Principles of Magnetic Resonance Imaging

1
Principles of Magnetic Resonance Imaging
J. Peter Mustonen (from David J.
Michalak) Presentation for Physics 250 05/01/2008
2
Outline

• Motivation
• Principles of NMR
• Interactions of spins in B0 field
• Principles of 1D-MRI
• Principles of 2D-MRI
• Summary

3
Motivation
Magnetic Resonance Imaging provides a
non-invasive imaging technique. Pros -No
injection of potentially dangerous elements
(radioactive dyes) -Only magnetic fields are
used for imaging no x-rays Cons -Current
geometries are expensive, and large/heavy
www.nlm.nih.gov
4
B0
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
M SMi
5
z
RF Pulse
B0
B0
M
y
x
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
M SMi
6
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
x
x
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
M precesses in the transverse plane. In the
absence of any disturbances, M continues to
rotate indefinitely in xy plane.
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
M SMi
7
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
Detector
x
x
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
M precesses in the transverse plane. In the
absence of any disturbances, M continues to
rotate indefinitely in xy plane.
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
M SMi
8
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
Detector
x
x
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
M precesses in the transverse plane. In the
absence of any disturbances, M continues to
rotate indefinitely in xy plane.
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane

• Assume
• All spins feel same B0.
• No other forces on Mi (including detection).

M SMi
9
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
Detector
x
x
(w0/2p)-1
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
signal, sr(t)
time, t
M SMi
10
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
Detector
x
x
(w0/2p)-1
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
sr(t)
t
FT
sr(w)
w
w0 2pf0
M SMi
11
z
z
RF Pulse
B0
B0
Time
exp-iw0t
B0
M
M
y
y
Detector
x
x
(w0/2p)-1
Application of prepolarizing magnetic field, B0,
aligns the spins in a sample to give a net
magnetization, M. M rotates about B0 at a Larmor
precession frequency, w0 gB0
Application of a rf pulse w02pf0 along the
x-axis will provide a torque that displaces M
from the z axis towards y axis. A certain pulse
length will put M right on xy plane
sr(t)
t
FT
sr(w)
w
w0 2pf0
Boring Spectrum!
M SMi
12
Complexity Makes Things Interesting

• In Reality
• Relaxation (Inherent even if B0 is homogeneous)
• T1 Spins move away from xy plane towards z.
• T2 Spins dephase from each other.
• B0 inhomogeneity.
• Chemical Shift.

13
T1 Spin Relaxation

• T1 Spin Relaxation return of the magnetization
  vector back to z-axis.
• Spin-Lattice Time Constant
• Energy exchange between spins and surrounding
  lattice.
• Fluctuations of B field (surrounding dipoles
  receivers) at w0 are important. Larger E
  exchange necessary for larger B0 ? longerT1.
• Math dM/dt -(Mz-M0)/T1
• Solution Mz M0 (Mz(0)-M0)exp(-t/T1)
• After 90 pulse Mz M0 1-exp(-t/T1)
• M0 net magnetization based on B0.
• Mz component of M0 along the z-axis.
• t time

14
T2 Spin Relaxation

• T2 Spin Relaxation Decay of transverse
  magnetization, Mxy.
• T1 plays a role, since as Mxy ? Mz, Mxy ? 0
• But dephasing also decreases Mxy T2 lt T1.
• T2 Spin-Spin Time Constant
• Variations in Bz with time and position.
• Pertinent fluctuations in Bz are those near dc
  frequencies (independent of B0) so that w0 is
  changed.
• Molecular motion around the spin of interest.
• Liquids High Temp more motion, less DB, high T2
• Solids slow fluctuations in Bz, extreme T2.
• Bio Tissues spins bound to large molecules vs.
  those free in solution.

