A Brief Introduction to Magnetic Resonance Imaging. Anthony Wolbarst, Kevin King, Lisa Lemen, Ron Price, Nathan Yanasak

Transcription

1 A Brief Introduction to Magnetic Resonance Imaging Anthony Wolbarst, Kevin King, Lisa Lemen, Ron Price, Nathan Yanasak

2 Outline for Today 1) Introduction Wolbarst MRI Case Study; Caveat!!! Quasi-Quantum Nuclear Magnetic Resonance 2) Net magnetization, m(x,t), of the voxel at x Lemen T1 Spin-Relaxation of m(x,t), T1 MRI of the 1D patient Sketch of the MRI Device 3) Classical Approach to NMR Yanasak FID Image Reconstruction, k-space 4) Spin-Echo Reconstruction Price T2 Spin-Relaxation T1-w, T2-w, and PD-w S-MRI Spin-Echo / Spin-Warp in 2D Some Important Topics Not Addressed Here

3 Magnetic Resonance Imaging: Mapping the Spatial Distribution of Spin-Relaxation Rates of Hydrogen Nuclei in Soft-Tissue Water and Lipids

4 µ Part 1: Quasi Quantum NMR and MRI Introduction MRI Case Study; Caveat!!! Quasi-Quantum NMR and MRI N S J

5 Introduction

6 Medical Imaging: differential interactions of probes with different tissues Beam of probes : particles or waves Differential transmission, reflection, emission, etc., of probes Image receptor

7 Forms of Contrast Between Good and Bad Apples produced and detected thru different bio/chemico/physical processes The Future: Contrast Mechanisms! Color Smell Taste Texture Holes etc.

8 How Imaging Modalities Detect Tissue Contrast Modality Probe / Signal Detector Source of Contrast: differences in Planar R/F X-ray CT Nuc Med, SPECT; PET X-rays passing through body Gamma-rays from body; 511 kev II+CCD, AMFPI; GdO, etc., array NaI single crystal; multiple NaI; LSO array µ( ρ, Z, kvp) ds Radiopharmaceutical uptake, concentration (e.g., 99m Tc), emission US MHz sound Piezoelectric transducer ρ, κ, µ US s MRI Magnet, RF AM radio receiver T1, T2, PD, [O], blood flow, water diffusion, chemical shift

9 CT vs. MRI Soft Tissue Contrast CT MRI Posterior reversible encephalopathy syndrome (PRES), edematous changes

10 Magnetic Resonance Imaging Multiple types of contrast, created through and reflecting different aspects of soft tissue biophysics. Selection of contrast type with pulse sequences Most reflect rotations, flow, etc., of water/lipid molecules; yield distinct, unique maps of anatomy, physiology No ionizing radiation Risks from intense magnetic fields, RF power Expensive Technology complex, challenging to learn; Much of biophysics little understood. But.

11 MRI Case Study; and Caveat!!! with T1, FLAIR, MRS, DTI, or f MRI contrast and MR-guided biopsy studies of a glioma

12 Case Study 57 year old medical physicist has had daily headaches for several months. Responds to ½ Advil. Physical examination 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 more information. Principal concern: Vision for reading.

13 Lesion: Right Posterior Temporo-Occipital Region, adjacent to occipital horn of right lateral ventricle T1-w FLAIR No enhancement with Gd contrast. f MRI visual stimulus DTI

14 Chemical Shift and Non-invasive MRS acetic acid COOH CH ppm Astrocytoma

15 MRI-Guided Needle Biopsy Grade 1-2 Astrocytoma with scattered cellular pleomorphism and nuclear atypia

16 Caveat!!! Extreme Danger

17 How Not to Clean a Magnet strong gradient fields outside bore

18 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

19 MRI Safety First and Foremost, Restrict 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 O 2 bottle, IV pole scissors, pen, phone, laptop tool, tool chest wheelchair, gurney ax, fire extinguisher gun, handcuffs cleaning bucket, mop

20 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

21 Two Approaches to Proton NMR/MRI (incompatible!) quantum spin-state function for hydrogen nucleus start with this ψ Simple QM, transitions between spin-up-, spin-down states Classical Bloch Eqs. expectation values in voxel precession, nutation of net voxel magnetization, m(t) f Larmor, m 0, T1 oversimplified; like Bohr atom f Larmor, T2, 2D, k-space exact, for expectation values; from full QM

22 Open MRI Magnet principal magnetic field B 0 defines z-axis z S B 0 z B 0 x N

23 Moving Charge Produces Magnetic Field (e.g., mag. dipole); Magnetic Dipole Tends to Align in External Field µ N e B z µ S N S T mechanical relaxation J simplified QM picture proton

24 Energy to Flip Over Needle with Magnetic Moment μ in B z B z along z-axis z θ N µ º E = μ B z = μ B z cos θ ΔE 180º = ± 2 μ B z

