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
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
Øyvind Evju
(c) Saccular/
bifurcation
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.
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
often 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 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
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
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
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
1
+ u · ∇u = ∇ · 2ν(u) − ∇p
∂t
ρ
∇·u=0








for x ∈ Ω







u(x, 0) = 0
p(x, 0) = 0
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 = ∇ · 2ν(u) − ρ1 ∇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
An exact solution
Fully developed, steady state flow in a straight channel/cylinder.
Yields the exact solutions for velocity and WSS:
r 2 − a2 dp
4µ dx
a dp τw = 2 dx u=
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
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
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
Øyvind Evju
(c) Zoom-in
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
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%.
Appliying 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 predicted change in average WSS: 2.24%.
Largest predicted 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.
Predicted change in average WSS range from -3.2% to 5.2%.
Significantly higher average WSS is predicted (mean=1.56,
P=0.026).
Predicted change 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.
Predicted average WSS and maximum WSS 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 predicted change in average WSS: 36.15%.
Largest predicted 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
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 ur 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