Try fixing figur and eq numb

2 T1 Mapping/2-1 Inversion Recovery/01-introduction.md

@@ -22,6 +22,6 @@ Widely considered the gold standard for [T<sub>1</sub>](wiki:Spin–lattice_rela
 
 ```{figure} img/ir_pulsesequences.svg
 :label: irFig1
-:enumerator: 2.%s
+:enumerator: 2.1
 Pulse sequence of an inversion recovery experiment.
 ```

2 T1 Mapping/2-1 Inversion Recovery/02-IR_SignalModelling.md

@@ -18,6 +18,7 @@ The steady-state longitudinal magnetization of an [inversion recovery](wiki:Inve
 
 ```{math}
 :label: irEq1
+:enumerator:2.1
 \begin{equation}
 M_{z}(TI) = M_0 \frac{1-\text{cos}(\theta_{180})e^{- \frac{TR}{T_1}} -[1-\text{cos}(\theta_{180})]e^{- \frac{TI}{T_1}}}{1 - \text{cos}(\theta_{180}) \text{cos}(\theta_{90}) e^{- \frac{TR}{T_1}}}
 \end{equation}
@@ -27,15 +28,17 @@ where M<sub>z</sub> is the longitudinal magnetization prior to the θ<sub>90</su
 
 ```{math}
 :label: irEq2
+:enumerator:2.2
 \begin{equation}
 M_z(TI) = C(1-2e^{- \frac{TI}{T_1}} + e^{- \frac{TR}{T_1}})
 \end{equation}
 ```
 
-where the first three terms and the denominator of [](#irEq1) have been grouped together into the constant C. If the experiment is designed such that TR is long enough to allow for full relaxation of the magnetization (TR > 5T<sub>1</sub>), we can do an additional approximation by dropping the last term in [Equation 2.1](#irEq2):
+where the first three terms and the denominator of [](#irEq1) have been grouped together into the constant C. If the experiment is designed such that TR is long enough to allow for full relaxation of the magnetization (TR > 5T<sub>1</sub>), we can do an additional approximation by dropping the last term in [](#irEq2):
 
 ```{math}
 :label: irEq3
+:enumerator:2.3
 \begin{equation}
 M_z(TI) = C(1-2e^{- \frac{TI}{T_1}})
 \end{equation}
@@ -45,13 +48,15 @@ The simplicity of the signal model described by [Equation 2.3](#irEq3), both in
 
 :::{figure} #fig2p2cell
 :label: irPlot1
+:enumerator: 2.2
 Inversion recovery curves ([Equation 2.2](#irEq2)) for three different T1 values, approximating the main types of tissue in the brain.
 :::
 
 Practically, [Equation 2.1](#irEq1) is the better choice for simulating the signal of an [inversion recovery](wiki:Inversion_recovery) experiment, as the TRs are often chosen to be greater than 5T<sub>1</sub> of the tissue-of-interest, which rarely coincides with the longest T<sub>1</sub> present (e.g. TR may be sufficiently long for white matter, but not for CSF which could also be present in the volume). [Equation 2.3](#irEq3) also assumes ideal inversion pulses, which is rarely the case due to slice profile effects. [](#irPlot2) displays the [inversion recovery](wiki:Inversion_recovery) signal magnitude (complete relaxation normalized to 1) of an experiment with TR = 5 s and T<sub>1</sub> values ranging between 250 ms to 5 s, calculated using both equations.
 
 :::{figure} #fig2p3cell
 :label: irPlot2
+:enumerator: 2.3
 Signal recovery curves simulated using [Equation 2.3](#irEq3) (solid) and [Equation 2.1](#irEq1) (dotted) with a TR = 5 s for T1 values ranging between 0.25 to 5 s.
 :::
 

2 T1 Mapping/2-1 Inversion Recovery/03-IR_DataFitting.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 Several factors impact the choice of the [inversion recovery](wiki:Inversion_recovery) fitting algorithm.  If only magnitude images are available, then a polarity-inversion is often implemented to restore the non-exponential magnitude curves ([](#irPlot2)) into the [exponential](wiki:Exponential_function) form ([](#irPlot1)). This process is sensitive to noise due to the [Rician](wiki:Rice_distribution) noise creating a non-zero level at the signal null. If phase data is also available, then a phase term must be added to the fitting equation {cite:p}`Barral2010-qm`. [Equation 2.3](#irEq3) must only be used to fit data for the long TR regime (TR > 5T<sub>1</sub>), which in practice is rarely satisfied for all tissues in subjects.
@@ -28,7 +31,7 @@ where <i>a</i> and <i>b</i> are complex values. If magnitude-only data is availa
 
