Presentation
Sarntal 2005 Jan Götz
page
1
01.12.2013
Accelerated numerical Simulation of
B
loodflow in
Aneurysms using Lattice Boltzmann Methods and Multigrid
Sarntal 2005
Jan Götz
1.
Aneurysms
An an
eurysm is a local dilatation (ballooning)
of a blood
vessel. It is localize
d in
the brain or the aorta (near heart, or abdominal)
.
Most times the patient does not feel any
symptoms
; t
he
re might only be a pulsing
sensation. B
ut the aneurysm can cause pain, if it is pressing on internal organs.
In worst case a rupture of the aneu
rysm causes sudden pain and severe
internal
blood lost.
Most aneurysms occur from
arteriosclerotic diseases
. The rest is caused by
vessel infection, injuries, or it is born in (Marfan syndrome).
A
healthy lifestyle can prevent an aneurysm
.
There are the
following
possibilities to diagnose an aneurysm:
MRI (exact size and 3D shape)
CT (exact size and 3D shape)
Ultrasound (low cost, but imprecise)
X

Ray

Angiography (
exact size and 2D shape, is used during surgery)
p
hysical examination
An aneurysm can be t
reated with invasive
intervention
(bypass, clipping), or non

invasive intervention
(coils,
stents).
A
conservative treatment with medication
is
also possible
.
Note:
Routine surgery has a mortality rate of 2

5%
,
Surgery after rupture
has about 50%
Presentation
Sarntal 2005 Jan Götz
page
2
01.12.2013
pictures of a stent.

m
odel (left) and a real
stent (right)
2.
Numerical Basics
2.1 What is the Lattice Boltzmann method?
1.
can be imag
in
ed as a type of cellular automaton
2.
divide simulation region into a cartesian grid of
square/cubic cells
3.
each cell o
nly interacts with its direct
neighbourhood
4.
first order explicit discretization (in space and time)
of the Boltzmann equation in a discrete phase
space, w
h
ich describes all molecules with their
corresponding velocities
Example: D3Q19 is a
model for 3 di
mensions with 19 velocity

directions
2.2 Equations
1.
Collide
step
2.
S
tream step
)
,
(
)
,
(
1
)
,
(
)
,
(
*
i
i
eq
a
i
i
n
a
i
i
a
i
i
a
y
x
f
y
x
f
y
x
f
y
x
f
n
)
,
(
)
,
(
*
1
2
1
i
i
a
a
i
a
i
n
a
y
x
f
he
y
he
x
f
Presentation
Sarntal 2005 Jan Götz
page
3
01.12.2013
How to calcul
ate the equilibrium distribution
?
2.
3
Multigrid
We use the full approximation storage (FAS) for the nonlinear problem.
FAS e
quations
2.
4
Simplifications
a)
Blood
is a suspension of formed blood cells and some liquid particles in
an aqueous solution
At high shear rate
(
γ
<100 sec

1
)
blood can be treated as Newtonian
We focus on large vessels
→ here are high shear rat
es
b)
Fluid

structure interaction
W
e neglect the effect of elastic walls.
This is reasonable, because in
large
arteries the effect is quite minor
.
Additionally, we assume blood as homogenous and incompressible
.
3.
Simulation
3.
1 Goal of the Simulation
R
ecall:
Routine surgery has a mortality rate of 2

5%, but a surg
ery after rupture has
about 50%
!!!
And:
n
h
H
h
H
h
H
n
h
n
h
u
I
u
I
u
u
ˆ
1
1
3
/
1
i
for
w
i
7
,
2
18
/
1
i
for
w
i
19
,
8
36
/
1
i
for
w
i
)
(
ˆ
h
h
H
h
h
H
h
H
H
H
u
R
I
u
I
L
u
L
)
(
ˆ
h
h
H
h
h
H
h
H
H
H
u
R
I
u
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L
u
L
2
2
2
9
2
3
3
u
e
u
u
e
w
f
a
a
a
eq
a
a
eq
a
a
a
f
f
a
eq
a
a
a
a
a
f
e
f
e
u
Presentation
Sarntal 2005 Jan Götz
page
4
01.12.2013
The number one cause of death in a developed nation is a heart

or vascular
disease
We want to calculate a
time

independe
nt
incompressible velocity

field
and
use this as an initial guess for a p
eriodically forced
time

depe
nde
nt
velocity

field
3.
2 Why Lattice Boltzmann ?
1.
LBM results in an accurate reproduction of the Navier

Stokes

equations, so
why NOT ?
2.
very complex geometries are readily handled
3.
LB
M is simple to implement and modify
4.
changing the geometry during simulation is possible
5.
calculate pressure and other stresses locally in time and space
6.
very good parallelization, vectorization and cach

optimazation
3,3
The algorithm
1.
Collide step
2.
Str
eam step
3.
Relaxation
DH is called the defect correction
How to get the defect correction ???
1.
2.
)
,
(
)
,
(
1
)
,
(
)
,
(
*
i
i
eq
a
i
i
n
a
i
i
a
i
i
a
y
x
f
y
x
f
y
x
f
y
x
f
n
)
,
(
)
,
(
*
*
*
2
1
i
i
a
a
i
a
i
a
y
x
f
he
y
he
x
f
n
a
H
a
n
a
f
D
f
f
1
*
*
1
→
simulation
s
of hemodynamics
are very important
h
h
H
h
h
H
h
H
H
f
R
I
f
I
R
D
2
ˆ
)
,
(
1
)
,
(
1
1
)
,
(
2
1
2
1
*
a
i
a
i
eq
a
a
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a
i
a
i
i
a
h
he
y
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x
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R
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