diff --git a/2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.ipynb b/2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.ipynb
deleted file mode 100644
index f6ebffe..0000000
--- a/2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.ipynb
+++ /dev/null
@@ -1,562 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "id": "4c1c4b7f-505f-4112-8b4e-59fdd766eac2",
- "metadata": {
- "editable": true,
- "slideshow": {
- "slide_type": ""
- },
- "tags": []
- },
- "source": [
- "---\n",
- "title: Signal Modelling\n",
- "subtitle: Inversion Recovery\n",
- "date: 2024-07-25\n",
- "authors:\n",
- " - name: Mathieu Boudreau\n",
- " affiliations:\n",
- " - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada\n",
- "numbering:\n",
- " figure:\n",
- " template: Fig. %s\n",
- "---\n",
- "\n",
- "The steady-state longitudinal magnetization of an inversion recovery experiment can be derived from the Bloch equations for the pulse sequence {θ180 – TI – θ90 – (TR-TI)}, and is given by:\n",
- "\n",
- "\\begin{equation}\\label{eq:1}\n",
- "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}}}\n",
- "\\end{equation}\n",
- "\n",
- "where Mz is the longitudinal magnetization prior to the θ90 pulse. If the in-phase real signal is desired, it can be calculated by multiplying Eq. 1 by ksin(θ90)e-TE/T2, where k is a constant. This general equation can be simplified by grouping together the constants for each measurements regardless of their values (i.e. at each TI, same TE and θ90 are used) and assuming an ideal inversion pulse:\n",
- "\n",
- "\\begin{equation}\\label{eq:2}\n",
- "M_z(TI) = C(1-2e^{- \\frac{TI}{T_1}} + e^{- \\frac{TR}{T_1}})\n",
- "\\end{equation}\n",
- "\n",
- "where the first three terms and the denominator of Eq. 1 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 > 5T1), we can do an additional approximation by dropping the last term in Eq. 2:\n",
- "\n",
- "\\begin{equation}\\label{eq:3}\n",
- "M_z(TI) = C(1-2e^{- \\frac{TI}{T_1}})\n",
- "\\end{equation}\n",
- "\n",
- "The simplicity of the signal model described by Eq. 3, both in its equation and experimental implementation, has made it the most widely used equation to describe the signal evolution in an inversion recovery T1 mapping experiment. The magnetization curves are plotted in Figure 2 for approximate T1 values of three different tissues in the brain. Note that in many practical implementations, magnitude-only images are acquired, so the signal measured would be proportional to the absolute value of Eq. 3.\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "id": "c2475527-acd3-47cf-a65b-10902e7d5866",
- "metadata": {
- "editable": true,
- "slideshow": {
- "slide_type": ""
- },
- "tags": [
- "hide-output",
- "hide-input"
- ]
- },
- "outputs": [],
- "source": [
- "from repo2data.repo2data import Repo2Data\n",
- "import os \n",
- "import pickle\n",
- "import matplotlib.pyplot as plt\n",
- "import chart_studio.plotly as py\n",
- "import plotly.graph_objs as go\n",
- "import numpy as np\n",
- "from plotly import __version__\n",
- "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
- "from IPython.display import display, HTML\n",
- "from plotly import tools\n",
- "\n",
- "data_req_path = os.path.join(\"..\",\"..\", \"binder\", \"data_requirement.json\")\n",
- "repo2data = Repo2Data(data_req_path)\n",
- "DATA_ROOT = os.path.join(repo2data.install()[0],\"t1-book-neurolibre\")"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "id": "64331b70-ae8a-42a6-af7f-b275c36c4580",
- "metadata": {
- "editable": true,
- "slideshow": {
- "slide_type": ""
- },
- "tags": [
- "hide-input"
- ]
- },
- "outputs": [],
- "source": [
- "filename = os.path.join(DATA_ROOT,\"01\",'figure_2.pkl')\n",
- "\n",
- "with open(filename, 'rb') as f:\n",
- " params, signal_WM, signal_GM, signal_CSF = pickle.load(f)\n",
- "\n",
- "config={'showLink': False, 'displayModeBar': False}\n",
- "\n",
- "init_notebook_mode(connected=True)\n",
- "\n",
- "wm = go.Scatter(\n",
- " x = params[\"TI\"],\n",
- " y = signal_WM,\n",
- " name = 'T1 = 0.9 s (White Matter)',\n",
- " text = 'T1 = 0.9 s (White Matter)',\n",
- " hoverinfo = 'x+y+text'\n",
- ")\n",
- "\n",
- "gm = go.Scatter(\n",
- " x = params[\"TI\"],\n",
- " y = signal_GM,\n",
- " name = 'T1 = 1.5 s (Grey Matter)',\n",
- " text = 'T1 = 1.5 s (Grey Matter)',\n",
