NMR-0Fessler, Univ. of Michigan
Nuclear Magnetic Resonance Imaging
Jeffrey A. Fessler
EECS DepartmentThe University of Michigan
NSS-MIC: Fundamentals of Medical Imaging
Oct. 20, 2003
NMR-1Fessler, Univ. of Michigan
Outline
• Background• Basic physics• 4 magnetic fields• Bloch equation• Excitation• Signal equation and k-space• Pulse sequences• Image reconstruction• Summary
NMR-2Fessler, Univ. of Michigan
History
• 1946. NMR phenomenon discovered independently by◦ Felix Bloch (Stanford)◦ Edward Purcell (Harvard)
• 1952. Nobel prize in physics to Bloch and Purcell• 1966. Richard Ernst and W. Anderson develop Fourier transform
spectroscopy• ... NMR spectroscopy used in physics and chemistry• 1971. Ray Damadian discriminates malignant tumors from normal
tissue by NMR spectroscopy• 1973. Paul Lauterbur and Peter Mansfield
(independently) add field gradients, to make images• 1991. Nobel prize in chemistry to Ernst for NMR spectroscopy• 2002. Nobel prize in chemistry to Kurt Wuthrich for NMR spectro-
scopy to determine 3D structure of biological macromolecules• 2003. Nobel prize in medicine to Lauterbur and Sir Mansfield!
NMR-3Fessler, Univ. of Michigan
Advantages of NMR Imaging
• Nonionizing radiation• Good soft-tissue contrast• High spatial resolution• Flexible user control - many acquisition parameters• Minimal attenuation• Image in arbitrary plane (or volume)• Potential for chemically specified imaging• Flow imaging
NMR-4Fessler, Univ. of Michigan
Disadvantages of NMR Imaging
• High cost• Complicated “siting” due to large magnetic fields• Low sensitivity• Little signal from bone• Scan durations
NMR-5Fessler, Univ. of Michigan
Attenuation Considerations
10−4
10−2
100
10−10
10−8
10−6
10−4
10−2
100
Wavelength [Angstroms]
Tran
smis
sion
Transmission of EM Waves Through 25cm of Soft Tissue
DiagnosticX−rays
10−2
100
102
10−10
10−8
10−6
10−4
10−2
100
Wavelength [meters]
Tran
smis
sion
1.5 T MRI H2O63 MHz = TV 3
NMR-6Fessler, Univ. of Michigan
Nuclear Spins
The NMR phenomena is present in nuclei having an odd number ofprotons or neutrons and hence nuclear spin angular momentum.
Such nuclei often just called “spins”
Most abundant spin is 1H, a single proton, in water molecules.
NMR concerns the interaction of these spins with magnetic fields:1. Main field ~B0 (static)2. Local field effects◦ Effect of local orbiting electrons (chemical shift)◦ Differences in magnetic susceptibility
3. RF field ~B1(t) (user-controlled, amplitude-modulated pulse)4. Gradient fields ~G(t) (user-controlled, time-varying)
NMR-7Fessler, Univ. of Michigan
1. Main Field ~B0
1H has two energy states: parallel and anti-parallel to main field.
Zeeman Diagram:
B0
= 0
B0
Anti-Parallel
> 0
Higher Energy
ParallelLower Energy
z l ∆E = h γ2πB0
Thermal agitation causes random transitions between states.Equilibrium ratio of anti-parallel (n−) to parallel (n+) nuclei:
n−n+
= e−∆E/(kbT ) (Boltzmann distribution)
NMR-8Fessler, Univ. of Michigan
Precession (Classical Description)
Spins do not align exactly parallel or anti-parallel to z.A spin’s magnetic moment experiences a torque, causing precession.
B0
M> 0
(Anti-Parallel)Higher Energy State
(Parallel)Lower Energy State
0z
of MaterialNet Magnetization
NMR-9Fessler, Univ. of Michigan
Larmor Equation
Precession frequency is proportional to (local) field strength:
ω = γ |~B|
• γ is called the gyromagnetic ratio• γ/2π = 42.48 MHz/Tesla for 1H• At B0 = 1.5T, f0 = γ
2πB0 ≈ 63 MHz (TV Channel 3)
NMR-10Fessler, Univ. of Michigan
2.1 Local Field Effect: Chemical Shift
Orbital electrons surrounding a nuclei perturb the local magnetic field:
Beff(~r) = B0(1−σ(~r)),
where σ(~r) depends on local electron environment.
