Neural Fields for Continuous Periodic Motion Estimation in 4D Cardiovascular Imaging

TL;DR

This work tackles the limitation of static arterial walls in 4D flow MRI by modeling continuous periodic wall motion over the cardiac cycle using a trainable implicit neural representation $H_\theta$ for the time-dependent velocity field. The velocity field is embedded in an ODE to yield a diffeomorphic deformation vector field across time, with periodicity enforced through unit-circle time encoding and a cycle-consistency regularization term $R_{cycle}$. Evaluations on synthetic data, ECG-gated CT, and 4D flow MRI data show that incorporating time encoding and $R_{cycle}$ improves deformation accuracy and enables faithful visualization of wall motion, as reflected in metrics such as $HSD$ and $PSNR$. Overall, the approach provides a data-efficient, continuous, physiologically consistent framework for enhanced 4D cardiovascular motion analysis with potential clinical impact.

Abstract

Time-resolved three-dimensional flow MRI (4D flow MRI) provides a unique non-invasive solution to visualize and quantify hemodynamics in blood vessels such as the aortic arch. However, most current analysis methods for arterial 4D flow MRI use static artery walls because of the difficulty in obtaining a full cycle segmentation. To overcome this limitation, we propose a neural fields-based method that directly estimates continuous periodic wall deformations throughout the cardiac cycle. For a 3D + time imaging dataset, we optimize an implicit neural representation (INR) that represents a time-dependent velocity vector field (VVF). An ODE solver is used to integrate the VVF into a deformation vector field (DVF), that can deform images, segmentation masks, or meshes over time, thereby visualizing and quantifying local wall motion patterns. To properly reflect the periodic nature of 3D + time cardiovascular data, we impose periodicity in two ways. First, by periodically encoding the time input to the INR, and hence VVF. Second, by regularizing the DVF. We demonstrate the effectiveness of this approach on synthetic data with different periodic patterns, ECG-gated CT, and 4D flow MRI data. The obtained method could be used to improve 4D flow MRI analysis.

Figures (5)

• Figure 1: Overview of the proposed method. We train an implicit neural representation $H_{\theta}\colon \mathrm{\Omega}\times [0, T] \to \mathbb{R}^3$ with spatial domain $\Omega \subset \mathbb{R}^3$, time horizon $T$, and weights $\theta$ to optimize. For every spatial point $\hat{P} = (x,y,z) \in \mathbb{R}^3$ and time frame $t$, $H_{\theta}(\hat{P}, t)$ represents the time-dependent velocity field describing the periodic motion at any point in space and time. We obtain a DVF by embedding the VVF $H_\theta$ in an ODE. Applying the DVF to the first time frame mesh allows for a full-cycle deformation.
• Figure 2: Synthetic dataset of linear, exponential, and periodic growth patterns (a). Volume comparison of the sphere datasets for linear growth (b), exponential growth (c), and periodic growth (d) between predicted (orange) and reference (blue) lines.
• Figure 3: Point trajectories tracking target deformations for periodic sphere (a, b) and real 4D flow MRI data (c, d). For both datasets, the results without $R_{cycle}$ regularisation (a, c) and with $R_{cycle}$ (b, d) are shown. $R_{cycle}$ is required to enforce periodic motion.
• Figure 4: First 9 frames
• Figure 5: Volume comparison of the periodic sphere (a,e,i) and CT datasets (b-d,f-h,l-n) with no time encoding (a-d), time encoding (e-h), and time encoding coupled with $R_{cycle}$ (i-n) applied.