1. Principles of MRI
Physics and Engineering
Allen W. Song
Brain Imaging and Analysis Center
Duke University
2. What if the RF field is not synchronized?
Using the swingset example: now the driving force is no longer
synchronized with the swing frequency, thus the efficiency of
driving the swing is less.
In a real spin system, there is a term called “effective B1
field”,
given by ∆ω/γ B1eff
B1eff = B1 + ∆ω /γ
where B1
∆ω = ω o – ω e
5. What is image formation?
To define the spatial location of the proton
pools that contribute to the MR signal after
spin excitation.
6. A 3-D gradient field (dB/dx, dB/dy, dB/dz) would
allow a unique correspondence between the spatial
location and the magnetic field. Using this information,
we will be able to generate maps that contain spatial
information – images.
7. Gradient Coils
z z z
y y y
x x x
X gradient Y gradient Z gradient
Gradient coils generate varying magnetic field so that
spins at different location precess at frequencies unique
to their location, allowing us to reconstruct 2D or 3D
images.
12. 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
13. RF Field: Excitation Pulse
Fo
FT
0 t Fo Fo+1/ t
Time Frequency
Fo Fo
FT
∆F= 1/ t
t
14. 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.
Center
frequency f
60 KHz [63 MHz at 1.5 T]
Gx = 1 Gauss/cm = 10 mTesla/m
= strength of gradient field
x-axis
Left = –7 cm Right = +7 cm
15. 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
16. Exciting and Receiving Mxy in a Thin Slice of Tissue
Receive: RF coil
RF preamplifier
Filters
Analog-to-Digital Converter
Computer memory
19. Determining slice thickness
Resonance frequency range as the result
of slice-selective gradient:
∆F = γH * Gsl * dsl
The bandwidth of the RF excitation pulse:
∆ω/2π
Thus the slice thickness can be derived as
dsl = ∆ω / (γH * Gsl * 2π)
20. 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
22. 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 = γH (Bo + Gsl * Lsl )
Option (b) is usually used as it is not easy to change the
isocenter of a given gradient coil.
24. Readout Localization (frequency encoding)
♦ After RF pulse (B ) ends, acquisition (readout) of
1
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
25. Readout of the MR Signal
Constant Varying
Magnetic Magnetic
Field Field
w/o encoding w/ encoding
27. A typical diagram for MRI frequency encoding:
Gradient-echo imaging
Excitation
Slice
Selectio TE
n
Frequency
Encoding
readout
Readout
Data points collected during this
period corrspond to one-line in k-space
31. Image Resolution (in Plane)
♦ Spatial resolution depends on how well we can
separate frequencies in the data V(t)
• Resolution is proportional to ∆f = frequency accuracy
• Stronger gradients ⇒ nearby positions are better separated
in frequencies ⇒ resolution can be higher for fixed ∆f
• Longer readout times ⇒ can separate nearby frequencies
better in V(t) because phases of cos(f⋅t) and cos([f+∆f]⋅t)
will be more different
32. Calculation of the Field of View (FOV)
along frequency encoding direction
γ* Gf * FOVf = BW = 1/∆t
*
Which means FOVf = 1/ (γ Gf ∆t)
where BW is the bandwidth for the
receiver digitizer.
33. 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 φ 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
34. A typical diagram for MRI phase encoding:
Gradient-echo imaging
Excitation
Slice
Selectio
n
Frequency
Encoding
Phase
Encoding
readout
Readout
35. A typical diagram for MRI phase encoding:
Spin-echo imaging
Excitation
Slice
Selectio
n
Frequency
Encoding
Phase
Encoding
readout
Readout
36. Calculation of the Field of View (FOV)
along phase encoding direction
γ* Gp * FOVp = Np / Tp
*
Which means FOVp = 1/ (γ Gp Tp/Np)
= 1/ (γ Gp ∆t)
where Tp is the duration and Np the number
of the phase encoding gradients, Gp is the
maximum amplitude of the phase encoding
gradient.
