Magnetic Resonance Simulation of Effective Transverse Relaxation (T2*)

TL;DR

The paper tackles the challenge of efficiently simulating the MR effective transverse relaxation $T_2^{*}$, which comprises reversible $T_2^{\prime}$ and irreversible $T_2$ components. It introduces a continuous representation using a linear phase model to directly simulate the Lorentzian $T_2^{\prime}$ through frequency-domain derivatives, avoiding the need for 100+ isochromats per point. Key contributions are: (i) a Lorentzian representation via $f_{\omega}(\omega)=\ell(\omega;0,1/T_2^{\prime})\exp(i\omega\partial_{\omega}\phi_{\omega})$, (ii) analytic solutions for without-RF subsequences to compute $\partial_p\boldsymbol{M}$, and (iii) combined transitions with a 7-component state to accelerate RF-driven updates; each is supported by simple-case verification and large-scale phantom simulations. The results show realistic recovery of $T_2^{\prime}$ effects in circles-$T_2^{*}$ and brain-$T_2^{*}$ phantoms and demonstrate substantial runtime savings when using analytic solutions and combined transitions, while maintaining accuracy. This approach enables efficient, high-fidelity MR simulations of $T_2^{*}$ without prohibitive computational costs, enhancing MR sequence development and system evaluation.

Abstract

Purpose: To simulate effective transverse relaxation ($T_2^*$) as a part of MR simulation. $T_2^*$ consists of reversible ($T_2^{\prime}$) and irreversible ($T_2$) components. Whereas simulations of $T_2$ are easy, $T_2^{\prime}$ is not easily simulated if only magnetizations of individual isochromats are simulated. Theory and Methods: Efficient methods for simulating $T_2^{\prime}$ were proposed. To approximate the Lorentzian function of $T_2^{\prime}$ realistically, conventional simulators require 100+ isochromats. This approximation can be avoided by utilizing a linear phase model for simulating an entire Lorentzian function directly. To represent the linear phase model, the partial derivatives of the magnetizations with respect to the frequency axis were also simulated. To accelerate the simulations with these partial derivatives, the proposed methods introduced two techniques: analytic solutions, and combined transitions. For understanding the fundamental mechanism of the proposed method, a simple one-isochromat simulation was performed. For evaluating realistic cases, several pulse sequences were simulated using two phantoms with and without $T_2^{\prime}$ simulations. Results: The one-isochromat simulation demonstrated that $T_2^{\prime}$ simulations were possible. In the realistic cases, $T_2^{\prime}$ was recovered as expected without using 100+ isochromats for each point. The computational times with $T_2^{\prime}$ simulations were only 2.0 to 2.7 times longer than those without $T_2^{\prime}$ simulations. When the above-mentioned two techniques were utilized, the analytic solutions accelerated 19 times, and the combined transitions accelerated up to 17 times. Conclusion: Both theory and results showed that the proposed methods simulated $T_2^{\prime}$ efficiently by utilizing a linear model with a Lorentzian function, analytic solutions, and combined transitions.

Figures (7)

• Figure 1: Various $T_2^{\prime}$ models. (a) A set of the Dirac delta functions. (b) A set of the rectangular functions. (c) A Lorentzian function. Whereas the models (a) and (b) approximates a Lorentzian function to be simulated with a set of functions, the model (c) uses the Lorentzian function to be simulated directly.
• Figure 2: A continuous representation. (a) Relationship between a transverse magnetization and its partial derivative with respect to $\omega$. The transverse magnetization is orthogonal to its partial derivative. (b) A linear phase model. The transverse magnetization $M_{xy}$ is modeled as a constant magnitude with a phase linearly changed in the $\omega$ axis.
• Figure 3: An example case of approximating a Lorentzian function using a set of thin rectangular functions. The function approximated with 10 isochromats is far from the Lorentzian function. There are still spike-like errors in the function approximated with 100 isochromats. The function approximated with 1000 isochromats is similar to the Lorentzian function. These functions show that at least 100+ rectangular functions are required for better approximation.
• Figure 4: The layout of the circles-$T_2^{*}$ phantom. There were a large cylinder and 9 pairs of concentric cylinders. The $T_1$, $T_2$, and $T_2^{\prime}$ values of all cylinders are given in Table \ref{['tableCircles']}.
• Figure 5: Temporal dynamics of (a) the simulated pulse sequence, (b) the magnetization vector, (c) its partial derivatives, and (d) measured transverse magnetization using an ADC with and without $T_2^{\prime}$. As shown in (b), $M_y$ decayed monotonically with $T_2$. As shown in (c), $\partial_{\omega} M_x$ increased monotonically except the time around the refocusing pulse shown in (a). By simulating (b) only, $T_2$ decay was simulated. By simulating both (b) and (c), $T_2^{*}$ decay could be simulated.