25 Nuclear Zeeman Splitting for Proton: ΔE = ± 2 μ z B z μ points only along or against z South N E(B z ) High-energy state μ z B z 0 ΔE Low-energy state + μ z North B z

26 Swept-frequency NMR in a tissue voxel

27 Nuclear Magnetic Resonance proton ΔE 180 = 2 μ z B z photon E = hf Orientational Energy hf μ z +μ z spin flip hf Larmor (B z ) = 2 μ z B z 2πf Larmor ω = γ B z for water protons, f Larmor, H O = MHz T B z T f Larmor [MHz] MHz MHz H-1 P Energy (μev ) B z (x) = f Larmor (x) / B z (T)

28 I-1. For protons, f Larmor = MHz at 1 T. What is it at 1.5 T? 8% a MHz 2% 84% 3% 2% b MHz c MHz d MHz e. Cannot be determined from this info.

29 SAMs Q: I-1. For protons, f Larmor = MHz at 1 T. What is it at 1.5 T? (a) MHz (b) MHz (c) MHz (d) MHz (e) Cannot be determined from this info. Answer: (c). f Larmor = B z = MHz = MHz/T 1.5 T Ref: Medical Imaging: Essentials for Physicians, A.B. Wolbarst, P. Capasso, and A. Wyant. Wiley-Blackwell (2013), p (MPP booth)

30 Detected RF power NMR Gedanken-Experiment on Water, 1T monochromatic RF power absorption at (and only at) f Larmor, H O Antenna Transmitter H 2 O 1T principal z Antenna 2 magnet Power meter sweep the RF f Resonance at 1T: MHz Transmitter f (MHz)

31 I-2. If the field homogeneity of a 1.5 T scanner is measured to be one part per million (1 ppm), what will be the approximate spread in resonant frequencies? 16% a. 1.5 Hz 6% b MHz 8% c khz 11% d MHz 60% e Hz

32 SAMs Q: I-2. If the field homogeneity of a 1.5 T scanner is measured to be one part per million (ppm), what will be the approximate spread in resonant frequencies? (a) 1.5 Hz (b) MHz (c) khz (d) MHz (e) Hz Answer: (e). Δf Larmor = ( MHz/T) (1.5 T) (10 6 ) = Hz Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 319.

33 Proton-Density (PD) MRI in the 1D patient

34 z 1D Phantom in Principal B 0 and Gradient G x B z (x) B 0 = G x x G x db z (x)/dx Principal magnet Gradient magnet x = [B z (x) B 0 ] / G x B z (x) T T T B cm +10 cm x x (1.002T 0.998T) / 0.20m = 20 mt/m

35 1D PD-MRI: f Larmor = f Larmor [B z (x)] z 0. sweep the RF f B(x) x cm +10 cm 2. G x T T T x x = [B z (x) B 0 ] / G x f (x) MHz 1. B z (x) = f Larmor / PD MR Image x

36 Summary: PD MRI on 1D Patient A( f ) MHz x cm 1. f Larmor of NMR Peak Amplitude of NMR Peak, A( f ) B z (x) PD(x) 3. Voxel Position, x Pixel Brightness, PD 4'. MRI

37 Proton-Density MRI contrast from differences in PD

38 Two MRI Motion Artifacts respiration aortic pulsation

39 I-3. The net magnetic field B z (x) is measured to be T at x = 0.05 m and T at +5 cm. What is G x? 15% a T/m 29% 38% 13% 5% b T/m c. 20 mt/m d. 0.2 T/cm e. 0.2 T/m

40 SAMs Q: I-3. The net magnetic field B z (x) is measured to be T at x = 0.05 m and T at +5 cm. What is G x? (a) 0.01 T/m (b) 0.02 T/m (c) 20 mt/m (d) 0.2 T/cm (e) 0.2 T/m Answer: (c). 20 mt/m. ΔB z (x) / Δx = 2 mt / 0.1 m Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 318.

41 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 44

42 Net magnetization, m(x,t), of the voxel at x

43 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 B z > 0 m(x,t) Gentle agitation (noise energy)

44 Voxel s MRI Signal Proportional to Its Magnetization, m(x,t) i) What is the magnitude of voxel magnetization at dynamic thermal equilibrium, m 0? ii) How long does it take to get there (T1)? iii) What is the mechanism?

45 Switching B z (x) On at t = 0 Induces Magnetization m(x,t) voxel at x from protons in voxel at x t = 0 B z (x) = 0 m(x) = 0 t >> 0 B z > 0 m(x,t)

46 Filling Four Energy Levels of Marbles vs. Noise Level equilibrium from battle between energy and entropy (all slippery; black balls denser) Shaking (Noise) Energy: too much too little just right

47 Magnetization in Voxel at x, under Dynamic Equilibrium: m 0 (x,t) = [N (x,t) N + (x,t)] μ, t B z 0 tesla collective magnetic field produced by all the protons in it μ (Boltzmann) m T or t = 0+*»1.5 T ~ 0 N μ z 1.5 T t >> N μ z * after abruptly turning on B z, or after a 90º pulse.

