1. Principles of MRI Physics and Engineering
Allen W. Song
Brain Imaging and Analysis Center
Duke University

2. Part II.1
Image Formation

3. What is image formation?
Define the spatial location of the proton
pools that contribute to the MR signal.

4. 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

5. RF Field: Excitation PulseFo FT t Fo
Fo+1/ t
Time
Frequency Fo Fo FT
DF= 1/ t t

6. Gradient Fields: Spatially Nonuniform B:
Extra static magnetic fields (in addition to B0) that vary their intensity in a linear way across the subject
Precession frequency of M varies across subject
This is called frequency encoding — using a deliberately applied nonuniform field to make the precession frequency depend on location
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

7. Spin phase with x gradient on

8. Spin phase with y gradient on

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

11. Slice Selection

12. Slice Selection – along z

13. 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)

14. 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

15. Changing slice thickness
new slice
thickness

16. 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.

17. Selecting different slices
new slice
location

18.  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

19. Readout of the MR Signal
Constant
Magnetic
Field
Varying
Magnetic
Field
w/o encoding
w/ encoding

20. Readout of the MR Signal
Fourier Transform

21. A typical diagram for MRI frequency encoding: Gradient-echo imaging
Excitation
Slice
Selection TE
Frequency
Encoding
digitizer on
Readout

22. Phase History TE
Gradient
Phase
digitizer on

23. A typical diagram for MRI frequency encoding:
Spin-echo imaging
Excitation
Slice
Selection TE
Frequency
Encoding
digitizer on
Readout

24. Phase History
180o TE
Gradient
Phase

25. 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

26. Calculation of the Field of View (FOV) along frequency encoding direction
* Gf * FOVf = BW,
where BW is the bandwidth for the
receiver digitizer.

27.  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

28. A typical diagram for MRI phase encoding: Gradient-echo imaging
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
digitizer on
Readout

29. A typical diagram for MRI phase encoding:
Spin-echo imaging
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
digitizer on
Readout

30. Calculation of the Field of View (FOV) along phase encoding direction
* Gp * FOVp = Np / Tp
where Tp is the duration and Np the number
of the phase encoding gradients, Gp is the
maximum amplitude of the phase encoding
gradient.

31. Multi-slice acquisition
Total acquisition time =
Number of views * Number of excitations * TR
Is this the best we can do?
Interleaved excitation method

32. Part II.2 Introduction to k-space (a space of the spatial frequency)

33. Acquired MR Signal Mathematical Representation:
This equation is obtained by physically adding all the signals
from each voxel up under the gradients we use.
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.

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

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

36. A typical diagram for MRI frequency encoding:
A k-space perspective
Excitation
Slice
Selection
Frequency
Encoding
digitizer on
Readout
Exercise drawing its k-space representation

37. The k-space Trajectory

38. A typical diagram for MRI frequency encoding:
A k-space perspective
Excitation
Slice
Selection
Frequency
Encoding
digitizer on
Readout
Exercise drawing its k-space representation

39. The k-space Trajectory

40. A typical diagram for MRI phase encoding:
A k-space perspective
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
digitizer on
Readout
Exercise drawing its k-space representation

41. The k-space Trajectory

42. A typical diagram for MRI phase encoding:
A k-space perspective
Excitation
Slice
Selection
Frequency
Encoding
Phase
Encoding
digitizer on
Readout
Exercise drawing its k-space representation

43. The k-space Trajectory

44. . . . . . . . . . . Sampling in k-space Dk kmax Dk = 1 / FOV
2kmax = 1 / Dx
Link back to slides 26 and 30

45. . . . . . . . . . . . . . . . A B FOV: Pixel Size: FOV: 10 cm
Pixel Size: 1 cm

46. . . . . . . . . . . . . . . . A B FOV: 10 cm FOV: Pixel Size: 1 cm
FOV: 10 cm
Pixel Size: 1 cm
FOV:
Pixel Size:

47. A B
FOV:
Pixel Size:
FOV: 10 cm
Pixel Size: 1 cm

48. Examples of images and their k-space map