Magnetic resonance imaging (MRI) and magnetic resonance spectroscopy
(MRS) make use of the Larmor resonance signals emitted by
protons precessing in the local magnetic induction, which is
provided by an external static magnetic field modified by the local
invironment.
In fact, the MR signal is highly sensitive to the magnetic
properties of the matter surrounding the nuclei. Theoretical studies
of the resonance signal behaviour in situations of varying magnetic
susceptibilities and dynamic physiological processes as for instance
diffusion utilize the analytical magnetostatic solutions of specific
geometrical bodies. Models of structures like cells or blood vessels
can be built up from arrays of such simple magnetic bodies.
In particular prolate and oblate spheroids are used as such building
blocks to analyse the local magnetic field distribution in the
vicinity of blood cells in MRS of cells (Kuchel and Bulliman, 1989).
In that previous work the reaction fields are computed for spheroids
with the z-axis as the symmetry axis and a homogeneous static
external field of arbitrary direction. In the current work we derive
formulas in Cartesian coordinates for arbitrary directions of both
the symmetry axis and of the external magnetic field. This grants
still more freedom, flexibility and ease for building complex
structures composed of arbitrarily arranged spheroids. These
formulas are derived in the the report
Reaction
Fields of Homogeneous Magnetic Spheroids of Arbitrary Direction
in a Homogeneous Magnetic Field. A Toolbox for MRI and MRS of
Heterogeneous Tissue
ITPR-2011-21CorRev (January 2014}
given below under the link "Report" and in COMPEL (The international
journal for computation and mathematics in electrical and electronic
engineering)
v.23, #3(June 2013) 936 – 960:
Potential and Field of a
Homogeneous Magnetic Spheroid of Arbitrary Direction in a
Homogeneous Magnetic Field in Cartesian Coordinates
The corrected version of the final manuscript of the above paper can
be found below under the link "COMPEL2014".
A concise paper describing the subject is:
Potential of Spheroids in a Homogeneous Magnetic Field
in Cartesian Coordinates
in the Proc. of the
15th International IGTE Symposium
on Numerical Field Calculation in Electrical Engineering,
Graz, Austria, 16 to 19 September 2012,
pp. 310-314, ISBN 978-3-85125-258-3
. A copy of this paper with the corrections
added is given in the link "IGTE2012". Correction: This
paper contains four times the same misprint: In eqs.(33) to (35) and in the fifth
line after (35) the exponetial should read as
exp(-t/T_2) in place of exp(-T_2/t).
Some of the calculations are lengthy and involved. They were
performed in Mathematica.
The corresponding notebooks are given in the link "Mathematica
notebooks" below.
Reference:
Philip W. Kuchel and Brain T. Bulliman: "Perturbation of homogeneous
magnetic fields in isolated isolated single and confocal spheroids.
Implication for NMR spectroscopy of cells." NMR in biomedicine, 2:151- 160 (1989)