48 m 0 (x) and Signal from it Increase with B 0 p.s., trade-off: SNR, resolution, acquisition-time 1.5 T 7 T (+Gd)

49 T1 Spin-Relaxation of m(x,t)

50 Voxel s MRI Signal Proportional to Its Magnetization, m(x,t) i) What is the magnitude of voxel magnetization at thermal equilibrium, m 0? ii) How long does it take to get there (T1)? iii) What is the mechanism?

51 Disturbed System Moving toward Up-Down Equilibrium energy imparted to individual down spins from outside m(0) m(1) m(2) t = 0 ms t = 1 ms t = 2 ms m = 0 m = 2μ m = 4μ spin tickled down by f Larmor component of magnetic noise; emits RF photon, phonon. a few can be kicked up, as well.

52 n(t) n(0) Decay of Tc-99m Sample to Final Value, n( ) = 0 1 ½ 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) e λt dn/dt ~ [n(t) n( )] hr n( ) scint. decay (t) pop. growth (t) photon atten. (x) tracer conc. (t) cell killing (D) ultrasound atten. (x)

53 m z (t) Return of Voxel s m z (t) to Final Value, m z ( ) m 0 [m 0 m z (t)] m z ( ) m 0 t ms rate d [m 0 m z (t)] / dt = (1/T1) [m 0 m z (t)] m z (t)/m 0 = 1 e t / T1 Actually, dynamic equilibrium more complicated: Spins tickled down and up!

54 T1 MRI of the 1D Patient

55 T1 for 2 Voxels: Lipid vs. CSF both at f Larmor of H 2 O for local field lipid CSF RF transmit 10 0 cm +10 RF receive, FT T PD(x) f (x)

56 Recovery of m z (t) over Time, from m z (TR i ) Measurements longitudinal component of net voxel magnetization, m z (t) t = 0+ t = TR cm 1. First, do lipid at x = Strong burst of RF over range MHz ± 250 Hz causes t = TR 2 t = TR 3 N ~ N + 3. After TR 1 delay, sweep through f Larmor, record peak s amplitude. 4. Repeat for several other TR i values. 5. Plot.

57 m z (TR)/m 0 = (1 e TR / T1 ) curve-fit for set of {TR i } for voxel at x = 0 m z (t)/m TR 1 TR 2 TR 3 T1 TR 4 lipid: T1 ~ 250 ms t read (ms) T1 T1 MR image by convention for T1-imaging: pixel white for fast-t1 tissue Spritely Brightly

58 m z (x,t)/m 0 (x) = (1 e t / T1( x) ) repeat for CSF voxel at x = 5 cm, f Larmor = MHz m z (t) / m 0 Lipid: T1 = 250 ms x = 0 CSF: T1 = 2,000 ms x = +5 cm t T1 MR image T1 lipid: ~ 250 ms CSF: ~ 2,000 ms

59 T1-w MR Image CSF T1 ~ 2000 ms sub-cut. fat T1 ~ 260 ms Gd

60 Approximate Relaxation Times of Various Tissues Tissue PD T1, 1 T T1, 1.5 T T1, 3 T T2 p + /mm 3, rel. (ms) (ms) (ms) (ms) pure H brain CSF white matter gray matter edema glioma liver hepatoma muscle adipose

61 Voxel s MRI Signal Proportional to Its Magnetization, m(x,t) i) What is the magnitude of voxel magnetization at thermal equilibrium, m 0? ii) How long does it take to get there (T1)? iii) What is the mechanism?

62 In MRI, the only thing a proton is ever aware of, or reacts to, is the local magnetic field, B local (t).!!! But the source of f Larmor (B local (t)) can be either external (B RF ) or internal (e.g., moving partner)!!!!! 65

63 Random B RF (t) at f Larmor Causes T1-Transitions variations in proton magnetic dipole-dipole interactions H H O H T H O 1.5 Å each water proton produces magnetic field fluctuations of various frequencies, including local f Larmor, at its partner proton

64 Restrictions on Rotations of Water Molecules more or less hydration free bound layer

65 Probability / relative number Rotational Power Spectra, J( f ), of Water for several water populations Water on large biomolecule T1 spin flips f Larmor Water on small molecule Pure water Magnetic noise (water tumbling, etc.) frequency Slow motion Rapid

