eleg 479 lecture #12 magnetic resonance (mr) imaging
DESCRIPTION
ELEG 479 Lecture #12 Magnetic Resonance (MR) Imaging. Mark Mirotznik, Ph.D. Associate Professor The University of Delaware. Physics of Magnetic Resonance Summary. Protons and electrons have a property called spin that results in them looking like tiny magnets. . N. N. S. =. No Net - PowerPoint PPT PresentationTRANSCRIPT
ELEG 479Lecture #12
Magnetic Resonance (MR) Imaging
Mark Mirotznik, Ph.D.Associate Professor
The University of Delaware
Physics of Magnetic ResonanceSummary
Protons and electrons have a property called spin that results in them looking like tiny magnets.
S
NN
S
In the absence of an external magnetic field all the magnetic dipole are oriented randomly so we get zero net magnetic field when we add them all up.
RandomOrientation
=No Net
Magnetization
Physics of Magnetic ResonanceSummary
When we add a large external magnetic field we can get the protons to line up in 1 of 2 orientations (spin up or spin down) with a few more per million in one of the orientations than the other. This produces a net magnetization along the axis of the applied magnetic field.
When we add a large external magnetic field we cause a torque on the already spinning proton that causes it to precess like a top around the applied magnetic field. The frequency is precesses , called its Larmor frequency., is determined from Larmor’s equation.
Do
o PkT
BM4
22
z
x
y
z
xy
The net magnetization vector is the sum of all of these little magnetic moments added together. This is what we measure.
x
y
z
z
xy
x
y
z
z
xy
z
x
y
z
xy
x
y
z
z
xy
Net Magnetization Vector
zMtMtM
tzyxtM
zxy
N
nnnnn
ˆ)()(
),,,()(1
Physics of Magnetic ResonanceSummary from Last Lecture
Physics of Magnetic ResonanceSummary from Last Lecture
However since all the spinning protons are precessing out of phase with each other this results in zero net magnetization in the transverse plane.
zMtMtM zxy ˆ)()(
= 0
This is bad news since )(tM xy
is where are signal comes from!
Somehow we need to get these guys spinning together!
RF Excitationtime
B1
B1
To get them to all spin together we add a RF field whose frequency is the same as the Larmor resonant frequency of the proton and is oriented in the xy or transverse plane.
Physics of Magnetic ResonanceSummary from Last Lecture
a B1Dt
Tip Angle Amplitude of RFPulse
Time of Applicationof RF Pulse
Physics of Magnetic ResonanceSummary from Last Lecture
x
y
z
M
xyM
a
B1
Physics of Magnetic ResonanceSummary from Last Lecture
x
y
z
M
xyM
a
dttBe )(0 1
a
)(1 tBe
)(1 tBe= envelope of the RF signalIn general
Physics of Magnetic ResonanceSummary from Last Lecture
To get the signal out we place a coil near the sample. A time-varying transverse magnetic field will produce a voltage on the coil that can be digitized and stored for processing.
)sin()(
)()(
a
oo
xy
MKtV
tMdtdtV
Do
o PkT
BM4
22
recall
oo Band
Relaxation Processes
After the RF field is removed over time the spin system will return back to it’s equilibrium state due to several relaxation processes.
Physics of Magnetic ResonanceRelaxation
Physics of Magnetic ResonanceRelaxation
After the RF field is removed over time the spin system will return back to it’s equilibrium state due to several relaxation processes. These are:
(1) Spin-Spin relaxation (also called the T2 relaxation): Due to random processes in which neighboring proton spins effect each other spin system will lose coherence and Mxy will decay. This is an irreversible process.
(2) Spin-Lattice relaxation (also called T1 relaxation): Due to another random process the Mz will begin to recover back to it’s original equilibrium state. Also irreversible.
(3) T2* relaxation: Due to inhomogenities in the external Bo
field Mxy will decay much faster than T2. This is a reversible process.