 :::{figure} #fig2p4cell
 :label: irPlot3
-:enumerator: 1.4
+:enumerator: 2.4
 Fitting comparison of simulated data (blue markers) with T_1 = 1 s and TR = 1.5 to 5 s, using fitted using RD-NLS & [Equation 2.4](#irEq4) (green) and [Levenberg-Marquardt](wiki:Levenberg–Marquardt_algorithm) & [Equation 2.2](#irEq2) (orange, long TR approximation).
 :::
 

2 T1 Mapping/2-1 Inversion Recovery/04-IR_BenefitsAndPitfalls.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 The conventional [inversion recovery](wiki:Inversion_recovery) experiment is considered the gold standard T<sub>1</sub> mapping technique for several reasons: 

2 T1 Mapping/2-1 Inversion Recovery/05-IR_OtherMethods.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 Several variations of the [inversion recovery](wiki:Inversion_recovery) pulse sequence were developed to overcome challenges like those specified above. Amongst them, the Look-Locker technique {cite:p}`Look1970` stands out as one of the most widely used in practice. Instead of a single 90° acquisition per TR, a periodic train of small excitation pulses θ are applied after the inversion pulse, {θ<sub>180</sub> – 𝛕 – θ – 𝛕 – θ – ...}, where  𝛕 = TR/n and n is the number of sampling acquisitions. This pulse sequence samples the inversion time relaxation curve much more efficiently than conventional [inversion recovery](wiki:Inversion_recovery), but at a cost of lower SNR. However, because the magnetization state of each TI measurement depends on the previous series of θ excitation, it has higher sensitivity to B<sub>1</sub>-inhomogeneities and imperfect spoiling compared to [inversion recovery](wiki:Inversion_recovery) {cite:p}`Gai2013,Stikov2015`. Nonetheless, Look-Locker is widely used for rapid T<sub>1</sub> mapping applications, and variants like MOLLI (Modified Look-Locker Inversion recovery) and ShMOLLI (Shortened MOLLI) are widely used for cardiac T<sub>1</sub> mapping {cite:p}`Messroghli2004,Piechnik2010`.

2 T1 Mapping/2-2 Variable Flip Angle/01-Introduction.md

@@ -7,9 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
-  heading_2: false
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 ## Variable Flip Angle T<sub>1</sub> Mapping

2 T1 Mapping/2-2 Variable Flip Angle/02-VFA_SignalModelling.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 The steady-state longitudinal magnetization of an ideal variable flip angle experiment can be analytically solved from the [Bloch equations](wiki:Bloch_equations) for the spoiled [gradient echo](wiki:Gradient_echo) pulse sequence {<i>θ<sub>n</sub></i>–TR}:

2 T1 Mapping/2-2 Variable Flip Angle/03-VFA_DataFitting.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 At first glance, one could be tempted to fit VFA data using a [non-linear least squares](wiki:Non-linear_least_squares) fitting algorithm such as Levenberg-Marquardt with [Equation 2.5](#vfaEq1), which typically only has two free fitting variables ([T<sub>1</sub>](wiki:Spin–lattice_relaxation) and _M_<sub>0</sub>). Although this is a valid way of estimating [T<sub>1</sub>](wiki:Spin–lattice_relaxation) from VFA data, it is rarely done in practice because a simple refactoring of [Equation 2.5](#vfaEq1) allows [T<sub>1</sub>](wiki:Spin–lattice_relaxation) values to be estimated with a [linear least square](Linear_least_squares) fitting algorithm, which substantially reduces the processing time. Without any approximations, [Equation 2.5](#vfaEq1) can be rearranged into the form <b>y</b> = m<b>x</b>+b {cite:p}`Gupta1977`:

2 T1 Mapping/2-2 Variable Flip Angle/04-VFA_BenefitsAndPitfalls.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 It has been well reported in recent years that the accuracy of VFA [T<sub>1</sub>](wiki:Spin–lattice_relaxation) estimates is very sensitive to pulse sequence implementations {cite:p}`Baudrexel2017,Lutti2013,Stikov2015`, and as such is less robust than the gold standard inversion recovery technique. In particular, the signal bias resulting from insufficient spoiling can result in inaccurate [T<sub>1</sub>](wiki:Spin–lattice_relaxation) estimates of up to 30% relative to inversion recovery estimated values {cite:p}`Stikov2015`. VFA [T<sub>1</sub>](wiki:Spin–lattice_relaxation) map accuracy and precision is also strongly dependent on the quality of the measured B<sub>1</sub> map {cite:p}`Lee2017`, which can vary substantially between implementations {cite:p}`Boudreau2017`. Modern rapid B<sub>1</sub> mapping pulse sequences are not as widely available as VFA, resulting in some groups attempting alternative ways of removing the bias from the [T<sub>1</sub>](wiki:Spin–lattice_relaxation) maps like generating an artificial B<sub>1</sub> map through the use of image processing techniques {cite:p}`Liberman2013` or omitting B<sub>1</sub> correction altogether {cite:p}`Yuan2012`. The latter is not recommended, because most MRI scanners have default pulse sequences that, with careful protocol settings, can provide B<sub>1</sub> maps of sufficient quality very rapidly {cite:p}`Boudreau2017,Samson2006,Wang2005`.