- " hoverinfo = 'x+y+text'\n",
- ")\n",
- "\n",
- "csf = go.Scatter(\n",
- " x = params[\"TI\"],\n",
- " y = signal_CSF,\n",
- " name = 'T1 = 4.0 s (Cerebrospinal Fluid)',\n",
- " text = 'T1 = 4.0 s (Cerebrospinal Fluid)',\n",
- " hoverinfo = 'x+y+text'\n",
- ")\n",
- "\n",
- "data = [wm, gm, csf]\n",
- "\n",
- "layout = go.Layout(\n",
- " width=600,\n",
- " height=350,\n",
- " margin=go.layout.Margin(\n",
- " l=100,\n",
- " r=50,\n",
- " b=60,\n",
- " t=0,\n",
- " ),\n",
- " annotations=[\n",
- " dict(\n",
- " x=0.5004254919715793,\n",
- " y=-0.175,\n",
- " showarrow=False,\n",
- " text='Inversion Time – TI (ms)',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=22\n",
- " ),\n",
- " xref='paper',\n",
- " yref='paper'\n",
- " ),\n",
- " dict(\n",
- " x=-0.15,\n",
- " y=0.50,\n",
- " showarrow=False,\n",
- " text='Long. Magnetization (Mz)',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=22\n",
- " ),\n",
- " textangle=-90,\n",
- " xref='paper',\n",
- " yref='paper'\n",
- " ),\n",
- " ],\n",
- " xaxis=dict(\n",
- " showgrid=False,\n",
- " linecolor='black',\n",
- " linewidth=2\n",
- " ),\n",
- " yaxis=dict(\n",
- " showgrid=False,\n",
- " linecolor='black',\n",
- " linewidth=2\n",
- " ),\n",
- " legend=dict(\n",
- " x=0.55,\n",
- " y=0.15,\n",
- " traceorder='normal',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=12,\n",
- " color='#000'\n",
- " ),\n",
- " bordercolor='#000000',\n",
- " borderwidth=2\n",
- " ),\n",
- " plot_bgcolor='white'\n",
- ")\n",
- "\n",
- "fig = dict(data=data, layout=layout)\n",
- "\n",
- "plot(fig, filename = 'ir_fig_2.html', config = config)\n",
- "\n",
- "display(HTML('ir_fig_2.html'))"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "ab705653-9cf7-4495-865b-90e95951e3d2",
- "metadata": {
- "editable": true,
- "slideshow": {
- "slide_type": ""
- },
- "tags": []
- },
- "source": [
- "```{figure} img/plot1.png\n",
- ":label: irPlot1\n",
- "\n",
- "Inversion recovery curves (Eq. 2) for three different T1 values, approximating the main types of tissue in the brain.\n",
- "```\n",
- "\n",
- "Practically, Eq. 1 is the better choice for simulating the signal of an inversion recovery experiment, as the TRs are often chosen to be greater than 5T1 of the tissue-of-interest, which rarely coincides with the longest T1 present (e.g. TR may be sufficiently long for white matter, but not for CSF which could also be present in the volume). Equation 3 also assumes ideal inversion pulses, which is rarely the case due to slice profile effects. Figure 3 displays the inversion recovery signal magnitude (complete relaxation normalized to 1) of an experiment with TR = 5 s and T1 values ranging between 250 ms to 5 s, calculated using both equations."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "id": "2f86f9e3-8855-4312-be1f-9637120665e1",
- "metadata": {
- "editable": true,
- "slideshow": {
- "slide_type": ""
- },
- "tags": [
- "hide-input"
- ]
- },
- "outputs": [],
- "source": [
- "filename = os.path.join(DATA_ROOT,\"01\",'figure_3.pkl')\n",
- "\n",
- "with open(filename, 'rb') as f:\n",
- " T1_range, signal_T1_Eq1, signal_T1_Eq3 = pickle.load(f)\n",
- "\n",
- "config={'showLink': False, 'displayModeBar': False}\n",
- "\n",
- "init_notebook_mode(connected=True)\n",
- "\n",
- "data1 = [dict(\n",
- " visible = False,\n",
- " x = params[\"TI\"],\n",
- " y = abs(signal_T1_Eq3[ii]),\n",
- " name = '[Eq. 3] – Long TR approximation',\n",
- " text = '[Eq. 3] – Long TR approximation',\n",
- " hoverinfo = 'x+y+text') for ii in range(len(T1_range))]\n",
- "\n",
- "data1[3]['visible'] = True\n",
- "\n",
- "data2 = [dict(\n",
- " visible = False,\n",
- " x = params[\"TI\"],\n",
- " y = abs(signal_T1_Eq1[ii]),\n",
- " line = dict(\n",
- " color = ('rgb(22, 96, 167)'),\n",
- " dash = 'dash'),\n",
- " name = '[Eq. 1] – General Equation',\n",
- " text = '[Eq. 1] – General Equation',\n",
- " hoverinfo = 'x+y+text') for ii in range(len(T1_range))]\n",
- "\n",
- "data2[3]['visible'] = True\n",
- "\n",
- "data = data1 + data2\n",
- "\n",
- "steps = []\n",
- "for i in range(len(T1_range)):\n",
- " step = dict(\n",
- " method = 'restyle', \n",
- " args = ['visible', [False] * len(data1)],\n",
- " label = str(T1_range[i])\n",
- " )\n",
- " step['args'][1][i] = True # Toggle i'th trace to \"visible\"\n",
