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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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties Aim of this study

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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: ¡4-¿

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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: ¡4-¿

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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: ¡4-¿

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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/ sidewall (c) Saccular/ bifurcation

¡4-¿

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 every year.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 every year.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 every year.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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

• ften develop aneurysms.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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

• ften develop aneurysms.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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

• ften develop aneurysms.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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

• ften develop aneurysms.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties Aim of this study

Blood viscosity

The viscosity of blood is complex, and many models try to explain it.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Qualitative Analysis Quantitative Analysis Conclusion Aneurysms Blood Why simulate? Uncertainties 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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

The Navier-Stokes equations

∂u ∂t + u · ∇u = ∇ · 2νǫ(u) − 1 ρ∇p ∇ · u = 0 u(x, 0) = 0 p(x, 0) = 0

              

for x ∈ Ω u = 0 for x ∈ Γw u = u0 for x ∈ ΓI p = p0 for x ∈ ΓO

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

The Navier-Stokes equations

∂u ∂t + u · ∇u = ∇ · 2νǫ(u) − 1 ρ∇p ∇ · u = 0 u(x, 0) = 0 p(x, 0) = 0

              

for x ∈ Ω u = 0 for x ∈ Γw u = u0 for x ∈ ΓI p = p0 for x ∈ ΓO

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

The Navier-Stokes equations

∂u ∂t + u · ∇u = ∇ · 2νǫ(u) − 1 ρ∇p ∇ · u = 0 u(x, 0) = 0 p(x, 0) = 0

              

for x ∈ Ω u = 0 for x ∈ Γw u = u0 for x ∈ ΓI p = p0 for x ∈ ΓO

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

The Navier-Stokes equations

∂u ∂t + u · ∇u = ∇ · 2νǫ(u) − 1 ρ∇p ∇ · u = 0 u(x, 0) = 0 p(x, 0) = 0

              

for x ∈ Ω u = 0 for x ∈ Γw u = u0 for x ∈ ΓI p = p0 for x ∈ ΓO

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

Problem

∂u ∂t + u · ∇u

= ∇ · 2νǫ(u) − 1

ρ∇p

∇ · u = 0

• for x ∈ Ω.

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

Problem

∂u ∂t + u · ∇u

= ∇ · 2νǫ(u) − 1

ρ∇p

∇ · u = 0

• for x ∈ Ω.

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

Problem

∂u ∂t + u · ∇u

= ∇ · 2νǫ(u) − 1

ρ∇p

∇ · u = 0

• for x ∈ Ω.

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

Problem

∂u ∂t + u · ∇u

= ∇ · 2νǫ(u) − 1

ρ∇p

∇ · u = 0

• for x ∈ Ω.

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

An exact solution

Fully developed, steady state flow in a straight channel/cylinder. Yields the exact solutions for velocity and WSS: u = r 2 − a2 4µ dp dx τw =

• a

2 dp 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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

An exact solution

Fully developed, steady state flow in a straight channel/cylinder. Yields the exact solutions for velocity and WSS: u = r 2 − a2 4µ dp dx τw =

• a

2 dp 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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion The Mathematical Model Implementation Verification of implementation

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 Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Method

A single aneurysm was studied. The effects of three uncertainties were measured: Geometric effects Non-Newtonian effects Effects of different hematocrit levels

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Method

A single aneurysm was studied. The effects of three uncertainties were measured: Geometric effects Non-Newtonian effects Effects of different hematocrit levels

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

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 Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

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 Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

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 Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Simulation

Simulation of blood flow through aneurysm, at 1/5th of the speed.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Results

None of the three changes studied seem to have any significant effect on the overall flow pattern within the aneurysm. Locally, the effects on the velocity could be quite large. Areas of high flow velocity were less affected by the changes than areas of low velocity. Areas of high WSS were less affected by the changes than areas of low WSS. The effects were most prominent at systole.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Results

None of the three changes studied seem to have any significant effect on the overall flow pattern within the aneurysm. Locally, the effects on the velocity could be quite large. Areas of high flow velocity were less affected by the changes than areas of low velocity. Areas of high WSS were less affected by the changes than areas of low WSS. The effects were most prominent at systole.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Results

None of the three changes studied seem to have any significant effect on the overall flow pattern within the aneurysm. Locally, the effects on the velocity could be quite large. Areas of high flow velocity were less affected by the changes than areas of low velocity. Areas of high WSS were less affected by the changes than areas of low WSS. The effects were most prominent at systole.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Results

None of the three changes studied seem to have any significant effect on the overall flow pattern within the aneurysm. Locally, the effects on the velocity could be quite large. Areas of high flow velocity were less affected by the changes than areas of low velocity. Areas of high WSS were less affected by the changes than areas of low WSS. The effects were most prominent at systole.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Method Simulation Results

Results

None of the three changes studied seem to have any significant effect on the overall flow pattern within the aneurysm. Locally, the effects on the velocity could be quite large. Areas of high flow velocity were less affected by the changes than areas of low velocity. Areas of high WSS were less affected by the changes than areas of low WSS. The effects were most prominent at systole.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

