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.

翻译：目的：作为磁共振模拟的一部分，模拟有效横向弛豫($T_2^*$)。$T_2^*$由可逆($T_2^{\prime}$)和不可逆($T_2$)分量组成。虽然$T_2$的模拟较为容易，但如果仅模拟单个等色团的磁化强度，$T_2^{\prime}$则不易模拟。理论与方法：提出了模拟$T_2^{\prime}$的高效方法。为了真实地逼近$T_2^{\prime}$的洛伦兹函数，传统模拟器需要100个以上的等色团。通过利用线性相位模型直接模拟整个洛伦兹函数，可以避免这种近似。为了表示线性相位模型，还模拟了磁化强度相对于频率轴的偏导数。为了加速这些偏导数的模拟，所提出的方法引入了两种技术：解析解和组合跃迁。为了理解所提出方法的基本机制，进行了简单的单等色团模拟。为了评估实际情况，使用两个体模（分别进行和不进行$T_2^{\prime}$模拟）模拟了若干脉冲序列。结果：单等色团模拟证明$T_2^{\prime}$模拟是可行的。在实际情况下，无需为每个点使用100个以上的等色团，即可按预期恢复$T_2^{\prime}$。进行$T_2^{\prime}$模拟的计算时间仅为不进行$T_2^{\prime}$模拟时的2.0至2.7倍。当利用上述两种技术时，解析解加速了19倍，组合跃迁加速了高达17倍。结论：理论和结果均表明，所提出的方法通过利用具有洛伦兹函数的线性模型、解析解和组合跃迁，高效地模拟了$T_2^{\prime}$。