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C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Applications of Spin Echo and Gradient Echo:
Diffusion and Susceptibility Contrast
Chunlei Liu, PhD
Department of Electrical Engineering & Computer Sciences
and Helen Wills Neuroscience Institute
University of California, Berkeley, CA
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R.F.
Gz
Gy
Gx
90º 180º
Readout
𝑇𝐸2
𝑇𝐸2
90º 180º
Readout
𝑇𝑅
R.F.
Gz
Gy
Gx
θ
Readout
𝑇𝐸
𝑇𝑅
θ
Readout
Spin Echo
Grad Echo
Review of Spin Echo and Gradient Echo
2
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Outline• Spin echo: diffusion contrast and quantification
• Diffusion-weighted imaging (DWI)
• Diffusion-tensor imaging (DTI)
• Diffusion fiber tractography
• Gradient echo: magnetic susceptibility contrast and quantification
• T2* weighting and blood oxygen level dependent (BOLD) contrast
• Susceptibility weighted imaging (SWI)
• Quantitative susceptibility mapping (QSM)
3
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
It’s all about phase!!!!
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
1.1 Diffusion-Weighted Imaging
Spin Echo
4
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
75% in skeletal muscle
78% in brain
Water molecules are at constant
random movement; described by a
diffusion coefficient D.
2 2x Dt
Water in Brain and Muscle
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R.F.
Gz
Gy
Gx
90º 180º
Start End
m(0) m(b)
large diffusion coefficient
small diffusion coefficient
2 2x Dt
G
𝑏 = 𝛾2𝐺2𝛿2(∆ − 𝛿 3)𝑚 𝑏 = 𝑚 0 exp(−𝑏𝐷)
Diffusion Encoding with Single-Shot EPI
5
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R.F.
Gz
Gy
Gx
90ºx 180ºx
M0
x
y
z
Equilibrium
M0
Excitation Refocusing Rephasing
Spin Echo
m
x
y
z
x
y
m
x
y
z
y
x
Field inhomogeneity:
Dephasing
x
y
M0
Static spin sees the same
field inhomogeneity:
Spin Echo Without Diffusion Encoding
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
90o Spin Echo
No Diffusion: running at constant speed
180o
Spin Echo Without Diffusion Encoding
spin1
spin2
spin3
6
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
90o Spin Echo
No Diffusion: running at constant speed
180o
Spin Echo Without Diffusion Encoding
spin1
spin2
spin3
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R.F.
Gz
Gy
Gx
90ºx 180ºx
M0
x
y
z
M0 m
x
y
z
y
x
Equilibrium Excitation Dephasing Refocusing
Static spins: Rephasing
Moving spins: Dephasing
Spin Echo
m
x
y
z
x
y
x
y
M0
x
y
Spin Echo With Diffusion Encoding
7
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
90o Spin EchoG G
With Diffusion: running when drunk
180o
Spin Echo With Diffusion Encoding
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
90o G Spin EchoG
With Diffusion: running when drunk
180o
Spin Echo With Diffusion Encoding
8
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2 2 2 1( )3
( ) (0)exp
b G
m b m bD
2C D Ct
23 01 2
2 1
M MM MD
t T T
i jMM B k M
Fick’s Second Law
C is spin density
Each spin carries magnetic moment.
Magnetization is proportional to spin density.
This equation can be solved using standard methods for solving partial
differential equations. For a spin echo sequence, the solution for transverse
magnetization is given by
Derive Diffusion Signal with Bloch Equation
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
90o 180o
1 ( )left
t dt G r
2 1( )
0 0( ) exp( )jM M e p d M b D r r
G G
b: b-value D: diffusion coefficient
2
21
( )4
r
Dtp eDt
r
Spin Echo
B0+G·r
Probability Distribution Function of diffusion
2 ( )right
t dt G r
Derive Diffusion Signal with Statistics
9
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
b = 0
No diffusion weighting
b = 1000 s/mm2
Diffusion weighting
D (mm2/s)
Computed diffusion coefficient
•Need a minimal of 2 measurements at 2 different b-values to computed D.
•Diffusion coefficient is commonly referred to as “apparent diffusion coefficient” (ADC)
•ADC of free water at room temperature: 2.2x10-3 mm2/s
•ADC of brain tissue around 1.0x10-3 mm2/s
Diffusion-Weighted Imaging
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Why Single-Shot EPI?Gy
Gx
K-Space
iFFT
DWI measures molecular diffusion ~ 10 µm during imaging window;
Bulk motion (body motion, breathing, cardiac, brain pulsation) ~ 1 mm, introducing
more phase than diffusion; varies from TR to TR.
