Fast approximate solution of Bloch equation for simulation of
RF artifacts in Magnetic Resonance Imaging
Stéphane Balac
Institut Camille Jordan - UMR CNRS 5208
INSA de Lyon
69621 Villeurbanne, France
Laurent Chupin
Institut Camille Jordan - UMR CNRS 5208
INSA de Lyon
69621 Villeurbanne, France
Abstract
The technique used to spot information in Magnetic Resonance Imaging
(MRI) uses electromagnetic fields. Even minor perturbations of these
magnetic fields can disturb the imaging process and may render clinical
images inaccurate or useless. Modelling and numerical simulation of the
effects of static field inhomogeneities are now well established. Less
attention has been paid to mathematical modeling of the effects of
radio-frequency (RF) field inhomogeneities in the imaging process. When
considering RF field inhomogeneities, the major difficulty is
that the mathematical expression of the magnetisation vector is not
anymore explicitly known contrarily to the unperturbed case. Indeed,
the Bloch equation becomes an ordinary differential equation with non
constant coefficients that cannot be solved analytically. The use of
standard numerical schemes for ordinary differential equations to
compute the magnetisation vector appears to be costly and not well
suited for MRI image simulation. In this paper, we present an original
method for solving the Bloch equation based on a truncated series
expansion of the solution. The computational cost of the method reduces
to the computation of the eigen-elements of a block tridiagonal matrix
of a very small size.
Keywords
Bloch equation, magnetic resonance imaging, Fourier series expansion,
Floquet theory