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1Challenge the future
Modeling Electromagnetic Fields in Strongly Inhomogeneous MediaAn Application in MRIKirsten Koolstra, September 24th, 2015
2Challenge the future
IntroductionMagnetic Resonance Imaging (MRI)
www.neurensics.com/technische-specificaties
3Challenge the future
Introduction
𝐵0=1.5T
RF Interference in MRI
𝜆∝1𝐵0
𝐵0=3.0T
Brink et al., JMRI (2015)
4Challenge the future
5Challenge the future
IntroductionThe Effect of Dielectric Pads
De Heer et al., Magn Res Med (2012)
6Challenge the future
Introduction
Without pad With pad
With padWithout pad
The Effect of Dielectric Pads
De Heer et al., Magn Res Med (2012)
Brink et al., Invest Rad (2014)
7Challenge the future
IntroductionDesign Procedure: Numerical Modeling
Brink and Webb, Magn Res Med (2013)
8Challenge the future
Challenges
• Strong (localized) inhomogeneities in medium parameters
• Large computational domain due to the body model
• Accurate for low resolution!
• Fast!
• Take into account the boundary conditions
In Numerical Modeling
9Challenge the future
Goal
Obtain a solution that is
1. accurate
2. obtained within short computation time
Approach:
• Compare different discretization schemes for a simple test case
• Compare two iterative solvers, GMRES and IDR(s), to solve the discretized system
• Verify the results by performing human body simulations
10Challenge the future
The Volume Integral Equation
𝐄=𝐄 inc+(kb2+𝛻𝛻 ∙)𝐒 ( 𝜒 𝑒𝐄 )
𝐒( 𝐉)=∫Ω
❑
𝑔 (𝐱 ′ −𝐱 ) 𝐉 (𝐱 )d 𝐱
𝐄 inc
𝐄sc
11Challenge the future
Different Formulations
EVIE:
DVIE:
12Challenge the future
The Volume Integral Equation
𝐄 inc=𝐄− (kb2+𝛻𝛻 ∙)𝐒 ( 𝜒𝑒𝐄)
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
2𝐷
• The - and -components of the electric field are coupled via the operator.
• The vector potential depends on the material parameters.
13Challenge the future
The Method of Moments
1 2 3 4
𝑛
5
6
1 2 3 4
𝑛
5
6
𝐴𝐱=𝐛ℒ𝑢= 𝑓
14Challenge the future
The Method of MomentsApproximation of a Function
1 2 3 4 5 6 7 8 9
𝑓 (𝑥)
𝑥𝑖
𝑓 (𝑥 )=∑𝑖=1
9
𝑓 𝑖𝜑𝑖(𝑥 )
1. Specify
2. Find for all
3. Reconstruct
15Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
16Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
is solved via expanding
17Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
is solved via expanding
18Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
[𝒆𝒙
𝒆𝒚 ]=[𝒃𝒙
𝒃𝒚 ]• How do we incorporate the operator in the matrix ?
• How do we deal with the derivative terms?
𝐴𝛻𝛻 ∙𝐒=[ 𝜕
𝜕 𝑥𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑥
𝜕𝜕 𝑦
𝑆 𝑦
𝜕𝜕 𝑦
𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑦
𝜕𝜕 𝑦
𝑆𝑦 ]
19Challenge the future
The Method of MomentsFast Fourier Transform
Remember,
𝐒(𝜒𝑒𝐄)(𝐱′ )=∫Ω
❑
𝑔 (𝐱′−𝐱 ) 𝜒𝑒(𝐱)𝐄 (𝐱 )d 𝐱¿𝑔∗ 𝜒𝑒𝐄
¿ℱ {𝑔}ℱ {𝜒𝑒𝐄 }ℱ {𝐒 }=ℱ {𝑔∗ 𝜒 𝑒𝐄 }And
⟹𝐒=ℱ− 1 {ℱ {𝑔 }ℱ {𝜒𝑒𝐄} } .
So, use fast Fourier transform (FFT) algorithms to incorporate in the matrix !
20Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
[𝒆𝒙
𝒆𝒚 ]=[𝒃𝒙
𝒃𝒚 ]• How do we incorporate the operator in the matrix ?
• How do we deal with the derivative terms?
𝐴𝛻𝛻 ∙𝐒=[ 𝜕
𝜕 𝑥𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑥
𝜕𝜕 𝑦
𝑆 𝑦
𝜕𝜕 𝑦
𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑦
𝜕𝜕 𝑦
𝑆𝑦 ]
21Challenge the future
The Method of Moments
𝑥 𝑦
𝑥𝑦
Basis Functions: Rooftop 𝐸𝑥 (𝒙 )=∑𝑖=1
𝑛
𝑒𝑖𝑥𝜓 𝑖
𝑥 (𝒙 )
22Challenge the future
The Method of Moments
[𝐸𝑥i nc
𝐸𝑦i nc ]=[𝐸𝑥
𝐸 𝑦 ]− (kb2+𝛻𝛻 ∙) [𝑆𝑥(𝜒 𝑒𝐸𝑥)𝑆𝑦 (𝜒 𝑒𝐸 𝑦)]
Expansions
[𝒆𝒙
𝒆𝒚 ]=[𝒃𝒙
𝒃𝒚 ]• How do we incorporate the operator in the matrix ?