10
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
If Acquire One k-Space Line per TR
𝑚 𝐫, 𝑏 = 𝑚(𝐫, 0)𝑒−𝑏𝐷𝑒𝑗𝜑1(𝐫,TR1)
Gy
Gx
Single-Shot
…
𝑚 𝐫, 𝑏 = 𝑚(𝐫, 0)𝑒−𝑏𝐷𝑒𝑗𝜑2(𝐫,TR1)
𝑚 𝐫, 𝑏 = 𝑚(𝐫, 0)𝑒−𝑏𝐷𝑒𝑗𝜑3(𝐫,TR1)
Inconsistent k-space data causing aliasing
Spatial varying signal cancellationHow to address it?
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
1.2 Diffusion-Tensor Imaging and Tractography
Spin Echo
11
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R.F.
Gz
Gy
90º 180º
Gx
Diffusion encoding gradients can be applied in either one of the three axis or a
combination of axis. The gradients are represented by a vector (Gx Gy Gz).
(1 1 0) (1 -1 0) 0.10
1.00
0 200 400 600 800 1000 1200
b(s/mm2)
log
(s)
(a) right splenium corpus callosum
(b) right splenium corpus callosum
D = 1.5x10-3
mm2/s
D = 0.55x10-3
mm2/s
Anisotropic Diffusion: Orientation Dependent
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
xx xy xz
xy yy yz
xz yz zz
D D D
D D D
D D D
Scalar D
similar molecular
displacements in all directions
greater molecular displacement
along cylinders than across
Isotropic Anisotropic
Mathematical Models of Diffusion
12
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Diffusion Tensor Signal Model
𝑏 = 𝛾2𝐺2𝛿2(∆ − 𝛿 3)
𝑚 𝑏 = 𝑚 0 exp(−𝑏𝐷)
Scalar Diffusion
𝑏𝑖𝑗 = 𝛾2𝐺𝑖𝐺𝑗𝛿
2(∆ − 𝛿 3)
𝑚 𝑏𝑖𝑗 = 𝑚 0 exp(−𝑏𝑖𝑗𝐷𝑖𝑗)
Tensor Diffusion
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
1 2 1 2
( ) (0)exp i i i im b m b D
Cerebral
Spinal
FluidInstead of a diffusion coefficient, we have a diagonal diffusion tensor, a 3x3 matrix
Probability Density Function
Isotropic Diffusion
𝐷𝑥𝑥 0 00 𝐷𝑦𝑦 0
0 0 𝐷𝑧𝑧
13
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Anisotropic Diffusion
1 2 1 2
( ) (0)exp i i i im b m b D
xx xy xz
xy yy yz
xz yz zz
D D D
D D D
D D D
Instead of a diffusion coefficient, we have a diffusion tensor, a 3x3 matrix
Probability Density Function
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Bloch Equation with Diffusion Term
2 2 1( ),3
( ) (0)exp ,
ij i j ij ji
ij ij ij ji
b G G b b
m b m b D D D symmetric positive definite
ij ij
CD C
t
3 01 2
2 1
ij ij
M MM MD
t T T
i jMM B k M
Fick’s Second Law
C is spin density; Einstein
summation rule.
Each spin carries magnetic moment.
Magnetization is proportional to spin density.
This equation can be solved using standard methods for solving partial
differential equations. For a spin echo sequence, the solution for transverse
magnetization is given by
14
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2 1( )
0 0
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
,
( ) exp( )
,
Tensor Product
j
ij ij
ij ij ij ij
i j
M M e p d M b D
b b b D D D
b b b D D D
b b b D D D
b D b D
r r
b D
b : D
2
3 2
1( )
(4 )
covariance matrix: =2 t
tp et
Tr Dr
rD
Σ D
Probability Distribution Function of anisotropic diffusion
For diffusion-weighted spin-echo sequence, echo amplitude is
Statistical Interpretation for Anisotropic Diffusion
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2 2 2
11 12 21 22
11 12 22
1 21
1 2 , ( )3
0
1 2 1 2 0
1 2 1 2 0
0 0 0
( )
( 2 )
G b G
b
b D D D D
b D D D
G
b
b : D
Example
15
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Determine Diffusion Tensor Experimentally
11 11 12 12 13 31 22 22 23 23 33 33
(0)ln 2 2 2
( )ij ij
ms b D b D b D b D b D b D b D
m b
11
(1) (1) (1) (1) (1) (1)(1)1211 12 13 22 23 33
(2) (2) (2) (2) (2) (2)(2)1311 12 13 22 23 33
22
( ) ( ) ( ) ( ) ( ) ( )( )2311 12 13 22 23 33
33
2 2 2
2 2 2
2 2 2n n n n n nn
D
Db b b b b bs
Db b b b b bs
D
Db b b b b bs
D
M M M M M MM
D is a symmetric tensor. It has six unknowns. A minimal of six non-colinear
measurements are required to determine a diffusion tensor. Different
measurements are achieved by varying the diffusion encoding gradients
including both amplitude and direction.
