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A robust comparison approach of velocity data between MRI and CFD based on divergence-free space projection

TL;DR: An aortic phantom study has been set-up under fully controlled laminar flow conditions with helical flow patterns to validate the proposed flow regularization approach and analyse the robustness of applied pre-processings including the denoising of MRI data and the decomposition of velocity vector field.
Abstract: Recent achievements in 4D flow MRI increased the interest of CFD-MRI studies, which require comparison of velocity fields from both approaches for validation purposes. A novel flow regularization approach is proposed to provide a ground truth and to perform robust, mathematically reasonable comparisons between CFD and MRI. Our suggested method projects the measured and denoised data into the same space as the computational domain and applies the Helmholtz-Hodge theorem to recover the divergence-free property of the flow field by decomposing the velocity field into its divergence-free, curl-free and harmonic components. Furthermore, an aortic phantom study has been set-up under fully controlled laminar flow conditions with helical flow patterns to validate the proposed method using phase-contrast MRI measurements, whereas a dynamic stenosed case was used under turbulent flow conditions to analyse the robustness of applied pre-processings including the denoising of MRI data and the decomposition of velocity vector field.

Summary (2 min read)

1. INTRODUCTION

  • Recently, 4D flow MRI combining 3D spatial encoding with three-directional velocity-encoding has revealed great potential [1].
  • Such measurements are limited by many factors such as acquisition times, signal-to-noise ratio and resolution depending on the set-up and region of interest.
  • Numerical phantoms play an important role in the assessment and validation of hemodynamics.
  • A reasonable comparison method has not yet been developed.
  • This work was supported by the Swiss National Science Foundation grant 320030-149567.

2. EXPERIMENTAL SETTING

  • A silicon replica of a healthy human aortic arch (Elastrat, Switzerland) (B) was connected to a centrifugal pump (A) (BG-GP 636, Einhell Germany AG, Germany, maximum pressure 3.9 bar) via PVC tubing with 19 mm inner diameter with a total length of 20 m.
  • Inlet and outlet were connected to a reservoir (D) resulting in an open circuit.
  • The flow rate was monitored in 10 min intervals by PhaseContrast MRI.
  • For the acquisition of the flow field a 3D spoiled Gradient-Echo sequence with flow encoding gradients was used.
  • Linear Phase Correction was applied to compensate for the eddy-current induced background phase.

3. PRE-PROCESSINGS OF THE NUMERICAL PHANTOM

  • Flow reconstruction yields both the proton density images and three-directional velocity data, denoted by uMRI.
  • The phantom aorta was segmented semi-automatically with snake evolution methods using ITK-SNAP [11], see Fig. 2 (left).
  • The surface mesh was smoothed with VMTK [12].
  • The mesh was built using snappyHexMesh, an OpenFOAM utility for creating hexahedral meshes.

3.1. Denoising of MRI data

  • The three-dimensional velocity data was composed with the segmentation binary mask, such that only the original values were kept within the surface of the segmented aorta.
  • Values outside the surface were set to zero, such that they are not considered by the latter interpolation of measurements onto the mesh domain.
  • After these processes, one is still left with the noise within the aorta, for which fast, accurate and automatic post-processings are required to get an appropriate input for numerical computations.
  • This problem has been extensively studied in the existing literature [13].
  • In order to use the MRI acquisitions as initial conditions for the numerical simulations, the denoised data were projected onto the mesh domain using the linear interpolation provided by ITK [14].

3.2. Projection onto the divergence-free space

  • In what follows, the authors look for the divergence-free component of the denoised velocity, simply denoted by ũ⋆ = ũ+ u⋆.
  • The authors use no-slip boundary condition at the aortic wall Γ5 to ensure the well-posedness of the problem, while homogeneous Neumann boundary condition is considered on the remaining boundaries.
  • Let n denote the outward unit normal vector on the boundary.

4.1. Navier-Stokes equations for Newtonian fluids

  • Ω where u, p are the computed velocity and pressure fields respectively, σ(u, p) represents the Cauchy stress tensor, and the gravity is neglected.
  • Being consistent with the experiment, corresponding parameters are used for water at 29◦C.
  • To ensure that the problem Pfd is well-posed, reasonable boundary conditions are required.
  • The aortic wall is assumed to be rigid and a no-slip boundary condition is imposed on Γ5, meaning that frictional forces will create a boundary layer along the wall.