Because of the Larmor relationship, this field shift causes a shift in thelocal resonant frequency:
ωeff = ω0(1−σ).
Example:the resonant frequency of 1H in fat is about 3.5 ppm lower than inwater.
Chemical shift causes artifacts in usual “frequency encoded” imaging.
NMR-11Fessler, Univ. of Michigan
2.2 Local Field Effect: Susceptibility
Differences in the magnetic susceptibility of different materials,particularly air and water, cause local perturbations of the magneticfield that also cause local shifts in the resonance frequency.
This effect can cause undesirable signal loss.
It can also be a source of contrast, such as the BOLD effect in fMRI.
NMR-12Fessler, Univ. of Michigan
3. RF Field ~B1(t)
The first step in any NMR experiment is to perturb the spins awayfrom equilibrium using an RF pulse.
Amplitude-modulated RF pulsecircularly polarized in x,y plane:
~B1(t) = B1
cosω0t−sinω0t
0
p(t).
Spins now precess about the total time-varying magnetic field:~B(t) = ~B0 +~B1(t).
|~B1| � |~B0|
“RF excitation” puts energy into the system.
NMR-13Fessler, Univ. of Michigan
RF Excitation Example
Non-selective excitation:
-
t
6p(t)
t0
|~B1|
Time evolution of local magnetization ~M(t)
Laboratory Frame Rotating Frame
y
z
Mx
θ
z
MM
0
y’
B1
x’
NMR-14Fessler, Univ. of Michigan
Relaxation
After RF excitation, the magnetization ~M(t) returns to its equilibriumexponentially, releasing (some of) the energy put in by the excitation.
0 0.5 1 1.5 2 2.5 3−1
−0.5
0
0.5
1
Time [sec]
Rel
ativ
e M
agne
tizat
ion
Mz(t
) / M
0
Longitudinal Relaxation
Mz(t) = M0(1 − e−t/T1) + Mz(0) e−t/T
1
T1 ∼ 100−1000 msec
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
Time [sec]
Rel
ativ
e M
agne
tizat
ion
|M(t
)| / |
M(0
)|
Transverse Relaxation
|M(t)| = |M(0)| e−t/T2
T2 ∼ 40−100 msec
• T1: spin-lattice time constant. Long T1 slows imaging :-(• T2: spin-spin time constant• T1 6= T2 ⇒ |~M(t)| not constant
NMR-15Fessler, Univ. of Michigan
Relaxation of Transverse Magnetization
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
Mx
My
Transverse Relaxation
NMR-16Fessler, Univ. of Michigan
Signal!
As spins return to equilibrium, measurable energy is released.
By Faraday’s law, precessing magnetic moment induces currentin RF receive coil:
sr(t) =∫
volbcoil(~r)
∂∂t
M(~r, t)d~r,
• transverse magnetization: M(~r, t)4= MX(~r, t)+ ıMY(~r, t)
• Transverse RF coil sensitivity pattern: bcoil(~r).• Higher spin density or larger ω0 ⇒ more signal.
Spatial localization? Not much!
Small coils have a somewhat localized spatial sensitivity pattern. Thisproperty has been exploited recently in “sensitivity encoded” (SENSE)imaging, to reduce scan times.
NMR-17Fessler, Univ. of Michigan
Free-Induction Decay
0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
s(t)
Time t [sec]
Free Induction Decay
(Oscillations not to scale)
T2 = 100 msec
NMR-18Fessler, Univ. of Michigan
NMR Spectroscopy
Material with 1H having several relaxation rates and chemical shifts.
Received FID signal: sr(t) =N
∑n=1
ρn e−t/T2,n eıωnt , t ≥ 0
• ρn: spin density of nth species• T2,n: relaxation time of nth species• ωn: resonant frequency of nth species
Fourier transform: Sr(ω) =N
∑n=1
ρn1
1+ ı(ω−ωn)T2,n
−15 0 150
∆ω [ppm]
|Sr(ω
)|
NMR-19Fessler, Univ. of Michigan
3. Gradient Fields
For a uniform main field ~B0, all spins have (almost) the same resonantfrequency. For imaging we must “encode” spin spatial position.
