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Colóquio IFSC - Março/2011
MR Imaging:k-Space formalism
A. Tannús – 11/2006IFSC - USP
Colóquio IFSC - Março/2011
1952 (1946): Physics : F. Bloch & E. Purcell:
Precision measurement of Nuclear Magnetism
1942 (1930): Physics: I. Rabbi:Resonant method for measuringmagnetic properties of atomic nuclei.
Nobel prizes:NMR as a source of insight
Colóquio IFSC - Março/2011
Nobel prizes in MR 1992 (1966): Chemistry: R. Ernst: High Resolution Pulsed Magnetic Resonance - Spectroscopy.
2003 (1973): Medicine:
P. Mansfield & P. C. Lauterbur
Magnetic Resonance Imaging.
Colóquio IFSC - Março/2011
MRI temporal and spatial resolution
Improve
Improve
Colóquio IFSC - Março/2011
NMR Phenomena
Quantum Mechanical approach:Easy for spin ½;Gets complex when dealing with
different nuclear species in a system. Classical Approach.
Explain almost completely the development of Imaging methodologies.
To QM..
Colóquio IFSC - Março/2011
Classical Approach
μBγ μdtd
L
Fundamental properties of nuclei
Evolution described by an equation of a precessing rotor
LB
Colóquio IFSC - Março/2011
Spinning Top in a gravitational field:a very bad example…
“Spinning Top”
Reaction from base
L=angular momentum
“Spinning nucleus”
= magnetic momentL=angular momentum
t = magnetically induced torque = - x B0
t = torque produced by the binary forces:Weight and reaction at contact point
Weight force
B0
Colóquio IFSC - Março/2011
Relaxation
2
,,
1
0
)( )(
dtd
))(( )(dtd
TtM
tM
TMtMtM
yxyx
zz
T1 and T2 are determined based on experimental results!
(Phenomenology)
M Macroscopic Magnetization
Colóquio IFSC - Março/2011
Bloch Phenomenological Equations:
1
0
2
2
))(()()( )(dtd
)()()()(
dtd
)()()()(dtd
TMtMtBtMtM
TtM
tBtMtM
TtMtBtMtM
zzz
yyy
xxx
Colóquio IFSC - Março/2011
Excitation/Detection Scheme
z
x y M
e.m.f
V(t)
B o a)
B1
e.m.f
Colóquio IFSC - Março/2011
Detected signal
t
FIDInduced e.m.f.
Colóquio IFSC - Março/2011
x
x
y
y
z
z
x
x
x x
y
y
y y
z
z
z z
M L
M L
MT
ML
ML=Mo
MT=Mo
Exc ita tion
MT = M0 e-t/T2
ML = M0 ( 1 - e-t/T1)
tT2 T1<
Colóquio IFSC - Março/2011
Imaging Scanner Overview:HardwareFully digital, multichannel now!
Work in progress at our group
Colóquio IFSC - Março/2011
Magnet: superconducting,axial access.
Colóquio IFSC - Março/2011
Other Magnet TypesPermanent magnets, e.g. light weight rare earth magnets, <0.3T
“H” type, transverse access
“C” type, transverse access (open systems)
Colóquio IFSC - Março/2011
Other Magnet Types
“H” and “C” mixed type, transverse
access(open systems)
Colóquio IFSC - Março/2011
Other Magnet Types
Electromagnet <0.3T
Colóquio IFSC - Março/2011
Field is aligned to subject;Other designs than solenoidal must be used.
Saddle coil allows axial access. Efficiency is low, and homogeneity is poor
RF CoilsRemember:
Brf (B1) must be orthogonal to B0 !!
Colóquio IFSC - Março/2011
By mapping the spins according to their position. How?Using their frequency/position correspondence
(r) =B0(r)
Now that we have an NMR signal, how to get an image?
x
y
z
Gy
x
y
z
Gx
x
y
z
Gz
Imaging basic principles: encoding
Colóquio IFSC - Março/2011
A bit of history…First 2D NMR image:
came from an annoyance for spectroscopists!!!
P. C. Lauterbur - (1973) State University - New York
z
B0
Gz
x y
G
Projection/Reconstructionmethod(same as in CT)
Colóquio IFSC - Março/2011
10 years later…
Colóquio IFSC - Março/2011
Encoding inmore than one dimension:
solving the projection paradox.Magnetic field gradients add as vectors, giving a newly oriented
gradient!!
Colóquio IFSC - Março/2011
Magnetic Field Gradients
t
zo
z
t
zo
y
t
zo
x
zB(t)= G
, y
B(t)= G
, x
B(t)= G
Now,
gradients
are
time
dependent!!
