On the role of demagnetizing tensors in arbitrary orientations of general ellipsoids and implications for MRI safety assessment

Abstract

Background: Demagnetizing tensors describe the shape anisotropic contribution of magnetic susceptibility, that is, how an object's shape and orientation affect its internal magnetization. Ellipsoids possess a unique geometric property by exhibiting homogeneous internal magnetization, enabling a purely geometrical characterization. A general description of the demagnetizing tensor under arbitrary rotations would broaden their applicability across various fields, most notably in magnetic resonance safety evaluation. Purpose: Demagnetizing tensor is a well-defined concept in the ellipsoids principal frame, though its transformation under three-dimensional reorientation is often overlooked, a justifiable omission for spheroidal solutions of the Poisson equation. However, this does not hold for general ellipsoids under arbitrary reorientation. This work is motivated by the concerns in magnetic resonance imaging safety and its practical evaluation in clinical environments, aiming to extend and simplify the process. Methods: This work demonstrates the validity of directly rotating the orthogonal basis solutions, derived from Poisson's equation, and uses the procedure to evaluate a practical approximation, based on orthogonal area-projections. The approach is also applied to generalize force and torque calculations for ellipsoids under three-dimensional reorientation. Results: The method shows an exact match with the numerical solution and agrees with the standard scalar expressions for translation force and torque. Additionally, a unique connection to the MRI magic angle is found as the point of convergence for prolate spheroid aspect ratios under rotation. The area-projection approximation was demonstrated to perform fairly across prolate and oblate spheroids. Similar approximations might extend to irregular shapes, but numerical approaches remain preferable due to the complexity of internal field distributions. Conclusion: The presented approach offers an alternative method for force and torque calculations, such as those provided by ASTM, also generalizing the conventional approach. The area-projection approximation via SVD offers a straightforward extension to the original method. Finally, the connection between demagnetizing tensor rotation and the magic angle provides a new perspective on the phenomenon.