Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall

TL;DR

This paper tackles the high computational cost of CFD for patient-specific hemodynamics by introducing GEM-GCN, a SE(3)-equivariant, gauge-equivariant mesh convolutional network that operates directly on artery-wall meshes to predict time-resolved hemodynamic fields. By leveraging an encoder–decoder with three pooling levels and GEM convolution that respects intrinsic surface geometry, the approach achieves accurate directional WSS and low NMAE while delivering predictions orders of magnitude faster than CFD. The method is evaluated on large synthetic datasets of single and bifurcating coronary arteries, including steady and pulsatile flows, and demonstrates robustness to rotations and some extrapolation across boundary conditions, with noted sensitivity to mesh remeshing. The results highlight GEM-GCN as a practical, data-efficient surrogate for CFD in personalized hemodynamics, with potential for rapid prototyping and integration into clinically oriented workflows, pending further validation and uncertainty quantification.

Abstract

Computational fluid dynamics (CFD) is a valuable asset for patient-specific cardiovascular-disease diagnosis and prognosis, but its high computational demands hamper its adoption in practice. Machine-learning methods that estimate blood flow in individual patients could accelerate or replace CFD simulation to overcome these limitations. In this work, we consider the estimation of vector-valued quantities on the wall of three-dimensional geometric artery models. We employ group equivariant graph convolution in an end-to-end SE(3)-equivariant neural network that operates directly on triangular surface meshes and makes efficient use of training data. We run experiments on a large dataset of synthetic coronary arteries and find that our method estimates directional wall shear stress (WSS) with an approximation error of 7.6% and normalised mean absolute error (NMAE) of 0.4% while up to two orders of magnitude faster than CFD. Furthermore, we show that our method is powerful enough to accurately predict transient, vector-valued WSS over the cardiac cycle while conditioned on a range of different inflow boundary conditions. These results demonstrate the potential of our proposed method as a plugin replacement for CFD in the personalised prediction of hemodynamic vector and scalar fields.

Figures (11)

• Figure 1: Overview. We propose a gauge-equivariant mesh-graph convolutional network (GEM-GCN) to estimate discrete hemodynamic fields mapped to the vertices of a surface mesh of the artery wall. The GCN is powered by anisotropic (spatially-oriented) gauge-equivariant mesh (GEM) convolution with high filter expressivity. The combination of GEM convolution with appropriate input features leads to an end-to-end $\mathrm{SE}(3)$-equivariant neural network.
• Figure 2: Artery datasets. We develop and evaluate our method using two distinct classes of geometric models: synthetic single arteries (left) and bifurcating arteries modelled after the left main bifurcation of the coronary artery tree (right). The single arteries contain flow extensions to let the flow fully develop from a uniform inflow boundary condition. The bifurcating arteries are simulated with parabolic inflow and thus without flow extensions. They consist of the proximal main vessel (PMV) that branches into distal main vessel (DMV) and side branch (SB). Each bifurcation can be described by the angles $\beta$ and $\beta'$.
• Figure 3: Pulsatile-flow waveform adapted from BeierOrmiston2016. We linearly scale this waveform for the simulations with varying (average) coronary blood flow boundary condition.
• Figure 4: Network architecture. Our mesh-based GCN outputs time-discretised, pulsatile hemodynamic fields $f^\text{out} \colon \mathcal{V} \to \mathbb{R}^{T \times c_\text{out}}$, where $|\mathcal{V}| = N$, subject to a (scalar) coronary blood flow parameter, given an input consisting of artery-wall mesh and vertex-wise geodesic distance to the artery inlet. A large receptive field is efficiently obtained using a three-level pooling scheme. To enable deep networks, we employ residual blocks consisting of two convolution modules and skip connection. The per-vertex colour of the signal before and after residual blocks corresponds to the scalar activation mapped to the vertices.
• Figure 5: Filter comparison. Isotropic, attention-scaled, and GEM convolution use kernels, in comparison to PointNet++ message passing. While attention-scaled convolution and PointNet++ both learn to distinguish neighbouring vertices through an attention mechanism, GEM convolution is equipped with a notion of direction.

Theorems & Definitions (7)

• Definition 1: Anisotropy
• Proposition 1
• Proposition 2
• proof
• Remark 1
• Corollary 1
• proof