A fast calculation method for the magnetic field distribution due to (dynamic) changes in susceptibility may allow real-time interventional applications. Here it is shown that a direct relationship can be obtained between the magnetic field perturbation and the susceptibility distribution inside the MR magnet using a first order perturbation approach to Maxwell's magneto-static equations, combined with the Fourier transformation technique to solve partial derivative equations. The mathematical formalism does not involve any limitation with respect to shape or homogeneity of the susceptibility field. A first order approximation is sufficient if the susceptibility range does not exceed 10 Ϫ4 (or 100 ppm). The formalism allows fast numerical calculations using 3D matrices. A few seconds computation time on a PC is sufficient for a 128 ϫ 128 ϫ 128 matrix size. Predicted phase maps fitted both analytical and experimental data within 1% precision.

Respiratory motion introduces substantial uncertainties in abdominal radiotherapy for which traditionally large margins are used. The MR-Linac will open up the opportunity to acquire high resolution MR images just prior to radiation and during treatment. However, volumetric MRI time series are not able to characterize 3D tumor and organ-at-risk motion with sufficient temporal resolution. In this study we propose a method to estimate 3D deformation vector fields (DVFs) with high spatial and temporal resolution based on fast 2D imaging and a subject-specific motion model based on respiratory correlated MRI. In a pre-beam phase, a retrospectively sorted 4D-MRI is acquired, from which the motion is parameterized using a principal component analysis. This motion model is used in combination with fast 2D cine-MR images, which are acquired during radiation, to generate full field-of-view 3D DVFs with a temporal resolution of 476 ms. The geometrical accuracies of the input data (4D-MRI and 2D multi-slice acquisitions) and the fitting procedure were determined using an MR-compatible motion phantom and found to be 1.0-1.5 mm on average. The framework was tested on seven healthy volunteers for both the pancreas and the kidney. The calculated motion was independently validated using one of the 2D slices, with an average error of 1.45 mm. The calculated 3D DVFs can be used retrospectively for treatment simulations, plan evaluations, or to determine the accumulated dose for both the tumor and organs-at-risk on a subject-specific basis in MR-guided radiotherapy.