| title | T2 Mapping | ||||||||||
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| date | 2024-10-07 | ||||||||||
| label | t2MappingChapter | ||||||||||
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T{sub}2 relaxation, also known as the transverse or spin-spin relaxation, is characterized by the dephasing of spins, leading to a reduction of the total magnetization in the x-y plane. In practical terms, the T{sub}2 value represents the upper limit of the signal decay time under ideal imaging conditions. In practice, magnetic field inhomogeneities cause the transverse magnetization to decay faster than what is captured by T{sub}2 relaxation. These inhomogeneities can be macroscopic, caused by factors such as metallic implants or air-tissue interfaces, or microscopic, resulting from differences in magnetic susceptibility between tissues [@Chavhan2009-gk;@Cohen-Adad2014-uf] . When considering this phenomenon, we refer to the transverse relaxation as T{sub}2{sup}*.
T{sub}2 mapping, a quantitative magnetic resonance imaging method, offers images of T{sub}2 relaxation times (called T{sub}2 maps) providing valuable insights into tissue composition. Given that T{sub}2 relaxation is sensitive to specific microstructural changes associated with diseases, such as iron accumulation, myelination and inflammation [@Dortch2020-nq], it is considered a promising modality for clinical and research applications. Commonly used T{sub}2-mapping techniques are split into two categories: the mono-exponential methods, which consider that the voxel’s signal arises from a single tissue compartment, and multi-exponential methods, where various tissues that contribute to a single voxel’s signal are considered. Recently, novel techniques for T{sub}2 mapping have emerged beyond the conventional techniques that were developed for NMR, such as MR fingerprinting (Ma et al., 2013), a fast relaxation mapping technique that uses a single image acquisition to quantify multiple parameters [@Hamilton2017-pd].
In the following sections, we will introduce the current state-of-the-art signal modeling and data fitting theory for both mono-exponential and multi-exponential T{sub}2 mapping, as well as the benefits and pitfalls of both approaches. An overview of some common applications of T{sub}2 mapping (e.g., myelin water fraction (MWF) imaging) will also be presented. We will also cover the fundamentals of T{sub}2{sup}* mapping and explore how its signal decay curve can differ from that of T{sub}2 relaxation.
The standard practice in clinical MRI is T{sub}2-weighted imaging (T2w), i.e. images in which the different tissues produce signals according to their T{sub}2 relaxation times. T2w images are considered qualitative, as they rely on differences in T{sub}2 relaxation of various tissues to provide contrast rather than actual quantitative tissue measurements, thus making their interpretation dependent on the observer. Meanwhile, T{sub}2 mapping is considered a quantitative technique, as it directly measures the T{sub}2 relaxation time of tissues by fitting multiple data points to a decay curve to calculate the T{sub}2 constant. The objective nature of T{sub}2 mapping makes it a promising technique for various applications in both clinical practice and research, including disease progression monitoring and the identification of novel biomarkers. Another advantage of T{sub}2 mapping is that it allows for consistent comparisons across different imaging protocols, thereby enhancing reproducibility and making it ideal for longitudinal and multi-institutional studies [@Cheng2012-in]. shows several examples of T2w images acquired at different echo times, compared to the T{sub}2 map generated by fitting the decay curve.
:::{figure} #t2Fig1jn
:label: t2Plot1
:enumerator: 3.1
T{sub}2-weighted images at different TE, compared to T{sub}2 map following data fitting of the T{sub}2 decay curve.
