Magnetic Resonance Imaging
Ho Kyung Kim
Pusan National University
Introduction to Medical Engineering (Medical Imaging)
Suetens 4
• The Nobel Prize in Medicine or Physiology in 2003
– Paul C. Lauterbur – the first NMR image in 1973
– Peter Mansfield – the math theory for fast scanning & image reconstruction in 1974
• MRI
– Measures a magnetic property of tissue
– Based on the nuclear magnetic resonance (NMR)
– NMR studies the behavior of atomic nuclei with spin angular momentum (JJJJ) and associated
magnetic moment (µµµµ) in an external magnetic field (BBBB)
– NMR (i.e., the property of spin angular momentum) can be described by the quantum
electrodynamics (= the special theory of relativity + the quantum mechanics)
• What happens when human tissue, which contains a huge quantity of particles, is placed in
an external magnetic field?
2
Spin
• Classical mechanics (i.e., the laws of
Newton and Maxwell) can describe an
orbital angular momentum
• Quantum electrodynamics can only
describe a spin angular momentum
(shortly, spin) with an associated magnetic
moment
– electrons, protons, neutrons
– net spin is the vector sum
3
Nucleus Spin �
��(MHz/T)
H��
H��
C���
C���
N��
N��
O���
O��
P����
S����
Ca����
1/2
1
0
1/2
1
1/2
0
5/2
1/2
3/2
7/2
42.57
6.54
10.71
3.08
-4.31
-5.77
17.23
3.27
-2.86
No spin, no NMR sensitivity
• � = ��– � = gyromagnetic ratio (constant)
Magnetic moment
• The interaction between � and � yields a
precession motion and a potential energy
•��
��= � × γ�
• Solution:
– Transverse term: ��� � = ��� 0 !"#$�
– Longitudinal term: �% � = �% 0
– &' = �('
• The motion of � is a precession about the
z-axis with precession freq. &'
• In a rotating reference frame, the effective
� perceived by � is zero
4
) = � × � (torque = distance × force)
d�
d�= )
= (0,0, (')
Space quantization
• Potential energy
– . = −� · � = −�(' cos 4 = −�5(' cos 4
– Minimal if � || �
– Classical mechanics says that 5% ∈ [−5,+5]
• Quantum mechanics says that the outcome of a measurement of a physical variable is a
"multiple of a basic amount (quantum)," so-called quantization
– . = −:�ℏ(' with : = −<,−< + 1,… , < − 1, <
• ℏ = ℎ/2B
• < = spin quantum number
– e.g., proton (nucleus of H�� ) with < = 1/2
• .↑ = −�
��ℏ('
• .↓ = +�
��ℏ('
• Proton can occupy only two energy states!
– Spin-up state: �% � > 0
– Spin-down state: �% � < 0
5
• A proton in the state.↑ can switch to .↓ by
absorbing an energy:
– .↓ − .↑ = ℏ�('
• Resonance condition
– &GH = �(' = &', called the Larmor
(angular) frequency
• Depends on molecular structure
– e.g., If (' = 1 T, the Larmor freq. is
approximately 42.6 MHz for H��
– Radio-frequency (RF) waves suffice the
typical resonance condition
• MRI visualizes hydrogen-containing tissues
– muscles, brain, kidney, CSF, edema, fat,
bone marrow, etc.
6
Dynamic equilibrium
• For IJ spins in a voxel, the net macroscopic magnetization vector in a voxel is given by
– K' = ∑ �"MN"O�
– More the spin-up states, more net polarization in the direction of �
– Larger �, larger K' & signal
– On average, the net transverse magnetization of P��= 0; hence K' = (0,0,P')
• Longitudinal P' cannot be measured
• Transvers P�� can be measured
• The net magnetization precession about the axis of �:
–�K$
��= K' × γ�
7
Disturbing the dynamic equilibrium
• Apply RF wave (EM wave with the Larmor freq.) by sending AC current along the x and y
axes
– Transverse component: (��� � = (� !"#$�
– Longitudinal term: (�% � = 0
–�K
��= K× γ(� + �� � )
• In the rotating reference frame, K precesses about �� (not �) with precession freq. &� =�(�
8
• Flip angle:
– Q = R �(�dS�
'= �(�� = &��
– Any flip angle with an appropriate choice of (� & �
– Halving the up-time of the RF field requires 2× (� or 4× AC power!
