Accuracy and performance analysis for Bloch and Bloch-McConnell simulation methods

Christina Graf, Armin Rund, Christoph Stefan Aigner, Rudolf Stollberger*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number107011
JournalJournal of Magnetic Resonance
Publication statusPublished - Aug 2021


  • Bloch equations
  • Bloch-McConnell equations
  • Exact solution
  • Magnetic resonance imaging
  • Numerical solution
  • Operator splitting

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Nuclear and High Energy Physics
  • Biophysics
  • Biochemistry


Dive into the research topics of 'Accuracy and performance analysis for Bloch and Bloch-McConnell simulation methods'. Together they form a unique fingerprint.

Cite this