Mxy
z
B0DB(r,t)
y
x
15
T1/T2 Spin Relaxation
Comparison of T1 and T2 Spin Relaxation
Tissue T1 (ms) T2 (ms)
Gray Matter 950 100
White Matter 600 80
Muscle 900 50
Fat 250 60
Blood 1200 100-200
200 for arterial blood, 100 for venous blood. B0
1.5 T, 37 degC (Body Temp) Magnetic Resonance
Imaging Physical Principles and Sequence Design,
Haacke E.M. et al., Wiley New York, 1999.
z
B0DB(r,t)
y
Math dM/dt -Mxy/T2 After 90 pulse Mxy M0
exp(-t/T2)
x
16
T1/T2 Spin Relaxation
Comparison of T1 and T2 Spin Relaxation
Tissue T1 (ms) T2 (ms)
Gray Matter 950 100
White Matter 600 80
Muscle 900 50
Fat 250 60
Blood 1200 100-200
200 for arterial blood, 100 for venous
blood. Magnetic Resonance Imaging Physical
Principles and Sequence Design, Haacke E.M. et
al., Wiley New York, 1999.
z
T2 ltlt T1 Mxy decays exp(-t/T2)
B0DB(r,t)
Because T2 is independent of B0, higher B0 gives
better resolution
y
2/T2
FT
sr(t)
x
t
w0
Detector
FID
Spectrum
17
T1/T2 Spin Relaxation

• Inclusion of T1 and T2 Spin Relaxation
• Inclusion of mathematical expression
• Bloch Equation

z
g gyromagnetic ratio T1 Spin-Lattice
(longitudinal-z) relaxation time constant T2
Spin-Spin (transverse-x/y) relaxation time
constant M0 Equilibrium Magnetization due to B0
field. i, j, k Unit vectors in x, y, z
directions respectively.
B0DB(r,t)
y
x
18
T1/T2 Spin Relaxation

• Inclusion of T1 and T2 Spin Relaxation
• Inclusion of mathematical expression
• Bloch Equation

Precession
Transverse Decay
Longitudinal Growth
z
g gyromagnetic ratio T1 Spin-Lattice
(longitudinal-z) relaxation time constant T2
Spin-Spin (transverse-x/y) relaxation time
constant M0 Equilibrium Magnetization due to B0
field. i, j, k Unit vectors in x, y, z
directions respectively.
B0DB(r,t)
y
x
Net magnetization is not necessarily constant
e.g., very short T2, long T1.
19
Chemical Shift

• Chemical Shift Nuclei are shielded (slightly)
  from B0 by the presence of their electron clouds.
• Effective field felt by a nuclear spin is
  B0(1-s).
• Larmor precession freq, w gB0(1-s).
• Shift is often in the ppm range.
• 500,000 precessions before Mxy 0
• Chemical environment determines amount of s.
• H2O vs. Fat (fat about 3.5 ppm lower w0)

z
2d-
B0(1-s)
O
C
y
d
d
H
H
H
H
x
Discrete Shift
Less Shielding
More Shielding
Detector
20
Chemical Shift
Chemical Shift Nuclei are shielded (slightly)
from B0 by the presence of their electron clouds.
Ability to resolve nuclei in different chemical
environments is key to NMR
2/T2
z
w0
w0(1-s)
B0(1-s)
Because T2 is independent of B0, higher B0 gives
better resolution
y
x
Discrete Shift
Detector
21
Field Inhomogeneity

• T2 B0 Inhomogeneity Additional decay of Mxy.
• In addition to T2, which leads to Mxy decay even
  in a constant B0, application of dB0(x, y, z, t)
  will cause increased dephasing 1/T2 1/T2
  1/T, where T is the dephasing due only to
  dB0(x, y, z, t).
• T2 lt T2, and depends on dB0(x, y, z, t).
• Additional loss of resolution between peaks.

time, t
z
B0dB(r,t)
y
x
22
Field Inhomogeneity

• T2 B0 Inhomogeneity Additional decay of Mxy.
• In addition to T2, which leads to Mxy decay even
  in a constant B0, application of dB0(x, y, z, t)
  will cause increased dephasing 1/T2 1/T2
  1/T, where T is the dephasing due only to
  dB0(x, y, z, t).
• T2 lt T2, and depends on dB0(x, y, z, t).
• Additional loss of resolution between peaks.
• If dB0(x, y, z) is not time dependent, then it
  can be corrected by an echo pulse.

time, t
z
B0dB(r,t)
y
x
23
Field Inhomogeneity

• T2 B0 Inhomogeneity Additional decay of Mxy.
• In addition to T2, which leads to Mxy decay even
  in a constant B0, application of dB0(x, y, z, t)
  will cause increased dephasing 1/T2 1/T2
  1/T, where T is the dephasing due only to
  dB0(x, y, z, t).
• T2 lt T2, and depends on dB0(x, y, z, t).
• Additional loss of resolution between peaks.
• If dB0(x, y, z) is not time dependent, then it
  can be corrected by an echo pulse.

time, t
z
z
z
B0dB(r,t)
B0dB(r,t)
B0dB(r,t)
y
y
y
time, t
180x pulse (x ? x, y ? y)
x
x
x
Echo!
24
Field Inhomogeneity

• T2 B0 Inhomogeneity Additional decay of Mxy.
• If echo pulse applied at time, t, then echo
  appears at 2t.
• Only T can be reversed by echo pulsing, T2
  cannot be echoed as the field inhomogeneities
  that lead to T2 are not constant in time or
  space.
• 4) Signal after various echo pulsed displayed
  below.

