Outline for Today 1. 2. 3. Introduction to MRI Quantum NMR and MRI in 0D Magnetization, m(x,t), in a Voxel Proton T1 Spin Relaxation in a Voxel Proton Density MRI in 1D MRI Case Study, and Caveat Sketch of the MRI Device Classical NMR in a Voxel Free Induction Decay in 1D T2 Spin-Relaxation Spin-Echo Reconstruction in 1D Tissue Contrast-Weighting in SE Spin-Echo / Spin-Warp in 2D
Sketch of the MRI Device
Major Components of a Superconducting MRI System fringe fields partially shielded B 0 G x B RF z y x RF transmitter RF receiver y x DAC MIX z Pulse Programmer ADC PACS Operator console
Cylindrical Superconducting Magnets B 0 : 1.5 T, 3.0 T, (7 T) 50 km Nb-Ti wire in Cu Homogeneity: <10 ppm B 0 Shielding: active, passive, Cryogen: 0.1 liter He/y Weight: 4 tons z
External Magnetic Fields G x ΔB z (x)/δ x x B 0 z
x-gradient Magnet Winding for Superconducting Magnet B 0 z One layer of multi-layer x-gradient coil x-gradient db z /dx 20 60 mt/ m Rise time 0.3 ms (to reach 10 mt/ m) Slew rate 50 200 mt/m/ms
Artifact: Gradient Non-Linearity correctable 7
RF Coils B RF : 20 μt Pulse on-time: 3 msec RF power: ~15 kw SAR: ~2W/ kg
Parallel RF Receiving Coils for Much Faster Imaging transmit parallel coils in the works
Classical NMR in a Voxel
The Two Approaches to NMR/MRI quantum state function ψ now this Simple QM, transitions between spin-up, spin-down states Classical Bloch Eqs. for expectation values precession, nutation of voxel magnetization, m(t) f Larmor, m 0, T1 oversimplified f Larmor, T2, k-space exact; from full QM
Normal Mode, at v normal relaxation time T f normal f normal f normal
A Normal Mode of a 2-D Pendulum f normal
Normal Mode Precession about External Gravitational Field J(t) ( dp/dt = F ) v precession dj / dt = τ (torque) J(t): Angular momentum Precession at v normal r ττ mg
Packet of Protons in Voxel Acts Like Classical m(t) Vector μ μ μ μμμ μ μμμμμμ μμ B z Think classical: m(t)!!!!!
Normal Mode Precession of Voxel s m(x,t) in B 0 can be derived rigorously from quantum mechanics z v normal = v Larmor (x) B z (x) longitudinal m z (t) magnetization m(t) = [m x (t) + m y (t) + m z (t)] y x m xy (t) = [m x (t) + m y (t)] transverse magnetization
Bloch Equations of Motion for m(t,x) in B z (x) dj /dt = τ but μ = γ J, so d(μ/γ) /dt = τ Lorentz torque on spins with magnetic moment μ in B z : τ = μ B z Equation of motion becomes: dμ(t) /dt = γ μ(t) B z. Sum/average over all protons in bundle: dm(t)/dt = γ m(t) B z (vector cross product) With T1 relaxation along z-axis: dm(t)/dt = γ m(t) B z + [m(t) m 0 ]ẑ / T1 Expectation Value, m(t), behaves classically
Precession of m(t) about B z (x) at v Larmor = (γ/2π) B z (x) as seen from: Fixed Frame Rotating at v Larmor (x) z as if B 0 = 0 z' m(t) y m(x,t) y' x x' The ponies don t advance when you re on the carousel.
Resonance Energy Transfer when v driving = v normal F res F res F res
Resonance and Nutation of a 2-D Pendulum net v resonance power input Precession Nutation v driving = v normal v normal
Resonance and Nutation of a Gyroscope net v resonance power input string Cone of Precession Nutation at Resonance
Nutation of the Voxel s Magnetization, m(x,t) B RF /B 0 ~ 10 5 10 4 z v Nutation = (γ/2π) B RF m v Larmor = (γ/2π)b z (x) B z (x) m B RF y x precesses at v Lar = (γ/2π) B z (x) nutates at v nut = (γ/2π) B RF (always!) (only when B RF is on!)
Free Induction Decay in 1D
Component amplitude Reminder: Fourier Decomposition of Periodic Signal S(t) ~ ½ + (2/π){sin (2π v 1 t) + ⅓ sin (6π v 1 t) + 1/5 sin (10π v 1 t) + } v 1 Hz (cycles/sec) 1/v1 orthonormal basis Components vectors S(t) t ½ 2/π 2 Fourier spectrum 1 st harmonic fundamental v 1 0 1v 1 3v 1 5v 1 7v 1 v Hz v 3 2/3π 3 rd harmonic 0 v 1 v 3 v 5 v 7 v v 5 5 th FT 0 2/5π (2 /7π) v
FID: m(x,t) for a Single Voxel at x, following a 90º pulse, precessing in the x-y plane Tissue RF RF detect. coil (Faraday) RF m xy (x,t) RF transmit coil
In MRI, the only signal the detector ever sees comes from the set {m(x,t)}, all precessing in the x-y plane!!!