40. Acquired MR Signal
Mathematical Representation:
− i 2π ( k x x + k y y )
S (k x , k y ) = ∫ ∫ I ( x, y )e dxdy
Kx = γ/2π ∫0t Gx(t) dt
Ky = γ/2π ∫0t Gy(t) dt
41. Image Space
K-Space
.. . . . . .. .
.. . . . . .. .
+Gy
.. . . . . .. .
.. . . . . .. .
.. . . . . .. .
.. . . . . .. .
0
.. . . . . .. .
.. . . . . .. .
-Gy
.. . . . . .. .
-Gx 0 +Gx
Figure 4.7. 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. We have indicated, for four sample k-space
points, which gradient vectors contribute at different image space locations to the k-space data.
42. Acquired MR Signal
By physically adding all the signals from each voxel up under
the gradients we use.
− i 2π ( k x x + k y y )
S (k x , k y ) = ∫ ∫ I ( x, y )e 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.
43. Two Spaces
k-space Image space
ky y
IFT
kx x
FT
Acquired Data Final Image
45. The k-space Trajectory
Equations that govern k-space trajectory:
Kx = γ/2π ∫0t Gx(t) dt
Ky = γ/2π ∫0t Gy(t) dt
Gx (amplitude)
Kx (area)
time
0 t
46. A typical diagram for MRI frequency encoding:
A k-space perspective
90o
Excitation
Slice
Selectio
n
Frequency
Encoding
readout
Readout
Exercise drawing its k-space representation
48. A typical diagram for MRI frequency encoding:
A k-space perspective
90o 180o
Excitation
Slice
Selectio
n
Frequency
Encoding
readout
Readout
Exercise drawing its k-space representation
50. A typical diagram for MRI phase encoding:
A k-space perspective
90o
Excitation
Slice
Selectio
n
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation
52. A typical diagram for MRI phase encoding:
A k-space perspective
90 o 180o
Excitation
Slice
Selectio
n
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation
57. A
..........
.......... B
.......... ..........
.........
..........
..........
.
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
..........
.......... FOV:
Pixel Size:
FOV: 10 cm
Pixel Size: 1 cm
58. ..........
..........
.......... . . . . .
.......... .....
..... . . . . .
..........
.......... .....
.......... ..... . . . . .
.......... ..... . . . . .
..........
.......... . . . . .
Figure 4.16. Effects of sampling in k-space upon the resulting images. Field of view and resolution have an inverse
relation between image space and k-space. Shown in (A) is a schematic representation of densely sampled k-space with a
wide field of view, resulting in the high-resolution image below. If only the center of k-space is sampled (B), albeit with
the same sampling density, then the resulting image below has the same field of view, but does not have as high of spatial
resolution. Conversely, if k-space is sampled across a wide field of view but with limited sampling rate (C), the resulting
image will have a small field of view but high resolution.
59. Original image K-space trajectory Displaced image
Spatial Displacement
Original image Distorted image
Figure 4.17. Spatial and intensity distortions due to magnetic field inhomogeneities during readout. If there is a
systematic change in the spin frequency over time, the resulting image may be spatially displaced (B). Another possible
type of distortion results from local field inhomogeneities, which in turn cause the resonant frequency to vary slightly
across spatial locations, affecting the intensity of the image over space.
60. Original image K-space trajectory Distorted Image
Figure 4.18. Image distortions caused by gradient problems during readout. Each row shows the ideal image, the problem with acquisition in k-
space, and the resulting distorted image. Distortions along the x-gradient will affect the length of the trajectory in k-space, resulting in an image that
appears stretched (A). Distortions along the y-gradient will affect the path taken through k-space over time, resulting in a skewed image (B). Distortions
along the z-gradient will affect the match of excitation pulse and slice selection gradient, influencing the signal intensity (C).