66 Relaxation Time, T1 Average Water T1 Determined by Noise Power Spectrum, J( f Larmor ) T1 Water on intermediate sized molecule Water on large, slow biomolecule Pure water 1.5 T f Larmor Magnetic noise (water tumbling, etc.) frequency Slow motions Rapid

67 MRI: Water, Lipids of Soft Tissues their Hydrogen Nuclei. Spin-Relaxation Times (T1, T2) their Spatial Distributions!

68 Sketch of the MRI Device

69 Damadian s Indominable, 1977 (Smithsonian) Nobel Prize, 2003 Paul Lauterbur Peter Mansfield 72

70 Major Components of a Superconducting MRI System y x z G x B 0 B RF RF RF fringe fields partially shielded MIX DAC ADC Operator console PACS

71 B 0 z MRI Magnets x y Open (most of this presentation) electromagnetic or permanent Superconducting e.g., niobium-titanium wire B 0 : < 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 B 0

72 Three External Magnetic Fields in Open Magnet MRI z B 0, G z, and G x all point along z! B RF B 0 G z ΔB z (z) /Δ z G x ΔB z (x)/δ x x

73 x-gradient Magnet Winding for Superconducting Magnet B 0 z one layer of 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

74 Artifact: Gradient Non-Linearity 77

75 II-4 Gradient coils affect the strength of the magnetic field that, at all locations in space, point : a. along the x direction of the scanner b. along the y direction of 40% the scanner c. along the z direction of the scanner d. along all directions of the scanner 5% 3% not enough information given here to answer a. b. c. d. 52%

76 SAMs Q: II-4 Gradient coils affect the strength of the magnetic field that, at all locations in space, point : (a) along the x direction of the scanner (b) along the y direction of the scanner (c) along the z direction of the scanner (d) along all directions of the scanner (e ) not enough information given here to answer Answer: (c). Along the principal magnetic field, B 0, hence the z-axis Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 316.

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

78 Parallel RF Receiving Coils for Much Faster Imaging transmit coils coming

79 II-5 The acronym used to describe the amount of RF radiation uptake in patient tissue is: 4% a. REM 90% 1% 3% 2% b. SAR c. CTDI d. STIR e. RARE

80 SAMs Q: II-5. The acronym used to describe the amount of RF radiation uptake in patient tissue is: (a) REM (b) SAR (c) CTDI (d) STIR (e) RARE Answer: (b). Specific Absorption Rate Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 349.

81 T1W MRI Part 3: Classical MRI Classical Approach to NMR FID Image Reconstruction, k-space

82 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

83 The Two Approaches to NMR/MRI (incompatible!) 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

84 Normal Modes, Resonance, Precession, Nutation

85 Normal Mode, at f normal f normal f normal f normal

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

87 Normal Mode Precession about External Gravitational Field J(t) J(t): Angular Momentum With torque, τ: dj / dt = τ Equation of Motion f precession Angular acceleration is just like dp / dt = F but here, J(t) changes direction only: Precession at f normal r τ mg

88 Normal Mode Precession of Voxel m(x,t) in Magnetic Field can be derived rigorously from quantum mechanics z f normal = f 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

89 Classical Bloch Equations of Motion for m(x,t) in B 0 Equation of motion for spinning body d J /dt = τ d( μ / γ) / dt = τ (external torque) recall: μ = γ J Lorentz torque on spins at x with magnetic moment μ in B z (x): τ = μ B z (x) (vector cross product) Equation of motion becomes: d μ(x,t) /dt = γ μ(x,t) B z (x). Sum/average over all protons in voxel: d m(x,t) /dt = γ m(x,t) B z (x) Expectation Value, m(x,t), behaves classically With T1 relaxation along z-axis: d m(x,t) /dt = γ m(x,t) B z (x) + [ m(x,t) m 0 (x)]ẑ / T1

90 Precession of m(x,t) about B 0 at f Larmor = (γ / 2π) B z (x) as seen from a frame that is Fixed z Rotating at f Larmor (x) as if B 0 = 0 z' m(x,t) x m(x,t) x' y y' The ponies don t advance when you re on the carousel; It s as if B 0 = 0!

91 Resonance Energy Transfer when f driving = f normal F res F res F res

92 Normal Mode and Resonance of a 2-D Pendulum Precession Nutation f driving = f normal f normal

93 And Now for Something Completely Different: Nutation string Cone of Precession Nutation at Resonance

94 Nutation of the Voxel s Magnetization, m(x,t) B RF /B 0 ~ z f Nutation = (γ/2π) B RF m f Larmor = (γ/2π) B 0 B z (x) m B RF y x precesses at f Lar = (γ /2π) B z (x) nutates at f nut = (γ /2π) B RF (always!) (only when B RF is on!)

