Nuclear Magnetic Resonance Imaging Jeffrey A. Fessler EECS Department The University of Michigan NSS-MIC: Fundamentals of Medical Imaging Oct. 20, 2003 NMR-0
Background Basic physics 4 magnetic fields Bloch equation Excitation Signal equation and k-space Pulse sequences Image reconstruction Summary Outline NMR-1
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 Wüthrich for NMR spectroscopy to determine 3D structure of biological macromolecules 2003. Nobel prize in medicine to Lauterbur and Sir Mansfield! NMR-2
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-3
Disadvantages of NMR Imaging High cost Complicated siting due to large magnetic fields Low sensitivity Little signal from bone Scan durations NMR-4
10 0 Attenuation Considerations Transmission of EM Waves Through 25cm of Soft Tissue 10 0 10 2 10 2 Transmission 10 4 10 6 Transmission 10 4 10 6 10 8 Diagnostic X rays 10 8 1.5 T MRI H 2 O 63 MHz = TV 3 10 10 10 10 10 4 10 2 10 0 Wavelength [Angstroms] 10 2 10 0 10 2 Wavelength [meters] NMR-5
Nuclear Spins The NMR phenomena is present in nuclei having an odd number of protons or neutrons and hence nuclear spin angular momentum. Such nuclei often just called spins Most abundant spin is 1 H, a single proton, in water molecules. NMR concerns the interaction of these spins with magnetic fields: 1. Main field B 0 (static) 2. Local field effects Effect of local orbiting electrons (chemical shift) Differences in magnetic susceptibility 3. RF field B 1 (t) (user-controlled, amplitude-modulated pulse) 4. Gradient fields G(t) (user-controlled, time-varying) NMR-6
1. Main Field B 0 1 H has two energy states: parallel and anti-parallel to main field. Zeeman Diagram: Higher Energy Anti-Parallel z B 0 > 0 E = h γ 2π B 0 B 0 = 0 Lower Energy Parallel Thermal agitation causes random transitions between states. Equilibrium ratio of anti-parallel (n ) to parallel (n + ) nuclei: n = e E/(k bt ) (Boltzmann distribution) n + NMR-7
Precession (Classical Description) Spins do not align exactly parallel or anti-parallel to z. A spin s magnetic moment experiences a torque, causing precession. Lower Energy State (Parallel) z B 0 > 0 M 0 Net Magnetization of Material Higher Energy State (Anti-Parallel) NMR-8
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 1 H At B 0 = 1.5T, f 0 = γ 2π B 0 63 MHz (TV Channel 3) NMR-9
2.1 Local Field Effect: Chemical Shift Orbital electrons surrounding a nuclei perturb the local magnetic field: B eff ( r) = B 0 (1 σ( r)), where σ( r) depends on local electron environment. Because of the Larmor relationship, this field shift causes a shift in the local resonant frequency: ω eff = ω 0 (1 σ). Example: the resonant frequency of 1 H in fat is about 3.5 ppm lower than in water. Chemical shift causes artifacts in usual frequency encoded imaging. NMR-10
2.2 Local Field Effect: Susceptibility Differences in the magnetic susceptibility of different materials, particularly air and water, cause local perturbations of the magnetic field 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-11
3. RF Field B 1 (t) The first step in any NMR experiment is to perturb the spins away from equilibrium using an RF pulse. cosω Amplitude-modulated RF pulse 0 t B circularly polarized in x, y plane: 1 (t) = B 1 sinω 0 t p(t). 0 Spins now precess about the total time-varying magnetic field: B(t) = B 0 + B 1 (t). B 1 B 0 RF excitation puts energy into the system. NMR-12