- " steps.append(step)\n",
- "\n",
- "sliders = [dict(\n",
- " x = 0,\n",
- " y = -0.0,\n",
- " active = 3,\n",
- " currentvalue = {\"prefix\": \"T1 value (s): \"},\n",
- " pad = {\"t\": 50, \"b\": 10},\n",
- " steps = steps\n",
- ")]\n",
- "\n",
- "layout = go.Layout(\n",
- " width=580,\n",
- " height=400,\n",
- " margin=go.layout.Margin(\n",
- " l=80,\n",
- " r=40,\n",
- " b=60,\n",
- " t=10,\n",
- " ),\n",
- " annotations=[\n",
- " dict(\n",
- " x=0.5004254919715793,\n",
- " y=-0.2,\n",
- " showarrow=False,\n",
- " text='Inversion Time – TI (ms)',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=22\n",
- " ),\n",
- " xref='paper',\n",
- " yref='paper'\n",
- " ),\n",
- " dict(\n",
- " x=-0.14,\n",
- " y=0.5,\n",
- " showarrow=False,\n",
- " text='Signal (magnitude)',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=22\n",
- " ),\n",
- " textangle=-90,\n",
- " xref='paper',\n",
- " yref='paper'\n",
- " ),\n",
- " ],\n",
- " xaxis=dict(\n",
- " autorange=False,\n",
- " range=[0, 5],\n",
- " showgrid=False,\n",
- " linecolor='black',\n",
- " linewidth=2\n",
- " ),\n",
- " yaxis=dict(\n",
- " autorange=False,\n",
- " range=[0, 1],\n",
- " showgrid=False,\n",
- " linecolor='black',\n",
- " linewidth=2\n",
- " ),\n",
- " legend=dict(\n",
- " x=0.5,\n",
- " y=0.5,\n",
- " traceorder='normal',\n",
- " font=dict(\n",
- " family='Times New Roman',\n",
- " size=12,\n",
- " color='#000'\n",
- " ),\n",
- " bordercolor='#000000',\n",
- " borderwidth=2\n",
- " ), \n",
- " sliders=sliders,\n",
- " plot_bgcolor='white'\n",
- ")\n",
- "\n",
- "fig = dict(data=data, layout=layout)\n",
- "\n",
- "plot(fig, filename = 'ir_fig_3.html', config = config)\n",
- "display(HTML('ir_fig_3.html'))"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "00b03f12",
- "metadata": {},
- "source": [
- "```{figure} img/plot2.png\n",
- ":label: irPlot2\n",
- "\n",
- "Signal recovery curves simulated using Eq. 3 (solid) and Eq. 1 (dotted) with a TR = 5 s for T1 values ranging between 0.25 to 5 s.\n",
- "```"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "3405434d",
- "metadata": {
- "editable": false,
- "slideshow": {
- "slide_type": ""
- },
- "tags": []
- },
- "source": [
- "\n",
- "````{admonition} Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 2.\n",
- ":class: tip, dropdown\n",
- "\n",
- "```octave\n",
- "\n",
- "% Verbosity level 0 overrides the disp function and supresses warnings.\n",
- "% Once executed, they cannot be restored in this session\n",
- "% (kernel needs to be restarted or a new notebook opened.)\n",
- "VERBOSITY_LEVEL = 0;\n",
- "\n",
- "if VERBOSITY_LEVEL==0\n",
- " % This hack was used to supress outputs from external tools\n",
- " % in the Jupyter Book.\n",
- " function disp(x)\n",
- " end\n",
- " warning('off','all')\n",
- "end\n",
- "\n",
- "try\n",
- " cd qMRLab\n",
- "catch\n",
- " try\n",
- " cd ../../../qMRLab\n",
- " catch\n",
- " cd ../qMRLab\n",
- " end\n",
- "end\n",
- "\n",
- "startup\n",
- "clear all\n",
- "\n",
- "%% Setup parameters\n",
- "% All times are in seconds\n",
- "% All flip angles are in degrees\n",
- "\n",
- "params.TR = 5.0;\n",
- "params.TI = linspace(0.001, params.TR, 1000);\n",
- " \n",
- "params.TE = 0.004;\n",
- "params.T2 = 0.040;\n",
- " \n",
- "params.EXC_FA = 90; % Excitation flip angle\n",
- "params.INV_FA = 180; % Inversion flip angle\n",
- "\n",
- "params.signalConstant = 1;\n",
- "\n",
- "%% Calculate signals\n",
- "%\n",
- "% The option 'GRE-IR' selects the analytical equations for the\n",
- "% gradient echo readout inversion recovery experiment The option\n",
- "% '4' is a flag that selects the long TR approximation of the \n",
- "% analytical solution (TR>>T1), Eq. 3 of the blog post.\n",
- "%\n",
- "% To see all the options available, run:\n",
- "% `help inversion_recovery.analytical_solution`\n",
- "\n",
- "% White matter\n",
- "params.T1 = 0.900; % in seconds\n",
- "\n",
- "signal_WM = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);\n",
- "\n",
- "% Grey matter\n",
- "params.T1 = 1.500; % in seconds\n",
- "signal_GM = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);\n",
- "\n",
- "% CSF\n",
- "params.T1 = 4.000; % in seconds\n",
- "signal_CSF = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);\n",
- "```\n",
- "\n",
- "````"
- ]
- },
- {
- "cell_type": "markdown",
- "id": "0a19cc2f",