T1 Relaxation
T2 Relaxation (FID)
T2 Decay
RF
RF
T2* Relaxation
T2* Decay: Dephasing due to field inhomogeneity
x'
y'
z'
Mxy = 0
T2* relaxation is dephasing of transverse magnetization too
but it turns out to be reversible
Animation of T2* Dephasing
Spin- Echo
Spin Echo
Summary of Relaxation Processes
MRI Image An MRI image is determined by two things
three intrinsic properties of the tissue. These are: T1, T2 and Pd. (two relaxation time constants and the density of protons)
the details of the external magnetic fields (Bo, B1 and the gradient magnetics (have not talked about these yet)). How they are configured and how we turn them on and off (pulse sequence) effects what the image looks like.
By varying the pulse sequence we can control which of the intrinsic properties to emphasize in the image.
Tissue Contrast
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
0.5TE
0.5TE
TE
0.5TE
0.5TE
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses CASE I: TR>>T1 , TE<T2
What do we measure?
TE
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
CASE I: TR>>T1 , TE<T2
Short TE means that the signal has not decayed much due to T2relaxation. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?
TE
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
CASE I: TR>>T1 , TE<T2
Short TE means that the signal has not decayed much due to T2relaxation. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?
PD Weighted imaging! TE
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
TE
CASE II: TR>>T1 , TE~T2
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
TE
CASE II: TR>>T1 , TE~T2
TE on the order of T2means that the signal is proportional to the T2relaxation constant. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?
White matterT1=813 msT2=101 ms
TR
TE
90 degreeRF pulses
180 degreeRF pulses
TE
CASE II: TR>>T1 , TE~T2
TE on the order of T2means that the signal is proportional to the T2relaxation constant. Long TR means that by the next pulse the system is back at equilibrium (Mz due to T1 relaxation has fully recovered) So what are we measuring?
T2 Weighted imaging!
TR
90 degreeRF pulses
180 degreeRF pulses
TE
CASE III: TR~T1 , TE<T2
What do we measure?
TR
90 degreeRF pulses
180 degreeRF pulses
TE
CASE III: TR~T1 , TE<T2
TE is shorter than T2means that the signal is not heavily weighted on the T2relaxation constant. TR on the order of T1 means that by the next pulse the system is not back at equilibrium (Mz due to T1 relaxation has not fully recovered) So what are we measuring?
TR
90 degreeRF pulses
180 degreeRF pulses
TE
CASE III: TR~T1 , TE<T2
TE is shorter than T2means that the signal is not heavily weighted on the T2relaxation constant. TR on the order of T1 means that by the next pulse the system is not back at equilibrium (Mz due to T1 relaxation has not fully recovered) So what are we measuring?
T1 Weighted imaging!
Tissue ContrastSummary
TE TR
PD weighted TE<T2 (short TE) TR>>T1 (long TR)
T1weighted TE<T2 (short TE) TR~T1 (short TR)
T2 weighted TE~T2 (long TE) TR>>T1 (long TR)
Bloch Equations
Full Bloch equation including relaxation
12 Tz))((
T)y(t)x(t)(
(t)Bγ)(dt
d ozyx MtMMMtMM
transverse magnetization
precession,RF excitation longitudinal
magnetization
)(B(t) B 1o tB
includes Bo and B1
Example: Solve for the transverse components of M after a 90 degree pulse.
12 Tz))((
T)y(t)x(t)(
(t)Bγ)(dt
d ozyx MtMMMtMM
-(t)1
(t)1
(t)1
)()(M)()(M)()(M)(M-)()(M)(M
γ)()()(
dtd
1
2
2
yx
zx
zy
oz
y
x
xy
xo
yo
z
y
x
MMT
MT
MT
tBttBttBtBttBtBt
tMtMtM
Example: Solve for the transverse components of M after a 90 degree pulse.