Fourier component MR signal FID: Precession, Reception, Fourier Analysis (single voxel) n.b. detect induced V(t), not power absorption (as before) y m(t) x antenna t (µs) FT v (MHz) 63.87 MHz x = 0 cm
FID: Selecting the z-slice that Contains the x-row B z z y y x G z x B z z z-slice m xy (z,t)
90º pulse PD(x) FID: 90 &G z for z-slice Selection, then G x for x-encoding/readout of Voxels water ½ water 10 5 0 5 10 cm G z on for slice z-selection x t B z (x) y G(t) G z G x G x on for voxel x-encoding/readout t x
Fourier spectrum S(t) Voltages FID: Following 90º Pulse, Precession at v Larmor (x) m(x,t) = m(x,0)e 2πi v(x) t z m xy (x=0) y m xy (x=5) x (μs) Separate RF signals temporal FT FID PD MRI (MHz) (μs) Detected FID signal Larmor Eq. x = v L (x)/(γ/2π)g x
To Summarize What We Have Done So Far with FID follow temporal FT with isomorphism of A(v) to real-space, R(x) A(v) = S(t) e 2πi v(t) t dt t v FT Signal spectrum A(v) 63.87 MHz 63.96 x = v L (x) / (γ/2π)g x PD(x) v Real-space representations 0 cm +5 x
New: Keep on Going, Closing the Loop S(t) MRI Signal t temporal FT S(ν) Spectrum v 63.87 63.93 MHz k x (t) (γ/2π) G x (t) t x = ν L (x) / (γ/2π)g x (t) S(k x ) PD(x) k x spatial FT 1 k-space representation 0 cm +5 x Real-space representation
S(t) S(k x ) Try Going the Other Way. k-space Approach! MRI Signal t necessary for 2D, 3D but also k x (t) (γ/2π) G x (t) t PD(x) = S(x) e +2πi k x x dx k x k x spatial frequency, 1/λ. spatial FT PD(x) k-space representation x 0 cm 0 +5 +5 Real-space representation
During Readout, k x Increases Linearly with t Signal is sampled sequentially 256 or 512 times spaced Δt apart. t n = n Δt is the exact sampling time after G x is turned on. k x (t) for all voxels increases linearly with t while the echo signal is being received and read: k n = [G x γ / 2π] t n. Larger magnitude k-values correspond to greater spatial frequencies! k = k max k 0 = 0 k max SE reads out from k n = k max to k = 0 to k = k max
Herringbone Artifact noise spike during data acquisition Zhou/Gullapalli
T2 Spin Relaxation
T2 Relaxation T2 Relaxation refers to the rate at which the transverse magnetization, m xy (t), and the Echo signal it generates, Decay. T2 relaxation results from T1-Events, plus those from Non-Static, Random, Non-Reversible Proton-Proton Dipole Interactions. Both Contribute to the Rate 1/T2! Dipole interactions: 1) Proton fields overlap, alter v Larmor ; 2) Exchange of spin neither involves an energy transfer
Secular Component of T2 Relaxation Mechanism quasi-static spin-spin interactions not spin flips. H B C C H C H H H O H
Phase Loss: T2 De-coherence of Proton Packets in Voxel m z (t) B 0 m xy (t) of proton packet 39
m xy (t) / m xy (0) Exponential T2-Caused De-Phasing of m xy (x,t) in x-y Plane 1 0 20 40 60 ms t dm xy (t) / dt = (1/T2) m xy (t) m xy (t) /m xy (0) = e t /T2 dm(x,t)/dt = γ m(x,t) B z (x) [m z (x,t) m 0 (x)]ẑ [m x x + m y ŷ] T1 T2
T2: Loss of Phase of Voxel Packets in the xy-plane #3: m xy (t) /m xy (0) = e t / T2 41
Typical T1 and T2 Relaxation Times relaxation rates: 1/T2 ~ 10 (1/ T1) Tissue PD T1, 1 T T1, 1.5T T1, 3T T2 p + /mm 3, rel. (ms) (ms) (ms) (ms) pure H 2 0 1 4000 4000 4000 brain CSF 0.95 2500 2500 2500 200 white matter 0.6 700 800 850 90 gray matter 0.7 800 900 1300 100 edema 1100 110 glioma 930 1000 110 liver 500 40 hepatoma 1100 85 muscle 0.9 700 900 1800 45 adipose 0.95 240 260 60
One Last Member of the Spin-Relaxation Family Tree: T2* fastest T2* FID, Gradient-Echo T2 Spin-Echo not with spin-echo! T1 Discrete spin dephasing, involves ΔE Continuous spin dephasing, no ΔE!