- "metadata": {
- "editable": false,
- "slideshow": {
- "slide_type": ""
- },
- "tags": []
- },
- "source": [
- "\n",
- "```{admonition} Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 3.\n",
- ":class: tip, dropdown\n",
- "\n",
- "```octave\n",
- "% Verbosity level 0 overrides the disp function and supresses warnings.\n",
- "% Once executed, they cannot be restored in this session\n",
- "% (kernel needs to be restarted or a new notebook opened.)\n",
- "VERBOSITY_LEVEL = 0;\n",
- "\n",
- "if VERBOSITY_LEVEL==0\n",
- " % This hack was used to supress outputs from external tools\n",
- " % in the Jupyter Book.\n",
- " function disp(x)\n",
- " end\n",
- " warning('off','all')\n",
- "end\n",
- "\n",
- "try\n",
- " cd qMRLab\n",
- "catch\n",
- " try\n",
- " cd ../../../qMRLab\n",
- " catch\n",
- " cd ../qMRLab\n",
- " end\n",
- "end\n",
- "\n",
- "startup\n",
- "clear all\n",
- "\n",
- "%% Setup parameters\n",
- "% All times are in seconds\n",
- "% All flip angles are in degrees\n",
- "\n",
- "params.TR = 5.0;\n",
- "params.TI = linspace(0.001, params.TR, 1000);\n",
- " \n",
- "params.TE = 0.004;\n",
- "params.T2 = 0.040;\n",
- " \n",
- "params.EXC_FA = 90; % Excitation flip angle\n",
- "params.INV_FA = 180; % Inversion flip angle\n",
- "\n",
- "params.signalConstant = 1;\n",
- "\n",
- "T1_range = 0.25:0.25:5; % in seconds\n",
- "\n",
- "%% Calculate signals\n",
- "%\n",
- "% The option 'GRE-IR' selects the analytical equations for the\n",
- "% gradient echo readout inversion recovery experiment. The option\n",
- "% '1' is a flag that selects full analytical solution equation \n",
- "% (no approximation), Eq. 1 of the blog post. The option '4' is a\n",
- "% flag that selects the long TR approximation of the analytical \n",
- "% solution (TR>>T1), Eq. 3 of the blog post.\n",
- "%\n",
- "% To see all the options available, run:\n",
- "% `help inversion_recovery.analytical_solution`\n",
- "\n",
- "for ii = 1:length(T1_range)\n",
- " params.T1 = T1_range(ii);\n",
- " \n",
- " signal_T1_Eq1{ii} = inversion_recovery.analytical_solution(params, 'GRE-IR', 1);\n",
- "\n",
- " signal_T1_Eq3{ii} = inversion_recovery.analytical_solution(params, 'GRE-IR', 4);\n",
- "end\n",
- "```\n",
- "\n",
- "```"
- ]
- }
- ],
- "metadata": {
- "celltoolbar": "Tags",
- "kernelspec": {
- "display_name": "Python 3 (ipykernel)",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.8.18"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 5
-}
diff --git a/2 T1 Mapping/Inversion Recovery T1 Mapping/myst-02-IR_SignalModelling.md b/2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.md
similarity index 99%
rename from 2 T1 Mapping/Inversion Recovery T1 Mapping/myst-02-IR_SignalModelling.md
rename to 2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.md
index 3088981..7e723d2 100644
--- a/2 T1 Mapping/Inversion Recovery T1 Mapping/myst-02-IR_SignalModelling.md
+++ b/2 T1 Mapping/Inversion Recovery T1 Mapping/02-IR_SignalModelling.md
@@ -11,7 +11,6 @@ numbering:
template: Fig. %s
---
-## Articles with code vs articles from code
The steady-state longitudinal magnetization of an inversion recovery experiment can be derived from the Bloch equations for the pulse sequence {θ180 – TI – θ90 – (TR-TI)}, and is given by:
\begin{equation}\label{eq:1}
diff --git a/2 T1 Mapping/Inversion Recovery T1 Mapping/03-IR_DataFitting.md b/2 T1 Mapping/Inversion Recovery T1 Mapping/03-IR_DataFitting.md
new file mode 100644
index 0000000..09e7e78
--- /dev/null
+++ b/2 T1 Mapping/Inversion Recovery T1 Mapping/03-IR_DataFitting.md
@@ -0,0 +1,194 @@
+---
+title: Data Fitting
+subtitle: Inversion Recovery
+date: 2024-07-25
+authors:
+ - name: Mathieu Boudreau
+ affiliations:
+ - NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada
+numbering:
+ figure:
+ template: Fig. %s
+---
+
+Several factors impact the choice of the inversion recovery fitting algorithm. If only magnitude images are available, then a polarity-inversion is often implemented to restore the non-exponential magnitude curves (Figure 3) into the exponential form (Figure 2). This process is sensitive to noise due to the Rician 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 3 must only be used to fit data for the long TR regime (TR > 5T1), which in practice is rarely satisfied for all tissues in subjects.