-(t)1
(t)1
(t)1
)()(M)()(M)()(M)(M-)()(M)(M
γ)()()(
dtd
1
2
2
yx
zx
zy
oz
y
x
xy
xo
yo
z
y
x
MMT
MT
MT
tBttBttBtBttBtBt
tMtMtM
After 90 degree pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
Example: Solve for the transverse components of M after a 90 degree pulse.
After 90 degree pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
Initial conditions for 90 degree pulse:
0)0(
)0(0)0(
z
oy
x
M
MMM
Example: Solve for the transverse components of M after a 90 degree pulse.
After 90 degree pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
)1()( 1Tt
oz eMtM
2
2
)cos()(
)sin()(
Tt
ooy
Tt
oox
etMtM
etMtM
Solutions
Example: Solve for the transverse components of M after a 90 degree pulse.
After 90 degree pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
)1()( 1Tt
oz eMtM
2)(
)()()(
Tt
tjoxy
yxxy
eeMtM
tjMtMtM
o
Solutions
Example: Solve for the transverse components of M after an arbitrary flip angle (a)
-(t)1
(t)1
(t)1
)()(M)()(M)()(M)(M-)()(M)(M
γ)()()(
dtd
1
2
2
yx
zx
zy
oz
y
x
xy
xz
yz
z
y
x
MMT
MT
MT
tBttBttBtBttBtBt
tMtMtM
After an arbitrary RF pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
Example: Solve for the transverse components of M after an arbitrary flip angle (a)
After an arbitrary RF pulse the RF field is shut down and only Bo is non-zero
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
Initial conditions for 90 degree pulse:
)cos()0(
)sin()0(0)0(
a
a
oz
oy
x
MM
MMM
Example: Solve for the transverse components of M after an arbitrary flip angle (a)
-(t)1
(t)1
(t)1
0)(M-)(M
γ)()()(
dtd
1
2
2
0x
0y
oz
y
x
z
y
x
MMT
MT
MT
BtBt
tMtMtM
11
11
2
2
)cos()1(
)0()1()(
)sin(
)0()(
Tt
oTt
o
Tt
zTt
oz
Tttjo
TttBjxyxy
yxxy
eMeM
eMeMtM
eeM
eeMtM
jMMM
o
o
a
a
Solutions
Solve full Bloch equation with only B=Bo
Solution for transverse components Mx and My
)sin()0()0(
)0(
)0()(2
2
a
zxy
Tttjxy
TttBjxyxy
MM
eeM
eeMtMo
o
Where a is the flip angle after RF excitation
11 )cos()1()( Tt
oTt
oz eMeMtM
a
Signal Detection
Signal Detection via RF coil
Signal Detection via RF coil
max
min
max
min
max
min
2 ),,(/)0,,,()(z
z
y
y
x
x
zyxTttBjxy dxdydzeezyxMAts o
Coils oriented as shown above will only respond to changes in the transverse magnetic field (this is what we want)
Assuming the magnetic fields are homogenous the signal will be a weighted integration of all the protons within the coil.
The waiting will be based on the total magnetization at location x,y,z at the start of the pulse (Mxy(x,y,z,0)) and the tissue decay time T2(x,y,z)
This is not an image!!
Transverse magnetization at t=0.
Signal Detection via RF coil
max
min
max
min
max
min
2 ),,(/)0,,,()(z
z
y
y
x
x
zyxTttBjxy dxdydzeezyxMAts o
max
min
max
min
max
min
2
max
min
max
min
max
min
2
),,(/
),,(/
)0,,,()(
)0,,,()(
z
z
y
y
x
x
zyxTtxy
tj
z
z
y
y
x
x
zyxTttjxy
dxdydzezyxMAets
dxdydzeezyxMAts
o
o
After demodulation:
max
min
max
min
max
min
2 ),,(/)0,,,()(z
z
y
y
x
x
zyxTtxyo dxdydzezyxMAts
Creating an Image
Creating an ImageTo create an image using NMR we need to figure out a way to encode the proton spins spatially in three dimensions.
