Applications of Spin Echo and Gradient Echo: Diffusion and ...ee225e/sp16/notes/... · Guest...

1

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



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

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C. Liu

Guest Lecture



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)

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It’s all about phase!!!!

C. Liu

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1.1 Diffusion-Weighted Imaging

Spin Echo

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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



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

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Guest Lecture



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



90o Spin Echo

No Diffusion: running at constant speed

180o


spin1

spin2

spin3

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90o Spin Echo

No Diffusion: running at constant speed

180o


spin1

spin2

spin3

C. Liu

Guest Lecture



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

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Guest Lecture



90o Spin EchoG G

With Diffusion: running when drunk

180o


C. Liu

Guest Lecture



90o G Spin EchoG

With Diffusion: running when drunk

180o


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Guest Lecture



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



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

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Guest Lecture



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



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.

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C. Liu

Guest Lecture



If Acquire One k-Space Line per TR

𝑚 𝐫, 𝑏 = 𝑚(𝐫, 0)𝑒−𝑏𝐷𝑒𝑗𝜑1(𝐫,TR1)

Gy

Gx

Single-Shot

…



Inconsistent k-space data causing aliasing

Spatial varying signal cancellationHow to address it?

C. Liu

Guest Lecture



1.2 Diffusion-Tensor Imaging and Tractography

Spin Echo

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Guest Lecture



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



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

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Guest Lecture



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



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

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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



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

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C. Liu

Guest Lecture



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



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

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Guest Lecture



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



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)

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Guest Lecture



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



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

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C. Liu

Guest Lecture



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



• 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

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C. Liu

Guest Lecture



2.1 T2*-Weighting and BOLD

Gradient Echo

R.F.

Gz

Gy

Gx

θ

Readout

𝑇𝐸

𝑇𝑅

θ

Readout

C. Liu

Guest Lecture



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.

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C. Liu

Guest Lecture



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



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

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C. Liu

Guest Lecture



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



Susceptibility Induced Tissue Contrast

RFEcho 1 Echo 2 Echo n……

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Guest Lecture



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

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Magnitude: T2* Decay

2

0( )t T

S t S e

OR

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Guest Lecture



R2* Mapping and Contrast

R2* 0 40 Hz

T2* 25 ms

White matter

Blood vessels

Globus pallidusSubstatia nigra

Red nucleus

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Blood-Oxygen-Level-Dependent Signal

Diamagnetic Hb

Paramagnetic Hbr

Ogawa S. et al, 1992

C. Liu

Guest Lecture



2.2 Phase Images of Gradient Echo

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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



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

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Guest Lecture



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



Filter Background: Sphere mean value property

Harmonic function

Mean phase over a sphere

Phase at the center

S

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Summary of Phase Processing

unwrapping Filtering

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Guest Lecture



2.3 Susceptibility Weighted Imaging (SWI)

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Susceptibility Weighted Imaging

GRE Phase GRE Magn

Unwrapping

Filtering Phase Mask

X

SWI = Magn*(PhaseMask)4

SWI

SWI MIP

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Guest Lecture



Susceptibility Weighted Imaging

Courtesy of Juergen Reichenbach

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Guest Lecture



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



2.4 Quantitative Susceptibility Mapping (QSM)

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Guest Lecture



m

B

Magnetic Field ChangeSusceptibility Source

0m H

B0

C. Liu

Guest Lecture



B0

m

B


0m H

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?

B0

m

B

C. Liu

Guest Lecture



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

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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



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

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C. Liu

Guest Lecture



Step 3: Solve A Deconvolution Problem

Convolution Deconvolution

Measurements

Unknown

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Guest Lecture



kx

ky

kz

Dividing by zero!

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Quantitative Susceptibility Mapping

Raw Phase Unwrapped Phase Tissue Phase Susceptibility

(ppm)

Filter Background Phase

SolveInverse Problem

C. Liu

Guest Lecture



Tissue Phase Susceptibility

3 T

-0.02 0.02ppm

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Susceptibility Is Orientation Dependent

B0

C. Liu

Guest Lecture



Magnetic Susceptibility Is AnisotropicSusceptibility at Orientation #

1 2 3 4

Magnetic susceptibility is orientation dependent

C. Liu, MRM 2010; 63: 1471-1477

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Guest Lecture



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



Susceptibility Tensor Imaging

=

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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

Applications of Spin Echo and Gradient Echo: Diffusion and ...ee225e/sp16/notes/... · Guest...

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Transcript of Applications of Spin Echo and Gradient Echo: Diffusion and ...ee225e/sp16/notes/... · Guest...