:::
:class: tip, dropdown
```matlab
%% Requirements
% qMRLab must be installed: git clone https://www.github.com/qMRLab/qMRLab.git
% The mooc chapter branch must be checked out: git checkout mooc-03-T2
% qMRLab must be added to the path inside the MATLAB session: startup
% Define the model
Model = mono_t2;
% Load data into environment, and rotate mask to be aligned with IR data
data = struct;
data.SEdata = double(load_nii_data('../data/mono_t2/SEdata.nii.gz'));
data.Mask = double(load_nii_data('../data/mono_t2/Mask.nii.gz'));
% Define fitting parameters
EchoTime = [12.8000; 25.6000; 38.4000; 51.2000; 64.0000; 76.8000; 89.6000; 102.4000; 115.2000; 128.0000; 140.8000; 153.6000; 166.4000; 179.2000; 192.0000; 204.8000; 217.6000; 230.4000; 243.2000; 256.0000; 268.8000; 281.6000; 294.4000; 307.2000; 320.0000; 332.8000; 345.6000; 358.4000; 371.2000; 384.0000];
Model.Prot.SEdata.Mat = [ EchoTime ];
FitResults = FitData(data,Model,0);
% T2w MRI data at different TE values
TE_1 = imrotate(squeeze(data.SEdata(:,:,:,1).*data.Mask),-90);
TE_2 = imrotate(squeeze(data.SEdata(:,:,:,10).*data.Mask),-90);
TE_3 = imrotate(squeeze(data.SEdata(:,:,:,20).*data.Mask),-90);
TE_4 = imrotate(squeeze(data.SEdata(:,:,:,30).*data.Mask),-90);
% Extract the T2 map from FitResults and rotate it
T2_map = imrotate(squeeze(FitResults.T2.*data.Mask), -90);
% Plotting the images
figure;
subplot(2, 3, 1);
imagesc(TE_1);
colormap(gray);
colorbar;
axis image;
title('TE = 12.80 ms');
xlabel('X-axis');
ylabel('Y-axis');
caxis([0, 2500]);
subplot(2, 3, 2);
imagesc(TE_2);
colormap(gray);
colorbar;
axis image;
title('TE = 128.00 ms');
xlabel('X-axis');
ylabel('Y-axis');
caxis([0, 2500]);
subplot(2, 3, 3);
imagesc(TE_3);
colormap(gray);
colorbar;
axis image;
title('TE = 256.00 ms');
xlabel('X-axis');
ylabel('Y-axis');
caxis([0, 2500]);
subplot(2, 3, 4);
imagesc(TE_4);
T2_map = imrotate(squeeze(FitResults.T2.*data.Mask),-90);
colormap(gray);
colorbar;
axis image;
title('TE = 384.00 ms');
xlabel('X-axis');
ylabel('Y-axis');
caxis([0, 2500]);
subplot(2, 3, 5);
imagesc(T2_map);
colormap(gray);
colorbar;
axis image;
title('T2 map');
xlabel('X-axis');
ylabel('Y-axis');
caxis([80, 150]);
%% Export
EchoTimes = [EchoTime(1), EchoTime(10), EchoTime(20), EchoTime(30)]
save("T2w_vs_T2map.mat", "T2_map", "TE_1", "TE_2", "TE_3", "TE_4", "EchoTimes")
```
The simplest technique for performing T{sub}2-mapping involves the acquisition of multiple images using a single-echo sequence with different echo times (TE). This set of data can then be used to fit the T{sub}2 from the decaying signal curve [@Milford2015-ef]. However, this method is typically not employed in-vivo due to its prohibitively long acquisition times.
Multi-spin echo (MSE) sequences, based on the Carr-Purcell-Meiboom-Gill (CPMG) pulse train [@Carr1954-dd;@Meiboom1958-ur], employ multiple 180-degree refocusing pulses to generate multiple echoes within a single acquisition () [@Fatemi2020-tm;@Milford2015-ef]. This substantially reduces the scan time relative to acquiring multiple echoes using separate single-SE acquisitions, making MSE the sequence of choice in clinical settings.
The original Carr-Purcell method, which involves applying a 90-degree pulse followed by a series of 180-degree pulses along the same axis (e.g., x-axis), was designed to mitigate diffusion effects by preventing phase accumulation. However, this approach can lead to the accumulation of imperfections in the 180-degree pulses, resulting in a faster than expected loss of transverse magnetization (Mxy) and inaccurate T{sub}2 measurements [@Brown2014-bj]. The Meiboom-Gill improvement addresses this issue by applying the initial 90-degree pulse along one axis (e.g., x-axis) and the subsequent 180-degree pulses along a perpendicular axis (e.g., y-axis). This phase cycling distributes errors such that they average out over time, producing a more stable and accurate echo train [@Brown2014-bj].
:label: t2seq
:enumerator: 3.2
Simplified illustration of a multi-spin echo sequence for _T_{sub}`2` mapping based on the Carr-Purcell-Meiboom-Gill method
To fit the data using the mono-exponential model, homogeneity within each voxel is assumed implicitly, implying a consistent T{sub}2 decay time across the tissue. This results in a single fitted T{sub}2 value for each voxel in the image. Although it is widely used, the mono-exponential model has been shown to be insufficient for proper estimation of tissue T{sub}2 relaxation times, given that the signal in a voxel often arises from multiple tissue components with different T{sub}2 values [@Graham1996-sr]. In a later section of this chapter, we will cover the multi-exponential method and discuss its use in applications such as myelin water fraction (MWF) imaging.
The decay of the transverse magnetization (Mxy) is exponential and can be derived from the transverse component of the Bloch equations:
:label: t2Eq1
:enumerator:3.1
\begin{equation}
\textit{M}_{xy}\left ( TE \right ) = Mz\left ( 0^{-} \right )e^{-TE/T_{2}}
\end{equation}
where Mz(0-) is the longitudinal magnetization immediately preceding the 90 degree excitation pulse. By using this equation, we make the assumption that the measured signal is proportional to the transverse magnetization (Mxy), and that Mz(0-) remains constant regardless of echo time (TE) [@Dortch2020-nq].
shows transverse relaxation curves for T{sub}2 and T{sub}2{sup}* values for white matter and gray matter, using the relaxation times from [@Siemonsen2008-bc].