+
+Early implementations of inversion recovery fitting algorithms were designed around the computational power available at the time. These included the “null method” {cite:p}`Pykett1983`, assuming that each T1 value has unique zero-crossings (see Figure 2), and linear fitting of a rearranged version of Eq. 3 on a semi-log plot {cite:p}`Fukushima1981`. Nowadays, a non-linear least-squares fitting algorithm (e.g. Levenberg-Marquardt) is more appropriate, and can be applied to either approximate or general forms of the signal model (Eq. 3 or Eq. 1). More recent work {cite:p}`Barral2010-qm` demonstrated that T1 maps can also be fitted much faster (up to 75 times compared to Levenberg-Marquardt) to fit Eq. 1 – without a precision penalty – by using a reduced-dimension non-linear least squares (RD-NLS) algorithm. It was demonstrated that the following simplified 5-parameter equation can be sufficient for accurate T1 mapping:
+
+\begin{equation}\label{eq:4}
+S(TI) = a + be^{- \frac{TI}{T_1}}
+\end{equation}
+
+where a and b are complex values. If magnitude-only data is available, a 3-parameter model can be sufficient by taking the absolute value of Eq. 4. While the RD-NLS algorithms are too complex to be presented here (the reader is referred to the paper, (Barral et al. 2010)), the code for these algorithms [was released open-source](http://www-mrsrl.stanford.edu/~jbarral/t1map.html) along with the original publication, and is also available as a [qMRLab](https://github.com/qMRLab/qMRLab) T1 mapping model. One important thing to note about Eq. 4 is that it is general – no assumption is made about TR – and is thus as robust as Eq. 1 as long as all pulse sequence parameters other than TI are kept constant between each measurement. Figure 4 compares simulated data (Eq. 1) using a range of TRs (1.5T1 to 5T1) fitted using either RD-NLS & Eq. 4 or a Levenberg-Marquardt fit of Eq. 2.
+
+:::{figure} #fig2p4cell
+:label: irPlot3
+Fitting comparison of simulated data (blue markers) with T1 = 1 s and TR = 1.5 to 5 s, using fitted using RD-NLS & Eq. 4 (green) and Levenberg-Marquardt & Eq. 2 (orange, long TR approximation).
+:::
+
+
+Figure 5 displays an example brain dataset from an inversion recovery experiment, along with the T1 map fitted using the RD-NLS technique.
+
+
+:::{figure} #fig2p5cell
+:label: irPlot4
+Example inversion recovery dataset of a healthy adult brain (left). Inversion times used to acquire this magnitude image dataset were 30 ms, 530 ms, 1030 ms, and 1530 ms, and the TR used was 1550 ms. The T1 map (right) was fitted using a RD-NLS algorithm.
+:::
+
+
+```{admonition} Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 4.
+:class: tip, dropdown
+
+
+```octave
+
+% Verbosity level 0 overrides the disp function and supresses warnings.
+% Once executed, they cannot be restored in this session
+% (kernel needs to be restarted or a new notebook opened.)
+VERBOSITY_LEVEL = 0;
+
+if VERBOSITY_LEVEL==0
+ % This hack was used to supress outputs from external tools
+ % in the Jupyter Book.
+ function disp(x)
+ end
+ warning('off','all')
+end
+
+try
+ cd qMRLab
+catch
+ try
+ cd ../../../qMRLab
+ catch
+ cd ../qMRLab
+ end
+end
+
+startup
+clear all
+
+%% Setup parameters
+% All times are in milliseconds
+% All flip angles are in degrees
+
+params.TI = 50:50:1500;
+TR_range = 1500:50:5000;
+
+params.EXC_FA = 90;
+params.INV_FA = 180;
+
+params.T1 = 1000;
+
+%% Calculate signals
+%
+% The option 'GRE-IR' selects the analytical equations for the gradient echo readout inversion recovery experiment
+% The option '1' is a flag that selects full analytical solution equation (no approximation), Eq. 1 of the blog post.
+%
+% To see all the options available, run `help inversion_recovery.analytical_solution`
+
+for ii = 1:length(TR_range)
+ params.TR = TR_range(ii);
+ Mz_analytical(ii,:) = inversion_recovery.analytical_solution(params, 'GRE-IR', 1);
+end
+
+%% Fit data using Levenberg-Marquardt with the long TR approximation equation
+%
+% The option '4' is a flag that selects the long TR approximation of the analytical solution (TR>>T1), Eq. 3 of the blog post.
+%
+% To see all the options available, run `help inversion_recovery.fit_lm`
+
+
+for ii=1:length(TR_range)
+ fitOutput_lm{ii} = inversion_recovery.fit_lm(Mz_analytical(ii,:), params, 4);
+ T1_lm(ii) = fitOutput_lm{ii}.T1;
+end
+
+%% Fit data using the RDLS method (Barral), Eq. 4 of the blog post.
+%
+
+% Create a qMRLab inversion recovery model object and load protocol values
+irObj = inversion_recovery();
+irObj.Prot.IRData.Mat = params.TI';
+
+for ii=1:length(TR_range)
+
+ data.IRData = Mz_analytical(ii,:);
+
+ fitOutput_barral{ii} = irObj.fit(data);
+
+ T1_barral(ii) = fitOutput_barral{ii}.T1;
+
+end
+```
+
+```
+
+
+```{admonition} Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 5.
+:class: tip, dropdown
+
+```octave
+% Verbosity level 0 overrides the disp function and supresses warnings.
+% Once executed, they cannot be restored in this session
+% (kernel needs to be restarted or a new notebook opened.)