Non-selective excitation: RF Excitation Example B 1 p(t) t 0 t Time evolution of local magnetization M(t) Laboratory Frame z Rotating Frame z M 0 θ M y x M y x B 1 NMR-13
Relaxation After RF excitation, the magnetization M(t) returns to its equilibrium exponentially, releasing (some of) the energy put in by the excitation. Longitudinal Relaxation Transverse Relaxation Relative Magnetization M z (t) / M 0 1 0.5 0 0.5 T 100 1000 msec 1 M (t) = M (1 e t/t 1) + M (0) e t/t 1 1 z 0 z 0 0.5 1 1.5 2 2.5 3 Time [sec] Relative Magnetization M(t) / M(0) 1 0.8 0.6 M(t) = M(0) e t/t 2 0.4 T 40 100 msec 2 0.2 0 0 0.5 1 1.5 Time [sec] 2 2.5 3 T 1 : spin-lattice time constant. Long T 1 slows imaging :-( T 2 : spin-spin time constant T 1 T 2 M(t) not constant NMR-14
Relaxation of Transverse Magnetization 1 Transverse Relaxation 0.5 M y 0 0.5 1 1 0.5 0 0.5 1 Mx NMR-15
Signal! As spins return to equilibrium, measurable energy is released. By Faraday s law, precessing magnetic moment induces current in RF receive coil: s r (t) = vol b coil ( r) t M( r,t)d r, transverse magnetization: M( r,t) = M X ( r,t) + ım Y ( r,t) Transverse RF coil sensitivity pattern: b coil ( r). Higher spin density or larger ω 0 more signal. Spatial localization? Not much! Small coils have a somewhat localized spatial sensitivity pattern. This property has been exploited recently in sensitivity encoded (SENSE) imaging, to reduce scan times. NMR-16
Free-Induction Decay 1 Free Induction Decay 0.8 0.6 0.4 0.2 s(t) 0 0.2 0.4 0.6 T 2 = 100 msec (Oscillations not to scale) 0.8 1 0 0.2 0.4 0.6 0.8 1 Time t [sec] NMR-17
NMR Spectroscopy Material with 1 H having several relaxation rates and chemical shifts. Received FID signal: s r (t) = N n=1 ρ n : spin density of nth species T 2,n : relaxation time of nth species ω n : resonant frequency of nth species Fourier transform: S r (ω) = ρ n e t/t 2,n e ıω nt, t 0 N 1 ρ n n=1 1 + ı(ω ω n )T 2,n S r (ω) 0 15 0 15 ω [ppm] NMR-18
3. Gradient Fields For a uniform main field B 0, all spins have (almost) the same resonant frequency. For imaging we must encode spin spatial position. Frequency encoding relates spatial location to (temporal) frequency by using gradient coils to induce a field gradient, e.g., along x: a gradient along the x direction: B(x,y,z) = 0 0 B 0 + xg X = (B 0 + xg X ) k. Here, x location becomes encoded through the Larmor relationship: ( ω(x,y,z) = γ B(x,y,z) = γ(b 0 + xg X ) = 2π f 0 + x f ) f, x x = γ 2π G X. B 0 10 4 Gauss f 0 60 MHz G X 1 Gauss/cm f x 6 KHz / cm NMR-19
Effect of gradient fields: B(0, y, z) No Gradient 2 1 y 0 1 2 2 2 1 0 1 2 Z Gradient 1 y 0 1 2 2 2 1 0 1 2 Y Gradient 1 y 0 1 2 2 1 0 1 2 z [cm] NMR-20
Bloch Equation Phenomenological description of time evolution of local magnetization M( r,t): d M dt = M γ B M X i + M Y j T 2 (M Z M 0 ) k T 1 Precession Relaxation Equilibrium Driving term B( r,t) includes Main field B 0 RF field B 1 (t) Field gradients r G(t) = xg X (t) + yg Y (t) + zg Z (t) B( r,t) = B 0 + B 1 (t) + r G(t) k NMR-21
Excitation Considerations NMR-22
Selective (for slice selection) Excitation Non-selective (affects entire volume) Design parameters (functions!): RF amplitude signal B 1 (t) Gradient strengths G(t) NMR-23
