Universität Stuttgart - Institut für Wasser und
Umweltsystemmodellierung
Lehrstuhl für Hydromechanik und
Hydrosystemmodellierung
Prof. Dr.-Ing. Rainer Helmig
Diplomarbeit
Inspecting endovascular aneurysm
treatments with porous medium flow
simulations and the use of a statistical
framework
Submitted by
Sebastian Warmbrunn
Matrikelnummer 2425845
Stuttgart, October 31th, 2012
Examiner:
Prof. Dr.-Ing. R. Helmig
Tutors:
Dipl.-Ing. K. Baber
Dr. Kent-Andre Mardal
Øyvind Evju, M.Sc.
Preface
This thesis was written during the period of May 2012 - November 1st 2012.
It is the final project for the degree program “Umweltschutztechnik” at the
University of Stuttgart.
Most parts of this thesis were written at the Simula Research Laboratory,
Oslo. I am very grateful for the possibility that was given to me, to work
abroad in such a great working environment, with so many open-minded
people.
First of all I want to thank Prof. Rainer Helmig, who gave me this opportunity. I want to thank Kent-Andre Mardal, who was my supervisor in
Oslo. He was always very helpful and supportive during my stay at Simula.
Thanks to Øyvind Evju, who was my second supervisor in Oslo helped me
when needed.
Another thanks goes to Katherina Baber. She was my supervisor in Germany and helped me, especially during the last month of finalizing my work.
A five year long study now comes to an end and I would like to thank my
friends and the people who were surrounding me during this period of my
life.
Last, but not the least, I want to thank my parents and my sisters for their
love and support.
Contents
1 Introduction
10
2 Medical Background
12
2.1 The clinical picture . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 The treatment methods . . . . . . . . . . . . . . . . . . . . . 13
3 Physical Model
3.1 Simplifying the fluid . . . . . . . . . . . . . . . . . . . . . . .
3.2 Simplifying the structure and boundaries . . . . . . . . . . . .
3.3 Generation of the aneurysm model . . . . . . . . . . . . . . .
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4 Mathematical Model
4.1 Mathematical approach for the free flow . . . . . .
4.2 Mathematical approach for the porous medium flow
4.3 Coupling of the approaches . . . . . . . . . . . . . .
4.4 The surrogate management framework . . . . . . .
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5 Numerical Model
5.1 The finite element method . . . . .
5.2 The incremental pressure correction
5.3 Implementation in FEniCS . . . . .
5.4 Verification of the implementation .
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scheme
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6 Results
6.1 Two dimensional flow field . . . . . . . . . .
6.2 Aneurysm geometry - three design variables
6.3 Aneurysm geometry - stent . . . . . . . . . .
6.4 Mesh convergence . . . . . . . . . . . . . . .
6.5 Discussion . . . . . . . . . . . . . . . . . . .
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List of abbreviations
CT computed tomography
FEM finite element method
FSI fluid structure interaction
GDC Guglielmi detachable coil
IPCS incremental pressure correction scheme
LHS latin hypercube sampling
PDE partial differential equation
REA represatentive elementary area
SMF surrogate management framework
vmtk vascular modeling toolkit
WSS wall shear stress
6
List of Figures
1
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4
5
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7
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9
10
11
12
13
14
15
Different treatment methods for the endovascular treatment
of an aneurysm. Top left: many coils, lower left: less coils,
top right: stent, lower right: flow diverter . . . . . . . . . . .
The two considered types of a cerebral aneurysm. A sidewall
aneurysm on the left side and a bifurcation aneurysm on the
right side. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A very simplified scheme of the Circle of Willis. The arrows
represent the blood flow from the heart. . . . . . . . . . . .
Illustration of the procedure to define the representative elementary area . . . . . . . . . . . . . . . . . . . . . . . . . .
The porosity of the coil configuration with a shrinking area.
The user can drag the three planes along their normals to
investigate the cerebral vessel system. The structures with a
higher contrast are bones, the ones with a lower contrast are
vessels. The red encircled part is an aneurysm. . . . . . . . .
The fast marching method with the start point on the left
image, the end point on the center image and the result on
the right image. . . . . . . . . . . . . . . . . . . . . . . . . .
The two points for the colliding front method on the top image,
the resulting vessel structure on the bottom image. . . . . .
A smoothed aneurysm geometry with added flow extensions.
The aneurysm geometry with the size information for the generation of the mesh cells. Red is a small mesh size and dark
blue a coarse mesh size. Those parameters can be scaled. . .
The aneurysm geometry with a small sphere, representing a
flow diverter and a bigger sphere representing a coiling. The
white line marks the degree of freedom. . . . . . . . . . . . .
Illustrations of the surrogate management framework. On the
top, the initial points for the calculation of the first Kriging
surrogate are shown. The picture in the middle shows a first
mesh, with new chosen points. On the bottom, the mesh was
refined and the poll step is initiated. . . . . . . . . . . . . .
The surrogate management framework as a flow chart . . . .
The domain on which the Poisson equation is solved. The
Dirichlet condition continues on the green axes. . . . . . . .
The results for the pressure drop of the Poisson equation . .
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Illustration of the example problem . . . . . . . . . . . . . .
Pressure distributions for the two different models . . . . . .
The velocity distribution for the third surrogate management
framework (SMF) run . . . . . . . . . . . . . . . . . . . . .
Different placement possibilities for the spheres and the corresponding porous medium areas (red) . . . . . . . . . . . .
The geometry of the sidewall aneurysm . . . . . . . . . . . .
Streamlines for the three results of the SMF runs and without
coil placement . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity inside the aneurysm for the three results of the SMF
runs and without coil placement . . . . . . . . . . . . . . . .
The geometry of the bifurcation aneurysm . . . . . . . . . .
Streamlines for the three results of the SMF runs and without
coil placement . . . . . . . . . . . . . . . . . . . . . . . . . .
Velocity inside the aneurysm. No coiling on the very left,
following run01 - run03 from the left to the right . . . . . . .
The sidewall aneurysm geometry with a spherical shaped disk,
representing the aneurysm. . . . . . . . . . . . . . . . . . . .
The change of the volume Vkinen , used for the calculation of
the kinetic energy measure Ekin . . . . . . . . . . . . . . . .
The aneurysm with the coiled area (red) and the point of the
velocity measurement (white dot) . . . . . . . . . . . . . . .
The pressure drop from inlet to outlet with increasing mesh
resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The kinetic energy inside the aneurysm with increasing mesh
resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The velocity at a measurement point with increasing mesh
resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Tables
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The thresholds for the design variables. First row: lower
threshold. Second row: upper threshold . . . . . . . . . . . .
The results for the SMF runs of the example problem . . . .
The results for the design variables and cost functions for the
sidewall geometry . . . . . . . . . . . . . . . . . . . . . . . .
The results for the SMF runs on the bifurcation geometry .
The thresholds of the design variables for the SMF . . . . .
The comparision between stent and coils on the sidewall geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The components of the goal function for a stent and a coil
situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The components of the goal function with another Vkinen . .
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1
Introduction
In fact, the interrelation between medical science and engeneering already
exists quite a long time. The design of surgical instruments, the building of
medical devices like an ultrasound device or the inspection of material properties of tissues are all examples where engeneering experience and knowledge
from clinicians fuse together. With the constant development of new technologies, especially in the field of computer science, this interrelation grew
stronger during the past decades.
Therefore, it is not surprising that also computer simulations contribute more
and more to the understanding of medical processes.
In the context of this work, a computer simulation begins with setting up
a simplified physical model for a certain phenomenon or process in nature.
The physical model is then described by mathematical equations. Most of
the time those mathematical equations are too complex to be solved without
the help of numerical tools and fast computers.
Figure 1: Different treatment methods for the endovascular treatment of an
aneurysm. Top left: many coils, lower left: less coils, top right: stent,
lower right: flow diverter
In this case, we are interested in the blood flow through an aneurysm. Different variants of treatment are considered and compared with respect to the
developing flow. Figure 1 gives an idea of how an aneurysm and the different
treatments look like. Chapter 2 describes the clinical picture of an aneurysm
and the relevant treatment methods more detailed. The physical model is
10
discussed with all its simplifications in Chapter 3. To find a good treatment
solution for a certain aneurysm, the different variants are compared. Therefore a mathematical model to describe the flow is set up. Together with the
help of a statistical framework, a good treatment variant for the aneurysm is
expected to be found. Both the mathematical flow equations and the statistical framework are covered in Chapter 4. The equations have to be solved
with the help of numerical schemes, which are summarized in Chapter 5.
Finally the results of the executed simulations are presented in Chapter 6
and a discussion about the possible improvements of the method is given.
11
2
Medical Background
A short introduction to the medical aspects of the topic is given in this
chapter. This is not meant to be an examination of all the clinical details,
but a brief summary of the most important facts. First of all, the clinical
picture of a cerebral aneurysm is described, so that the reader gets a basic
understanding of the disease. The second part is focused on the possible
treatment methods, emphasizing endovascular treatment, as it is the main
subject of this thesis.