Rows have to be independent.
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
One Simple Encoding Scheme
(1 1 0)
(1 0 1)
(0 1 1)
(1 -1 0)
(1 0 -1)
(0 1 -1) x
y
z
R.F.
Gz
Gy
Gx
90º 180º
(1 1 0)
16
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Eigen Decomposition
1 2 3
1
2 1 2 3
3
1 2 3
, , ,
0 0
0 0 , , ,
0 0
3
U U U are eigenvectors
are eigenvalues
mean diffusivity
T
1 2 3
D = UΛU
U = U U U
Λ
D is coordinate system dependent. If the subject rotates in the magnet,
the measured diffusion tensor will be different.
Eigen decomposition defines rotation invariant quantities.
Diffusion Ellipsoid
U2
U1
U3
1
Matlab function: eig().
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Fractional Anisotropy (FA)
2 2 2
1 2 3
2 2 2
1 2 3
3(( ) ( ) ( ) )
2( )FA
x
y
z
Fractional Anisotropy (FA): a measure of diffusion anisotropy, 0
17
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
DTI Fiber Tractography
x
y
z
0 0
0 0
0 0
x
y
zFiber Tractography: a representation of 3D white matter fiber structure.
Fractional Anisotropy (FA)
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
• Diffusion-weighted imaging is created by applying diffusion encoding gradients
• Tissue contrast is based difference in diffusion coefficient
• Diffusion-tensor imaging measures the orientation dependent diffusion coefficient
• Major eigenvector of a diffusion tensor is parallel to white matter fiber
Summary
18
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2.1 T2*-Weighting and BOLD
Gradient Echo
R.F.
Gz
Gy
Gx
θ
Readout
𝑇𝐸
𝑇𝑅
θ
Readout
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnitude and Phase of Gradient Echo
abs(image) angle(image)
Phase is due to offset in Larmor frequency. Different voxel has different frequency,
consequently, accumulates different phase angle over time. This frequency offset is
mainly due to field inhomogeneity caused by magnetic susceptibility variations.
19
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
What is Magnetic Susceptibility?Magnetic susceptibility is a physical quantity that measures the extent to which a
material is magnetized by an applied magnetic field.
M – magnetization vector
H – Magnetic field vector
χ – volume magnetic susceptibility (unitless in SI units)
B – magnetic flux density vector, or magnetic induction
μ0 – vacuum permeability
H
M
M = χ H
H
B
B = μ0 (1+χ) H
appliedapplied
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Paramagnetic vs. DiamagneticH
M
M = χ H
H
B
B = μ0 (1+χ) H
H
M
M = χ H
H
B
B = μ0 (1+χ) H
Paramagnetic
χ > 0
Diamagnetic
χ < 0
20
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnetic Susceptibility in MRI
B0=0H0
B0 is perturbed by local magnetization induced by susceptibility
0m H
m B
magnetization susceptibility
0 B B B
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Induced Tissue Contrast
RFEcho 1 Echo 2 Echo n……
21
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
𝑚 𝐤 = 𝑚 𝐫 𝑒−𝑖2𝜋𝐤∙𝐫𝑑𝐫Ideal Case
𝑚 𝐤 = 𝑚 𝐫 𝑒−𝑖𝛾𝛿𝐵(𝐫)(𝑇𝐸+𝑡)𝑒−𝑖2𝜋𝐤∙𝐫𝑑𝐫Inhomogeneity
t is k dependent
𝑚 𝐫 = 𝑚 𝐤 𝑒𝑖2𝜋𝐤∙𝐫𝑑𝐤Image Recon
Susceptibility Induces Field Inhomogeneity
θReadout
𝑇𝐸
𝑡
𝑚 𝐫 = 𝑚(𝐫) 𝑒−𝑖𝛾𝛿𝐵(𝐫)𝑇𝐸If t
22
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnitude: T2* Decay
2
0( )t T
S t S e
OR
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
R2* Mapping and Contrast
R2* 0 40 Hz
T2* 25 ms
White matter
Blood vessels
Globus pallidusSubstatia nigra
Red nucleus
23
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Blood-Oxygen-Level-Dependent Signal
Diamagnetic Hb
Paramagnetic Hbr
Ogawa S. et al, 1992
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2.2 Phase Images of Gradient Echo
24
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
What Is in the Phase?
