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mOOC
qMRLab

Signal modelling

Monoexponential T2 Mapping

NeuroPoly Lab, Polytechnique Montreal, Quebec, Canada

The decay of the transverse magnetization (Mxy) is exponential and can be derived from the transverse component of the Bloch equations:

Mxy(TE)=Mz(0−)e−TE/T2\textit{M}_{xy}\left ( TE \right ) = Mz\left ( 0^{-} \right )e^{-TE/T_{2}}
(3.1)

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) Dortch, 2020.

Figure 3.3 shows transverse relaxation curves for T2 and T2* values for white matter and gray matter, using the relaxation times from Siemonsen et al., 2008.

Source:Jupyter Notebook
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Figure 3.3:Transverse relaxation decay curves for T2 and T2* values in white matter and gray matter. The T2 and T2* constants were taken from Siemonsen et al., 2008.

In NMR physics, it has been shown that T2 relaxation times must be equal to or shorter than 2 T1 Levitt, 2008; however, it has been demonstrated that T2 can exceed T1 in very rare cases Traficante, 1991. In living organisms however, T2 is always shorter than T1.

Click here to view the qMRLab (MATLAB/Octave) code that generated Figure 3.3.
%% 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")
References
  1. Dortch, R. D. (2020). Quantitative T2 and T2* Mapping. In Advances in Magnetic Resonance Technology and Applications (pp. 47–64). Elsevier.
  2. Siemonsen, S., Fitting, T., Thomalla, G., Horn, P., Finsterbusch, J., Summers, P., Saager, C., Kucinski, T., & Fiehler, J. (2008). T2’ imaging predicts infarct growth beyond the acute diffusion-weighted imaging lesion in acute stroke. Radiology, 248(3), 979–986.
  3. Levitt, M. H. (2008). Spin Dynamics: Basics Nuclear Magnetic Resonance.
  4. Traficante, D. D. (1991). Relaxation. Can T₂, be longer than T₁? Concepts Magn. Reson., 3(3), 171–177.
Monoexponential T2 Mapping
Introduction
Data Fitting
Data Fitting