But how?
Frequency and Phase Are Our Friends in MR Imaging
q
q = t
The spatial information of the proton pools contributingMR signal is determined by the spatial frequency andphase of their magnetization.
Gradient Coils
Gradient coils generate spatially varying magnetic field so that spins at different location precess at frequencies unique to their location, allowing us to reconstruct 2D or 3D images.
X gradient Y gradient Z gradient
x
y
z
x
z z
x
y y
Gradient Coils
Sounds generated during imaging due to mechanical stress within gradient coils.
Purpose: Spatially alter magnitude of B0 (not direction)
Vector Notation
rGBzyxB
azayaxr
aGaGaGG
oz
zyx
zzyyxx
),,(
ˆˆˆ
ˆˆˆ
Larmor frequency within a gradient field
rGBr
rGBzyxB
azayaxr
aGaGaGG
o
oz
zyx
zzyyxx
)(
),,(
ˆˆˆ
ˆˆˆ
Slice Selection
Slice Selection Gradient
BG
Coil 1
Coil 2
Helmholtz Coils
zGzzGBz
z
zo
0)()(
Z-Gradient Fields
By adding a z-gradient field we cause a variation in theresonant frequency from head to toe.
A sample is put inside a 1.5T magnet. A z-gradient of 3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?
Example
A sample is put inside a 1.5T magnet. A z-gradient of3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?
TB
TB
z
zGBrGBzyxB
z
z
zooz
509144.154.21210000
35.1
490856.154.21210000
35.1
1000035.1
),,(
max
min
Example
A sample is put inside a 1.5T magnet. A z-gradient of3 gauss/cm is applied. If we wish to image a 2 ft in length section of a person what is the range of resonant frequencies we will encounter?
MHzMHzteslaMHzzBz
TB
TB
z
z
26.64509144.158.4248.63490856.158.42
/58.42),()(
509144.154.21210000
35.1
490856.154.21210000
35.1
max
min
max
min
Example
A Field Gradient Makes the Larmor Frequency Depend upon Position
B(Z) B G Zo Z *
(z) B(z)
64,260,000 Hz63,480,000 Hz
1.500 T 1.501 T
Z
B0
Gradient in Z
Slice Selection
64 MHz
65 MHz
66 MHz
63 MHz
62 MHz
G
(-)
(+)
Slice SelectionHow do we determine the slice width and center?
z
x
Bo
After z selectiongradient and excitationz-gradient
zDz (slice width)(slice center)
Determining slice thicknessResonance frequency range as the result of slice-selective gradient:
z
oo
z
o
z
o
zo
GBBzzz
GBz
GBz
zGBz
D
minmaxminmax
maxmax
minmin ,
)(
zGz
D
D
Changing slice thickness
There are two ways to do this:
(a) Change the slope of the slice selection gradient
(b) Change the bandwidth of the RF excitation pulse
Both are used in practice, with (a) being more popular
zGz
D
D
Suppose we wish to have a slice thickness of 2 mm and we are using a z-gradient of 1.0 G/cm ? What range of RF frequencies should we use?
Example
Suppose we wish to have a slice thickness of 2 mm and we are using a z-gradient of 1 G/cm ? What range of RF frequencies should we use?
