Spectrally optimized implicit neural representations mitigate acquisition-related image quality trade-offs in intracranial 4D Flow MRI

1 Introduction

• Intracranial 4D Flow MRI must balance image quality and effective resolution against scan duration, with small vessels and complex flows raising demands on acquisition.VENC selection also trades velocity sensitivity against phase-wrapping risk.Source: §1 · pp. 2–3
• Many PINN formulations have focused on simplified or model-dependent settings, reduced-dimensional assumptions, or single geometries requiring retraining for each subject.Source: §1 · p. 3
• INR reconstruction remains little explored in 4D Flow MRI; prior aortic work used a single spectral setting, leaving spectral-bias optimization and complex cerebrovascular generalization untested [25].Source: §1 · p. 3
• The study introduces WIRE-based INRs to examine spectral-bias tuning, its balance of fine detail and noise, and generalization across patient cases.It compares a fast data-driven model with a more computationally demanding physics-informed extension.Source: §1 · pp. 3–4

2.1 Study design and evaluation overview

• Nine cases were evaluated using CFD-derived high-resolution references and MR-realistic synthetic low-resolution inputs, alongside qualitative direct fitting to acquired in-vivo data.The models included data-driven and physics-informed variants, with spectral-bias, noise, aliasing, and computational-cost analyses.Source: §2.1 · pp. 4–5

2.2 Study cohort and MRI acquisitions

• The cohort comprised three healthy volunteers and six ICAD patients, split equally between moderate (50%) and severe (70%) stenosis groups.All underwent 3T dual-VENC 4D Flow MRI; TOF angiography was acquired for arterial segmentation.Source: §2.2 · p. 5

2.3 CFD reference simulations

• Subject-specific CFD simulations generated high-resolution velocity references from reconstructed vascular geometries, using MRI-derived inlet velocities and outlet flow splits based on cross-sectional area.The simulations modeled incompressible Newtonian blood and mapped results to 0.5-mm isotropic image grids at 20-ms temporal resolution.Source: §2.3 · p. 6

2.4 Synthetic 4D Flow MRI data

• Synthetic low-resolution inputs were generated from CFD fields using acquisition-mimicking phase and magnitude images, k-space cropping, and added complex Gaussian noise.The primary benchmark used target SNR 10 and case-specific VENCs selected to avoid phase wrapping; temporal super-resolution used alternate-frame removal, a pragmatic approximation that omits scanner-level temporal averaging and k-space effects.Source: §2.4 · pp. 6–7

• The INR maps four spatiotemporal coordinates (x, y, z, t) to three velocity components through an MLP.Source: §2.5 · pp. 7–8
• WIRE uses a real-valued Gabor activation with ω0 controlling oscillation frequency and s0 controlling Gaussian-window width.The real-valued formulation is used because the physics-informed model repeatedly differentiates network outputs to evaluate Navier–Stokes residuals.Source: §2.5 · pp. 8–9
• The data-driven INR uses periodic anti-aliasing loss, while the physics-informed INR adds incompressible Navier–Stokes regularization to the same architecture and loss.Source: Figure 2 · p. 8
• The sinusoidal component promotes fine-scale oscillatory features, while Gaussian localization improves optimization stability and mitigates ringing artifacts associated with purely sinusoidal INRs.Source: §2.5 · p. 9
• Input voxel coordinates are standardized for numerical stability, and target velocity components are normalized by a fixed characteristic velocity U.Source: §2.5 · p. 9
• Template-based normalization uses fixed spatial coordinates representing a 20 cm field of view at 1 mm spacing and a 10 s temporal axis at 100 ms spacing.The same spatial factors apply to x, y, and z; this avoids case-specific coordinate scaling changes and supports shared spectral-bias hyperparameters.Source: §2.5 · p. 9

2.6 Data fidelity and anti-aliasing loss

• The periodic cosine anti-aliasing loss matches velocities modulo phase-encoding periodicity, allowing aliased voxels to be fitted without penalizing physically correct predictions.Velocity differences of 2k VENC yield zero penalty for integer k.Source: §2.6 · pp. 9–10
• Optimization is restricted to voxels inside a binary lumen mask, excluding background and requiring vascular segmentation to define the fluid domain.Source: §2.6 · p. 10

2.7 Physics-informed regularization

• The physics-informed variant regularizes velocity with incompressible Navier–Stokes momentum and continuity equations, extending the network output to include pressure.Source: §2.7 · p. 10
• The physics loss uses non-dimensional residuals evaluated at collocation points sampled within the fluid domain, with 1.5 × 10^6 jittered points providing off-grid coverage.The characteristic scales are fixed at U = 2 m/s and L = 5 mm; the fluid is modeled as incompressible and Newtonian.Source: §2.7 · pp. 10–11
• The data-loss weight is adjusted using the relative gradient magnitudes of data and physics losses.Source: §2.7 · p. 11