95 Free Induction Decay (FID) in a voxel, without the decay e.g., very slow T1 relaxation

96 m(x,t) for a Single Voxel at x and Precessing in the x-y Plane following a 90º pulse Tissue RF RF detect. coil (Faraday) RF m xy (x,t) RF transmit coil

97 Fourier component MR signal FID Precession, Reception, Fourier Analysis (single voxel) n.b. measure induced V(t), not power absorption (as before) y m(t) x antenna t (µs) F f (MHz) MHz x = 0 cm

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

99 III-6. The rate of magnetization nutation depends upon: a. the magnitude of the magnetization, m(t) 69% b. the strength of the principal magnetic field, B 0 c. strength of the x-gradient field, x G x d. the strength of the RF magnetic field, B RF (t) e. T1 12% 8% 8% 3% a. b. c. d. e.

100 SAMs Q: III-6. The rate of magnetization nutation depends upon: a) the magnitude of the magnetization, m(t) b) the strength of the principal magnetic field, B 0 c) strength of the x-gradient field, x G x d) the strength of the RF magnetic field, B RF (t) e) T1 Answer: (d). Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 328.

101 Untangling the FID RF Signal from Two Voxels: temporal Fourier series approach

102 Wave Interference in Time or Space positive (in-phase) interference negative interference

103 Fourier components Composite beat signal Separate signals Wave Interference and Spectral Analysis time or space interference t (sec) x (cm) decomposition via Fourier analysis, F f (Hz) k (cm 1 )

104 Component amplitude Fourier Decomposition of Periodic Temporal Signal S(t) ~ ½ + (2/π){sin (2π f 1 t) + ⅓ sin (6π f 1 t) + 1/5 sin (10π f 1 t) + } fundamental: f 1 Hz (cycles/sec) orthonormal basis vectors Components S(t) 1/f 1 t 22 /π spectrum ½ 1 st harmonic fundamental f 1 0 1f 1 3f 1 5f 1 7f 1 f Hz f 3 2 /3π 3 rd harmonic 0 f 1 f 3 f 5 f 7 f 5 th f 5 F 0 2 /5π (2 /7π) f

105 FID imaging of the two-voxel 1D patient

106 f Larmor (x) = (γ/2π) (x G x ) (in rotating frame) e.g., FID from 2 water slices, at x = 0 and 5 cm; little decay! B RF RF receive RF transmit 10 0 cm x B(x) T T T MHz

107 G x 90º pulse PD(x) FID Following Wide-Bandwidth 90 RF Pulse covers f Larmor (x) for all N voxels while G x is being applied! * water ½ water x cm t z B 0 y * while gradient is on t x

108 Fourier spectrum S(t ) Voltages FID: After Wide-Band RF 90º Pulse Centered on f Larmor (B 0 ) n.b.: m(x,t) = m(x,0)e 2πi f (x) t z m xy (x=0) y m xy (x=5) x (μs) Separate RF signals Temporal F (μs) Detected FID signal (MHz) FID PD MRI x = f L (x) / (γ/2π) G x

109 R(x) S(t ) A( f ) To Summarize What We Have Done So Far with FID follow temporal F with isomorphism of A( f ) to real-space, R(x) MRI signal t t f F Signal spectrum A( f ) = S(t) e 2πi f (t) t dt MHz f x = 2π f (x) / γg x Real-space representations 0 cm +5 x

110 FID Imaging via k-space; Spatial FT

111 R(x) S(t ) A( f ) Again, What We Have Done So Far: Detected FID signal Temporal: t f F Signal spectrum t sec MHz f x = 2π f Larmor (x) / γ G x x 0 +5 cm Real-space representation

112 Component amplitude Now Consider 1D Spatial Fourier Analysis fundamental: k 1 (cycles / mm) ½ 2/π k 1 3k 1 5k 1 7k 1 k x 2/3π S(x) 1 1/k 1 x F 0 2/5π 0 k 1 k 3 k 5 k 7 S(x) ~ ½ + (2/π) [sin (2π k 1 x) + ⅓ sin (6π k 1 x) + 1/5 sin (10π k 1 x) + ]

113 α(k) R(x) S(t ) A( f ) Experiment: Continue on with a FT from x- to k x -Space Detected FID signal Temporal: t f F Signal spectrum t sec MHz f α(k) = R(x) e 2πi k x x dx x = 2π f Larmor (x) / γ G x k k-space representation F Spatial: k x x x 0 +5 cm Real-space representation

114 α(k) R(x) S(t ) A( f ) And Complete the Loop Linking t-, f-, x- and k x -Spaces Detected FID signal Temporal: t f F Signal spectrum t sec MHz f k x (t) γ G x t /2π x = 2π f Larmor (x) / γ G x k k-space representation F Spatial: k x x x 0 +5 cm Real-space representation