4.2. Multiscale coupling with zero-dimensional model

  • Reduced order modeling represent a useful formalism that can provide partial but accurate information about the arterial hemodynamics.
  • During the past decades, there has been significant developments using several techniques of multiscale modeling, where reduced order models, also referred to as lumped-parameter models, are coupled with multidimensional description of the cardiovascular system [16].
  • Such reduced order models provide boundary conditions to be coupled with the detailed three-dimensional model.
  • Using the flow rate QΓ provided by the fluid solver, i.e.
  • In the presented work, the coupled problem is solved until the steady state, where a threshold tolerance ǫ is reached.

5. NUMERICAL EXAMPLES AND VALIDATION

  • The numerical phantom was implemented using icoFoam, an OpenFOAM [18] solver for incompressible, laminar Navier-Stokes equations using the PISO algorithm.
  • The spatial accuracy is studied by computing the error in L2, denoted by ‖.‖0,2,Ω, and L ∞, denoted by ‖.‖0,∞,Ω, norms with respect to the exact solution, respectively for several finite element polynomial approximations.
  • Notice that div ũ⋆ < 0.05s−1. In the second experiment, the authors performed numerical computations under laminar flow conditions with helical flow patterns using the denoised data as depicted in the graph in Section 4.2.
  • Comparisons in Fig. 6 show that the velocity magnitudes turned out to be almost identical in the defined planes P1, P2 and P3 across the aorta, see Fig.
  • HFI measures the alignment between the local velocity u and the vorticity w vectors, and it is given by the normalized helicity density HFI = u ·w/(|u| |w|).

6. CONCLUSION

  • A novel approach has been proposed to perform reliable comparisons between CFD and MRI.
  • Phantom experiments of laminar flow under helical patterns have been performed.
  • Numerical computations were compared against MRI acquisitions, and results show good agreement.
  • Stability and the numerical issues will be provided in a forthcoming work.

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ETH Library
A robust comparison approach of
velocity data between MRI and
CFD based on divergence-free
space projection
Conference Paper
Author(s):
Koltukluoglu, Taha Sabri; Hirsch, Sven; Binter, Christian; Kozerke, Sebastian; Székely, Gábor; Laadhari, Aymen
Publication date:
2015
Permanent link:
https://doi.org/10.3929/ethz-a-010542673
Rights / license:
In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection.
For more information, please consult the Terms of use.

