NerT-CA: Efficient 4D Coronary Reconstruction

• The paper introduces a hybrid framework combining low-rank tensorial fields for static structures with sparse neural fields for dynamic vessel modeling, reducing training time from 361 to 37 minutes and boosting DSC from 0.41 to 0.75.
• NerT-CA addresses the ill-posed 4D reconstruction problem by explicitly decomposing the continuous attenuation field into static background and temporally varying, contrast-enhanced vessels.
• The approach leverages physically-informed rendering and targeted regularization techniques to achieve enhanced computational efficiency, improved image quality, and clearer vessel delineation.

NerT-CA (Neural and Tensorial representations for Coronary Angiography) is a hybrid learning-based framework for rapid, automated 4D (three-dimensional plus time) reconstruction of coronary arteries from extremely sparse X-ray projection data. By combining low-rank tensorial fields for static background and sparse neural fields for dynamic, contrast-filled vessel structures, NerT-CA achieves marked improvements in both reconstruction quality and computational efficiency in the clinically challenging regime of sparse-view coronary angiography (Maas et al., 25 Jul 2025).

1. Sparse-View CA Reconstruction: Problem Landscape

In X-ray coronary angiography (CA), only 2–4 C-arm projections (200×200 pixels each) of a beating heart are acquired due to radiation dose constraints, resulting in an ill-posed, depth-ambiguous 4D inverse problem. The anatomical sparsity—vessels occupy a minute fraction of the total volume—together with view scarcity and strong intra-scan cardiac motion, presents formidable challenges. Traditional tomographic reconstructions and segmentation-based pipelines either require extensive manual interaction or fail when the number of available projections is limited. Prior neural field work (NeRF-CA) utilized dual MLPs to model static and dynamic components but incurred prohibitive (~6 hours/case) training times and limited scalability for interactive or on-table clinical use. NerT-CA is designed to address these obstacles through architectural and representational innovations.

2. Hybrid Low-Rank and Sparse Modeling

Central to NerT-CA is an explicit decomposition of the continuous attenuation field

$$H(x, y, z, t) = T_{\text{static}}(x, y, z) + F_{\text{dynamic}}(x, y, z, t)$$

where $T_{\text{static}}$ captures background anatomy via a low-rank tensorial representation, and $F_{\text{dynamic}}$ models the sparse, temporally varying contrast agent distribution in coronary vessels.

• Tensorial Field for Low-Rank Component: The static term $T_{\text{static}}$ is discretized onto a $D \times D \times D$ grid (with $D=48$) and parameterized using a vector–matrix (VM) decomposition:

$$G = \sum_{r=1}^{R} \left( v_r^X \circ M_r^{Y,Z} + v_r^Y \circ M_r^{X,Z} + v_r^Z \circ M_r^{X,Y} \right)$$

for rank $R=3$, yielding memory- and computation-efficient modeling of large-scale, slowly varying anatomical structures.

• Neural Field for Sparse, Dynamic Vessels: $F_{\text{dynamic}}$ is represented by a compact MLP mapping a windowed positional encoding $\gamma_a(x)$ and a learnable cardiac phase code $T_{\text{static}}$0 to a positive density $T_{\text{static}}$1. The architecture employs 4 hidden layers of 128 ReLU units with a Softplus output, and is responsible for capturing sparse, fine-scale dynamic vessel content.

This hybrid approach ensures both expressive modeling of sharply localized, moving vascular features and highly efficient capture of static background.

3. Physically-Informed Rendering and Optimization

Forward projections are computed using the Beer–Lambert law for discrete X-ray ray integration:

$T_{\text{static}}$2

where $T_{\text{static}}$3 sample the path of each X-ray, $T_{\text{static}}$4 (static attenuation) derives from the tensorial field, and $T_{\text{static}}$5 (dynamic attenuation) from the neural field with phase modulation.

The training loss comprises three key terms:

• $T_{\text{static}}$6: Photometric reconstruction error (MSE) between rendered and ground truth projections
• $T_{\text{static}}$7: Total variation regularization on all VM factors to enforce global smoothness in the static background
• $T_{\text{static}}$8: Occlusion minimization, suppressing unphysical "floating" densities in front of the source

Weighted coefficients $T_{\text{static}}$9 and $F_{\text{dynamic}}$0 are empirically tuned.

