Accelerated Numerical Simulation of Bloodflow in Aneurysms Using Lattice Boltzmann Methods and Multi

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Accelerated Numerical Simulation of Bloodflow in
Aneurysms Using Lattice Boltzmann Methods and
Multigrid

• Sarntal 2005

18.09.2005 Jan Götz
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Outline

• What are aneurysms?
• Numerical Basics
• Simulation

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What are aneurysms?

• Definition Description
• Symptoms
• Causes Prevention
• Diagnostics
• Treatment

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Definition Description 1

• Greek Dilatation
• An aneurysm is a local dilatation (balooning) of
  a blood vessel
• Localisation larger arteries in soft tissue
• brain
• aorta
• 1. near heart
• 2. abdominal

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Definition Description 2

Bernoulli-Principle Decrease in velocity occurs
simultaneously with increase in pressure

• What are aneurysms ?
• Numerical Basics
• Simulation

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• 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

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Causes Prevention

• 80 are arteriosclerotic diseases
• rest vessel infection, injuries or borne in
  (e.g. Marfan syndrome)
• healthy lifestyle can prevent most aneurysm

• What are aneurysms ?
• Numerical Basics
• Simulation

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• 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

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Diagnostics 2

Example Angiography

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Diagnostics 3

Example physical examination

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Diagnostics 4

Case study 12 year old girl with headaches and
vision problems

• What are aneurysms ?
• Numerical Basics
• Simulation

Angliogram
Rotational angliogram in 3D
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• Treatment 1

• invasive intervention clipping /bypass
• non-invasive intervention
• coils (GDC)
• stents

• What are aneurysms ?
• Numerical Basics
• Simulation

• conservative treatment with medication

mortality rates preventive surgery
2-5 surgery after rupture 50
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• Treatment 2

• Example stents

• What are aneurysms ?
• Numerical Basics
• Simulation

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Numerical Basics

• Lattice Boltzmann
• Multigrid
• Simplifications

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Lattice Boltzmann 1

• You only need to know where you are, everything
  else is not important
• microscopic Hamiltons-equations
• mesoscopic Lattice-Boltzmann
• macroscopic Navier-Stokes

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Lattice Boltzmann 2

• What is the Lattice Boltzmann method?
• can be imaged as a type of cellular automaton
• divide simulation region into a Cartesian grid of
  square/cubic cells
• each cell only interacts with its direct
  neighbourhood
• 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

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Lattice Boltzmann 3

D3Q19 model for 3 dimensions with 19 discrete
velocity-directions

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Lattice Boltzmann 4

Evolution equation to be computed
The Boltzmann equation

• What are aneurysms ?
• Numerical Basics
• Simulation

BGK
Discretisation
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• Lattice Boltzmann 5

We can do this in two steps

• What are aneurysms ?
• Numerical Basics
• Simulation

collision
advection/streaming
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• Lattice Boltzmann 6

collision and streaming

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Lattice Boltzmann 7

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

• What are aneurysms ?
• Numerical Basics
• Simulation

velocity
density
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• Multigrid 1

Why no normal multigrid ???

• What are aneurysms ?
• Numerical Basics
• Simulation

exact solution
approx. solution
This does not work here, because Lh is not linear
in our case!
? We need another approach for our nonlinear
problem
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• Multigrid 2

• Options for nonlinear MG
• 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
  e.g. Newtons method

• What are aneurysms ?
• Numerical Basics
• Simulation

for LB-application FAS is better
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• Multigrid 3

equations
This operator is the direct injection

• What are aneurysms ?
• Numerical Basics
• Simulation

term on RHS is called defect correction
correction of uh
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• Simplifications 1

• Blood is a suspension of
• formed blood cells (red, white, etc)
• some liquid particles
• an aqueous solution (plasma)

• What are aneurysms ?
• Numerical Basics
• Simulation

At high shear rate (?lt100 sec-1) blood can be
treated as Newtonian
We focus on large vessels ? high shear rates
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• Simplifications 2

Fluid-structure interaction
To first approximation we neglect the effect of
elastic walls. This is reasonable, because in
large arteries the effect is quite minor

• What are aneurysms ?
• Numerical Basics
• Simulation

Additionally, we assume blood as homogenous and
incompressible
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Simulation

• Goal of the Simulation
• Why Lattice Boltzmann ?
• performance FAS vs. linear
• The algorithm
• Example

• What are aneurysms ?
• Numerical Basics
• Simulation

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• 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

• What are aneurysms ?
• Numerical Basics
• Simulation

? simulations of hemodynamics (blood flow,
including flow and pressure perturbations and
vessel wall loading) are very important
clinical applications need fast simulations
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• Goal of the Simulation 2

• 1. preliminary time-independent incompressible
  velocity-field

use as initial condition

• What are aneurysms ?
• Numerical Basics
• Simulation

2. periodically forced time-dependant (pulsating)
velocity-field
? faster convergence / stability
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• Why Lattice Boltzmann ?

• LBM results in an accurate reproduction of the
  Navier-Stokes-equations, so why NOT ?
• very complex geometries are readily handled
• LBM is simple to implement and modify
• changing the geometry during simulation is
  possible
• calculate pressure and other stresses locally in
  time and space
• very good parallelization, vectorisation and
  cache-optimisation

• What are aneurysms ?
• Numerical Basics
• Simulation

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performance FAS vs. linear

• What are aneurysms ?
• Numerical Basics
• Simulation

? We already have an existing non- linear LBE
method, so lets use it
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• The algorithm 1

collision

• What are aneurysms ?
• Numerical Basics
• Simulation

advection
This equation is new!
relaxation
DH is called the defect correction
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• The algorithm 2

How to get the defect correction ???
1.

• What are aneurysms ?
• Numerical Basics
• Simulation

2.
In a standard FAS the defect correction would be
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• The algorithm 3

Why do we have no standard ???
In a MG scheme, we have changing h resulting in a
changing t, but we run into problems if ? is
smaller than 0,5. And For smaller? the
convergence rate of the LBM-scheme degrades

• What are aneurysms ?
• Numerical Basics
• Simulation

? Trick Use a constant ?, but rescale the
correction generated by the coarse grid
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• Example

Flowlines in a saccular aneurysm

• What are aneurysms ?
• Numerical Basics
• Simulation

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• Links useful stuff

• A Multigrid Tutorial (Second Edition), Briggs,
  Henson, McCormick
• Multigrid Solution of the Steady-State Lattice
  Boltzmann Equation, Mavriplis
• Interactive Free Surface Fluids with the Lattice
  Boltzmann Method, Thürey, Rüde, Körner
• http//en.wikipedia.org
• http//www.medical.siemens.com (CT, MRI,
  Angiography)
• http//www.healthscout.com
• http//www.merck.com (pharmaceutical company)
• http//www.prosper-hospital.de

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• Download

This presentation can be downloaded
from www.sarntal.erlangen-rockt.de