Medical Physics: Tools for Modelling Tissues

Markus Kraiger and Bernhard Schnizer


General description of the research

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 biomedicine2:151- 160 (1989)

  1. Report
  2. Mathematica notebooks
  3. IGTE2012
  4. COMPEL2014