:::{figure} #t2Fig2jn
:label: t2Plot2
:enumerator: 3.3
Transverse relaxation decay curves for T{sub}2 and T{sub}2{sup}* values in white matter and gray matter. The T{sub}2 and T{sub}2{sup}* constants were taken from [@Siemonsen2008-bc].
:::
In NMR physics, it has been shown that T{sub}2 relaxation times must be equal to or shorter than 2 T{sub}1 [@Levitt2008-zk]; however, it has been demonstrated that T{sub}2 can exceed T{sub}1 in very rare cases [@Traficante1991-es]. In living organisms however, T{sub}2 is always shorter than T{sub}1.
:class: tip, dropdown
```matlab
%% Requirements
% qMRLab must be installed: git clone https://www.github.com/qMRLab/qMRLab.git
% The mooc chapter branch must be checked out: git checkout mooc-03-T2
% qMRLab must be added to the path inside the MATLAB session: startup
%% T2 and T2* decay curves
% Script to display T2 and T2* relaxometry curves for different tissues
% Simulation parameters
params.TE = linspace(0, 300, 100); % Echo times (in ms)
% Define T2 values for different tissues
params.T2_WM = 109.77; % T2 of white matter (in ms)
params.T2_GM = 96.07; % T2 of gray matter (in ms)
% Define T2* values for different tissues
params.T2star_WM = 67.63; % T2* of white matter (in ms)
params.T2star_GM = 48.48; % T2* of gray matter (in ms)
% Generate T2 and T2* decay signals
signal_WM_T2 = exp(-params.TE / params.T2_WM);
signal_GM_T2 = exp(-params.TE / params.T2_GM);
signal_WM_T2star = exp(-params.TE / params.T2star_WM);
signal_GM_T2star = exp(-params.TE / params.T2star_GM);
% Plot the T2 and T2* signals
figure;
hold on;
plot(params.TE, signal_WM_T2, '-b', 'DisplayName', 'T2 = 109.77 ms (white matter)');
plot(params.TE, signal_GM_T2, '-r', 'DisplayName', 'T2 = 96.07 ms (gray matter)');
plot(params.TE, signal_WM_T2star, '--b', 'DisplayName', 'T2* = 67.63 ms (white matter)');
plot(params.TE, signal_GM_T2star, '--r', 'DisplayName', 'T2* = 48.48 ms (gray matter)');
xlabel('Echo Time - TE (ms)');
ylabel('Transverse Magnetization (Mxy)');
legend();
title('T2 and T2* Decay Signals');
%% Export
TE = squeeze(params.TE)
save("t2_and_t2star_curvs.mat", "signal_WM_T2", "signal_GM_T2", "signal_WM_T2star", "signal_GM_T2star", "params")
```
The T{sub}2 signal decay for the mono-exponential model is described mathematically as :
:label: t2Eq2
:enumerator:3.2
\begin{equation}
\textit{S}\left ( TE \right ) = S_{0}e^{-TE/T_{2}}
\end{equation}
where S0 is the signal intensity immediately following the excitation pulse [@Dortch2020-nq;@Milford2015-ef].
In practice, B{sub}1 inhomogeneities and RF pulse imperfections can influence the T{sub}2 signal decay curve and result in inaccurate T{sub}2 estimations. This may cause refocusing pulses to deviate from the ideal 180 degrees, generating additional echoes known as stimulated or spurious echoes. These unwanted echoes can contaminate the signal decay, resulting in erroneous T{sub}2 estimations [@McPhee2018-wd]. To account for these stimulated echoes, some studies have shown that T{sub}2 fitting accuracy can be improved either by using only even-numbered echoes [@Focke2011-xh;@Kim2009-yf], or by discarding the first echo [@Biasiolli2013-vy;@Milford2015-ef].
(t2T2star)=
Until now, we have assumed that the transverse signal (Mxy) decays exponentially with the echo time divided by the T{sub}2 constant (see ). However, in practice, other factors such as B{sub}0 inhomogeneities can cause a more rapid loss of the transverse signal; this results in a faster transverse decay, which is referred to as T{sub}2{sup}* relaxation (see ).
