1. Accelerated Numerical Simulation of Bloodflow in Aneurysms Using Lattice Boltzmann Methods and Multigrid Sarntal 2005 18.09.2005 Jan Götz

2. Outline 1. What are aneurysms? 2. Numerical Basics 3. Simulation

3. What are aneurysms? Definition & Description Symptoms Causes & Prevention Diagnostics Treatment 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

4. 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 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

5. Bernoulli-Principle: „Decrease in velocity occurs simultaneously with increase in pressure“ Definition & Description 2 High velocity, low pressure Low velocity, high pressure 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

6. 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

7. Causes & Prevention 80 % are arteriosclerotic diseases rest: vessel infection, injuries or borne in (e.g. Marfan syndrome) healthy lifestyle can prevent most aneurysm 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

8. 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

9. Example: Angiography Diagnostics 2 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

10. Example: physical examination Diagnostics 3 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

11. Case study: 12 year old girl with headaches and vision problems Diagnostics 4 AngliogramRotational angliogram in 3D 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

12. Treatment 1 invasive intervention: clipping /bypass non-invasive intervention: – coils (GDC) – stents conservative treatment with medication 1.What are aneurysms ? 2.Numerical Basics 3.Simulation mortality rates: preventive surgery: 2-5% surgery after rupture: 50%

13. Example: stents Treatment 2 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

14. Lattice Boltzmann Multigrid Simplifications Numerical Basics 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

15. „You only need to know where you are, everything else is not important“ 1.microscopic: Hamilton‘s-equations 2.mesoscopic: Lattice-Boltzmann 3.macroscopic: Navier-Stokes Lattice Boltzmann 1 1.What are aneurysms ? 2.Numerical Basics 3.Simulation size

16. 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 square/cubic 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

17. D3Q19: model for 3 dimensions with 19 discrete velocity-directions Lattice Boltzmann 3 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

18. Evolution equation to be computed: The Boltzmann equation Lattice Boltzmann 4 BGK Discretisation 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

19. We can do this in two steps: collision advection/streaming Lattice Boltzmann 5 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

20. collision and streaming Lattice Boltzmann 6 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

21. Equilibrium distribution: truncated (small- velocity) version of the „shifted Maxwellian“ (the equilibrium in standard Boltzmann theory) Lattice Boltzmann 7 1.What are aneurysms ? 2.Numerical Basics 3.Simulation density velocity

22. Why no „normal“ multigrid ??? This does not work here, because L h is not linear in our case! → We need another approach for our nonlinear problem Multigrid 1 1.What are aneurysms ? 2.Numerical Basics 3.Simulation exact solution:approx. solution:

23. 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 e.g. Newton‘s method Multigrid 2 for LB-application FAS is better 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

24. equations Multigrid 3 term on RHS is called defect correction This operator is the direct injection 1.What are aneurysms ? 2.Numerical Basics 3.Simulation correction of u h :

25. Blood is a suspension of: formed blood cells (red, white, etc) some liquid particles an aqueous solution (plasma) Simplifications 1 At high shear rate (γ<100 sec -1 ) blood can be treated as Newtonian We focus on large vessels → high shear rates 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

26. 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

27. Simulation Goal of the Simulation Why Lattice Boltzmann ? performance: FAS vs. linear The algorithm Example 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

28. 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 → simulations of hemodynamics (blood flow, including flow and pressure perturbations and vessel wall loading) are very important clinical applications need fast simulations

29. Goal of the Simulation 2 use as initial condition → 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

30. 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

31. 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: 75other terms:36 advection: 0under-relaxation:27 total: 93total: 209 →We already have an existing non- linear LBE method, so lets use it

32. The algorithm 1 collision advection relaxation This equation is new! D H is called the defect correction 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

33. The algorithm 2 How to get the defect correction ??? 1. 2. 1.What are aneurysms ? 2.Numerical Basics 3.Simulation In a standard FAS the defect correction would be

34. The algorithm 3 Why do we have no „standard“ ??? 1.What are aneurysms ? 2.Numerical Basics 3.Simulation In a MG scheme, we have changing h resulting in a changing τ, but we run into problems if  is smaller than 0,5. And: For smaller  the convergence rate of the LBM-scheme degrades → Trick: Use a constant , but rescale the correction generated by the coarse grid

35. Example Flowlines in a saccular aneurysm 1.What are aneurysms ? 2.Numerical Basics 3.Simulation

36. 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.medical.siemens.com http://www.healthscout.com http://www.merck.com (pharmaceutical company) http://www.merck.com http://www.prosper-hospital.de

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