Segmentation of the coronary vessel wall in intravascular ultrasound is a fundamental step in guiding coronary intervention. However, it is an challenging task, even for highly skilled cardiologists, due to image artefacts and shadowed regions caused by calcified plaque, guide wires and vessel side branches. Recently, dense-based neural networks have been applied to this task, however, they often fail to predict anatomically plausible contours in these low-signal areas. We propose a novel methodology called Polygon-based Contour Refiner (POLYCORE) that addresses topological error in dense-based segmentation networks using a relational inductive bias through higher-order connections between vertices to learn anatomically rational contours. Our approach remedies the over-smoothing phenomena common in polygon networks by introducing a new vector field refinement module which enables pixel-level detail to be added in an iterative process. POLYCORE is enhanced with augmented polygon aggregation which we show is more effective than typical dense-based test-time augmentation strategies. We achieve state-of-the-art results on two diverse datasets, observing particular improvements when segmenting the lumen structure and in topologically-challenging regions containing shadow artefacts. Our source code is available here: http://orcid.org/https://github.com/kitbransby/POLYCORE.