Open Journal of Medical Imaging, 2012, 2, 23-28 Published Online March 2012 (
Computational Fluid Dynamics and Its Impact on Flow
Measurements Using Phase-Contrast MR-Angiograph y
Claus Kiefer*, Frauke Kellner-Weldon
Support Center for Advanced Neuroimaging, Institute of Diagnostic and Interventional Neuroradiology,
University of Bern (Inselspital), Bern, Switzerland
Email: {*claus.kiefer, frauke.kellner-weldon}
Received December 14, 2011; revised December 19, 2011; accepted January 6, 2012
Rationale and Objectives: Computational fluid dynamic (CFD) simulations are discussed with respect to their poten-
tial for quality assurance of flow quantification using commercial software for the evaluation of magnetic resonance
phase contrast angiography (PCA) data. Materials and Methods: Magnetic resonance phase contrast angiography data
was evaluated with the Nova software. CFD simulations were performed on that part of the vessel system where the
flow behavior was unexpected or non-reliable. The CFD simulations were performed with in-house written software.
Results: The numerical CFD calculations demonstrated that under reasonable bou ndary co nditions, defined b y the PCA
velocity values, the flow behavior within the critical parts of the vessel system can be correctly reproduced. Conclusion:
CFD simulations are an important ex tension to commercial flow quantification tools with regard to quality assurance.
Keywords: CFD; Flow; Phase Contrast; Angiography; Quality Assurance
1. Introduction
Magnetic resonance phase contrast angiography mean-
while is an established technique to measure the velocity
and the direction of flow [1]. The major drawbacks of
this approach are often related to misaligned encoding
gradients (venc), a too low SNR or high susceptibility
gradients nearby (small) vessels. In each case the relative
phase are not unique and the absolute velocity values are
wrong. At this point the co rrect answer to the flow in the
critical regions can only be derived on the base of physi-
cal models and computational flow dynamics (CFD).
Almost all CFD-problems are associated with the solu-
tion of the Navier-Stokes equations [2,3], named after
Claude-Louis Navier and George Gabriel Stokes, which
describe the motion of fluid substances. The calculations
were performed on incompressible fluids within human
blood vessels. The idea is to selectively model the rele-
vant vessel system and to include the known velocity
values from the PCA measurements within the big ves-
sels as boundary conditions. These velocity conditions
are sufficient in most cases because the velocities tune in
to the given pressure conditions. The intrinsic symmetry
of the vessels furthermore can be used to reduce the di-
mensionality of the problem from three to two, which is
necessary to use the software on commercial computers
with limited working memory and CPU power—on ly the
relative orientation of the v essels and their diameters (e.g.
due to ste noses) have to be consi dered.
The spectrum of potential applications span the simu-
lation of pressure and force distributions near the neck of
aneurysms, the velocity and direction of flow within com-
plex vessel systems with or without stenoses and pres-
sure and vorticity distributions near atherosclerotic pla-
In this technical note we will concentrate on a few
representative examples in order to demonstrate the im-
portance of quality assurance in addition to commercial
flow evaluation software.
2. Methods
2.1. Flow Measurement
Magnetic resonance phase contrast angiography meas-
urements of flow were performed on a 3 Tesla scanner.
The FLASH sequence parameters were as follows: Base
resolution 256, Phase resolution 60%, Flip angle 25 deg,
TR 113.20 ms, TE 4.66 ms, Voxel size: 0.9 × 0.5 × 4.0
mm, TA: 1:06 min, 1st Signal/Mode Pulse/Retro, calcu-
lated phases 12, segments 5, slices 1. Each slice was
planned on the base of a time-of-flight (TOF) measure-
ment using the Nova software (,
which guarantees the exact orthogonal positioning rela-
tive to the course of the vessel. The flow measurements
were performed for a number of selected vessels that
*Correspondi n g a uthor.
opyright © 2012 SciRes. OJMI
define a part of the simulated vessel system. Figure 1
shows a typical Nova rendered 3D-presentation of the
vessel system of a TOF measurement. The velocities are
measured at the positions given by the small slice car-
toons perpendicularly oriented to the vessel. The most
important aspects which are essential for the choice of
these locations are the distance to bifurcations and ste-
noses because often turbulences may occur in their im-
mediate vicinity. Nova currently provides no option for
the separate presentation of systolic and diastolic phases
but merely shows the mean volumetric flow rate values
on the diagrams (Figure 1(b)). The mean velocity values,
necessary for the simulations, are not shown on the Nova
diagram, but are stored in a text file.