• Increase temperature in tissue
– The 90° pulse
• K = (0,P', 0)
• K rotates clockwise in the transverse plane in the stationary reference frame
– The 180° or inverse pulse
• K = (0,0,−P')
• K rotates about –z axis; all the individual spins rotate in phase (phase coherence)
• Relaxation: return to dynamic equilibrium when the RF field is switched off
9
Spin-spin relaxation
• Causes dephasing process (i.e., disappearance of the transverse component of the net
magnetization vector)
– P�� � = P' sin Q !�/VW
– P' sin Q = the value of transverse component immediately after the RF pulse
– X� = the spin-spin relaxation time
• dependent considerably upon the tissue
• X� ≈ 100 ms for fat while X� ≈ 2000 ms for cerebrospinal fluid (CSF)
10
Spin-lattice relaxation
• Causes the longitudinal component of the net magnetization vector to increase from
P' cos Q (the value of longitudinal component immediately after the RF pulse)
– P% � = P' cos Q !�/VZ +P'(1 − !�/VZ)
– X� = the spin-lattice relaxation time
• dependent considerably upon the tissue type & proportional to �
• X� ≈ 200 ms for fat while X� ≈ 3000 ms for cerebrospinal fluid (CSF) at 1.5 T
• for the same tissue, always X� > X�
11
Summary
12
The RF pulse creates a net
transverse magnetization due to
energy absorption and phase
coherence. After the RF pulse, two
distinct relaxation phenomena
ensure that the dynamic (thermal)
equilibrium is reached again.
Inversion recovery (IR)
• For an inversion pulse, the longitudinal
magnetization becomes "null" after the
inversion time (TI)
– TI = 70% of X�
• Proper choice of TI can suppress the signal
of particular tissue type
– STIR (short TI inversion recovery)
• suppression of fatty tissue
• short TI
– FLAIR (fluid attenuated inversion recovery)
• suppression of fluid (e.g., CSF)
• long TI
13
Signal detection
• P�� in each voxel rotates clockwise at the precession freq. in the stationary reference
frame and induces an AC current in an antenna (coil)
14
• For Q = 90°– Detected signal in the stationary reference frame
• ] � = ]� � + ^]� � = P' !�/VW !"#$�
– Detected signal in the rotating reference frame
• ] � = P' !�/VW
• If the experiment is repeated after a repetition time TR,
– P% TR = P'(1 − !ab/VZ)
• After a new excitation with a 90° pulse,
– ] � = P'(1 − !ab/VZ) !�/VW
• Tissue-dependent parameters: the amount of spins, X�, X�• System- or operator-dependent parameters: (', TR, �
15
Slice or volume selection
• Superimposing a linear magnetic field gradient along the c-axis onto the main �:
– d = e� , e� , e% = 0,0,fghf%
in dimension of millitesla/meter
• 1000× smaller than �
– Larmor frequency: & c = �((' + e%c)
16
• Thickness of the selected slice or slab (volume)
– ∆c =∆#
�jh=
kl
�jh
• RF pulse bandwidth BW = ∆& = �e%∆c
• Table motion is not required for the slice selection!
• Limitations for very thin ∆c– e% < 50–80 mT/m for safety
– Difficulty in the realization of a very small BW
– Small SNR in a thin slice (due to few spins)
– ∆c (FWHM) = 2 mm or 1 mm for 1.5 T or 3 T
Position encoding
• After a 90° pulse, the transverse component of the net magnetization stands still:
– P�� o, p, � = P'(o, p)(1 − !ab/VZ) !�/VW
• If e� is applied, P�� rotates with a temporal freq. &(o) = �e�o
17
• For � ≥ TE (i.e., moment of the measurement):
– P�� o, p, � = P'(o, p)(1 − !ab/VZ) !�/VW !"�js(�!at)
• Receiver measures a signal from the excited spins in the whole op plane:
– ] � = ∬ v o, p (1 − !ab/VZ) !�/VW !"�js(�!at)dodpw
!w
• v o, p = net magnetization density in (o, p) at time � = 0 ∝ the spin or proton density
• ](�) describes a trajectory in the Fourier domain of the image y(o, p) to be reconstructed:
– ] � = ℱ y(o, p) = {(|� , 0)
• |� =�
��e�(� − TE)
• y o, p = v o, p (1 − !ab/VZ) !at/VW, the weighted spin density
