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