Key MRI Concepts and Equations#

Background Material#

MRI Math Concepts

Electricity and Magnetism

Signals and Systems

MR Spin Physics#

Resonance - nuclear spins in a magnetic field precess at a frequency proportional to the magnetic field strength

\[f = \bar{\gamma} \|\vec{B}\|\]

Polarization - equilibrium magnetization

\[M_0(\vec{r}) = \frac{N(\vec{r}) \bar{\gamma}^2 h^2 I_Z (I_Z +1) B_0}{3 k T}\]

Net Magnetization at Equilibrium

\[\begin{split}\vec{M}(\vec{r},0) = \begin{bmatrix} 0 \\ 0 \\ M_0(\vec{r}) \end{bmatrix}\end{split}\]

Excitation

  • Apply magnetic field at resonant frequency to rotate net magnetization out of alignment with static magnetic field

Relaxation

\[M_{XY}(\vec{r},t) = M_{XY}(\vec{r},0) e^{-t/T_2(\vec{r})}\]
\[M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r})(1- e^{-t/T_1(\vec{r})})\]

MRI System#

  1. Main magnet - \(B_0\)

  2. Radiofrequency (RF) coils

    • transmit RF coil - \(B_1^+(\vec{r},t)\): provide homogeneous excitation

    • receive RF coil - \(B_1^-(\vec{r},t)\): detect signal with high sensitivity

  3. Magnetic field gradient coils - \(\vec{G}(t)\)

MRI Experiment#

  1. Polarization

  2. Excitation

  3. Signal Acquisition

  • Gradients during Excitation and Acquisition for spatial encoding

  • Repeat Excitation and Acquisition as needed

Experiment described by a “Pulse Sequence”

MR Contrasts#

Contrast weightings

  • T1-weighted - short TE, short TR

  • T2-weighted - long TE, long TR

  • Proton Density (PD)-weighted - short TE, long TR

spoiled GRE contrast

\[S \propto M_0 \sin(\theta) \exp(-TE/T_2) \frac{1- \exp(-TR/T_1)}{1- \cos(\theta) \exp(-TR/T_1)}\]

Ernst angle - flip angle for maximum SNR

\[\theta_{optimal} = \cos^{-1}(\exp(-TR/T_1))\]

Magnetization Preparation: Inversion Recovery

\[S_{IR} \propto M_0 \exp(-TE/T_2) (1 - 2\exp(-TI/T_1) + \exp(-TR/T_1) )\]

In Vivo Spin Physics#

Magnetic susceptibility effects

  • magnetic susceptibility is inherent property of materials

  • differences in magnetic susceptibility lead to distortions of the magnetic field

  • in vivo sources include: iron, oxygenated versus deoxygenated blood

Chemical Shift

  • chemcial environment of an atom creates variations in local magnetic field

  • in vivo consideration: “fat”, assumed to have a -3.5 ppm chemcial shift from water protons

In Vivo Contrasts#

Phase - chemical shift and off-resonance (e.g. magnetic susceptibility effects) create phase differences in MR signal

T2*

  • intra-voxel dephasing due to magnetic field inhomogeneity

  • largely driven by magnetic susceptibility

  • eliminate with spin-echo

  • create susceptibility contrast

Fat

  • fat/water imaging - separate fat and water images based on multiple echo times

  • fat suppression - spectrally-selective RF pulses and/or inversion recovery

Contrast Agents

  • Gadolinium (Gd)-based contrast agents - most common, primarily shortens \(T_1\)

  • Iron-based contrast agents - less common, shortens \(T_1\) but also can shorten \(T_2\)

RF Pulses#

Pulse Characteristics

  • pulse profile - approximately proportional to the Fourier Transform of the pulse shape

  • flip angle

\[\theta = \gamma \int_0^{T_{rf}} b_1(\tau) d\tau \]

  • Time-bandwidth product - constant for a given pulse shape

\[ TBW = T_{rf} \cdot BW_{rf} \]