Example
HzMHz
cm
Gz
z
6.85110516.810000
158.422.0
4 D
D
DD
Selecting different slices
z
o
z
o
z
o
z
o
z
o
zo
GGBz
GBzzz
GBz
GBz
zGBz
2/2
22
,
)(
minmax
minmaxminmax
maxmax
minmin
z
o
Gz
Selecting different slicesIn theory, there are two ways to select different slices:
(a) Change the position of the zero point of the slice selection gradient with respect to isocenter
(b) Change the center frequency of the RF to correspond to a resonance frequency at the desired slice
Option (b) is usually used as it is not easy to change theisocenter of a given gradient coil.
z
o
Gz
RF Excitation (RF Pulse)
0 t
Fo
Fo Fo+1/ t
Time Frequency
t
Fo Fo
DF= 1/ t
FT
FT
RF Excitation (RF Pulse)
t
Fo
D
FT
tjettAts
2)sin()(DD
D
2
)()(
12 vvv
vrectAS
D
A
21
RF Excitation: Flip angle
t
Fo
D
FT
tjettAts
2)sin()(DD
D
2
)()(
12 vvv
vrectAS
D
A
21
max
min
)(1
t
t
e dttBa
Envelope of the pulse
RF Excitation: Flip angle
t
Fo
D
FT
tjettAts
2)sin()(DD
D
2
)()(
12 vvv
vrectAS
D
A
21
max
min
max
min
21 )()(
t
t
tjt
t
e dtetsdttB a
RF Excitation: Flip angle
t
Fo
D
FT
tjettAts
2)sin()(DD
D
2
)()(
12 vvv
vrectAS
D
A
21
)()(
)sin()(max
min
1
zzzrectAvrectA
dtttAdttB
t
t
e
D
D
DD
D
a
a
RF Excitation: Flip angle (truncated sinc)
)()sin()(~ 2
p
tj trectettAts
DD
D
))((sin)( zzGczzzrectA zpp *
D
a
DD
D2
2
)sin(p
p
dtttA
a
2/p 2/p
A potential problemWait a minute! Dr. M I remember you telling us that if the magnetic field varied from place to place (inhomogeneous) then we would get rapid dephasing of spins. Since some spins are spinning faster than others they quickly get out of phase. That was the whole reason behind the spin echo stuff!
Won’t that happen again?
Wait a minute! Dr. M I remember you telling us that if the magnetic field varied from place to place (inhomogeneous) then we would get rapid dephasing of spins. Since some spins are spinning faster than others they quickly get out of phase. That was the whole reason behind the spin echo stuff! Won’t that happen again?
Spinning fast
Spinning slow
Why yes it will! It is called gradient dephasingGood question!
Spinning fast
Spinning slow
Why yes it will! It is called gradient dephasing. It will quickly kill our signal much faster than T2 or even T2*
Any ideas on how to get around this?
Localization in xy plane
Lets Start with a Simple Flat Person(only xz plane)
z
x
Bo
Lets Start with a Simple Flat Person(only xz plane)
z
x
Bo
After z selectiongradient and excitation
z-gradient
Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis
max
min
max
min
max
min
2 ),,(/)0,,,()(z
z
y
y
x
x
zyxTttBjxy dxdydzeezyxMAts o
Recall that the signal we measure is given by:
Now we have selected only a single slice in z (z=zo) and we have no y dependence (flat person)
dxexMAets xTtxy
tj o )(/2 2)0,()(
After demodulation (envelope detection)
dxexMAetsts xTtxy
tjo
o )(/2 2)0,()()(
Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis
After demodulation (envelope detection)
dxexMAetsts xTtxy
tjo
o )(/2 2)0,()()(
Let )(/ 2)0,()( xTtxy exMAxf
dxxftso )()(
This is what we want to image (called the effective proton density)
Lets Start with a Simple Flat Person(frequency encoding using x-gradient)
z
x
Bo
After z selectiongradient and excitation
Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis
Now lets apply a gradient in the x direction (Gx)
)(/ 2)0,()( xTtxy exMAxf
dxeexMAts xTttxGjxy
xo )(/2 2)0,()(
dxeexMAets xTttxGjxy
tj xo )(/22 2)0,()(
After demodulation (envelope detection)
dxeexMAts xTttxGjxyo
x )(/2 2)0,()(
dxexfts txGjo
x2)()( What does this look like?
Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis
After demodulation (envelope detection)
dxexfts txGjo
x2)()(
Let tGu x
dxexfGusuF uxj
xo
2)()()(
The received signal is related to the Fourier transform (THIS IS THE KEY!)
Lets Start with a Simple Flat PersonFrequency Encoding Mathematical Analysis
)(/ 2)0,()( xTtxy exMAxf
After demodulation (envelope detection)
dxexfts txGjo
x2)()(
Let tGu x
dxexfGusuF uxj
xo
2)()()(
We can now find our image as a function of x by taking an inverse Fourier Transform
A Simple Example of Spatial Encoding with Frequency Encoding
w/o encoding w/ encoding
ConstantMagnetic Field
VaryingMagnetic Field
FrequencyDecomposition
A Simple Example of Spatial Encoding with Frequency Encoding
Decays faster than T2*
Extend this to a full 3D person
x
y
Extend this to a full 3D personAfter slice selection we need to image in xy plane
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
Now lets apply a gradient in the x direction (Gx)
),(/ 2)0,,(),( yxTtxy eyxMAyxf
dydxeeyxMAts yxTttxGjxy
xo
),(/2 2)0,,()(
After demodulation (envelope detection)
dydxeyxfts txGjo
x2),()(
dydxeeyxMAts yxTtxtGjxyo
x
),(/2 2)0,,()(
Effective proton density
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
),(/ 2)0,,(),( yxTtxy eyxMAyxf
After demodulation (envelope detection)
dydxeyxfts txGjo
x2),()(
dydxeeyxMAts yxTtxtGjxyo
x
),(/2 2)0,,()(
Let tGu x
dydxeyxfGutsvuF uxj
xo
2),()()0,(
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
dydxeyxfts txGjo
x2),()(
Let tGu x
dydxeyxfGutsvuF uxj
xo
2),()()0,(
Corresponds to a single line or trajectory in the uv plane
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
Now lets apply gradients in both the x direction (Gx) and y direction (Gy)
After demodulation (envelope detection)
dydxeeeyxMAts yxTtytGjxtGjxyo
yx
),(/22 2)0,,()(
),(/ 2)0,,(),( yxTtxy eyxMAyxf
dydxeeyxfts tyGjtxGjo
yx 22),()(
Let
Effective proton density
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
Now lets apply gradients in both the x direction (Gx) and y direction (Gy)
dydxeeyxfts tyGjtxGjo
yx 22),()(
Let tGvtGu yx ,
dydxeeyxfvuF vxjuxj 22),(),(
Spatial Encoding in xy planeFrequency Encoding Mathematical Analysis
Let tGvtGu yx ,
dydxeeyxfvuF vxjuxj 22),(),(
Polar Scanning
Gradient Echo(A brief detour)
Ts/2
Spatial Encoding in xy planeGradient Echo Mathematical Analysis
Spins will dephase very quickly (quicker than T2*) due to the
gradient fields.
