From 26f6aa5b7918d513e19536597dc0f8a074be9ae7 Mon Sep 17 00:00:00 2001
From: agentmess Partial Fourier imaging, also known as fractional NEX or partial k-space, utilizes the conjugate symmetry property of the Fourier Transform. This property states that the Fourier Transform of a real-valued function has conjugate symmetry. If our image, \(m(\vec{r})\), is real-valued, then the k-space data satisfies In this case, only half of k-space is required, and the other half can be filled in by this property. In practice, there is typically low spatial frequency phase components in the image so the initial assumption of a real-valued image is violated. However, the data acquisition can still be accelerated by acquiring slightly more than half of k-space, and the fully sampled center of k-space can provide sufficient information about the low spatial frequency phase to still allow for accelerationg based on this conjugate symmetry property. For parallel imaging (PI), we need to consider the coil sensitivity profiles, \(\mathbf{C}_q\), for each RF coil into encoding matrix along with a Fourier Transform encoding matrix, \(\mathbf{F}\), as well as a k-space sub-sampling operator, \(\mathbf{S}\), for the measurements from each RF coil, \(\mathbf{y}_q\): next MRI Notation and Terminology Key MRI Concepts and Equations
Contents
General Formulation of MRI Reconstruction
\[ \hat{x} = \mathbf{F^{H}} y \]
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Partial Fourier Imaging#
+Parallel Imaging#
Reference
Reference
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Spatial Resolution\(k_{max}\) as:
For example, \(\delta_x = \frac{1}{2 k_{x,max}}\)
+For example, \(\delta_x = \frac{1}{2 k_{x,max}}\).
+For symmetric sampling in k-space, this can also be defined based on the width of the k-space sampling, \(W_k = 2 k_{max}\):
+% rectangular object to demonstrate resolution
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SEE ALSO MRI Math Concepts
+Larmor Frequency
-M0
-Polarization
+Electricity and Magnetism
+Signals and Systems
+Resonance +$\(f = \bar{\gamma} \|\vec{B}\|\)$
+Polarization and Net Magnetization +$\(\vec{M}(\vec{r},0) = +\begin{bmatrix} +0 \\ +0 \\ +M_0(\vec{r}) +\end{bmatrix}\)\( +\)\(M_0(\vec{r}) = \frac{N(\vec{r}) \bar{\gamma}^2 h^2 I_Z (I_Z +1) B_0}{3 k T}\)$
+Excitation
+Apply magnetic field at resonant frequency to rotate net magnetization out of alignment with static magnetic field
Relaxation
T2/T2*
-T1
+Main magnet - \(B_0\)
Radiofrequency (RF) coils, including a transmit RF coil - \(B_1^+(\vec{r},t)\) - and a receive RF coil - \(B_1^-(\vec{r},t)\)
Magnetic field gradient coils - \(\vec{G}(t)\)
Polarization
Excitation
Signal Acquisition
Gradients during Excitation and Acquisition for spatial encoding
Repeat Excitation and Acquisition as needed
Experiment described by a “Pulse Sequence”
+spoiled GRE contrast
Contrast weightings: T1w, T2w, PDw
+Magnetization Preparation:
Inversion Recovery $\(S_{IR} \propto M_0 \exp(-TE/T_2) (1 - 2\exp(-TI/T_1) + \exp(-TR/T_1) )\)$
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
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
+Gd-based contrast agents - most common, primarily shortens \(T_1\)
iron-basec contrast agents - less common, shortens \(T_1\) but also can shorten \(T_2\)
flip angle
-SAR
-TBW = BW_RF T_{RF}
-Slice thickness
-slice shifting
+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 +$\( f_{off} = \bar{\gamma} G_{Z,SS} \ z_{off} \)$
k-space
-Frequency encoding - turn on gradient during data acquisition to map frequency to position
Phase encoding - perform step-wise frequency encoding, which appears in the phase versus position of the signals. This measurement is repeated for \(n = 1, \ldots, N_{PE}\)
k-space - define spatial encoding based on the cumulative sum of the gradients (i.e. gradient areas) applied after excitation
Formulates image reconstruction as an inverse Fourier Transform
describes all MRI acquisitions including frequency and phase encoding
effects of gradients can be refocused
supports 2D and 3D imaging
SNR
-FOV/resolution - in general, in Cartesian 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
+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
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 and Deep Learning Reconstructions
+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 Reconstructions
+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.
Scan times
-effective TE
+See Artifacts for high-level comparison
previous
-Accelerated Imaging Methods
+Key MRI Concepts and Equations
Typically the 2nd (and optionally 3rd) dimensions of the object are encoded using “phase encoding”. This means that, after RF excitation but before the frequency encoding gradient, a pulsed gradient is applied such that the location is encoded in the phase of the next magnetization:
This measurement is repeated for \(n = 1, \ldots, N_{PE}\). \(G_{yp}\) is the maximum phase encoding gradient strength, \(G_{yi}\) is the phase encoding gradient amplitude increment, and \(t_y\) is the phase encoding gradient duration. Note that \(2 G_{yp} = (N_{PE} - 1) G_{yi}\).
These additional dimensions are fully encoded by repeating this pulsed gradient with different amplitudes. This is equivalent to taking different samples of a frequency encoding gradient.