Transcript Principles of MRI: Image Formation

Principles of MRI:
Image Formation
Allen W. Song
Brain Imaging and Analysis Center
Duke University
What is image formation?
To define the spatial location of the sources
that contribute to the detected signal.
But MRI does not use projection, reflection, or refraction
mechanisms commonly used in optical imaging methods
to form image. So how are the MR images formed?
Frequency and Phase Are Our Friends in MR Imaging
w
q = wt
q
The spatial information of the proton pools contributing
MR signal is determined by the spatial frequency and
phase of their magnetization.
Gradient Coils
z
z
z
y
y
x
x
X gradient
y
Y gradient
x
Z gradient
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.
0.8
A Simple Example of Spatial Encoding
Constant
Magnetic
Field
w/o encoding
Varying
Magnetic
Field
w/ encoding
Spatial Decoding of the MR Signal
Frequency
Decomposition
Steps in 3D Localization
 Can only detect total RF signal from inside the “RF
coil” (the detecting antenna)
 Excite and receive Mxy in a thin (2D) slice of the
subject
 The
RF signal we detect must come from this slice
 Reduce dimension from 3D down to 2D
 Deliberately make magnetic field strength B depend
on location within slice

Frequency of RF signal will depend on where it comes from
 Breaking total signal into frequency components will provide
more localization information
 Make RF signal phase depend on location within slice
 Exciting and Receiving Mxy in a Thin Slice of Tissue
Excite:
Source of RF frequency on resonance
Addition of small frequency variation
Amplitude modulation with “sinc” function
RF power amplifier
RF coil
Electromagnetic Excitation Pulse (RF Pulse)
Fo
FT
0
Fo Fo+1/ t
t
Time
Frequency
Fo
Fo
FT
t
DF= 1/ t
Gradient Fields: Spatially Nonuniform B:
During readout (image acquisition) period, turning on gradient field is
called frequency encoding --- using a deliberately applied nonuniform
field to make the precession frequency depend on location
 Before readout (image acquisition) period, turning on gradient field is
called phase encoding --- during the readout (image acquisition) period,
the effect of gradient field is no longer time-varying, rather it is a fixed
phase accumulation determined by the amplitude and duration of the
phase encoding gradient.
60 KHz
Center
frequency
[63 MHz at 1.5 T]
f
Gx = 1 Gauss/cm = 10 mTesla/m
= strength of gradient field
Left = –7 cm
x-axis
Right = +7 cm
Exciting and Receiving Mxy in a Thin Slice of Tissue
Receive:
RF coil
RF preamplifier
Filters
Analog-to-Digital Converter
Computer memory
Slice Selection
Slice Selection – along z
z
Determining slice thickness
Resonance frequency range as the result
of slice-selective gradient:
DF = gH * Gsl * dsl
The bandwidth of the RF excitation pulse:
Dw/2p
Thus the slice thickness can be derived as
dsl = Dw / (gH * Gsl * 2p)
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
Changing slice thickness
new slice
thickness
Selecting different slices
In 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
F = gH (Bo + Gsl * Lsl )
Option (b) is usually used as it is not easy to change the
isocenter of a given gradient coil.
Selecting different slices
new slice
location
 Readout Localization (frequency encoding)
 After RF pulse (B1) ends, acquisition (readout) of
NMR RF signal begins
 During readout, gradient field perpendicular to slice
selection gradient is turned on
 Signal is sampled about once every few microseconds,
digitized, and stored in a computer
• Readout window ranges from 5–100 milliseconds (can’t be longer
than about 2T2*, since signal dies away after that)
 Computer breaks measured signal V(t) into frequency
components v(f ) — using the Fourier transform
 Since frequency f varies across subject in a known way, we
can assign each component v(f ) to the place it comes from
Spatial Encoding of the MR Signal
Constant
Magnetic
Field
w/o encoding
Varying
Magnetic
Field
w/ encoding
It’d be easy if we image with only 2 voxels …
But often times we have imaging matrix at 256 or higher.
More Complex Spatial Encoding
x gradient
More Complex Spatial Encoding
y gradient
A 9×9 case
Physical Space
MR data space
After Frequency Encoding
(x gradient)
So each data point contains information from all the voxels
Before Encoding
A typical diagram for MRI frequency encoding:
Gradient-echo imaging
Excitation
Slice
Selection
TE
Frequency
Encoding
Readout
Time point #1
readout
………
Time point #9
Data points collected during this
period corrspond to one-line in k-space
Phase Evolution of MR Data
TE
Gradient
Phases of spins
Time point #1
………
digitizer on
Time point #9
A typical diagram for MRI frequency encoding:
Spin-echo imaging
Excitation
Slice
Selection
TE
Frequency
Encoding
readout
………
Readout
Phase History
180o
TE
Gradient
Phase
………
digitizer on
Image Resolution (in Plane)
 Spatial resolution depends on how well we can
separate frequencies in the data V(t)
 Resolution is proportional to Df = frequency accuracy
 Stronger gradients  nearby positions are better separated
in frequencies  resolution can be higher for fixed Df
 Longer readout times  can separate nearby frequencies
better in V(t) because phases of cos(ft) and cos([f+Df]t)
will be more different
Calculation of the Field of View (FOV)
along frequency encoding direction
g* Gf * FOVf = BW = 1/Dt
Which means FOVf = 1/ (g Gf Dt)
where BW is the bandwidth for the
receiver digitizer.
 The Second Dimension: Phase Encoding
 Slice excitation provides one localization dimension
 Frequency encoding provides second dimension
 The third dimension is provided by phase encoding:
 We make the phase of Mxy (its angle in the xy-plane) signal
depend on location in the third direction
 This is done by applying a gradient field in the third
direction ( to both slice select and frequency encode)
 Fourier transform measures phase  of each v(f )
component of V(t), as well as the frequency f
 By collecting data with many different amounts of phase
encoding strength, can break each v(f ) into phase
components, and so assign them to spatial locations in 3D
A 9×9 case
Physical Space
MR data space
Before Encoding After Frequency Encoding After Phase Encoding
y gradient
x gradient
So each point contains information from all the voxels
A typical diagram for MRI phase encoding:
Gradient-echo imaging
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
………
Readout
A typical diagram for MRI phase encoding:
Spin-echo imaging
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
………
Readout
Calculation of the Field of View (FOV)
along phase encoding direction
g* Gp * FOVp = Np / Tp
Which means FOVp = 1/ (g Gp Tp/Np)
= 1/ (g Gp Dt)
where Tp is the duration and Np the number
of the phase encoding gradients, Gp is the
maximum amplitude of the phase encoding
gradient.
Part II.2 Introduction to k-space
(MR data space)
Image
k-space
Phase
Encode
Step 1
Time
point #1
Time
point #2
Time
point #3
Phase
Encode
Step 2
Time
point #1
Time
point #2
Time
point #3
Phase
Encode
Step 3
Time
point #1
Time
point #2
Time
point #3
……..
……..
……..
……..
Physical Space
K-Space
+Gy
0
-Gy
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-Gx
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Contributions of different image locations to the raw k-space data. Each
data point in k-space (shown in yellow) consists of the summation of MR
signal from all voxels in image space under corresponding gradient fields.
Acquired MR Signal
For a given data point in k-space, say (kx, ky), its signal S(kx,
ky) is the sum of all the little signal from each voxel I(x,y) in the
physical space, under the gradient field at that particular moment
S (k x , k y )  

