1. ELEG 479 Lecture #12 Magnetic Resonance (MR) Imaging
Mark Mirotznik, Ph.D.
Associate Professor
The University of Delaware

2. Physics of Magnetic Resonance Summary
Protons and electrons have a property called spin that results in them looking like tiny magnets. S N
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.
Random
Orientation =
No Net
Magnetization

3. Physics of Magnetic Resonance Summary
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.

4. Physics of Magnetic Resonance Summary from Last Lecture
The net magnetization vector is the sum of all of these little magnetic moments added together. This is what we measure. x y z x y z x y z z z x y x y
Net Magnetization Vector

5. Physics of Magnetic Resonance Summary 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.
= 0
This is bad news since
is where are signal comes from!
Somehow we need to get these guys spinning together!

6. Physics of Magnetic Resonance Summary from Last Lecture
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. B1
RF Excitation
time B1

7. Physics of Magnetic Resonance Summary from Last Lecturex y z B1 a
a B1Dt
Time of Application
of RF Pulse
Tip Angle
Amplitude of RF
Pulse

8. Physics of Magnetic Resonance Summary from Last Lecturex y z a
= envelope of the RF signal
In general

9. Physics of Magnetic Resonance Summary 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.
recall
and

10. Relaxation Processes

11. Physics of Magnetic Resonance Relaxation
After the RF field is removed over time the spin system will return back to it’s equilibrium state due to several relaxation processes.

12. Physics of Magnetic Resonance Relaxation
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:
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.
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.
T2* relaxation: Due to inhomogenities in the external Bo field Mxy will decay much faster than T2. This is a reversible process.

13. T1 Relaxation

14. T2 Relaxation (FID)

15. T2 Decay RF RF

16. T2* Relaxation

17. T2* Decay: Dephasing due to field inhomogeneityz' y'
Mxy
= 0 x'
T2* relaxation is dephasing of transverse magnetization too but it turns out to be reversible

18. Animation of T2* Dephasing

19. Spin- Echo

21. Spin Echo

22. Summary of Relaxation Processes

23. 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.

25. Tissue Contrast

29. TR TE TE 0.5TE 0.5TE 0.5TE 0.5TE 180 degree RF pulses 90 degree
White matter
T1=813 ms
T2=101 ms

30. TR TE TE CASE I: TR>>T1 , TE<T2 What do we measure?
180 degree
RF pulses
CASE I: TR>>T1 , TE<T2
What do we measure?
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

31. 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? TR
180 degree
RF pulses
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

32. 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! TR
180 degree
RF pulses
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

33. CASE II: TR>>T1 , TE~T2
180 degree
RF pulses
CASE II: TR>>T1 , TE~T2
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

34. 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?
180 degree
RF pulses
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

35. 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
180 degree
RF pulses
90 degree
RF pulses TE TE
White matter
T1=813 ms
T2=101 ms

36. TR TE CASE III: TR~T1 , TE<T2 What do we measure? 180 degree
RF pulses
90 degree
RF pulses TE

37. TR 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
180 degree
RF pulses
90 degree
RF pulses TE

38. TR TE T1 Weighted imaging! 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! TR
180 degree
RF pulses
90 degree
RF pulses TE

39. Tissue Contrast Summary TE TR PD weighted T1weighted T2 weighted
TE<T2 (short TE)
TR>>T1 (long TR)
T1weighted
TR~T1 (short TR)
T2 weighted
TE~T2 (long TE)

40. Bloch Equations

41. Full Bloch equation including relaxation
precession,
RF excitation
transverse
magnetization
longitudinal
magnetization
includes Bo and B1

42. Example: Solve for the transverse components of M after a 90 degree pulse.

43. 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

44. 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
Initial conditions for 90 degree pulse:

45. 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
Solutions

46. 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
Solutions

47. 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

48. 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
Initial conditions for 90 degree pulse:

49. Example: Solve for the transverse components of M after an arbitrary flip angle (a)
Solutions

50. Solve full Bloch equation with only B=Bo
Solution for transverse components Mx and My
Where a is the flip angle after RF excitation

51. Signal Detection

52. Signal Detection via RF coil

53. Signal Detection via RF coil
Transverse magnetization at t=0.
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!!

54. Signal Detection via RF coil
After demodulation:

55. Creating an Image

56. Creating an Image
To create an image using NMR we need to figure out a way to encode the proton spins spatially in three dimensions.
But how?

57. 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.

58. Gradient Coils
X gradient Y gradient Z gradient x y z
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.

59. Gradient Coils
Purpose: Spatially alter magnitude of B0 (not direction)
Sounds generated during imaging due to mechanical stress within gradient coils.

61. Vector Notation

62. Larmor frequency within a gradient field

64. Slice Selection

65. Slice Selection GradientBG
Coil 1
Coil 2

66. Helmholtz Coils

68. Z-Gradient Fields
By adding a z-gradient field we cause a variation in the
resonant frequency from head to toe.

69. Example
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?

70. Example 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?

71. Example 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?

72. A Field Gradient Makes the Larmor Frequency Depend upon PositionB0
63,480,000 Hz
64,260,000 Hz Z
Gradient in Z
B(Z) = B + G * Z o Z =
n(z) g
B(z)

73. Slice Selection
(-)
62 MHz
63 MHz
64 MHz G
65 MHz
66 MHz
(+)

74. How do we determine the slice width and center?
Slice Selection
How do we determine the slice width and center? z
After z selection
gradient and excitation
z-gradient Bo
(slice center)
(slice width) x

75. Determining slice thickness
Resonance frequency range as the result of slice-selective gradient:

76. 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

77. Example
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?

78. 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?

