Add dual echo

5 B0 Mapping/2 Dual echo B0 mapping/01-Introduction.md

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+---
+title: Introduction
+subtitle: Dual echo B0 mapping
+date: 2024-07-25
+authors:
+  - name:  Alexandre Dastous
+    affiliations:
+      - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
+numbering:
+  heading_2: false
+  figure:
+    template: Fig. %s
+---
+
+B0 mapping estimates the B0 field from the expected field for every voxel. These B0 maps can be used to perform prospective B0 shimming to minimize B0 inhomogeneities [1], they can be used to retrospectively correct for geometric distortions (FSL FUGUE [13], [14]) (e.g.: for EPI), or to perform retrospective correction for k-space readout trajectory (e.g.: for spiral readout). Moreover, they can be used for retrospective recovery of enhanced signal decay [15], [16], for T2* mapping and they are also vital to quantitative susceptibility mapping (QSM) where the goal is to map the susceptibility of the subject.
+
+One of the most simple and widely adopted techniques used to perform B0 mapping is the 2-echo phase difference technique. This technique is faster and simpler than most other alternatives. Before we dive into the technique, let's dip our toes in some theory.

5 B0 Mapping/2 Dual echo B0 mapping/02-Signal Theory.md

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+---
+title: Signal Theory
+subtitle: Dual echo B0 mapping
+date: 2024-07-25
+authors:
+  - name:  Alexandre Dastous
+    affiliations:
+      - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
+numbering:
+  heading_2: false
+  figure:
+    template: Fig. %s
+---
+
+In the ideal case, spins rotate at the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession), shown in blue in Fig. 1. In the presence of field inhomogeneities, the frequency of the spins (shown in red) is different and is proportional to the field inhomogeneities. Both the laboratory and rotating frame of reference are shown. Importantly, note that the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) phase appears stationary in the rotating frame of reference. 
+
+:::{figure} #fig5p6cell
+:label: b0Plot6
+:enumerator: 5.6
+Two spins rotating (one at the Larmor frequency (f0), one at a lower frequency). A view of the spins in the transverse plane (left) and of their phase (right) is shown. A dropdown is available to select between the laboratory frame and the rotating frame of reference.
+:::
+
+The phase () evolution follows the following equation (not considering transient effects such as [eddy currents](https://en.wikipedia.org/wiki/Eddy_current)) in the rotating frame of reference.
+
+```{math}
+:label: b0Eq3
+:enumerator:5.3
+\begin{equation}
+\phi\left(\textbf{r},t \right) = \phi_{0}\left( \textbf{r} \right)+\gamma\Delta B_{0}\left( \textbf{r} \right)\cdot t
+\end{equation}
+```
+
+where x,y,z are the coordinate locations, t is time,  is the gyromagnetic ratio, B0 is the B0 field offset (T) and 0 is an initial constant phase offset (e.g.: coil induced, material induced through local conductivity/permittivity). We can observe phase evolution through time in Fig. 2 by looking at phase data acquired in the brain at progressively longer echo times. The phase at a single voxel changes linearly (not considering transient effects). Note that the sharp variations forming vertical lines in the previous figure are called phase wraps and occur because the phase is defined over - to . Phase-wrapping effects will be discussed in more detail in the following chapter. Wraps can also occur spatially as sharp variations as seen in the following figure. Note that the longer the echo times, the more wraps there are.
+
+:::{figure} #fig5p7cell
+:label: b0Plot7
+:enumerator: 5.7
+Phase shown at different echo times. The slider can be used to show the phase that would be acquired at different echo times.
+:::
+
+MRI manufacturers do not all output phase data by default. It should be possible to toggle the output of phase data on all MRI systems. It can also be computed from real/imaginary data using Eq. 2.
+
+```{math}
+:label: b0Eq4
+:enumerator:5.4
+\begin{equation}
+\angle \left( \text{real}+i\text{ imaginary} \right) =\text{arctan2}\left( \text{imaginary, real} \right)
+\end{equation}
+```
+
+where ∠ is the phase operator. 
+
+As phase changes linearly with time (t) and with the field offset (B0), it is possible to acquire two phase images at two different echo times and compute B0(x,y,z).
+
+
+```{math}
+:label: b0Eq5
+:enumerator:5.5
+\begin{equation}
+\Delta B_{0}\left( \textbf{r} \right)=\frac{\phi\left( \textbf{r}, \text{TE}_{2} \right)-\phi\left( \textbf{r}, \text{TE}_{1} \right)}{\gamma\cdot \left( \text{TE}_{2}-\text{TE}_{1}  \right)}=\frac{\Delta\phi\left( \textbf{r}\text{,}\Delta \text{TE}\right)}{\gamma\Delta\text{TE}}
+\end{equation}
+```
+
+where TE1 and TE2 are the echo times, and  TE = TE2- TE1. To compute the phase offset , phase subtraction is necessary. The complex difference can be used to keep the phase between - to , although other phase difference techniques are also possible.
+
+```{math}
+:label: b0Eq5
+:enumerator:5.5
+\begin{equation}
+\Delta\phi\left( \textbf{r},\Delta\text{TE} \right)=\angle \left( \text{e}^{i\phi\left( \textbf{r}\text{,TE}_{2} \right)} \right)\cdot \left( \text{e}^{-i\phi\left( \textbf{r}\text{,TE}_{1} \right)} \right)
+\end{equation}
+```
+
+In some sequences, the phase images are exported as a single phase difference image (x,y,z, TE).

