The complex transverse water proton magnetization subject to diffusion‐encoding magnetic field gradient pulses in a heterogeneous medium such as brain tissue can be modeled by the Bloch‐Torrey partial differential equation. The spatial integral of the solution of this equation in realistic geometry provides a gold‐standard reference model for the diffusion MRI signal arising from different tissue micro‐structures of interest. A closed form representation of this reference diffusion MRI signal called matrix formalism, which makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal, was derived 20 years ago. In addition, once the Laplace eigendecomposition has been computed and saved, the diffusion MRI signal can be calculated for arbitrary diffusion‐encoding sequences and b‐values at negligible additional cost. Up to now, this representation, though mathematically elegant, has not been often used as a practical model of the diffusion MRI signal, due to the difficulties of calculating the Laplace eigendecomposition in complicated geometries. In this paper, we present a simulation framework that we have implemented inside the MATLAB‐based diffusion MRI simulator SpinDoctor that efficiently computes the matrix formalism representation for realistic neurons using the finite element method. We show that the matrix formalism representation requires a few hundred eigenmodes to match the reference signal computed by solving the Bloch‐Torrey equation when the cell geometry originates from realistic neurons. As expected, the number of eigenmodes required to match the reference signal increases with smaller diffusion time and higher b‐values. We also convert the eigenvalues to a length scale and illustrate the link between the length scale and the oscillation frequency of the eigenmode in the cell geometry. We give the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch‐Torrey operator and compute the Bloch‐Torrey eigenfunctions and eigenvalues. This work is another step in bringing advanced mathematical tools and numerical method development to the simulation and modeling of diffusion MRI.

中文翻译：

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基于拉普拉斯特征函数的真实神经元扩散 MRI 信号的实际计算。

受扩散编码磁场梯度脉冲影响的复杂横向水质子磁化强度可以通过 Bloch-Torrey 偏微分方程建模。该方程在现实几何中解的空间积分为来自不同组织微结构的扩散 MRI 信号提供了黄金标准参考模型。这个参考扩散 MRI 信号的封闭形式表示称为矩阵形式，它明确了生物细胞的拉普拉斯特征值和特征函数与其扩散 MRI 信号之间的联系，是在 20 年前推导出来的。此外，一旦计算并保存了拉普拉斯特征分解，就可以计算任意扩散编码序列的扩散 MRI 信号，并且乙- 附加成本可忽略不计的价值。到目前为止，这种表示虽然在数学上很优雅，但由于难以计算复杂几何结构中的拉普拉斯特征分解，因此并未经常用作扩散 MRI 信号的实用模型。在本文中，我们提出了一个模拟框架，我们已经在基于 MATLAB 的扩散 MRI 模拟器 SpinDoctor 中实现了该框架，该框架使用有限元方法有效地计算了真实神经元的矩阵形式表示。我们表明，当细胞几何结构源自真实神经元时，矩阵形式化表示需要几百个特征模式来匹配通过求解 Bloch-Torrey 方程计算出的参考信号。正如预期的那样，b 值。我们还将特征值转换为长度尺度，并说明长度尺度与单元几何中特征模式的振荡频率之间的联系。我们给出将拉普拉斯特征函数与 Bloch-Torrey 算子的特征函数联系起来的变换，并计算 Bloch-Torrey 特征函数和特征值。这项工作是将先进的数学工具和数值方法开发引入扩散 MRI 的模拟和建模的又一步。