Accelerated numerical Simulation of Bloodflow in Aneurysms using Lattice Boltzmann Methods and Multigrid

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
I
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
i
a
i
a
i
i
a
h
he
y
he
x
f
he
y
he
x
f
y
x
f
f
R