A ROBUST COMPARISON APPROACH OF VELOCITY DATA BETWEEN MRI AND CFD
BASED ON DIVERGENCE-FREE SPACE PROJECTION
Taha Sabri Koltukluoglu
Sven Hirsch
Christian Binter
Sebastian Kozerke
G
´
abor Sz
´
ekely
Aymen Laadhari
Computer Vision Laboratory, Swiss Federal Institute of Technology, CH-8092 Z¨urich
Institute for Biomedical Engin eering, Swiss Federal Institute of Technology, CH-8092 Z¨urich
Institute o f Applied Simulation, Zurich University of Applied Sciences, CH-882 0 W¨a denswil
ABSTRACT
Recent achievem ents in 4D flow MRI increased the interest
of CFD-MRI studies, which require comparison of veloc-
ity fields from both approaches for validation purposes. A
novel flow regularization approach is proposed to pr ovide
a ground truth and to pe rform robust, math ematically rea-
sonable comparisons between CFD and MRI. Our suggested
method projects the measured and denoised data into the
same space as th e computational do main and applies the
Helmholtz-Hodge theorem to recover the divergence-free
property of the flow field by decomposing the velocity field
into its divergence-free, curl-free and harmonic components.
Furthermore, an aortic phantom study has been set-up under
fully controlled laminar flow c onditions with helical flow
patterns to validate the proposed method using phase-contrast
MRI measurements, whereas a dynamic stenosed case was
used under turbulent flow cond itions to analyse the robustness
of ap plied pre-processings inc luding the denoising of MRI
data an d the d e composition of velocity vector field.
1. INTRODUCTION
Recently, 4D flow MRI combining 3D spatial encoding with
three-dir ectional velocity-encoding has revealed great poten-
tial [1]. However, such measu rements are limited by many
factors such as acquisition time s, signal-to-noise ratio and res-
olution depending on the set-up and region of interest. Nu-
merical phantom s play an important role in the assessment
and validation of hem odynamics. However, due to the d if-
ferent spatial representation of the vector fields between the
image a nd computational domain, the validation is difficult.
Although such co mbined studies are under extensive research,
a reasonable co mparison method has not yet been developed.
Most of these works do either not describe in detail how the
compariso ns were achieved exactly or perform visual inter-
pretation [2, 3, 4], whe reas some other works compare ve-
locity ma gnitude profiles and/or peak-velocities [5, 6], lin-
ear correlation co e fficients [7] or volumetric flow rates [8].
This work was supported by the Swiss National Science Foundation
grant 320030-149567.
Most of these comparisons are performed on 2D planes, do
not define any norm or do not consider flow properties. Due
to their limitation s, su ch comparisons can often lead to m is-
leading arguments. The p resented work proposes the devel-
opment of a divergence-free gro und tru th within the co mputa-
tional doma in for numerical phan toms to achieve a meaning-
ful comparison of computations and the me asurements within
the same space based on L
1
, L
2
or L
norms. These proper-
ties, to the authors best knowledge, have not been considered
yet by any other combined CFD and MRI studies. Deno is-
ing is achieved by utilizing norma lized med ia n test [9]. The
feasibility of the proposed approach is tested and validate d
using in-vitro experiments of an aortic phantom under lami-
nar flow conditions with helical flow patter ns. Furth e rmore,
the applied pre-processings were analysed for their good ness
using in-vitro experiments of a stenosed phantom under dy-
namic and turbulent cond itions. The latter case has not been
used for numerical c omputations.
2. EXPERIMENTAL SETTING
Fig. 1. Ex perimental phan tom setup.
A silicon replica of a healthy hum an aortic arch (Elas-
trat, Switzerland) (B) was connected to a centrifugal pump
(A) (BG-GP 636, Einhell Germany AG, Germany, max imum
pressure 3.9 bar) via PVC tubing with 1 9 mm inne r diameter
with a total length of 20 m. Inlet and outlet were connected
to a reservoir (D) resulting in an open circuit. A schematic of
the setup can be seen in Fig. 1. T he flow rate was controlled