2.1
The clinical picture
A cerebral aneurysm is a local dilatation of an artery in the head. There are
four main artery branches in the neck, leading to the brain, two in the front
and two in the back. They fuse in the Circle of Willis (see Figure 3), from
where several branches spread into the brain, to supply it with blood. This
is also the most common region for cerebral aneurysms to occur, and will be
the area where most of our discussed aneurysms are located.
The diameters of the vessels in the Circle of Willis range from < 0.1mm up to
3mm. [22] The size of a cerebral aneurysm itself varies from such aneurysms,
which are barely distinguishable from the surrounding vessels of the Circle
of Willis, to aneurysms with diameters of 50mm and above. Figure 2 shows
the types of aneurysms, which are inspected in this study.
Cerebral aneurysms are a frequent disease. A reasonable assumption for the
average prevalence in the general population is 2% [10,20] . Once present, a
severe danger about aneurysms is the risk of rupture, associated with a subarachnoid hemmorhage.
Although it is not exactly understood why aneurysms develop and rupture,
certain risk factors like older age, atherosclerosis, hypertension and smoking
are known [10,24,11] .
Also, mechanical stresses on the blood vessel walls are considered to influence
the development of a cerbral aneurysm. So it is interesting to look at the
flow patterns, flow velocities and pressure conditions inside aneurysms, to
draw conclusions from the impact of the flow on the blood vessel wall. It is
difficult to get high resolution data for these flow informations though, which
12
Figure 2: The two considered types of a cerebral aneurysm. A sidewall aneurysm
on the left side and a bifurcation aneurysm on the right side.
is why computational models are used.
If unruptured aneurysms are found incidentally, it is not easy to decide
whether to perform a treatment or not. The risk of rupture may be very
low in various cases, depending on the size, geometry and location of the
aneurysm and the general physical condition of the patient. If an invasive
treatment seems necessary though, there are several options from which the
clinician can choose.
2.2
The treatment methods
There are two general options for the invasive treatment of a cerebral aneurysm:
craniotromy with clip ligation (clipping) and endovascular treatment. [5]
Endovascular treatment has become the method of choice for many cases.
Especially if the patient is already in a bad condition (e.g. a brain bleeding already occured), it is desireable not to impose more physiological stress
through a craniotomy.
Interventional neuroradiologist Guido Guglielmi invented the Guglielmi detachable coil (GDC), which has become the standard device for endovascular
treatment of cerebral aneurysms. Coils can be imagined as very thin, preshaped wires. In January 1991 the first surgery with GDCs was performed
and in 1995 the Food and Drug Administration approved commercial sale
for the GDC. [23] GDCs appear in a vast variety of length and cross sectional
shapes.
13
Anterior Cerebral Artery
Anterior Communicating Artery
Internal Carotid Artery
Middle Cerebral Artery
Posterior Communicating Artery
Posterior Cerebral Artery
Basilar Artery
Vertebral Artery
Figure 3: A very simplified scheme of the Circle of Willis. The arrows represent
the blood flow from the heart.
14
In the procedure of an endovascular treatment, a microcatheter is injected in
the thigh and guided up to the location of the aneurysm. With the help of
this microcatheter, the detachable coils are introduced into the lumen of the
aneurysm. In most of the cases, several coils are used to clog the aneurysm.
The coils are preshaped, such that they interlace when released into the
aneurysm and create a twine. This constellation is supposed to prevent high
velocity blood flow inside the aneurysm.
Another way to reduce the flow velocity inside the aneurysm is the use of
a stent. Stents are tube shaped meshes, which can be placed in the vessel
adjacent to the aneurysm using a microcatheter. Thereby the flow is guided
past the aneurysm.
There is also an active development of new possibilities to prevent high flow
velocities in aneurysms. Here, we also want to consider denser structures,
which could be placed at the inflow of the aneurysm (possibly with the help
of a stent). Those structures are called flow diverter in the following, unlike
in some literature, where stents are sometimes called flow diverter.
Figure 1 illustrates different treatment variants.
It is not always clear for clinicians, which treatment method is the best for a
specific aneurysm geometry. Many of the desicions for a particular treatment
method are based on experience today. In this work, several treatment variants with coils, stents and flow diverters are inspected and compared with
respect to the resulting flow field. We use flow simulations and a computational model to perform this procedure.
15
3
Physical Model
Models are by definition simplifications of the processes occuring in nature.
An investigation of the blood flow inside an aneurysm is a problem with
highly complex geometries and system properties. Therefor, we have to make
some assumptions. The descriptions of our simplifications and assumptions
are divided in two parts. First, the simplifications of the fluid are described,
followed by the simplifications of the structural parts of our model (including the vessel walls, inflow and outflow conditions and the porous medium
approach).
The final part of this section describes how the model areas of the aneurysms
were generated for the further use in our computational model.
3.1
Simplifying the fluid
Blood is a suspension of various particles (∼ 45%) — mainly red blood cells
(∼ 97%), white blood cells, and platelets — in plasma (∼ 55%). Plasma
in turn consists mostly (∼ 92%) of water. The concentration level of the
suspended particles influence the behaviour of blood. Due to the high concentration of red cells, they have the most influence on the mechanical properties of blood. [9]
As can be seen in the following sections, the blood is described as simple as
possible in our models. Many assumption in this section follow the explanations given in [9,27,26] .
Continuum approach for fluids
Fluids are aggregations of molecules, widely spaced for a gas, closely spaced
for a liquid [27] . We want to look at fluids in an averaged way, excluding
molecular movement and effects. A fixed volume of a certain size is assumed.
The interchange of molecules over the boundaries is so large, that it can be
assumed to be nearly constant. Hence, we can for example assume a density
without fluctuations for this volume, which is the mass of molecules times the
molecules per volume unit. The limiting volume for the continuum approach
is about 10−9 mm3 for all liquids.
The volumes we take into account are in the mm − cm range. That means
we can assume the continuum approach for fluids of our problems and use
averaged quantities such as the density or viscosity.
16
Incompressibilty
Liquids are mostly considered as incompressible in flow simulations. The density of water, for example, changes one percent if the pressure is increased
by a factor of 220 [27] .
Blood can be described as an incompressible fluid in our simulations as well.
kg
We assume blood to have a constant density ρ = 1060 m
3 for our simulations.
Viscosity and surface stress model
We know from experience, that fluids deform while they are in motion. This
deformation behaviour varies from fluid to fluid. A heavy oil, for example,
has a higher resistance to flow than water.
The viscosity is a measure to describe this resistance of a fluid to its movement. In general, viscosity relates the fluid strain rate to the shear stress
which is applied. For some common fluids as water, the applied shear σ is
proportional to the velocity gradient of the fluid [27] . If this linear relationship
is observed, we call the fluid a Newtonian fluid.
For a one-dimensional case, with fluid flow in the x-direction only, the proportionality can be expressed by
σ=µ
du
,
dy
(3.1)
where µ is the constant viscosity coefficient, sometimes also called dynamic
viscosity, and du
describes the velocity change in y-direction.
dy
Blood does in fact not show a Newtonian fluid behaviour. The concentration
of red blood cells, or hematocrit level, influences the flow behaviour. It is
a shear-thinning fluid, which means that it has a lower (apparent) viscosity
with an increasing rate of deformation. [9] Especially in smaller vessels like
capillaries, or if local fluid behaviour is in focus, it is important to take those
effects into account.
We are not really interested in the local behaviour of the blood flow, but
more in the system behaviour as a whole. So to keep our model as simple
as possible, we assume blood to have a Newtonian behaviour and assign a
constant value to µ.
Generalizing (3.1) to three dimensions, σ becomes a nine-component tensor
σ ∗ = µ(∇u + ∇u> )
17
(3.2)
σ ∗ describes the viscous stresses acting on a fluid that arise from motion
with velocity gradients.
In addition to that, the hydrostatic pressure contributes to the surface stresses
in the main directions
σ = −pI + µ(∇u + ∇u> )
(3.3)
where p is the scalar value of pressure at a specific position and I the identity
matrix. σ is also called the Cauchy stress tensor.
This is the final equation for the surface stresses acting in the blood flow. We
will include this tensor later in the mathematical model (see Section 4.1).
3.2
Simplifying the structure and boundaries
Boundaries
We assume rigid vessel walls and do not include the complexity of fluid structure interaction (FSI). FSI is a highly complex model approach and it would
be too time consuming to include it in this work. Also, the vessel walls
are assumed impermeable, as the interchange over the vessel wall only takes
place in capillaries.
The blood flow through the body is pulsatile. In our models, we use an averaged, stationary inflow velocity. This is probably one of the most drastic
simplifications in our model. Pulsatile flow can lead to turbulences inside the
aneurysm, which in turn leads to a high kinetic energy inside the aneurysm.
At the outflow of the aneurysm, a zero pressure condition is imposed. The
outlet profiles are placed far away from the aneurysm, such that the pressure drop in this region should not be influenced by the boundary condition.
Nevertheless it is an assumption and due to a lack of time we could not check
for every aneurysm if the distance is sufficient.
