A novel Flux Limited Splitting (FLS) non-linear Finite Volume (FV) method for families of linear Control Volume Distributed Multi Point Flux Approximation (CVD-MPFA) schemes is presented. The new formulation imposes a local discrete maximum principal (LDMP) which ensures that the discrete solution is free of spurious oscillations. The FLS scheme can be seen as a natural extension of the M-Matrix Flux Splitting method that splits the MPFA flux components in terms of the Two-Point Flux Approximation (TPFA) flux and Cross Diffusion Terms (CDT), with the addition of a dynamically computed relaxation parameter to the CDT that identifies and locally corrects the regions where the LDMP is violated. Moreover, the whole non-linear procedure was devised as a series of simple straightforward matrix operations. The methodology is presented considering the Multi-Point Flux Approximation with a Diamond (MPFA-D) in what we call the FLS + MPFA-D formulation which is tested using a series of challenging benchmark problems. For all test cases, the FLS repair technique imposes the LDMP and eliminates the spurious oscillations induced by the original MPFA-D method.

An automated method for registering B-mode ultrasound (US) and magnetic resonance imaging (MRI) of the carotid arteries is proposed. The registration uses geometric features, namely, lumen centerlines and lumen segmentations, which are extracted fully automatically from the images after manual annotation of three seed points in US and MRI. The registration procedure starts with alignment of the lumen centerlines using a point-based registration algorithm. The resulting rigid transformation is used to initialize a rigid and subsequent non-rigid registration procedure that jointly aligns centerlines and segmentations by minimizing a weighted sum of the Euclidean distance between centerlines and the dissimilarity between segmentations. The method was evaluated in 28 carotid arteries from eight patients and six healthy volunteers. First, the automated US lumen segmentation method was validated and optimized in a cross-validation experiment. Next, the effect of the weighting parameter of the proposed registration dissimilarity metric and the control point spacing in the non-rigid registration was evaluated. Finally, the proposed registration method was evaluated in comparison to an existing intensity-and-point-based method, a registration using only the centerlines and a registration using manual US lumen segmentations. Registration accuracy was measured in terms of the mean surface distance between manual US segmentations and the registered MRI segmentations. The average mean surface distance was 0.78 ± 0.34 mm for all subjects, 0.65 ± 0.09 mm for healthy volunteers and 0.87 ± 0.42 mm for patients. The results for the complete set were significantly better (Wilcoxon test, p <0.01) than the results for the intensity-and-point-based method and the centerline-based registration method. We conclude that the proposed method can robustly and accurately register US and MR images of the carotid artery, allowing multimodal analysis of the carotid plaque to improve plaque assessment.