The aneurysms

(a) M1 (b) M2 (c) M3 (d) M5 (e) M8 (f) M9 (g) M11 (h) M12 (i) M15 (j) M16 (k) M18 (l) M20

Large variation in sizes and types. Used in a previous study at Simula.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects studied

The four different effects which sensitivity analysis has been performed on are Neglecting the shear thinning (non-Newtonian) behaviour of blood. An increase in the hematocrit level from 38% to 40%. Increasing the inlet flux by 33%. Applying a different set of outlet boundary conditions for the pressure.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects studied

The four different effects which sensitivity analysis has been performed on are Neglecting the shear thinning (non-Newtonian) behaviour of blood. An increase in the hematocrit level from 38% to 40%. Increasing the inlet flux by 33%. Applying a different set of outlet boundary conditions for the pressure.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects studied

The four different effects which sensitivity analysis has been performed on are Neglecting the shear thinning (non-Newtonian) behaviour of blood. An increase in the hematocrit level from 38% to 40%. Increasing the inlet flux by 33%. Applying a different set of outlet boundary conditions for the pressure.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects studied

The four different effects which sensitivity analysis has been performed on are Neglecting the shear thinning (non-Newtonian) behaviour of blood. An increase in the hematocrit level from 38% to 40%. Increasing the inlet flux by 33%. Applying a different set of outlet boundary conditions for the pressure.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Non-Newtonian effects

Method: Comparing a Casson viscosity model to a reference Newtonian viscosity model with the same asymptotic viscosity at the limit of infinite shear rate.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Non-Newtonian effects - Results

Large differences between the aneurysms:

Aneurysms with a high average shear rate shows little non-Newtonian effects. Differences seems to be largest at diastole and early systole.

Largest change in average WSS: 2.24%. Largest change in maximum WSS: -7.14%. Including non-Newtonian effects predicts a significantly lower maximum WSS (mean=-2.31%, P=0.0091).

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Non-Newtonian effects - Results

Large differences between the aneurysms:

Aneurysms with a high average shear rate shows little non-Newtonian effects. Differences seems to be largest at diastole and early systole.

Largest change in average WSS: 2.24%. Largest change in maximum WSS: -7.14%. Including non-Newtonian effects predicts a significantly lower maximum WSS (mean=-2.31%, P=0.0091).

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Non-Newtonian effects - Results

Large differences between the aneurysms:

Aneurysms with a high average shear rate shows little non-Newtonian effects. Differences seems to be largest at diastole and early systole.

Largest change in average WSS: 2.24%. Largest change in maximum WSS: -7.14%. Including non-Newtonian effects predicts a significantly lower maximum WSS (mean=-2.31%, P=0.0091).

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Non-Newtonian effects - Results

Large differences between the aneurysms:

Aneurysms with a high average shear rate shows little non-Newtonian effects. Differences seems to be largest at diastole and early systole.

Largest change in average WSS: 2.24%. Largest change in maximum WSS: -7.14%. Including non-Newtonian effects predicts a significantly lower maximum WSS (mean=-2.31%, P=0.0091).

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased hematocrit

Physiological motivation: The increase of two percentage points is an average increase seen in women going through menopause. The average age of menopause for women is 51.7 years, and the average age of aneurysm rupture is 52 years. This triggers a hypothesis of a correlation. Method: Using the Casson viscosity model (which incorporates the hematocrit level) the hematocrit is increased from 38% to 40%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased hematocrit - Results

Large differences between the aneurysms. Changes in average WSS range from -3.2% to 5.2%. Significantly higher average WSS is predicted (mean=1.56, P=0.026). Changes in maximum WSS range from -12.7% to 5.7%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased hematocrit - Results

Large differences between the aneurysms. Changes in average WSS range from -3.2% to 5.2%. Significantly higher average WSS is predicted (mean=1.56, P=0.026). Changes in maximum WSS range from -12.7% to 5.7%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased hematocrit - Results

Large differences between the aneurysms. Changes in average WSS range from -3.2% to 5.2%. Significantly higher average WSS is predicted (mean=1.56, P=0.026). Changes in maximum WSS range from -12.7% to 5.7%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased hematocrit - Results

Large differences between the aneurysms. Changes in average WSS range from -3.2% to 5.2%. Significantly higher average WSS is predicted (mean=1.56, P=0.026). Changes in maximum WSS range from -12.7% to 5.7%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow

Method: Adjusting the inlet spatial peak velocity from an average of 535mm/s to 695mm/s. This corresponds to an increase in inlet flux of 33%.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow - Results

The changes in WSS deviate highly from an expected linear relation between inlet flux and WSS. Measured average WSS and maximum WSS is increased by an average of 72.8% and 73.6% respectively. The changes in WSS seems greater within the aneurysm than in the surrounding arteries. All changes are highly significant. All aneurysms showed the same tendencies.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow - Results