4. Training Protocol, Data Regimes, and Convergence

Training samples $F_{\text{dynamic}}$1 rays per batch, with $F_{\text{dynamic}}$2 samples per ray, across $F_{\text{dynamic}}$3 cardiac phases. VM decomposition (tensorial field) optimization proceeds first, with neural MLP updates delayed until iteration 1,500. Positional encoding frequencies are annealed over $F_{\text{dynamic}}$4 iterations to facilitate progressive capture of high-frequency geometry.

On a single RTX A5000, typical total training time is $F_{\text{dynamic}}$537 minutes—an order-of-magnitude reduction compared to NeRF-CA. Training is robust across 3, 4, or 9 projection regime; in the 3-view scenario, the projection with maximal vessel overlap is intentionally discarded to alleviate depth ambiguity. Convergence is monitored through DSC plateau, photometric loss stabilization, and visual assessment of vessel integrity.

5. Quantitative Performance: Accuracy and Speed

Evaluation spans two digital phantom datasets: XCAT (0.5 mm³ voxels) and MAGIX (0.4×0.4×2 mm³). Reconstruction quality is quantified by:

• Vessel Dice Score (DSC): Segmentation overlap following MIP-thresholding
• PSNR and SSIM: Fidelity of rendered projections relative to ground truth
• Training Time and Inference FPS

Table: Comparative Results (XCAT, 3/4/9 Views, 40 Views Averaged Across 10 Phases) | Method | DSC | PSNR | SSIM | Training Time, Inference | |---------------|----------|--------|--------|---------------------------| | SAX-NeRF | 0.00 | ~13.5 | 0.58 | -- | | X-Gaussian | 0.05 | 12.6 | 0.49 | minutes | | R2-Gaussian | 0.01 | 13.0 | 0.57 | minutes | | NeRF-CA | 0.41 | 11.7 | 0.65 | 361 min, 0.5 FPS | | NerT-CA | 0.75 | 15.3 | 0.79 | 37 min, 2 FPS |

Compared to NeRF-CA, NerT-CA achieves a 10-fold speedup in training and 4× inference throughput, with 0.75 (NerT-CA) vs. 0.41 (NeRF-CA) DSC in the 3-view regime. Similar trends are observed across 4- and 9-view settings. On the MAGIX phantom, sharper vessel delineation and background suppression are evident.

6. Regularization, Ablation, and Qualitative Analysis

Ablation experiments reveal that all three regularizers—total variation ($F_{\text{dynamic}}$6), occlusion ($F_{\text{dynamic}}$7), and windowed positional encoding—are necessary for robust dynamic vessel recovery, particularly in the severely ill-posed 3-view case. For instance, on XCAT:

• Without $F_{\text{dynamic}}$8 and $F_{\text{dynamic}}$9, DSC = 0.09
• Adding $T_{\text{static}}$0 or $T_{\text{static}}$1, DSC increases to 0.58
• With windowed encoding, DSC achieves 0.75

Qualitatively, NerT-CA recovers higher-order vessel branches missed by NeRF-CA and achieves cleaner boundaries with less background artifact.

7. Limitations and Clinical Trajectory

All evaluations to date use digital phantoms with ideal, noise-free projections. The framework currently omits real-world degradation sources—including C-arm noise, X-ray scatter, and calibration errors. The spatial grid resolution (48³) does not resolve the smallest vessels; upgrades (larger D or multiscale schemes) are possible. While 2 FPS is approaching clinical real-time, integration of faster rendering strategies (e.g., 3D Gaussian splatting) could be beneficial.

Clinically, NerT-CA's ability to deliver nearly segmentation-free, dynamic 4D vascular reconstructions from minimal projection data in under 40 minutes positions it for significant impact in interventional workflow. The explicit low-rank/sparse decomposition isolates static from dynamic content, facilitating both anatomical context and real-time perfusion assessment (Maas et al., 25 Jul 2025). Automated, interactive angiographic mapping from sparse, low-dose data could assist in catheter navigation, guidewire planning, and possibly noninvasive stenosis grading.

• "NerT-CA: Efficient Dynamic Reconstruction from Sparse-view X-ray Coronary Angiography" (Maas et al., 25 Jul 2025).