diff --git a/Notebooks/FOV and Resolution.ipynb b/Notebooks/FOV and Resolution.ipynb index 54ad93b..7a2cb47 100644 --- a/Notebooks/FOV and Resolution.ipynb +++ b/Notebooks/FOV and Resolution.ipynb @@ -114,7 +114,11 @@ "\n", "$$ \\delta = \\frac{1}{2 k_{max}}$$\n", "\n", - "For example, $\\delta_x = \\frac{1}{2 k_{x,max}}$" + "For example, $\\delta_x = \\frac{1}{2 k_{x,max}}$.\n", + "\n", + "For symmetric sampling in k-space, this can also be defined based on the width of the k-space sampling, $W_k = 2 k_{max}$:\n", + "\n", + "$$ \\delta = \\frac{1}{W_k}$$" ] }, { diff --git a/Notebooks/Key MRI Concepts.md b/Notebooks/Key MRI Concepts.md index 8c0b830..d59c400 100644 --- a/Notebooks/Key MRI Concepts.md +++ b/Notebooks/Key MRI Concepts.md @@ -1,19 +1,29 @@ # Key MRI Concepts and Equations -SEE ALSO MRI Math Concepts +## Background Material [MRI Math Concepts](./MRI%20Math%20Concepts.ipynb) -## Spin Physics +Electricity and Magnetism -Larmor Frequency +Signals and Systems -Polarization and Net Magnetization +## MR Spin Physics + +Resonance +$$f = \bar{\gamma} \|\vec{B}\|$$ -M0 +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 -* RF Pulses +* Apply magnetic field at resonant frequency to rotate net magnetization out of alignment with static magnetic field Relaxation @@ -23,23 +33,21 @@ $$M_Z(\vec{r},t) = M_Z(\vec{r},0)e^{-t/T_1} + M_0(\vec{r})(1- e^{-t/T_1(\vec{r}) ## MRI System -Main magnet - -RF coils - -Gradient coils +1. Main magnet - $B_0$ +1. Radiofrequency (RF) coils, including a transmit RF coil - $B_1^+(\vec{r},t)$ - and a receive RF coil - $B_1^-(\vec{r},t)$ +1. Magnetic field gradient coils - $\vec{G}(t)$ ## MRI Experiment 1. Polarization 1. Excitation 1. Signal Acquisition -* Spatial Encoding during Excitation and Acquisition +* Gradients during Excitation and Acquisition for spatial encoding * Repeat Excitation and Acquisition as needed -Described by a "Pulse Sequence" +Experiment described by a "Pulse Sequence" -## Contrast +## MR Contrasts spoiled GRE contrast @@ -52,74 +60,132 @@ Magnetization Preparation: Inversion Recovery $$S_{IR} \propto M_0 \exp(-TE/T_2) (1 - 2\exp(-TI/T_1) + \exp(-TR/T_1) )$$ +## In vivo spin physics + +Magnetic susceptibility effects +* magnetic susceptibility is inherent property of materials +* differences in magnetic susceptibility lead to distortions of the magnetic field +* in vivo sources include: iron, oxygenated versus deoxygenated blood + +Chemical Shift +* chemcial environment of an atom creates variations in local magnetic field +* in vivo consideration: "fat", assumed to have a -3.5 ppm chemcial shift from water protons + +## In vivo contrasts + +T2* +* intra-voxel dephasing due to magnetic field inhomogeneity +* largely driven by magnetic susceptibility +* eliminate with spin-echo +* create susceptibility contrast + +fat +* fat/water imaging - separate fat and water images based on multiple echo times +* fat suppression - spectrally-selective RF pulses and/or inversion recovery + +Contrast Agents +* Gd-based contrast agents - most common, primarily shortens $T_1$ +* iron-basec contrast agents - less common, shortens $T_1$ but also can shorten $T_2$ + + ## RF Pulses Pulse Characteristics +* pulse profile - approximately proportional to the Fourier Transform of the pulse shape * flip angle +$$\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} $$ * SAR -* TBW = BW_RF T_{RF} +$$ 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} $$ ## Spatial Encoding -Frequency encoding +Frequency encoding - turn on gradient during data acquisition to map frequency to position -Phase encoding +$$ x = \frac{f}{\bar \gamma G_{xr}}$$ -k-space - -$$\vec{k}(t) = \frac{\gamma}{2\pi} \int_0^t \vec{G}(\tau) d\tau$$ +Phase encoding - perform step-wise frequency encoding, which appears in the phase versus position of the signals. This measurement is repeated for $n = 1, \ldots, N_{PE}$ -$$M_{XY}(\vec{r}, t) = M_{XY}(\vec{r}, 0) e^{ -i 2 \pi \vec{k}(t) \cdot \vec{r} }$$ +$$ \Phi(n) = \gamma (-G_{y,PE} + (n-1) G_{yi} ) t_y y$$ -$$\begin{align} -s(t) & = \int_\mathrm{Volume} M_{XY}(\vec{r},t) \ d\vec{r} \\ - & = \int_{\textrm{Volume}} M_{XY}(\vec{r},0) \exp(-i2\pi \vec{k}(t) \cdot \vec{r}) \ d\vec{r} - \end{align}$$ +k-space - define spatial encoding based on the cumulative sum of the gradients (i.e. gradient areas) applied after excitation -$$s(t) = \int m(\vec{r})\ e^{-i 2 \pi \vec{k}(t) \cdot \vec{r}} \ d\vec{r}$$ +$$\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))$$ +* describes all MRI acquisitions including frequency and phase encoding +* effects of gradients can be refocused +* supports 2D and 3D imaging ## Image Characeristics -SNR - -resolution +$$SNR \propto f_{seq}\ \mathrm{Voxel\ Volume}\ \sqrt{T_{meas}}$$ -FOV +$$ FOV = \frac{1}{\Delta k}$$ -## 2D FT Imaging Sequence +$$ \delta = \frac{1}{2 k_{max}}$$ -Show sequence +## FT Imaging Sequence -Parameters +Typical acquisition uses frequency and phase encoding. -FOV, resolution +See [Pulse Sequence](./Pulse%20Sequence.ipynb) for a typical 2D gradient-echo sequence -Scan Time +Can convert between sequence parameters (e.g. timings, gradient amplitudes) and the FOV, resolution and scan time ## 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-echoes +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 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. ## Artifacts +See [Artifacts](./Artifacts.ipynb) for high-level comparison \ No newline at end of file diff --git a/Notebooks/Spatial Encoding.ipynb b/Notebooks/Spatial Encoding.ipynb index 6ddc6de..ae634e1 100644 --- a/Notebooks/Spatial Encoding.ipynb +++ b/Notebooks/Spatial Encoding.ipynb @@ -236,7 +236,9 @@ "\n", "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:\n", "\n", - "$$ \\Phi = \\gamma G_{yp} y t_y$$ \n", + "$$ \\Phi(n) = \\gamma (-G_{yp} + (n-1) G_{yi} ) t_y y$$ \n", + "\n", + "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 $2 G_{yp} = (N_{PE} - 1) G_{yi}$.\n", "\n", "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." ] diff --git a/Notebooks/_toc.yml b/Notebooks/_toc.yml index 5cb555b..85b8b9c 100644 --- a/Notebooks/_toc.yml +++ b/Notebooks/_toc.yml @@ -34,6 +34,9 @@ parts: chapters: - file: "Fast Imaging Pulse Sequences" - file: "Accelerated Imaging Methods" + - caption: Summary + chapters: + - file: "Key MRI Concepts" - caption: Reference Material chapters: - file: "MRI Notation" diff --git a/Notebooks/images/artifacts/abdomen_aliasing.png b/Notebooks/images/artifacts/abdomen_aliasing.png new file mode 100644 index 0000000..e9d3cee Binary files /dev/null and b/Notebooks/images/artifacts/abdomen_aliasing.png differ diff --git a/Notebooks/images/artifacts/abdomen_motion.png b/Notebooks/images/artifacts/abdomen_motion.png new file mode 100644 index 0000000..3feaf51 Binary files /dev/null and b/Notebooks/images/artifacts/abdomen_motion.png differ diff --git a/Notebooks/images/artifacts/brain_coronal_eyes_artifact.jpg b/Notebooks/images/artifacts/brain_coronal_eyes_artifact.jpg new file mode 100644 index 0000000..5dd4817 Binary files /dev/null and b/Notebooks/images/artifacts/brain_coronal_eyes_artifact.jpg differ diff --git a/Notebooks/images/artifacts/brain_motion_artifact.jpg b/Notebooks/images/artifacts/brain_motion_artifact.jpg new file mode 100644 index 0000000..6cb739e Binary files /dev/null and b/Notebooks/images/artifacts/brain_motion_artifact.jpg differ diff --git a/Notebooks/images/artifacts/epi_T2star_artifact.png b/Notebooks/images/artifacts/epi_T2star_artifact.png new file mode 100644 index 0000000..7d0f647 Binary files /dev/null and b/Notebooks/images/artifacts/epi_T2star_artifact.png differ diff --git a/Notebooks/images/artifacts/epi_distortion_artifact_changes.jpg b/Notebooks/images/artifacts/epi_distortion_artifact_changes.jpg new file mode 100644 index 0000000..837a57b Binary files /dev/null and b/Notebooks/images/artifacts/epi_distortion_artifact_changes.jpg differ diff --git a/Notebooks/images/artifacts/epi_fat_artifact.jpg b/Notebooks/images/artifacts/epi_fat_artifact.jpg new file mode 100644 index 0000000..b0bb120 Binary files /dev/null and b/Notebooks/images/artifacts/epi_fat_artifact.jpg differ