2.2. Preparation of CFD
2D-projections from individual TOF maximum intensity
projection maps are co-registered to a template (see e.g.
sions/Circle_of_Willis, binar y mask in Figure 2(a)) while
preserving the individual properties (e.g. stenoses, mal-
formations, relative orientation). The final mask was used
to define the coordinates and nodes essential for the
mesh2d Matlab program, which calculates the nodes and
elements (Figure 2(b)) necessary for the numerical simu-
2.3. Numerical Algorithms
The numerical code for the flow simulations is an in-house
development, written in Matlab. The program simulates
the motion of incompressible fluids via numerical solu-
tions of the 2D, unsteady Navier-Stokes equations. The
CFD-program is a vertex centered Finite Volume (FV)
code that uses the median dual mesh to form the control
volumes about each vertex by connecting the centroids of
each cell to their edge midpoints. The time and spatial
linearizations are dealt with separately. Fully 2nd order
piecewise linear reconstructions are used to develop
sparse operators, which approximate the continuous dif-
ferential terms. These methods can be found in detail in
[4,5] described by terms like Green’s theorem and its
application to face averaged gradients (Laplacian, un-
weighted least squares) or upwind piece-wise linear ex-
trapolation with the HQUICK slope limiter (non-linear
The time integration of the equations is based on the
well-known fractional step method [5,6]. The non-staggered
pressure/velocity arrangement was shown to support the
common mesh scale pressure oscillation and hence the
standard 2nd order incremental pressure correction me-
thod [7,8] was modified to incorpor ate Rhie-Chow stabi-
lization [9]. This reduced the order of the time integra-
tion to 1st/2nd. Unlike many other Navier-Stokes solvers
the CFD-program makes use of a complete LU factorize-
tion for the Poisson pressure equation. Instead of using
standard iterative approaches like preconditioned bicon-
jugate gradients stabilized method (bicgstab ( M at lab)) we
decided on including the fill reducing UMFPACK algo-
rithm [10], an approach which was shown to be far more
efficient in this case. The momentum equations are solved
in a semi-implicit manner. The non-linear terms are eva-
luated explicitly via a 2nd order TVD Runge-Kutta me-
thod [11]. In order to solve the linear systems that arise
due to the implicit treatment of the viscous terms, the
bicgstab method was favored. Minimization of errors
associated with the numerical solution was managed ac-
cording to the concepts of Roache et al. [12].
Figure 1. A typical 3D-plot (a) of a NOVA flow planning
system, illustrating the rendered vessel system according to
the TOF measurement. The velocities are measured at the
positions given as small slice cartoons perpendicularly ori-
ented to the vessel (here the right ACA). In (b) one possible
schematic Nova diagram of a part of the circle-of-willis
system is shown, which contains information about the ves-
sel names, the direction of flow (green arrow) and volumet-
ric flow rate values (ml/min).
Copyright © 2012 SciRes. OJMI
2.4. CPU Parameters
The CFD simulations were performed on a personal com-
puter with Intel® Xeon processor, 2.4 GHz, 12 GB RAM,
4 CPUs, NVIDIA Quadro FX 580, Video RAM 512 MB,
GPU frequency 450 MHz, Ubuntu 10.04.
2.5. Program Features
The mesh2d program of Matlab was used for the 2D
mesh (triangles) generation. This type of mesh allows the
automatic treatment of complex geometries. The mesh2d
algorithm optimizes the mesh quality which is important
to guarantee correct results. The CFD program solves the
Navier-Stokes equations for the velocity and pressure
fields of incompressible fluids. The time step size and the
stability of the integration are adjustable. The velocity
values measured by PCA were used as the boundary
conditions (at the nodes). The program includes a module
that calculates the [x,y] forces applied to a boundary due
to pressure and skin friction effects and can be used in
this example to calculate the lift and drag forces on an
aneurysm neck.
2.6. Patient
High grade stenosis of the left ICA and occlusion of the
right ICA.
3. Results
The total time needed for the preparation and simulation
is in the range of 3 min which is actually feasible for
emergency cases. The computer, however, should at least
have 4 GB working memory and processors of the new
generation (Pentium IV, Xeon) with at least 2.4 GHz.
