A Brief Introduction to
Magnetic Resonance Imaging
Anthony Wolbarst, Kevin King, Lisa Lemen,
Ron Price, Nathan Yanasak
Outline for Today
1)
Introduction
MRI Case Study; Caveat!!!
“Quasi-Quantum” NMR and MRI
Wolbarst
2)
Net magnetization, m(x,t), of the voxel at x
T1 Spin-Relaxation of m(x,t),
T1 MRI of the 1D patient
Sketch of the MRI Device
Lemen
3)
‘Classical’ Approach to NMR
FID Image Reconstruction, k-Space
Yanasak
4)
Spin-Echo Reconstruction
T2 Spin-Relaxation
T1-w, T2-w, and PD-w S-MRI
Spin-Echo / Spin-Warp in 2D
Some Important Topics Not Addressed Here
Price
Magnetic Resonance Imaging:
Mapping the Spatial Distribution
of Spin-Relaxation Rates
of Hydrogen Nuclei
in Soft-Tissue Water and Lipids
µ
Part 1:
“Quasi Quantum”
NMR and MRI
N
++
+
++
S
J
Introduction
MRI Case Study; Caveat!!!
“Quasi-Quantum” NMR and MRI
Introduction
Medical Imaging:
differential transmission, etc., of probes by different tissues
Beam of ‘probes’:
particles or waves
Differential
transmission, reflection,
emission of probes
Image receptor
How Imaging Modalities Detect Tissue Contrast…
Modality
Probe / Signal
Detector
Source of Contrast:
Differences in…
Planar R/F X-ray X-rays passing
CT
through body
II+CCD, AMFPI;
GdO, etc., array
µ( ρ, Z, kVp), x
Nuc Med,
SPECT;
PET
Gamma-rays
from body;
511 keV
NaI single crystal;
multiple NaI;
LSO array
Radiopharmaceutical
uptake, concentration
(e.g., 99mTc), emission
US
MHz sound
Piezoelectric transducer
ρ, κ, µUS
MRI
Magnet, RF
AM radio receiver
T1, T2, PD, [O], blood
flow, water diffusion,
chemical shift
Digital Display: Voxels to Pixels
tissue property parameterized by α (µ, [18FDG], κ,T1, …)
α
Measures of Image Quality
Contras
Resolution
Image
Quality
Artifacts
Noise
Forms of Contrast Between Good and Bad Apples
produced and detected thru different bio/chemico/physical processes
Color
Smell
Texture
Holes
Firmness
Shape
Taste
etc.
CT vs. MRI Soft Tissue Contrast
CT
MRI
Posterior reversible encephalopathy syndrome (PRES), edematous changes
Magnetic Resonance Imaging
Multiple types of contrast, created through and reflecting
different aspects of biophysics.
Most detect rotations, flow, etc., of water/lipid molecules,
yielding unique maps of tissue anatomy, physiology:
vessels, nerve trunks, cartilage, tumors,…
No ionizing radiation
Risks from intense magnetic fields, RF power
Expensive
Technology complex, challenging to learn
MRI Case Study; and Caveat!!!
with T1, FLAIR, MRS, DTI, f MRI,
and MR-guided biopsy studies of a glioma
Case Study
57 year old ♀ medical physicist has had daily headaches
for several months. Responds fully to ½ Advil.
Physical examination at local clinic unremarkable.
Patient appears to be in good general health, apart from
mild hypertension, controlled by medication.
Good diet, exercises moderately. Patient reports no
major stresses, anxieties.
CT indicates a lesion in the right posterior temporooccipital region, adjacent to occipital horn of right
lateral ventricle. MRI for better contrast.
Principal concern: Vision for reading.
Right Posterior Temporo-Occipital Region,
adjacent to occipital horn of right lateral ventricle
T1-w
FLAIR
No enhancement
with Gd contrast.
f MRI
optical stimulus
DTI
Chemical Shift and Non-invasive MRS
fLarmor (MHz)
= 42.58 (1 + σ) B0 (T)
CH3
COOH
1.56 ppm
Astrocytoma
MRI-Guided Needle Biopsy
Grade 1-2 Astrocytoma
with scattered cellular pleomorphism and nuclear atypia
Caveat!!!
Extreme Danger
Strong Gradient Fields Outside Bore
www.simplyphysics.com
How Not to Clean the Device
www.simplyphysics.com
MRI Safety
fringes of principal magnetic field
rapidly switching (gradient) fields
high RF power
FDA: ~ 40 MRI-related accidents/year
70% RF burns, 10% metallic “missiles”
RF Specific Absorption Rate (SAR): dE/dm (W/kg)
no magnetic anything in or entering MRI room
restricted access, safe zones, prominent warnings
accompany all patients, visitors, non-MRI staff
accurate medical history; double-check for metal
training for any staff (e.g., cleaners)
possible gadolinium risk: nephrogenic systemic fibrosis (NSF)
pregnant patients generally should not have Gd-contrast
MRI Safety – Prevent Entry!
