1. Principles of MRI: Image Formation
Allen W. Song
Brain Imaging and Analysis Center
Duke University

2. What is image formation?
To define the spatial location of the sources
that contribute to the detected signal.

3. 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?

4. Frequency and Phase Are Our Friends in MR Imagingw
q = wt
The spatial information of the proton pools contributing
MR signal is determined by the spatial frequency and
phase of their magnetization.

5. Gradient Coils z z z y y y x x x
X gradient Y gradient 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.

6. A Simple Example of Spatial Encoding
0.8
A Simple Example of Spatial Encoding
Constant
Magnetic
Field
Varying
Magnetic
Field
w/o encoding
w/ encoding

7. Spatial Decoding of the MR Signal
Frequency
Decomposition

8. 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

9.  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

10. Electromagnetic Excitation Pulse (RF Pulse)Fo FT t Fo
Fo+1/ t
Time
Frequency Fo Fo FT
DF= 1/ t t

11. 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
[63 MHz at 1.5 T] f
60 KHz
Gx = 1 Gauss/cm = 10 mTesla/m
= strength of gradient field
x-axis
Left = –7 cm
Right = +7 cm

12. Exciting and Receiving Mxy in a Thin Slice of Tissue
Receive:
RF coil
RF preamplifier
Filters
Analog-to-Digital Converter
Computer memory

13. Slice Selection

14. Slice Selection – along z

15. 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)

16. Changing slice thickness
There are two ways to do this:
Change the slope of the slice selection gradient
Change the bandwidth of the RF excitation pulse
Both are used in practice, with (a) being more popular

17. Changing slice thickness
new slice
thickness

18. Selecting different slices
In theory, there are two ways to select different slices:
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.

19. Selecting different slices
new slice
location

20.  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

21. Spatial Encoding of the MR Signal
Constant
Magnetic
Field
Varying
Magnetic
Field
w/o encoding
w/ encoding

22. It’d be easy if we image with only 2 voxels …
But often times we have imaging matrix at 256 or higher.

23. More Complex Spatial Encoding
x gradient

24. More Complex Spatial Encoding
y gradient

25. After Frequency Encoding
A 9×9 case
Physical Space
MR data space
Before Encoding
After Frequency Encoding
(x gradient)
So each data point contains information from all the voxels

26. A typical diagram for MRI frequency encoding: Gradient-echo imaging
Excitation
Slice
Selection TE
Frequency
Encoding
readout
………
Time point #1
Time point #9
Readout
Data points collected during this
period corrspond to one-line in k-space

27. ……… Phase Evolution of MR Data TE Gradient Phases of spins
digitizer on
Phases of spins
Gradient TE
………
Time point #1
Time point #9

28. A typical diagram for MRI frequency encoding:
Spin-echo imaging
Excitation
Slice
Selection TE
Frequency
Encoding
readout
………
Readout

29. Phase History
180o TE
Phase
Gradient
………
digitizer on

30. 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

31. Calculation of the Field of View (FOV) along frequency encoding direction
* Gf * FOVf = BW = 1/Dt
Which means FOVf = 1/ (g Gf Dt)
where BW is the bandwidth for the
receiver digitizer.

32.  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

33. After Frequency Encoding
A 9×9 case
Physical Space
MR data space
Before Encoding
After Frequency Encoding
x gradient
After Phase Encoding
y gradient
So each point contains information from all the voxels

34. A typical diagram for MRI phase encoding: Gradient-echo imaging
readout
Excitation
Slice
Selection
Frequency
Encoding
Phase
Readout
………

35. A typical diagram for MRI phase encoding:
Spin-echo imaging
readout
Excitation
Slice
Selection
Frequency
Encoding
Phase
Readout
………

36. Calculation of the Field of View (FOV) along phase encoding direction
* 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.

37. Part II.2 Introduction to k-space (MR data space)
Image
k-space

38. …….. …….. …….. …….. Phase Encode Time Time Time Step 1 point #1

39. .
+Gx
-Gx
+Gy
-Gy
Physical Space
K-Space
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.

40. Acquired MR Signal Kx = g/2p 0t Gx(t) dt Ky = g/2p 0t Gy(t) dt
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
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

41. Two Spaces k-space Image space ky y IFT kx x FT Final Image
Acquired Data
Image space x y
Final Image
IFT FT

42. Image K
High
Signal

43. Full k-space Lower k-space Higher k-space Full Image
Intensity-Heavy Image
Detail-Heavy Image

44. The k-space Trajectory
Equations that govern k-space trajectory:
Kx = g/2p 0t Gx(t) dt
Ky = g/2p 0t Gy(t) dt
time t
Gx (amplitude)
Kx (area)

45. 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

46. The k-space Trajectory

47. 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

48. The k-space Trajectory

49. A typical diagram for MRI phase encoding:
A k-space perspective
90o
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation

50. The k-space Trajectory

51. A typical diagram for MRI phase encoding:
A k-space perspective
180o
90o
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
readout
Readout
Exercise drawing its k-space representation

52. The k-space Trajectory

53. Sampling in k-space
Dk = gGDt
kmax
Dk = 1 / FOV

55. . . . . . . . . . . . . . . . A B FOV: 10 cm Pixel Size: 2 cm

56. A B
FOV: 10 cm
Pixel Size: 1 cm
FOV:
Pixel Size:
5 cm
1 cm

57. A B
FOV:
Pixel Size:
20 cm
2 cm
FOV: 10 cm
Pixel Size: 1 cm

58. K-space can also help explain imaging distortions:
Original image
K-space trajectory
Distorted Image