The relation between T{sub}2 and T{sub}2{sup}* is described as follows [@Brown2014-bj]:
:label: t2Eq3
:enumerator:3.3
\begin{equation}
\frac{1}{T_{2}^{*}} = \frac{1}{T_{2}} + \frac{1}{T_{2}^{'}}
\end{equation}
Where T{sub}2′ quantifies the portion of relaxation which is due to magnetic field inhomogeneities. Some studies suggest that T{sub}2′ mapping, which can be performed by removing the T{sub}2 relaxation effect from T{sub}2{sup}*, could offer valuable insights for brain disease diagnosis, notably by quantifying blood oxygenation levels [@Lee2014-cq] and to predict infarct growth in acute stroke patients [@Siemonsen2008-bc].
The T{sub}2{sup}* decay can then be calculated using the same methods as T{sub}2, where can be rewritten as follows :
:label: t2Eq4
:enumerator:3.4
\begin{equation}
\textit{S}\left ( TE \right ) = S_{0}e^{-TE/T_{2}}
\end{equation}
Unlike T{sub}2 mapping, which uses spin echo type sequences, T{sub}2{sup}* mapping is performed using gradient echo sequences (GRE), as they are sensitive to magnetic field inhomogeneities [@Cohen-Adad2014-uf;@Markl2012-oi]. Nonetheless, there are specialized sequences such as the spin and gradient echo (SAGE) sequence [@Stokes2014-zn] that enable the simultaneous acquisition of both T{sub}2 and T{sub}2{sup}*.
In MRI, noise in the data can make it harder to accurately fit the T{sub}2 decay curve, which is problematic given the necessity for highly precise T{sub}2 values in clinical contexts. This issue is particularly pronounced when using pixel-wise T{sub}2 mapping, as the signal-to-noise (SNR) is much lower compared to region-of-interest (ROI) T{sub}2 mapping approaches [@Sandino2015-pz]. shows how varying the level of noise in the acquired data can influence the fitting of the T{sub}2 relaxation curve and the resulting T{sub}2 constant. As observed in this figure, a low SNR can have a considerable impact on the T{sub}2 fitting process.
:::{figure} #t2Fig3jn
:label: t2Plot3
:enumerator: 3.4
Impact of noise on T{sub}2 relaxometry fitting. The figure shows a single voxel fit with a true T{sub}2 relaxation time of 109 ms. As the noise level increases, the accuracy of the T{sub}2 fitting decreases, leading to deviations in the estimated T{sub}2 relaxation time from the true value. This demonstrates how higher noise levels can adversely affect the reliability of T{sub}2 measurements and may result in inaccurate representations of tissue relaxation properties.
:::
The number of echoes used in T{sub}2 relaxometry is influenced by several factors, including the need for adequate spacing between echoes, the potential risk of heating the sample, and the challenges associated with processing data from samples with low signal-to-noise ratios. Therefore, selecting an optimal number of echoes is crucial for achieving accurate and reliable results while addressing these constraints [@Shrager1998-du]. The Cramer-Rao lower-bound (CRLB) method is a statistical tool that can be used in the context of T{sub}2 relaxometry to estimate the smallest possible variance, known as the lower bound, of an unbiased estimator given the noise present in the data [@Cavassila2001-si]. Using the lower bounds, the optimal number of echoes needed to accurately fit the T{sub}2 decay curve can be determined, ensuring more robust T{sub}2 mapping [@Jones1996-na]. In their work, [@Shrager1998-du] introduced another method for optimizing the selection of echo time points to improve the accuracy of T{sub}2 value estimates based on a predetermined range of expected T{sub}2 values. Their approach demonstrated superior accuracy compared to conventional methods that use uniformly-spaced echo times, suggesting that these methods are not optimal for T{sub}2 curve fitting accuracy.
