Medical Physics: Tools for Modelling Tissues

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 environment. 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 a report ITPR-2011-21CorRev (January 2014} and in a paper published in COMPEL. They are used to model trabecular bone structures.