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RUPRECHT-KARLS-UNIVERSITY HEIDELBERG

Computerunterstützte Klin. Medizin

Dr. Friedrich Wetterling11/24/2011 | /27

Hochschule Mannheim

RF Methoden und BildgebungLehrstuhl für Computerunterstützte Klinische MedizinMedizinische Fakultät Mannheim, Universität HeidelbergTheodor-Kutzer-Ufer 1-3D-68167 Mannheim, DeutschlandFriedrich.Wetterling@MedMa.Uni-Heidelberg.dewww.ma.uni-heidelberg.de/inst/cbtm/ckm/

Bildgebende Systeme in der Medizin

Magnet Resonanz Tomographie III:

Der k-Raum

Dr. Friedrich Wetterling



Dr. Friedrich Wetterling11/24/2011 | Page 2/27

- phase encoding gradient includes a

spatial dependency of the spin

phase according to:

φp = – γ · Gx · x · tx

= – kx · x

- sequence has to be repeated

N-times !

Gx

Phase Encoding: Principle

source: Reiser and Semmler. “Magnetresonanztomographie” 2002

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Movie: Signal Phase

© Plewes DB, Plewes B, Kucharczyk W. The Animated Physics of MRI, University Toronto, Canada




- in both encoding techniques the

transversal magnetization of all voxels

of the excited slice contribute to the

detected FID-signal

- the spatial information is encoded in

the phase difference which has been

developed during the phase encoding

gradient

Frequency and Phase Encoding

source: Reiser and Semmler. “Magnetresonanztomographie” 2002

frequency encoding

phase encoding

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k-Raum

the k-space construction is a relation between spatial encoding

(phase and frequency encoding)

and the Fourier transformation

frequency encoded signal:

∫∞

∞−

⋅⋅⋅⋅−⋅⋅= dxextS

txGi xγρ )()(

tGk xx ⋅⋅⋅

=π

γ

2

∫∞

∞−

⋅⋅⋅⋅−⋅⋅= dxexkS

xkix

xπρ 2)()(

K-Space: Definition




S(kx) is defined only for a limited numberof measuring points in the k-space

k-space coordinates of measured points define the so called trajectory in k-space

K-Space: Note

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kx

ky

φ

kx

ky

⋅⋅=

⋅⋅=

tGk

tGk

yy

xx

γ

γ

⋅=

⋅=

φ

φ

sin

cos

kk

kk

y

x

=

+⋅⋅=⋅⋅=

x

y

yxfe

G

G

GGttGk

arctan

22

φ

γγ

( )( )

−⋅⋅=

−⋅⋅=

TEtGk

TEtGk

yy

xx

γ

γ

frequency encoded FID frequency encoded echo

Sampling Trajectories




- the k-space sampling trajectory of a frequency encoded signal is a straight

line if a temporally constant gradient is used for encoding

- in general:

)(tGG fefe =

ττγ dGtk

t

fe ⋅⋅= ∫0

)()(rr

Sampling Trajectories: General

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Radial Trajectory

ky

kx

sequence diagram k-space trajectory

Glover and Pauly. MRM 1992

radial readout

slice selection




K-Space: Gridding

gridding

- non-rectilinear k-space trajectories

data points not on equally spaced grid points

- necessary to adjust positions of

data points before FFT

- most commonly used method

called "gridding“: sinc, density pre-

compensation, etc.

density pre-compensation

- sample data points are

interpolated onto grid points using

frequency-limited kernel (Kaiser-Bessel window function)

- amplitude density compensation

- FFT

- divide by inverse of interpolation

kernel

O‘Sullivan. IEEE Trans Med Imaging 1985 Jackson et al. IEEE Trans Med Imaging 1991

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� Cartesian Imaging

Spiral Trajectory

k-space gradient design

FAT-SAT RF-pulse α spoiler

time

volunteer: kidney-MRAProband: Herz, TrueFISP

� spiral imaging




acqxx TttGk ≤≤⋅⋅= 0γ

FID

trajectory starts at kx = 0 und ends at tGk xx ⋅⋅= γ

Frequency Encoding: FID

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gradient-echo

20 acqxx TttGk ≤≤⋅⋅−= γpreparation:

( )

2)(

)(

22

acqx

acqx

acqxacqxx

TTEtTEtG

TtG

TtGTGk

≤−−⋅⋅=

−⋅⋅=

−⋅⋅+⋅⋅−=

γ

γ

γγacquisition:

Frequency Encoding: Gradient-Echo

- trajectory is a symmetrical line through k-space origin during acquisition:

starts at and ends at2/acqxx TGk ⋅⋅−= γ 2/acqxx TGk ⋅⋅=γ




- trajectory the same as with gradient-echo

20 acqxx TttGk ≤≤⋅⋅= γpreparation:

acquisition: ( )

2)(

)(

22

acqx

acqx

acqxacqxx

TTEtTEtG

TtG

TtGTGk

≤−−⋅⋅=

−⋅⋅=

−⋅⋅+⋅⋅−=

γ

γ

γγ

180°-pulse: 22 acqxxacqxx TGkTGk ⋅⋅−=→⋅⋅= γγ

Frequency Encoding: Spin-Echo

spin-echo

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ti

Obj

TrGierdertS pepe ⋅⋅−⋅⋅⋅⋅−

⋅

⋅⋅= ∫ 03

)()(ωγ

ρrr

r

after demodulation:

∫ ⋅⋅= ⋅⋅⋅⋅−

Obj

rkirderkS

32)()(

rrrr

πρpepe TGk ⋅⋅=

rrγ

as a function of has the same form

as for frequency encodingkr

)(kSr

Phase Encoding Trajectory




note:

- with respect to frequency and phase encoding a measured time signal

is represented in different way in k-space:

phase encoding:

frequency encoding:

has a fixed value for a given Gpe and Tpekr

is always a function of timekr

Frequency and Phase Encoding: Note

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( )

( )dxexI

dxdydzezyxtS

TEtGi

TEtGi

x

x

−⋅⋅⋅−

∞

∞−

∞

∞−

∞

∞−

∞

∞−

−⋅⋅⋅−

⋅=

⋅=

∫

∫ ∫ ∫

γ

γρ

)(

),,()(

2acqTTEt ≤−

spin-echo signal:

dydzzyxxI ⋅= ∫ ∫∞

∞−

∞

∞−

),,()( ρwanted image function:

1D MR Imaging Sequence

spin-echo




Movie: 1D K-Space


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( )TEtGk xx −⋅⋅= γ

using the substitution:

dxexIkSxki

xx ⋅⋅⋅⋅−

∞

∞−

⋅= ∫π2

)()(

1D imaging equation

1D Imaging Equation




( )( )0

0

ttGnk

ttGk

yy

x

−⋅∆⋅⋅=

−⋅⋅=

γ

γ

200 acqTttt +≤≤

- interval between 90°- and 180°-pulse

(phase encoding interval)

( )peypexA TGnTG ⋅∆⋅⋅⋅⋅= γγ ,k

AB kk −=

- 180°-pulse:

- point A:

2acqTTEt ≤−

( )

peyy

x

TGnk

TEtGk

⋅∆⋅⋅−=

−⋅⋅=

γ

γacquisition:


source: Liang and Lauterbur. “Principles of Magnetic Resonance Imaging” 2000

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Movie: 2D K-Space X





Movie: 2D K-Space Y


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Movie: 2D K-Space X and Y





Movie: 2D K-Space Signal Encoding


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( )dzdxdyezyxIkkkS

zkykxki

zyxzyx∫ ∫ ∫

∞

∞−

∞

∞−

⋅+⋅+⋅⋅⋅⋅−∞

∞−

⋅=π2

),,(),,(

( )

pezz

peyy

xx

TGnk

TGmk

TEtGk

⋅∆⋅⋅=

⋅∆⋅⋅=

−⋅⋅=

γ

γ

γ

2acqTTEt ≤−

- acquisition:






constant gradient results

in a straight trajectory

change in polarity

inverts the trajectory

refocusing pulse (180°)

inverts phase

(mirroring at origin)

read

phase

read

phase RF180°-pulse

Surfing through K-Space

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k-space image-space

K-Space and Image-Space I

hologramfrequency distribution

imagedensity distribution




- definition:

( ) ( ) ( )dydxeyxkkS

ykxki

yx

yx +∞

∞−

∞

∞−

∫ ∫=π

ρ2

,,

- image of spin density distribution is calculated by inverse Fourier Transformation (FT):

( ) ( ) ( )∫ ∫∞

∞−

∞

∞−

+−= yx

ykxki

yx dkdkekkSyx yxπρ

2,,

FT

tGk xx ⋅⋅⋅

=π

γ

2pepey TGk ⋅⋅

⋅=

π

γ

2

Fourier Transformation (FT)

Jean Baptiste Joseph Fourier (1768–1830)

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Point Spread Function: PSF

optics / :

PSF = image of a point-like radiation /

MRI

magnetization source

image k-space

Fourier

transformation




PSF: K-Space Inhomogeneity

k-space

15 echoes

∆TE = 10 msT2 = 50 ms

position [pixel]

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1973

Paul Lauterbur

• second scanner: collecting many points at once.

• the improved method was based on the principle of back projection.

• magnetic field gradients were used to realize the projections.

Nature 1973;242:190-191

Richard R. Ernst

• 2D Fourier transform MRI

1974

Zurich

© Yves De Deene. University of Gent, Belgium

NMR History: Imaging II




ky = γ Gp t

kx = γ Gr t

y = ω/(γ Gp)

x = ω/(γ Gr)

FT

k-space image-space

K-Space and Image-Space II

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reference

K-Space Properties

k-space image




Movie: K-Space FT Point


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Movie: K-Space FT Line




Dr. Friedrich Wetterling11/24/2011 | Page 36/27 k-Raum-Darstellung

????

K-Space Quiz

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K-Space: Mona Lisa




K-Space: Non Locality

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k-space

image

Fouriertransformation

kx

ky

y

x

hologram

K-Space: Summary

density




x

xW

k1

≤∆

y

yW

k1

≤∆

samplingtheorem:

⋅∆⋅=∆

∆⋅⋅=∆

peyy

xx

TGk

tGk

γ

γ


K-Space: Sampling Requirements I

Gx : frequency encoding gradient∆t : frequency encoding interval∆Gy : phase encoding incrementTpe : phase encoding interval

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yNTWTG

xNGWGt

ypeype

y

xxxx

∆⋅⋅⋅

⋅=

⋅⋅

⋅≤∆

∆⋅⋅⋅

⋅=

⋅⋅

⋅≤∆

γ

π

γ

π

γ

π

γ

π

22

22

K-Space: Sampling Requirements II

Nyquist - interval




- during a MR measurement the signal S(t) is discretely sampled

(frequency encoding interval ∆t) in a

total acquisition time taq (typical 5 - 30 ms)

→ number of measuring points N = taq / ∆t

S(∆t), S(2∆t), ... S(N∆t)

⇒ spatial resolution ∆x is limited by:

tNGN

Wx

xxx

x

∆⋅⋅⋅==∆

γ

π2

∆x = 1.593 mm

Wx = N · ∆x = 50 cm⇒

example: Nx = 256

∆t = 30 µs

Gx = 1,566 mT/m

Example

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Nobel Prizes NMR

1944 Nobel prize in physicsIsidor Rabispin of nuclei (1939)

1952 Nobel prize in physicsFelix Bloch and Edward Purcell discovery of NMR (1946)

1991 Nobel prize in chemistryRichard ErnstFourier transformation, MRS (1966)

2002 Nobel prize in chemistryKurt Wüthrich3D structure of proteins, MRS (1982)

2003 Nobel prize in medicinePaul Lauterbur and Peter MansfieldMR-imaging, MRI (1973)

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