• Similarly, nonzero p (thus, |�) component signal can be reconstructed by applying a
gradient in the p-direction
18
-theorem
• For 3D functions
– Angular frequency: &(}, �) = �d � · }(�)
– Measured signal: ] � = ∭ v o, p, c (1 − !ab/VZ) !�/VW !"� R d � ·}(�)���$ dodpdc
w
!w
• The �-theorem states that the time signal ](�) is equivalent to the Fourier transform of the
image y(o, p, c) to be reconstructed:
– ] � = ℱ y(o, p, c) = {(|� , |�, |%)
• �(�) =�
��R d S �S�
'
• y o, p, c = v o, p, c (1 − !ab/VZ) !at/VW, weighted spin or proton density distribution
– Weights
» (1 − !ab/VZ) = the growth of longitudinal component
» !at/VW = the decay of transverse component
» Short TR ⇒ X�-weighted images
» Long TE ⇒ X�-weighted images
» Long TR & short TE ⇒ v-weighted or proton density weighted images
19
20
Dephasing
• Breakdown of phase coherence due to different spin vectors of individual magnetic
moments with different Larmor frequencies, hence resulting in a small and noisy signal in
the receiver
– Dephasing by spin-spin interactions (irreversible)
– Dephasing by magnetic field inhomogenieties (reversible)
– Dephasing by magnetic field gradients (reversible)
21
Undo dephsing of inhomogenieties
• Applying a 180° pulse at � = TE/2, thereby creating the spin-echo (SE) signal at � = TE
22
Undo dephsing of gradients
• Applying another gradient with the same duration but with opposite polarity to make the
phase shift [Φ TE = R �d � · }(�)d�at
'] be zero, thereby creating the gradient-echo (GE)
signal at � = TE
23
Spin-echo pulse sequence
• 2D Fourier transform SE imaging is the mainstay of clinical MRI
• SE pulse sequence (to sample the �-space)
– Apply a slice selection gradient e% with a 90° & a 180° RF pulse
• To avoid dephasing of the first e%, use the longer second e% (the same effect of using the
negative first e%)
– Apply a phase-encoding gradient e� (= :��) with a temporal phase shift � p = �e�pX��, and
which results in |� =�
��:��X��
• Dephasing of e� implies position encoding
– Apply a frequency-encoding gradient e� to measure the signal ](�)
• To avoid dephasing of the e�, apply a compensating gradient before the 180° pulse
• Perform the inverse FT
• Note that the gradients encode by means of the angular frequency and initial phase of the
magnetization vector during measurements
– e� causes an initial phase shift dependent on p, � p
– e� yields an angular frequency & that depends on o
24
25
ky
Gradient-echo pulse sequence
• SE imaging requires long imaging times
• Primarily used for fast 2D & 3D acquisition of X�-weighted images
• Difference of the GE pulse sequence compared with the SE sequence
– Use a flip angle Q ≤ 90°
– No spin-echo because there is no 180° pulse
• Rephasing is done by means of gradient reversal only
• Signal characteristics are influence by X�∗
26
3D imaging
• Further encode the z-position by a second phase-encoding gradient ladder I�%, hence
� p, c = �(:��pX�� + I�%cXJJ)
27
Chemical shift imaging
• The Larmor frequency slightly depends on the molecular structure the proton belong to
• This frequency difference is called the chemical shift: &J ≡ 2ByJ• Perform multiple imaging for different frequencies yJ ⇒ chemical shift imaging (CSI)
• Require two phase-encoding gradient ladders for e� & e� in 2D and three ladders for e�,
e�, & e% in 3D imaging
• Acquisition time for CSI is an order of magnitude larger than for regular imaging
28
Acquisition time
• Acquisition time TA = # excitations × interval between two successive excitations
– TA�� = ���TR
• ��� = # in-plane phase-encoding steps
– TA�� = ����JJTR
• �JJ = # phase-encoding steps in the slab-selection direction
• e.g., X�-weighted 3D SE imaging with TR = 2000 ms
– TA�� = 256 32 2000 ≅ 4.6 hours!!!