Porous media approach
Instead of modeling the geometry of every single coil which is introduced
into the aneurysm, the coils are considered a porous medium. This allows
us to describe the area occupied with the coils with a few parameters in an
averaged way. Thereby we prevent computationally demanding complex flow
fields around accurately modeled coils.
18
(a) Aan ∼
= 78.5
(b) Aan ∼
= 50.3
(c) Aan ∼
= 19.6
(d) Aan ∼
= 3.1
Figure 4: Illustration of the procedure to define the representative elementary
area
There is a variety of porous media to be found in technical applications as
well as in nature. Defining a porous medium is not an easy task and an
agreed-upon definition does not exist. However, in general one can say that
a porous medium is a solid material which includes a pore space. A good
example for a porous medium is a sponge. We argue in the following paragraphs, why it is reasonable in our case to apply the porous medium approach.
According to [3,4] , one part of a porous medium is a persistent solid phase,
called the solid matrix. In our case, the intertwisted coils represent the solid
matrix. The other part is at least one gas or fluid phase, which is present
in the spaces between the solid matrix. Those spaces are referred to as void
space. We only have the blood as a fluid phase present.
The void space should be interconnected, which is assumed to be true in our
case. There is no realistic scenario, where the coils are configured in a way
that the flow is guided only through seperated pores.
Another criterium for a material to be described as a porous medium is the
definition of a continuum. This means, that the material properties can be
described with few effective parameters. To be able to do that, a material
needs to fulfill certain geometrical properties, which are examined for coils
in the following.
We do this with an analysis of the geometrical proportions of coils and an
aneurysm for a representative case. A Python script was written for this
purpose. It includes the Polygon [19] package for Python, which adds the
ability to handle polygonal shapes in 2D.
A circle for the aneurysm is generated with smaller circles inside, representing
19
0.7
Porosity [-]
0.6
0.5
0.4
0.30
10
20
30
40
Area [mm^2]
50
60
70
Figure 5: The porosity of the coil configuration with a shrinking area.
the coils.
We want to take a look at the porosity of this configuration, which is defined
in our case as
φ=1−
Acoils
Aan
(3.4)
where φ is the porosity. Acoils and Aan is the area of the coils and the
aneurysm respectively. We take approximately the measurements from [12]
for the aneurysm lumen and radius of the coils. The radius of the aneurysm
is set to 5mm, the radius of the coils to 0.127mm and the porosity to 0.7.
The script then generates the circles necessary to reach the porosity.
We now keep the configuration of the coils constant and start shrinking Aan
stepwise. For every step we calculate the porosity, such that we can create a
graph like shown in Figure 5. We recognize that the porosity starts fluctu20
ating as we approach Aan = 0. The point where the fluctuation is starting
can be defined as the represatentive elementary area (REA). The REA is
the minimal area which can be used for the porous medium approach with
the current coil size. This means that for areas larger than the REA, we can
describe the geometry averaged and use the porosity as a suitable parameter.
This is obviously only a rough approximation, as our problems are three
dimensional. Nevertheless it shows, that the geometrical proportions of a
typical coil configuration are in the right range to be considered a porous
medium.
Another parameter, which is widely used to describe the properties of a
porous medium is the permeability. It can be imagined as the resistance of
the material against the flow. In general, the permeability is not only dependent on the porosity. There are model approaches though, which connect
the porosity with the permeability directly. We follow the capillary theory
of Kozeny and set up a linear relationship between the porosity and the
permeability, as described in Section 4.2.
21
3.3
Generation of the aneurysm model
Patient specific image data is used in our procedure, to create the geometry models of aneurysms. We obtained images from Rikshospitalet in Oslo.
The images were produced by computed tomography (CT). In this section,
we briefly describe our main procedure of using those images to create a
three dimensional model for the flow simulations. Besides that, we want to
describe how the coil area i.e. the porous medium area is defined in our
model. We used the software tools ParaView [18] and the vascular modeling
toolkit (vmtk) [25] for those tasks.
This is only a general, compact summary of the workflow. It is recommended
to visit the websites of the software packages to get more insight into the program features.
Generating the surface
The first thing we have to do is to generate a surface from the CT-images.
A surface is a representation of the vessel, and is the limiting surface of the
volumes, which are obtained from the CT-images.
The starting point for the generation is a three dimensional set of CT-images
of the patients head. Using vmtk, the user can scroll layer-wise along three
main axes and see the images of the head. This way it is possible to get
an overview of the vessel system. The images have different contrast values
for the various biological structures in the head. This is used by vmtk to
seperate the vessels from the surrounding structures (see Figure 6).
To generate a surface, we extract another image set, called a level set. The
contrast value in the level set is set to small negative values for the vessels
and small positive values for the surrounding area. The mathematical description of the interface — i.e. where the contrast value is zero — is then
the surface.
In general, we use three methods to create the level sets for different kinds
of vessel structures. The fast marching method for the aneurysm lumen, the
colliding front method for bigger vessels and the seeds method for smaller vessel segments.
22
Figure 6: The user can drag the three planes along their normals to investigate
the cerebral vessel system. The structures with a higher contrast are
bones, the ones with a lower contrast are vessels. The red encircled
part is an aneurysm.
23
Figure 7: The fast marching method with the start point on the left image, the
end point on the center image and the result on the right image.
Fast marching method
Before the algorithm of this method is starting, the user has to set contrast
value thresholds, to tell vmtk the contrast range of the desired vessel structures. In addition to that, start points and end points are chosen on the
CT-images. Every start point emmits a spherical front with the same front
velocity. When one of the fronts reaches one of the endpoints, the algorithm
is stopped. Only the areas which are inside the spheres around the start
points and have an appropriate contrast value are then considered for the
level set.
We often use this method to model the aneurysm lumen by placing one start
point in the center of the lumen. For the end point we choose the farthest
point of the lumen wall. Figure 7 shows an example, where this method
creates a nice volume for the aneurysm lumen.
24
Figure 8: The two points for the colliding front method on the top image, the
resulting vessel structure on the bottom image.
Colliding front method
Here, we can only define two points on the image and again a threshold. A
front is propagated from each point in the direction of the other point and
the parts of the image within the threshold are recognized by the algorithm.
With a good choice for the thresholds, we often can use this method for big
vessels, like shown in Figure 8. Smaller vessels, especially if the contrast
value fluctuates a lot, are hard to model with this method.
Seeds method
This is the method of choice when it comes to modeling smaller vessel segments. We can create many points on the image and every point is modeled
as a part of the vessel segment independent of a threshold. This way allows
us to manually influence the model area.
25
Figure 9: A smoothed aneurysm geometry with added flow extensions.
Preparing the surface for meshing
The generated surface might need some smoothing and refinement, depending on the volume mesh, that we want to create from it later. For most of our
models, we use the recommended smoothing settings from the vmtk homepage. The different mesh resolution are generated from the same surface, so
that they are comparable, regarding this aspect.
Often, it is necessary to add pipe-shaped flow extensions to extend the model
area, if the artery system posterior to the aneurysm has too many bifurcations, which would lead to too much complexity for our system. The extension is necessary to be able to impose a zero pressure boundary condition
at the outlet, which does not influence the pressure drop at the important
locations e.g. the aneurysm neck. Sometimes, it is also desireable to extend
the inflow, such that the flow field at the inflow boundary is fully developed before it reaches the aneurysm. Figure 9 shows a smoothed aneurysm
geometry with added flow extensions.
Meshing the surface
The equations which describe the flow are solved in a discrete way. Hence, a
volume mesh is necessary to perform the flow simulations. Our meshes consist of irregular thetrahedal cells. We want to refine the mesh at locations
where the flow is expected to be complex. To achieve that, vmtk provides
a script to place spheres on the surface we created earlier. With adjustable
26
Figure 10: The aneurysm geometry with the size information for the generation
of the mesh cells. Red is a small mesh size and dark blue a coarse
mesh size. Those parameters can be scaled.
parameters, it is possible to refine the mesh proportional to the distance from
the spheres (see Figure 10).
From the generated surface with the refinement information we can create
the mesh and convert it to the desired format.
Placing the porous medium area
The porous medium area is in our case the representation of the coils in the
aneurysm. It is hard, even for trained clinicians, to decide where to place
coils. A case-specific evaluation has to be performed for every patient.
In Section 4.4, we present an automated method to find the placement of
the porous medium area i.e. the coils. This could be helpful in the future to
guide clinicians in their decision making.
We use ParaView to place a sphere over the aneurysm lumen. The area inside
the sphere is later defined as the porous medium. The coordinates of the
sphere center and the radius are used in the code for the flow simulations. It
is also possible to define a smaller and less conductive sphere inside the vessel
system to model the placement of a flow diverter. Also, we want to define a
vector inside the aneurysm. The coil positioning is variable in the automated
27
Figure 11: The aneurysm geometry with a small sphere, representing a flow diverter and a bigger sphere representing a coiling. The white line
marks the degree of freedom.
process, and this will be the degree of freedom for the coils. We use the line
tool in Paraview to define the direction of the vector. Figure 11 illustrates
the placement of a coil configuration and a flow diverter configuration with
the degree of freedom represented by the line.
