1 Challenge the future Modeling Electromagnetic Fields in Strongly Inhomogeneous Media An...

1Challenge the future

Modeling Electromagnetic Fields in Strongly Inhomogeneous MediaAn Application in MRIKirsten Koolstra, September 24th, 2015


IntroductionMagnetic Resonance Imaging (MRI)

www.neurensics.com/technische-specificaties


Introduction

𝐵0=1.5T

RF Interference in MRI

𝜆∝1𝐵0

𝐵0=3.0T

Brink et al., JMRI (2015)


IntroductionThe Effect of Dielectric Pads

De Heer et al., Magn Res Med (2012)


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)


IntroductionDesign Procedure: Numerical Modeling

Brink and Webb, Magn Res Med (2013)


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


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


The Volume Integral Equation

𝐄=𝐄 inc+(kb2+𝛻𝛻 ∙)𝐒 ( 𝜒 𝑒𝐄 )

𝐒( 𝐉)=∫Ω

❑

𝑔 (𝐱 ′ −𝐱 ) 𝐉 (𝐱 )d 𝐱

𝐄 inc

𝐄sc


Different Formulations

EVIE:

DVIE:


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.


The Method of Moments

1 2 3 4

𝑛

5

6

1 2 3 4

𝑛

5

6

𝐴𝐱=𝐛ℒ𝑢= 𝑓


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



[𝐸𝑥i nc



Expansions



[𝐸𝑥i nc



Expansions

is solved via expanding



[𝐸𝑥i nc



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 ]


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 !



[𝐸𝑥i nc



Expansions

[𝒆𝒙




𝐴𝛻𝛻 ∙𝐒=[ 𝜕

𝜕 𝑥𝜕𝜕𝑥


𝜕𝜕 𝑦

𝑆 𝑦

𝜕𝜕 𝑦

𝜕𝜕𝑥


𝜕𝜕 𝑦

𝑆𝑦 ]



𝑥 𝑦

𝑥𝑦

Basis Functions: Rooftop 𝐸𝑥 (𝒙 )=∑𝑖=1

𝑛

𝑒𝑖𝑥𝜓 𝑖

𝑥 (𝒙 )



[𝐸𝑥i nc



Expansions

[𝒆𝒙




𝐴𝛻𝛻 ∙𝐒=[ 𝜕

𝜕 𝑥𝜕𝜕𝑥


𝜕𝜕 𝑦

𝑆 𝑦

𝜕𝜕 𝑦

𝜕𝜕𝑥


𝜕𝜕 𝑦

𝑆𝑦 ]


Central Difference SchemesOn staggered and non-staggered grids

Non-staggered grid Staggered grid


𝑚 ,𝑛𝑚 ,𝑛

Central Difference SchemesOn staggered and non-staggered grids

Non-staggered grid Staggered grid

𝑚 ,𝑛


Benchmark Problem

• TE-polarization

• Plane wave incident field

• Muscle/fat tissue

Scattering on a Two-Layer Conducting Cylinder


Recap

• Equations:

• Method:

• Benchmark Problem:

The Ingredients

Model


ResultsScattering on a Two-Layer Conducting Cylinder


ResultsComparison of EVIE and DVIE


ResultsScattering on a Two-Layer Conducting Cylinder


Scattering on a Circle vs Square


Results

Circle Square

Scattering on a Circle vs on a Square


Central Difference Schemes

StaggeredNon-Staggered

2n

d o

rder

schem

e4

th o

rder

schem

e


ResultsGlobal Error Propagation


Error Reduction

Original With smoothing

Smoothing the Contrast


Ori

gin

al

Sm

ooth

ed

ResultsThe Effect of Smoothing the Contrast


Original With smoothing

ResultsThe Effect of Smoothing along the Axes


Overview

ℒ𝑢= 𝑓 A 𝐱=𝐛 𝐱

𝐱𝑖+1=𝐱 𝑖+𝛂𝑖

Method of

Moments

IterativeSolver

Finding a Solution

?


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




𝑛=¿


number of unknowns

GMRES IDR(s)


Work per iteration


Results

GMRES IDR(s)



Human Body SimulationsScattering on a Human Body with Dielectric Pad


Human Body Simulations

High resolution

Staggered grid Non-staggered grid

Comparison of the staggered and non-staggered grid

Low resolution Low resolution


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.


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.


Function Spaces

EVIE: DVIE: JVIE:

where


Simulation Parameters


Computation Times


Convergence


Contrast Dependence


Smoothing the Contrast

𝜀𝑚 ,𝑛= ∑𝑅 (𝑚 ,𝑛 )

116

𝜀𝑝 ,𝑞

A Matlab Filter


The Electric Fields


Scattering on a Two-Layer CylinderLow Resolution Results


The Electric FieldsScattering on a Square-Shaped Object

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