79. Selecting different slices

80. 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
Option (b) is usually used as it is not easy to change the
isocenter of a given gradient coil.

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

83. RF Excitation (RF Pulse)Fo FT A Dn t n1 n2

84. RF Excitation: Flip angleFo FT A Dn t n1 n2
Envelope of the pulse

85. RF Excitation: Flip angleFo FT A Dn t n1 n2

86. RF Excitation: Flip angleFo FT A Dn t n1 n2

88. RF Excitation: Flip angle (truncated sinc)

90. A potential problem
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?

91. Why yes it will! It is called gradient dephasing Good question!
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 slow
Spinning fast
Why yes it will! It is called gradient dephasing
Good question!

93. Any ideas on how to get around this?
Why yes it will! It is called gradient dephasing. It will quickly kill our signal much faster than T2 or even T2*
Spinning slow
Spinning fast
Any ideas on how to get around this?

96. Localization in xy plane

98. Lets Start with a Simple Flat Person
(only xz plane) z Bo x

99. Lets Start with a Simple Flat Person
(only xz plane) z
After z selection
gradient and excitation
z-gradient Bo x

100. Lets Start with a Simple Flat Person
Frequency Encoding Mathematical Analysis
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)
After demodulation (envelope detection)

101. Lets Start with a Simple Flat Person
Frequency Encoding Mathematical Analysis
After demodulation (envelope detection)
Let
This is what we want to image (called the effective proton density)

102. Lets Start with a Simple Flat Person
(frequency encoding using x-gradient) z
After z selection
gradient and excitation Bo x

103. Lets Start with a Simple Flat Person
Frequency Encoding Mathematical Analysis
Now lets apply a gradient in the x direction (Gx)
After demodulation (envelope detection)
What does this look like?

104. Lets Start with a Simple Flat Person
Frequency Encoding Mathematical Analysis
After demodulation (envelope detection)
Let
The received signal is related to the Fourier transform (THIS IS THE KEY!)

105. Lets Start with a Simple Flat Person
Frequency Encoding Mathematical Analysis
After demodulation (envelope detection)
Let
We can now find our image as a function of x by taking an inverse Fourier Transform

106. A Simple Example of Spatial Encoding with Frequency Encoding
Constant
Magnetic
Field
Varying
Magnetic
Field
w/o encoding
w/ encoding

107. A Simple Example of Spatial Encoding with Frequency Encoding
Decomposition

108. Decays faster than T2*

109. Extend this to a full 3D person

110. Extend this to a full 3D person
After slice selection we need to image in xy plane y x

111. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
Now lets apply a gradient in the x direction (Gx)
After demodulation (envelope detection)
Effective proton density

112. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
After demodulation (envelope detection)
Let

113. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
Let
Corresponds to a single line or trajectory in the uv plane

114. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
Now lets apply gradients in both the x direction (Gx) and y direction (Gy)
After demodulation (envelope detection)
Let
Effective proton density

115. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
Now lets apply gradients in both the x direction (Gx) and y direction (Gy)
Let

116. Spatial Encoding in xy plane Frequency Encoding Mathematical Analysis
Let
Polar Scanning

119. Gradient Echo
(A brief detour)

120. Ts/2

121. Spatial Encoding in xy plane Gradient Echo Mathematical Analysis
Spins will dephase very quickly (quicker than T2*) due to the gradient fields.
After negative x-gradient

122. Spatial Encoding in xy plane Gradient Echo Mathematical Analysis
After negative x-gradient
Now impose positive x-gradient for Ts

123. Spatial Encoding in xy plane Gradient Echo Mathematical Analysis
Now impose positive x-gradient for Ts

124. Phase Encoding

128. Pulse Repetition

140. Image Contrast

147. Image Quality

148. Field of View and Resolution in MRI
Fourier Plane
Spatial Domain Du Dx Dv
Vcoverage
FOVy Dy
Ucoverage
FOVx

149. 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)

150. Sampling in MRI Slice selection direction: sampling in z-direction
Slice thickness (Dz) controlled by RF excitation bandwidth (Dn)
To avoid aliasing
Where fmax,z is the highest spatial frequency in along the z-axis

151. 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

152. 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

153. 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

154. 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

155. Dv

157. X-gradient relates dimension x with Larmor freq n by
To avoid aliasing only frequency given below are measured

158. X-gradient relates dimension x with Larmor freq n by
To avoid aliasing only frequency given below are measured
Field of view in the x-direction (FOVx) is thus given by

159. Dependant on the phase encoding gradient Gy
Dependant on the phase encoding gradient Gy. The amount of phase change is given by
Field of view in y

160. 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

161. Fourier space coverage in MRI
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
In the spatial domain this is then given by the point spread function (PSF)

162. In the spatial domain this is then given by the PSF
The full width half max (FWHM) resolution is given by the width of the sinc function’s main lobe
Increasing the U,V (coverage area in Fourier space) reduces blurring.

164. Field of View and Resolution in MRI
Spatial Domain
Fourier Plane Du Dx Dv
Vcoverage
FOVy Dy
Ucoverage
FOVx