28
4
Mathematical Model
We apply mathematical models to describe the blood flow inside the aneurysm
and the surrounding vessels. The mathematical approach for the free flow
region is discussed in the first part of this section. For the region occupied
with the coils — thus the porous medium — a slight modification of the
free flow approach is necessary, which is covered in the second subsection.
Further we focus on how those two models are linked in a proper manner.
The final topic of this section is about the surrogate management framework.
It is a method, which uses statistical techniques and a search algorithm, to
find optimal parameters for our flow model.
4.1
Mathematical approach for the free flow
Using the principles of continuum mechanics is a common way to describe
fluid flow. Fluid properties, which may fluctuate on the molecular scale, are
averaged over a large number of molecules. The fluid flow is described with
the help of those averaged fluid properties (like density or viscosity) at a
fixed location. This approach of looking at a certain spatial location and
observing the changes in the fluid flow there, is also called Eulerian Method.
The set of equations, that are used to simulate the flow problems are called
the Navier-Stokes Equations. For isothermal problems, they are based on the
conservation of mass and momentum. A compact derivation of the equations
is given in the following, using [1,7] as a source.
Conservation of mass
We look at a fixed, infinitesimal small volume V of a fluid, with the boundary
∂V . To describe the total inflow and outflow across ∂V , we write
Z
(ρu) · n ds = 0,
(4.1)
∂V
where u is the velocity and n the outward pointing normal vector. Like
shown in Section 3.1, a constant density ρ for blood is used here. Applying
Gauss’s Theorem, equation 4.1 turns into
Z
Z
∇ · (ρu) dx = ρ∇ · u dx = 0.
(4.2)
V
V
29
Because infinitesimal small volumes are assumed, the integrand has to become zero. Thus, we get the continuity equation for incompressible fluids
∇ · u = 0.
(4.3)
Conservation of momentum
Consider Newton’s Second Law: ma = F . In this form the law states, that
the force F causes an acceleration a on a point mass m. Since we want to
look at the driving forces acting on the fluid which penetrates the same fixed
volume V as before, the law is adapted, such that
Z
d
ρu dx = F
(4.4)
dt V
Applying Reynolds transport theorem for the left hand side of the equation
Z
Z
∂
(ρu) dx +
(ρu)u · n ds = F
(4.5)
V ∂t
∂V
and using Gauss’s Theorem reveals
Z
∂u
+ ρ∇ · (uu) dx = F .
(4.6)
ρ
V ∂t
With the continuity equation, a reformulation of the second term of the
integrand is possible.
∇ · (uu) = u · ∇u + (∇ · u)u = u · ∇u,
Z
(4.7)
∂u
+ ρu · ∇u dx = F .
(4.8)
V ∂t
Now we want to take a closer look at the forces on the right hand side of the
equation. There are two kinds of forces acting on the fluid in motion, namely
volume forces and surface forces. Volume forces — like gravity — are here
denoted by the vector f and act on every mass unit of the fluid. The surface
force vector consists of the scalar product between the stress tensor σ and
the normal vector n. Equation (4.8) changes to
Z
Z
Z
∂u
ρ
+ ρu · ∇u dx =
σ · n ds + ρf dx.
(4.9)
V ∂t
∂V
V
ρ
30
We apply Gauss’s Theorem on the stress term
Z
Z
∂u
ρ
+ ρu · ∇u dx = ∇ · σ + ρf dx.
(4.10)
V ∂t
V
Once again this can be written without integrals, as infinitesimal small volumes are assumed. Also, we want to divide by ρ, and obtain
∂u
1
+ u · ∇u = ∇ · σ + f .
(4.11)
∂t
ρ
Following the definition in Section 3.1 we can reformulate the stress tensor
to get our final form of the momentum conversation of the Navier-Stokes
Equations for Newtonian fluids:

u





t




p
p
∂u
+ u · ∇u = −∇ + ν∆u + f . . . with

∂t
ρ
ρ





ν



f
4.2
velocity
time
pressure
(4.12)
density
kinematic viscosity
volume force
Mathematical approach for the porous medium flow
We derived the Navier-Stokes Equations in the previous subsection as follows
∂u
p
+ u · ∇u = −∇ + ν∆u + f
∂t
ρ
∇·u=0
(4.13)
This system of equations can be extended according to [12] , to describe the
flow in the coiled area. Resistance terms are introduced on the right hand
side of the momentum equation
∂u
p
νφ
φ2 CD
(4.14)
+ u · ∇u = −∇ + ν∆u + f −
u − √ u|u|
∂t
ρ
K
K
where φ represents the porosity, K the permeability of the porous medium
and CD a drag factor.
31
The porosity φ is defined by the ratio of the volume occupied by the pores
to the total volume of the aneurysm. In this case we can express this by
Vcoils
.
(4.15)
Van
The permeability K is related to the conductivity of a porous material. Approximating the porous medium as a solid material with straight parallel
tubes of the same diameter as the pores, we can write
φ=1−
φ3
.
(4.16)
cS 2
The Kozeny parameter c is dependend on how the porous medium is modeled.
For the chosen straight tubes c = 2.
S is the ratio of the surface area of the pores to the total volume of the
aneurysm. The surface area of the pores is assumed to equal the surface area
of the coils, hence
K=
Acoils
.
(4.17)
Van
The drag factor CD is related to the local Reynolds number inside the aneurysm

d
pore diameter




Uav averaged velocity in aneurysm
dUav ρ
(4.18)
. . . with
Re =
ρ
µ
density



µ
dynamic viscosity
S=
and can be taken from standard diagrams. [12]
Knowing the geometry of the coils which are introduced into the aneurysm,
we can now describe the porous medium and the corresponding flow equation.
4.3
Coupling of the approaches
As the equations for the two subsystems described above are of the same
mathematical order, a real coupling condition at the interface is not necessary. The pressure and velocity is assumed to be continous at the interface.
By setting φ = 1 and K = ”∞” the equations for the free flow region are
recovered.
32
This is used to easily model the porous medium. Later on, it is only necessary to mark a region in the parts of the model area, where the porous
medium equation is applied. For the implementation in the code this means
that only a boolean switch is neeeded to either consider the resistance terms
in the momentum equation, or not.
4.4
The surrogate management framework
The mathematical model to this point covers the description of the flow processes inside the aneurysm for one predefined coil configuration. Our goal is
to figure out what is a good coil setting for one particular aneurysm geometry. That is why we vary the properties of the coils with parameters of our
choice and compare the corresponding simulations.
The flow simulations can be very time-consuming though, depending on the
size of the mesh and the complexity of the flow pattern. To be able to make
a clever decision for the parameters we use a SMF, introduced in [15,21] , which
is described in the following on the basis of our particular problem.
First of all, some kind of measure is necessary to compare the numerous runs
of simulations. This measure has to be dependent on the setting of the coils.
The coil location, the size of the coiled area and its permeability are possible
parameters to describe the properties of the coil set-up. Those parameters
are called design variables in the following.
We now introduce the cost function J(x) and want to
minimize
subject to
J(x)
x,
(4.19)
where x represents the the design variables. The cost function J(x) can, for
instance, depend on the kinetic energy inside the aneurysm and a pressure
drop from the inlet to the outlet of the model area. The wall shear stress
inside the aneurysm or the velocity at one critical point could be alternatives.
J(x) is computed for every flow simulation run with one specific set of design
variables x.
To start the SMF algorithm, initial simulations have to be performed for
different design variables. The design variables of the inital simulations are
33
found by a method called latin hypercube sampling (LHS). This method is
generating a random sample of the space of our design variables [14] .
Latin Hypercube Sampling
Say we want to run k initial simulations and have two design variables. Each
of the dimension is divided into k non-overlapping intervals of the same size.
From each interval in each dimension, a point is sampled randomly and finally the points from the two dimensions are paired randomly, so we get k
pairs of design variables and can run our k simulations.
From the results of the initial runs, we get a discrete data set of the cost
function, from which a surrogate model is generated using Kriging. We give
a basic explanation of this procedure based on [2] .
The surrogate model
Kriging is a statistical function approximation method. It is very popular in
the field of geostatistics and was developed for the purpose of mining [2] . The
method uses spatial correlation functions to estimate unknown values from
a known data set.
∗
J (x) =
n
X
λi J(xi )
(4.20)
i=1
J ∗ (x) is the unknown value at the desired location x, which is to be found
through the linear combination of the known values J(xi ) weighted by λi .
As there are infinitely possibilities to weight the known values, we restrict
them by some conditions. The expected value for J(x) is assumed constant
for the domain D of the design variables
E[J(x)] = m ∀x ∈ D
(4.21)
.
From this we can write
∗
E[J (x)] =
n
X
λi E[J(xi )] = m,
i=1
so for the weights this means
34
(4.22)
n
X
λi = 1.