I ( x, y )e
 i 2p ( k x x  k y y )
dxdy
From this equation, it can be seen that the acquired MR signal,
which is also in a 2-D space (with kx, ky coordinates), is the
Fourier Transform of the imaged object.
Kx = g/2p 0t Gx(t) dt
Ky = g/2p 0t Gy(t) dt
Two Spaces
Image space
y
k-space
ky
IFT
kx
Acquired Data
x
FT
Final Image
Image
K
High
Signal
Full Image
Full k-space
Intensity-Heavy Image
Lower k-space
Detail-Heavy Image
Higher k-space
The k-space Trajectory
Equations that govern k-space trajectory:
Kx = g/2p 0t Gx(t) dt
Ky = g/2p 0t Gy(t) dt
Gx (amplitude)
Kx (area)
0
t
time
A typical diagram for MRI frequency encoding:
A k-space perspective
90o
Excitation
Slice
Selection
Frequency
Encoding
readout
Readout
Exercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI frequency encoding:
A k-space perspective
90o
180o
Excitation
Slice
Selection
Frequency
Encoding
readout
Readout
Exercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI phase encoding:
A k-space perspective
Excitation
90o
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation
The k-space Trajectory
A typical diagram for MRI phase encoding:
A k-space perspective
90o
180o
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation
The k-space Trajectory
Sampling in k-space
kmax
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..........
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..........
Dk = gGDt
Dk = 1 / FOV
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..........
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A
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..........
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FOV: 10 cm
Pixel Size: 1 cm
B
.....
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10 cm
FOV:
Pixel Size: 2 cm
B
A
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FOV: 10 cm
Pixel Size: 1 cm
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5 cm
FOV:
Pixel Size: 1 cm
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A
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..........
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FOV: 10 cm
Pixel Size: 1 cm
B
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..........
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..........
20 cm
FOV:
Pixel Size: 2 cm
K-space can also help explain imaging distortions:
Original image
K-space trajectory
Distorted Image