标题： 基于拉普拉斯本征函数的现实神经元扩散MRI信号的实际计算 显示英文标题

标题： Practical computation of the diffusion MRI signal of realistic neurons based on Laplace eigenfunctions

摘要： 在如脑组织这样的异质介质中，受到扩散编码磁场梯度脉冲作用的复杂横向水质子磁化强度可以通过Bloch-Torrey偏微分方程进行建模。 该方程解在现实几何中的空间积分提供了来自感兴趣的不同组织微结构的扩散MRI信号的黄金标准参考模型。 这种参考扩散MRI信号的闭合形式表示是在二十年前推导出来的，称为矩阵形式，它明确地建立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。 此外，一旦计算并保存了拉普拉斯特征分解，就可以以几乎可以忽略不计的额外成本计算任意扩散编码序列和b值的扩散MRI信号。 到目前为止，尽管这种表示在数学上很优雅，但由于在复杂几何中计算拉普拉斯特征分解的困难，它并未被广泛用作扩散MRI信号的实际模型。 在本文中，我们提出了一种模拟框架，我们在基于MATLAB的扩散MRI模拟器SpinDoctor内部实现了该框架，该框架使用有限元方法高效地计算现实神经元的矩阵形式表示。 我们展示了当细胞几何形状来自现实神经元时，矩阵形式表示需要大约几百个特征模态来匹配通过求解Bloch-Torrey方程计算出的参考信号。 正如预期的那样，为了匹配参考信号所需的特征模态数量随着更小的扩散时间和更高的b值而增加。 我们给出了将拉普拉斯特征函数与Bloch-Torrey算子的特征函数相关联的变换，并计算了Bloch-Torrey特征函数和特征值。 显示英文摘要

摘要： 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 has been derived twenty years ago, called Matrix Formalism that makes explicit the link between the Laplace eigenvalues and eigenfunctions of the biological cell and its diffusion MRI signal. 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 elements method. We show the Matrix Formalism representation requires around a few hundred eigenmodes to match the reference signal computed by solving the Bloch-Torrey equation when the cell geometry comes from realistic neurons. As expected, the number of required eigenmodes to match the reference signal increases with smaller diffusion time and higher b-values. We gave the transformation that links the Laplace eigenfunctions to the eigenfunctions of the Bloch-Torrey operator and computed the Bloch-Torrey eigenfunctions and eigenvalues.