Frequency encoding relates spatial location to (temporal) frequencyby using gradient coils to induce a field gradient, e.g., along x: agradient along the x direction:
~B(x,y,z) =
00
B0 + xGX
= (B0 + xGX)~k.
Here, x location becomes “encoded” through the Larmor relationship:
ω(x,y,z) = γ|~B(x,y,z)| = γ(B0 + xGX) = 2π(
f0 + x∂ f∂x
)
,∂ f∂x
=γ
2πGX.
• B0 ≈ 104 Gauss ⇒ f0 ≈ 60 MHz• GX ≈ 1 Gauss/cm ⇒ ∂ f
∂x ≈ 6 KHz / cm
NMR-20Fessler, Univ. of Michigan
Effect of gradient fields: ~B(0,y,z)
−2 −1 0 1 2−2
−1
0
1
2
y
No Gradient
−2 −1 0 1 2−2
−1
0
1
2
yZ Gradient
−2 −1 0 1 2−2
−1
0
1
2
z [cm]
y
Y Gradient
NMR-21Fessler, Univ. of Michigan
Bloch Equation
Phenomenological description of time evolution of local magnetization~M(~r, t):
d ~Mdt
= ~M× γ~B −MX
~i+MY~j
T2−
(MZ −M0)~kT1
Precession ↑Relaxation ↑ ↑Equilibrium ↑
Driving term ~B(~r, t) includes• Main field ~B0
• RF field ~B1(t)
• Field gradients~r · ~G(t) = xGX(t)+ yGY(t)+ zGZ(t)
~B(~r, t) = ~B0 +~B1(t)+~r · ~G(t)~k
NMR-22Fessler, Univ. of Michigan
Excitation Considerations
NMR-23Fessler, Univ. of Michigan
Excitation
• Selective (for slice selection)• Non-selective (affects entire volume)
Design parameters (functions!):• RF amplitude signal B1(t)
• Gradient strengths ~G(t)
NMR-24Fessler, Univ. of Michigan
Non-Selective Excitation
• No gradients: excite entire volume• ~B1(t) = B1(t)(~icosω0t −~j sinω0t) (circularly polarized)• Ignore relaxation
Solution to Bloch equation:
~Mrot(t) = RX
(
∫ t
0ω1(s)ds
)
~Mrot(0)
ω1(t) = γB1(t)
RX(θ) =
1 0 00 cosθ sinθ0 −sinθ cosθ
θ
z
MM
0
y’
B1
x’
Typically θ = 90◦ or θ = 180◦.
NMR-25Fessler, Univ. of Michigan
Selective Excitation
Ideal slice profile:
z z
z zM M
MM
Before Excitation After 90 Excitationo
xy xy
zz
Requires non-zero gradient so the resonant frequencies vary with z.
NMR-26Fessler, Univ. of Michigan
90◦ Selective Excitation
−8 −6 −4 −2 0 2 4 6 8−5
0
5Before Excitation (Equilibrium)
−8 −6 −4 −2 0 2 4 6 8−5
0
5
z
y
After Ideal Slab Excitation
NMR-27Fessler, Univ. of Michigan
RF Pulse Design: Small Tip-Angle Approximation
Assumptions:• θ < 30◦ (weak RF pulse)• MZ(t) ≈ M0
• d/dtMZ(t) ≈ 0
• Constant gradient in z direction
(Approximate) solution to Bloch equation:
M(z) ∝ F1{B1(t)}∣
∣
∣
f=z γ2πGZ
For a rectangular slice profilewe a need sinc-like RF pulse:
RF
τ tτ2
NMR-28Fessler, Univ. of Michigan
Receive Considerations
NMR-29Fessler, Univ. of Michigan
Fundamental Equation of MR
After excitation, magnetization continues to evolve via Bloch equation.
Solve Bloch equation for RF=0 to find evolution oftransverse magnetization: M(~r, t) = MX(~r, t)+ ıMY(~r, t).
Fundamental Equation of MR Imaging
M(~r, t) = m0(~r) e−ıω0t e−t/T2(~r) e−ıφ(~r,t)
Post-excitation ↑Precession ↑Relaxation ↑User-controlled phase term ↑
φ(~r, t) = γ∫ t
0~r · ~G(τ)dτ.
• Key design parameters are gradients: ~G(t).• How to manipulate the gradients to make an image of m0(~r)?
