Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Sensitivity Analysis of Simulated Blood Flow in Cerebral Aneurysms Øyvind Evju August 19, 2011 Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study What are aneurysms? An aneurysm is an abnormal bulge of a blood vessel. Most common in the Circle of Willis, part of the brains blood supply. These are called cerebral aneurysms . Large variations in size, up to above 50 mm in diameter. Several types of aneurysms: (a) Fuseform (b) Saccular/ (c) Saccular/ sidewall bifurcation ¡4-¿ Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Rupture of cerebral aneurysms Common cause of subarachnoid hemorrhage. Often leads to serious brain damage or death. In a population of 100,000, about 10-11 cases of aneurysm rupture is expected. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Risk factors and cause Some factors have been identified to increase proneness for aneurysm development and rupture: Environmental factors such as smoking, alcoholism and hypertension. Women are more prone to aneurysm rupture than men. People with an asymmetric or incomplete Circle of Willis more often develop aneurysms. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Wall shear stress From a mechanical point of view, especially high values of wall shear stress (WSS) is indentified as a possible factor of aneurysm development. A surface force working tangential to the vessel wall. Induced by the blood flow. Calculated from the stress tensor. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Blood Blood flow is complicated to simulate. Blood behaves as a non-Newtonian fluid. It is a heterogenous fluid, consisting mainly of blood cells (45%) and plasma (55%). It shows clear shear thinning properties, arising from concentration of red blood cells in the middle of the blood vessel. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Blood viscosity The viscosity of is complex, and many models try to explain it. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Why simulate? Better resolution than any measurement methods available. Minimal disturbance to the patient. Easily change physical parameters. Increased computational power yields greater accuracy. Assist medical personnel in making prognoses and determening treatment. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Uncertainties There are many sources of errors present: Poor resolution of medical images. Little exact patient specific data available. Several simplifications and assumptions are made on the model. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Aneurysms Numerical Methods Blood Qualitative Analysis Why simulate? Quantitative Analysis Uncertainties Conclusion Aim of this study Aim of this study Assess qualitative and quantitative effects of several common simplifications and assumptions. Viscosity Geometry Boundary conditions Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion Main assumptions Rigid walls. Body forces such as gravity are negligible. Incompressibility. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion The Navier-Stokes equations ∂ u ∂ t + u · ∇ u = ∇ · 2 νǫ ( u ) − 1 ρ ∇ p ∇ · u = 0 for x ∈ Ω u ( x , 0 ) = 0 p ( x , 0 ) = 0 u = 0 for x ∈ Γ w u = u 0 for x ∈ Γ I p = p 0 for x ∈ Γ O Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion Problem = ∇ · 2 νǫ ( u ) − 1 � ∂ u ∂ t + u · ∇ u ρ ∇ p for x ∈ Ω . ∇ · u = 0 Several difficulties: Nonlinear. Combination of a hyperbolic and a parabolic term. Two unknowns. Exact solutions exist only to simple problems. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion Implementation The Incremental Pressure Correction Scheme was used. Implementation done using the finite element method, and the software library FEniCS. Source code modified from a previous project ( nsbench ). Meshes are built using tetrahedral cells. The solution is approximated by using polynomials at each cell. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion An exact solution Fully developed, steady state flow in a straight channel/cylinder. Yields the exact solutions for velocity and WSS: u = r 2 − a 2 dp 4 µ dx � a dp � � � τ w = � � 2 dx � � where dp dx is determined from the average flow velocity applied at the inlet. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion Method A common set of parameters, modelling the middle cerebral artery (MCA). Tests were performed in both 2D and 3D. A range of different time steps were tested. Comparisons were made between a quadratic and a linear approximation to the velocity. The exact solution of the WSS was used as reference. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods The Mathematical Model Qualitative Analysis Implementation Quantitative Analysis Verification of implementation Conclusion Resulting choices A timestep of 0.00125s was chosen. A linear approximation of the velocity was preferred to a quadratic approximation. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods Method Qualitative Analysis Simulation Quantitative Analysis Results Conclusion Method A single aneurysm was studied. The effects of three uncertainties were measured: Geometric effects Non-Newtonian effects Effects of different hematocrit levels Segmented out an aneurysm from CT-images using VMTK. (a) Cross section (b) Isosurface (c) Zoom-in Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods Method Qualitative Analysis Simulation Quantitative Analysis Results Conclusion Method Three different meshes were created, all with about 1,300,000 cells. A pulsatile flow profile was set at inlet, with a heart rate of 75bpm. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods Method Qualitative Analysis Simulation Quantitative Analysis Results Conclusion Simulation Simulation of blood flow through aneurysm at 75 bpm. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
Background And Motivation Numerical Methods Method Qualitative Analysis Simulation Quantitative Analysis Results Conclusion Simulation Simulation of blood flow through aneurysm, at 1/5th of the speed. Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms
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