115 α(k) R(x) S(t ) A( f ) So, We 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 Detected FID signal Signal spectrum F k x (t) γ G x t /2π t sec f Larmor (x) t = k x (t) x MHz x = 2π f Larmor (x) / γ G x f k k-space representation F 1 x 0 +5 cm Real-space representation

116 α(k) R(x) S(t ) k-space Procedure for FID in 1D Phantom S(t) comprised of contributions from the m(x) for all x Detected FID signal t m(x,t) = m(x,0)e 2πi f (x) t = m(x,0) e 2πi k x (t) x S(t) = S(k(t)) ~ m(x) e 2πi k x (t) x dx. m(x) = S(t) e +2πi k(t) x d 3 k k x (t) γ G x t /2π R(x) = α(k) e +2πi kx dk Therefore, taking the inverse FT, F : 1 k x k k-space representation F 1 x 0 +5 cm Real-space representation

117 Spatial Waves, and Points in k-space

118 Point in k-space and Its Associated Wave in Real Space k y k x k-space k-space

119 Points in k-space and Their Waves in Real Space k y k x k-space

120 Composite Patterns in k-space and Real Space linearity + =

121 Fourier Transform of k-space Pattern to Real Space k-space Real Space k y y k x x F

122 Fourier Transform of MRI Data in k-space to Real Space F

123 F Fourier Transform from Parts of k-space to Real Space

124 Herringbone Artifact noise spike during data acquisition Zhou/Gullapalli

125 III-7 Information about the structural detail in an image is: a. dependent on the Fourier transform. b. contained in the periphery of k-space. c. blurred by increasing separation of lines in k- space. d. contained in the center of k-space. e. related to gradient duration. a. b. c. d. e. 7% 60% 3% 28% 2%

126 SAMs Q: III-7 Information about the structural detail in an image is: a) dependent on the Fourier transform. b) contained in the periphery of k-space. c) blurred by increasing separation of lines in k-space. d) contained in the center of k-space. e) related to gradient duration. Answer: (b). Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 337.

127 Part 4: Spin-Warp; 2D Spin-Echo Reconstruction (SE vs. FID) T2 Spin-Relaxation (Static vs. Random Dephasing) T1-w, T2-w, and PD-w S-E MR Imaging Spin-Echo / Spin-Warp in 2D (2X2 illustration)

128 Spin-Echo Reconstruction Spin Echo vs. FID

129 m xy (x,t) / m xy (x,0) Static Field de-p hasing of FID in x-y Plane of Spins in Each Voxel SFd-P from static-field inhomogeneities & susceptibility effects This de-phasing is REVERSABLE!!!! m xy (x,t) t=0 1 0 FID m xy (x,t) ~ e t / T SFd-P Rate of SFd-P decay: 1/ T SFd-P = κ ΔB 0 + κ' Δχ t Δ f Larmor ~ ΔB 0 n.b., m xy (x,t). but still 1D phantom!

130 Kentucky Turtle-Teleportation Derby Illustration of Spin-Echo i t = 0 ii iii t = ½ TE iv v t = TE

131 Spin-Echo Generates an Echo and Eliminates Effects of SFd-P Local magnetic environment for each spin packet is static. So this de-phasing can be REVERSED!!! T2 dephasing, by contrast, is random over time, and can NOT be un-done!! (a) (b) (c) (d) m(x,t) 90º y x 180º t = 0 t = ½ TE (e) t = TE

132 RF pulse S-E Sequence Using 180º RF Pulse and Readout Gradient with 256 voxels, e.g., sample/digitize echo signal times 90º 180º t S(t) Echo t G x f Larmor (B 0 ± G x ½ FOV) readout t

133 α(k) R(x) S(t ) k-space (spatial frequency) Procedure for SE in 1D Phantom Sampled SE signal t S(t) = S(k(t)) ~ m(x) e 2πi k x (t) x dx S(t) made up of contributions from m(x) for all x; m(x) can be extracted from S(t) with k x x F : k x = (γ/2π) G x t m(x) = S(t) e 2πi k(t) x d 3 k k x k-space representation α(k): Amplitude and phase of spatial frequency (k) F 1 k x x R(x) = α(k) e +ikx dk x 0 cm 5 Real-space representation

134 During Readout, k x Increases Linearly with t S(t) = S(k(t)) ~ m(x) e 2πi k x (t) x dx. k x (t) for all voxels increases linearly with t while the echo signal is being received and read. With signal sampled at 512 adjacent instances spaced t apart, t represents the exact sampling time: the n th sample is taken a time t n = n t after G x was turned on. Different values of k correspond to different sampling times. In particular, for t n, k n = [G x γ / 2π] t n Larger k-values correspond to greater spatial frequencies! 141

135 During Readout, k x Increases Linearly with t Signal is sampled sequentially at times spaced t apart; t n is the exact sampling time after G x is turned on. k(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. Later sampling times (and larger k-values) correspond to greater spatial frequencies. S-E reads out from k n = k max to k = 0 to k = k max k = k max k = 0 k max t n t 512

136 T2 Spin-Relaxation T2 Relaxation refers to the rate at which the transverse magnetization decays (disappears). T2 relaxation results from T1-Events and Non-Static, Random, Non-Reversible Proton-Proton Dipole Interactions Both Contribute to the decay rate (1/T2) (Specifically, any process that either reduces the number of transverse spins or their relative phase relationship will add to T2 relaxation.)