using a ball be a ring valve (C) placed 1.5 m downstream of the
pump. Working fluid was H
2
O with a temperature of 29
C.
The flow rate was monitore d in 10 min intervals by Phase-
Contrast MRI.
First, the flow was adopted in such a way, that the flow were
static laminar under helical flow patterns without stenosis.
Secondly, a rigid model of the aortic valve was designed us-
ing the valve orifice geom e try of an aortic steno sis patient
obtained by 2D MRI as a te mplate. The orifice area was
scaled to be 0.75 mm
2
, and a cone sha ped inlet was applied.
This geometry was then 3D printed (Dime nsion Elite, Strata -
sys Ltd., Eden Prairie, MN, USA) using Acrylonitrile buta-
diene styrene (ABS) as a source material and used to ob-
tain dynam ic turbulent flow under stenosed condition. For
the acquisition of the flow field a 3D spoiled Gradient-Echo
sequence with flow encoding gradients was used. In order
to achieve a higher velocity-to-noise ratio a Bayesian Multi-
Point approach [10] with three different encoding velocities
(280, 93 and 40 cm/s) per direction was cho sen. Flip angle
was set to 10 degrees, the voxel size was 1 mm isotropic with
a field of view of 250 x 155 x 60 mm
3
and TE/TR were 5.9
ms and 10 m s, re spectively. Total scan time was 15.5 min.
All scans were performed using a 6-element cardiac coil on
a 3T Philips scanner (Achieva, Ph ilips Health c are, Best, The
Netherland s) . Linear Phase Correction was applied to com-
pensate for the eddy-c urrent induced background phase.
3. PRE-P ROCESSINGS OF THE NUMERICAL
PHANTOM
Flow reconstruction yields both the proton density images and
three-dir ectional velocity data, denoted by u
MRI
. The phan-
tom aorta was segmented sem i-automatically with snake evo-
lution methods using IT K -SNAP [11], see Fig. 2 (left). The
surface mesh was smo othed with VMTK [12]. The mesh was
built using snappyHexMesh, an OpenFOAM utility for creat-
ing hexahedral meshes.
3.1. Denoising of MRI data
The three-dimensiona l velocity data was composed with the
segmentation binary mask, such that only the original values
were kept within the surface of the segmented aorta. Values
outside the surface were set to zero, such that th ey are not con-
sidered by the latter interpolation of measurements onto the
mesh domain. After these processes, one is still left with the
noise within the aorta, for w hich fast, accurate and automatic
post-processings a re required to get an appropriate input for
numerical computations. Outlier de te c tion tech niques should
also avoid heavy computational cost. This problem has been
extensively studied in the existing literatur e [13]. We have ap-
plied the normalized median test proposed by [9]. In or der to
use the MRI acquisitions as initial conditions for the numer-
ical simulations, the denoised data were projected onto the
mesh domain using the lin ear interpolation provided by ITK
[14]. The resulting projection of denoised velocities, denoted
by
¯
u
MRI
, d oes not respect the incompressibility constraint,
and therefore a projection onto the diverg e nce-free space were
performed to add ress this issue.
3.2. Project ion onto the divergence-free space
The Helmholtz-Hodge decompo sition [15] is adopted. For
regularity reasons, it is supposed that
¯
u
MRI
is quadratically
integrable, and the aortic domain, , is assumed to be a
bounded, simply-connec te d and Lipschitz subdomain of R
3
.
Under such assumptio ns, the following space splitting holds:
L
2
(Ω)
3
= H
div,0
(Ω) H
curl,0
(Ω) H
har
(Ω) ,
where H
div,0
is the Sobolev space of squar e integrable vec-
tor fields with a square integrable divergence and a free-
divergence , H
curl,0
is the Sobolev space of squar e integrable
vectors with a square integrable curl and a free- c url , and H
har
is the space of harmonic scalar functions q H
1
(Ω) having
zero Laplac ia n. As a matter of fact, the vector
¯
u
MRI
can be
uniquely decomposed into the sum
¯
u
MRI
=
˜
u + u
+ u
,
where
˜
u H
div,0
(Ω), u
H
curl,0
(Ω) and u
H
har
(Ω).
In what follows, we look for the divergence-free component
of the denoised velocity, simply denoted by
˜
u
=
˜
u + u
.
Since the space H
curl,0
coincides with the gradient of potential
space, it exists a potential q H
1
0
(Ω) such that u
= q. Let
us assume that f = div
¯
u
MRI
H
1
(Ω), where H
1
(Ω)
is the dual space of H
1
0
(Ω). We use no-slip boundary condi-
tion at the aortic wall Γ
5
to ensure the well-posedness of the
problem, while homogeneous Neumann boundary condition
is considered on the remaining boundaries. Let n denote the
outward unit normal vector on the boundary. By applying the
divergence operator, the projection problem reads
P
: nd
˜
u
=
¯
u
MRI
q H
div
(Ω) such that
q = f in , q = 0 on Γ
5
and
n
q = 0 on \Γ
5
.
4. MATHEMATICAL MODEL
4.1. Navier-Stokes equations for N ewtonian fluids
The flow is modelled as homogene ous, incompressible, and
Newtonian fluid governed by the Navier-Stokes equation s
by solving the following steady state problem, P
fd
: for
T > 0 and t (0, T ), find u C
0
(0, T ), L
2
(Ω)
3
L
2
(0, T ),
H
1
0
(Ω)
3
and p L
2
(0, T ), L
2
0
(Ω)
such
that
ρ
u
dt
+ u · u
div σ (u, p) = 0 in (0, T ) ×
div u = 0 in (0, T ) ×
where u, p are the computed velocity and pressure fields
respectively, σ(u , p) represents the Cauchy stress tensor,