2.8 Training and optimization

• The data-driven INR trains for 8,000 Adam iterations on mini-batches of 20,000 fluid-mask points, with an exponentially decayed learning rate.All models start at learning rate 10^-4.Source: §2.8 · p. 11
• The physics-informed INR uses 10,000 Adam iterations followed by 5,000 L-BFGS iterations, with batches containing 6,000 data and 6,000 collocation points.The L-BFGS stage uses learning rate 5 × 10^-2 and a strong-Wolfe line search.Source: §2.8 · p. 12

2.9 Evaluation metrics

• After training, models are queried on the high-resolution spatiotemporal grid and compared voxel-wise within the fluid mask using MAE, DE, MRE, and vNRMSE.Source: §2.9 · p. 12

2.10 Comparator methods

• The INR models are compared with pretrained 4DFlowNet and GAN-Gen, whose training pipelines differ in downsampling or VENC setup from the present study.Both CNN baselines provide two-fold spatial super-resolution but not temporal super-resolution, so they are evaluated only at acquired temporal frames.Source: §2.10 · pp. 12–13

2.11 Spectral bias selection, robustness analyses, and in-vivo fitting

• Spectral-bias sensitivity was assessed by sweeping WIRE (ω0, s0) from 0–100 in stride-10 increments, training each configuration with full optimization and evaluating end-of-training voxel-wise error against the HR reference.Noise and VENC-dependent phase-wrapping analyses used three representative cases: one healthy control, one moderate ICAD case, and one severe ICAD case; in-vivo INR models were fitted directly to representative acquired 4D Flow MRI cases and evaluated qualitatively.Source: §2.11 · p. 13

2.12 Empirical neural tangent kernel visualization

• The empirical NTK at initialization diagnoses frequency-dependent inductive bias, with dominant eigenvectors reshaped as images to compare low-, intermediate-, and high-spectral-bias configurations.The visualization compares both SIREN and WIRE architectures.Source: §2.12 · pp. 13–14

2.13 Meta learning

• MAML learns shared parameters across subject tasks so a small number of gradient updates can adapt the network to a new case.Six subjects are used for meta-training and three held-out subjects for validation.Source: §2.13 · p. 14
• Each task’s support set drives inner-loop adaptation, while its query set updates the shared initialization in the outer loop.The model uses 10 inner-loop steps; physics-informed meta-learning applies the physics loss after task adaptation because including it in the inner loop is computationally infeasible.Source: §2.13 · pp. 14–15
• The meta-learned initialization is fine-tuned on held-out subjects and compared with random initialization using cohort-optimal spectral-bias settings.Early training and reference losses are reported as lower for meta-learned initialization in both INR variants.Source: §2.13 · p. 15

3.1 Shared spectral bias settings generalize across healthy and diseased cerebrovascular flow

• Cohort-optimal settings increased relative error by only 0.41±0.60 percentage points for data-driven INR and 0.26±0.29 for physics-informed INR versus subject-specific optima.Low bias yielded overly smooth reconstructions, while high bias increased high-frequency fluctuations; intermediate settings balanced detail recovery and artifacts.Source: Figure 3 · p. 16, §3.1 · p. 17

3.2 Implicit neural representations recover cerebrovascular flow structure across anatomies and disease severity

• INR reconstructions reproduced main high-resolution flow patterns and showed reduced voxel-level noise, with clearer localized high-velocity flow near stenoses than the low-resolution input.Representative cases included healthy, moderate ICAD, and severe ICAD subjects.Source: §3.2 · pp. 18–19, Figure 4 · p. 18
• Physics-informed INR reduced average DE from 5.3% to 3.9% and MRE from 33.5% to 32.6% versus data-driven INR, but required 125–182 minutes rather than 1.5–2.3 minutes per subject.Cohort-averaged MAE differed by 0.2 cm/s (10.4 vs. 10.2 cm/s).Source: §3.2 · p. 19
• Cohort-averaged data-driven INR MAE was 0.100 vs 0.143 for tricubic interpolation, 0.145 for 4DFlowNet, and 0.113 for GAN-Gen; vNRMSE was 0.081 vs 0.111, 0.112, and 0.096, respectively.The data-driven INR gains over 4DFlowNet were larger in moderate and severe stenotic groups than in healthy cases.Source: §3.2 · p. 20, Table 4 · p. 20