T2
T2
sr(t)
t
t 180 pulse applied
t 180 pulse applied
2(t-t) Echo
2t Echo
t 0 90 pulse
25
Single B0 No Spatial Information
Measured response is from all spins in the sample
volume. Detector coil probes all space with equal
intensity
B0
B0
B0
time
90 pulse
Detector coil
If only B0 is present (and homogeneous) all spins
remain in phase during precession (as drawn). -
B(x, y, z, t) B0 thus, w(x, y, z) w0 gB0
2/T2
FT
No Spatial Information (Volume integral)
sr(t)
t
w0
FID
Spectrum
26
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Field strength indicated by line thickness
Gz dBz/dz integrate BzGzz It follows
that B(z0)B0
27
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Selective 90 pulse wrfw0gGzz
Field strength indicated by line thickness
Gz dBz/dz integrate BzGzz It follows
that B(z0)B0
28
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz

• Larmor Precession frequency is z-dependent
• w(z) gB(z)
• w(z) g(B0 Gzz)
• w(z) w0 gGzz

Gz
Selective 90 pulse wrfw0gGzz
Field strength indicated by line thickness

1. Excite only one plane of z Dz by using only one
   excitation frequency for the 90 pulse. For
   example, using B0 for excitation only spins at
   z0 get excited. All other spins are off
   resonance and are not tipped into the transverse
   plane.

Gz dBz/dz integrate BzGzz It follows
that B(z0)B0
29
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Selective 90 pulse wrfw0gGzz

• In practice, you must bandwidth match the
  frequency of the 90 pulse with the desired
  thickness (Dz) of the z-slice. (i.e., with a
  linear gradient, the Larmor precession of spins
  within z 0 Dz oscillate with frequency w0
  gGzDz. Thus, BW 2gGzDz.)
• 4) To apply a boxcar of frequencies w gGzDz,
  we need the 90 deg excitation profile to be a
  sinc function in time.
• FT(sinc) rect

