Abdominal aortic aneurysms (AAAs) are progressive focal dilatations of the abdominal aorta. AAAs may rupture, with fatal consequences in >80% of cases. Current clinical guidelines recommend elective surgical repair when the maximum AAA diameter exceeds 55 mm in men or 50 mm in women. Patients that do not meet these criteria are periodically monitored, with surveillance intervals based on the maximum AAA diameter. However, this diameter does not take into account the complex relation between the 3D AAA shape and its growth, making standardized intervals potentially unfit. Personalized AAA growth predictions could improve monitoring strategies. We propose to use an SE(3)-symmetric transformer model to predict AAA growth directly on the vascular model surface enriched with local, multi-physical features. In contrast to other works which have parameterized the AAA shape, this representation preserves the vascular surface's anatomical structure and geometric fidelity. We train our model using a longitudinal dataset of 113 computed tomography angiography (CTA) scans of 24 AAA patients at irregularly sampled intervals. After training, our model predicts AAA growth to the next scan moment with a median diameter error of 1.18 mm. We further demonstrate our model's utility to identify whether a patient will become eligible for elective repair within two years (acc = 0.93). Finally, we evaluate our model's generalization on an external validation set consisting of 25 CTAs from 7 AAA patients from a different hospital. Our results show that local directional AAA growth prediction from the vascular surface is feasible and may contribute to personalized surveillance strategies.

Hemodynamic quantities are valuable biomedical risk factors for cardiovascular pathology such as atherosclerosis. Non-invasive, in-vivo measurement of these quantities can only be performed using a select number of modalities that are not widely available, such as 4D flow magnetic resonance imaging (MRI). In this work, we create a surrogate model for hemodynamic flow field estimation, powered by machine learning. We train graph neural networks that include priors about the underlying symmetries and physics, limiting the amount of data required for training. This allows us to train the model using moderately-sized, in-vivo 4D flow MRI datasets, instead of large in-silico datasets obtained by computational fluid dynamics (CFD), as is the current standard. We create an efficient, equivariant neural network by combining the popular PointNet++ architecture with group-steerable layers. To incorporate the physics-informed priors, we derive an efficient discretisation scheme for the involved differential operators. We perform extensive experiments in carotid arteries and show that our model can accurately estimate low-noise hemodynamic flow fields in the carotid artery. Moreover, we show how the learned relation between geometry and hemodynamic quantities transfers to 3D vascular models obtained using a different imaging modality than the training data. This shows that physics-informed graph neural networks can be trained using 4D flow MRI data to estimate blood flow in unseen carotid artery geometries.

The orientation of a blood vessel as visualized in 3D medical images is an important descriptor of its geometry that can be used for centerline extraction and subsequent segmentation, labeling, and visualization. Blood vessels appear at multiple scales and levels of tortuosity, and determining the exact orientation of a vessel is a challenging problem. Recent works have used 3D convolutional neural networks (CNNs) for this purpose, but CNNs are sensitive to variations in vessel size and orientation. We present SIRE: a scale-invariant rotation-equivariant estimator for local vessel orientation. SIRE is modular and has strongly generalizing properties due to symmetry preservations. SIRE consists of a gauge equivariant mesh CNN (GEM-CNN) that operates in parallel on multiple nested spherical meshes with different sizes. The features on each mesh are a projection of image intensities within the corresponding sphere. These features are intrinsic to the sphere and, in combination with the gauge equivariant properties of GEM-CNN, lead to SO(3) rotation equivariance. Approximate scale invariance is achieved by weight sharing and use of a symmetric maximum aggregation function to combine predictions at multiple scales. Hence, SIRE can be trained with arbitrarily oriented vessels with varying radii to generalize to vessels with a wide range of calibres and tortuosity. We demonstrate the efficacy of SIRE using three datasets containing vessels of varying scales; the vascular model repository (VMR), the ASOCA coronary artery set, and an in-house set of abdominal aortic aneurysms (AAAs). We embed SIRE in a centerline tracker which accurately tracks large calibre AAAs, regardless of the data SIRE is trained with. Moreover, a tracker can use SIRE to track small-calibre tortuous coronary arteries, even when trained only with large-calibre, non-tortuous AAAs. Additional experiments are performed to verify the rotational equivariant and scale invariant properties of SIRE. In conclusion, by incorporating SO(3) and scale symmetries, SIRE can be used to determine orientations of vessels outside of the training domain, offering a robust and data-efficient solution to geometric analysis of blood vessels in 3D medical images.

Data-driven cell tracking and segmentation methods in biomedical imaging require diverse and information-rich training data. In cases where the number of training samples is limited, synthetic computer-generated data sets can be used to improve these methods. This requires the synthesis of cell shapes as well as corresponding microscopy images using generative models. To synthesize realistic living cell shapes, the shape representation used by the generative model should be able to accurately represent fine details and changes in topology, which are common in cells. These requirements are not met by 3D voxel masks, which are restricted in resolution, and polygon meshes, which do not easily model processes like cell growth and mitosis. In this work, we propose to represent living cell shapes as level sets of signed distance functions (SDFs) which are estimated by neural networks. We optimize a fully-connected neural network to provide an implicit representation of the SDF value at any point in a 3D+time domain, conditioned on a learned latent code that is disentangled from the rotation of the cell shape. We demonstrate the effectiveness of this approach on cells that exhibit rapid deformations (Platynereis dumerilii), cells that grow and divide (C. elegans), and cells that have growing and branching filopodial protrusions (A549 human lung carcinoma cells). A quantitative evaluation using shape features and Dice similarity coefficients of real and synthetic cell shapes shows that our model can generate topologically plausible complex cell shapes in 3D+time with high similarity to real living cell shapes. Finally, we show how microscopy images of living cells that correspond to our generated cell shapes can be synthesized using an image-to-image model.