Sources of phase
Receiver coil
Objects outside the FOV
Objects inside the FOV
Phase wraps
background
tissue
background
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Phase Unwrapping
Extensively researched; perfect solution still lacking
Examples:
3D SRNCP or BP-ASLH. Abdul-Rahamn, et al, "Fast And Robust Three-Dimensional Best Path Phase Unwrapping
Algorithm", Applied Optics, Vol. 46, No. 26, pp. 6623-6635, 2007
FSL PRELUDE, University of Oxford
25
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Laplacian “Phase Unwrapping”
Totally automatic; Fast; Guarantee continuity
Remove phase originated from sources outside FOV
Li W. et al, NeuroImage 2011; 55: 1645-1656
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Filter Background: Sphere mean value property
Harmonic function
Mean phase over a sphere
Phase at the center
S
26
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Summary of Phase Processing
unwrapping Filtering
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2.3 Susceptibility Weighted Imaging (SWI)
27
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Weighted Imaging
GRE Phase GRE Magn
Unwrapping
Filtering Phase Mask
X
SWI = Magn*(PhaseMask)4
SWI
SWI MIP
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Weighted Imaging
Courtesy of Juergen Reichenbach
28
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Pitfalls of SWI
high sensitivity (micro-lesions)
low specificity (hypointense)
venous blood, iron, calcium
qualitative, not quantitative
Is this bleeding?
Bo
Bo
Phase is orientation dependent
Phase
SWI
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
2.4 Quantitative Susceptibility Mapping (QSM)
29
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
m
B
Magnetic Field ChangeSusceptibility Source
0m H
B0
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
B0
m
B
Magnetic Field ChangeSusceptibility Source
0m H
30
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnetic Field ChangeSusceptibility Source
?
B0
m
B
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Find the Demagnetizing Field h
0 0 (1 )( ) 0B χ H h
( ) 0 H• • χ •h
0 0 •B
H0
h
?
𝐡 = −𝐹𝑇−1 𝐤𝑘𝑧𝑘2
𝜒 𝐤 𝐻0
Magnetic flux density distribution in a first order approximation
31
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnetic Field Observed By a Spin
Magnetic flux density seen by a spin
B
H0
h
Susceptibility inclusion
Magnetic susceptibility
Applied field vector
Demagnetizing field vector
𝐁 = 𝜇0(1 +13𝜒)(𝐇0 + 𝐡)
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnetic Field Observed By a Spin𝐵𝑧 = 𝜇0 1 +
13𝜒 𝐻0𝑧 + ℎ𝑧
= 𝜇0 1 +1
3𝜒 𝐻0 − 𝐹𝑇
−1 𝑘𝑧2
𝑘2𝜒 𝐤 𝐻0
≅ 𝜇0 𝐻0 +1
3𝜒𝜇0𝐻0 − 𝜇0𝐹𝑇
−1 𝑘𝑧2
𝑘2𝜒 𝐤 𝐻0
𝛿𝐵𝑧(𝐫) = 𝐵𝑧 − 𝜇0𝐻0
= 𝐹𝑇−1 (13− 𝑘𝑧
2
𝑘2)𝜒 𝐤 𝜇0𝐻0
𝛿𝐵𝑧(𝐤) = (13− 𝑘𝑧
2
𝑘2)𝜒 𝐤 𝐵0
For simplicity, write 𝛿𝐵𝑧 as 𝛿B𝐵0 = 𝜇0 𝐻0
32
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Step 3: Solve A Deconvolution Problem
Convolution Deconvolution
Measurements
Unknown
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
kx
ky
kz
Dividing by zero!
33
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Quantitative Susceptibility Mapping
Raw Phase Unwrapped Phase Tissue Phase Susceptibility
(ppm)
Filter Background Phase
SolveInverse Problem
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Tissue Phase Susceptibility
3 T
-0.02 0.02ppm
34
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Is Orientation Dependent
B0
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Magnetic Susceptibility Is AnisotropicSusceptibility at Orientation #
1 2 3 4
Magnetic susceptibility is orientation dependent
C. Liu, MRM 2010; 63: 1471-1477
35
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Tensor Imaging
0 2
ˆ1 ( )ˆ ˆ ˆ( ) ( )3
Htk
TT k χ k H
k H χ k H H k
Susceptibility tensor is symmetric; 6 unknowns
H0
kx
ky
kz
11 12 13
21 22 23
31 32 33
k
C. Liu, MRM 2010; 63: 1471-1477
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Susceptibility Tensor Imaging
=
36
C. Liu
Guest Lecture
EE C225E, Spring 2016
Principles of Magnetic Resonance Imaging
Summary• Magnetic susceptibility causes field perturbation
• Field perturbation results in frequency shift that can be measured by gradient echo phase images
• High-passed filtered phase is used to generate SWI images
• Background phase can be removed with sphere mean value filter
• The relation between field perturbation and susceptibility is a convolution
• QSM solves the deconvolution problem
• STI treats susceptibility as a tensor instead of a scalar