K-Raum - Medizinische Fakultät Mannheim

Transcription

RUPRECHT-KARLSUNIVERSITY HEIDELBERG
Computerunterstützte Klin. Medizin
Dr. Friedrich Wetterling
11/24/2011 | Page 1/27
Hochschule Mannheim
Bildgebende Systeme in der Medizin
Magnet Resonanz Tomographie III:
Der k-Raum
RF Methoden und Bildgebung
Lehrstuhl für Computerunterstützte Klinische Medizin
Medizinische Fakultät Mannheim, Universität Heidelberg
Theodor-Kutzer-Ufer 1-3
D-68167 Mannheim, Deutschland
[email protected]
www.ma.uni-heidelberg.de/inst/cbtm/ckm/
11/24/2011 | Page 2/27
Phase Encoding: Principle
Gx
- 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 !
source: Reiser and Semmler. “Magnetresonanztomographie” 2002
Seite
11/24/2011 | Page 3/27
Movie: Signal Phase
© Plewes DB, Plewes B, Kucharczyk W. The Animated Physics of MRI, University Toronto, Canada
11/24/2011 | Page 4/27
Frequency and Phase Encoding
frequency encoding
- in both encoding techniques the
transversal magnetization of all voxels
of the excited slice contribute to the
detected FID-signal
phase encoding
- the spatial information is encoded in
the phase difference which has been
developed during the phase encoding
gradient
source: Reiser and Semmler. “Magnetresonanztomographie” 2002
Seite
11/24/2011 | Page 5/27
k-Raum
K-Space: Definition
the k-space construction is a relation between spatial encoding
(phase and frequency encoding)
and the Fourier transformation
frequency encoded signal:
∞
S (t ) =
∫ ρ ( x) ⋅ e
− i ⋅γ ⋅G x ⋅ x ⋅t
⋅ dx
−∞
kx =
γ
2 ⋅π
⋅Gx ⋅t
∞
S (k x ) =
∫ ρ ( x) ⋅ e
− i ⋅ 2 ⋅π ⋅ k x ⋅ x
⋅ dx
−∞
11/24/2011 | Page 6/27
K-Space: Note
S(kx) is defined only for a limited number
of measuring points in the k-space
k-space coordinates of measured points
define the so called trajectory in k-space
Seite
11/24/2011 | Page 7/27
Sampling Trajectories
frequency encoded FID
frequency encoded echo
ky
ky
φ
kx
kx
 k x = γ ⋅ G x ⋅ (t − TE )

 k y = γ ⋅ G y ⋅ (t − TE )
k x = γ ⋅ G x ⋅ t

k y = γ ⋅ G y ⋅ t
 k x = k ⋅ cos φ

 k y = k ⋅ sin φ
k = γ ⋅ G ⋅ t = γ ⋅ t ⋅ G 2 + G 2
fe
x
y


 Gy 

φ = arctan 


 Gx 
11/24/2011 | Page 8/27
Sampling Trajectories: General
- 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:
G fe = G fe (t )
t
r
r
k ( t ) = γ ⋅ G fe (τ ) ⋅ d τ
∫
0
Seite
11/24/2011 | Page 9/27
Radial Trajectory
ky
slice selection
kx
radial readout
sequence diagram
k-space trajectory
Glover and Pauly. MRM 1992
11/24/2011 | Page 10/27
K-Space: Gridding
gridding
density pre-compensation
-
non-rectilinear k-space trajectories
data points not on equally spaced
grid points
-
-
necessary to adjust positions of
data points before FFT
sample data points are
interpolated onto grid points using
frequency-limited kernel (KaiserBessel window function)
-
amplitude density compensation
-
FFT
-
divide by inverse of interpolation
kernel
-
most commonly used method
called "gridding“: sinc, density precompensation, etc.
O‘Sullivan. IEEE Trans Med Imaging 1985
Jackson et al. IEEE Trans Med Imaging 1991
Seite
11/24/2011 | Page 11/27
Spiral Trajectory
Cartesian
Imaging
spiral imaging
FAT-SAT
RF-pulse α
spoiler
time
k-space
volunteer: kidney-MRA
Proband: Herz, TrueFISP
gradient design
11/24/2011 | Page 12/27
Frequency Encoding: FID
FID
kx = γ ⋅Gx ⋅t
0 ≤ t ≤ T acq
trajectory starts at kx = 0 und ends at
Seite
kx = γ ⋅Gx ⋅t
11/24/2011 | Page 13/27
Frequency Encoding: Gradient-Echo
gradient-echo
preparation:
k x = −γ ⋅ G x ⋅ t
acquisition:
k x = −γ ⋅ G x ⋅ T acq 2 + γ ⋅ G x ⋅ t − T acq 2
0 ≤ t ≤ T acq 2
(
)
= γ ⋅ G x ⋅ ( t − T acq )
= γ ⋅ G x ⋅ ( t − TE )
t − TE ≤ T acq 2
- trajectory is a symmetrical line through k-space origin during acquisition:
starts at kx = −γ ⋅ Gx ⋅Tacq / 2 and ends at kx = γ ⋅ Gx ⋅ Tacq / 2
11/24/2011 | Page 14/27
Frequency Encoding: Spin-Echo
spin-echo
preparation:
kx = γ ⋅Gx ⋅t
180°-pulse:
k x = γ ⋅ G x ⋅ T acq 2 →
acquisition:
k x = −γ ⋅ G x ⋅ T acq 2 + γ ⋅ G x ⋅ t − T acq 2
0 ≤ t ≤ T acq 2
k x = − γ ⋅ G x ⋅ T acq 2
(
)
= γ ⋅ G x ⋅ ( t − T acq )
= γ ⋅ G x ⋅ ( t − TE )
- trajectory the same as with gradient-echo
Seite
Phase Encoding Trajectory
11/24/2011 | Page 15/27
r r


