Geometry adaptive waveformer for cardio-vascular modeling

TL;DR

The paper tackles the computational burden of patient-specific cardiovascular simulations by introducing a geometry-adaptive waveformer that maps irregular vascular geometries to a regular latent space for operator learning. It fuses a graph-operator-based geometry encoder/decoder with a waveformer that operates in latent space, augmented by a reduced-2D latent variant to improve efficiency. The framework is validated on multiple cardiovascular geometries (healthy pulmonary, healthy aorta, and coarctation cases), demonstrating high accuracy in predicting time-dependent pressure and flow fields and enabling uncertainty propagation analyses. The approach promises significant acceleration of cardiovascular analyses while preserving dynamical fidelity, with potential clinical impact for rapid, personalized cardiovascular assessment.

Abstract

Modeling cardiovascular anatomies poses a significant challenge due to their complex, irregular structures and inherent pathological conditions. Numerical simulations, while accurate, are often computationally expensive, limiting their practicality in clinical settings. Traditional machine learning methods, on the other hand, often struggle with some major hurdles, including high dimensionality of the inputs, inability to effectively work with irregular grids, and preserving the time dependencies of responses in dynamic problems. In response to these challenges, we propose a geometry adaptive waveformer model to predict blood flow dynamics in the cardiovascular system. The framework is primarily composed of three components: a geometry encoder, a geometry decoder, and a waveformer. The encoder transforms input defined on the irregular domain to a regular domain using a graph operator-based network and signed distance functions. The waveformer operates on the transformed field on the irregular grid. Finally, the decoder reverses this process, transforming the output from the regular grid back to the physical space. We evaluate the efficacy of the approach on different sets of cardiovascular data.

Figures (25)

• Figure 1: Schematic representing a patient-specific cardiovascular analysis involving computational modeling, post-processing, inference, and diagnosis
• Figure 2: A diagrammatic representation of the waveformer architecture. Firstly, inputs are uplifted by passing through local transformation $P$. The integral layer consists of two separate branches. In the first branch, wavelet decomposition is performed on the uplifted images and then is passed through a transformer. In the second branch, the inputs are directly fed to the other transformer. An activation is applied in the resultant output obtained by summing the outputs of the two branches. Then, the outputs are downlifted by passing through the transformation $Q$, which yields the prediction u(x). Here, the local transformations $P$ and $Q$ are modeled as fully connected neural networks of one hidden layer.
• Figure 3: Schematic of the geometry-adaptive waveformer, which consists of three components: a geometry encoder, a waveformer, and a geometry decoder. The initial input, $a_0(x)$, is passed through the geometry encoder, which transforms the input into a regular latent grid. The waveformer then operates on this latent space, and finally, the geometry decoder maps the output back to the original space to predict the response at the subsequent time step $u_{k+1}$.
• Figure 4: Schematic of modified geometry-adaptive waveformer, which consists of three components: a geometry encoder, a waveformer, and a geometry decoder. The initial input, $a_0(x)$, is passed through the geometry encoder, which transforms the input into a regular latent grid. The reduction block reduces the dimension of the 3D latent space to 2D space. The waveformer then operates on this 2D latent space and, subsequently, transforms it into a 3D latent space employing an expansion block. Finally, the geometry decoder maps the output back to the original space to predict the response at the subsequent time step $u_{k+1}$.
• Figure 5: Predictions results of pressure field measured in $mmHg$ for the dataset 1: (Top to bottom) Geometry adaptive waveformer prediction, ground truth response, and $L_1$ error.

Theorems & Definitions (1)

• Remark