Table of Contents


1. Introduction to MRI

1.1 A Stratospheric, Warp-Speed Sketch of NMR and MRI
The nuclear magnetic moment is proportional to the nuclear spin
Some Notes on Vectors
First picture of NMR: quasi-quantum mechanical (spin-up/spin-down along Bz and the z-axis)
MRI of a 1D patient
Second picture of NMR: classical (magnetization precessing in x-y plane)
Proton spin relaxation
Longitudinal proton spin relaxation time, T1
Transverse proton spin relaxation time, T2
T1-weighted and T2-w clinical images
1.2 Uniqueness of MRI
1.3 A Real MRI Case Study
Computed tomography (CT)
MRI: T1-w, T2-w, and FLAIR imaging
Magnetic resonance spectroscopy (MRS)
Functional MRI (f MRI)
Diffusion tensor MR imaging (DTI)
MRI-guided fine-needle biopsy
Positron emission tomography?
Treatment guidance and follow-up
1.4 A Brief History of MRI: Bloch, Purcell, Damadian, Lauterbur, et. al.
1.5 A Clinical Caveat

2. Quasi-Quantum, Two-State Picture of Proton NMR in a Single Voxel

2.1 A Brief Review of Electromagnetism
Fields, Maxwell’s equations, and the wave equation
2.2 Magnetic Fields within Matter: Magnetic Susceptibility
Applied main, gradient, and RF fields at a voxel
Spontaneous magnetic fields that affect T1 and T2
Magnetic susceptibility: diamagnetism, paramagnetism, and ferromagnetism induced by B0
2.3 Nuclear Magnetic Dipole Moment
Proton-proton dipole interaction
2.4 Proton Spin-Up and Spin-Down States
2.5 NMR at the Larmor Resonance Frequency
2.6 Preliminary Demonstration of the NMR Phenomenon in Water

3. Proton Density MR Study of a 1D Multi-Voxel Patient and the Quality of the Image

3.1 The Three Distinct Applied Magnetic Fields in 1D MRI
3.2 Encoding of x-Position with an x-Gradient, and the Local NMR Resonance Frequency
3.3 Proton Density (PD) MRI of a Multi-Voxel, One-Dimensional Patient
3.4 Measures of Image Quality: Contrast, Resolution, and the MTF
Contrast (C)
Resolution (R)
Glimpse of the modulation transfer function (MTF)
3.5 Stochastic Noise, Signal-to-Noise Ratio, Contrast-Detail Curves, and the DQE
Stochastic noise
Signal-to-noise ratio (SNR)
Contrast-detail curves
Detective quantum efficiency (DQE)
3.6 Judging Performance with ROC Curves
True positives (TP), false negatives (FN), etc.
Sensitivity, selectivity, and accuracy
Receiver operating characteristic (ROC) curves
Bayes’ theorem
3.7 Some MR Artifacts
Nonlinear gradient-field artifact
Chemical shift artifact
Susceptibility artifact

4. Magnetization of a Voxel

4.1 The Magnetization, m(x,t), of the Voxel at Location x
4.2 Thermal Equilibrium: The Battle between Energy and Entropy
4.3 The Magnitude of m(t) at Thermal Equilibrium, m0 ≡ m(∞), and Its Impact on Image Quality
4.4 Effect of Field Strength, |B0|, on Image Quality
4.5 Spin Population Dynamics
4.6 Stimulated Radiative Transitions and Spin System Saturation
4.7 Stimulated Non-Radiative Spin Transitions and T1 Relaxation

