B0 chapter figs and eqs crossregs

5 B0 Mapping/1 B0 Inhomogeneities/1-Introduction.md

@@ -16,7 +16,7 @@ numbering:
 
 The main magnetic field, also called the B0 field, plays a crucial role in MRI. It dictates the precessional frequency of the spins and sets-up the bulk magnetization, which plays an important role in the image signal-to-noise ratio. Moreover, the radio frequency coils, tuned to the B0 field, are responsible for flipping the spins in the transverse plane and for acquiring the signal. However, imaging reconstruction techniques assume a perfectly homogeneous B0 field to reconstruct the signal from k-space data. An inhomogeneous B0 field can lead to image artifacts such as signal loss, distortions [1], poor fat saturation [2] and many other image artifacts. In extreme cases, it can completely hinder the ability to create an image. B0 inhomogeneities are also problematic for MR spectroscopy (MRS), because they widen the spectral linewidth.
 
-When a subject is introduced in the scanner, the static B0 field can be rendered more homogeneous through a technique called active shimming. Active shimming sends the appropriate amount of current through specific gradient and shim coils, in order to generate a magnetic field that will compensate for the existing (inhomogeneous) magnetic field. This procedure requires precise and accurate mapping of the B0 field. B0 maps show the difference between the current field and the expected field, and are typically displayed in units of magnetic field strength (Tesla [T]), precessional frequency (Hertz [Hz]) or in parts per million (ppm). Eq. 1 can be used to convert from the different units. 
+When a subject is introduced in the scanner, the static B0 field can be rendered more homogeneous through a technique called active shimming. Active shimming sends the appropriate amount of current through specific gradient and shim coils, in order to generate a magnetic field that will compensate for the existing (inhomogeneous) magnetic field. This procedure requires precise and accurate mapping of the B0 field. B0 maps show the difference between the current field and the expected field, and are typically displayed in units of magnetic field strength (Tesla [T]), precessional frequency (Hertz [Hz]) or in parts per million (ppm). [](#b0Eq1) can be used to convert from the different units. 
 
 ```{math}
 :label: b0Eq1
@@ -28,7 +28,7 @@ When a subject is introduced in the scanner, the static B0 field can be rendered
 
 where f, f0 and f represent the actual precessional frequency, the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) and the difference between current and the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) (f-f0=f) respectively. B, B0 and B all have similar interpretations as  f, f0 and f but are in units of field strength (T). The relationship between frequency and field strength can be found using the well known [Larmor equation](https://en.wikipedia.org/wiki/Larmor_precession) (f=B2). ppm is the field offset in ppm.
 
-Fig. 1 shows typical brain magnitude, phase, and B0 field map images of a brain in a 3 T scanner.
+[](#fig5p1cell) shows typical brain magnitude, phase, and B0 field map images of a brain in a 3 T scanner.
 
 :::{figure} #fig5p1cell
 :label: b0Plot1

5 B0 Mapping/1 B0 Inhomogeneities/2-Sources/02-Magnetic susceptibility.md

@@ -24,7 +24,7 @@ Materials have a property called magnetic susceptibility () that reflects their
 \end{equation}
 ```
 
-The dipole kernel (d) is illustrated in Fig. 2 along with the dipole kernel (D) in the k-space domain often used in QSM.
+The dipole kernel (d) is illustrated in [](#fig5p2cell) along with the dipole kernel (D) in the k-space domain often used in QSM.
 
 :::{figure} #fig5p2cell
 :label: b0Plot2
@@ -34,7 +34,7 @@ Dipole kernel (d) in the image domain as well as in the k-space domain (D).
 
 When a subject is introduced in the scanner, it interacts with the B0 field and distorts it. Therefore, a perfectly homogeneous field in an empty bore will usually have an inhomogeneous B0 field once a patient is introduced. This is the reason why active shimming is required when a patient is introduced in the scanner. Although these inhomogeneities happen everywhere in the body, stronger field variations occur at the boundaries of strong susceptibility differences such as air (slightly paramagnetic: +) and water/tissue (diamagnetic: -). 
 