Non-Selective Excitation No gradients: excite entire volume B 1 (t) = B 1 (t)( icosω 0 t j sinω 0 t) (circularly polarized) Ignore relaxation Solution to Bloch equation: M rot (t) = R X ( t 0 ) ω 1 (s)ds ω 1 (t) = γb 1 (t) 1 0 0 R X (θ) = 0 cosθ sinθ 0 sinθ cosθ M rot (0) M 0 z θ M y x B 1 Typically θ = 90 or θ = 180. NMR-24
Selective Excitation Ideal slice profile: Before Excitation o After 90 Excitation M z M z M xy z M xy z z z Requires non-zero gradient so the resonant frequencies vary with z. NMR-25
90 Selective Excitation 5 Before Excitation (Equilibrium) 0 5 8 6 4 2 0 2 4 6 8 5 After Ideal Slab Excitation y 0 5 8 6 4 2 0 2 4 6 8 z NMR-26
RF Pulse Design: Small Tip-Angle Approximation Assumptions: θ < 30 (weak RF pulse) M Z (t) M 0 d/dtm Z (t) 0 Constant gradient in z direction (Approximate) solution to Bloch equation: M(z) F 1 {B 1 (t)} f =z γ 2π G Z For a rectangular slice profile we a need sinc-like RF pulse: RF τ 2 τ t NMR-27
Receive Considerations NMR-28
Fundamental Equation of MR After excitation, magnetization continues to evolve via Bloch equation. Solve Bloch equation for RF=0 to find evolution of transverse magnetization: M( r,t) = M X ( r,t) + ım Y ( r,t). Fundamental Equation of MR Imaging M( r,t) = m 0 ( r) e ıω 0t 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 m 0 ( r)? NMR-29
Received Signal From Faraday s law (assuming uniform receive coil sensitivity): s r (t) = vol t M( r,t)d r. Using the Fundamental Equation of MR and converting to baseband: s(t) = s r (t)e ıω 0t = Disregarding T 2 decay: where s(t) = vol φ( r,t) = γ vol m 0 ( r)e t/t 2( r) e ıφ( r,t) d r. m 0 ( r)e ıφ( r,t) d r, t 0 r G(τ)dτ.... The baseband signal is a phase-weighted volume integral of m 0 ( r). NMR-30
Rewriting baseband signal: s(t) = Signal Equation m(x,y,z)e ı2π[xk X(t)+yk Y (t)+zk Z (t)] dxdydz where the k-space trajectory is defined in terms of the gradients: k X (t) = k Y (t) = k Z (t) = γ 2π γ 2π γ 2π t 0 t 0 t 0 G X (τ)dτ G Y (τ)dτ G Z (τ)dτ. MRI measures samples of the FT of the object s magnetization. (Ignoring relaxation and off-resonance effects.) NMR-31
2D Projection-Reconstruction Pulse Sequence RF k-space k Y Gz t=t1 Gx g cos θ θ k X Gy g sin θ start: t=0 cements Signal t t 0 0 t 1 NMR-32
Gridding k-space Gridding k Y k X Followed by inverse 2D FFT to reconstruct image of magnetization. Alternatively, one can apply the filtered backprojection algorithm. NMR-33
Spin-Warp Imaging (Cartesian k-space) RF k y Gz k-space Gx t 1 t 2 t 3 Gy k x Signal start: t=0 0 t1 t 2 t 3 NMR-34
Spiral Fast Imaging Methods Square Spiral Echo Planar NMR-35
Off-Resonance Effects Main field inhomogeneity, susceptibility variations, chemical shift. z M 0 x ω y x y x y x y x y Dephasing causes destructive interference and signal loss. NMR-36
Spin Echo Time Time 180 RF Pulse F S S Time Time F Spin Echo NMR-37
Spin Echo and Signal RF 180 180 90 Signal T 2 Decay 1st Echo 2nd Echo t T* Decay 2 T E τ τ 1 1 τ τ 2 2 NMR-38
2DFT Spin-Echo Spin-Warp Pulse Sequence RF 90 180 Gz Slice Select Gx Gy Spoiler Readout Phase Encode k-space k y t 1 Signal 0 k x T 0 t1 τ t 2 E t 3 t 2 t 3 T E NMR-39
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) = Not a Fourier transform! m 0 ( r)e ıω( r)t e ı2π k(t) r d r NMR-40
Summary NMR images represent the local magnetic environment of 1 H 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 T 1 and T 2 contrast Slides and lecture notes available from: http://www.eecs.umich.edu/ fessler NMR-41
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)... NMR-42