• e.g., X�-weighted 3D GE imaging with TR = 40 ms
– TA�� = 256 32 40 ≅ 5.5 minutes (acceptable)
29
Very fast imaging sequences
• Multiple echoes per excitation & sampling within the same excitation
– TA�� =���ab
ta�
• ETL = the echo train length (i.e., # echoes per excitation)
– To reduce TA��:
① Decreasing TR (e.g., GE pulse sequences)
② Decreasing ��� (e.g., truncated & half-Fourier imaging)
③ Increasing ETL (> 1)
– Filtered version of signal �� |� , |� = � |� , |� �(|� , |�) due to the dephasing effect
» �(|� , |�) = the signal with ETL = 1 (without dephasing)
– Degrading the spatial resolution
30
• Examples
– TurboSE & TurboGE
• e.g., TurboSE sequences for 256 × 256 X�-weighted brain imaging with 4 echos/expiation
– TA�� =���ab
ta�=
�����''��
�= 160 seconds
– Echo planar imaging (EPI)
• The fastest 2D imaging sequence without 180° pulses
• Typ. 128 × 128 image size & TA with 100 ms or lower
• Functional MRI
• Diffusion and perfusion imaging
31
Imaging of spin motions
Motion type Velocity range
Diffusion
Perfusion
CSF flow
Venous flow
Arterial flow
Stenotic flow
10 µm/s – 0.1 mm/s
0.1 mm/s – 1 mm/s
1 mm/s – 1 cm/s
1 cm/s – 10 cm/s
10 cm/s – 1 m/s
1 m/s – 10 m/s
32
• In practice, the spins move due to various
human body motion
• Motions in the human body (see the Table) can
be visualized by imaging spin motions
• Since moving spins experience a change in (,
the total phase shift and signal respectively
given by
– &(}, �) = R �d � · }(�)dS�
'spin position depends on time
– ] � = R v∗(}) !" } ·¡Z � ¢£ } ·¡W � ¢⋯ !"}·¡$ � �}}
due to spin motion
• v∗(}) = v } (1 − !ab/VZ) !at/VW
• ¡¥ � ≡ R �d ��¦
¥!dS
�
', ¨ = 0,1,2, … the ¨th order gradient moment
• Motion-induced dephasing
– Smaller and noisier signal
– Position artifact (e.g., ghosting) if phase shift is small and coherent within a single voxel
Magnetic resonance angiography
• Obtain hyperintense vessel signals for blood flowing at a constant velocity by rephasing the
motion-induced dephasing:
– ] � = R v∗(}) !" } ·¡Z � ¢£ } ·¡W � ¢⋯ !"}·¡$ � �}}
• Time-of-flight (TOF) MRA
– GE-based sequences
– Enhance vascular contrast using the signal difference between the inflowing spins of the blood
and the stationary spins of the tissues
33
• Phase-contrast MRA
– Additional bipolar pulse and reversed bipolar pulse sequences
– Derive the blood velocity from a phase difference image of moving spins by subtracting the phase
images of the two subsequent acquisitions
34
• Contrast-enhanced MRA
– 3D GE sequence with short TE & TR
– Use a contrast agent in the blood
• Paramagnetic, superparamagnetic, & ferromagnetic substances
– e.g., chelates of the rare earth metal gadolinium (superparamagnetic)
• Disturb the local magnetic field
• Decrease X�∗
– Hypointense for a X�∗-weighted sequence
– Hyperintense for a X�-weighted sequence
35
Diffusion
• Spin-echo EPI sequences (or pulsed gradient spin-echo, PGSE)
• Visualize molecular Brownian motion by emphasizing the
dephasing caused by random thermal motion of spins in a gradient
field
– � © = �' !ª�
• �' = signal when no diffusion
• © = ��«� ∆ −¬
�e�
• e = the gradient amplitude
• « = the on-time of each of the gradients
• ∆ = the time between the application of the two gradients
• = the diffusion coefficient (mathematically, a tensor)
– Covariance matrix describing the displacement of the
Brownian random motion in each direction
• Diffusion tensor imaging (DTI)
– Visualize both the principal (diffusion) direction and its anisotropy by
color coding the hue and brightness respectively
36
Perfusion
• Blood perfusion of tissues refers to the activity of the capillary network, where exchanges
between blood and tissues are optimized
• Investigate perfusion by visualizing blood flow using a contrast agent such as gadolinium
chelate
• X� or X�∗ sensitive EPI sequences
37
Functional imaging
• Visualize the brain function using the dependence of brain tissue relaxation on the
oxygenation level in the blood
• BOLD (blood oxygenation-level dependent) effect
– Influences the MR signal
– Oxyhemoglobin
• Oxygen-rich hemoglobin in the arteries
• Diamagnetic
– Deoxyhemoglobin
• Oxygen-poor hemoglobin in the capillaries
• Paramagnetic (causing magnetic field inhomogenieties)
