Accelerated Numerical Simulation of Bloodflow in Aneurysms Using Lattice

Boltzmann Methods and Multigrid

Sarntal 2005

18092005

Jan Goumltz

Outline

1 What are aneurysms

2 Numerical Basics

3 Simulation

What are aneurysms

Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment

1 What are aneurysms

2 Numerical Basics

3 Simulation

Definition amp Description 1

Greek bdquoDilatationldquo

An aneurysm is a local dilatation (balooning) of a blood vessel

Localisation larger arteries in soft tissuendash brainndash aorta

1 near heart

2 abdominal

1 What are aneurysms

2 Numerical Basics

3 Simulation

Outline

1 What are aneurysms

2 Numerical Basics

3 Simulation

What are aneurysms

Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment

1 What are aneurysms

2 Numerical Basics

3 Simulation

Definition amp Description 1

Greek bdquoDilatationldquo

An aneurysm is a local dilatation (balooning) of a blood vessel

Localisation larger arteries in soft tissuendash brainndash aorta

1 near heart

2 abdominal

1 What are aneurysms

2 Numerical Basics

3 Simulation

What are aneurysms

Definition amp Description Symptoms Causes amp Prevention Diagnostics Treatment

1 What are aneurysms

2 Numerical Basics

3 Simulation

Definition amp Description 1

Greek bdquoDilatationldquo

An aneurysm is a local dilatation (balooning) of a blood vessel

Localisation larger arteries in soft tissuendash brainndash aorta

1 near heart

2 abdominal

1 What are aneurysms

2 Numerical Basics

3 Simulation

Definition amp Description 1

Greek bdquoDilatationldquo

An aneurysm is a local dilatation (balooning) of a blood vessel

Localisation larger arteries in soft tissuendash brainndash aorta

1 near heart

2 abdominal

1 What are aneurysms

2 Numerical Basics

3 Simulation

Bernoulli-Principle

bdquoDecrease in velocity occurs simultaneously with increase in pressureldquo

Definition amp Description 2

High velocity low pressure

Low velocity high pressure

1 What are aneurysms

2 Numerical Basics

3 Simulation

Symptoms

often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on

internal organs or nerves rupture causes sudden pain and

severe internal bleeding objective diagnosis from X-ray-

angiography or computer tomography

1 What are aneurysms

2 Numerical Basics

3 Simulation

Causes amp Prevention

80 are arteriosclerotic diseases rest vessel infection injuries or borne

in (eg Marfan syndrome)

healthy lifestyle can prevent most aneurysm

1 What are aneurysms

2 Numerical Basics

3 Simulation

Symptoms

often no symptoms felt by the patient pulsing sensation pain if aneurysm is pressing on

internal organs or nerves rupture causes sudden pain and

severe internal bleeding objective diagnosis from X-ray-

angiography or computer tomography

1 What are aneurysms

2 Numerical Basics

3 Simulation

Causes amp Prevention

80 are arteriosclerotic diseases rest vessel infection injuries or borne

in (eg Marfan syndrome)

healthy lifestyle can prevent most aneurysm

1 What are aneurysms

2 Numerical Basics

3 Simulation

Causes amp Prevention

80 are arteriosclerotic diseases rest vessel infection injuries or borne

in (eg Marfan syndrome)

healthy lifestyle can prevent most aneurysm

1 What are aneurysms

2 Numerical Basics

3 Simulation

Diagnostics 1

X-Ray-Angiography (exact size 2D shape is used also during surgery)

MRI (for exact size 3D shape) CT (for exact size 3D shape) Ultrasound (low cost imprecise) physical examination