3.3 Implicit neural representations improve robustness to noise and VENC-dependent aliasing

• At SNR=2, physics-informed and data-driven INR average errors were 36.0% and 38.4%, versus 45.6% for GAN-Gen, 52.9% for interpolation, and 55.1% for 4DFlowNet.At SNR=10, the corresponding errors were 32.2% and 32.9% for the INR variants.Source: §3.3 · pp. 20–21, Figure 5 · p. 21
• At VENC = 0.3 m/s, INR errors were 34.5% and 32.6% for moderate ICAD, and 36.5% and 36.1% for severe ICAD; aliasing-related errors remained in reconstructed slices.The values are for data-driven and physics-informed INR, respectively; at VENC = 0.5 m/s, data-driven INR corrected phase-wrapping artifacts in the LR input.Source: Figure 5 · p. 21, §3.3 · p. 22

3.4 Meta-learned initialization accelerates patient-specific adaptation

• After 500 iterations on three held-out subjects, meta-initialization reduced MAE from 0.123 to 0.111 for data-driven INR and from 0.167 to 0.120 for physics-informed INR.It also reduced vNRMSE from 0.102 to 0.095 and from 0.134 to 0.101, respectively; physics-informed adaptation remained slower.Source: Table 5 · p. 22, Figure 6 · p. 23, §3.4 · p. 24

3.5 Implicit neural representations enable direct in-vivo cerebrovascular reconstruction

• Direct in-vivo fitting produced smoother, more spatially coherent fields, but velocity recovery within stenotic segments remained limited where native low-resolution data had poor coverage.Without noise-free high-resolution in-vivo references, this evaluation remained qualitative.Source: §3.5 · pp. 24–26, Figure 7 · p. 25

4 Discussion

• Both INR variants improved velocity reconstruction across healthy and stenotic configurations; the physics-informed model achieved the strongest quantitative performance, with modest gains over the data-driven model at substantially higher computational cost.The data-driven INR achieved near-comparable accuracy with a fraction of the training time.Source: §4 · pp. 26, 28, 29
• Intermediate spectral-bias settings balanced detail recovery and denoising, while substantially overlapping optimal settings across subjects support cohort-level tuning without per-subject hyperparameter adjustment.Low settings produced overly smooth fields; high settings increased sensitivity to high-frequency fluctuations and noise-like patterns.Source: §4 · p. 26
• Implicit models maintained lower relative error than alternative reference approaches, particularly under high noise; periodic anti-aliasing improved tolerance to moderate phase wrapping, but very low VENC still left aliasing errors.At high VENC, reduced phase sensitivity creates a separate trade-off; reliable velocity information cannot be fully recovered when acquired phase data are too strongly corrupted.Source: §4 · pp. 26–27
• Unlike supervised CNN approaches requiring large, diverse training datasets, the INR is fitted to each subject’s measured field and represents it as a continuous coordinate-based function.The discussion also positions this work beyond prior PINN applications that depended on CFD meshes, simplified flow assumptions, or idealized in-silico settings.Source: §4 · p. 27
• Compared with the closest prior 4D Flow INR work in localized aortic imaging, this study examines complex neurovasculature, spectral-bias dependence, data-driven versus physics-informed formulations, and acquisition-degradation robustness.The prior study used a SIREN-based representation.Source: §4 · pp. 27–28
• The nine-subject cohort and synthetic benchmark constrain population-level and clinical inference; the authors also note that synthetic data cannot capture all in-vivo variability and that clinical added value remains future work.The quantitative evaluation combines denoising and resolution enhancement, and the physics-informed variant remains computationally expensive.Source: §4 · pp. 27–28

7 Author contributions

• Author roles span conceptualization, implementation, methodology, analysis, evaluation, computational modeling, data acquisition, writing, and supervision.O.W.O. and D.M. are credited with conceptualization, methodology, analysis, evaluation, and original-draft writing; several authors contributed to data acquisition or supervision.Source: §7 · p. 29

• Supplementary Fig. S1 compares low-resolution, data-driven INR reconstructions queried on the high-resolution grid, and high-resolution references at peak systole across healthy, moderate ICAD, and severe ICAD cases.Magnified insets show stenosis regions in the moderate and severe cases.Source: S1 Additional reconstruction examples · p. 34, Figure S1 · p. 35
• Supplementary Fig. S2 compares acquired native 4D Flow MRI with direct data-driven INR and physics-informed INR reconstructions at peak systole across healthy, moderate ICAD, and severe ICAD cases.Magnified insets show stenosis regions in the moderate and severe cases.Source: S1 Additional reconstruction examples · p. 34, Figure S2 · p. 36