Field strength indicated by line thickness
Gz dBz/dz integrate BzGzz It follows
that B(z0)B0
FT
90
z Dz
w
t
sinc (sinx)/x
30
Slice Selection z-Gradient
Gradient Echo Pulse. Gradient Echo pulse restores
all spins to have the same phase within the slice
Dz.
B(z) B0 Gzz
Before Gradient Echo t t
z
Gz
Selective 90 pulse
w0gGzDz
w0
w0-gGzDz
Spins out of phase on xy plane
Pulse Sequence
RF
Gradient Echo
Gz
t
3t/2
time
0
31
Slice Selection z-Gradient
Gradient Echo Pulse. Gradient Echo pulse restores
all spins to have the same phase within the slice
Dz.
B(z) B0 Gzz
Before Gradient Echo t t
z
Gz
Selective 90 pulse
w0gGzDz
w0
w0-gGzDz
Spins out of phase on xy plane
Pulse Sequence
Top View of xy plane
w0
w0gGzDz
w0-gGzDz
RF
tt
Gradient Echo
Gz
t
3t/2
time
0
32
Slice Selection z-Gradient
Gradient Echo Pulse. Gradient Echo pulse restores
all spins to have the same phase within the slice
Dz.
B(z) B0 Gzz
Before Gradient Echo t t
z
Gz
Selective 90 pulse
w0gGzDz
w0
w0-gGzDz
Spins out of phase on xy plane
Pulse Sequence
Top View of xy plane
w0
w0gGzDz
w0-gGzDz
RF
After Gradient Echo t 3t/2
tt
Gradient Echo
z
Gz
t3t/2
t
3t/2
time
0
Spins all IN phase
33
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Selective 90 pulse
34
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Selective 90 pulse
time
Detector coil
35
Slice Selection z-Gradient
Slice selection along z-axis. Gradient in z and
selective excitation allows detection of a single
slice.
B(z) B0 Gzz
Gz
Selective 90 pulse
time
Detector coil
exp(-t/T2)
No x, y Information, but only spins from the z
Dz slice contribute to the signal.
2/T2
FT
sr(t)
t
FID
w0
Spectrum
If we can encode along x and y dimensions, we can
iterate for each z slice.
36
Frequency Encoding
Perform z-slice. Now only look at 2D plane from
now on. Use Gradient along x to generate
different Larmor frequencies vs. x-position.
y
y
z
z
z
Selective 90 pulse in z Dz
time
x
x
x
2Dz
Bz(x) - B0
Apply x-Gradient Gx dBz/dx Precession Frequency
varies with x
37
Frequency Encoding
Perform z-slice. Now only look at 2D plane from
now on. Use Gradient along x to generate
different Larmor frequencies vs. x-position.
y
y
z
z
z
Selective 90 pulse in z Dz
time
x
x
x
2Dz
Bz(x) - B0
w(x)
w0
w0 gGxx
w0 - gGxx
Frequency Encoding along x
Apply x-Gradient Gx dBz/dx Precession Frequency
varies with x
38
Frequency Encoding
Perform z-slice. Now only look at 2D plane from
now on. Use Gradient along x to generate
different Larmor frequencies vs. x-position.
z
Pulse Sequence
RF
x
Gz
Bz(x) - B0
Detector coil
Gx
w(x)
time
0
w0
w0 gGxx
w0 - gGxx
Detect Signal readout Gx on while detecting
39
Frequency Encoding
Perform z-slice. Now only look at 2D plane from
now on. Use Gradient along x to generate
different Larmor frequencies vs. x-position.
z
T2 is based on the intentionally applied
gradient.
exp(-t/T2)
sr(t)
t
x
FT
FID
Bz(x) - B0
Detector coil
w(x)
w0
w0 gGxx
w0 - gGxx
2/T2
Apply x-Gradient DURING acquisition. Precession
Frequency varies with x.
w0
w0 - gGxx
w0 gGxx
40
Frequency Encoding
Perform z-slice. Now only look at 2D plane from
now on. Use Gradient along x to generate
different Larmor frequencies vs. x-position.
z
T2 is based on the intentionally applied
gradient.
exp(-t/T2)
sr(t)
t
x
FT
FID
Bz(x) - B0
Detector coil
w(x)
w0 gGxx
w0
w0 - gGxx
2/T2
Apply x-Gradient DURING acquisition. Precession
Frequency varies with x. Spins at various x
positions in space are encoded to a different
precession frequency
w0
w0 - gGxx
w0 gGxx
41
Imaging Example
Two Microfluidic Channels. Water only exists in
two microfluic channels as shown.
y
y
Application of Gx
z
z
z
90 pulse
time
x
x
x
Dz
Bz(x)
42
Imaging Example
Two Microfluidic Channels. Water only exists in
two microfluic channels as shown.
y
y
Application of Gx
z
z
z
90 pulse
time
x
x
x
Dz
Bz(x)

1. No spins exist at x0 where Gx0 (w0) FT of
   signal has no intensity at w0.
2. Signal is the line integral along y. (Still no
   info about y distribution of spins.)

