent Statement:
It only involved a retrospective analysis of clinically indicated proce-
dures; therefore, informed consent was not required.
Data Availability Statement:
The data presented in this study are available on request from the
corresponding author. The data are not publicly available.
Conflicts of Interest: The authors declare no conflict of interest.
Nomenclatures
AA Aortic Artery
BC Boundary Condition
CFD Computational Fluid Dynamics
CTA Computed Tomography Angiography
DUS Doppler Ultrasound
FOSM First-Order Second Moment
GPU Graphic Processing Unit
ICHD Image-Based Computational Hemodynamics
LBM Lattice Boltzmann Method
LRA Left Renal Artery
MAP Mean Arterial Pressure
N-S Navier-Stokes
RRA Right Renal Artery
TSPG Trans-Stenotic Pressure Gradient
UQ Uncertainty Quantification
VLBM Volumetric Lattice Boltzmann Method
WK3 Three-Element Windkessel Model
WSS Wall-Shear Stress

Fluids 2022,7, 30 14 of 15
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