:class: tip, dropdown
```matlab
%% Requirements
% qMRLab must be installed: git clone https://www.github.com/qMRLab/qMRLab.git
% The mooc chapter branch must be checked out: git checkout mooc-03-T2
% qMRLab must be added to the path inside the MATLAB session: startup
%% T2 and T2* decay curves
close all
clear all
clc
% Define model
Model = mono_t2;
params.TE = linspace(0, 300, 100); % Echo times (in ms)
% Define signal parameters for different tissues
x = struct;
x.M0 = 1000;
x.T2 = 109; % (in ms)
% Define the signal-to-noise ratio
Opt1.SNR = 10;
Opt2.SNR = 50;
Opt3.SNR = 90;
Opt4.SNR = 130;
% Run the simulation for T2 and T2* decay curves
[FitResult_SNR10, data_SNR10] = Model.Sim_Single_Voxel_Curve(x, Opt1);
[FitResult_SNR50, data_SNR50] = Model.Sim_Single_Voxel_Curve(x, Opt2);
[FitResult_SNR90, data_SNR90] = Model.Sim_Single_Voxel_Curve(x, Opt3);
[FitResult_SNR130, data_SNR130] = Model.Sim_Single_Voxel_Curve(x, Opt4);
% T2 constants
T2_SNR10 = FitResult_SNR10.T2;
T2_SNR50 = FitResult_SNR50.T2;
T2_SNR90 = FitResult_SNR90.T2;
T2_SNR130 = FitResult_SNR130.T2;
% T2 decay curves
signal_SNR10 = FitResult_SNR10.M0/1000 * exp(-params.TE / FitResult_SNR10.T2);
signal_SNR50 = FitResult_SNR50.M0/1000 * exp(-params.TE / FitResult_SNR50.T2);
signal_SNR90 = FitResult_SNR90.M0/1000 * exp(-params.TE / FitResult_SNR90.T2);
signal_SNR130 = FitResult_SNR130.M0/1000 * exp(-params.TE / FitResult_SNR130.T2);
% Noisy data points
EchoTimes = [12.8000; 25.6000; 38.4000; 51.2000; 64.0000; 76.8000; 89.6000; 102.4000; 115.2000; 128.0000; 140.8000; 153.6000; 166.4000; 179.2000; 192.0000; 204.8000; 217.6000; 230.4000; 243.2000; 256.0000; 268.8000; 281.6000; 294.4000; 307.2000; 320.0000; 332.8000; 345.6000; 358.4000; 371.2000; 384.0000];
SEdata_SNR10 = data_SNR10.SEdata/1000;
SEdata_SNR50 = data_SNR50.SEdata/1000;
SEdata_SNR90 = data_SNR90.SEdata/1000;
SEdata_SNR130 = data_SNR130.SEdata/1000;
%% Export
disp(EchoTimes)
disp(params.TE)
save("t2_noise_simulation.mat", "params", "signal_SNR10", "signal_SNR50", "signal_SNR90", "signal_SNR130", "T2_SNR10", "T2_SNR50", "T2_SNR90", "T2_SNR130", "SEdata_SNR10", "SEdata_SNR50", "SEdata_SNR90", "SEdata_SNR130", "EchoTimes")
```
The main benefit of mono-exponential T{sub}2 mapping is its simplicity and straightforward implementation, making it a convenient and efficient method for T{sub}2 fitting. Additionally, as mentioned previously, the use of multi-echo spin echo (MESE) sequences significantly reduces the acquisition time, further enhancing its practicality [@Fatemi2020-tm;@Milford2015-ef].
Despite these advantages, mono-exponential methods have certain drawbacks. First, by assuming a single T{sub}2 relaxation constant per voxel, the mono-exponential method tends to over-simplify the tissue microstructure, potentially leading to inaccurate T{sub}2 estimations. This limitation can be particularly problematic when studying tissues that have a complex microstructure, where a single voxel may contain components with different T{sub}2 relaxation times. Furthermore, it has been shown that MESE sequences are sensitive to imperfections in the radiofrequency pulses. For instance, factors such as B{sub}1 inhomogeneities and reduced flip angles have been shown to overestimate T{sub}2 times when using mono-exponential methods [@Fatemi2020-tm].
(t2Multiexpo)=
By using the mono-exponential curve described in the previous sections, we use a single compartment tissue model. This means that we assume that all tissue components contained in a voxel have the same T{sub}2 relaxation time. However, in practice, the assumption of T{sub}2 decay uniformity within a voxel can result in inaccurate fittings, as a voxel may contain different tissue compartments with different T{sub}2 times. For example, a voxel at a tissue boundary inside the brain can contain both cerebrospinal fluid and gray matter, a phenomenon which is also commonly referred to as the partial volume effect. In such cases, it would be preferable to compartmentalize different tissues inside a single voxel: this is made possible with multi-exponential T{sub}2 mapping, where we consider the T{sub}2 relaxation contribution of each tissue compartment within a voxel. The multi-exponential T{sub}2 mapping method, which will be described in this section, can be useful in many applications, such as myelin water fraction imaging which will be explained further in the section on applications.
For multiexponential T{sub}2 mapping, the transverse magnetization (Mxy) acquired at different echo times (TE) can be modeled as a sum of exponential decays :
:label: t2Eq5
:enumerator:3.5
\begin{equation}
\textit{M}_{xy}\left ( TE \right ) = \sum_{i=1}^{N}M_{z,i}\left ( 0^-{} \right )e^{-TE/T_{2,i}}
\end{equation}
where each term of the summation represents the contribution of the ith tissue component to the overall transverse magnetization decay [@Collewet2022-wj;@Dortch2020-nq].
presents a single-voxel simulation of T{sub}2 relaxation curves of myelin water (MW) and intra/extracellular water (IEW) using mono-exponential T{sub}2 fitting, compared to a multi-exponential fitting for both MW and IEW. In this example, we see that using a multi-exponential model rather than mono-exponential for complex tissues like myelin enables more precise quantification of the T{sub}2 relaxation time within each voxel.
