Study On Fractional Order Equation Based MRI Model

Fractional calculus has been a useful tool for a long time.We can use it to generalize the derivatives from integer order to fractional order.MRI is the most common imaging technique nowadays.The most critical model describing MR signal attenuation is based on the Bloch equations.Researchers have shown,when the MR signal attenuation is influenced badly by the diffusion of water molecules,the Bloch equations can’t describe the signal attenuation well.Then researchers suggest to use the Bloch-Torrey equations,which basically,is the Bloch equations coupled with the diffusion part.In order to obtain more accurate medical image,researchers have extend the classical Bloch equations and the classical Bloch-Torrey equations to the fractional order Bloch equations and the fractional order Bloch-Torrey equations.They have shown there exists significant differences between the integer model and the fractional model and someone might propose a new method or build a new model to obtain more accurate MRI images based on these differences.This article mainly involves two research contents,one is the numerical solution of fractional order Bloch equations.The other is the numerical solution of fractional order Bloch-Torrey equations.First of all,we introduce two schemes which are used in the numerical solution of partial differential equations of fractional order.Secondly,we use these two schemes for the numerical solution of fractional differential equations.Next,we rewrite them by QTT decomposition,and then solve the fractional Bloch equations and the fractional Bloch-Torrey equations with them.At the last part,a series of numerical examples are presented,here we mainly base on the scheme of the second-order accuracy in space to obeserve the temporal evolution of the numerical solution of the fractional Bloch equations and Bloch-Torrey equations.One of the inovation points of this article is that we use the QTT format to solve the fractional order Bloch equations.Besides,we deduce the numerical format for the twodimensional Bloch-Torrey equations,and a numerical simulation of one dimensional case is presented.