+VERBOSITY_LEVEL = 0;
+
+if VERBOSITY_LEVEL==0
+ % This hack was used to supress outputs from external tools
+ % in the Jupyter Book.
+ function disp(x)
+ end
+ warning('off','all')
+end
+
+try
+ cd qMRLab
+catch
+ try
+ cd ../../../qMRLab
+ catch
+ cd ../qMRLab
+ end
+end
+
+startup
+
+clear all
+
+%% MATLAB/OCTAVE CODE
+
+% Load data into environment, and rotate mask to be aligned with IR data
+load('data/ir_dataset/IRData.mat');
+load('data/ir_dataset/IRMask.mat');
+
+IRData = data;
+Mask = imrotate(Mask,180);
+clear data
+
+% Format qMRLab inversion_recovery model parameters, and load them into the Model object
+Model = inversion_recovery;
+TI = [30; 530; 1030; 1530];
+Model.Prot.IRData.Mat = [TI];
+
+% Format data structure so that they may be fit by the model
+data = struct();
+data.IRData= double(IRData);
+data.Mask= double(Mask);
+
+FitResults = FitData(data,Model,0); % The '0' flag is so that no wait bar is shown.
+
+% Code used to re-orient the images to make pretty figures, and to assign variables with the axis lengths.
+
+T1_map = imrotate(FitResults.T1.*Mask,-90);
+xAxis = [0:size(T1_map,2)-1];
+yAxis = [0:size(T1_map,1)-1];
+
+% Raw MRI data at different TI values
+TI_0030 = imrotate(squeeze(IRData(:,:,:,1).*Mask),-90);
+TI_0530 = imrotate(squeeze(IRData(:,:,:,2).*Mask),-90);
+TI_1030 = imrotate(squeeze(IRData(:,:,:,3).*Mask),-90);
+TI_1530 = imrotate(squeeze(IRData(:,:,:,4).*Mask),-90);
+```
+
+```
\ No newline at end of file
diff --git a/figures/2 T1 mapping/figure_2-4.ipynb b/figures/2 T1 mapping/figure_2-4.ipynb
new file mode 100644
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+ "#| label: fig2p4cell\n",
+ "from repo2data.repo2data import Repo2Data\n",
+ "import os \n",
+ "import pickle\n",
+ "import matplotlib.pyplot as plt\n",
+ "import chart_studio.plotly as py\n",
+ "import plotly.graph_objs as go\n",
+ "import numpy as np\n",
+ "from plotly import __version__\n",
+ "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
+ "from IPython.display import display, HTML\n",
+ "from plotly import tools\n",
+ "\n",
+ "from contextlib import contextmanager\n",
+ "import sys, os\n",
+ "\n",
+ "@contextmanager\n",
+ "def suppress_stdout():\n",
+ " with open(os.devnull, \"w\") as devnull:\n",
+ " old_stdout = sys.stdout\n",
+ " sys.stdout = devnull\n",
+ " try: \n",
+ " yield\n",
+ " finally:\n",
+ " sys.stdout = old_stdout\n",
+ "\n",
+ "with suppress_stdout():\n",
+ " data_req_path = os.path.join(\"..\",\"..\", \"binder\", \"data_requirement.json\")\n",
+ " repo2data = Repo2Data(data_req_path)\n",
+ " DATA_ROOT = os.path.join(repo2data.install()[0],\"t1-book-neurolibre\")\n",
+ " filename = os.path.join(DATA_ROOT,\"01\",'figure_4.pkl')\n",
+ "\n",
+ "with open(filename, 'rb') as f:\n",
+ " params, Mz_analytical, fitOutput_lm, fitOutput_barral, TR_range = pickle.load(f)\n",
+ "\n",
+ "config={'showLink': False, 'displayModeBar': False}\n",
+ "\n",
+ "init_notebook_mode(connected=True)\n",
+ "\n",
+ "data1 = [dict(\n",
+ " visible = False,\n",
+ " mode = 'markers',\n",
+ " x = params[\"TI\"],\n",
+ " y = abs(np.squeeze(np.asarray(Mz_analytical[ii]))),\n",
+ " name = 'Simulated data',\n",
+ " text = 'Simulated data',\n",
+ " hoverinfo = 'x+y+text') for ii in range(len(TR_range))]\n",
+ "\n",
+ "data1[10]['visible'] = True\n",
+ "\n",
+ "data2 = [dict(\n",
+ " visible = False,\n",
+ " mode = 'lines',\n",
+ " x = params[\"TI\"],\n",
+ " y = abs(fitOutput_lm[ii]['c'] * (1 - 2*np.exp(-params['TI']/fitOutput_lm[ii]['T1']))),\n",
+ " name = '[C(1-2e-TI/T1)] Fitted T1: ' + str(round(fitOutput_lm[ii]['T1'])) + ' ms',\n",
+ " text = '[C(1-2e-TI/T1)]',\n",
+ " hoverinfo = 'x+y+text') for ii in range(len(TR_range))]\n",
+ "\n",
+ "data2[10]['visible'] = True\n",
+ "\n",
+ "data3 = [dict(\n",
+ " visible = False,\n",
+ " mode = 'lines',\n",
+ " x = params[\"TI\"],\n",
+ " y = abs((fitOutput_barral[ii]['ra']+fitOutput_barral[ii]['rb']*np.exp(-params['TI']/fitOutput_barral[ii]['T1']))),\n",
