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

Oliver Welin Odeback; Nerea González-Aranceta; Sebastián Jofre; Javier Bisbal; Edward Ferdian; Alistair A. Young; Pia Callmer; Sameer A. Ansari

eess.IVOct 23, 2026

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

Authors: Oliver Welin Odeback, Nerea González-Aranceta, Sebastián Jofre, Javier Bisbal, Edward Ferdian, Alistair A. Young, Pia Callmer, Sameer A. Ansari,

Organizations: Dept. Molecular Medicine and Surgery, Karolinska Institutet, Karolinska Universitetssjukhuset Solna (L1:00), Stockholm, 171 76, Sweden. · Department of Physics, University of Greifswald, Felix-Hausdorff-Str. 6, Greifswald, 174 89, Germany. · Biomedical Imaging Center, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, 7820436, Chile. · School of Electrical Engineering and Informatics, Bandung Institute of Technology, Jl. Ganesha No. 10, Bandung, 40132, West Java, Indonesia. · School of Biomedical Engineering & Imaging Sciences, King’s College London, 1 Lambeth Palace Rd, South Bank, London, SE1 7EU, UK. · Department of Radiology, Neurology, and Neurological Surgery, Feinberg School of Medicine, Northwestern University, Chicago, IL, 60611, USA. · Department of Biomedical Engineering, University of Michigan, 2200 Bonisteel Blvd, 1107 Carl A. Gerstacker Building, Ann Arbor, MI, 48109-2099, USA. · Department of Neuroradiology, Karolinska University Hospital, Hälsovägen 13, O42, Stockholm, 141 86, Sweden. · Department of Clinical Physiology, Karolinska University Hospital, Eugeniavägen 3, A8:01, Solna, 171 64, Sweden. · Institute for Medical Engineering and Science, Massachusetts Institute of Technology, 45 Carleton St, Cambridge, 02142, MA, USA.

Abstract

4D Flow MRI promises opportunities for non-invasive mapping of neurovascular hemodynamics, but its clinical utility faces trade-offs between resolution, acquisition time, and noise. Deep learning-based super-resolution offers a post-scan solution, but supervised approaches require paired or domain-specific data, while patient-specific physics-informed models can be computationally demanding. This creates a need for unsupervised reconstruction models that can adapt to individual acquisitions without paired high-resolution training data, while remaining robust across anatomy, disease severity, and acquisition quality. To address this, we introduce a spectrally tuned implicit neural representation (INR) framework for intracranial 4D Flow MRI, representing the velocity field as a continuous coordinate-based neural field using WIRE activation functions with tunable spectral bias. Data-driven and physics-informed variants were evaluated in nine cerebrovascular cases spanning healthy controls and moderate-to-severe intracranial atherosclerotic disease, using CFD-derived high-resolution reference fields, MR-realistic synthetic low-resolution inputs, and direct fitting to acquired in-vivo data. Cohort-optimal spectral-bias settings generalized across subjects with only small increases in relative error compared with subject-specific optima, reducing the need for case-specific hyperparameter tuning. Physics-informed regularization provided modest additional accuracy but required substantially longer optimization, whereas the data-driven INR achieved near-comparable reconstruction performance within minutes. The framework also remained robust to acquisition-realistic noise and moderate VENC-dependent aliasing. These findings establish spectrally optimized INRs as a flexible unsupervised approach for mitigating acquisition-related trade-offs in intracranial 4D Flow MRI.

Figures & tables

Figure 1: Study overview. To establish a spectrally optimized implicit neural representation (INR) framework, healthy participants and patients with intracranial atherosclerotic disease (ICAD) underwent cerebrovascular 4D Flow MRI and time-of-flight (TOF) angiography. The INR framework was evaluated by direct in-vivo fitting of acquired low-resolution data and quantitative in-silico benchmarking using subject-specific CFD simulations. Data-driven and physics-informed INR reconstructions on a super-resolution grid were then evaluated across subjects, with network spectral bias controlled through the tunable activation function $\sigma(x)$ and characterized using neural tangent kernel (NTK) analysis.