38
Contrast
• Signal for a SE sequence (with Q = 90°) is proportional to v(1 − !ab/VZ) !at/VW
• Parameters affecting the image contrast:
– Tissue-dependent parameters
• Relaxation times X� & X�• Spin or proton density v (actually net magnetization density)
– Technical parameters
• Repetition time TR
• Echo time TE
39
Type TR TE
v-weighted
X�-weighted
X�-weighted
long
short
long
short
short
long
• Signal for a GE sequence (with Q < 90°, e.g., FLASH sequence) is proportional to
v !at/VW∗ (�!®¯°±/²Z) �³´ µ
�!®¯°±/²Z ¶·� µ
Resolution
• In the Fourier space
40
o p
Nyquist theorem ∆|� ≤1
2o¸¹�=
1
FOV�∆|� ≤
1
2p̧ ¹�=
1
FOV�
�-theorem ∆|� =�
2Be�∆� ∆|� =
�
2B��X��
Resultant restriction e�∆� ≤2B
�FOV���X�� ≤
2B
�FOV�
• In the image space
– Note that "the PSF defines the highest
frequency |¸¹� available in the signal"
– |�,¸¹� ≤�
��e�
�s∆�
�
– |�,¸¹� ≤�
�������
V��
�
Noise
• I↑ − I↓ ≈ IJ�ℏg$�¼½V
= 3.3 × 10!�IJ
– IJ = I↑ + I↓– I↑ & I↓ = the number of spins with energy .↑ & .↓, respectively
– |g = Boltzmann's constant
– X = the absolute temp. of an object
• P ≈(ℏ�)WMNg$
�¼½V
– Typically quite small (vulnerable to noise!)
• e.g., 1-L water at X = 310 K & (' = 1 T ⇒ IJ ≈ 6.7 × 10�� & P ≈ 3 × 10!� J/T (very small
value)
• Thermal noise in the patient and in the receiver part of the MR imaging system
41
Artifacts
• Due to 1) technical imperfections, 2) inaccurate assumptions about the data, & 3)
numerical approximations
– System failure, inappropriate shielding of the magnet room or interaction with unshielded
monitoring equipment
– Assumption that � is homogeneous (to avoid unnecessary dephasing, which causes signal loss
and geometric deformations)
• In real, inhomogeneous �⇒ inhomogeneous RF field ⇒ spatially varying Q⇒ low-
frequency signal intensity modulation (called the bias field)
42
43
– Assumption that the data are independent of X�• If this fails (e.g., multiple echoes per excitation),
the spatial resolution decreases
– Assumption that tissues are stationary
• Motion yields dephasing artifact
– The magnetic susceptibility of tissues or foreign
particles & implants yields dephasing
– Truncation errors during digital image reconstruction
may produce visual artifacts
• Truncated FT yields ripples at high-contrast
boundaries (known as the Gibbs artifact or ringing
artifact)
• Inadequate sampling yields aliasing, known as the
wrap-around artifact
44
– Phase cancellation artifact
• Dephasing in voxels that contain both water and fat elements (due to the chemical shift
between water and fat)
45
– Chemical shift artifact
• Mutual spatial misregistration due to a phase difference between water and fat
46
MRI systems
47
• Magnets
– Desirable with compact designs with higher field homogeneities
– Superconducting magnets
• Higher field strengths, higher SNR
– Permanent & resistive magnets
• Lower field strength (poor homogeneity), lower SNR
48
• Interventional MRI
– Open MR systems for MR-guided procedures (e.g.,
surgery or therapy)
– All surgical instruments must use MR-compatible
materials
– RF radiation from electronic equipment must be
shielded from the RF of the MR system and vice
versa
– Electrical leads with the RF field can produce hot
spot; hence causing skin burns (preferred to use
fiberoptic technology)
• Gradient system
– Linearity for correct phase-encoding
– Maximum amplitude & its rise time for fast imaging
• RF system
– For sensitivity & in-plane homogeneity of signal detection
49
Clinical use
• Anatomical imaging
– All parts of the human body that contain hydrogen (e.g., soft tissue, cerebrospinal fluid, edema, …)
50
– Better contrast (using a various v-, X�-, & X�-weighted images) between different soft tissues than
with CT
51
– Tissue characterization due to the availability of v-, X�-, & X�-weighted images
52
– Imaging with contrast agents
• Gadolinium compounds (not captured by cells)
• Iron oxide (taken up by specific cells)
53
– Statistical image analysis
54
– Perfusion imaging
• e.g., after brain tumor resection
to exclude tumor residue or
recurrence
• e.g., after myocardial infarction
to assess tissue viability
55
56
– Diffusion imaging
• To investigate microscopically small
displacements of hydrogen-
containing fluid
57
Top Related