Figure 2(a) shows a typical bitmap of the circle-of-
willis which is then transformed to a 2D-mesh with
nodes and elements (Figure 2(b)) using the mesh2d Mat-
lab program. As a typical example for the application of
the CFD-simulations, the data of a patient before and
after stenting of the left ICA was used. The velocity val-
ues at the boundaries were given by the PCA measure-
ments. The simulations of the corresponding vessel sys-
tem before (Figures 3(a)-(c)) and after stenting (Figures
4(a)-(c)) confirm the change in direction within the
LPCOM vessel (Figure 4(b)) with respect to the situa-
tion before stenting (Figure 3(b)): it is opposite to the
flow within the RPCOM (Figures 3(c) and 4(c)). The as-
sociated Nova diagrams are shown in Figure 3(d) (be-
fore stent) and Figure 4(d) (after stent). The measured
mean velocity values are shown in Table 1.
In order to demonstrate the potential of the CFD
simulation technique, an example for such an (artificial)
aneurysm is given in Figure 5, for which the velocity
(Figure 5(a)) and pressure distributions (Figure 5(b))
Figure 2. The bitmap of the circle-of-willis vessel system (a)
and the mesh nodes and elements for a part of it (b), calcu-
lated by the mesh2d Matlab program. The mesh serves as
the base for the CFD simulations. Potential boundaries
which may have non-zero in- or outflow are marked with
red cycles.
Copyright © 2012 SciRes. OJMI
Figure 3. Simulated CFD data of the circle-of-willis vessel
system, shown in Figure 2, if the boundary conditions are
given by the velocity values of a real PCA measurement be-
fore stenting: note the equal flow directions in the LPCOM
(b) and RPCOM (c) which coincides with the results of the
Nova data (d).
Figure 4. Simulated CFD data of the circle-of-willis vessel
system, shown in Figure 2, if the boundary conditions are
given by the velocity values of a real PCA measurement
after stenting: note the opposite flow directions in the
LPCOM (b) and RPCOM (c) and the change in direction
within the LPCOM vessel (Figure 4(b)) with respect to the
situation before stenting (Figure 3(b)) which coincides with
the results of the Nova data (d).
Copyright © 2012 SciRes. OJMI
Figure 5. CFD simulations of an artificial aneurysm. (a) The
pressure distribution (b) and the forces (in dependence of
time [sec]) (c) at the marked position (red arrow in a)) de-
monstrate, that under special conditions the local load can
achieve critical values, followed by a disrupture of the an-
eurysm neck.
Table 1. Mean velocities within selected vessels in cm/sec.
Before42.6113.357.93 42.77 26.3230.47
Stent 55.7029.2910.71 27.62 30.8829.62
are shown if physiologically reasonable boundary condi-
tions are presumed. At the border of the area, indicated
by the red arrow, the forces (Figure 5(c)) in x—(blue)
and y—direction (red) are shown in dependence of time.
The absolute values are not relevant in this case but
merely the relative pressure and force enhancements at
the neck of the aneurysm which finally may lead to a
4. Discussion
CFD simulations were used to verify unexpected flow
patterns that occur after stenting in the LPCOM vessel
and to demonstrate the critical pressure and force distri-
butions at the neck of an artificial aneurysm. The data
shows that CFD simulations are feasible on comer- cially
available PCs if the simulations are restricted to the es-
sential vessel systems and if intrinsic symmetries are
used to reduce the dimensionality of the problem. CFD
data may also provide clinically useful information re-
garding post-steno tic vessel wall alteration and progress-
sion of atherosclerotic disease. More generally, wall shear
stress is believed to play an important role in the evolu-
tion of atherosclerosis and other arterial diseases, such as
At the current stage only vorticity but no turbulence
model is implemented. Further CFD program options are
different boundary conditions such as to extrapolate the
boundary values of the pressure from the interior nodes or
just to fix the value of the pressure at the outflow to zero.
The step from the TOF-MIP to a two-dimensional bit-
map is currently not fully automated but necessitates ma-
nual intervention by an experienced radiologist.
In future developments, a separate evaluation of dif-
ferent heart phases will be possible as well. This deficit,
however, does not lower the impact of the results, espe-
cially because the flow directions before and after stent-
ing do not dep end on the heart phase.
In summarize, the intention of this work was to dem-
onstrate, that the simulation of model based fluid dyna-
mics provides important clinical information which is
currently not available in most commercial software
packages for flow quantification.
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