1 What are aneurysms

2 Numerical Basics

3 Simulation

Example Angiography

Diagnostics 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

Example physical examination

Diagnostics 3

1 What are aneurysms

2 Numerical Basics

3 Simulation

Example physical examination

Diagnostics 3

1 What are aneurysms

2 Numerical Basics

3 Simulation

Case study 12 year old girl with headaches and vision problems

Diagnostics 4

Angliogram Rotational angliogram in 3D

1 What are aneurysms

2 Numerical Basics

3 Simulation

Treatment 1

invasive intervention clipping bypass non-invasive intervention

ndash coils (GDC) ndash stents

conservative treatment with medication1 What are

aneurysms

2 Numerical Basics

3 Simulation mortality rates

preventive surgery 2-5

surgery after rupture 50

Example stents

Treatment 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

Lattice Boltzmann Multigrid Simplifications

Numerical Basics

1 What are aneurysms

2 Numerical Basics

3 Simulation

bdquoYou only need to know where you are everything else is not importantldquo

1 microscopic Hamiltonlsquos-equations

2 mesoscopic Lattice-Boltzmann

3 macroscopic Navier-Stokes

Lattice Boltzmann 1

1 What are aneurysms

2 Numerical Basics

3 Simulation

size

What is the Lattice Boltzmann method

1 can be imaged as a type of cellular automaton

2 divide simulation region into a Cartesian grid of squarecubic cells

3 each cell only interacts with its direct neighbourhood

4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities

Lattice Boltzmann 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

Lattice Boltzmann Multigrid Simplifications

Numerical Basics

1 What are aneurysms

2 Numerical Basics

3 Simulation

bdquoYou only need to know where you are everything else is not importantldquo

1 microscopic Hamiltonlsquos-equations

2 mesoscopic Lattice-Boltzmann

3 macroscopic Navier-Stokes

Lattice Boltzmann 1

1 What are aneurysms

2 Numerical Basics

3 Simulation

size

What is the Lattice Boltzmann method

1 can be imaged as a type of cellular automaton

2 divide simulation region into a Cartesian grid of squarecubic cells

3 each cell only interacts with its direct neighbourhood

4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities

Lattice Boltzmann 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

bdquoYou only need to know where you are everything else is not importantldquo

1 microscopic Hamiltonlsquos-equations

2 mesoscopic Lattice-Boltzmann

3 macroscopic Navier-Stokes

Lattice Boltzmann 1

1 What are aneurysms

2 Numerical Basics

3 Simulation

size

What is the Lattice Boltzmann method

1 can be imaged as a type of cellular automaton

2 divide simulation region into a Cartesian grid of squarecubic cells

3 each cell only interacts with its direct neighbourhood

4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities

Lattice Boltzmann 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

What is the Lattice Boltzmann method

1 can be imaged as a type of cellular automaton

2 divide simulation region into a Cartesian grid of squarecubic cells

3 each cell only interacts with its direct neighbourhood

4 first order explicit discretisation (in space and time) of the Boltzmann equation in a discrete phase space which describes all molecules with their corresponding velocities

Lattice Boltzmann 2

1 What are aneurysms

2 Numerical Basics

3 Simulation

D3Q19 model for 3 dimensions with 19 discrete velocity-directions

Lattice Boltzmann 3

1 What are aneurysms

2 Numerical Basics

3 Simulation

Evolution equation to be computed

)(t

fff

)()(1

txftxftxf eqaaaa

ttxftxf

tttxf

aa

aa

)(

)(

The Boltzmann equation

Lattice Boltzmann 4

BGK

Discretisation

1 What are aneurysms

2 Numerical Basics

3 Simulation

We can do this in two steps

)()(1

)()(ii

eqaii

naii

naiia yxfyxfyxfyxf

n

)()( 1

21 iiaaiaina yxfheyhexf

collision

advectionstreaming

Lattice Boltzmann 5

1 What are aneurysms

2 Numerical Basics

3 Simulation

collision and streaming

Lattice Boltzmann 6

1 What are aneurysms

2 Numerical Basics

3 Simulation

collision and streaming

Lattice Boltzmann 6

1 What are aneurysms

2 Numerical Basics

3 Simulation

Equilibrium distribution truncated (small-velocity) version of the bdquoshifted Maxwellianldquo (the equilibrium in standard Boltzmann theory)

Lattice Boltzmann 7

1 What are aneurysms

2 Numerical Basics

3 Simulation

22

2

9

2

33 ueuuewf aaa

eqa

131 iforwi72181 iforwi198361 iforwi

a

af a

aa feu

density velocity

Why no bdquonormalldquo multigrid

hhh fuL

hhhhhhh uLfuLuL ~~

hhhhhh uLfuuL ~)~( This does not work here because Lh is not linear in our case

rarr We need another approach for our nonlinear problem

Multigrid 1

1 What are aneurysms

2 Numerical Basics

3 Simulation

exact solution approx solution hu~

hu

Options for nonlinear MG

1 local linearization compute the original nonlinear operator (full approximation scheme FAS) but assume corrections are small and can be linearised