5. Mathematical Machinations

5.1 Two Simple but Essential Functions
Exponential decay and growth
Sinusoidal oscillation
5.2 Scalar (Dot) Product of Vectors and Functions
Spectral analysis of real vectors
Into the complex plane
Orthonormal sinusoidal basis vectors
5.3 Fourier Series Expansion of a Function Periodic in Time or Space
Basis vectors, and the Dirac delta function
Series truncation
5.4 Fourier Integrals and Transforms
FT [LSF(x)] = MTF(v)
Patient motion artifacts and the Fourier Shift Theorem
5.5 Vectors in a 1D k-Space
5.6 Digital Representations of Symbols and Images
Pixel size and grayscale
Data compression
5.7 Noise, Statistics, and Probability Distribution Functions
Elementary statistical notions
Notes on the Poisson distribution
Rician Distribution
5.8 Sampling an Analog Signal
Shannon-Nyquist sampling theorem
Aliasing artifact
5.9 Resolution vs. Detectability
Detectability requires enough size, sufficiently high contrast, and low noise

6. Classical Approach to Proton MRI in a Voxel

6.1 Normal Modes of Oscillation
6.2 Resonance at the Normal Mode Frequency
Precession and nutation of a gyroscope
6.3 In a Voxel, Classical Precession of m(t) about B0 at vLarmor; The Bloch Equations
6.4 Classical View of NMR
6.5 Grand Entrance of the 90° Pulse
6.6 Laboratory Frame of Reference vs. One Rotating at vLarmor

7. Free Induction Decay Imaging of a 1D Patient (without the Decay)

7.1 Saturation-Recovery in a Single Voxel, Assuming Unrealistically Slow Spin Relaxation
7.2 Improving the SNR with Quadrature Detection
7.3 Net Nuclear Magnetization from All the Voxels in the 1D Phantom Together
7.4 Selecting the z-Slice (or in 1D, the Row of Voxels) for Imaging
7.5 Signal from the 1D Phantom
7.6 FID Imaging of the 1D Phantom: Encoding Voxel x-Position Again with an x-Gradient
7.7 1D Image Reconstruction by Way of Frequency-Space
7.8 1D Image Reconstruction by Way of k-Space
7.9 Connections between Image Space and k-Space
Resolution in real space is inversely proportional to the width of the FOVk in k-space
7.10 Phase Waves

8. MRI Instrumentation

8.1 Main Magnet
MRI superconducting magnets
Fringe fields
Permanent magnets and electromagnets
8.2 Gradient Coils
8.3 The MRI RF System Is Like an AM Radio Transmitter/Receiver Pair
Super-heterodyne AM radio
New MRI machines employ direct detection, not an IF stage
Signal Bandwidth
8.4 Antennae/RF Coils
Signal-to-noise ratio, again
8.5 Phased Array of Receiver Coils

9. T1 (Longitudinal, Spin-Lattice) Relaxation

9.1 The Longitudinal Proton Spin-Relaxation Time, T1, in a Single Voxel
9.2 Exponential Regrowth of mz(t) toward m0 at the Rate 1/T1: m(t) = m0 (1 – e(–t/T1))
9.3 The Bloch Equations, Including T1 Relaxation
9.4 Biophysical Mechanism of Proton T1 Relaxation: Spontaneous, Random, vLarmor Fluctuations
in the Local Bz(t), Created by the Tissues Themselves, Can Induce Spin-State Transitions
9.5 Tumbling Frequencies of Water Molecules in Three Aqueous Environments
9.6 T1 Increases with the Strength of the Main Field, B0
9.7 Determining T1 in a Voxel
9.8 T1 MRI in a 1D Patient
9.9 Speeding Things Up: Short TRs and Small Flip Angles

10. T2 (Transverse, Spin-Spin) Relaxation

10.1 Transverse (Spin-Spin) Proton Relaxation: Spin De-Phasing in the x-y Plane with Characteristic Time T2
10.2 Biophysics of Proton T2 Relaxation
10.3 Static Field de-Phasing, at the Rate 1/TSFd-P , and the Spin-Echo Sequence that Removes It
10.4 1/T2* is 1/T2 Speeded Up by 1/TSFd-P
10.5 Shape of an NM Resonance Line
10.6 Magnetization Transfer (MT) Imaging
10.7 Contrast Agents such as Gadolinium Chelates