Q
Γ
P
Γ
R
p
P
R
d
C
Fig. 2. Left: No menclature used for the boundaries and the
cross-sectional planes through the aorta. Right: Schematic
representation of the Windkessel model.
and the gravity is neglected. Being consistent with the
experiment, corresponding parameters are u sed for wa-
ter at 29
C. These are ρ = 10
3
kg/m
3
for density and
µ = 8.01 · 10
4
kg/(sm) for kinematic viscosity. To ensure
that the problem P
fd
is well-posed , reasonable boundar y
conditions are required. The aortic wall is assumed to be
rigid and a no-slip boundar y condition is imposed on Γ
5
,
meaning th at frictional forces will create a boundary layer
along the wall. Post-processed and proje cted velocity data,
˜
u
, is prescribed on the inlet, Γ
0
. The remaining boundary
conditions are modelled using multiscale couplings.
4.2. Multiscale coupling with zero-dimensional model
Reduced order modeling represent a useful formalism that
can provide partial but accurate information about the arte-
rial hemodynamics. During the past decades, there has been
significant developments using several techniques of multi-
scale mod e ling, where reduced order models, also referred
to as lumped-parameter models, are coupled with multi-
dimensional de scription of the cardiovascular system [16].
Such r educed order models provide boundary co nditions to
be coupled with the detailed three- dimensional model. In this
way, flow rate and pressure may be exchange d between the
models of different complexity. In this work, a re duced circu-
lation model allows to describe th e systemic hemodynamics
and provides a physiologica l pressure load at the downstream
boundary of the descending aorta [17].
Among the existing models, the three-element Wind-
kessel model is represented by an analog electrical circuit
scheme, and it accounts for the vessel wall compliance and
the fluid viscosity through a capacitor C and two resistances
R
p
and R
d
, see Fig. 2. The following second order differen-
tial problem holds: P
0d
: nd P
Γ
such that
P
Γ
P
+ CR
d
dP
Γ
dt
= (R
1
+ R
2
) Q
Γ
+ CR
1
R
2
dQ
Γ
dt
.
Using the flow rate Q
Γ
provided b y the fluid solver, i.e. P
fd
,
as inp ut, the reduced model allows to get a physiological
pressure load ap plied as bounda ry condition on the outlets
Γ
i
, i {1, 2, 3, 4}. The parameters of the reduced mode l are
tuned in a dyna mic case to obtain physiologically relevant
results. In the pre sented work, the coupled problem is solved
until the steady state, where a thresh old tolerance ǫ is reached .
Fig. 3 reveals the entire workflow.
normalized
median test
4D flow
MRI:
u
MRI
replace-
ment scheme
linear
interpolation
solve P
magnitude
images
segmentation
in vitro experiment
set initial
condition
set BCs on
Γ
0
Γ
5
u
(k)
, p
(k)
solve P
0d
solve P
fd
k k + 1
err
k
< ǫ
compute
|u
˜
u
|
0,2,
err
k
=
|u
(k+1)
u
(k)
|
1,2,
|u
k
|
1,2,
+
|p
(k+1)
p
(k)
|
0,2,
|p
k
|
0,2,
pre-processing
CFD simulations
¯
u
MRI
˜
u
no
yes
Fig. 3. Workflow: Pre-processing and simulation.
5. NUMERICAL EXAMPLES AND VALIDATION
The numerical phantom was implemented using icoFoam,
an O penFOAM [18] so lver for incompressible, laminar
Navier-Stokes equations using the PISO algorithm. The
first experim ent is concerned with numeric al validation of
the divergence-free projection. A validation case was im-
plemented in 2D where the exact solution is known. The
problem was solved with Rheolef [19] finite element frame-
work and the error was compared between the computed and
the exact solutions for several polynomial approxim a tions.
Let us consider the initial vector u =
˜
u + u
+ u
such that
˜
u =
sin
2
(2πx) sin(4πy), sin
2
(2πy) sin(4πx)
T
,
u
=
2.2 sin(4πx) sin
2
(2πy), 2.2 sin(4πy) sin
2
(2πx)
T
and
u
=
1.7 log
x
2
+ y
2
, 1.7 log
x
2
+ y
2

T
. Given u,
this test co nsists in finding the divergence-free component
˜
u
=
˜
u + u
. The spa tial accuracy is studied by compu ting
the error in L
2
, denoted by k.k
0,2,
, and L
, denoted by
k.k
0,,
, norms with respect to the exact solution, respec-
tively for several finite element polynomial approximations.
By obser ving the slope in logarithm ic scale, Fig. 4 depicts
that the error evolution shows similar convergence rates to
the expected theoretical errors. In particular, the L
2
error has
a convergence rate equal to k when using P
k,k1
polynoms.