:::{figure} #t2Fig4jn
:label: t2Plot4
:enumerator: 3.5
Comparison of mono-exponential and multi-exponential T{sub}2 fitting. This figure contrasts mono-exponential and multi-exponential fitting approaches for a single voxel containing myelin water (MW) and intra/extracellular water (IEW). The green and orange curves represent mono-exponential fittings for MW and IEW, respectively. The dotted purple curve illustrates the multi-exponential fitting, which combines both MW and IEW components.
:::
:::{figure} #t2Fig6jn
:label: t2Plot7
:enumerator: 3.6
Comparison of mono-exponential and multi-exponential T{sub}2 fitting. This figure contrasts mono-exponential and multi-exponential fitting approaches for a single voxel containing myelin water (MW) and intra/extracellular water (IEW). The green and orange curves represent mono-exponential fittings for MW and IEW, respectively. The dotted purple curve illustrates the multi-exponential fitting, which combines both MW and IEW components.
:::
:class: tip, dropdown
```matlab
%% Requirements
% qMRLab must be installed: git clone https://www.github.com/qMRLab/qMRLab.git
% The mooc chapter branch must be checked out: git checkout mooc-03-T2
% qMRLab must be added to the path inside the MATLAB session: startup
close all
clear all
Model = mwf
% Define initial MWF and T2 times of myelin water and intra- and extracellular water
x = struct;
x.MWF = 50;
x.T2MW = 20;
x.T2IEW = 120;
% Define echo times
params.TE = linspace(0, 300, 100);
% Set simulation options
Opt.SNR = 120;
Opt.T2Spectrumvariance_Myelin = 5;
Opt.T2Spectrumvariance_IEIntraExtracellularWater = 20;
% Run simulation
figure('Name','Single Voxel Curve Simulation');
FitResult = Model.Sim_Single_Voxel_Curve(x,Opt);
% T2 relaxation curves for myelin water and intra/extracellular water
% (using a mono-exponential curve)
signal_mono_MW = exp(-params.TE / FitResult.T2MW);
signal_mono_IEW = exp(-params.TE / FitResult.T2IEW);
% T2 relaxation curve for multi-expo model
signal_multi_MWF = (FitResult.MWF/100)*signal_mono_MW + (1 - FitResult.MWF/100)*signal_mono_IEW;
%% Export
TE = squeeze(params.TE)
save("multiexpo_T2_curves.mat", "signal_mono_MW", "signal_mono_IEW", "signal_multi_MWF", "TE", "FitResult", "params", "x", "Opt")
```
To fit the data for multi-exponential T{sub}2 mapping, can be rewritten to express the signal decay in terms of the initial amplitude (Si) for each tissue component. The multi-exponential T{sub}2 signal decay is then given by :
:label: t2Eq6
:enumerator:3.6
\begin{equation}
\textit{S}\left ( TE \right ) = \sum_{i=1}^{N}S_{i}e^{-TE/T_{2,i}}
\end{equation}
Where the term Si corresponds to the initial signal amplitude of the ith tissue component, and T{sub}2,iis the T{sub}2 relaxation time of that component.
For example, the multi-exponential signal from can be expressed as the combination of the signals from both myelin water (MW) and intra/extracellular water (IEW), as follows :
:label: t2Eq7
:enumerator:3.7
\begin{equation}
\textit{S}\left ( TE \right )_{MWI} = S_{MW}e^{-TE/T_{2,MW}}+S_{IEW}e^{-TE/T_{2,IEW}}
\end{equation}
presents an example of multi-exponential T{sub}2 fitting applied to an image of the spinal cord. The figure shows the myelin water fraction (MWF) map, which highlights the distribution of myelin water across the spinal cord, as well as the T{sub}2 maps for intra/extracellular water and myelin water. The multi-exponential fitting approach allows for the separation of these components, enhancing tissue characterization in complex structures compared to mono-exponential models. In the following section, we will explain why MWF imaging is a key application of multi-exponential T{sub}2 mapping, and how it provides valuable insights into the microstructure of the spinal cord, highlighting its potential benefits for understanding and diagnosing neurological conditions.
:::{figure} #t2Fig5jn
:label: t2Plot5
:enumerator: 3.7
Multi-exponential T{sub}2 mapping example of the spinal cord. The left image displays multi-exponential (ME) T2w data acquired at different echo times (TE) used for the data fitting. The right image presents the resulting myelin water fraction (MWF) map and T{sub}2 relaxation maps for myelin water (MW) and intra/extracellular water (IEW).
