Mathematical design of a magnetic resonance imaging sequence based on Bloch NMR flow equations and Bessel functions

Abstract

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.