2)2,(
2,),(
sx
sp
spppx
t
x
TxGTx
TttxGxdtGtxp
After negative x-gradient
Spatial Encoding in xy planeGradient Echo Mathematical Analysis
2)2,(
2,),(
sx
sp
spppx
t
x
TxGTx
TttxGxdtGtxp
After negative x-gradient
Now impose positive x-gradient for Ts
23
2,),(
222),(2
sp
sppxsx
t
T
spx
sxx
sx
TtTtxGxTGtx
TtxGTxGxdtGTxGtxs
p
Spatial Encoding in xy planeGradient Echo Mathematical Analysis
Now impose positive x-gradient for Ts
0),(
23
2,),(
pspxsxsp
sp
sppxsx
TxGxTGTtx
TtTtxGxTGtx
Phase Encoding
Pulse Repetition
Image Contrast
Image Quality
Dv
Du
Vcoverage
Ucoverage
Dx
Dy
FOVx
FOVy
Fourier PlaneSpatial Domain
Field of View and Resolution in MRI
Nyquist Sampling Theorem: Review
• Assume we have a continuous signal with maximum frequency of fmax
• To avoid aliasing we must sample the signal at a sampling frequency of fs>=2 fmax
• The sampling interval T=1/ fs
• fmax<=1/(2T)
Sampling in MRI
• Slice selection direction: sampling in z-directionSlice thickness (Dz) controlled by RF excitation
bandwidth (D)To avoid aliasing
Where fmax,z is the highest spatial frequency in along the z-axis
zz f
zfz max,
max, 2121
DD
Sampling in MRI
• Within each slice: sampling in xy plan We sample in the Fourier domain (u,v)
(called k-space in MRI literature, kx=u, ky=v) Rectilinear Scan
Du depends on sampling interval T during readout (ADC)
Du depends on sampling interval during this time
Sampling in MRI• Within each slice: sampling in xy plan
We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Rectilinear Scan
Dv depends on spacing between phase encoding
Dv depends on the integrated phase shift here
Sampling in MRI• Within each slice: sampling in xy plan
We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Polar Scan
Angle scan depends on steps in Gy/Gx
Angle scan depends on steps in Gy/Gx
Sampling in MRI• Within each slice: sampling in xy plan
We sample in the Fourier domain (u,v) (called k-space in MRI literature, kx=u, ky=v) Polar Scan
Rho spacing depends on sampling interval T during readout
r spacing depends on sampling interval during this time
Dv
X-gradient relates dimension x with Larmor freq byxGx xo )(
To avoid aliasing only frequency given below are measured
222)(
2s
xss
os
ofxGffxf
X-gradient relates dimension x with Larmor freq byxGx xo )(
To avoid aliasing only frequency given below are measured
x
s
x
s
sx
sso
so
Gfx
Gf
fxGffxf
22
222)(
2
Field of view in the x-direction (FOVx) is thus given by
uTGFOV
TGGf
Gf
GfFOV
xxFOV
xx
xx
s
x
s
x
sx
x
D
11
1)2
(2
minmax
Dependant on the phase encoding gradient Gy. The amount of phase change is given by
PEyTGv DDD
Field of view in y
vTGyyFOV
PEyy D
11
minmax
While FOV is limited by the sampling interval in the UV plane (Fourier plane) the resolution is limited by the total extent of the UV plane being sampled. If we ignore high spatial frequency content we will have lower resolution (blur our image). Since MRI scans cover only a finite area of the Fourier space we can expect a finite resolution.
Fourier space coverage in MRI
PEyyyerage
xxxerage
TGNvNV
TGNuNU
DD
D
cov
cov
Fourier space coverage in MRI
PEyyyerage
xxxerage
TGNvNV
TGNuNU
DD
D
cov
cov
Outside of this range we assume the contributions to be zero. This is equivalent to passing the actual image through a low-pass filter in the uv plane whose transfer function is given by
)()(),(covcov erageerage Vvrect
UurectvuH
In the spatial domain this is then given by the point spread function (PSF)
)(sin)(sin),( covcovcovcov yVcxUcVUyxh erageerageerageerage
In the spatial domain this is then given by the PSF
)(sin)(sin),( covcovcovcov yVcxUcVUyxh erageerageerageerage
The full width half max (FWHM) resolution is given by the width of the sinc function’s main lobe
PEyyyeragey
xxxeragex
TGNvNVFWHM
TGNuNUFWHM
D
D
D
111
111
cov
cov
Increasing the U,V (coverage area in Fourier space) reduces blurring.
erage
y
erage
x
VvNNFOV
y
UuMMFOVx
cov
cov
11
11
D
D
D
D
Dv
Du
Vcoverage
Ucoverage
Dx
Dy
FOVx
FOVy
Fourier Plane Spatial Domain
Field of View and Resolution in MRI
eragey
eragex
VFWHM
UFWHM
cov
cov
1
1
vFOV
uFOV
y
x
D
D
1
1