Technical Note: Harmonic analysis applied to MR image distortion fields specific to arbitrarily shaped volumes

Magnetic resonance imaging is expected to play a more important role in radiation therapy given the recent developments in MR-guided technologies. MR images need to consistently show high spatial accuracy to facilitate RT-specific tasks such as treatment planning and in-room guidance. The present study investigates a new harmonic analysis method for the characterization of complex three-dimensional (3D) fields derived from MR images affected by system-related distortions.

An interior Dirichlet problem based on solving the Laplace equation with boundary conditions (BCs) was formulated for the case of a 3D distortion field. The second-order boundary value problem (BVP) was solved using a finite elements method (FEM) for several quadratic geometries - that is, sphere, cylinder, cuboid, D-shaped, and ellipsoid. To stress-test the method and generalize it, the BVP was also solved for more complex surfaces such as a Reuleaux 9-gon and the MR imaging volume of a scanner featuring a high degree of surface irregularities. The BCs were formatted from reference experimental data collected with a linearity phantom featuring a volumetric grid structure. The method was validated by comparing the harmonic analysis results with the corresponding experimental reference fields.

The harmonic fields were found to be in good agreement with the baseline experimental data for all geometries investigated. In the case of quadratic domains, the percentage of sampling points with residual values larger than 1 mm was 0.5% and 0.2% for the axial components and vector magnitude, respectively. For the general case of a domain defined by the available MR imaging field of view, the reference data showed a peak distortion of about 1 mm and 79% of the sampling points carried a distortion magnitude larger than 1 mm (tolerance intrinsic to the experimental data). The upper limits of the residual values after comparison with the harmonic fields showed max and mean of 1.4 and 0.25 mm, respectively, with only 1.5% of sampling points exceeding 1 mm.

A novel harmonic analysis approach relying on finite element methods was introduced and validated for multiple volumes with surface shape functions ranging from simple to highly complex. Since a boundary value problem is solved the method requires input data from only the surface of the desired domain of interest. It is believed that the harmonic method will facilitate (a) the design of new phantoms dedicated for the quantitation of MR image distortions in large volumes and (b) an integrative approach of combining multiple imaging tests specific to radiotherapy into a single test object for routine imaging quality control.