Analytical Solution to Diffusion-Advection Equation in Spherical Coordinate Based on the Fundamental Bloch NMR Flow Equations

摘要

A general solution for transverse magnetization, the nuclear magnetic resonance (NMR) signals for diffusion-advection equation with spatially varying velocity and diffusion coefficients, which is based on the fundamental Bloch NMR flow equations, was obtained using the method of separation of variable. It assumed that the velocity component is proportional to the coordinate and that the diffusion coefficient is proportional to the square of the corresponding component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient. After some assumptions, the 3-D equation degenerates to a 2-D problem. The solution to this equation is useful in describing physical phenomenon such as transport of materials in a fluid. The result obtained in this study can have applications in functional magnetic resonance imaging (fMRI) with more accurate information.