within / with patient, others
aneurysm clip, shrapnel
cochlear implant, prostheses
artificial heart valve
stent, permanent denture
defibrillator, pacemaker,
electrodes, nerve stimulator
medical infusion pump
drug-delivery patch, tattoo
in / into imaging suite
hemostat, scalpel, syringe
O2 bottle, IV pole
scissors, pen, phone, laptop
tool, tool chest
wheelchair, gurney
ax, fire extinguisher
gun, handcuffs
cleaning bucket, mop
“Quasi-Quantum” NMR and MRI
Swept-frequency NMR in a tissue voxel
Proton-Density (PD) MRI in the 1D patient
Magnetization, m(x,t), in the voxel at x
Two Approaches to Proton NMR/MRI (incompatible!)
quantum spin-state function for hydrogen nucleus
|ψ
start with this
Simple QM
|↑ , |↓
transitions between
spin-up-, spin-down states
fLarmor , m0 , T1
oversimplified
Classical Bloch Eqs.
expectation values in voxel
precession, nutation of
net voxel magnetization, m(t)
fLarmor , T2 , 2D, k-space
exact; from full QM
‘Open’ MRI Magnet
principal magnetic field B0 defines z-axis
z
S
B0
z
B0
x
N
Moving Charge Produces Magnetic Field (e.g., mag. dipole);
Magnetic Dipole Tends to Align in External Field
µ
µ
e
N
N
Bz
++
+
++
S
S
T
mechanical relaxation
J
simplified QM picture
proton
Energy to Flip Over Needle with Magnetic Moment
z
Bz along
z-axis
θ
µ
N
º
E = – μ
•
Bz
= – μ Bz cos θ
ΔE180º = ± 2 μ Bz
μ in Bz
Swept-frequency NMR in a tissue voxel
Nuclear Zeeman Splitting: ΔE = ± 2 μz Bz
μ points only along or against z
South
E(Bz)
N
ΔE
0
Bz
High-energy
state
–μz
Low-energy
state
+ μz
North
Bz
Nuclear Magnetic Resonance
– μz
Orientational
Energy
ΔE180° = 2 μz Bz
photon E = hf
spin
flip
hf
+μ z
fLarmor(Bz) = (2/h) μz Bz
63.87 MHz
fLarmor, H O = 42.58 MHz
×Bz
T
2
n.b.
Bz(x) = fLarmor(x) / 42.58
40
T 20
0.2
P-31
0.5
1
0.1
1.5
(μeV )
H-1
42.58 MHz
Energy
for water protons,
60
[MHz]
γ Bz
fLarmor
2πfLarmor ≡ ω =
Bz (T)
Proton Spin Transitions: µeV
therapy x-rays, electrons: MeV
radioactivity: keV – MeV
diagnostic x-rays: 30 – 120 keV
visible light, chemical reactions: eV
MRI: µeV (~100 MHz)
NMR Gedanken-Experiment on Water
monochromatic RF power absorption at (and only at) fLarmor, H O
2
z
Antenna
Transmitter
principal
Antenna
H2O
magnet
Power meter
Detected
RF power
sweep the RF f
Resonance at 1 T:
42.58 MHz
Transmitter f (MHz)
Proton-Density (PD) MRI in the 1D patient
1D Phantom in Principal B0 and Gradient Gx
x = [Bz(x) − B0] / Gx
Principal magnet
Gradient magnet
z
Gx = dBz(x)/dx
Bz(x) = B0 + Gx×x
B0
x
Bz(x)
(1.002T ̶ 0.998T) / 0.20m
= 20 mT/m
1.002 T
1.000 T
0.998 T
x
–10
0 cm
+10 cm
fLarmor = fLarmor[Bz(x)]
1D PD-MRI:
z
0.
sweep the RF f
x
–10
B(x)
0 cm
+10 cm
Gx
1.002 T
1.000 T
0.998 T
3.
2.
x = [Bz(x) − B0] / Gx
x
Bz(x) = fLarmor / 42.58
4.
f (x)
3. 4.
42.58
42.62
MHz
1.
PD MR Image
x
Summary: PD MRI on 1D Patient
A( f )
42.58
42.62 MHz
x
0
1.
fLarmor of
NMR Peak
2.
Bz (x)
3.
+5
10 cm
Amplitude of
NMR Peak, A( f )
4.
PD(x)
Voxel Position, x Pixel Brightness, PD
MRI
4'.
Matrix Size, Resolution, and Field of View
Nx = voxels (matrix dimension) along x-axis
(e.g., 256)
z
B0
1 2 3….
Δx
(e.g., ½ mm)
Nx
x
FOVx = Nx × Δx
(e.g., ~25 cm)
In-plane resolution: 2Δx ~ 1 lp/mm ~ 1 mm
(but may see object ≤ Δx)
Imaging plane: ~ 1 mm thick.
Proton-Density MRI
contrast from differences in PD
Two MRI Motion Artifacts
respiration
aortic pulsation
Part 2:
Magnetization
& Relaxation
Net magnetization, m(x,t), of the voxel at x
T1 Spin-Relaxation of m(x,t),
T1 MRI of the 1D Patient
Sketch of the MRI Device
40
Net magnetization, m(x,t), of the voxel at x
Net Magnetization, m(x,t), for the Single Voxel at Position x:
magnetic field from the ensemble of protons or needles themselves
Single voxel at position x
Bz > 0
m(x,t)
Gentle agitation
(noise energy)
Voxel’s MRI Signal Proportional to Its Magnetization, m(x,t)
i) What is the magnitude of voxel magnetization
at dynamic thermal equilibrium, m0?
ii) How long does it take to get there (T1)?
iii) What is the mechanism?