P
rt
✂❡s s
P
rt
✁❡s s
Fig. 4. Convergence properties of th e div-free projection.
raw vector u
MRI
denoised vector
¯
u
MRI
divergence-free vector
˜
u
Fig. 5. Pre-processings shown under stenosed condition.
Top: Velocity vector. Bottom: Divergence of the velocity.
Followin g the preprocessing steps described in Section 3,
we perform the deno ising and divergence-free projection in
the case of a dynamic and stenotic phantom. Results in Fig.
5 clearly show the accuracy of the method under turbulent
conditions, where the outliers are re moved and the final ve-
locity respects the incompressibility constraint. Notice that
div
˜
u
< 0.05s
1
.
In the second experiment, we performed n umerical com-
putations under laminar flow conditions with helical flow pat-
terns using the denoised data as depicted in the graph in Sec-
tion 4.2. Four d efined planes are considered across the aor ta,
see Fig. 2(left), where a quantitative comparison of the veloc-
ity magnitude is performed. Comparisons in Fig. 6 show that
the velocity magnitud es turned out to be almost identical in
pre-processed data [m
3
/s] computations [m
3
/s]
Γ
1
2.798 × 10
5
5.409 × 10
5
Γ
2
1.984 × 10
5
3.362 × 10
5
Γ
3
4.287 × 10
5
5.512 × 10
5
Γ
4
9.041 × 10
5
8.322 × 10
5
Table 1. Comparison between initial and computed fluxes.
˜
u
u
˜
HFI
HFI
P1 P2 P3 P4
Streamlines of u
Fig. 6. Comparison between d a ta and computations under
laminar flow conditions.
the defined pla nes P1, P2 and P3 acr oss the aorta, see Fig. 2.
However, a slight difference is observed in P4, which corre-
sponds to the brachiocepha lic artery. A potential explanation
is that this error orig inates from the sma ll lengths of the arches
and from the low accuracy of the MRI acqu isitions in these
small arches. Moreover, the same observation holds when
we evaluate the flow rates on the outlets. Unlike the flow
rate on the descending aorta, Table 1 shows less similarities
with respect to th e experimental acquisitions in the arche s.
Therefore, the normalized error between the computed solu-
tion an d the ground truth in the entire compu tational domain
is
Z
|u
˜
u
|/
Z
|
˜
u
| 0.112 9.
Since th e flow considered in the aorta has a helical pattern, see
streamlines in Fig. 6, we chose the helical flow ind ex (HFI)
as an indicator to quantify the helicity [20]. HFI measures
the alignment between the local velocity u and the vorticity
w vecto rs, and it is given by the normalize d helicity density
HFI = u · w/(|u| |w|). Accordingly, HFI is given by the co-
sine of the angle between u and w. Therefore, HFI is close to
1 if the flow is purely helical, while the sign of HFI gives the
direction of the rotation. The index is calculated on data and
computational results, referred to as
˜
HFI
and HFI respec-
tively (see Fig. 6), and it shows the zones of almost purely
helical flow where |HFI| 1.
6. CONC LUSION
A novel approach has been proposed to perform reliable com-
parisons between CFD and MRI. Phan tom experiments of
laminar flow under helical patterns have b een performed. Nu-
merical computations were compared against MRI acquisi-
tions, and results show good agreement. As a future exten-
sion, the present strategies will be used to simulate a dynamic
flow considering a stenotic valve upstream the domain u sing
a blood mimicking flow instead of H
2
O. Stability and the nu-
merical issues will be provided in a forthcoming work.