5 B0 Mapping/2 Dual echo B0 mapping/03-Single Frequency Population.md

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+---
+title: Single Frequency Population
+subtitle: Dual echo B0 mapping
+date: 2024-07-25
+authors:
+  - name:  Alexandre Dastous
+    affiliations:
+      - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
+numbering:
+  heading_2: false
+  figure:
+    template: Fig. %s
+---
+
+To build intuition about field maps, let us imagine a sample at a constant offset frequency from f0 . Note that this simplistic representation of the field typically does not occur due to how the susceptibilities of the neighboring regions interact with one another to create the B0 field offset (see Chapter 4.1), but is shown as such for learning purposes. The sample is shown as a circle in Fig. 3. As the frequency is not at the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession), phase accumulation is observed at the different echo times and a phase difference map can be computed. The B0 field map is then calculated using the echo times. Note that if TE is too long, the phase could make more than a half revolution between the two echo times resulting in an erroneous B0 field estimation. This is because phase is defined over - to  and the sampled points should respect the [Nyquist criteria](https://en.wikipedia.org/wiki/Nyquist_frequency). In practice, this example field (constant offset) could easily be corrected by adjusting the transmit and receive frequency of the scanner.
+
+:::{figure} #fig5p8cell
+:label: b0Plot8
+:enumerator: 5.8
+Different images of a homogeneous cylinder field offset showing a simulated phase at two echo times, the calculated phase difference image and the computed B0 field map.
+:::

5 B0 Mapping/2 Dual echo B0 mapping/04-Multi-Frequency Population.md

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+---
+title: Multi-Frequency Population
+subtitle: Dual echo B0 mapping
+date: 2024-07-25
+authors:
+  - name:  Alexandre Dastous
+    affiliations:
+      - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
+numbering:
+  heading_2: false
+  figure:
+    template: Fig. %s
+---
+
+A brain dataset is used to show a concrete example of a field map that could be acquired in practice. Fig 4. shows two phase images where phase accumulation is shown due to frequency offsets that vary spatially. As mentioned previously, phase wraps are visible where phase transitions from - to  and will be discussed in more detail in the next chapter. The phase difference and B0 field maps are also shown. Note that taking the phase difference eliminates the wraps in this example, however, there could be residual wraps when the field is more inhomogeneous. 
+
+:::{figure} #fig5p9cell
+:label: b0Plot9
+:enumerator: 5.9
+Two phase images, a phase difference and a B0 field map. Phase wraps are visible where the phase transitions from - to 
+:::