Filling Four Energy Levels of Marbles vs. Noise Level
battle between energy and entropy
(all slippery; black balls denser)
Shaking (Noise) Energy:
too much
too little
just right
Switching Bz(x) On at t = 0 Induces Magnetization m(x,t)
voxel at x
from protons in voxel at x
t=0
m(x) = 0
Bz(x) = 0
t >> 0
Bz > 0
m(x,t)
Magnetization in Voxel at x: m(x,t) = [N–(x,t) – N+(x,t)]×μ
collective magnetic field produced by all the protons in it
Bz
μ
N ≡ (N– + N+) protons/voxel
(Boltzmann)
m
0
0 tesla
0.01 T
~0
or
t = 0+*
N μz
»1.5 T
1.5 T
t >> 0
5×10– 6 N μz
* after abruptly turning on Bz , or after a 90º pulse.
|m0(x)| and Signal from it Increase with B0
p.s., trade-off: SNR, resolution, acquisition-time
1.5 T
7 T (+Gd)
T1 Spin-Relaxation of m(x,t)
Voxel’s MRI Signal Proportional to Its Magnetization, m(x,t)
i) What is the magnitude of voxel magnetization
at thermal equilibrium, m0?
ii) How long does it take to get there (T1)?
iii) What is the mechanism?
Disturbed System Moving toward Up-Down Equilibrium
energy imparted to individual spins from ‘outside’
m(2)
m(1)
m(0)
t = 0 ms
m = 0
t = 1 ms
m = 2μ
t = 2 ms
m = 4μ
spin ‘tickled’ down by fLarmor component of magnetic noise;
emits RF photon, phonon
Decay of Tc-99m Sample to Final Value, n(∞) = 0
n(0) radionuclei at initially
n(t) remaining after time t.
n(∞) at ‘equilibrium’
n(∞) = 0
dn/dt(t) =
λ n(t)
n(t)
n(0)
n(t) = n(0) e λt
1
½
0
dn/dt ~ [n(t)

6
12
18
n(∞)]
n(∞)
hr
scint. decay (t)
tracer conc. (t)
pop. growth (t)
cell killing (D)
photon atten. (x)
ultrasound atten. (x)
Return of Voxel’s mz(t) to Final Value, mz(∞) ≡ m0
[m0 – mz(t)]
mz(t)
mz(∞) ≡ m0
0

250
500
t
750 ms
rate
d[m0 – mz(t)] / dt =  (1/T1) [m0 – mz(t)]
mz(t)/m0 = 1  e t / T1
T1 MRI of the 1D Patient
T1 for 2 Voxels at 1 tesla: Lipid vs. CSF
both at fLarmor of H2O for local field
lipid
CSF
RF
receive,
FT
RF
transmit
–10
0 cm
0.998
1.000
+10
1.002 T
f (x)
PD(x)
42.58
42.62
Recovery of mz(t) over Time: mz(TRi) Measurements
longitudinal component of net voxel magnetization, mz(t)
t = 0+
0
t = TR1
5 cm
1. First, do lipid at x = 0.
2. Strong burst of RF over range
42.58 MHz ± 250 Hz causes
N– ~ N+
t = TR2
3. After TR1 delay, sweep through
fLarmor , record amplitude.
4. Repeat for several TRi values.
t = TR3
5. Plot.
mz(TR)/m0 = (1 − e − TR / T1)
curve-fit for set of {TRi}
1
mz(t)/m0
0.63
 TR2
TR
1
0


250
TR3
 TR
4
lipid: T1 ~ 250 ms
T1
500
750
tread (ms)
T1
T1 MR image
by convention for T1-imaging: Spritely Brightly
mz(x,t)/m0(x) = (1 − e− t / T1)
repeat for CSF voxel at x = 5 cm, fLarmor = 42.62 MHz…
mz(t) / m0
Lipid: T1 = 250 ms
x=0
CSF: T1 = 2,000 ms
x = +5 cm
t
T1
T1 MR image
lipid: ~ 250 ms
CSF: ~ 2,000 ms
T1-w MR Image
CSF
T1 ~ 2000 ms
sub-cut. fat
T1 ~ 260 ms
Gd
Approximate Relaxation Times of Various Tissues
Tissue
PD
p+/mm3, rel.
pure H20
1
brain
CSF
0.95
white matter 0.6
gray matter 0.7
edema
glioma
liver
hepatoma
T1, 1 T
(ms)
T1, 1.5 T
(ms)
4000
2000
700
800
930
2000
800
900
1100
1000
T1, 3 T
T2
(ms)
(ms)
4000
4000
2000
850
1300
200
90
100
110
110
500
1100
muscle
0.9
700
900
adipose
0.95
240
260
40
85
1800
45
60
Voxel’s MRI Signal Proportional to Its Magnetization, m(x,t)
i) What is the magnitude of voxel magnetization
at thermal equilibrium, m0?
ii) How long does it take to get there (T1)?
iii) What is the mechanism?
In MRI, the only thing a proton
is ever aware of, or reacts to, is the
local magnetic field, Blocal(t).
!!!
But the source of fLarmor(Blocal(t)) can be either
external (BRF) or internal (e.g., moving partner)
!!!!!
