SIRE: Scale-invariant, rotation-equivariant estimation of artery orientations using graph neural networks.

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.