Group	Case	Sex	Age	Voxel size [mm]	$N_{\mathrm{cardiac}}$	$\Delta t$ [ms]	Scanner	Stenosis
Healthy	H1	F	25	(0.9821, 0.9821, 1.0)	7	82.6	Prisma	–
	H2	M	31	(0.9821, 0.9821, 1.0)	6	82.6	Prisma	–
	H3	M	29	(0.9821, 0.9821, 1.0)	7	82.6	Prisma	–
Moderate	M2	M	35	(0.9821, 0.9821, 1.0)	9	86.8	Skyra	RMCA
	M1	M	61	(0.9821, 0.9821, 1.0)	7	86.8	Skyra	RMCA
	M3	M	65	(1.0417, 1.0417, 1.0)	13	42.7	Skyra	BA

Table 1: Subject-level demographics and acquisition characteristics for the included sub-cohort. Voxel size is reported as

(\Delta x,\Delta y,\Delta z)

in mm.

N_{\mathrm{cardiac}}

denotes the number of acquired cardiac phases. For ICAD cases, the clinically reported primary lesion location is listed.

Case	$U_{\max}$ [m/s]	VENC [m/s]	$N_{\mathrm{t}}^{\mathrm{HR}}$	$N_{\mathrm{t}}^{\mathrm{LR}}$	$N_{\mathrm{d}}^{\mathrm{HR}}$	$N_{\mathrm{d}}^{\mathrm{LR}}$
H1	1.62	1.7	25	13	19,109	3,107
H2	1.21	1.3	21	11	19,371	3,185
H3	0.99	1.2	25	13	20,951	3,422
M1	1.29	1.3	35	18	22,672	3,687
M2	1.28	1.3	27	14	14,037	2,371
M3	5.06	5.1	26	13	26,040	4,192

Table 2: Case-specific parameters for synthetic 4D Flow MRI generation. All synthetic inputs were generated with target SNR = 10 and two-fold spatial and temporal downsampling.

Figure 2: Coordinate-based INR reconstruction framework. The 4D Flow MRI velocity field is represented as a continuous function of spatiotemporal coordinates using a WIRE-based multilayer perceptron with tunable Gabor activation parameters $(\omega_{0},s_{0})$ . The data-driven INR is optimized using only a periodic anti-aliasing loss, while the physics-informed INR uses the same coordinate-based architecture and loss with additional incompressible Navier-Stokes regularization.

Figure 3: Spectral bias selection and robustness for data-driven and physics-informed INR. a) Representative low-resolution (LR) input and high-resolution (HR) reference slices, reconstructed velocity slices for low ( $(\omega_{0},s_{0})=(5,5)$ ), optimal ( $(20,20)$ for data-driven INR and $(60,30)$ for the physics-informed INR (PINN)), and high ( $(90,90)$ ) spectral bias settings, and neural tangent kernel (NTK) visualizations. The NTK visualizations are generated at initialization for networks using both pure sinusoidal activation functions (SIREN) and Gabor activation functions (WIRE). b) End-of-training relative error across the $(\omega_{0},s_{0})$ grid for all subjects, shown separately for the data-driven INR and PINN. Circles indicate subject-specific optima and crosses indicate cohort-optimal settings. c) Training dynamics for representative low-, optimal-, and high-bias settings. Solid lines denote loss evaluated on the LR training data and dashed lines denote loss against the HR reference. For the physics-informed model, the right panel shows the data loss, physics loss, and the adaptive data-weight $\lambda_{D}$ for the optimal setting. Vertical dotted lines indicate the transition from Adam to L-BFGS optimization.

Figure 4: Representative velocity reconstructions across disease severity. a) Three-dimensional velocity magnitude maps at peak systole for representative healthy, moderate ICAD, and severe ICAD cases (H1, M1, and S1), comparing low-resolution (LR) data, data-driven INR reconstruction queried on the HR grid, and high-resolution reference (HR) at peak systole. Insets show magnified regions around the stenosis in the moderate and severe subjects. b) Representative cross-sectional, uni-directional velocity slices in axial and sagittal orientations for the same cases, with INR reconstructions queried on a grid with two-fold higher spatial sampling than the HR grid.