Colóquio IFSC - Março/2011
Spatially encoded frequency and phase:more than one dimension?
t t
oo
o
(t')dt'Gγrt + ω,t')dt' = rω(,t)=rΘ(
tr(t).Gr + γ,t) = ωrω(
(t),Gr,t) = BrB(
thenis and at phase The
then0
Colóquio IFSC - Março/2011
Generalizing the definition of k(t)
(t)krΘ(r,t)=
so
(t')dt'G(t) = γkt
o
Colóquio IFSC - Março/2011
The 3D Image Equation!!
vo
(t)kri dv)r(M(t))kS( e
3D Signal 3D Image
Colóquio IFSC - Março/2011
K-Space properties
velocity(t)Gγ
trajectory(t)k
(t')dt'G(t) = γkt
o
:
Colóquio IFSC - Março/2011
Only spins inside this band are excited
Steps to NMR ImagingSelective excitation Absorption line broadening Narrow bandwidth RF pulses
Gz
B0
z
x
y e.m.f.
Gz RF
Colóquio IFSC - Março/2011
Selective excitation Absorption line broadening Narrow bandwidth RF pulses
Gz
B0
z
x
y e.m.f
Gy
Phase encoding
Encoding in this dimension is done through the initial phase.
Gy
Principles of NMR Imaging
Colóquio IFSC - Março/2011
Selective excitation Absorption line broadening Narrow bandwidth RF pulses
Gz
Gx
Frequency encoding
B0 z
x
y e.m.f
Gx
Phase encoding
Encoding in this dimension is done through the spatially dependent frequency.
Gy
Principles of NMR Imaging
Colóquio IFSC - Março/2011
Selective excitation Absorption line broadening Narrow bandwidth RF pulses
Gz
Gx Gy
Frequency encodingPhase encoding
dxdyeyxM
dxdyeyxMkkSyx
yxx
iykixk
yGitxGiyx
),(
),(),( t
Principles of NMR Imaging
Colóquio IFSC - Março/2011
Acquisition sequences and image formation :
• Spin Echo ( SE )• Echo Planar Imaging ( EPI )• Gradient Recalled Echo ( GRE )
Spin Echo ( SE )
FID
Signal
tA
Gy A
ECO
tC 2t tB
Gx
Preparation
C’
A’
B’
A’’
C’’ B’’ Acquisition
C
Gz
Gx Gy
0
RF p/2
kX
ky
0
Gz
p
t
p
B
Principles of NMR Imaging
Colóquio IFSC - Março/2011
k-space
k-space is the raw data space before Fourier transformation into the image
2D image will be represented by a 2D array of data points spread throughout k-space(it could be 3D!!)
Changing the k-space trajectory will alter image contrast
Colóquio IFSC - Março/2011
k-space
k-space must be sampled in equally spaced intervals in order to allow 2D FFT.
As a consequence the image is also presented in equally spaced sampled values.
All concepts of discrete Fourier formalism applies.
Colóquio IFSC - Março/2011
Image vs. k-space data
(r) S(k)k(t)=
/2pG(t)dt
Colóquio IFSC - Março/2011
(r) S(k)k(t)= /2pG(t)dt
Image vs. k-space data
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
Image vs. k-space data
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
Image vs. k-space data
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
FFT
Image vs. k-space data
Colóquio IFSC - Março/2011
GE k-space trajectory
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE k-space trajectory
Colóquio IFSC - Março/2011
Why does MRI take so long Answer
Only one phase encode line acquired per excitation
Spin Echo: 256*3s for T2, 256*0.6s for T1 Gradient Echo: 256*35ms (but have to do 3D
Solution get more phase encode lines per excitation
Colóquio IFSC - Março/2011
Presenting Echo Planar Imaging (EPI)Main tool for Neurosciences
Fastest imaging method Typical Acquisition times: 30-100ms Lower RF deposition Very fast gradient switching Highly demanding on MRI hardware
B0 homogeneitygradient switching
P. Mansfield
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
(r) S(k)k(t)=
/2pG(t)dt
RF
G S
G R
G P
S(t)
-kr +kr
-kp
+kp
GE-EPI k-space trajectory
Colóquio IFSC - Março/2011
Localized Spectroscopy (d)
Following metabolites and biochemical process inside the body
Spectralinformation
Spatialinformation
ppm020406080100120140160180200
ppm020406080100120140160180200
ppm020406080100120140160180200
13C, 31P
Colóquio IFSC - Março/2011
Angiography (V)Transverse: Head Coronal: Neck-Chest
Aneurisms
Colóquio IFSC - Março/2011
functional MRI (T2*)
Mapping of a finger tapping experiment