基于布洛克磁共振流动方程和贝塞尔函数的磁共振成像序列数学设计(英文)

Mathematical design of a magnetic resonance imaging sequence based on bloch NMR flow equations and bessel functions

Bloch方程是NMR/MRI计算、模拟和实验的基础,但通常在不加特定的绝热和非绝热条件的前提下获得Bloch流动方程的解析解是非常困难的。流动方程的一般解析解可以为理解NMR/MRI的基本概念提供额外的信息,而又不需要通常的指数方程。作者的目的是通过贝塞尔函数及其特性得到与时间无关的NMR流动方程的解析解。在不需要主观添加弥散项的前提下利用贝塞尔函数及其特性从NMR流动方程中获得了Stejskal-Tanner公式。这证实了弥散是Bloch流动方程的内在属性并可以通过如贝塞尔函数的适当数学函数提取出来。从解析解得到的非高斯行为的弥散信号在如脑白质的各项异性组织环境中是非常有意义的。发现弥散系数是与T1和T2弛豫参数直接相关的,因此通过对大量已有的贝塞尔函数进行合适利用可以在四个分离的缓存内采集MRI信号(实部和虚部,相位和绝对值)。能够利用MRI监测药物对于不同组织尤其是脑部功能活动的效果。

Bloch NMR equations are fundamental to all NMR/MRI computations,simulations and experiments.It has been very difficult to solve the Bloch NMR flow equations analytically without imposing specific adiabatic and non adiabatic conditions.General analytical solutions of the flow equations can easily provide additional information to understand the basic concept of NMR/MRI without the usual exponential functions.The goal of this report is to present analytical solutions to the time independent NMR flow equation using the Bessel functions and properties.We derived the Stejskal-Tanner formula from the NMR flow equations using the Bessel functions and properties without the need to arbitrarily add the diffusion term.This confirms that diffusion is an intrinsic property embedded in the Bloch NMR flow equation and can be extracted by the use of appropriate mathematical functions such as Bessel functions and properties.The analytical solutions result in a non-Gaussian behavior of the diffusion signal which may be very useful when tissue environment is anisotropic such as in white matter of the brain.It is exciting to note that the diffusion coefficient is directly related to the T 1 and T 2 relaxation parameters.The abundantly available Bessel functions and properties can then be appropriately applied to acquire MRI signals in four separate buffers（real and imaginary parts as well as phase and absolute value）.We may be able to monitor the effects of drugs on the functional activities of different tissues especially the brain by means of magnetic resonance Imaging.