Severity	Subject	Model	Train time	MAE [m/s]	DE [%]	MRE [%]	vNRMSE
Healthy	H1	INR	1.5 min	0.111	$3.5$	$31.52$	$0.084$
	H1	PINN	126 min	0.111	3.0	31.12	0.083
	H2	INR	1.7 min	$0.105$	$2.8$	$31.01$	0.109
	H2	PINN	135 min	0.104	2.4	30.73	0.109
	H3	INR	1.5 min	0.085	$2.9$	$30.71$	0.111
	H3	PINN	125 min	0.085	2.6	30.66	0.111

Table 3: Performance of data-driven INR and physics-informed INR (PINN) models. Metrics are computed and averaged over the full cardiac cycle, with mean absolute error (MAE), directional error (DE), mean relative error (MRE) and velocity normalized root mean square error (vNRMSE) reported. Bold values indicate the best-performing model for each subject and metric.

Severity	Model	Train time	Inference	MAE [m/s]	DE [%]	MRE [%]	vNRMSE
Healthy	Interpolation	–	–	$0.117$	$5.1$	$34.64$	$0.128$
	4DFlowNet	30 h	11.8 s	$0.128$	$6.3$	$36.44$	$0.129$
	GAN-Gen	49 h	12.1 s	$0.112$	$4.0$	$32.71$	$0.127$
	INR	1.6 min	1.6 min	0.103	$3.2$	$31.19$	$0.097$
	PINN	129 min	129 min	$0.104$	2.8	30.95	0.094
Moderate	Interpolation	-	-	$0.146$	$10.2$	$41.86$	$0.090$

Table 4: Performance comparison between data-driven INR and physics-informed INR (PINN) models and CNN-based models (4DFlowNet, GAN-Gen), with tricubic interpolation as a baseline. Metrics are computed voxel-wise over the full cardiac cycle and then averaged across subjects within each severity group, with mean absolute error (MAE), directional error (DE), mean relative error (MRE) and velocity normalized root mean square error (vNRMSE) reported. Bold values indicate the best-performing method within each severity group and metric.

Figure 5: Robustness to noise and VENC-dependent aliasing. a) Relative error as a function of added measurement noise, expressed by signal-to-noise ratio (SNR), for representative healthy, moderate ICAD, and severe ICAD cases (H1, M1, and S1). Curves compare 4DFlowNet, GAN-Gen, tricubic interpolation, data-driven INR, and physics-informed INR (PINN). Representative cross-sectional velocity slices for H1 show LR and data-driven INR reconstructions (SR) at SNR = 2, 6, and 10, with the HR reference shown for comparison. b) Relative error as a function of velocity encoding (VENC) for the same representative cases. Dashed vertical lines indicate the maximum velocity in each case. Representative cross-sectional velocity slices for H1 show LR and data-driven INR reconstructions (SR) at VENC = 0.2, 0.3, and 0.5 m/s.

Figure 6: Meta-learned initialization accelerates early adaptation. a) Cross-sectional velocity reconstructions for a held-out validation (H3) subject after 10, 50, 100, 250, 500, and 1000 fine-tuning iterations. Results are shown for the data-driven INR and physics-informed INR (PINN), comparing meta-learned initialization (META) with standard initialization using the cohort-optimal spectral bias setting (OPTIMAL). Low-resolution input (LR) and high-resolution reference (HR) are shown for comparison. b) Training dynamics for the same validation subject over the first 1000 iterations, with zoomed views over the first 10 iterations. Solid lines denote loss evaluated on the LR training data and dashed lines denote loss against the HR reference. For the PINN, data loss, physics loss, and adaptive data-weight $\lambda_{D}$ are also shown for the corresponding iterations.

Subject	Model	Initialization	Train time	MAE [m/s]	DE [%]	MRE [%]	vNRMSE
H3	INR	OPTIMAL	0.5 min	0.093	2.6	32.39	0.122
	INR	META	0.5 min	0.089	3.0	31.70	0.116
	PINN	OPTIMAL	4.2 min	0.146	5.4	43.49	0.183
	PINN	META	4.2 min	0.095	3.1	32.69	0.128
M3	INR	OPTIMAL	0.5 min	0.162	8.0	38.06	0.070
	INR	META	0.5 min	0.144	9.1	35.33	0.066

Table 5: Performance comparison between meta-learned and standard initialization at early training. Metrics are reported for each held-out validation subject (H3, M3, S3) after 500 iterations for the data-driven INR and the physics-informed INR (PINN). Metrics are computed voxel-wise over the full cardiac cycle, with mean absolute error (MAE), directional error (DE), mean relative error (MRE) and velocity normalized root mean square error (vNRMSE) reported. META denotes fine-tuning from the meta-learned initialization, and OPTIMAL denotes standard initialization using the cohort-optimal spectral-bias setting. Bold values indicate the best-performing method within each severity group, INR model and error metric.

