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Lecture 2010-03-16

• K-space

• Intro to k-space sampling (chap 3)

– Frequency encoding and phase encoding

• Discrete sampling (chap 2)

– Point Spread Function

– K-space properties

• K-space sampling principles (chap 3)

• Basic Contrast mechanism (chap 4)

• K-space

• Intro to k-space sampling

– Frequency encoding and phase encoding

• Discrete sampling

– Point Spread Function

– K-space properties

• K-space sampling – Pulse Sequence

• Basic Contrast mechanisms

k-space

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k = γ G(τ )0

t

∫ dτ =

kx

ky

kz

Limiting discussion to a slice (2D – xy plane), magnetization distribution is given by the

2-dimensional Fourier transform of the spin distribution across the slice

MT (t) = ρ(r)⋅ exp − jk⋅ r( )drslice

∫∫

ρ(r) is obtained from the inverse Fourier transform of MT(t) under the influence of a

known gradient configuration

ρ(x,y) =1

2πMT(kx ,ky )exp j kxx + kyy( )( )dkxdky

ky

∫kx

∫

k-space = visualization of the distribution of spatial frequencies in the image.

k-space = Fourier transform of the MR image

FT

{FT}-1

FT

{FT}-1

FT

{FT}-1

FT

{FT}-1

( )( )∫ ∫ +=

x yk k

yxyxyxT dkdkykxkjkkMyx exp),(2

1),(

πρ

MT(t)

== ∫z

y

xt

k

k

k

dττγ0

)(Gk

k-space illustrations

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Gy

Combination of Gx and Gy to “rotate” the total gradient

orientation � reconstruction by back projection

slice selectionGz

Gx

echo

Use of gradients to make an image

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Zeugmatography

Relationship between a three-dimensional object, its two-dimensional projection along the Y-axis,

and four one-dimensional projections at 45° intervals in the XZ-plane. The arrows indicate the gradient directions.

Lauterbur PC. Image formation by induced local interactions: examples of employing nuclear magnetic resonance. Nature 1973; 242: 190-

191.

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• K-space

• Intro to k-space sampling

– Frequency encoding and phase encoding

• Discrete sampling

– Point Spread Function

– K-space properties

• K-space sampling – Pulse Sequence

• Basic Contrast mechanisms

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Sum of waves with

different frequencies

Frequency information

Difference of phases

between the 2 sumsSame frequencies as

above

What describes waves (signal)

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SE magnetization evolution

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ki = γ Gi t( )0

′ t

∫ dt

The phase angle of a spin in a slice at a time t is

given by:

( ) txGtyGdttyxxyn

t

γγω +=∫′

0,,

Gradient y “on”

Definition of k:

pulsed field gradient along x

tx

Gx

(in the direction i) surface

The total transverse magnetisation is a function of kx, ky and the position in the

slice: MT(kx, ky)

Image reconstruction:

2D Fourier Transform

FYS-KJEM 4740 10

ρ(x,y) =1

2πMT(kx ,ky )exp j kxx + kyy( )( )dkxdky

ky

∫kx

∫

dephasing

sampling of the signal which

contains frequency information

(x-axis)

Spin Echo: freq. encoding

slice selection

Read-out direction

(frequency encoding)

Gz

Gx

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t=0 t=TE/2 t=TE

echo

phase encoding

Spatial information in

“y” direction

Gy

Phase encoding

Gzslice selection

Read-out direction

(frequency encoding)

Gz

Gx

acquisition of a profile

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t=0 t=TE/2 t=TE

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Phase encoding:

needs multiple

echo acquisitions

Frequency and Phase encoding

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acquisition of a profile

signal acquisition

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profile

Digital signal sampling

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Discrete samplingDiscrete sampling

• K-space

• Intro to k-space sampling

– Frequency encoding and phase encoding

• Discrete sampling

– Point Spread Function

– K-space properties

• K-space sampling – Pulse Sequence

• Basic Contrast mechanisms

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Sampling interval: U(t) = 1 if t ∈ [-Tread/2, Tread/2] and 0 elsewhere

Signal sampling is modulated by a Block Function U(t)

Tread = N.ts

In frequency domain, this translates to:

Tread

U(t)

MR-signal (MT)

Discrete sampling

PSF(x) = FFT U(t)( ) = U(t)e(− iγGx .x.t )dt

−∞

+∞

∫17FYS-KJEM 4740 18FYS-KJM 4740

PSF (x) = A.e(−iγGx .x.t )dt

−Tread / 2

+Tread / 2

∫

PSF(x) =−A

iγGxxe

(−iγGx .x .t )[ ]−Tread / 2

Tread / 2

PSF(x) =A sin(γGx .x.Tread / 2)