(4.23)
i=1
Further, we want to minimize the estimation variance σ 2 (x) which is the
variance between the real value J(x) and the estimated value J ∗ (x). This
variance can be reformulated with the help of the coviariance function C(h)
where h is the distance between two locations.
σ 2 (x) = Var[J(x) − J ∗ (x)] =
n
X
∗
2
2
E[(J(x) − J (x)) ] = E (J(x) −
λi J(xi ))
i=1
n X
n
n
X
X
2
= E J(x) +
λi λj J(xi )J(xj ) − 2
λi J(xi )J(x)
= C(0) +
i=1 j=1
n
n
XX
n
X
i=1 j=1
i=1
(4.24)
i=1
λi λj C(xi − xj ) − 2
λi C(xi − x)
Our goal is to minimize σ 2 (x) with the constraint from equation 4.23 and
thereby finding the weights λi . The Lagrange multiplier ζ is introduced for
this purpose and
2
σ (x) − 2ζ
X
n
λi − 1
(4.25)
i=1
is minimized. The partial derivatives of 4.25 with respect to the weights λi
and the Lagrange multiplier ζ are set equal to zero, to find the minimum.
This results in the linear equation system
n
X
λj C(xi − xj ) − ζ = C(xi − x) i = 1, . . . , n
j=1
n
X
(4.26)
λj = 1
j=1
.
Solving this system, we get the weights and can calculate the estimated value
at the desired location x with the help of equation 4.20.
35
Using a Kriging algorithm, the space of the design variables can be explored,
and points which are likely to improve the cost function J(x) can be selected.
This is further on called the SEARCH of the SMF. All the points, where J(x)
is evaluated are restricted to lie on a mesh. After the evaluation, the surrogate is updated and the mesh for the evaluation can be refined. The first
three pictures on Figure 12 illustrate the SEARCH step for two design variables.
The SEARCH step provides means for local and global exploration of the parameter space, but is not strictly required for convergence [15] .
The exploratory advantages of the surrogate model are combined with a
method that garantuees convergence.
The poll
In the poll step, or simply POLL, the neighbouring points of the current best
found solution by the SEARCH are evaluated on the mesh. Therefore, basis
vectors are generated. We want to evaluate n + 1 points, where n is the
dimension of the design variable space. For example, in a two dimensional
space we take the basis vectors (1, 0), (0, 1) as the first two searching directions and (−1, −1) as the negative linear combination of both. The last
picture on Figure 12 shows the POLL step for a two dimensional case.
The framework
We want to summarize the framework with Figure 13. The figure was generated from the explanation of the SEARCH and the POLL step in [15] . Mk is the
mesh at iteration step k, Tk is a set of points on Mk and xk is the current
best point of design variables. κk is the set of poll points around xk .
36
permeability
permeability
position
permeability
position
position
Figure 12: Illustrations of the surrogate management framework. On the top,
the initial points for the calculation of the first Kriging surrogate
are shown. The picture in the middle shows a first mesh, with new
chosen points. On the bottom, the mesh was refined and the poll step
is initiated.
37
Find initial points with
LHS
Calculate J(x) for initial
points
Generate / update
surrogate
Identify Tk on Mk
and evaluate J(x) ∀x ∈ Tk ⊂ Mk
Yes
Increment k
SEARCH
J(x) < J(xk )?
No, increment k
Search for J(xpoll ) < J(xk )
with xpoll ∈ κk
POLL
Yes
Increment k
J(xpoll ) < J(xk )?
No
No
Increment k
Converged?
Yes
End
Figure 13: The surrogate management framework as a flow chart
38
5
Numerical Model
In this section, we want to outline the numerical schemes which were used
to discretize the continous partial differential equation (PDE) system, which
describes our flow problems (see Sections 4.1 - 4.3). This discretization is
necessary, as the governing equations can not be solved analytically. We first
focus on the methodology of the finite element method and later on discuss
the discretization of the continous governing equations with the incremental
pressure correction scheme (IPCS). For a more detailed discussion, especially
on the implementation of those schemes in FEniCS, see [13] , which was also
used as the main source for the following explanations.
5.1
The finite element method
We use the finite element method (FEM) to approximate the solutions of
the Navier-Stokes Equations in space. The FEM is a broadly used method
to solve all kinds of differential equations in various fields of science and engineering.
To keep things simple and focus on the method itself, we want to explain it
with an easier equation based on the groundwater flow equation
∂h
= −∇ · (−K∇h) + f.
(5.1)
∂t
This PDE solves for the hydraulic head h. Ss is the storage ability of the soil
in the groundwater aquifer, K is the permeability tensor and f is a source
or sink term.
We assume a stationary problem, ∂h/∂t = 0. The soil has isotropic properties
and is uniform over the domain. Hence, the permeability tensor K becomes
a scalar constant k and equation 5.1 can be reformulated as
Ss
−k∆h = f.
(5.2)
For k = 1 the Poisson’s equation is recovered. It is a partial differential
equation of second order and it was published in 1813 by Siméon Denis Poisson [13] . It is applied to model many physical problems.
39
While discussing the FEM, we provide small snippets of sample Python-code
which can be used to solve the problem with the help of FEniCS.
The system we want to solve with suitable boundary conditions may be
written as follows:
−∆h = f
h = h0
−∂n h = g
in Ω,
on ΓD ⊂ ∂Ω,
on ΓN ⊂ ∂Ω.
(5.3)
In this set of equations, h = h(x) ∈ Ω ⊂ Rd represents the pressure head field,
which is influenced by the function f = f (x). f is sometimes also referred
to as a source. Ω is the domain on which the problem is solved, bounded by
∂Ω. The boundary is split into the Dirichlet part ΓD and the Neumann part
ΓN = ∂Ω \ ΓD . Boundary conditions are imposed on the system accordingly.
A fixed value of h on ΓD and a normal flux on ΓN resulting from the normal
derivative of h.
This example is two dimensional, d = 2, and standard cartesian coordinates
are used, so the Laplace operator ∆ can be written as
∆h =
∂ 2h ∂ 2h
+
.
∂x2 ∂y 2
(5.4)
Let us consider a rectangle as a simple two dimensional domain with the
boundary conditions defined as seen in Figure 14.
To do that in our code, the functions from the DOLFIN [6] library are imported, which are needed to solve the problem. It is an interface which
communicates the Python code with the underlying code written in C++.
Further, we use a function to generate the mesh.
from dolfin import *
mesh = UnitSquare(32,32)
UnitSquare(32,32) generates 1024 uniform rectangles, which are then divided in triangles.
40
Neumann no flow
Dirichlet
x[1]
x[0]
Figure 14: The domain on which the Poisson equation is solved. The Dirichlet
condition continues on the green axes.
41
To solve the partial differential equation 5.3 on this mesh, the equation has to
be discretized on those triangles. Multiplying equation 5.3 by a test function
v and integrating by parts reveals
Z
Z
Z
∇h · ∇v dx −
∂n h v ds =
f v dx.
(5.5)
Ω
∂Ω
Ω
h is chosen from a function space, which is suitable to discretize the problem
on the given mesh. It is the same as the test function space v is chosen from.
The only difference is, that the test function space is shifted by the Dirichlet
boundary condition, such that it equals zero on those boundaries. From that
we obtain the variational problem: Find h from a suitable function space
such that
Z
Z
Z
∇h · ∇v dx =
f v dx −
gv ds.
(5.6)
Ω
Ω
ΓN
The Neumann boundary condition is left in our equation, while the Dirichlet
boundary condition is defined by the choice of our test function space. This
is why we have to define the Dirichlet boundary on the right hand side of
our domain by
def boundary(x):
return x[0] > 1 - DOLFIN_EPS
To define the function space with Lagrange elements of first order we simply
write
V = FunctionSpace(mesh, "Lagrange", 1)
From this function space, we choose a basis {Φj }N
j=1 for the trial function
and {Φ̂i }N
i=1 for the test function, where N is the dimension of those spaces.
An Ansatz to solve hΦ in terms of the basis functions for the trial space can
be done
hΦ (x) =
N
X
Hj Φj (x).
(5.7)
j=1
Hj is the vector with the desired values for hΦ at the locations x of the
mesh. Inserting this into 5.6 and using the same Ansatz for the test function
42
(only that its basis is shifted by the Dirichlet boundary condition) we get the
equation
N
X
j=1
Z
Z
∇Φj · ∇Φ̂i dx =
Hj
Ω
Z
f Φ̂i dx −
Ω
g Φ̂i ds
(5.8)
ΓN
which is a linear system and can be written in the form
AH = b.
(5.9)
To get the functions from the function space it is sufficient to write
h = TrialFunction(V)
v = TestFunction(V)
Now we impose the Dirichlet boundary condition, and define source term
and the no flow condition for the remaining Neumann boundaries
#Set value for Dirichlet condition and impose it on the system
h_bound = Constant(10)
bc = DirichletBC(V, h_bound, DC_boundary)
#Define source term and set the Neumann condition to no flow
f = Expression("10*1/pow(3.14*0.01,0.5)*exp(-(pow(x[0]-0.5,2)
+pow(x[1]-0.5,2))/0.01)")
g = Constant(0)
To finish the code, we have to write down the equation for the variational
problem and solve it.