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01 Jan 2018
TL;DR: Three novel techniques that combine CFD with 4D flow MRI measurements in order to address the resolution and noise issues of MR flow imaging are presented.
Abstract: Novel Algorithms for Merging Computational Fluid Dynamics and 4D Flow MRI by Ali Bakhshinejad The University of Wisconsin–Milwaukee, 2018 Under the Supervision of Professors Roshan M. D’Souza And Vitaliy L. Rayz Time-resolved three-dimensional spatial encoding combined with three-directional velocity-encoded phase contrast magnetic resonance imaging (termed as 4D flow MRI), can provide valuable information for diagnosis, treatment, and monitoring of vascular diseases. The accuracy of this technique, however, is limited by errors in flow estimation due to acquisition noise as well as systematic errors. Furthermore, available spatial resolution is limited to 1.5mm 3mm and temporal resolution is limited to 30-40ms. This is often grossly inadequate to resolve flow details in small arteries, such as those in cerebral circulation. Recently, there have been efforts to address the limitations of the spatial and temporal resolution of MR flow imaging through the use of computational fluid dynamics (CFD). While CFD is capable of providing essentially unlimited spatial and temporal resolution, numerical results are very sensitive to errors in estimation of the flow boundary conditions. In this work, we present three novel techniques that combine CFD with 4D flow MRI measurements in order to address the resolution and noise issues. The first technique is a variant of the Kalman Filter state estimator called the Ensemble Kalman Filter (EnKF). In this technique, an ensemble of patient-specific CFD solutions are used to compute filter gains. These gains are then used in a predictor-corrector scheme to not only denoise the data but also increase its temporal and spatial resolution. The second technique is based on proper orthogonal decomposition and ridge regression ii (POD-rr). The POD method is typically used to generate reduced order models (ROMs) in closed control applications of large degree of freedom systems that result from discretization of governing partial differential equations (PDE). The POD-rr process results in a set of basis functions (vectors), that capture the local space of solutions of the PDE in question. In our application, the basis functions are generated from an ensemble of patient-specific CFD solutions whose boundary conditions are estimated from 4D flow MRI data. The CFD solution that should be most closely representing the actual flow is generated by projecting 4D flow MRI data onto the basis vectors followed by reconstruction in both MRI and CFD resolution. The rr algorithm was used for between resolution mapping. Despite the accuracy of using rr as the mapping step, due to manual adjustment of a coefficient in the algorithm we developed the third algorithm. In this step, the rr algorithm was substituted with a dynamic mode decomposition algorithm to preserve the robustness. These algorithms have been implemented and tested using a numerical model of the flow in a cerebral aneurysm. Solutions at time intervals corresponding to the 4D flow MRI temporal resolution were collected and downsampled to the spatial resolution of the imaging data. A simulated acquisition noise was then added in k-space. Finally, the simulated data affected by noise were used as an input to the merging algorithms. Rigorous comparison to state-of-the-art techniques were conducted to assess the accuracy and performance of the proposed method. The results provided denoised flow fields with less than 1% overall error for different signal-to-noise ratios. At the end, a small cohort of three patients were corrected and the data were reconstructed using different methods, the wall shear stress (WSS) was calculated using different reconstructed data and the results were compared. As it has been shown in chapter 5, the calculated WSS using different methods results in mutual high and low shear

2 citations


Additional excerpts

  • ...[74] proposed an algorithm using HelmholtzHodge theorem....

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Journal ArticleDOI
TL;DR: In this paper , a new mathematical and computational model to determine the quality of 4D Flow MRI is presented, which is derived by assuming the true velocity satisfies the incompressible Navier-Stokes equations and that can be decomposed by the measurements u→meas plus an extra field w→ .
Abstract: 4D Flow Magnetic Resonance Imaging (MRI) is the state‐of‐the‐art technique to comprehensively measure the complex spatio‐temporal and multidirectional patterns of blood flow. However, it is subject to artifacts such as noise and aliasing, which due to the 3D and dynamic structure is difficult to detect in clinical practice. In this work, a new mathematical and computational model to determine the quality of 4D Flow MRI is presented. The model is derived by assuming the true velocity satisfies the incompressible Navier–Stokes equations and that can be decomposed by the measurements u→meas plus an extra field w→ . Therefore, a non‐linear problem with w→ as unknown arises, which serves as a measure of data quality. A stabilized finite element formulation tailored to this problem is proposed and analyzed. Then, extensive numerical examples—using synthetic 4D Flow MRI data as well as real measurements on experimental phantom and subjects—illustrate the ability to use w→ for assessing the quality of 4D Flow MRI measurements over space and time.

