Health Research, Arlington Innovation Center, Virginia Polytechnic Institute and State University Arlington, VA, USA.
Front Integr Neurosci. 2013 Apr 2;7:18. doi: 10.3389/fnint.2013.00018. eCollection 2013.
The foundation for an accurate and unifying Fourier-based theory of diffusion weighted magnetic resonance imaging (DW-MRI) is constructed by carefully re-examining the first principles of DW-MRI signal formation and deriving its mathematical model from scratch. The derivations are specifically obtained for DW-MRI signal by including all of its elements (e.g., imaging gradients) using complex values. Particle methods are utilized in contrast to conventional partial differential equations approach. The signal is shown to be the Fourier transform of the joint distribution of number of the magnetic moments (at a given location at the initial time) and magnetic moment displacement integrals. In effect, the k-space is augmented by three more dimensions, corresponding to the frequency variables dual to displacement integral vectors. The joint distribution function is recovered by applying the Fourier transform to the complete high-dimensional data set. In the process, to obtain a physically meaningful real valued distribution function, phase corrections are applied for the re-establishment of Hermitian symmetry in the signal. Consequently, the method is fully unconstrained and directly presents the distribution of displacement integrals without any assumptions such as symmetry or Markovian property. The joint distribution function is visualized with isosurfaces, which describe the displacement integrals, overlaid on the distribution map of the number of magnetic moments with low mobility. The model provides an accurate description of the molecular motion measurements via DW-MRI. The improvement of the characterization of tissue microstructure leads to a better localization, detection and assessment of biological properties such as white matter integrity. The results are demonstrated on the experimental data obtained from an ex vivo baboon brain.
通过仔细重新审视扩散加权磁共振成像（DW-MRI）信号形成的基本原则，并从头开始推导出其数学模型，为基于傅里叶的精确统一 DW-MRI 理论奠定了基础。这些推导专门通过使用复数来包括 DW-MRI 信号的所有元素（例如，成像梯度）来获得。与传统的偏微分方程方法相比，使用了粒子方法。结果表明，信号是磁矩数量（在初始时间的给定位置）和磁矩位移积分的联合分布的傅里叶变换。实际上，通过将三个附加维度对应于位移积分向量的频域变量，来扩展 k 空间。通过对完整的高维数据集应用傅里叶变换，恢复联合分布函数。在此过程中，为了重建信号的厄米对称性，应用相位校正以获得具有物理意义的实值分布函数。因此，该方法是完全无约束的，直接呈现位移积分的分布，而无需任何假设，例如对称性或马尔可夫性质。通过将描述位移积分的等位面叠加在低迁移率的磁矩数量分布图上，可视化联合分布函数。该模型通过 DW-MRI 提供了对分子运动测量的精确描述。组织微结构特征的改善导致更好地定位、检测和评估生物特性，例如白质完整性。结果在从体外狒狒大脑获得的实验数据上进行了演示。
