TY - JOUR
T1 - Accuracy and performance analysis for Bloch and Bloch-McConnell simulation methods
AU - Graf, Christina
AU - Rund, Armin
AU - Aigner, Christoph Stefan
AU - Stollberger, Rudolf
N1 - Copyright © 2021 The Authors. Published by Elsevier Inc. All rights reserved.
PY - 2021/8
Y1 - 2021/8
N2 - PURPOSE: To introduce new solution methods for the Bloch and Bloch-McConnell equations and compare them quantitatively to different known approaches.THEORY AND METHODS: A new exact solution per time step is derived by means of eigenvalues and generalized eigenvectors. Fast numerical solution methods based on asymmetric and symmetric operator splitting, which are already known for the Bloch equations, are extended to the Bloch-McConnell equations. Those methods are compared to other numerical methods including spin domain, one-step and multi-step methods, and matrix exponential. Error metrics are introduced based on the exact solution method, which allows to assess the accuracy of each solution method quantitatively for arbitrary example data.RESULTS: Accuracy and performance properties for nine different solution methods are analyzed and compared in extensive numerical experiments including various examples for non-selective and slice-selective MR imaging applications. The accuracy of the methods heavily varies, in particular for short relaxation times and long pulse durations.CONCLUSION: In absence of relaxation effects, the numerical results confirm the rotation matrices approach as accurate and computationally efficient Bloch solution method. Otherwise, as well as for the Bloch-McConnell equations, symmetric operator splitting methods are recommended due to their excellent numerical accuracy paired with efficient run time.
AB - PURPOSE: To introduce new solution methods for the Bloch and Bloch-McConnell equations and compare them quantitatively to different known approaches.THEORY AND METHODS: A new exact solution per time step is derived by means of eigenvalues and generalized eigenvectors. Fast numerical solution methods based on asymmetric and symmetric operator splitting, which are already known for the Bloch equations, are extended to the Bloch-McConnell equations. Those methods are compared to other numerical methods including spin domain, one-step and multi-step methods, and matrix exponential. Error metrics are introduced based on the exact solution method, which allows to assess the accuracy of each solution method quantitatively for arbitrary example data.RESULTS: Accuracy and performance properties for nine different solution methods are analyzed and compared in extensive numerical experiments including various examples for non-selective and slice-selective MR imaging applications. The accuracy of the methods heavily varies, in particular for short relaxation times and long pulse durations.CONCLUSION: In absence of relaxation effects, the numerical results confirm the rotation matrices approach as accurate and computationally efficient Bloch solution method. Otherwise, as well as for the Bloch-McConnell equations, symmetric operator splitting methods are recommended due to their excellent numerical accuracy paired with efficient run time.
KW - Bloch equations
KW - Bloch-McConnell equations
KW - Exact solution
KW - Magnetic resonance imaging
KW - Numerical solution
KW - Operator splitting
UR - http://www.scopus.com/inward/record.url?scp=85107950290&partnerID=8YFLogxK
U2 - 10.1016/j.jmr.2021.107011
DO - 10.1016/j.jmr.2021.107011
M3 - Article
C2 - 34147025
SN - 1090-7807
VL - 329
JO - Journal of Magnetic Resonance
JF - Journal of Magnetic Resonance
M1 - 107011
ER -