61
Nuclear Magnetic Resonance vs. T1 Relaxation!
fLarmor =
42.58 Bz(x)
NMR: Magnetic field of Larmor-frequency photon can couple
to magnetic dipole moment of proton in lower-energy spin state,
transferring energy hfLarmor to it, flipping it “up.”
T1 Relaxation: Any magnetic field varying at fLarmor can ‘tickle’
a proton down from the higher-energy spin state, with the proton
giving off a photon (RF) or phonon (heat) of energy hfLarmor .
Spontaneous fLarmor-BRF(t) Causes T1-Transitions
variations in proton magnetic dipole-dipole interactions
H
O
☻
☻
O
H
H
H
4×10 – 4 T
1.5 Å
each water proton produces magnetic field fluctuations
of various frequencies, including local fLarmor , at its partner proton
Restrictions on Rotations of Water Molecules
more or less
bound
hydration
layer
free
Rotational Power Spectra, J( f ), of Water
Probability / relative number
for several water populations
Water on large
biomolecule
T1 spin flips
Water on small molecule
fLarmor
Pure water
Magnetic noise (water tumbling, etc.) frequency
Slow motion
Rapid
Average Water T1 Determined by J( fLarmor)
Relaxation Time, T1
T1
Water on
intermediate
sized molecule
Water on large,
slow biomolecule
Pure water
1.5 T
fLarmor
Magnetic noise (water tumbling, etc.) frequency
Slow motion
Rapid
MRI:
Water, Lipids of Soft Tissues ̶
their Hydrogen Nuclei.
Spin-Relaxation Times (T1, T2) ̶
their Spatial Distributions!
Sketch of the MRI Device
Damadian’s Indominable, 1977
(Smithsonian)
Nobel Prize, 2003
Paul Lauterbur
Peter Mansfield
69
Major Components of a Superconducting MRI System
y
Gx
x
B0
z
BRF
fringe
fields
partially
shielded
RF
MIX
RF
DAC
ADC
Operator
console
PACS
z
MRI Magnets
Open (most of this presentation)
B0
electromagnetic or permanent
y
Superconducting
e.g., niobium-titanium
x
B0 : < 0.5T, 1.5 T, 3.0 T, (7 T)
Homogeneity: 10 ppm
Shielding: passive and active
Cryogen: 0.1 liter He/y
Weight: 4 tons (supercond.)
z
B0
Three External Magnetic Fields in Open Magnet MRI
z
B0 , Gz , and Gx all point along z!
B0
Gz ≡
ΔBz(z)/Δ z
BRF
Gx ≡ ΔBz(x)/Δ x
x
x-Gradient Magnet Winding for Superconducting Magnet
B0
z
one layer of x-gradient coil
x-Gradient dBz /dx
Rise time
Slew rate
20 − 60 mT/ m
0.3 ms (to reach 10 mT/ m)
50 − 200 mT/m/ms
Artifact: Gradient Non-Linearity
74
RF Coils
BRF : 20 μT
Pulse on-time: 3 msec
RF power: 15 – 25 kW
SAR: 2 – 20 W/ kg
‘Parallel’ RF Receiving Coils for Much Faster Imaging
transmit coils coming
T1W MRI
Part 3:
“Classical” MRI
‘Classical’ Approach to NMR
FID Image Reconstruction, k-Space
‘Classical’ Approach to NMR;
FID Image Reconstruction, k-Space
Normal modes, resonance, precession, nutation
Free Induction Decay (FID) in a voxel, without the decay
e.g., very slow T1 relaxation
Untangling the FID RF signals from a row of voxels:
Temporal Fourier series approach
FID imaging of the two-voxel 1D patient
FID Imaging via k-Space; Spatial FT
The Two Approaches to NMR/MRI (incompatible!)
quantum state function
|ψ
now this
Simple QM
|↑ , |↓
transitions between
spin-up, spin-down states
fLarmor , m0 , T1
oversimplified
Classical Bloch Eqs.
for expectation values
precession, nutation of
voxel magnetization, m(t)
fLarmor , T2 , k-space
exact; from full QM
Normal Modes, Resonance, Precession, Nutation
Normal Mode, at fnormal
fnormal
fnormal
fnormal
A Normal Mode of a 2-D Pendulum
fnormal
Normal Mode Precession about External Gravitational Field
J(t)
J(t): Angular Momentum
With torque, τ:
fprecession
dJ/dt = τ
Equation of Motion
Angular acceleration is just like
dp / dt = F
but here, J(t) changes direction only:
Precession at fnormal
r
mg
ττ
Normal Mode Precession of Voxel m(x,t) in Magnetic Field
can be derived rigorously from quantum mechanics
z
fnormal =
fLarmor(x)
longitudinal mz(t)
magnetization
Bz(x)
m(t) =
[mx(t) + my(t) + mz (t)]
y
x
mxy(t) = [mx(t) + my(t)]
transverse
magnetization
Classical Bloch Equations of Motion for m(x,t) in B0
Equation of motion for spinning body
dJ /dt = τ (external torque)
recall: μ = γ J
d( μ/γ) / dt = τ
Lorentz torque on spins at x with magnetic moment μ in Bz(x):
τ = μ×Bz(x)
(vector cross product)
Equation of motion becomes:
dμ(x,t) /dt = γ μ(x,t)×Bz(x) .