137 Thus, the Spin-Echo Signal Intensity S(t) Does Decay, and decays much Faster than 1/T1! Spin-Echo Spin De-Phasing Is Caused by T1 Events as well as By Random, Proton-Proton Dipole Interactions 90º º 180º 180º RF t Spin Multi-Echo: we might expect S(t) However, Harsh Reality! m xy (x,t) ~ e t / T2

138 Spin-Spin (Secular) De-Phasing in Bound Water with its protons precessing in the x-y plane H B C C H C H H H O H Quasi-static spin-spin interactions does not involve exchange of energy like T1 relaxation!

139 m xy (t) / m xy (0) Exponential T2-Caused De-Phasing of m xy (t) in x-y Plane ms t rate 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(x,t) m 0 (x)] ẑ [m x x + m y ŷ] classical Bloch Equation T1 T2

140 One Last Member of the Spin-Relaxation Family Tree: T2* fastest T2* FID, Gradient-Echo G-E generally much faster than S-E T2 T2 spin-echo (T SFd P ) T1 Discrete spin-dephasing: involves Δ E Continuous spin-dephasing: no Δ E!

141 T2 relaxation refers to: 7% 11% 7% 72% 3% a. The rate at which longitudinal magnetization recovers. b. The rate at which longitudinal magnetization disappears. c. The rate at which transverse magnetization recovers. d. The rate at which transverse magnetization disappears. e. The rate at which tissue is magnetized.

142 T2 relaxation refers to: a) the rate at which longitudinal magnetization recovers. b) the rate at which longitudinal magnetization disappears. c) the rate at which transverse magnetization recovers. d) the rate at which transverse magnetization disappears. e) the rate at which tissue is magnetized. Answer: (d). Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 364.

143 Tissue Contrast Weighting (w) in S-E MRI T1-w, T2-w, and PD-w

144 Typical T1 and T2 Relaxation Times Relaxation Rates: 1/T2 ~ 10 (1/ T1) Tissue PD T1, 1 T T1, 1.5 T T1, 3 T T2 p + /mm 3, rel. (ms) (ms) (ms) (ms) pure H brain CSF white matter gray matter edema glioma liver hepatoma muscle adipose

145 Three Different Forms of MRI Contrast created by, and reflecting, three quite different physical properties T1 T2 PD

146 Multiple Spin-Echo Pulse Sequence i ii RF 90º 180º 90º 180º G x readout

147 MRI Signal Strength at t = TE Depends on. Inherent Parameters T1 T2 PD Operator Settings TR TE time since previous S-E sequence S(t = TE) ~ PD (1 e TR / T1 ) e TE / T2 proton density prior regrowth along z-axis bright regions: m xy large at t = TE current dephasing in x-y plane

148 83% 1% 2% 2% In a spin-echo pulse sequence, a short TE will: 13% a. Minimize T1 image contrast b. Eliminate T2 effects c. Result in a lower SNR d. Increase the effects of static-field de-phasing e. Enhance tissue susceptibility differences

149 In a spin-echo pulse sequence, a short TE will: (a) Minimize T1 image contrast (b) Eliminate T2 effects (c ) Result in a lower SNR (d) Increase the effects of static-field dephasing (e) Enhance tissue susceptibility differences Answer: (b). Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 368.

150 T1-weighted Short TE to Eliminate T2 Contribution TR ~ T1 av to maximize contrast Short TE to minimize T2 impact S(t~TE) ~ PD (1 e TR / T1 ) e TE / T2 proton density prior regrowth along z-axis current de-phasing in x-y plane T1-w lipid CSF T1short Short-T1 tissues bright (sprightly brightly)

151 T1-w Short TE to Eliminate T2 Contribution TR i TR ii TR m 0 m z (TR) m xy (TE) lipid CSF lipid m z (t) CSF lipid CSF TE Short (20 ms); TR ~ T1 av (500 ms) 90 TR 500 ms TE 90 time Short T1 Bright lipid CSF 20 ms

152 For T2-w imaging T2-w Long TR to Eliminate T1 Contribution Long TR: eliminate T1 impact TE at mid-t2: maximize contrast S(t~TE) ~ PD (1 e TR / T1 ) e TE / T2 proton density prior regrowth along z-axis current de-phasing in x-y plane T2-w lipid CSF T2 long Long-T2 tissues bright

