Solving the Bloch equation with the Magnus expansion

Abstract

In this paper we will study the use of the Magnus expansion in solving the Bloch equation. Solving the Bloch equation is a vital step in quantitative MRI methods such as MR-STAT and MR-fingerprinting and is also necessary for other applications, such as RF pulse design. These applications require an efficient algorithm for finding a numerical solution of the Bloch equation. Numerical schemes for approximating the fundamental solution of the linear differential equation $\dot{X}=A(t)X(t)$ are given in a paper by Blanes et al. "Improved high order integrators based on the Magnus expansion" and are based on the Magnus expansion. Two of these methods, which we will call the Taylor scheme and the Gauss-Legendre scheme, will be covered in this thesis. We will explore the applicability of these methods to solving the Bloch equation and in particular, to the spinorized Bloch equation (SBE). We will investigate how these methods perform with respect to approximation errors and computation time, and how they compare to the commonly used piecewise constant method. Results indicate that the Magnus series methods require fewer time steps and less computing time than the piecewise constant method to reach the same level of accuracy.