• How do we deal with the derivative terms?
𝐴𝛻𝛻 ∙𝐒=[ 𝜕
𝜕 𝑥𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑥
𝜕𝜕 𝑦
𝑆 𝑦
𝜕𝜕 𝑦
𝜕𝜕𝑥
𝑆𝑥+𝜕𝜕 𝑦
𝜕𝜕 𝑦
𝑆𝑦 ]
23Challenge the future
Central Difference SchemesOn staggered and non-staggered grids
Non-staggered grid Staggered grid
24Challenge the future
𝑚 ,𝑛𝑚 ,𝑛
Central Difference SchemesOn staggered and non-staggered grids
Non-staggered grid Staggered grid
𝑚 ,𝑛
25Challenge the future
Benchmark Problem
• TE-polarization
• Plane wave incident field
• Muscle/fat tissue
Scattering on a Two-Layer Conducting Cylinder
26Challenge the future
Recap
• Equations:
• Method:
• Benchmark Problem:
The Ingredients
Model
27Challenge the future
ResultsScattering on a Two-Layer Conducting Cylinder
28Challenge the future
ResultsComparison of EVIE and DVIE
29Challenge the future
ResultsScattering on a Two-Layer Conducting Cylinder
30Challenge the future
Scattering on a Circle vs Square
31Challenge the future
Results
Circle Square
Scattering on a Circle vs on a Square
32Challenge the future
Central Difference Schemes
StaggeredNon-Staggered
2n
d o
rder
schem
e4
th o
rder
schem
e
33Challenge the future
ResultsGlobal Error Propagation
34Challenge the future
Error Reduction
Original With smoothing
Smoothing the Contrast
35Challenge the future
Ori
gin
al
Sm
ooth
ed
ResultsThe Effect of Smoothing the Contrast
36Challenge the future
Original With smoothing
ResultsThe Effect of Smoothing along the Axes
37Challenge the future
Overview
ℒ𝑢= 𝑓 A 𝐱=𝐛 𝐱
𝐱𝑖+1=𝐱 𝑖+𝛂𝑖
Method of
Moments
IterativeSolver
Finding a Solution
?
38Challenge the future
Comparison of GMRES and IDR(s)
Properties of the Iterative Solver
𝑛=¿
𝑖=¿iteration number
number of unknowns
GMRES IDR(s)
Iterations until convergence
Work per iteration
39Challenge the future
Comparison of GMRES and IDR(s)
Properties of the Iterative Solver
𝑛=¿
𝑖=¿iteration number
number of unknowns
GMRES IDR(s)
Iterations until convergence
Work per iteration
40Challenge the future
Comparison of GMRES and IDR(s)
Properties of the Iterative Solver
𝑛=¿
𝑖=¿iteration number
number of unknowns
GMRES IDR(s)
Iterations until convergence
Work per iteration
41Challenge the future
Results
GMRES IDR(s)
Comparison of GMRES and IDR(s)
42Challenge the future
Human Body SimulationsScattering on a Human Body with Dielectric Pad
43Challenge the future
Human Body Simulations
High resolution
Staggered grid Non-staggered grid
Comparison of the staggered and non-staggered grid
Low resolution Low resolution
44Challenge the future
Conclusions
• Factors that influence the accuracy are the geometry and the mixed derivative terms.
• Smoothing improves the geometrical inaccuracies with the cost of computation time.
• The mixed derivative term has a large effect on the accuracy and is best approximated on a staggered grid.
• IDR(s) reduces the computation time considerably.
• Human body simulations are in agreement with the cylider test case simulations: the DVIE method on a staggered grid results in the most accurate solution on low resolution.
45Challenge the future
AbstractModeling electromagnetic fields in MRI involves two
main challenges: the solution has to be accurate and it
has to be obtained within short computation time.
The method of moments is used to discretize different
formulations of the volume integral equation
corresponding to Maxwell's equations.
The good performance of a staggered grid with respect
to a non-staggered grid shows that the way of treating
the mixed derivative terms is of great importance.
The performance of a higher order derivative scheme
on a non-staggered grid is close to the performance of
a staggered grid.
IDR(s) shows excellent performance in reducing the
computation time that is obtained with GMRES.
46Challenge the future
Function Spaces
EVIE: DVIE: JVIE:
where
47Challenge the future
Simulation Parameters
48Challenge the future
Computation Times
49Challenge the future
Convergence
50Challenge the future
Contrast Dependence
51Challenge the future
Smoothing the Contrast
𝜀𝑚 ,𝑛= ∑𝑅 (𝑚 ,𝑛 )
116
𝜀𝑝 ,𝑞
A Matlab Filter
52Challenge the future
The Electric Fields
53Challenge the future
The Electric Fields
54Challenge the future
Scattering on a Two-Layer CylinderLow Resolution Results
55Challenge the future
The Electric FieldsScattering on a Square-Shaped Object
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