2 T1 Mapping/2-3 MP2RAGE/01-Abstract.md

@@ -7,9 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
-  heading_2: false
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 Dictionary-based MRI techniques capable of generating T<sub>1</sub> maps are increasing in popularity, due to their growing availability on clinical scanners, rapid scan times, and fast post-processing computation time, thus making quantitative T<sub>1</sub> mapping accessible for clinical applications. Generally speaking, dictionary-based quantitative MRI techniques use numerical dictionaries—databases of pre-calculated signal values simulated for a wide range of tissue and protocol combinations—during the image reconstruction or post-processing stages. Popular examples of dictionary-based techniques that have been applied to T<sub>1</sub> mapping are MR Fingerprinting (MRF) (Ma et al. 2013), certain flavours of compressed sensing (CS) (Doneva et al. 2010; Li et al. 2012), and Magnetization Prepared 2 Rapid Acquisition Gradient Echoes (MP2RAGE) (Marques et al. 2010). Dictionary-based techniques can usually be classified into one of two categories: techniques that use information redundancy from parametric data to assist in accelerated imaging (e.g. CS, MRF), or those that use dictionaries to estimate quantitative maps using the MR images after reconstruction. Because MP2RAGE is a technique implemented primarily for T<sub>1</sub> mapping, and it is becoming increasingly available as a standard pulse sequence on many MRI systems, the remainder of this section will focus solely on this technique. However, many concepts discussed are shared by other dictionary-based techniques.

2 T1 Mapping/2-3 MP2RAGE/02-SignalModelling.md

@@ -7,9 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
-  heading_2: false
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 Prior to considering the full signal equations, we will first introduce the equation for the MP2RAGE parameter (<i>S</i><sub>MP2RAGE</sub>) that is calculated in addition to the T<sub>1</sub> map. For complex data (magnitude and phase, or real and imaginary), the MP2RAGE signal (<i>S</i><sub>MP2RAGE</sub>) is calculated from the images acquired at two TIs (<i>S</i><sub>GRE,TI1</sub> and <i>S</i><sub>GRE,TI2</sub>) using the following expression (Marques et al. 2010):

2 T1 Mapping/2-3 MP2RAGE/03-DataFitting.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---
 
 Dictionary-based techniques such as MP2RAGE do not typically use conventional minimization algorithms (e.g. Levenberg-Marquardt) to fit signal equations to observed data. Instead, the MP2RAGE technique uses pre-calculated signal values for a wide range of parameter values (e.g. T<sub>1</sub>), and then interpolation is done within this dictionary of values to estimate the T<sub>1</sub> value that matches the observed signal. This approach results in rapid post-processing times because the dictionaries can be simulated/generated prior to scanning and interpolating between these values is much faster than most fitting algorithms. This means that the quantitative image can be produced and displayed directly on the MRI scanner console rather than needing to be fitted offline.
@@ -17,6 +20,7 @@ MP2RAGE is an extension of the conventional MPRAGE pulse sequence widely used in
 
 ```{figure} #mp2rageFig1cell
 :label: mp2rageplot1
+:enumerator: 2.14
 
 T<sub>1</sub> lookup table as a function of B<sub>1</sub> and <i>S</i><sub>MP2RAGE</sub> value. Inversion times used to acquire this magnitude image dataset were 800 ms and 2700 ms, the flip angles were 4° and 5° (respectively),  TR<sub>MP2RAGE</sub> = 6000 ms, and TR = 6.7 ms. The code that was used were shared open sourced by the authors of the original MP2RAGE paper (<a href="url">https://github.com/JosePMarques/MP2RAGE-related-scripts</a>).
 ```
@@ -26,6 +30,7 @@ To produce T<sub>1</sub> maps with good accuracy and precision using dictionary-
 
 ```{figure} #mp2rageFig2cell
 :label: mp2rageplot2
+:enumerator: 2.15
 Example MP2RAGE dataset of a healthy adult brain at 7T and T<sub>1</sub> map. Inversion times used to acquire this magnitude image dataset were 800 ms and 2700 ms, the flip angles were 4° and 5° (respectively),  TR<sub>MP2RAGE</sub> = 6000 ms, and TR = 6.7 ms. The dataset and code that was used were shared open sourced by the authors of the original MP2RAGE paper (<a href="url">https://github.com/JosePMarques/MP2RAGE-related-scripts</a>).
 ```
 

2 T1 Mapping/2-3 MP2RAGE/04-BenefitsAndPitfalls.md

@@ -7,8 +7,11 @@ authors:
     affiliations:
       - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
 numbering:
+  heading_2: true
   figure:
-    template: Fig. %s
+    template: Figure 2.%s
+  equation:
+    template: Eq. 2.%s
 ---