r
− i ⋅γ ⋅G pe ⋅ r ⋅T pe
3 

S (t ) =
ρ (r ) ⋅ e
⋅ d r ⋅ e − i ⋅ω 0 ⋅t


 Obj

∫
after demodulation:
r
S (k ) =
∫
r r
r
− i ⋅ 2 ⋅π ⋅ k ⋅ r
ρ (r ) ⋅ e
⋅ d 3r
r
r
k = γ ⋅ G pe ⋅ T pe
Obj
r
r
S (k ) as a function of k has the same form
as for frequency encoding
11/24/2011 | Page 16/27
Frequency and Phase Encoding: Note
note:
- with respect to frequency and phase encoding a measured time signal
is represented in different way in k-space:
phase encoding:
r
k has a fixed value for a given Gpe and Tpe
frequency encoding:
r
k is always a function of time
Seite
11/24/2011 | Page 17/27
1D MR Imaging Sequence
∞
wanted image function:
∞
∫ ∫ ρ ( x , y , z ) ⋅ dydz
I ( x) =
−∞ −∞
spin-echo
∞
spin-echo signal:
S (t ) =
∞
∞
∫ ∫ ∫ ρ ( x, y, z ) ⋅ e
− i ⋅γ ⋅G x ⋅(t − TE )
dxdydz
−∞ −∞ −∞
∞
=
∫ I ( x) ⋅ e
− i ⋅γ ⋅G x ⋅ (t − TE )
dx
−∞
11/24/2011 | Page 18/27
Movie: 1D K-Space
Seite
11/24/2011 | Page 19/27
1D Imaging Equation
using the substitution:
k x = γ ⋅ G x ⋅ (t − TE )
∞
S (k x ) =
∫
I ( x ) ⋅ e − i ⋅ 2 ⋅π ⋅ k x ⋅ x dx
−∞
1D imaging equation
11/24/2011 | Page 20/27
- interval between 90°- and 180°-pulse
(phase encoding interval)
k x = γ ⋅ G ⋅ (t − t 0 )
k y = γ ⋅ n ⋅ ∆ G y ⋅ (t − t 0 )
- point A:
(
t 0 ≤ t ≤ t 0 + T acq 2
k A = γ ⋅ G x ⋅ T pe , γ ⋅ n ⋅ ∆ G y ⋅ T pe
- 180°-pulse:
k B = −k A
acquisition: k x = γ ⋅ G ⋅ (t − TE )
k y = −γ ⋅ n ⋅ ∆ G y ⋅ T pe
source: Liang and Lauterbur. “Principles of Magnetic Resonance Imaging” 2000
Seite
)
11/24/2011 | Page 21/27
Movie: 2D K-Space X
11/24/2011 | Page 22/27
Movie: 2D K-Space Y
Seite
11/24/2011 | Page 23/27
Movie: 2D K-Space X and Y
11/24/2011 | Page 24/27
Movie: 2D K-Space Signal Encoding
Seite
11/24/2011 | Page 25/27
- acquisition:
k x = γ ⋅ G x ⋅ (t − TE )
k y = γ ⋅ m ⋅ ∆ G y ⋅ T pe
k z = γ ⋅ n ⋅ ∆ G z ⋅ T pe
∞
S (k x , k y , k z ) =
∞
∞
∫∫∫
I ( x, y, z ) ⋅ e
(
− i ⋅ 2 ⋅π ⋅ k x ⋅ x + k y ⋅ y + k z ⋅ z
)dxdy dz
−∞ −∞ −∞
11/24/2011 | Page 26/27
Surfing through K-Space
read
read
phase
phase
constant gradient results
in a straight trajectory
180°-pulse
change in polarity
inverts the trajectory
Seite
RF
refocusing pulse (180°)
inverts phase
(mirroring at origin)
K-Space and Image-Space I
11/24/2011 | Page 27/27
k-space
image-space
hologram
image
frequency distribution
density distribution
11/24/2011 | Page 28/27
Fourier Transformation (FT)
- definition:
kx =
γ
2 ⋅π
S (k x , k y ) =
⋅Gx ⋅t
ky =
∞ ∞
∫
∫ ρ (x , y ) e
γ
2 ⋅π
(
⋅ G pe ⋅ T pe
i 2π k x x + k y y
)
dx dy
− ∞− ∞
FT
- image of spin density distribution is calculated by
inverse Fourier Transformation (FT):
Jean Baptiste Joseph Fourier (1768–1830)
∞ ∞
ρ (x , y ) =
∫
∫ S (k x , k y ) e
−∞−∞
Seite
(
− i 2π k x x + k y y
)
dk x dk y
Point Spread Function: PSF
11/24/2011 | Page 29/27
k-space
image
Fourier
transformation
optics / MRI :
PSF = image of a point-like radiation / magnetization source
11/24/2011 | Page 30/27
PSF: K-Space Inhomogeneity
k-space
15 echoes
∆TE = 10 ms
T2 = 50 ms
position [pixel]
Seite
11/24/2011 | Page 31/27
NMR History: Imaging II
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.
1973
Nature 1973;242:190-191
Richard R. Ernst
Zurich
• 2D Fourier transform MRI
1974
© Yves De Deene. University of Gent, Belgium
11/24/2011 | Page 32/27
K-Space and Image-Space II
k-space
image-space
k y = γ Gp t
y = ω/(γ Gp)
FT
k x = γ Gr t
Seite
x = ω/(γ Gr)
11/24/2011 | Page 33/27
K-Space Properties
reference
k-space
11/24/2011 | Page 34/27
Movie: K-Space FT Point
Seite
image
11/24/2011 | Page 35/27
Movie: K-Space FT Line
11/24/2011 | Page 36/27
K-Space
Quiz
k-Raum-Darstellung
????
Seite
11/24/2011 | Page 37/27
K-Space: Mona Lisa
11/24/2011 | Page 38/27
K-Space: Non Locality
Seite
11/24/2011 | Page 39/27
K-Space: Summary
density
y
image
Fourier
transformation
x
ky
k-space
kx
hologram
11/24/2011 | Page 40/27
sampling
theorem:
K-Space: Sampling Requirements I
∆k x ≤
1
Wx
∆k x = γ ⋅ G x ⋅ ∆t