-The following figure shows different susceptibility distributions in ppm for a homogeneous cylinder within a larger homogeneous cylinder placed in a homogeneous background (top) and a brain (bottom). The corresponding B0 field maps are simulated at 7 T and shown in Fig. 3. In the brain, the B0 field inhomogeneities are dominated by air-tissue boundaries. On the right-hand panel, the slow varying spatial variations (also called background field) were removed to show the local field variations.
+The following figure shows different susceptibility distributions in ppm for a homogeneous cylinder within a larger homogeneous cylinder placed in a homogeneous background (top) and a brain (bottom). The corresponding B0 field maps are simulated at 7 T and shown in [](#fig5p3cell). In the brain, the B0 field inhomogeneities are dominated by air-tissue boundaries. On the right-hand panel, the slow varying spatial variations (also called background field) were removed to show the local field variations.
 
 :::{figure} #fig5p3cell
 :label: b0Plot3

5 B0 Mapping/1 B0 Inhomogeneities/3-Effects on signal.md

@@ -15,7 +15,7 @@ numbering:
 ---
 To excite the spins in the transverse plane, a carrier frequency tuned to the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) is used by the transmit coil. If the frequency of the spins does not match the excitation frequency, it results in a suboptimal tip of the spins in the transverse plane. If the frequency of the spins varies across the ROI, the flip angle is affected differently across the image [12].
 
-When a signal is acquired, it is demodulated to remove the carrier frequency ([Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession)) from the signal. An example of a FID is shown in Fig. 4. The number of species represent the number of isochromats in the simulation. An isochromat represents an ensemble of spins with the same properties rotating at the same [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession). For a single isochromat, if the acquired signal and demodulation frequency perfectly match, the T2 signal can be recovered. If the carrier frequency is different from the expected frequency (such as when there are inhomogeneities), the demodulation introduces low-frequency variations. A non-homogeneous sample is also shown featuring many isochromats. Alternatively, a homogeneous sample with a non-homogeneous B0 field could be simulated as well and would have a similar shape as the one with multiple species. In that case, the difference from the T2 curve would reflect T2* (1T2*=1T2+1T2') effects. During the relaxation process, spins precessing at different frequencies, due to the presence  of B0 inhomogeneities, will give rise to phase offsets between the spins within a voxel. This intravoxel phase dispersion leads to signal decay. 
+When a signal is acquired, it is demodulated to remove the carrier frequency ([Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession)) from the signal. An example of a FID is shown in [](#fig5p4cell). The number of species represent the number of isochromats in the simulation. An isochromat represents an ensemble of spins with the same properties rotating at the same [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession). For a single isochromat, if the acquired signal and demodulation frequency perfectly match, the T2 signal can be recovered. If the carrier frequency is different from the expected frequency (such as when there are inhomogeneities), the demodulation introduces low-frequency variations. A non-homogeneous sample is also shown featuring many isochromats. Alternatively, a homogeneous sample with a non-homogeneous B0 field could be simulated as well and would have a similar shape as the one with multiple species. In that case, the difference from the T2 curve would reflect T2* (1T2*=1T2+1T2') effects. During the relaxation process, spins precessing at different frequencies, due to the presence  of B0 inhomogeneities, will give rise to phase offsets between the spins within a voxel. This intravoxel phase dispersion leads to signal decay. 
 
 
 :::{figure} #fig5p4cell
@@ -24,7 +24,7 @@ When a signal is acquired, it is demodulated to remove the carrier frequency ([L
 FID curves with signal demodulation at [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) (single species), at two different frequencies ([Larmor](https://en.wikipedia.org/wiki/Larmor_precession) and offset frequency, two species) and at multiple frequencies ([Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession) and many other offset frequencies, multiple species). The resulting shape of the graphs depends on the relative amplitudes and frequencies.
 :::
 