2 global linearization use standard MG but for a linearised system obtained from the original by eg Newtonlsquos method

Multigrid 2

for LB-application FAS is better

1 What are aneurysms

2 Numerical Basics

3 Simulation

equations

Multigrid 3

term on RHS is called defect correction

)(ˆhh

Hhh

HhHHH uRIuILuL

)(ˆhh

Hhh

HhHHH uRIuILuL

nhHhH

hH

nh

nh uIuIuu ˆ1

This operator is the direct injection

1 What are aneurysms

2 Numerical Basics

3 Simulation

correction of uh

Blood is a suspension of

bull formed blood cells (red white etc)

bull some liquid particles

bull an aqueous solution (plasma)

Simplifications 1

At high shear rate (γlt100 sec-1) blood can be treated as Newtonian

We focus on large vessels rarr high shear rates

1 What are aneurysms

2 Numerical Basics

3 Simulation

equations

Multigrid 3

term on RHS is called defect correction

)(ˆhh

Hhh

HhHHH uRIuILuL

)(ˆhh

Hhh

HhHHH uRIuILuL

nhHhH

hH

nh

nh uIuIuu ˆ1

This operator is the direct injection

1 What are aneurysms

2 Numerical Basics

3 Simulation

correction of uh

Blood is a suspension of

bull formed blood cells (red white etc)

bull some liquid particles

bull an aqueous solution (plasma)

Simplifications 1

At high shear rate (γlt100 sec-1) blood can be treated as Newtonian

We focus on large vessels rarr high shear rates

1 What are aneurysms

2 Numerical Basics

3 Simulation

Blood is a suspension of

bull formed blood cells (red white etc)

bull some liquid particles

bull an aqueous solution (plasma)

Simplifications 1

At high shear rate (γlt100 sec-1) blood can be treated as Newtonian

We focus on large vessels rarr high shear rates

1 What are aneurysms

2 Numerical Basics

3 Simulation

Fluid-structure interaction

Simplifications 2

To first approximation we 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

1 What are aneurysms

2 Numerical Basics

3 Simulation

Simulation

bull Goal of the Simulationbull Why Lattice Boltzmann bull performance FAS vs linearbull The algorithmbull Example

1 What are aneurysms

2 Numerical Basics

3 Simulation

Goal of the Simulation 1

Recall

Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50

And

The number one cause of death in a developed nation is a heart- or vascular disease

1 What are aneurysms

2 Numerical Basics

3 Simulation

rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important

clinical applications need fast simulations

Goal of the Simulation 2

use as initial condition

rarr faster convergence stability

1 What are aneurysms

2 Numerical Basics

3 Simulation

1 preliminary time-independent incompressible velocity-field

2 periodically forced time-dependant (pulsating) velocity-field

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 LBM 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 vectorisation and cache-optimisation

1 What are aneurysms

2 Numerical Basics

3 Simulation

Goal of the Simulation 1

Recall

Routine surgery has a mortality rate of 2-5 but a surgery after rupture has about 50

And

The number one cause of death in a developed nation is a heart- or vascular disease

1 What are aneurysms

2 Numerical Basics

3 Simulation

rarr simulations of hemodynamics (blood flow including flow and pressure perturbations and vessel wall loading) are very important

clinical applications need fast simulations

Goal of the Simulation 2

use as initial condition

rarr faster convergence stability

1 What are aneurysms

2 Numerical Basics

3 Simulation

1 preliminary time-independent incompressible velocity-field

2 periodically forced time-dependant (pulsating) velocity-field

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 LBM 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 vectorisation and cache-optimisation

1 What are aneurysms

2 Numerical Basics

3 Simulation

Goal of the Simulation 2

use as initial condition

rarr faster convergence stability

1 What are aneurysms

2 Numerical Basics

3 Simulation

1 preliminary time-independent incompressible velocity-field

2 periodically forced time-dependant (pulsating) velocity-field

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 LBM 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 vectorisation and cache-optimisation

1 What are aneurysms

2 Numerical Basics

3 Simulation

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 LBM 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 vectorisation and cache-optimisation