Sum/average over all protons in voxel:
d m(x,t) /dt = γ m(x,t) ×Bz(x)
With T1 relaxation along z-axis:
Expectation
Value,
m(x,t) ,
behaves
classically
d m(x,t) /dt = γ m(x,t) ×Bz(x) + [ m(x,t) ̶ m0(x)]ẑ / T1
Precession of m(x,t) about B0 at fLarmor = (γ /2π) Bz(x)
as seen from a frame that is
Fixed
Rotating at fLarmor(x)
z
as if B0 = 0
m(x,t)
z'
m(x,t)
x
y
x'
y'
The ponies don’t advance when you’re on the carousel;
It’s as if B0 = 0 !
Resonance Energy Transfer when fdriving = fnormal
Fres
Fres
Fres
Normal Mode and Resonance of a 2-D Pendulum
Precession
Nutation
fdriving = fnormal
fnormal
And Now for Something Completely Different: Nutation
string
Cone of
Precession
Nutation at Resonance
Nutation of the Voxel’s Magnetization, m(x,t)
BRF /B0 ~ 10 5 ̶ 10 4
z
m
fNutation =
fLarmor =
(γ/2π) BRF
y
Bz(x)
m
BRF
x
precesses at
(always!)
fLar = (γ/2π) Bz(x)
nutates at
fnut = (γ/2π) BRF
(only when BRF is on!)
(γ/2π) B0
Free Induction Decay in a voxel, without the decay
e.g., very slow T1 relaxation
m(x,t) for a Single Voxel at x and Precessing in the x-y Plane
following a 90º pulse
Tissue
RF
RF
RF transmit coil
RF detect. coil
(Faraday)
mxy(x,t)
FID Precession, Reception, Fourier Analysis
(single voxel)
n.b. measure induced V(t), not power absorption (as before)
x
F
Fourier
component
MR
signal
m(t)
y
antenna
t (µs)
f
42.58 MHz
x = 0 cm
(MHz)
In MRI, the only signal
you ever see comes from
the set {m(x,t)}
all precessing in the x-y plane
!!!
Untangling the FID RF Signal from Two Voxels:
temporal Fourier series approach
Wave Interference in Time or Space
positive
(in-phase)
interference
negative
interference
Separate
signals
Wave Interference and Spectral Analysis
time
or
space
Composite
beat signal
interference
t (sec)
x (cm)
Fourier
components
decomposition
via Fourier analysis, F
f (Hz)
k (cm1)
Fourier Decomposition of Periodic Temporal Signal
S(t) ~ ½ + (2/π){sin (2π f1t) + ⅓ sin (6π f1t) + 1/5 sin (10π f1t) + …}
fundamental: f1 Hz (cycles/sec)
S(t)
orthonormal
basis
vectors
Components
1/f1
spectrum
t
22/π
½
harmonic
fundamental
f1
Component amplitude
1st
f3
3rd harmonic
f5
5th
0 1f1
5f1
7f1
f5
f7
f
Hz
2 /3π
0 f1
0
F
3f1
f3
f
2 /5π
(2 /7π)
f
FID imaging of the two-voxel 1D patient
fLarmor(x) = (γ/2π) (x×Gx)
(in rotating frame)
e.g., FID from 2 water slices, at x = 0 and 5 cm; little decay!
RF
receive
BRF
RF
transmit
–10
0 cm
5
B(x)
1.002 T
1.000 T
0.998 T
42.58
42.62 MHz
+10
x
* 90° RF Pulse
FID Following Wide-Bandwidth*
PD(x)
covers fLarmor(x) for all N voxels while Gx is being applied!
½ water
water
x
5
0
5
10 cm
Gx
10
y
z
90º pulse
t
**
t
while gradient is on…
B0
x
FID: Wide-Band RF Pulse, Centered on fLarmor(B0)
z
n.b.: m(x,t) = m(x,0)e– 2πi f (x) t
mxy(x=0)
y
mxy(x=5)
F
Fourier
spectrum
Temporal
S(t )
Voltages
x
FID PD MRI
(μs)
Separate
RF signals
(μs)
Detected
FID signal
(MHz)
x = fL(x) / (γ/2π) Gx
To Summarize What We Have Done So Far with FID
MRI signal
t↔f
t
A( f )
Signal spectrum
F
A( f ) = S(t) e– 2πi f (t) t dt
f
42.58 MHz
42.62
x = 2π f (x) / γGx
R(x)
S(t )
follow temporal F with isomorphism of A( f ) to real-space, R(x)
Real-space
representations
x
0 cm
+5
FID Imaging via k-Space; Spatial FT
Again, What We Have Done So Far:
Temporal:
S(t )
t↔f
Signal spectrum
A( f )
Detected FID signal
F
tsec
42.58 MHz
42.62
f
R(x)
x = 2π fLarmor(x) / γ Gx
x
0
+5 cm
Real-space representation
Now Consider 1D Spatial Fourier Analysis
fundamental: k1 (cycles/mm)
2/π
Component amplitude
½
kx
k1 3k1 5k1 7k1 …
2/3π
2/5π
0
S(x)
F
1/k1
1
x
0 k1
k3
k5 k7 …
S(x) ~ ½ + (2/π) [sin (2π k1 x)
+ ⅓ sin (6π k1 x) + 1/5 sin (10π k1 x) + … ]
Detected FID signal
Temporal:
S(t )
t↔f
A( f )
New Approach: Continue on with a FT from x- to kx-Space
F
tsec
42.58 MHz
F
k
k-space representation
x
Spatial:
kx ↔ x
f
x = 2π fLarmor(x) / γ Gx
R(x) e–2πi kx x dx
α(k) =
42.62
R(x)
α(k)
Signal spectrum
0
+5 cm
Real-space representation
Detected FID signal
Temporal:
S(t )
t↔f
A( f )
And Complete the Ring Linking t-, f-, x- and kx-Spaces
Signal spectrum
F
tsec
42.58 MHz
kx(t) ≡ γ Gx t /2π
R(x)
α(k)
k-space representation
f
x = 2π fLarmor(x) / γ Gx
F
k
42.62
x
Spatial:
kx ↔ x
0
+5 cm
Real-space representation
So, Can Get from S(t) to R(x) in Either of Two Ways!
