1. Modeling Electromagnetic Fields
An Application in MRI Kirsten Koolstra
April 22nd, 2015

2. Introduction
Magnetic Resonance Imaging (MRI)

3. Introduction
RF Interference in MRI
𝜆∝ 1 𝐵 0
𝐵 0 =1.5 𝑇
𝐵 0 =3.0 𝑇

5. Introduction
The Effect of Dielectric Pads

6. Introduction Without pad With pad Without pad With pad
The Effect of Dielectric Pads
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7. Introduction
Design Procedure: Numerical Modeling

8. Challenges Strong (localised) inhomogeneities in medium parameters
Large computational domain due to the body model
Accurate for low resolution!
Fast!
Take into account the boundary conditions

9. The Volume Integral Equation (VIE)
𝑬 𝑖𝑛𝑐 =𝑬− 𝑘 𝑏 2 +𝛻𝛻∙ 𝑺 𝜒 𝑒 𝑬
𝑺(𝑱)= Ω 𝑔 𝒙′−𝒙 𝑱 𝒙 𝑑𝒙

10. Different Formulations
𝑫 𝑐 = 𝜀 𝑐 𝑬 𝑱= 𝜂 0 𝜒 𝑒 𝑬
𝒩= 𝑘 𝑏 2 +𝛻𝛻∙
EVIE: 𝑬 𝑖𝑛𝑐 = 𝑬 − 𝒩𝑺 𝜒 𝑒 𝑬 DVIE: 𝑬 𝑖𝑛𝑐 = 1 𝜀 𝑐 𝑫 𝑐 − 𝒩𝑺( 𝜒 𝑒 𝜀 𝑐 𝑫 𝑐 ) JVIE: 𝜂 0 𝜒 𝑒 𝑬 𝑖𝑛𝑐 = 𝑱 − 𝒩𝑺(𝑱) 𝜒 𝑒

11. The Method of Moments ℒ𝑢=𝑓 Derive weak form ℒ𝑢,𝜂 = 𝑓,𝜂 .
Define a mesh.
Expand the unknown 𝑢 through basis functions φ 𝑖 .
Choose test functions 𝜂 𝑖 .
Calculate/approximate the integrals.
Define the discretised system
𝑎 𝑥𝑥 𝑎 𝑥𝑦 𝑎 𝑦𝑥 𝑎 𝑦𝑦 𝒆 𝑥 𝒆 𝑦 = 𝒃 𝑥 𝒃 𝑦 .

12. Test Functions Callocation Method: 𝜂 𝑖 𝒙 = 𝛿 𝒙− 𝒙 𝑖
Galerkin’s Method: 𝜂 𝑖 𝒙 = 𝜑 𝑖 (𝒙)

13. Basis Functions
Rooftop 𝑥 𝑦 𝑥 𝑦

14. Basis Functions

15. A Closer Look EVIE Formulation
EVIE: 𝑬 𝑖𝑛𝑐 = 𝑬 − 𝒩𝑺 𝜒 𝑒 𝑬 DVIE: 𝑬 𝑖𝑛𝑐 = 1 𝜀 𝑐 𝑫 𝑐 − 𝒩𝑺( 𝜒 𝑒 𝜀 𝑐 𝑫 𝑐 ) JVIE: 𝜂 0 𝜒 𝑒 𝑬 𝑖𝑛𝑐 = 𝑱 − 𝒩𝑺(𝑱) 𝜒 𝑒

16. A Closer Look 𝑥 𝑦 Scattering on a Two-Layered Cylinder TE-polarisation
𝐸 𝑖𝑛𝑐 =− 𝑒 −𝑖 𝑘 𝑏 𝑦
𝑓=128 𝐻𝑧 𝑥 𝑦

17. A Closer Look
Scattering on a Two-Layered Cylinder

18. Some Observations Contrast: Low contrast vs high contrast
Basis functions: Rooftop vs 2 x linear
Geometry: Cylinder vs square

19. Observation 1
Low Contrast vs High Contrast

20. Observation 1
Low Contrast vs High Contrast

21. Observation 2
Rooftop Expansion vs Linear Expansion

22. Observation 2
Rooftop Expansion vs Linear Expansion

23. Observation 3
Two-Layered Cylinder vs Two-Layered Square

24. Observation 3
Two-Layered Cylinder vs Two-Layered Square

25. What is the cause of these jumps?
Basis functions?
Formulation?
Geometry?

26. Approach
Compare the EVIE formulation with the DVIE and JVIE formulations.
Analysing contrast dependency.
Find out what happens for different geometries.
Compare the performance of GMRES and IDR(s) methods.
Accurately
Fast

27. Function Spaces
EVIE: 𝐻 𝑐𝑢𝑟𝑙, ℝ 3 ⟼ 𝐻 𝑐𝑢𝑟𝑙, ℝ 3 DVIE: 𝐻 𝑑𝑖𝑣, ℝ 3 ⟼ 𝐻 𝑐𝑢𝑟𝑙, ℝ 3 JVIE: 𝐿 2 ℝ 3 3 ⟼ 𝐿 2 ℝ 3 3 where 𝐻 𝑐𝑢𝑟𝑙, ℝ 3 = 𝑓 𝑓∈ 𝐿 2 ℝ 3 ∧ 𝛻×𝑓∈ 𝐿 2 ℝ 3 𝐻 𝑑𝑖𝑣, ℝ 3 = 𝑓 𝑓∈ 𝐿 2 ℝ 3 ∧ 𝛻∙𝑓∈ 𝐿 2 ℝ 3