γGxx

PSF(x) = Tread

sinγGxxTread

2

γGxxTread

2

periodic function (See eq. 2-24)

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PSF(x)

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P…

δ function

(point object)

image representation

of the point object

We can calculate the Full Width at Half Height (FWHH)

∆x =π

γ ⋅ Gx ⋅ Tread

λ = 2π / kxwavelength

Field of View (FOV)

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ρ(x,y) =1

2πMT(kx ,ky )exp j kxx + kyy( )( )dkxdky

ky

∫kx

∫ ρ(x,y) =1

2πMT(kx )exp j kxx( )( )dkx

kx

∫

kx, max = γGx⋅ Tread / 2 (smallest wavelength)

Tread = N.ts

λmax =2π

kx, min

=2π

γGx ts

= FOVx

Gx =2π

γ ⋅ FOVx ⋅ ts

=2π

γ

fs

FOVx

Gx is automatically calculated by the scanner

FOVx is entered by the user (see Eq. 2-28)

FOVy

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Same definition can be done for the phase encoding direction

FOVy =2π

ky, min

=2π

γ Gy, n −Gy, n−1( )Ty

π phase difference at the edge of the FOVy

Consider a square matrix: N2 = Nx.Ny

Gy,n − Gy,n−1 = ∆Gy We need to have ∆Gy ⋅N

2= Gy, max

Gy, max =πN y

γTyFOVy

(See Eq. 2-33)

K-space properties

sxx tNGx

γ

πδ

2=

Resolution (x):

Field of view (x):

x

sxx

x FoVtGk

===γ

ππλ

22

min,

max,

Field of view (y):

yy

y

yyTG

NFoV

max_

max,γ

πλ ==

Maximum frequency in read-out (x) direction

2/max xx FoVGγω ±=±

πγ 2//1 xxs FoVGt ≥

Min sampling rate (x):

yyyyFoVTGN

max_γ=

‘Sampling rate ‘ (y):

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FT

{FT}-1

∆x ∆kx

F(k)S(r)

FoVx

FoV

y

∆y ∆ky

Nx∆kx

Ny∆ky

ts

ky

kx

-ωmax = -γGxFoVx/2 ωmax = γGxFoVx/2

.ρ(x,y)

kx_max

( )( )∫ ∫ +=

x yk k

yxyxyxT dkdkykxkjkkMyx exp),(2

1),(

πρ

== ∫z

y

xt

k

k

k

dττγ0

)(Gk

K-space vs image space

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Object FoV

Image FoV

Increase 1/ts, Nx

discardeddiscarded

Back-folding / x direction

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Image F

oV

y

Obje

ct F

oV

y

Back-folding / y direction

Fold-Over artefact

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FFT 1D: Truncation Artefact

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FFT 2D: Truncation artefact

Ringing- (or truncation) artifacts in regions with high spatial frequencies (edges)

in a phantom. The artifacts are more evident in the right image due to a lower matrix (N=112, vs N=256 in the left image).

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Truncation artifact

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Fold-Over artefact

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• K-space

• Intro to k-space sampling

– Frequency encoding and phase encoding

• Discrete sampling

– Point Spread Function

– K-space properties

• K-space sampling – Pulse Sequence

• Basic Contrast mechanisms

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Phase encoding:

needs multiple

echo acquisitions

Frequency and Phase encoding

FYS-KJEM 4740 31

acquisition of a profile

Pulse sequence introduction

Field gradients � Spatial coding

• frequency encoding (gradient x “on” during

signal acquisition)

• Phase encoding (gradient y “on” before

signal acquisition, repeated at different

amplitudes)

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Spatially encoded Echoes

• Spin-Echo: SE

Use of gradient in a spin-echo experiment to

induce spatially dependent “dephasing” and “re-

phasing”

• Gradient Echo: GRE

Gradient “induced” echo

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dephasing

sampling of the signal which

contains frequency information

(x-axis)

Spin Echo: freq. encoding

Read-out direction

(frequency encoding)

Gx

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t=0 t=TE/2 t=TE

echo

90° 180°

z

x y

z

x y

z

x y

echo

z

x y

z

x y

M

0

Gx

TE

Gradient Echo90°

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Gradient Echo

kx =γ

2πGx _ rewdt +

0

Ty

∫ Gx _ rewdtTy

Ty +Tread / 2

∫

= 0⇒ Gx _ rewTy + Gx _ r

Tread

2= 0

T2* relaxation (FID)