but only the approach via k-space works in 2D and 3D
A( f )
S(t )
Detected FID signal
Signal spectrum
F
tsec
42.58 MHz
kx(t) ≡ γ Gx t /2π
̶1
R(x)
α(k)
k-space representation
f
x = 2π fLarmor(x) / γ Gx
F
k
42.62
x
0
+5 cm
Real-space representation
k-Space Procedure for FID in 1D Phantom
S(t) comprised of contributions from the m(x) for all x
S(t )
Detected FID signal
m(x,t) = m(x,0)e– 2πi f (x) t= m(x,0) e– 2πi kx(t) x
S(t) = S(k(t)) ~
m(x) e– 2πi kx(t) x dx .
Therefore, taking the inverse F ̶ 1 :
m(x) =
kx(t) ≡ γ Gx t /2π
R(x) = α(k) e+2πi kx dk
F
kx
k
k-space representation
̶1
R(x)
α(k)
t
S(t) e+2πi k(t)·x d 3k
x
0
+5 cm
Real-space representation
Spatial Waves, and Points in k-Space
Point in k-Space and Its Associated Wave in Real Space
ky
kx
k-Space
k-Space
Points in k-Space and Their Waves in Real Space
ky
kx
k-Space
Composite Patterns in k-Space and Real Space
linearity
+
=
Fourier Transform of k-Space Pattern to Real Space
k-Space
Real Space
y
ky
kx
F
x
Fourier Transform of MRI Data in k-Space to Real Space
F
Fourier Transform from Parts of k-Space to Real Space
F
Herringbone Artifact
noise spike during data acquisition
Zhou/Gullapalli
Part 4:
Spin-Warp; 2D
Spin-Echo Reconstruction
T2 Spin-Relaxation
T1-w, T2-w, and PD-w S-E MRI
Spin-Echo / Spin-Warp in 2D
Spin-Echo Reconstruction
Static Field de-Phasing in x-y Plane of Spins in a Voxel
SFd-P from static-field inhomogeneities & susceptibility effects
This de-phasing is REVERSABLE!!!!
mxy(x,t)
t=0
mxy(x,t) / mxy(x,0)
1
mxy(x,t) ~ e– t / T SFd-P
Rate of SFd-P decay:
1/ TSFd-P = κ ΔB0 + κ' Δχ
0
t
n.b., mxy(x,t)….
but still 1D phantom!
Kentucky Turtle-Teleportation Derby
i
t = 0
ii
iii
t = ½ TE
iv
v
t = TE
Spin-Echo Generates an Echo and Eliminates SFd-P Effects
m(x,t)
(a)
Local magnetic environment
for each spin packet is static.
So this de-phasing
can be REVERSED!!!
y
90º
t = 0
x
(b)
(c)
t = ½ TE
180º
(T2 dephasing, by contrast,
is random over time,
and can NOT be un-done!!)
(d)
(e)
t = TE
S-E Pulse Sequence
RF pulse
with 256 voxels, e.g., sample/digitize echo signal 512 times
S(t)
90º
180º
fLarmor(B0 ̶ Gx × ½ FOV) to
fLarmor(B0 + Gx × ½ FOV)
t
Echo
t
Gx
readout
t
Again: k-Space Procedure for FID in 1D
S(t) comprised of contributions
Phantomfrom the m(x) for all x
Detected FID signal
S(t )
m(x,t) = m(x,0)e– 2πi f (x) t= m(x,0) e– 2πi kx(t) x
S(t) = S(k(t)) ~
m(x) e– 2πi kx(t) x dx .
Therefore, taking the inverse, F ̶ 1 …
m(x) =
kx(t) ≡ γ Gx t /2π
R(x) = α(k) e+2πi kx dk
F
kx
k
k-space representation
̶1
R(x)
α(k)
t
S(t) e+2πi k(t)·x d 3k
x
0
+5 cm
Real-space representation
During Readout, kx Increases Linearly with t
Signal is sampled sequentially at 512 times, tn , after Gx is turned on.
k(t) for all voxels increases linearly with t while the echo signal is
being received and read: kn = [Gx γ / 2π] tn. Later sampling times,
hence larger k-values, correspond to greater spatial frequencies!