+ " name = '[a+be-TI/T1] Fitted T1: ' + str(round(fitOutput_barral[ii]['T1'])) + ' ms',\n",
+ " text = '[a+be-TI/T1]',\n",
+ " hoverinfo = 'x+y+text') for ii in range(len(TR_range))]\n",
+ "\n",
+ "data3[10]['visible'] = True\n",
+ "\n",
+ "\n",
+ "\n",
+ "data = data1 + data2 + data3\n",
+ "\n",
+ "steps = []\n",
+ "for i in range(len(TR_range)):\n",
+ " step = dict(\n",
+ " method = 'restyle', \n",
+ " args = ['visible', [False] * len(data1)],\n",
+ " label = str(TR_range[i])\n",
+ " )\n",
+ " step['args'][1][i] = True # Toggle i'th trace to \"visible\"\n",
+ " steps.append(step)\n",
+ "\n",
+ "sliders = [dict(\n",
+ " x = 0,\n",
+ " y = -0.02,\n",
+ " active = 10,\n",
+ " currentvalue = {\"prefix\": \"TR value (ms): \"},\n",
+ " pad = {\"t\": 50, \"b\": 10},\n",
+ " steps = steps\n",
+ ")]\n",
+ "\n",
+ "layout = go.Layout(\n",
+ " width=580,\n",
+ " height=450,\n",
+ " margin=go.layout.Margin(\n",
+ " l=80,\n",
+ " r=40,\n",
+ " b=60,\n",
+ " t=10,\n",
+ " ),\n",
+ " annotations=[\n",
+ " dict(\n",
+ " x=0.5004254919715793,\n",
+ " y=-0.18,\n",
+ " showarrow=False,\n",
+ " text='Inversion Time – TI (ms)',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=22\n",
+ " ),\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " dict(\n",
+ " x=-0.14,\n",
+ " y=0.5,\n",
+ " showarrow=False,\n",
+ " text='Signal (magnitude)',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=22\n",
+ " ),\n",
+ " textangle=-90,\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " ],\n",
+ " xaxis=dict(\n",
+ " autorange=False,\n",
+ " range=[0, params['TI'][-1]],\n",
+ " showgrid=False,\n",
+ " linecolor='black',\n",
+ " linewidth=2\n",
+ " ),\n",
+ " yaxis=dict(\n",
+ " autorange=False,\n",
+ " range=[0, 1],\n",
+ " showgrid=False,\n",
+ " linecolor='black',\n",
+ " linewidth=2\n",
+ " ),\n",
+ " legend=dict(\n",
+ " x=0.2,\n",
+ " y=0.9,\n",
+ " traceorder='normal',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=12,\n",
+ " color='#000'\n",
+ " ),\n",
+ " bordercolor='#000000',\n",
+ " borderwidth=2\n",
+ " ), \n",
+ " sliders=sliders,\n",
+ " plot_bgcolor='white'\n",
+ ")\n",
+ "\n",
+ "fig = dict(data=data, layout=layout)\n",
+ "\n",
+ "iplot(fig, filename = 'ir_fig_4.html', config = config)\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
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+}
diff --git a/figures/2 T1 mapping/figure_2-5.ipynb b/figures/2 T1 mapping/figure_2-5.ipynb
new file mode 100644
index 0000000..b3dd38f
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+ "source": [
+ "#| label: fig2p5cell\n",
+ "from repo2data.repo2data import Repo2Data\n",
+ "import os \n",
+ "import pickle\n",
+ "import matplotlib.pyplot as plt\n",
+ "import chart_studio.plotly as py\n",
+ "import plotly.graph_objs as go\n",
+ "import numpy as np\n",
+ "from plotly import __version__\n",
+ "from plotly.offline import download_plotlyjs, init_notebook_mode, plot, iplot\n",
+ "from IPython.display import display, HTML\n",
+ "from plotly import tools\n",
+ "\n",
+ "from contextlib import contextmanager\n",
+ "import sys, os\n",
+ "\n",
+ "@contextmanager\n",
+ "def suppress_stdout():\n",
+ " with open(os.devnull, \"w\") as devnull:\n",
+ " old_stdout = sys.stdout\n",
+ " sys.stdout = devnull\n",
+ " try: \n",
+ " yield\n",
+ " finally:\n",
+ " sys.stdout = old_stdout\n",
+ "\n",
+ "with suppress_stdout():\n",
+ " data_req_path = os.path.join(\"..\",\"..\", \"binder\", \"data_requirement.json\")\n",
+ " repo2data = Repo2Data(data_req_path)\n",
+ " DATA_ROOT = os.path.join(repo2data.install()[0],\"t1-book-neurolibre\")\n",
+ " filename = os.path.join(DATA_ROOT,\"01\",'figure_5.pkl')\n",
+ "\n",
+ "with open(filename, 'rb') as f:\n",
+ " T1_map, TI_0030, TI_0530, TI_1030, TI_1530, xAxis, yAxis = pickle.load(f)\n",
+ "\n",
+ "config={'showLink': False, 'displayModeBar': False}\n",
+ "\n",