#Write the equation system
a = inner(grad(h), grad(v))*dx
L = f*v*dx - g*v*ds
# Compute solution
h = Function(V)
solve(a == L, h, bc)
43
Figure 15: The results for the pressure drop of the Poisson equation
We save the result file in the .pvd format, which can be viewed in ParaView.
Figure 15 shows a plot of the result.
# Save solution in pvd format
file = File("poisson.pvd")
file << h
44
5.2
The incremental pressure correction scheme
Our results presented in Section 6 all have stationary inflow conditions, so a
time discretization is not absolutetely necessary. Nevertheless, we calculate
some timesteps, until the flow field is fully developed inside the aneurysm geometries. Also, to do further investigations on this topic, it is recommended
to use time dependent inflow conditions, as the stationary inflow is only a
rough averaging of the actually pulsating blood flow. The following explanations are a summary of the explanations in [13] .
We use the IPCS to solve the Navier-Stokes equations
u̇ + u · ∇u = ∇ · σ + f
∇·u=0
(5.10)
where u̇ = ∂u/∂t and a unit fluid density is used. For the weak form, the
multiplication with a test function is done and the stress term is integrated
by parts. The momentum equation in integral form becomes
∂Ω
Ω
Ω
(σ ·n)v+
σ ·∇v dx+
(u·∇u)v dx = −
u̇v dx+
Ω
Z
Z
Z
Z
Z
f v dx, (5.11)
Ω
where Ω is the model area and ∂Ω the boundary. The idea of the IPCS is to
first calculate a tentative velocity with the momentum equation, using the
pressure and the velocity of the previous timestep. The corrected pressure of
the current time-step is then obtained with the help of the tentative velocity
by solving the Darcy problem
unh − u?h
+ ∇(pnh − pn−1
h ) = 0,
kn
∇ · unh = 0.
(5.12)
unh , pnh denote the velocity and the pressure of the current time step n discretized on the used mesh. u?h denote the tentative velocity and pn−1
the
h
pressure of the previous time step. kn is the local time step size tn − tn−1 .
Equation system 5.12 can be simplified to a Poisson problem
−∆(pnh
−
phn−1 )
45
∇ · u?h
=
.
kn
(5.13)
With the corrected pressure pnh and the tentative velocity u?h , the corrected
velocity unh can be calculated by inserting into the first equation of 5.12.
The steps of the IPCS are summarized in the scheme below [13] .
h· , ·i marks the inner product of two components. Note that the stress
n− 12
tensor is evaluated at uh
0.5(∇v + ∇v > ).
= (u?h + un−1
h )/2. (v) is the symmetric gradient
1. Compute the tentative velocity u?h by solving
n− 12
n−1
h(u?h − un−1
· ∇uhn−1 , vi + hσ(uh
h )/kn , vi + huh
, phn−1 ) , (v)i
n− 12 >
+hpn−1
h n , vi∂Ω − hµ · (∇uh
(5.14)
) , vi∂Ω = hf n , vi.
2. Compute the corrected pressure pnh by solving
h∇pnh , ∇qi = h∇pn−1
, ∇qi − h∇ · u?h , qi/kn .
h
(5.15)
3. Compute the corrected velocity unh by solving
hunh , vi = hu?h , vi − kn h∇(pnh − pn−1
h ) , vi
5.3
(5.16)
Implementation in FEniCS
Headflow [8] provides the IPCS scheme discussed in the previous section.
Headflow is a collection of Python scripts, which the FEniCS to solve flow
related problems. Headflow has the following main directories
~/headflow/data
/problems
/results
/solvers
The meshes for the computation are stored in the /data directory. The
/problems directory contains the scripts which include information like inflow, outflow- and boundaryconditions and problem specific parameters. The
solvers which generate a timestepping algorithm (see Section 5) with the
help of FEniCS are located in the /solvers directory. The generated results
are written to /results.
46
To solve our problems, we use the IPCS written in /solvers/ipcs.py and
adapt it for our purpose.
The IPCS scheme in the original file:
# Tentative velocity step
u_mean = 0.5 * (u + u0)
F_u_tent = ((1/k) * inner(v, u - u0) * dx
+ inner(v, grad(u0)*u0) * dx
+ inner(epsilon(v), sigma(u_mean, p0, nu)) * dx
- nu * inner(grad(u_mean).T*n, v) * ds
+ inner(v, p0*n) * ds
- inner(v, f) * dx)
a_u_tent = lhs(F_u_tent)
L_u_tent = rhs(F_u_tent)
# Pressure correction
a_p_corr = inner(grad(q), grad(p))*dx
L_p_corr = inner(grad(q), grad(p0))*dx - (1/k)*q*div(u1)*dx
# Velocity correction
a_u_corr = inner(v, u)*dx
L_u_corr = inner(v, u1)*dx - k*inner(v, grad(p1-p0))*dx
The terms for the description of the porous medium flow are added in the
step of the tentative velocity calculation
# Resistance constants for porous media
K1 = problem.K1
K2 = problem.K2
# Tentative velocity step
u_mean = 0.5 * (u + u0)
u_mag = sqrt(inner(u0, u0))
F_u_tent = ( ((1/k)) * inner(v, u - u0) * dx
+ inner(v, grad(u0)*u) * dx
+ inner(epsilon(v), sigma(u_mean, p0, nu)) * dx
47
+
+
+
nu * inner(grad(u_mean).T*n, v) * ds
inner(v, p0*n) * ds
inner(v, f) * dx
inner(v, K1*u_mean) * dx
inner(v, K2*u_mean*u_mag) * dx)
The resistance constants K1 and K2 are defined in a problem file inside the
/problems directory:
class ResistanceTerms(Expression):
def __init__(self, PMVol, term, nu, porosity, diameter):
self.por = porosity
self.PMVol = PMVol
self.term = term
self.d = diameter
self.nu = nu
def eval(self, v, x):
VolPM = self.insidePMVol
por = self.por
#Porous media default parameter...
#c1: Drag factor, koz: Kozeny coefficient
#K: coillength dependent on porosity
c1 = 2.2
koz = 2
K = (1-por)*VolPM/(pi*((self.d/2)**2))
if por >= 1:
v[0] = 0
else:
#Permeability dependent on porosity
per = (por**3)/(koz*(1000*pi*self.d*K/VolPM)**2)
#Terms
term1 = (por*self.nu/per)
term2 = ((por**2)*c1)/sqrt(per)
48
if self.term == 1:
v[0] = term1
elif self.term == 2:
v[0] = term2
self.K1 = ResistanceTerms(PMVol, 1, self.nu, poros, 0.25)*
inside_pm
self.K2 = ResistanceTerms(PMVol, 2, self.nu, poros, 0.25)*
inside_pm
\
\
K1 and K2 are dependent on the size of the porous medium area, its porosity
and the kinematic viscosity of the fluid.
5.4
Verification of the implementation
To verify the used equations for the description of the flow and their implementation in FEniCS a small example was calculated and compared with a
code implementation of DuMux [16] . DuMux is a framework, which is used at
the Department of Hydromechanics and Modelling of Hydrosystems, University of Stuttgart.
A problem on the unit square was solved with the model properties illustrated in Figure 16. With a constant inflow velocity at the top boundary of
the domain, the resulting pressure distribution and the pressure head at the
inflow of both implementations are compared.
Figure 17 summarizes the results of the simulations. The pressure distributions are very similar. The pressure drop starts to develop in the porous
medium part of the domain. The pressure at the inflow is higher for the
DuMux code compared to the the FEniCs code.
The difference in the pressure values is explained by the use of different
equation systems. The used DuMux code is based on a coupling approach,
where the Stokes equation is describing the free flow, while the porous medium
flow is described with the Darcy equation. The nonlinear convective term
of the Navier-Stokes equations is missing, if Stokes flow is considered. This
49
1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
constant y−velocity
velocity = 0
pressure = 0
porous medium
area
Figure 16: Illustration of the example problem
pressure [kPa]
DuMux
FEniCS
1290
pressure [kPa]
1725
0
0
Figure 17: Pressure distributions for the two different models
is why a higher pressure drop is necessary for the development of the same
velocity.
50
6
Results
In this section the generated results from various runs of the SMF are discussed. First, we give a simple two dimensional example of a flow field,
without the use of an aneurysm geometry to present the functionality of
the SMF framework in the combination with FEniCS. Next, we present the
results for the use of the SMF with three design variables on aneurysm geometries following a section with the discussion of a stent variant. A mesh
convergence test is described, finishing with a discussion of the results and a
short outlook.
6.1
Two dimensional flow field
This is a simple example, which is supposed to show the functionality of the
SMF. It solves the stationary Stokes equation on the unit square domain
with the resolution of 800 triangular cells. A resistance term is added to the
standard Stokes equation in the parts where the porous medium is considered.
So the equation
−∇ · (∇u + pI)) − k0 u = f
(6.1)
is solved on the porous medium domain. k0 is the resistance constant.