S-E reads out from kn= ̶ kmax to k = 0 to k = kmax
k = ̶ kmax
k=0
tn
kmax
t512
T2 Spin-Relaxation
T1-Events
and Non-Static, Random, Non-Reversible
Proton-Proton Dipole Interactions
Both Contribute to 1/T2
Specifically,
any process that decreases either the number
of transverse spins or their phase coherence
will add to T2 relaxation.
Thus, Spin-Echo Signal Does Decay,
and Much Faster than 1/T1!
90º
180
180º
180º
180º
RF
t
S(t)
Spin Multi-Echo:
we might expect
mxy(x,t) ~ e– t/ T
º
However,
Harsh
Reality
!!!!!
º
º
º
º
So…
Something BIG
must be going on
in addition to
the T1 Process
!!!!!
Another Form of Relaxation, As Well
!
Call it T2
!
Even with S-E, Spin De-Phasing Is Caused by T1 Events and
by Non-Static, Random, Proton-Proton Dipole Interactions
mxy(x,t) / mxy(x,0)
1
0
mxy(x,t) ~ e– t / T2
t
Spin-Spin (Secular) De-Phasing in Bound Water
with its protons precessing in the x-y plane at different rates
H
B
C
O
H
C
H H
C
H
H
Quasi-static spin-spin interactions
do not involve exchange of energy!
~
10 × (1/T1)
T1, 1 T
T1, 1.5 T
1/T2
Tissue
PD
p+/mm3, rel.
pure H20
1
brain
CSF
0.95
white matter 0.6
gray matter 0.7
edema
glioma
liver
hepatoma
(ms)
(ms)
4000
2500
700
800
930
2500
800
900
1100
1000
T1, 3 T
T2
(ms)
(ms)
4000
4000
2500
850
1300
200
90
100
110
110
500
1100
muscle
0.9
700
900
adipose
0.95
240
260
40
85
1800
45
60
mxy(t) / mxy(0)
Exponential, T2-Caused De-Phasing of mxy(x,t) in x-y Plane
1
0
20
40
60 ms
t
rate
dmxy (t) / dt =  (1/T2) mxy (t)
mxy (t) /mxy(0) = e t / T2
dm(x,t)/dt = γ m(x,t)×Bz(x) ̶ [m(x,t) ̶ m0(x)]ẑ ̶ [mx x + my ŷ]
classical Bloch Equation
T1
T2
1/T2 = 1/(2 T1) + 1/T2secular
faster
T2
spin-echo
T1
Discrete
spin-dephasing;
involves ΔE
Continuous
spin-dephasing;
no ΔE!
One Last Member of the Spin-Relaxation Family Tree: T2*
fastest
T2*
Gradient-Echo
G-E generally much
faster than S-E
T2
(TSFd–P)
T1
spin-echo
T1-w, T2-w, and PD-w S-E MRI
Three Different Forms of MRI Contrast
created by, and reflecting, three quite different physical processes
T1
T2
PD
Multiple Spin-Echo Pulse Sequence
i
RF
Gx
readout
90º
180º
ii
90º
180º
MRI Signal Strength at t = TE Depends on….
Inherent Parameters
Operator Settings
T1
T2
PD
S(t = TE)
TR
TE
time since
previous
S-E
sequence
~ PD (1  e−TR / T1 ) e−TE / T2
proton
density
prior
regrowth
along z-axis
bright regions:
mxy large at t = TE
current
dephasing
in x-y plane
T1-weighted – Short TE to Eliminate T2 Contribution
TR ~ T1av
to maximize
contrast
S(t~TE)
~ PD (1  e−TR / T1 ) e−TE / T2
proton
density
T1-w
Short TE
to minimize
T2 impact
lipid
prior
regrowth
along z-axis
current
de-phasing
in x-y plane
CSF
T1short
Short-T1 tissues bright
(sprightly brightly)
T1-w – Short TE to Eliminate T2 Contribution
TR
m0
i
TR
ii
TR
mxy(TE)
mz(TR)
TE Short;
TR ~ T1av
lipid
lipid
mz(t)
CSF
CSF
t
Short T1
Bright
TR
500 ms
90°
90° 180°
20 ms
TE
90°
lipid
CSF
T2-w – Long TR to Eliminate T1 Contribution
Long TR:
eliminate
T1 impact
For T2-w
imaging
S(t~TE)
~ PD (1  e−TR / T1 ) e−TE / T2
proton
density
T2-w
TE at mid-T2:
maximize
contrast
lipid
prior
regrowth
along z-axis
current
de-phasing
in x-y plane
CSF
T2 long
Long-T2 tissues bright
T2-w – Long TR to Eliminate T1 Contribution
TE ~ mid-T2 to maximize T2 contrast
i
TR
lipid
Mz(t)
ii
TR
Mxy(TR+)
CSF
lipid
Mxy(TE)
CSF
Mxy(t)
time
TR
90°
2000 ms
90° 180° TE
½ TE
~ 80-100 ms
PD-, T1-, & T2-Weighted Spin-Echo Images
T1-w
T2-w
PD-w
TR
(ms)
mid- (~T1av)
300 − 700
long
1,500 − 3,500
long
1,500 − 3,500
TE
(ms)
short
0 − 25
mid- (~T2av)
60 − 150
short
10 − 25
Bright
short T1
long T2
high PD
lower
best
SNR
good
Magnetic Resonance Imaging:
Mapping the Spatial Distribution
of Spin-Relaxation Rates
of Hydrogen Nuclei
in Soft-Tissue Water and Lipids
Spin-Echo / Spin-Warp in 2D
Sensitive Point (Earliest) Reconstruction
Oscillating Gradient Fields
net field stable in one voxel at a time;
move sensitive point around.
cannot be done with CT.