153 TR T2-w Long TR to Eliminate T1 Contribution TE ~ mid-t2 to maximize T2 contrast i TR ii lipid CSF M z (t) M xy (TR+) M xy (t) lipid CSF TE Long (80 ms); TR Long (2000 ms) M xy (TE) time 90 TR 2000 ms TE TE ~ ms

154 PD-, T1-, & T2-Weighted Spin-Echo Images (1.5T) T1-w T2-w PD-w TR mid- (~T1 av ) long long (ms) ,500 3,500 1,500 3,500 TE short mid- (~T2 av ) short (ms) Bright short T1 long T2 high PD SNR good lower best

155 Which of the following would appear bright on a T2-weighted image of the brain? 87% a. CSF 7% 0% 6% 0% b. Fat c. Bone d. White Matter e. Air

156 Which of the following would appear bright on a T2-weighted image of the brain? (a) CSF (b) Fat (c) Bone (d) White matter (e) Air Answer: (a) cerebral spinal fluid. Ref: Medical Imaging, A.B. Wolbarst et al., Wiley-Blackwell (2013), p. 368.

157 2D Images using Spin-Echo/Spin-Warp One of the earliest 2D imaging methods was Sensitive Point Reconstruction. This method used oscillating gradient fields to produce a net field stable in only one voxel at a time. That sensitive point was then scanned in space.

158 In 2-D MRI, phase is as important as frequency and is used to add a second spatial dimension. a 3 components b 2 different superimpositions

159 2D Spin-Warp 2X2 Matrix Illustration Assume: PD Map of Thin-Slice 2D, 4-Voxel Patient after 90º pulse drives M(t) into x-y plane z y x m(1) m(2) m(3) m(4) PD Map viewed from feet to head Pixel #

160 Spin-Echo, Spin-Warp Sequence for 2 2 Matrix involves 2 spin-echo pulse sequences first sequence second sequence slice selection phase encoding gradient RF G z G y 0 π 180º phase difference between rows 0º phase difference between rows TR TR Gx for S 0 for S π

161 e.g., No x- or y-gradients: G y = G x = 0 After (G y =0) phase encoding during (G x = 0) readout receiver G y G x 2 4 (Same Phase) 2 4 (Same frequency) S 0 (1,2,3,4) = 7 (e.g.) f Larmor F S 0 (1,2,3,4) F [S 0 (1,2,3,4)] 7 = m(1) + m(2) + m(3) + m(4)

162 1 st S-E Sequence: x-gradient Only, On During Readout G y = G x 1 3 G x 0 receiver 2 4 (Same phase) 2 4 (Different frequencies) S 0 (1,2) = 5 S 0 (3,4) = 2 F S 0 (1,2) + S 0 (3,4) 5 2 F [S 0 (1,2) + S 0 (3,4)] S 0 (1,2) = 5 = m(1) + m(2) S 0 (3,4) = 2 = m(3) + m(4)

163 2 nd S-E Sequence: y-gradient, then x-gradient φ + π φ G y 180º Δ. G x On During Readout 2 4 First, G y 0 G x 0 G y G 1 3 x (Different phases) 2 4 (Different frequencies) receiver + + S π (3,4) = 2 F S π (1,2) + S π (3,4) S π (1,2) = 3 S π (1,2) + S π (3,4) F S π (1,2) = 3 = m(1) m(2) S π (3,4) = 2 = m(3) m(4)

164 4 Equations in 4 Unknowns S 0 (1,2) + S 0 (3,4) F in phase: add S 0 (1,2) = 5 = m(1) + m(2) S 0 (3,4) = 2 = m(3) + m(4) S π (1,2) + S π (3,4) S 0 (1,2,3,4) F F º out of phase: subtract S π (1,2) = 3 = m(1) m(2) S π (3,4) = 2 = m(3) m(4) Solution: m(1) = 1 m(2) = 4 m(3) = 2 m(4) = 0 7 = m(1) + m(2) + m(3) + m(4)

165 Spin-Echo, Spin-Warp. ( matrix) 90 o pulse with G z on selects z-plane and initiates spin-echo Sample 512 G y 192 Free precession with G y on phase-encodes y-position G y (n) = n G y (1), n = 1, 2,, 192 Reading NMR signal with G x on frequency-encodes x-position Repeat, but with G y (n+1) n = 1, 2,, 192

166 Creation of an MRI Image in 2D k-space each line in k-space from data obtained during one M xy (TE) readout; different lines for different phase-encode gradient strength G phase #96 Amplitude G phase #1 G phase # 96 frequency (k x )

167 Multiple Important MR 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

168 A Brief Introduction to Magnetic Resonance Imaging Anthony Wolbarst, Kevin King, Lisa Lemen, Ron Price, Nathan Yanasak