-B0 inhomogeneities can lead to distorted k-space trajectories during the readout gradient. This effect is worse during further k-space traversal due to the compounding of the errors. When inhomogeneities are present, the frequencies of the spins are altered. The one-to-one relationship between frequency and spatial location (required to obtain accurate spatial correspondence) is broken. This leads to geometric distortions. Fig. 5 shows an animation of the filing of k-space of an EPI sequence using bi-polar readouts. A theoretical trajectory is shown as well as a trajectory where a constant parasite gradient in the phase encoding direction has been added. One can observe the trajectory differences.
+B0 inhomogeneities can lead to distorted k-space trajectories during the readout gradient. This effect is worse during further k-space traversal due to the compounding of the errors. When inhomogeneities are present, the frequencies of the spins are altered. The one-to-one relationship between frequency and spatial location (required to obtain accurate spatial correspondence) is broken. This leads to geometric distortions. [](#fig5p5cell) shows an animation of the filing of k-space of an EPI sequence using bi-polar readouts. A theoretical trajectory is shown as well as a trajectory where a constant parasite gradient in the phase encoding direction has been added. One can observe the trajectory differences.
 
 :::{figure} #fig5p5cell
 :label: b0Plot5

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

@@ -14,7 +14,7 @@ numbering:
     template: Eq. %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. 
+In the ideal case, spins rotate at the [Larmor frequency](https://en.wikipedia.org/wiki/Larmor_precession), shown in blue in [](#fig5p6cell). 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
@@ -32,15 +32,15 @@ The phase () evolution follows the following equation (not considering transient
 \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.
+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 [](#fig5p7cell) 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.
+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 [](#b0Eq4).
 
 ```{math}
 :label: b0Eq4

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

@@ -14,7 +14,7 @@ numbering:
     template: Eq. %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.
+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 [](#fig5p8cell). 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

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

@@ -14,7 +14,7 @@ numbering:
     template: Eq. %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. 
+A brain dataset is used to show a concrete example of a field map that could be acquired in practice. [](#fig5p9cell) 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

5 B0 Mapping/3 Phase Unwrapping/02-Temporal Unwrapping.md

@@ -29,11 +29,10 @@ Temporal unwrapping uses multiple time points (>=2) to unwrap the image. By acqu
 
 If we assume a maximum field offset f0,max of 1 ppm at 3T (~127 MHz), we can calculate a maximum field offset (f0,max) of 1 µT or 127 Hz. The [Nyquist criteria](https://en.wikipedia.org/wiki/Nyquist_frequency) can be used to calculate the maximum echo time difference (TE) required to satisfy the no-wrapping requirement in the phase difference image (TE=1 2f0,max). Table 1 shows TE for multiple field strength assuming inhomogeneities of 1 ppm. As shown, the echo spacing is B0 dependent, as higher field offsets are observed at higher field strengths.
 
-Table 1: Maximum echo time required to respect the Nyquist criteria for different field strengths for B0 of 1 ppm.
-
-```{list-table} This table title
+```{list-table}  Maximum echo time required to respect the Nyquist criteria for different field strengths for B0 of 1 ppm.
 :header-rows: 1
 :label: b0Table1
+:enumerator:5.1
 * - B0 (T)
   - deltaTE (ms)
 * - 0.064
@@ -52,7 +51,7 @@ Table 1: Maximum echo time required to respect the Nyquist criteria for differen
 
 If a longer TE is selected, or if the inhomogeneities are bigger than originally anticipated in some parts of the image, the phase difference image could also have wraps, and spatial unwrapping would be necessary.
 
-Temporal unwrapping can also be performed without phase difference. Fig. 1 shows the phase of a voxel acquired at four echo times in blue. Note that the last echo time is wrapped (i.e.: the phase rotated by more than 2 and “wrapped” to the positive side). With the assumption that phase does not vary by more than 2, we can unwrap the phase by, in this case, subtracting 2 from the acquired phase to recover the true phase (in red). A linear fit is shown in green. Note that the slope would represent the field map value.
+Temporal unwrapping can also be performed without phase difference. [](#fig5p10cell) shows the phase of a voxel acquired at four echo times in blue. Note that the last echo time is wrapped (i.e.: the phase rotated by more than 2 and “wrapped” to the positive side). With the assumption that phase does not vary by more than 2, we can unwrap the phase by, in this case, subtracting 2 from the acquired phase to recover the true phase (in red). A linear fit is shown in green. Note that the slope would represent the field map value.
 
 :::{figure} #fig5p10cell
 :label: b0Plot10