M xy

tTE

T2* relaxation (FID)

M xy

tTE

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Gradient Echo (GRE)

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“Travelling” in k-space

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kx

ky

RF

Gz

Gy

Gx

1 sample

kx

kyky

RF

Gz

Gy

Gx

RF

Gz

Gy

Gx

1 sample

Gx

Gradient Echo90°

TE

ky

kx

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Gx

Gradient Echo90° ky

kx

TE

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Gx

Gradient Echo90°

TE

ky

kx

ts

repeated Ny times

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Gx

Gradient Echo90°

TE

ky

kx

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read-out

phase encode

sliceselection

180º rf pulse

kx

ky

echo

Acquisition of a profile

Spin-Echo k-space travelling

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t=0 t=TE/2 t=TE

90° 180°t=

TE

/2

t=TE

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of a profile

Repeated Phase encoding

FYS-KJEM 4740 44

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of a profile

Repeated Phase encoding

FYS-KJEM 4740 45

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of a profile

Repeated Phase encoding

FYS-KJEM 4740 46

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of a profile

Repeated Phase encoding

FYS-KJEM 4740 47

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of a profile

Repeated Phase encoding

FYS-KJEM 4740 48

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Repeated Phase encoding

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

Acquisition of profile

FYS-KJEM 4740 49

Repeated Phase encoding

slice selection

Read-out direction

(frequency encoding)

phase encoding

Spatial information in “y”

direction

Gz

Gx

Gy

acquisition of all profiles

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Repeated acquisition of profiles

ky

kx

the field of view:

FOV

profiles

FOV depends on the

- gradient strengths- sampling time of a profile

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+128

-127

K0

phase

frequency

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+128

-127

K0

phase

frequency

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kx

frequency encoding

ky

phase encoding

2D FT

Signal intensity

distribution in the

selected slice

frequency ωx

ωy

m x, y( )=1

2πMT kx ,ky( )exp i kx x + ky y( )[ ]dkxdky

ky

∫kx

∫

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frequency ωx

ωy

IMAGE

x

y

Image generation

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• K-space

• Intro to k-space sampling

– Frequency encoding and phase encoding

• Discrete sampling

– Point Spread Function

– K-space properties

• K-space sampling – Pulse Sequence

• Basic Contrast mechanisms

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TE and TR

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TRTR

TETE

0time

100

Longitudinal and Transverse

Relaxation:

90o

Basic Contrast

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0time

100

Longitudinal and Transverse

Relaxation: at the same moment

90o

magnetisation

signal

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short T1

0time

100

90o

90o

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short T1

short T2

0time

100

90o

90o

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long T1

0time

100

90o

90o

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long T1

long T2

0time

100

90o

90o

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0time

100

90o

repetition time

TR90

o

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0time

100

90o

90o

echo time

TE

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0time

100

90o

90o

Short TE

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0time

100

90o

90o

Longer TE

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0time

100

90o

90o

2 different tissues with different T1/T2

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T1 contrast – brain example

0time

100

90o

White matter

Grey matter

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T1 contrast

0time

100

90o

90o

WM

GM

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T1 contrast

0time

100

90o

90o

WM

GM

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T1 contrast

0time

100

90o

90o

WM

GM

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T1 contrast

0time

100

90o

90o

WM

GM

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TE 10 ms TE 30 ms TE 40 ms

TR 400 ms

T1-weighted images

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T1 weighted images:

TR short (SE)

TE short

< 600 ms (can be as low as 1.5ms)

< 25 ms

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Fat bright

bone marrow

white matter

musclegrey

body fluids

bone

air black

grey matter

TR < 600 ms TE < 25 ms

i.v.contrast

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T1-w knee

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PD (ρ*, proton density) & T2 contrast

0time

100

90o

90o

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* Denoted rho contrast in the compendium

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PD & T2 contrast

0time

100

90o

90o

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PD & T2 contrast

0time

100

90o

90o

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PD contrast – long TR, short TE

0time

100

90o

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T2 contrast: TE increases

0time

100

90o

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T2 contrast, long TR, long TE

0time

100

90o

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PD & T2 weighted images

TR 2500 ms

TE 20 ms

TE 60 ms

TE 100 ms

TE 140 ms

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T2 weighted images:

TR long

TE long

> 1800 ms

> 80 ms

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Spine / summary

T1 weighted T2 weighted

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Table of relaxation times

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From: Greg J. Stanisz, Magnetic Resonance in Medicine 54:507–512 (2005)