Added complete Key MRI Concepts chapter

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",
+    "$$ \\delta = \\frac{1}{W_k}$$"
    ]
   },
   {

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

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."
    ]

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"