Image
m(x,y) spin density(x,y)
w0
w0 - gGxx
w0 gGxx
43
1DFT Math
Signal is the 1DFT of the line integral along y.
Homodyne the signal (from w0 to 0).
44
1DFT Math
Signal is the 1DFT of the line integral along y.
Homodyne the signal (from w0 to 0).
Let g(x) Line integral along y for a given x
position.
45
1DFT Math
Signal is the 1DFT of the line integral along y.
Homodyne the signal (from w0 to 0).
Let g(x) Line integral along y for a given x
position.
Spatial frequency Gxt kx
The homodyned signal is thus the Fourier
Transform (along x) of the line integral along y.
46
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
47
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
48
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
49
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
50
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
51
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
52
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
53
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
t1
Pulse Sequence
RF
time
Gz
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
54
k-vector perspective
Time Evolution of Spins in an x-Gradient. Spatial
frequency, k-vector, changes.
Homo- dyne s(t)
Spatial frequency encoded by phase
k-vector amount of spin warping over distance
SMi(x)
w0 gGxx
w0
w0 - gGxx
x
k0
t1
k one spatial period
Pulse Sequence
RF
time k
Gz
FID
Dephasing across x in time. Rotating frame w0 or
relative to x0
Gx
time
0
t1
t2
Each Point on FID is a different value of kx
55
2 Approaches to Understand FT
The imaging in 1D can be understood in 2
ways 1) From the received signal perspective
The spins, spatially separated along the
x-dimension, are distinguished by the application
of a gradient field that makes their Larmor
precession vary along x. The FT resolves the
difference in frequency and hence position. 2)
Homodyned (baseband) signal perspective As time
passes during the application of the gradient,
the spins dephase from each other. The amount of
dephasing can be represented as a spatial
frequency, kx, that increases with measurement
time.
Frequency Encoding Precession w(x)
2/T2
FT of FID (time) gives frequency, w. w depends on
position
w0
w0 - gGxx
w0 gGxx
Phase Encoding Phase(t) Spatial Freq.
FT of spatial frequency data, kx, data gives
position data, x. Different values of kx are
probed over time, t.
56
2DFT Principles
Again perform z-slice. Only look at 2D plane.
Want to now distinguish spins along y-direction
also.
y
y
Gy
z
z
x
x
90 pulsed Plane
Apply y-Gradient for time ty Gy dBz/dy Then Gy
turned off
57
2DFT Principles
Phase Encoding. Gy is turned on for a certain
time, ty, then off. This generates a difference
in phase over y.
y
y
y
Gy
z
z
z
All precess at w0
x
x
x
90 pulsed Plane
Apply y-Gradient for time ty Gy dBz/dy Then Gy
turned off
But spin warped along y by an amount determined
by Gyty single ky value
Phase encoding along y
58
2DFT Principles
Detect Using Gx. As usual detection occurs with
Gx.
y
y
y
Gy
z
z
z
All precess at w0
x
x
x
90 pulsed Plane
Apply y-Gradient for time ty Gy dBz/dy Then Gy
turned off
Detect with Gx
y
z
x
Bz(x)
Detector coil
Usual frequency encoding along x
59
2DFT Principles
Detect Using Gx. As usual detection occurs with
Gx.
y
y
y
Gy
z
z
z
All precess at w0
x
x
x
90 pulsed Plane
Apply y-Gradient for time ty Gy dBz/dy Then Gy
turned off
Detect with Gx
y
z
This time, the magnitude of the signal at each w
(x-position), corresponds to the intensity of the
spatial frequency, ky, encoded by Gy phase
encoding step. (for ky0 its the line integral)
x
Bz(x)
Detector coil
Frequency Encoding along x (Gxt) Phase Encoding
along y (Gyty)
60
2DFT Principles
Again perform z-slice. Only look at 2D plane.
Want to now distinguish spins along y-direction
also.
Pulse Sequence
z
RF
Phase Encoded
Gz
Gy
ty
x
Gx
Bz(x)
Detector coil
0
time
Detect Signal
Phase Encode
61
2DFT Principles
Again perform z-slice. Only look at 2D plane.
Want to now distinguish spins along y-direction
also.
Pulse Sequence
z
RF
Phase Encoded
Gz
Gy
ty
x
Gx
Bz(x)
Detector coil
0
time
Repeat experiment multiple times varying the Gy
gradient strength (or time ty) so that ky
receives the same sampling as kx (FID sampling
rate).
Detect Signal
Phase Encode
62
2DFT Math
Signal is the 2DFT of the image.
Baseband (Homodyned) signal.
Gx during recording of FID
Phase Encoding Step
For any given FID, ty is fixed and t is running
variable.
63
2DFT Math
Signal is the 2DFT of the image.
Baseband (Homodyned) signal.
Gx during recording of FID
Phase Encoding Step
For any given FID, ty is fixed and t is running
variable.
Intensities at each x correspond to intensity of