+ "init_notebook_mode(connected=True)\n",
+ "\n",
+ "trace1 = go.Heatmap(x = xAxis,\n",
+ " y = yAxis,\n",
+ " z=TI_0030,\n",
+ " colorscale='gray',\n",
+ " showscale = False,\n",
+ " visible=False,\n",
+ " name = 'Signal')\n",
+ "trace2 = go.Heatmap(x = xAxis,\n",
+ " y = yAxis,\n",
+ " z=TI_0530,\n",
+ " colorscale='gray',\n",
+ " showscale = False,\n",
+ " visible=False,\n",
+ " name = 'Signal')\n",
+ "trace3 = go.Heatmap(x = xAxis,\n",
+ " y = yAxis,\n",
+ " z=TI_1030,\n",
+ " colorscale='gray',\n",
+ " showscale = False,\n",
+ " visible=True,\n",
+ " name = 'Signal')\n",
+ "trace4 = go.Heatmap(x = xAxis,\n",
+ " y = yAxis,\n",
+ " z=TI_1530,\n",
+ " colorscale='gray',\n",
+ " visible=False,\n",
+ " showscale = False,\n",
+ " name = 'Signal')\n",
+ "trace5 = go.Heatmap(x = xAxis,\n",
+ " y = yAxis,\n",
+ " z=T1_map,\n",
+ " colorscale='Portland',\n",
+ " xaxis='x2',\n",
+ " yaxis='y2',\n",
+ " visible=True,\n",
+ " name = 'T1 values (ms)')\n",
+ "\n",
+ "data=[trace1, trace2, trace3, trace4, trace5]\n",
+ "\n",
+ "\n",
+ "updatemenus = list([\n",
+ " dict(active=2,\n",
+ " x = 0.12,\n",
+ " xanchor = 'left',\n",
+ " y = -0.15,\n",
+ " yanchor = 'bottom',\n",
+ " direction = 'up',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=16\n",
+ " ),\n",
+ " buttons=list([ \n",
+ " dict(label = '30 ms',\n",
+ " method = 'update',\n",
+ " args = [{'visible': [True, False, False, False, True]},\n",
+ " ]),\n",
+ " dict(label = '530 ms',\n",
+ " method = 'update',\n",
+ " args = [{'visible': [False, True, False, False, True]},\n",
+ " ]),\n",
+ " dict(label = '1030 ms',\n",
+ " method = 'update',\n",
+ " args = [{'visible': [False, False, True, False, True]},\n",
+ " ]),\n",
+ " dict(label = '1530 ms',\n",
+ " method = 'update',\n",
+ " args = [{'visible': [False,False, False, True, True]},\n",
+ " ])\n",
+ " ]),\n",
+ " )\n",
+ "])\n",
+ "\n",
+ "layout = dict(\n",
+ " width=560,\n",
+ " height=345,\n",
+ " margin = dict(\n",
+ " t=40,\n",
+ " r=50,\n",
+ " b=10,\n",
+ " l=50),\n",
+ " annotations=[\n",
+ " dict(\n",
+ " x=0.06,\n",
+ " y=1.15,\n",
+ " showarrow=False,\n",
+ " text='MR Image',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=26\n",
+ " ),\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " dict(\n",
+ " x=0.6,\n",
+ " y=1.15,\n",
+ " showarrow=False,\n",
+ " text='T1 map',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=26\n",
+ " ),\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " dict(\n",
+ " x=1.22,\n",
+ " y=1.15,\n",
+ " showarrow=False,\n",
+ " text='T1 (ms)',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=26\n",
+ " ),\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " dict(\n",
+ " x=0.02,\n",
+ " y=-0.15,\n",
+ " showarrow=False,\n",
+ " text='TI:',\n",
+ " font=dict(\n",
+ " family='Times New Roman',\n",
+ " size=22\n",
+ " ),\n",
+ " xref='paper',\n",
+ " yref='paper'\n",
+ " ),\n",
+ " ],\n",
+ " xaxis = dict(range = [0,127], autorange = False,\n",
+ " showgrid = False, zeroline = False, showticklabels = False,\n",
+ " ticks = '', domain=[0, 0.6]),\n",
+ " yaxis = dict(range = [0,127], autorange = False,\n",
+ " showgrid = False, zeroline = False, showticklabels = False,\n",
+ " ticks = '', domain=[0, 1]),\n",
+ " xaxis2 = dict(range = [0,127], autorange = False,\n",
+ " showgrid = False, zeroline = False, showticklabels = False,\n",
+ " ticks = '', domain=[0.4, 1]),\n",
+ " yaxis2 = dict(range = [0,127], autorange = False,\n",
+ " showgrid = False, zeroline = False, showticklabels = False,\n",
+ " ticks = '', domain=[0, 1], anchor='x2'),\n",
+ " showlegend = False,\n",
+ " autosize = False,\n",
+ " updatemenus=updatemenus,\n",
+ " plot_bgcolor='white'\n",
+ ")\n",
+ "\n",
+ "\n",
+ "fig = dict(data=data, layout=layout)\n",
+ "\n",
+ "iplot(fig, filename = 'ir_fig_5.html', config = config)\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.12.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}