1 What are aneurysms

2 Numerical Basics

3 Simulation

performance FAS vs linear

1 What are aneurysms

2 Numerical Basics

3 Simulation

Non-Linear LBE Time-Step

Linear LBE Jacobi Relaxation

computation of micro variables 18

9x9 matrix vector multiplications 146

collision 75 other terms36

advection 0 under-relaxation27

total 93 total 209

rarr We already have an existing non-linear LBE method so lets use it

The algorithm 1

)()(1

)()(ii

eqaii

naii

naiia yxfyxfyxfyxf

n

)()(

21 iiaaiaia yxfheyhexf

naHa

na fDff 11

collision

advection

relaxation

This equation is

new

DH is called the defect correction

1 What are aneurysms

2 Numerical Basics

3 Simulation

The algorithm 2

)(1

)(1

1)(

21

21

aiaieqa

aiaiaiiah

heyhexf

heyhexfyxffR

How to get the defect correction

hhHhh

HhHH fRIfIRD 2ˆ

1

2

1 What are aneurysms

2 Numerical Basics

3 Simulation

hhHhh

HhHH fRIfIRD ˆ

In a standard FAS the defect correction would be

The algorithm 3

Why do we have no bdquostandardldquo

1 What are aneurysms

2 Numerical Basics

3 Simulation

2

13

ch

In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05

And For smaller the convergence rate of the LBM-scheme degrades

rarr Trick Use a constant but rescale the correction generated by the coarse grid

The algorithm 1

)()(1

)()(ii

eqaii

naii

naiia yxfyxfyxfyxf

n

)()(

21 iiaaiaia yxfheyhexf

naHa

na fDff 11

collision

advection

relaxation

This equation is

new

DH is called the defect correction

1 What are aneurysms

2 Numerical Basics

3 Simulation

The algorithm 2

)(1

)(1

1)(

21

21

aiaieqa

aiaiaiiah

heyhexf

heyhexfyxffR

How to get the defect correction

hhHhh

HhHH fRIfIRD 2ˆ

1

2

1 What are aneurysms

2 Numerical Basics

3 Simulation

hhHhh

HhHH fRIfIRD ˆ

In a standard FAS the defect correction would be

The algorithm 3

Why do we have no bdquostandardldquo

1 What are aneurysms

2 Numerical Basics

3 Simulation

2

13

ch

In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05

And For smaller the convergence rate of the LBM-scheme degrades

rarr Trick Use a constant but rescale the correction generated by the coarse grid

The algorithm 2

)(1

)(1

1)(

21

21

aiaieqa

aiaiaiiah

heyhexf

heyhexfyxffR

How to get the defect correction

hhHhh

HhHH fRIfIRD 2ˆ

1

2

1 What are aneurysms

2 Numerical Basics

3 Simulation

hhHhh

HhHH fRIfIRD ˆ

In a standard FAS the defect correction would be

The algorithm 3

Why do we have no bdquostandardldquo

1 What are aneurysms

2 Numerical Basics

3 Simulation

2

13

ch

In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05

And For smaller the convergence rate of the LBM-scheme degrades

rarr Trick Use a constant but rescale the correction generated by the coarse grid

The algorithm 3

Why do we have no bdquostandardldquo

1 What are aneurysms

2 Numerical Basics

3 Simulation

2

13

ch

In a MG scheme we have changing h resulting in a changing τ but we run into problems if is smaller than 05

And For smaller the convergence rate of the LBM-scheme degrades

rarr Trick Use a constant but rescale the correction generated by the coarse grid

Example

Flowlines in a saccular aneurysm

1 What are aneurysms

2 Numerical Basics

3 Simulation

Links amp useful stuff

bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick

bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis

ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner

httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI

Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical

company) httpwwwprosper-hospitalde

Links amp useful stuff

bdquoA Multigrid Tutorialldquo (Second Edition) Briggs Henson McCormick

bdquoMultigrid Solution of the Steady-State Lattice Boltzmann Equationldquo Mavriplis

ldquoInteractive Free Surface Fluids with the Lattice Boltzmann Methodrdquo Thuumlrey Ruumlde Koumlrner

httpenwikipediaorg httpwwwmedicalsiemenscom (CT MRI

Angiography) httpwwwhealthscoutcom httpwwwmerckcom (pharmaceutical

company) httpwwwprosper-hospitalde

Download

This presentation can be downloaded from

wwwsarntalerlangen-rocktde