 ∆ k y = γ ⋅ ∆ G y ⋅ T pe
∆k y ≤
1
Wy
Gx :
∆t :
∆ Gy :
Tpe :
Seite
frequency encoding gradient
frequency encoding interval
phase encoding increment
phase encoding interval
11/24/2011 | Page 41/27
K-Space: Sampling Requirements II
∆t ≤
2 ⋅π
2 ⋅π
=
γ ⋅ G x ⋅W x γ ⋅ G x ⋅ N x ⋅ ∆x
∆G y ≤
2 ⋅π
2 ⋅π
=
γ ⋅ T pe ⋅ W y γ ⋅ T pe ⋅ N y ⋅ ∆ y
Nyquist - interval
11/24/2011 | Page 42/27
Example
- 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:
∆x =
Wx
2π
=
Nx
γ ⋅ G x ⋅ N x ⋅ ∆t
example: Nx = 256
∆t = 30 µs
Gx = 1,566 mT/m
∆x = 1.593 mm
⇒
W x = N · ∆x = 50 cm
Seite
11/24/2011 | Page 43/27
Nobel Prizes NMR
1944 Nobel prize in physics
Isidor Rabi
spin of nuclei (1939)
1952 Nobel prize in physics
Felix Bloch and Edward Purcell
discovery of NMR (1946)
1991 Nobel prize in chemistry
Richard Ernst
Fourier transformation, MRS (1966)
2002 Nobel prize in chemistry
Kurt Wüthrich
3D structure of proteins, MRS (1982)
2003 Nobel prize in medicine
Paul Lauterbur and Peter Mansfield
MR-imaging, MRI (1973)
Seite

K-Raum - Medizinische Fakultät Mannheim

Transcription

Similar documents

MRT Systeme - Medizinische Fakultät Mannheim

1. Basics of LASER Physics

Resonatoren - Medizinische Fakultät Mannheim