Open Magnet, 1D Phantom vs. Supercon, 2D Phantom
1-D
y
z
2-D
x
z
B0
B0
x
M(t)
In 2-D MRI, Phase Is Every Bit As Critical as Frequency
2 possible phases for the higher-frequency components
a
b
2 different superimpositions with the longer-λ wave
PD Map of Thin-Slice 2D, 4-Voxel Patient
after 90º pulse drives M(t) into x-y plane…
M(t) = m(1,t) + m(2,t) + m(3,t) + m(4,t)
y
m(1) m(3)
x
z
m(2) m(4)
viewed from feet to head
1
3
2
4
Spin-Echo, Spin-Warp Sequence for 2×2 Matrix
involves 2 spin-echo pulse sequences
first
sequence
second
sequence
RF
slice
selection
180º phase
difference
between rows
Gz
Gy
0º phase
difference
between rows
→π
→0
TR
for S0
TR
for Sπ
Gx
No x- or y-Gradients: Gy = Gx = 0
after phase encoding
Gy
during readout
1
3
1
3
1
3
1
3
2
4
2
4
Gx
S0(1,2,3,4) = 7 (e.g.)
receiver
Q: Are we
cheating here?
F
S0(1,2,3,4)
fLarmor
F [S0(1,2,3,4)]
7 = m(1) + m(2) + m(3) + m(4)
1st S-E Sequence: x-Gradient Only, On During Readout
Gy = 0
1
No
Gy
3
Gx
1
3
1
3
2
4
2
4
receiver
Hint: Is this a
S-E signal…
S0(1,2) = 5
S0(3,4) = 2
F [S0(1,2) + S0(3,4)]
F
S0(1,2) + S0(3,4)
S0(1,2) = 5 = m(1) + m(2)
S0(3,4) = 2 = m(3) + m(4)
2nd S-E Sequence: y-Gradient, then x-Gradient
Gy → 180º Δ . Gx On During Readout
φ+π
φ
1 1
First,
Gy
3 3
Gx
1
3
receiver
2
4
2
4
…or an FID?
Sπ(3,4) = 2
Sπ(1,2) = –3
Sπ(1,2) + Sπ(3,4)
F
F
Sπ(1,2) + Sπ(3,4)
Sπ(1,2) = –3 = m(1)  m(2)
Sπ(3,4) = 2 = m(3)  m(4)
4 Equations in 4 Unknowns
S0(1,2) + S0(3,4)
in phase: add
F S (1,2) = 5 = m(1) + m(2)
0
S0(3,4) = 2 = m(3) + m(4)
180º out of phase: subtract
F S (1,2) = –3 = m(1)  m(2)
π
Sπ(1,2) + Sπ(3,4)
Sπ(3,4) = 2 = m(3)  m(4)
Solution:
m(1) = 1
m(2) = 4
m(3) = 2
m(4) = 0
S0(1,2,3,4)
F
1
3
2
4
7 = m(1) + m(2) + m(3) + m(4) !
Spin-Echo, Spin-Warp….
(e.g., 256×192 matrix)
90o pulse with Gz on selects
z-plane and initiates spin-echo
Gy 192×
Free precession with Gy on
phase-encodes y-position for
current n, Gy(n) = n×Gy(1)
Reading NMR signal with Gx on
frequency-encodes x-position
Sample 512×
Repeat, but with Gy(n+1),
n = 1, 2,…, 192
Creation of an MRI ‘Image’ in 2D k-Space
each line in k-space from data obtained during one Mxy(TE) readout;
different lines for different phase-encode gradient strength
Gphase #96
Amplitude
Gphase #1
Gphase # ̶ 96
frequency (kx)
Creating a 3D Image
a
b
c
Interior of Bronchus
Some Important Topics Not Covered Today
Some Important Topics Not Addressed Here
2D, 3D, 4D Imaging
Inversion Recovery (STIR and FLAIR)
Fast S-E; Gradient Recovery Imaging (GRE, e.g., EPI)
Dynamic Contrast-Agent Enhancement (DCE)
Magnetization Transfer
CNR and Other Quantitative Measures of Image Quality
Parallel-Coil Receive, Transmit; Shim Coils
Magnetic Resonance Angiography (MRA)
Perfusion Imaging
Diffusion Tensor Imaging (DTI)
Functional MRI (f MRI)
Image QA, and ACR Accreditation
MRI/PET, MRI-Elastography, MRI/DSA, MRI/US
Highly Mobile MRI (e.g., for strokes)
Zero-Quantum Imaging
Artificial Intelligence Diagnosis
MRI-Guided Radiation Therapy: Tomorrow!
A Brief Introduction to
Magnetic Resonance Imaging
Anthony Wolbarst, Kevin King, Lisa Lemen,
Ron Price, Nathan Yanasak