Lagrange elements with a mixed finite element method are used, where the
velocity is approximated in second order and the pressure in first order.
The inflow boundary condition on the left side of the domain is set to a constant inflow rate with a quadratic velocity distribution. At the top and bottom boundaries, zero-velocity conditions are imposed. On the right boundary
a free outflow condition is given.
We take the value of the resistance constant k0 , the position of the porous
medium x0 , and the edgelength of the porous medium square c as design
variables. The position x0 is the position along the horizontal centerline of
the domian. Table 1 shows the thresholds for the design variables.
As a cost function for this problem,
Z
∇p · ∇pdΩ + 1000 ux (1.0, 0.5)
Ω
51
(6.2)
k0
100
10000
x0
0.1
1
c
0.1
0.5
Table 1: The thresholds for the design variables. First row: lower threshold.
Second row: upper threshold
run
01
02
03
04
k0
564.0625
564.0625
564.0625
718.7500
x0
0.9718
0.9500
0.9015
1.0000
c
0.1000
0.1000
0.1062
0.1000
cost function
113.4037
110.4510
107.9119
120.1772
Table 2: The results for the SMF runs of the example problem
is to be minimized. The pressure gradient ∇p in the whole domain and the
outflow velocity on the centerline ux (1.0, 0.5) are considered in the cost function. The SMF was run four times to check whether the resulting design
variables for the best found cost function are the same for all runs. Table
2 summarizes the resulting design variables and the cost function for every
run. Figure 18 shows the resulting velocity field for the third run.
One notes that the resulting optimal placement, size and permeability of the
porous medium square tends to be the same for all four runs of the SMF.
The fourth run results in a higher resistance value, but also in a higher cost
function. For this case, it seems to be a good solution to put a small square
with a relatively low resistance next to the point where the velocity for the
cost function is measured.
We conclude from this small example, that the SMF converges to a solution
for a particular problem. This does not necessarily mean that only one
solution exists for a more complex structure like an aneurysm. This is why
we run the SMF several times on the aneurysm geometries in the following
sections and compare the results afterwards.
52
Figure 18: The velocity distribution for the third SMF run
6.2
Aneurysm geometry - three design variables
The SMF with three design variables was applied to two different aneurysm
geometries.
A sphere is defined in the model area which represents the porous medium.
The porosity and the radius of the sphere would relate to the amount and
the preconditioned shape of the coil, which is introduced into the aneurysm.
Besides those two design variables, the position of the sphere along a vector
was chosen as the third design variable.
Figure 19 shows examples of sphere placements in the model area of a bifurcation geometry and the corresponding porous medium area. The configuration on the top is a coil placement with a neck remnant. Provided that
the porosity is very low, the configuration in the middle picture represents
the case of a flow diverter. Flow diverters are not used as a standard routine
for treating aneurysms. They could possibly be fixed by a stent during the
procedure. The configuration on the bottom shows a case, which would be
hardly executable in real life, because of the coils being pushed up into the
aneurysm by the blood flow. Nevertheless, it is a possible case in the SMF
and from an engeneering point of view it is interesting to know whether such
a case reveals a better solution for the problem.
53
Figure 19: Different placement possibilities for the spheres and the corresponding
porous medium areas (red)
54
Figure 20: The geometry of the sidewall aneurysm
To check the robustness of the method, the SMF was always executed three
times with the same model properties. The only differnce between the runs
is the choice of the design variables for the initial simulations of the SMF.
This was done by the LHS method as described in Section 4.4
As a cost function for the SMF, the sum of the pressure drop measure pD and
the kinetic energy measure Ekin should be minimized. Ekin is the squared
flow velocity integrated inside a defined aneurysm lumen Vkinen . pD is the
pressure drop from the inlet to the outlet multiplied with the density.
We scale those two measures with constants, such that none of them dominates the other. For the SMF runs of this section, the cost function
J(pD , Ekin ) = Ekin + 0.05pD
(6.3)
is minimized. This is a first try for the choice of a cost function and it is
suitable for the comparision of the simulation runs.
Sidewall aneurysm
The first examined geometry is a sidewall aneurysm, presented in Figure 20.
It is a giant aneurysm (∼ 1100mm3 ) located at the right internal carotid
artery of the patient.
55
Figure 21: Streamlines for the three results of the SMF runs and without coil
placement
The model mesh which was generated from the CT images of the patient
has about 220000 cells. At the inflow, a constant velocity of 300mm/s and
at the outflow, a zero pressure condition is chosen. The vessel walls are
assumed impermeable and a no-flow condition is imposed. We enlarge the
model area by adding flow extensions to the geometry, such that the flow
field is fully developed before it reaches the twisting part of the geometry
and the aneurysm. The flow extension at the outflow is needed, so that the
zero pressure condition at the outflow does not influence the flow pattern
near the aneurysm.
A vector from the aneurysm neck to the top of the aneurysm is defined,
along which the sphere - representing the porous medium - has one degree of
freedom. The position parameter for the SMF ranges from 0 to 1, where 0
is the lowest position in the aneurysm and 1 is the position at the top of the
aneurysm. The maximum radius of the sphere has to be large enough, that
the sphere at the top of the aneurysm neck covers the whole aneurysm (this
corresponds to a coiling without neck remnant). The minimum radius is of
the size, that the sphere at the aneurysm neck represents a flow diverter. For
this aneurysm geometry, the minimum radius threshold is 1mm and the maximum radius threshold 12mm. The porosity of the porous medium ranges
56
from 0.7 to 0.95.
Table 3 shows the resulting design variables, the pressure drop from inlet to
outlet and the cost function for the three runs. We found that the results
for the design variables are different for the three runs. The cost function is
in the same range in all three cases. This shows that there might be several
possibilities to place the coils.
The pressure drop for run01 is the lowest.
Looking at the position of the coils (grayscale) and the streamlines generated
in Figure 21 one can see that the flow pattern changes similarly for all three
result cases, compared to the situation without coils on the very left of the
figure.
The resistance against the flow in the aneurysm is increased in all three coiled
cases, which results in an acceleration of the flow beneath the aneurysm neck.
The location of the interface between the free flow region and the porous
medium region is approximately the same for all results. This shows that
the size of the neck remnant is determined, according to our simulation runs.
Figure 22 shows the velocity distribution inside the aneurysm lumen, which
was taken into account for the calculation of the kinetic energy measure. The
maximum threshold in the figures is set to 10mm/s, but this is not the maximum velocity. The flow in the aneurysm is prevented by all three coil settings.
Run number two and three are coil configurations which would be hard to
perform in real life, because the coils would be pushed up into the aneurysm.
From the three executed runs we can conclude that for this geometry, a coiling without neck remnant would be a good choice. Also, it is not neccesary
to fill the aneurysm lumen with many coils, as the porosity for the first run which has the lowest cost function and is performable in real life - is very high.
57
run
01
02
03
porosity [−]
0.95
0.7
0.7
position [−]
0.7695312
0.3125
0.4609375
radius [mm]
9.421875
4.093750
5.900781
cost function [−]
32248.99
33289.14
32745.46
p.-drop [mmHg]
4.52
4.55
4.59
Table 3: The results for the design variables and cost functions for the sidewall
geometry
Figure 22: Velocity inside the aneurysm for the three results of the SMF runs
and without coil placement
58
Figure 23: The geometry of the bifurcation aneurysm
Bifurcation aneurysm
The bifurcation aneurysm of this section is located at the middle cerebral
artery of the patient. It has a volume of ∼ 90mm3 and a wide aneurysm
neck.
Figure 23 illustrates the aneurysm from different perspectives. The mesh of
this aneurysm geometry has a size of about 120000 cells and the inflow velocity in this case was set to 800mm/s. Besides that, the boundary conditions
and the definition for the cost function are taken from the sidewall aneurysm
simulations. Flow extensions are added to this geometry, as well. The vector
along which the sphere moves is adapted to the geometry and the porosity
thresholds ranged from 0.7 to 0.95 as well. The radius for the porous medium
sphere ranges from 1 to 4.2 for this aneurysm geometry.
We performed the same procedure as with the sidewall aneurysm from the
previous part. The results of the three simulation runs are listed in Table 4.
The found design variables again differ from run to run, but the cost function
is in the same range.
The two first simulation runs have a similar location and sphere radius. The
porosity of the first run is low, while the porosity of the second run is rather
high. As both of the runs produce almost the same cost function, it can be
presumed that a smaller amount of coil is sufficient, to avoid the flow into
59
run
01
02
03
porosity [−]
0.703418
0.8875
0.8328125
position [−]
0.4921875
0.484375
0.5791016
radius [mm]
3.459375
3.45
3.903125
cost function [−]
127922.7
128422.4
124315.5
p.-drop [mmHg]
16.38
16.49
16.16
Table 4: The results for the SMF runs on the bifurcation geometry
Figure 24: Streamlines for the three results of the SMF runs and without coil
placement
the aneurysm sufficently.