  • SAR

\[ SAR \propto \int_0^{T_{rf}} |b_1(\tau)|^2 d\tau \]

Slice Selection

  • Slice thickness

\[ \Delta z = \frac{BW_{rf}}{\bar{\gamma} G_{Z,SS}} \]

  • Slice shifting

\[ f_{off} = \bar{\gamma} G_{Z,SS} \ z_{off} \]

Spatial Encoding#

Frequency encoding - turn on gradient during data acquisition to map frequency to position

\[ x = \frac{f}{\bar \gamma G_{xr}}\]

Phase encoding - perform step-wise frequency encoding, which appears in the phase versus position of the signals. This measurement is repeated for \(n = 1, \ldots, N_{PE}\)

\[ \Phi(n) = \gamma (-G_{y,PE} + (n-1) G_{yi} ) t_y y\]

k-space - define spatial encoding based on the cumulative sum of the gradients (i.e. gradient areas) applied after excitation

\[\vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) d\tau\]

  • Formulates image reconstruction as an inverse Fourier Transform

\[s(t) = \mathcal{F}\{m(\vec{r}) \} |_{\vec{k} = \vec{k}(t)} = M(\vec{k}(t))\]

  • describes all MRI acquisitions including frequency and phase encoding

  • effects of gradients can be refocused

  • supports 2D and 3D imaging

Image Characteristics#

\[SNR \propto f_{seq}\ \mathrm{Voxel\ Volume}\ \sqrt{T_{meas}}\]
\[ FOV = \frac{1}{\Delta k}\]
\[ \delta = \frac{1}{2 k_{max}}\]

Scan Time

\[ T_{scan} = \frac{ TR \cdot N_{PE,total} \cdot NEX}{ETL \cdot R}\]

FT Imaging Sequence#

Typical acquisition uses frequency and phase encoding

See Pulse Sequence for a typical 2D gradient-echo sequence

Can convert between sequence parameters (e.g. timings, gradient amplitudes) and the FOV, resolution and scan time, as well as predict relative SNR

Fast Imaging Pulse Sequences#

Volumetric coverage

  • 2D multislice imaging - interleave multiple slices within a single TR

  • 3D imaging - cover 3D k-space

EPI

  • k-space trajectory that covers multiple k-space lines per excitation

  • Echo spacing (\(t_{esp}\)), echo train length (ETL)

Multiple Spin-echo imaging (FSE/TSE/RARE)

  • multiple spin-echoes per excitation used to acquire different k-space lines

  • Echo spacing (\(t_{esp}\)), echo train length (ETL)

  • echo time, \(TE = TE_{eff}\), defined when data closest to center of k-space is acquired. Used to create different contrasts

Gradient Echo methods

  • Contrast can be changed based on whether transverse magnetization is available or refocused in a subsequent TR

  • Variations based on whether RF and/or gradient spoiling are used

Accelerated Imaging Methods#

Partial Fourier

  • Why does it work? MRI approximately satisfies conjugate symmetry property of k-space data

  • How does it work? Only sample slightly more than half of k-space

Parallel Imaging

  • Why does it work? RF coil arrays with different elements provide spatial encoding

  • How does it work? Skip k-space data in the direction(s) that have variation in RF coil element sensitivity profiles

  • Key variations: May require measurement of coil sensitivity maps, also autocalibrated methods

Simultaneous Multi-slice

  • Why does it work? RF coil arrays with different elements provide spatial encoding

  • How does it work? Excite multiple slices simultaneously

Compressed Sensing

  • Why does it work? MRI data has typical patterns that can be predicted are represented by sparse coefficients

  • How does it work? Skip k-space data with a pseudo-random pattern. Define a sparsity domain

Deep Learning Reconstruction

  • Why does it work? MRI data has typical patterns that can be learned

  • How does it work? Skip k-space data. Train a neural network using a large MRI dataset.

Artifacts#

See Artifacts for high-level comparison