the ky spatial frequency (applied during phase
encoding) at that x position. In other words,
the intensity corresponds to 1 pt on the FID
taken in the y direction
w0
w0 - gGxx
w0 gGxx
64
2DFT Principles
k-space perspective. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Set of data points sampled from the FID with a
phase encoding of a given ky (Gyty).
Set of data points along the kx axis corresponds
to the sampled FID taken with no Gy phase
encoding gradient.
kx
Change Gyty
Measure FID kx measured in time
65
2DFT Principles
k-space perspective. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Thus, it is evident that a column of data (at a
given x position) on the collection of points in
k-space represents the FT of the various Gy
values. The data along a line is the FT of the
signal in the y direction.
kx
66
2DFT Principles
k-space perspective. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Thus, it is evident that a column of data (at a
given x position) on the collection of points in
k-space represents the FT of the various Gy
values. The data along a line is the FT of the
signal in the y direction.
kx
Rotate for viewing
FID along y.
67
2DFT Principles
k-space perspective. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
Image
ky
What does this data look like?
y
Sinc function
kx
Rect function
x
Sinc function
Rect function
68
2DFT Principles
k-space perspective. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
Image
ky
What does this data look like?
y
kx
x
Jinc function Radially symmetric sinc)
Circle function (radially symmetric rect)
69
2DFT Principles
Updated Pulse Sequence. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Pulse Sequence
ty
RF
Gz
ty
Gy
kx
Gx
2ty
0
time
70
2DFT Principles
Updated Pulse Sequence. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Pulse Sequence
ty
RF
ty
Gz
ty
Gy
kx
Gx
0
time
71
2DFT Principles
Updated Pulse Sequence. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Pulse Sequence
ty
RF
ty
Gz
ty
Gy
kx
Gx
0
time
72
2DFT Principles
Updated Pulse Sequence. Want to map k-space then
take 2DFT. (Each FID samples line in k-space
along kx)
ky
Pulse Sequence
ty
RF
ty
Gz
Gy
kx
Gx
0
time
Representation
73
Discrete FT Imaging Issues
Sampling Rate Issues Real time FID is sampled
at various times of interval, Dt, which leads to
a sampling rate in the kx dimension of (Dkx).
Interval on Dky is determined by the change in
gradient area (DGyty) between different runs
ky
t, kx
We know that we need enough data to adequately
sample the FID in time (kx) dimension Same
principle applies for ky (Gyty) dimension
kx
Sampling rate of k-space
74
Field of View
Field of View Sampling rate of k-space
determines the field of view in the
object-oriented domain.
FOVx 1/(Dkx)
y
Dkx
ky
FOVy 1/(Dky)
Dky
kx
x
Sampling rate of k-space
FOV gt Image size! Prevent Aliasing
75
Aliasing Issues
Aliasing If sampling rate is not sufficient, the
Field of view will overlap.
y
FOVx 1/(Dkx)
Dkx
ky
Dky
FOVy 1/(Dky)
kx
x
Sampling rate of k-space
FOV gt Image size! Prevent Aliasing (Image Overlap)
76
Resolution
Resolution Resolution in the object-oriented
domain is determined by the extent of k-space
measured.
dx FOVx/Nread (DkxNread)-1
y
ky
DkxNread
dy FOVy/Npe (DkyNpe)-1
DkyNpe
kx
x
Sampling rate of k-space
Field of View/Resolution points need to
sample (e.g., 25.6 cm image, 1mm resolution 256
points/dimension, 65.5k points)
Nread of readout points during FID
Npe of phase encoding steps
77
Summary

1. MRI is based on the spatial encoding of spins
   either through a difference in phase (y) or a
   difference in Larmor frequency (x)
2. FID in the presence Gx, after a given phase
   encoding in y, gives a line of points in k-space.
   FIDs are repeated for a variety of ky values to
   fill up k-space.
3. 2DFT of k-space gives the image of spin density
   m(x,y)
4. Limitations.
5. Detection is based on the signal received in a
   coil.
6. Coil inductor has an impedance, Zcoil iwL,
   frequency. Thus significant voltage signals are
   observed only at high frequencies. (Mxy ? icoil.
   icoil vsignal/Zcoil.)
7. Requires Large Magnetic fields cryogenics,
   homogeneity.
8. Large Fields can lead to signal distortion.
   Samples containing metals cannot be imaged.

78
Acknowledgements David Michalak
Pines Group Alex Pines Chip Crawford Hattie
Ziegler Marcus Donaldson Thomas Theis All current
nuts
Budker Group Dmitry Budker Micah Ledbetter
Good Books 1) Principles of Magnetic Resonance
Imaging, Dwight G. Nishimura, Stanford
University 2) Magnetic Resonance Imaging
Physical Principles and Sequence Design, Haacke
E.M. et al., Wiley New York, 1999.