NMR-30Fessler, Univ. of Michigan
Received Signal
From Faraday’s law (assuming uniform receive coil sensitivity):
sr(t) =∫
vol
∂∂t
M(~r, t)d~r.
Using the Fundamental Equation of MR and converting to baseband:
s(t) = sr(t)e−ıω0t =∫
volm0(~r)e−t/T2(~r) e−ıφ(~r,t) d~r.
Disregarding T2 decay:
s(t) =∫
volm0(~r)e−ıφ(~r,t) d~r,
whereφ(~r, t) = γ
∫ t
0~r · ~G(τ)dτ.
... The baseband signal is a phase-weighted volume integral of m0(~r).
NMR-31Fessler, Univ. of Michigan
Signal Equation
Rewriting baseband signal:
s(t) =∫ ∫ ∫
m(x,y,z)e−ı2π[xkX(t)+ykY(t)+zkZ(t)] dxdydz
where the “k-space trajectory” is defined in terms of the gradients:
kX(t) =γ
2π
∫ t
0GX(τ)dτ
kY(t) =γ
2π
∫ t
0GY(τ)dτ
kZ(t) =γ
2π
∫ t
0GZ(τ)dτ.
⇒ MRI measures samples of the FT of the object’s magnetization.
(Ignoring relaxation and off-resonance effects.)
NMR-32Fessler, Univ. of Michigan
2D Projection-Reconstruction Pulse Sequence
Gz
RF
start: t=0
t=t
k-space
θ
1
Gxg cos θ
Gy
Signal
g sin θ
t1
t0
0
tPSfrag replacements
kX
kY
NMR-33Fessler, Univ. of Michigan
Gridding
Griddingk-space
Yk
kX
Followed by inverse 2D FFT to reconstruct image of magnetization.
Alternatively, one can apply the filtered backprojection algorithm.
NMR-34Fessler, Univ. of Michigan
Spin-Warp Imaging (Cartesian k-space)
t3t21t
1t t3t20
start: t=0
k-space
k
ky
x
Signal
Gy
Gx
RF
Gz
NMR-35Fessler, Univ. of Michigan
Fast Imaging Methods
Square SpiralSpiral Echo Planar
NMR-36Fessler, Univ. of Michigan
Off-Resonance Effects
Main field inhomogeneity, susceptibility variations, chemical shift.
z
M0
yx
yx
yx
ω
yx
yx
Dephasing causes destructive interference and signal loss.
NMR-37Fessler, Univ. of Michigan
Spin Echo
Time
Time Time
Time
F
S
180 RFPulse
Spin EchoF
S
NMR-38Fessler, Univ. of Michigan
Spin Echo and Signal
90
180
τ τ
Signal
RF
T* Decay2
TE
τ τ2 21 1
180
T Decay2 2nd Echo1st Echo
t
NMR-39Fessler, Univ. of Michigan
2DFT Spin-Echo Spin-Warp Pulse Sequence
1t t2 t3
1t
t2 t3
90RF
Gz
Gx
Gy
Signal
TE
Spoiler
180
0 τ
Readout
Phase Encode
Slice Select
kx
kyk-space
0
TE
NMR-40Fessler, Univ. of Michigan
Image Reconstruction Options
• Cartesian sampling patterns: inverse FFT• Non-Cartesian sampling patterns:
◦ gridding then inverse FFT◦ iterative reconstruction (regularized least squares)
• Iterative reconstruction with field inhomogeneity compensation:
s(t) =∫
m0(~r)e−ıω(~r)t e−ı2π~k(t)·~r d~r
Not a Fourier transform!
NMR-41Fessler, Univ. of Michigan
Summary
NMR images represent the local magnetic environment of 1H protons• Large main field establishes resonance• RF pulses and gradient fields perturb magnetization and phases• Measured signal related to Fourier transform of object• Reconstruction can be simple: inverse FFT• Time-varying gradients control k-space trajectory• Timing parameters control T1 and T2 contrast
Slides and lecture notes available from:http://www.eecs.umich.edu/∼fessler
NMR-42Fessler, Univ. of Michigan
Some of Many Omitted Topics• Spectroscopic imaging• Dynamic imaging• Functional imaging• Blood flow imaging• Multi-slice imaging• Main-field hardware (shimming, open magnets, ...)• RF coil design• Gradient nonlinearities• Signal to noise considerations• MR contrast agents• Interventional MR• Gating (cardiac and respiratory)• ...