Figure 24 shows streamlines for the uncoiled situation on the very left. The
remaining figures illustrate the coil locations of the three simulation runs
(grayscale) in combination with streamlines. Even though the location and
the radii of the three runs are not the same, the aneurysm is completely
filled with porous medium in each case, with approximately the same neck
remnant. This results in a reduction of the flow velocity inside the aneurysm.
Figure 25 shows the flow fields inside the aneurysm for the uncoiled case
and the three SMF runs. The pictured velocity field is a cut through the
aneurysm area which was taken into account for the calculation of the kinetic energy measure.
All three SMF runs on this aneurysm geometry result in a coil configuration
whithout neck remnant. The porosity (and from that, the permeability) does
60
Figure 25: Velocity inside the aneurysm. No coiling on the very left, following
run01 - run03 from the left to the right
not have a very strong influence on the cost function. In this case, a lower
porosity seems necessary to result in a good cost function, compared to the
sidewall aneurysm. The reason for that might be the bifurcation geometry
with a more direct flow into the aneurysm. Also the inflow velocity was
chosen higher in this case.
61
Figure 26: The sidewall aneurysm geometry with a spherical shaped disk, representing the aneurysm.
6.3
Aneurysm geometry - stent
From a discussion with a clinician from the Rikshospitalet in Oslo, we learned
that it is more likely for some aneurysm geometries to being treated with a
stent. The reason for that is, for example, a wide neck of the aneurysm
and, resulting from that, the risk of the coils to get loose and block the flow
through the vessel. A stent situation is not considered in our previous simulation runs.
In this section, we want to implement a stent situation in our SMF, but it is
also supposed to emphasize the sensitivity of the cost function. This means
we want to take a closer look on the problems that may occur by i.e. choosing
different weights for the components of the cost function.
We implement the stent situation using a spherical shaped disk, which marks
the porous medium area (see Figure 26). To compare the stent with a coiling
procedure, we perform seperate SMF simulation runs with two design variables for each situation.
For the coils, the position and the porosity of the porous medium area is
62
parameter
coil porosity [-]
coil position [-]
stent porosity [-]
stent thickness [mm]
min
0.65
0
0.8
0.3
max
0.97
1
0.95
0.8
Table 5: The thresholds of the design variables for the SMF
varied like in the previous section. This corresponds to the size of the neck
remnant and the amount of coil. The radius is kept constant, such that the
aneurysm is always filled completely except for the neck remnant.
For the stent situation, we vary the thickness of the disk and its permeability.
There are a lot of variants of stents and unfortunately, we could not find a
good source with technical properties, so we had to estimate the porosity
and thickness.
The cost function is changed in this case such that the pressure drop has less
weight. For these simulation runs
J(pD , Ekin ) = Ekin + 0.025pD
(6.4)
is minimized.
Three SMF runs are performed for each situation and compared afterwards.
The sidewall geometry from the previous section is used with the same boundary conditions and mesh resolution. Table 5 shows the thresholds for the SMF
design variables. The results of the two different situations is summed up in
Table 6.
The stent situation results in a much better cost function. This statement
should be reviewed critically though. We want to take a closer look at the
components of the goal function for one best solution of the two situations.
We take run01 for the stent situation and run02 for the coil situation.
Table 7 shows the kinetic energy measure Ekin and the weighted pressure
drop measure 0.25pD for the two runs.
We find that the stent situation has a much lower pD and Ekin , but also that
the pressure drop measure dominates the kinetic energy measure in both
cases. If we wanted to attribute more importance to the reduction of the kinetic energy inside the aneurysm, we could reduce the weighting of pD even
more, such that it only turns the scale.
63
stent
run
01
02
03
porosity [−]
0.8
0.809375
0.8
thickness [mm]
0.40156
0.40156
0.3625
cost function [−]
8492.45997651
8492.43277811
8404.60061028
coil
run
01
02
03
porosity [−]
0.6575
0.65
0.6525
position [−]
0
0
0
cost function [−]
19793.447378
19792.898641
19793.0778019
Table 6: The comparision between stent and coils on the sidewall geometry
situation
stent
coil
0.025pD [P a ∗ mm3 /g]
8068.45
15739.76
Ekin [mm2 /s2 ]
423.98
4053.14
Table 7: The components of the goal function for a stent and a coil situation
Figure 27: The change of the volume Vkinen , used for the calculation of the kinetic
energy measure Ekin
64
situation
stent
coil
0.00025pD [P a ∗ mm3 /g]
80.68
157.4
Ekin [mm2 /s2 ]
269.07
8.56
Table 8: The components of the goal function with another Vkinen
Also, we discovered that Ekin is very sensitive to the choice of the aneurysm
lumen Vkinen , which is taken to calculate the kinetic energy measure. We
changed Vkinen according to Figure 27. Both choices are reasonable for the
calculation of the kinetic energy inside the aneurysm, but the results for Ekin
- especially for the coiled case - change drastically (see Table 7 and 8).
Supposed that the design variables found by the SMF were the same with a
different pressure drop weight, we can see from Table 8 that the coil situation
could be considered the better choice for this aneurysm geometry. The pressure drop weight is still in a reasonable range, as the values for the measure
are not much lower than the values for the kinetic energy measure.
65
Figure 28: The aneurysm with the coiled area (red) and the point of the velocity
measurement (white dot)
6.4
Mesh convergence
A mesh convergence test was done to check if the mesh resolutions of the
meshes of the simulation runs are within a sufficient range. The bifurcation
geometry from Section 6.2 with one of the best found SMF solutions for the
coiled area was used (see Figure 28). The mesh resolutions range from about
100000 to five million cells. Besides the mesh resolution, the system had the
same properties as the one in Section 6.2.
Three parameters are plotted over the increasing mesh resolution in Figure 29
- 31. The horizontal axis of the plots is in a logarithmic scale.
All three parameters show a clear convergence behaviour. The pressure drop
from the inlet to the outlets decreases exponential-like with a threshold value
of about 14.3 mmHg. The kinetic energy inside the aneurysm and the velocity measure at one point show a minimum at about 150000 cells and then
start converging.
As we used the pressure drop and the kinetic energy in our cost function for
the SMF, the results could be influenced by the choice of our mesh resolution.
The kinetic energy is probably underestimated, while the pressure drop is
66
17
pressure drop [mmHg]
16.5
16
15.5
15
14.5
14
1.0e+05
1.0e+06
mesh resolution [cells]
1.0e+07
Figure 29: The pressure drop from inlet to outlet with increasing mesh resolution
overestimated. For the sake of time efficiency we had to use mesh resolutions
in the range where those parameters are not converged properly.
67
7.53e+07
kinetic energy measure [mm2/s2]
7.52e+07
7.51e+07
7.5e+07
7.49e+07
7.48e+07
7.47e+07
7.46e+07
1.0e+05
1.0e+06
mesh resolution [cells]
1.0e+07
Figure 30: The kinetic energy inside the aneurysm with increasing mesh resolution
68
815
810
805
velocity [mm/s]
800
795
790
785
780
775
770
1.0e+05
1.0e+06
mesh resolution [cells]
1.0e+07
Figure 31: The velocity at a measurement point with increasing mesh resolution
69
6.5
Discussion
In this work, a finite element flow simulator in combination with a statistical
framework was applied to compare possible endovascular treatment solutions
for an aneurysm. The equations for the porous medium flow, as described in
Section 4.2, were implemented in FEniCS. A short verification was done by
the compharision of the FEniCS code with a DuMux code. A small example
was calculated as described in Section 5.4.
The surrogate management framework (SMF), as described in Section 4.4,
was implemented as a combination of an existing Octave [17] framework (provided by the Marsden Research Group, University of California San Diego)
with the FEniCS code. The coupling was tested on a small example problem
(Section 6.1). Based on the results of that test, the framework was applied
to the aneurysm geometries.
For the application with three design variables (Section 6.2), deviations of
the design variables found by the several SMF runs were occuring. Nevertheless, we were able to make statements about the tendency of which treatment
would be preferable for the different aneurysm geometries.
In Section 6.3, two different treatment methods, namely coiling and the use
of a stent, were compared. The SMF was applied to both methods with two
design variables each. This section concentrates on the difficulties of choosing a right cost function. It was shown that the method, considered to be
the better solution for one cost function, proved to be worse for a slightly
different cost function.
The choice of the right cost function is considered to be one of the main
factors influencing the resulting solution found by the SMF. It is essential
to find a reasonable definition of the cost function and it is recommended to
further debate with clinicians about what could be best choice.
Besides the choice of the goal function, further improvements have to be done
regarding the mesh resolution (see Section 6.4).
Also, for the sake of time efficiency, stationary inflow velocities were used.
This is only a rough estimation, as with a pulsatile inflow velocity, different
flow patterns develop and turbulence might occur. This, in turn, might lead
70
to different cost function values.
The possibility of coupling a finite element flow simulator with the statistical framework was shown. Reasonable solutions were found for different
aneurysm geometries with respect to certain goal functions. Most of the
improvements outlined above are easy to accomplish. Before the described
method is applicable to guide clinicians in their decision making, it is essential to run many more simulations on different aneurysm geometries with
different cost functions and compare the results.
71
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