11. The Spin-Echo Pulse Sequence

11.1 Spin-Echo (SE) RF Pulse Sequence in a Single Voxel: 90°–180°– read
11.2 Put Another Way…
11.3 And More Realistically…
11.4 The Spin-Echo RF Pulse Sequence Eliminates the TSFd-P Effect
11.5 Selecting TE and TR for T1-Weighted SE
11.6 Optimizing the T1 Contrast for a Pair of Nearby Voxels
11.7 T2-weighted and PD-w for SE
11.8 SE Reconstruction of the Image of a 1D Patient from a Single Line in k-Space
11.9 Dealing with Unwanted Phase Changes in the Echo Signal Caused by Gradients
11.10 Following a Simple Trajectory through 1D k-Space
11.11 Inversion Recovery (IR) in a Voxel: 180°–90°–180°– read

12. Image Reconstruction in Two Dimensions

12.1 Selection of a Two-Dimensional Slice
12.2 Waves in 2D Real Space and Vectors in 2D k-space
12.3 Phase Encoding of Voxel y-Position
12.4 A 2D, 4-Voxel Spin-Echo/Spin-Warp Example
Preliminary study: Gy = Gx = 0
First S-E repetition: Gy = 0 throughout, but Gx on during readout
Second S-E repetition: Gy is on for the time tphase before readout
Solving the four equations for the four unknowns, mi
12.5 The General Spin-Echo, Spin-Warp Echo Data Acquisition Process

13. Fast Imaging

13.1 Time to Acquire a Slice-Image
13.2 Multiple Slice Imaging
On Calculating the Numbers of Batches and of Slices per Batch
13.3 Stack of 2D Slices vs. 3D Block of Tissue
13.4 Fast Spin Echo vs. Conventional Spin Echo—More Readouts per Excitation
13.5 Gradient Echo (GE) / Gradient Recall Echo (GRE)—Reducing the Repetition Time (TR)
13.6 Echo Planar Imaging (EPI)—Many More Readouts per Excitation
13.7 k-Space Has (Approximate) Conjugate Symmetry, and its Data are Redundant
13.8 Under-sampling of k-space in Combination with Phased Array Coils
13.9 Frequency- and Phase-Encoding Similarities
13.10 Cartesian vs. Non-Cartesian Readout

14. Fluid Motion in One Dimension

14.1 Water and Blood Flow
14.2 Magnetic Resonance Angiography (MRA)
14.3 Perfusion Imaging
14.4 Diffusion Tensor Imaging (DTI)
14.5 Functional MRI (fMRI) with the Blood Oxygenation Level Dependent (BOLD) Effect

15. MRI Device Quality Assurance and Safety

15.1 Daily/Weekly/Monthly Quality Assurance Checks by Technologists, and the Semi-Annual QA by a Medical Physicist
15.2 Accreditation by the American College of Radiology Every Three Years
15.3 MR Safety Requires Eternal Vigilance (Regulations, Restricted Access, Records…)
15.4 Danger from the Main Magnet and Its Fringe Fields: Beware of Aneurysm Clips and Pacemakers,
and of Flying Screwdrivers, Wheelchairs, and Oxygen Bottles
15.5 Biological Hazards from Switching Gradients and RF Fields, etc.
15.6 Contrast Agents and Injectors
15.7 On Siting, Installing, and Accepting an MRI Device

16. On the Horizon: Imaging with a Crystal Ball

16.1 Evolution: Ongoing Changes
Quantitative imaging, including MR fingerprinting
Compressed sensing
Inexpensive low-field MRI
High-field imaging
MRI on elements other than hydrogen; nuclear hyperpolarization
Helium-free MRI
Quiet imaging
Combining MRI with PET, NIR, US, etc.
Zero-quantum imaging
Radiation therapy treatment planning and follow-up
16.2 Revolution: Quantum Computers and Artificial Intelligence Quantum Artificial intelligence (AI)
16.3 Conclusion

Symbols and Abbreviations

Fundamental Equations

References and Sources