The changes in WSS deviate highly from an expected linear relation between inlet flux and WSS. Measured average WSS and maximum WSS is increased by an average of 72.8% and 73.6% respectively. The changes in WSS seems greater within the aneurysm than in the surrounding arteries. All changes are highly significant. All aneurysms showed the same tendencies.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow - Results

The changes in WSS deviate highly from an expected linear relation between inlet flux and WSS. Measured average WSS and maximum WSS is increased by an average of 72.8% and 73.6% respectively. The changes in WSS seems greater within the aneurysm than in the surrounding arteries. All changes are highly significant. All aneurysms showed the same tendencies.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow - Results

The changes in WSS deviate highly from an expected linear relation between inlet flux and WSS. Measured average WSS and maximum WSS is increased by an average of 72.8% and 73.6% respectively. The changes in WSS seems greater within the aneurysm than in the surrounding arteries. All changes are highly significant. All aneurysms showed the same tendencies.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of increased inflow - Results

The changes in WSS deviate highly from an expected linear relation between inlet flux and WSS. Measured average WSS and maximum WSS is increased by an average of 72.8% and 73.6% respectively. The changes in WSS seems greater within the aneurysm than in the surrounding arteries. All changes are highly significant. All aneurysms showed the same tendencies.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition

Method: Changing the outlet boundary conditions for pressure from a resistance boundary condition to a zero-pressure boundary condition.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition - Results

Differences in outlet flux of up to 231.6%. Significant changes in flow pattern in and after the bifurcation. Largest change in average WSS: 36.15%. Largest change in maximum WSS: 56.92%. Very large variation in the prediction of WSS.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition - Results

Differences in outlet flux of up to 231.6%. Significant changes in flow pattern in and after the bifurcation. Largest change in average WSS: 36.15%. Largest change in maximum WSS: 56.92%. Very large variation in the prediction of WSS.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition - Results

Differences in outlet flux of up to 231.6%. Significant changes in flow pattern in and after the bifurcation. Largest change in average WSS: 36.15%. Largest change in maximum WSS: 56.92%. Very large variation in the prediction of WSS.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition - Results

Differences in outlet flux of up to 231.6%. Significant changes in flow pattern in and after the bifurcation. Largest change in average WSS: 36.15%. Largest change in maximum WSS: 56.92%. Very large variation in the prediction of WSS.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion Non-Newtonian effects Effects of increased hematocrit Effects of increased inflow Effects of a different outlet boundary condition

Effects of a different outlet boundary condition - Results

Differences in outlet flux of up to 231.6%. Significant changes in flow pattern in and after the bifurcation. Largest change in average WSS: 36.15%. Largest change in maximum WSS: 56.92%. Very large variation in the prediction of WSS.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Layout

1

Background And Motivation

2

Numerical Methods

3

Qualitative Analysis

4

Quantitative Analysis

5

Conclusion

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Uncertainties regarding boundary conditions are far more important than uncertainties in flow parameters. Patient specific boundary data is absolutely necessary to accurately simulate cerebral blood flow accurately. Simulations might still be relevant even without patient specific boundary data. Non-Newtonian effects may safely be neglected if the boundary data is unknown or uncertain. The WSS is unexpectedly sensitive to changes in inlet flux.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Uncertainties regarding boundary conditions are far more important than uncertainties in flow parameters. Patient specific boundary data is absolutely necessary to accurately simulate cerebral blood flow accurately. Simulations might still be relevant even without patient specific boundary data. Non-Newtonian effects may safely be neglected if the boundary data is unknown or uncertain. The WSS is unexpectedly sensitive to changes in inlet flux.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Uncertainties regarding boundary conditions are far more important than uncertainties in flow parameters. Patient specific boundary data is absolutely necessary to accurately simulate cerebral blood flow accurately. Simulations might still be relevant even without patient specific boundary data. Non-Newtonian effects may safely be neglected if the boundary data is unknown or uncertain. The WSS is unexpectedly sensitive to changes in inlet flux.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Uncertainties regarding boundary conditions are far more important than uncertainties in flow parameters. Patient specific boundary data is absolutely necessary to accurately simulate cerebral blood flow accurately. Simulations might still be relevant even without patient specific boundary data. Non-Newtonian effects may safely be neglected if the boundary data is unknown or uncertain. The WSS is unexpectedly sensitive to changes in inlet flux.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Uncertainties regarding boundary conditions are far more important than uncertainties in flow parameters. Patient specific boundary data is absolutely necessary to accurately simulate cerebral blood flow accurately. Simulations might still be relevant even without patient specific boundary data. Non-Newtonian effects may safely be neglected if the boundary data is unknown or uncertain. The WSS is unexpectedly sensitive to changes in inlet flux.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms

Background And Motivation Numerical Methods Qualitative Analysis Quantitative Analysis Conclusion

Thank you.

Øyvind Evju SA of Simulated Blood Flow in Cerebral Aneurysms