The Circle of Willis (CoW) is an important network of arteries connecting major circulations of the brain. Its vascular architecture is believed to affect the risk, severity, and clinical outcome of serious neuro-vascular diseases. However, characterizing the highly variable CoW anatomy is still a manual and time-consuming expert task. The CoW is usually imaged by two angiographic imaging modalities, magnetic resonance angiography (MRA) and computed tomography angiography (CTA), but there exist limited public datasets with annotations on CoW anatomy, especially for CTA. Therefore we organized the TopCoW Challenge in 2023 with the release of an annotated CoW dataset. The TopCoW dataset was the first public dataset with voxel-level annotations for thirteen possible CoW vessel components, enabled by virtual-reality (VR) technology. It was also the first large dataset with paired MRA and CTA from the same patients. TopCoW challenge formalized the CoW characterization problem as a multiclass anatomical segmentation task with an emphasis on topological metrics. We invited submissions worldwide for the CoW segmentation task, which attracted over 140 registered participants from four continents. The top performing teams managed to segment many CoW components to Dice scores around 90%, but with lower scores for communicating arteries and rare variants. There were also topological mistakes for predictions with high Dice scores. Additional topological analysis revealed further areas for improvement in detecting certain CoW components and matching CoW variant topology accurately. TopCoW represented a first attempt at benchmarking the CoW anatomical segmentation task for MRA and CTA, both morphologically and topologically.

On general regular simplicial partitions $\mathcal{T}$ of bounded polytopal domains $\Omega \subset \mathbb{R}^d$, $d\in\{2,3\}$, we construct \emph{exact neural network (NN) emulations} of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical ``Raviart-Thomas element'', and the ``N\'{e}d\'{e}lec edge element''. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions $\mathcal{T}$ of $\Omega$ are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension $d\geq 2$. Our ``FE-Nets'' are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra $\Omega \subset \mathbb{R}^3$. They are thus an essential ingredient in the application of e.g., the methodology of ``physics-informed NNs'' or ``deep Ritz methods'' to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the ``Crouzeix-Raviart'' elements and Hybridized, Higher Order (HHO) methods.

Foundation models have taken over natural language processing and image generation domains due to the flexibility of prompting. With the recent introduction of the Segment Anything Model (SAM), this prompt-driven paradigm has entered image segmentation with a hitherto unexplored abundance of capabilities. The purpose of this paper is to conduct an initial evaluation of the out-of-the-box zero-shot capabilities of SAM for medical image segmentation, by evaluating its performance on an abdominal CT organ segmentation task, via point or bounding box based prompting. We show that SAM generalizes well to CT data, making it a potential catalyst for the advancement of semi-automatic segmentation tools for clinicians. We believe that this foundation model, while not reaching state-of-the-art segmentation performance in our investigations, can serve as a highly potent starting point for further adaptations of such models to the intricacies of the medical domain. Keywords: medical image segmentation, SAM, foundation models, zero-shot learning