1 citations

DissertationDOI
01 Jan 2016
TL;DR: Methods for the quantification of turbulent kinetic energy and blood flow velocities have been developed, validated and their clinical feasibility shown and errors due to assumptions underlying the signal models could be quantified.
Abstract: 4 validated for k-t SENSE and k-t PCA using pseudoreplica analysis. It is shown that temporal fidelity was better preserved in k-t PCA than in k-t SENSE. The gxf-metric is furthermore valuable to compare k-t methods with frame-byframe parallel imaging reconstruction techniques. The model for estimating TKE using Phase-Contrast MRI is based on a number of assumptions. In particular, a Gaussian distribution of velocities in a voxel is assumed and the spatial resolution is required to be sufficient to correctly distinguish between turbulence and velocity gradients of the mean field. Simulated Phase-Contrast MRI measurements based on Particle Tracking Velocimetry data revealed errors <15% for TKE estimation with scan settings feasible in clinical practice. In contrast, viscous losses, which have been proposed as an alternative marker of energy loss, were found to be significantly underestimated at clinically feasible spatial resolutions. In a study of 51 patients and 10 healthy volunteers the feasibility of TKE quantification in clinical routine is demonstrated. The study allowed for a direct comparison of TKE levels to Doppler echocardiography derived measures. TKE values were significantly elevated in patients with aortic dilatation and/or bicuspid aortic valves compared to patients with normal geometries. This distinction between the groups was not possible using Doppler echocardiography and promises additional diagnostic information. In conclusion, in this thesis methods for the quantification of turbulent kinetic energy and blood flow velocities have been developed, validated and their clinical feasibility shown. Errors due to assumptions underlying the signal models could be quantified and allowed for conclusions regarding spatial and temporal resolution. A larger study in patients revealed potential additional diagnostic information compared to Doppler echocardiography.

1 citations

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TL;DR: The results suggest that the differences in pressure wave forms are due to differences in reflections in the arterial tree and not secondary to Differences in cardiac function.
Abstract: SUMMARYThe relationship between the shape of the ascending aortic pressure wave form and aortic input impedance was studied in 18 patients who underwent elective cardiac catheterization but in whom no heart disease was found. Ascending aortic flow velocity and pressure were simultaneously recorded from a multisensor catheter with an electromagnetic velocity probe and a pressure sensor mounted at the same location. Another pressure sensor at the catheter tip provided left ventricular pressure or a second aortic pressure to determine pulse-wave velocity. Fick cardiac outputs were used to scale the velocity signal to instantaneous volumetric flow. Input impedance was calculated from 10 harmonics of aortic pressure and flow. For each patient, impedance moduli and phases from a minimum of 15 beats during a steady state were averaged. Peripheral resistance was 1137 ± 39 dyn-sec-cm-5 (± SEM) and characteristic impedance was 47 ±4 dyn-sec-cm-5; pulsewave velocity was 6.68 ± 0.32 m-sec-1. In all patients, a well-defined systolic inflection point divided the aortic pressure wave form into an early and late systolic phase. The patients were classified into three groups: group A (n = 7) patients whose late systolic pressure exceeded early systolic pressure; group B (n = 7) patients whose early and late systolic pressures were nearly equal; group C (n = 4) - patients whose early systolic pressure exceeded late systolic pressure. Group A and B patients all demonstrated oscillations of the impedance moduli about the characteristic impedance. Group C patients demonstrated flatter impedance spectra. Thus, a larger secondary rise in pressure was associated with a more oscillatory impedance spectrum. These results suggest that the differences in pressure wave forms are due to differences in reflections in the arterial tree and not secondary to differences in cardiac function. Using pulse-wave velocity, the \"effective\" reflection site distance was determined from both pressure (48 cm) and impedance (44 cm) data, implying that the region of the terminal abdominal aorta acts as the major reflection site in normal adult man.

1,013 citations


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Frequently Asked Questions (1)
Q1. What have the authors contributed in "A robust comparison approach of velocity data between mri and cfd based on divergence-free space projection" ?

In this paper, a flow regularization approach is proposed to provide a ground truth and to perform robust, mathematically reasonable comparisons between CFD and MRI.