:::
:class: tip, dropdown
```matlab
%% Requirements
% qMRLab must be installed: git clone https://www.github.com/qMRLab/qMRLab.git
% The mooc chapter branch must be checked out: git checkout mooc-03-T2
% qMRLab must be added to the path inside the MATLAB session: startup
% Define the model
Model = mwf;
% Load data into environment, and rotate mask to be aligned with IR data
data = struct;
load('../data/mwf/MET2data.mat');
load('../data/mwf/Mask.mat');
data.MET2data = double(MET2data);
data.Mask = double(Mask);
% Define fitting parameters
EchoTime = [12.8000; 25.6000; 38.4000; 51.2000; 64.0000; 76.8000; 89.6000; 102.4000; 115.2000; 128.0000; 140.8000; 153.6000; 166.4000; 179.2000; 192.0000; 204.8000; 217.6000; 230.4000; 243.2000; 256.0000; 268.8000; 281.6000; 294.4000; 307.2000; 320.0000; 332.8000; 345.6000; 358.4000; 371.2000; 384.0000];
Model.Prot.SEdata.Mat = [ EchoTime ];
% MET2w MRI data at different TE values
ME_TE_1 = imrotate(squeeze(data.MET2data(:,:,:,1).*data.Mask),-90);
ME_TE_2 = imrotate(squeeze(data.MET2data(:,:,:,10).*data.Mask),-90);
ME_TE_3 = imrotate(squeeze(data.MET2data(:,:,:,20).*data.Mask),-90);
ME_TE_4 = imrotate(squeeze(data.MET2data(:,:,:,30).*data.Mask),-90);
% Fit the data
FitResults_mwf = FitData(data,Model,0);
MWF = imrotate(squeeze(FitResults_mwf.MWF.*data.Mask), -90);
T2MW = imrotate(squeeze(FitResults_mwf.T2MW.*data.Mask), -90);
T2IEW = imrotate(squeeze(FitResults_mwf.T2IEW.*data.Mask), -90);
%% Export
save("multiexpo_T2_image.mat", "ME_TE_1", "ME_TE_2", "ME_TE_3", "ME_TE_4", "EchoTime", "FitResults_mwf", "MWF", "T2MW", "T2IEW")
```
(t2Applications)=
Myelin water fraction (MWF) imaging demonstrates the clinical relevance of multi-exponential T{sub}2 mapping [@Kumar2012-uv;@MacKay1994-yv], given that myelin quantification could help identify potential biomarkers for diseases such as multiple sclerosis (MS). Although MWF technically does not directly measure myelin but rather quantifies the amount of total water trapped between the myelin sheaths, it is argued that myelin can be conceptualized as water, as it constitutes its primary composition [@Alonso-Ortiz2015-yj].
Conventional T{sub}2-weighted images often display hyperintense lesions in MS patients. Unfortunately, these hyperintensities lack specificity for the disease and can be indistinguishable from other biological phenomena such as inflammation, edema and axonal loss [@Alonso-Ortiz2015-yj;@Lee2021-jk]. MWF holds promise as a potential biomarker for identifying early microstructural changes in myelin, which could improve our understanding of demyelinating diseases such as MS, facilitating both early diagnosis and monitoring of disease progression.
As described by , MWF represents the proportion of the MRI signal attributed to myelin water relative to the total water content in the brain. This total water content includes myelin water (MW) and intra/extracellular water (IEW), also referred to as axonal water [@Lee2021-jk].
:label: t2Eq8
:enumerator:3.8
\begin{equation}
\textit{MWF} = \frac{S_{MW}e^{-TE/T_{2,MW}}}{S_{MW}e^{-TE/T_{2,MW}} + S_{IEW}e^{-TE/T_{2,IEW}}}
\end{equation}
Direct imaging of myelin itself is challenging due to its very short T{sub}2 relaxation times. Instead, an alternative is to image myelin-associated water (MW), which has slightly longer T{sub}2 relaxation times, typically around 20 ms [@Lee2021-jk]. While these times are still short, they are measurable with standard spin-echo MRI sequences. Myelin Water Fraction (MWF) imaging takes advantage of this to differentiate and quantify the signal from myelin-associated water, providing a more feasible approach to studying myelin.
Quantitative susceptibility mapping (QSM) is another quantitative MRI technique, which measures variations in tissue magnetic susceptibility [@Ruetten2019-ox;@Wang2015-sm]. By assessing the differences in magnetic susceptibility between various tissues, QSM can accurately quantify paramagnetic substances like iron, calcium and oxygen, as well as diamagnetic substances such as myelin [@Ruetten2019-ox]. The ability to measure these substances has considerable clinical potential, as they provide valuable insights into tissue physiology and integrity. For instance, alterations in iron levels have been linked to neurodegenerative diseases like multiple sclerosis [@Stephenson2014-jz] and Parkinson’s disease [@Chen2019-ef], making QSM a promising technique for identifying biomarkers for these neurodegenerative disorders and emphasizing their importance in clinical research.
