Sketch of the MRI Device

Transcription

1 Outline for Today 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

2 Sketch of the MRI Device

3 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

4 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

5 External Magnetic Fields G x ΔB z (x)/δ x x B 0 z

6 x-gradient Magnet Winding for Superconducting Magnet B 0 z One layer of multi-layer x-gradient coil x-gradient db z /dx mt/ m Rise time 0.3 ms (to reach 10 mt/ m) Slew rate mt/m/ms

7 Artifact: Gradient Non-Linearity correctable 7

8 RF Coils B RF : 20 μt Pulse on-time: 3 msec RF power: ~15 kw SAR: ~2W/ kg

9 Parallel RF Receiving Coils for Much Faster Imaging transmit parallel coils in the works

10 Classical NMR in a Voxel

11 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

12 Normal Mode, at v normal relaxation time T f normal f normal f normal

13 A Normal Mode of a 2-D Pendulum f normal

14 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

15 Packet of Protons in Voxel Acts Like Classical m(t) Vector μ μ μ μμμ μ μμμμμμ μμ B z Think classical: m(t)!!!!!

16 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

17 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

18 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.

19 Resonance Energy Transfer when v driving = v normal F res F res F res

20 Resonance and Nutation of a 2-D Pendulum net v resonance power input Precession Nutation v driving = v normal v normal

21 Resonance and Nutation of a Gyroscope net v resonance power input string Cone of Precession Nutation at Resonance

22 Nutation of the Voxel s Magnetization, m(x,t) B RF /B 0 ~ 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!)

23 Free Induction Decay in 1D

24 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

25 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

26 In MRI, the only signal the detector ever sees comes from the set {m(x,t)}, all precessing in the x-y plane!!!

27 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) MHz x = 0 cm

28 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)

29 90º pulse PD(x) FID: 90 &G z for z-slice Selection, then G x for x-encoding/readout of Voxels water ½ water 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

30 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

31 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) MHz x = v L (x) / (γ/2π)g x PD(x) v Real-space representations 0 cm +5 x

32 New: Keep on Going, Closing the Loop S(t) MRI Signal t temporal FT S(ν) Spectrum v 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

33 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 Real-space representation

34 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

35 Herringbone Artifact noise spike during data acquisition Zhou/Gullapalli

36 T2 Spin Relaxation

37 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

38 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

39 Phase Loss: T2 De-coherence of Proton Packets in Voxel m z (t) B 0 m xy (t) of proton packet 39

40 m xy (t) / m xy (0) Exponential T2-Caused De-Phasing of m xy (x,t) in x-y Plane 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

41 T2: Loss of Phase of Voxel Packets in the xy-plane #3: m xy (t) /m xy (0) = e t / T2 41

42 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 brain CSF white matter gray matter edema glioma liver hepatoma muscle adipose

43 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!