Physics-informed graph neural networks for flow field estimation in carotid arteries

TL;DR

The work tackles rapid, patient-specific velocity-field estimation in carotid arteries from limited in-vivo data by building a physics-informed, SE(3)-equivariant graph neural network surrogate. It extends PointNet++ with steerable layers and employs mesh-free discretised Navier-Stokes residuals to regularise training, conditioned on boundary inflow. Key contributions include the SE-PointNet++ architecture, an efficient discretisation scheme for differential operators, demonstration of transfer to black-blood MRI geometries, and ablation insights on inflow conditioning and geometric features. Results show high directional accuracy and improved physical conformity (mass and momentum conservation), with competitive generalisation to unseen imaging modalities, offering a fast, data-efficient alternative to CFD for in-vivo hemodynamics. This approach could broaden clinical deployment by enabling rapid, noninvasive flow estimation across varied vascular geometries.

Abstract

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.

Figures (7)

• Figure 1: Overview. We represent the carotid artery by $n$ un-ordered points with $c_\text{in}$-dimensional input features, one of which is patient-specific inflow $u_\text{in}$ (shown in green). PointNet++ learns to map the input features to 3D velocity vectors using hierarchical down-sampling (pooling) and up-sampling (interpolation) layers. PointNet++ is comprised of learnable functions which we choose as vanilla multilayer perceptrons (MLP) or $\mathrm{E}(3)$-steerable BrandstetterHesselink2022 MLPs. We train PointNet++ with ground truth velocity fields obtained in-vivo via 4D flow MRI. Based on the Navier-Stokes equations for incompressible fluids that govern arterial blood flow, we regularise the training loss with residuals using discretised differential operators.
• Figure 2: Pooling and interpolation. We use message passing layers to pool clusters $\mathcal{C}(p)$ of fine-scale to coarse-scale features in the contracting pathway (left) and interpolation to expand coarse-scale features back to original resolution (right). For simplicity, we visualise interpolation in 2D based on three closest points.
• Figure 3: Operator discretisation. We visualise a) an element of a tetrahedral mesh used in our construction and b) colouring of a triangle using barycentric coordinates, as a visualisation of barycentric basis functions. Note that there is a linear gradient between just two colours along each of the three sides of the triangle.
• Figure 4: Results of neural network predictions ("ML") by PointNet++ compared with 4D flow MRI in the left and right carotid artery of subjects in the test split. We visualise the velocity field via 3D streamlines and render a projection image. Shown are examples of relatively a) good, b) average and c) poor performance. Additionally, we d) show an example where the ground truth is noisy, yet the model predicts a visually sound velocity field.
• Figure 5: Effect of group equivariance of SE-PointNet++ compared to PointNet++. We show the left and right carotid artery of subjects in the test split and visualise the velocity field via streamlines.