In MRI, the acquired signal is complex, consisting of a magnitude and phase component. While traditional contrasts such as T{sub}1, T{sub}2 and T{sub}2{sup}* only exploit the magnitude of the MRI signal, the phase component holds valuable information about tissue magnetic susceptibility. This is fundamental for QSM imaging.
As we have covered in a previous section, T{sub}2{sup}* accounts for both T{sub}2 relaxation times and the relaxation due to magnetic field inhomogeneities, characterized by T{sub}2’. These inhomogeneities induce signal dephasing in MRI. As differences in magnetic susceptibility between tissues are a primary cause of these inhomogeneities, phase information is crucial for measuring tissue susceptibility [@Ruetten2019-ox]. In QSM, the combination of both magnitude and phase data from T{sub}2{sup}* acquisitions enables the quantification and spatial mapping of magnetic susceptibility within tissues [@Shmueli2020-vt].
The following figure shows different susceptibility distributions in ppm a brain resuting from a B{sub}0 field map, simulated at 7 T. Two components of the B0 fields were seperated out: a high frequency component (left) and a low frequency component (middle).
:::{figure} #t2Fig7jn
:label: t2Plot8
:enumerator: 3.8
Susceptibility distributions in ppm a brain resuting from a B{sub}0 field map, simulated at 7 T. Two components of the B0 fields were seperated out: a high frequency component (left) and a low frequency component (middle). This figure was generated by reusing
:::
The primary advantage of multi-exponential T{sub}2 mapping lies in its improved accuracy in depicting the T{sub}2 relaxation of complex tissue microstructure. By considering each voxel as multi-compartmental with multiple tissues each having distinct T{sub}2 relaxation times, multiexponential T{sub}2 mapping has proven to be more accurate for capturing the T{sub}2 relaxation of complex, heterogeneous tissues. As we saw in the previous sections, this makes multi-exponential T{sub}2 mapping particularly advantageous in applications such as myelin water fraction imaging, where it is crucial to distinguish the fraction of water attributed to myelin to better understand demyelinating diseases such as MS [@Alonso-Ortiz2015-yj].
However, acquiring multi-exponential T{sub}2 mapping comes with its challenges. First, the increased complexity of multi-exponential mapping compared to mono-exponential models results in longer acquisition times [@Kumar2012-uv]. Additionally, multi-exponential methods are also sensitive to noise [@Dula2009-bj], which can make accurate fittings challenging.
In conclusion, the choice of mono-exponential versus multi-exponential will depend on the specific clinical or research application as well as the complexity of the tissues being studied. While mono-exponential T{sub}2 mapping offers simplicity and efficiency, multi-exponential T{sub}2 mapping provides a comprehensive and accurate characterization of tissue properties, particularly in heterogeneous or pathological tissues.
**a.** Using [](#t2Plot2), determine the approximate optimal echo time for _T_{sub}`2` contrast between white matter and gray matter.
**b.** Repeat for the _T_{sub}`2`{sup}`*` signal curves.
**c.** Observe [](#t2Plot1), which shows the signal at different TE values and the corresponding _T_{sub}`2` map. In practice, which TE out of these four images would you choose for a _T_{sub}`2`-weighted image and explain why that value coincides or doesn’t with the optimal TE you found above.
:label: t2Problem2
**a.** Using [](#t2Plot4), determine the echo time at which the signal myelin water (MW) signal has reduced by: 50%, 63%, and 90%.
**b.** Using the signal curve for intra/extra-cellular (IEW) water presented in the same figure, determine the echo time at which the signal has reduced by: 50%, 63%, and 90%.
**c.** Without fitting, estimate the _T_{sub}`2` value for MW and IEW in this figure.
**d.** Determine the (erroneous) mono-exponential _T_{sub}`2` value for voxels that would contain 50% MW and 50% IEW - is it closer to the _T_{sub}`2` value of MW or IEW?
Using [](#t2Eq8) and the MW/IEW T2 values estimated in [](#t2Problem2), calculate the myelin water fraction (MWF) values at the following TEs: 25, 50, 75, 100, 200, 300. Assume that there is 50% MW and 50% IEW.
**a.** The signal curves in [](#t2Plot3) were simulated for different SNR values, however these are relative to the maximum signal (i.e. TE=0). Using Binder or MATLAB and the notebook that was used to generate that figure, estimate the SNR of images acquired TE = 12.8, 128, 256, and 384 ms for a maximum signal SNR of 100.
**b.** Repeat the same calculations for a _T_{sub}`2`{sup}`*` of 30 ms.
**c.** Comment on these results and on how they would impact your imaging protocol planning. What’s the maximum TE you think is useful for each case?