Figure 7: Qualitative in-vivo reconstructions across disease severity. a) Three-dimensional velocity magnitude maps at peak systole for representative healthy, moderate ICAD, and severe ICAD cases (H1, M1, and S1), comparing acquired native 4D Flow MRI data with direct in-vivo reconstructions from the data-driven INR and physics-informed INR (PINN) at peak systole. Insets show magnified regions around the stenosis in the moderate and severe subjects. b) Representative cross-sectional, uni-directional velocity slices in axial and sagittal orientations for the same cases, with INR reconstructions queried on a grid with two-fold higher spatial sampling than the native grid.

Figure S1: Additional in-silico velocity reconstructions across disease severity. Three-dimensional velocity magnitude maps at peak systole for the remaining healthy, moderate ICAD, and severe ICAD cases, comparing low-resolution data, data-driven INR reconstruction queried on the HR grid, and high-resolution reference (HR) at peak systole. Insets show magnified regions around the stenosis in the moderate and severe subjects.

Figure S2: Additional in-vivo reconstructions across disease severity. Three-dimensional velocity magnitude maps at peak systole for the remaining healthy, moderate ICAD, and severe ICAD cases, comparing acquired native 4D Flow MRI data with direct in-vivo reconstructions from the data-driven INR and physics-informed INR (PINN) at peak systole. Insets show magnified regions around the stenosis in the moderate and severe subjects.

Medical Image Super-Resolution

Medical image super-resolution reconstructs fine spatial detail from low-resolution scans to support clearer anatomy and more useful clinical measurements.

See all 57 papers

Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction
Sep 29, 2026Mengyuan Xue, Bochun Mei4D Reconstruction Accelerated MRI Reconstruction
Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA
FlowMoDL: Model-Based Deep Learning with Conjugate-Gradient Data Consistency for Highly Accelerated 4D Flow MRI Reconstruction
Aug 26, 2026Tristan Gottwald, Michelle Bruch, Mubashir-Ul Hassan +6Accelerated MRI Reconstruction 4D Reconstruction
Institute of Information Processing (tnt), Leibniz University Hannover, Germany · L3S, Leibniz University Hannover, Hannover, Germany · Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany +2
Prior-Guided Implicit Neural Representations for Single-Subject Diffusion MRI Super-Resolution
Sep 1, 2026Abdulkader Ghandoura, Marsil Zakour, William Consagra +1Implicit Neural Representations Medical Image Super-Resolution
Psychiatry Neuroimaging Laboratory Brigham and Women’s Hospital Harvard Medical School Boston, MA, USA · School of Computation, Information and Technology Technical University of Munich Munich, Germany · Department of Statistics University of South Carolina Columbia, SC, USA

Sep 29, 2026J/K move · Enter open · S save

Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction

Mengyuan Xue, Bochun Mei

Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA

4D Reconstruction Accelerated MRI Reconstruction Magnetic Resonance Imaging Cardiac MRI

Open paper AskPDF arXiv

4D flow MRI measures time-resolved, three-directional blood velocity but requires long acquisition times, and its diagnostic signal is carried by the phase difference between velocity encodings, not by image magnitude. Recent work has developed a per-encoding variational network to address image reconstruction in this field. In this work, we incorporate a joint supervised and self-supervised training regime and utilize both magnitude and velocity data during supervision. At the same time, we add multiple acquisition-robust and conditioning strategies based on the acceleration factors. On the CMRx4DFlow~2026 aortic dataset (1.5 and 3T), our model lowers RelErr by

38

50\%

and AngErr by

7.0

8.7^\circ

against a training-matched baseline across

R{=}10

50

, improving on every held-out subject at every acceleration. Our model also shows strong generalization ability to transfer on out-of-distribution data by employing the joint training scheme, with an increase of

7.2\%

in SSIM and decrease of

38\%

and

31\%

in AngErr and RelErr respectively.

Abstract

Medical Image Super-Resolution

Explore similar work

Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction

FlowMoDL: Model-Based Deep Learning with Conjugate-Gradient Data Consistency for Highly Accelerated 4D Flow MRI Reconstruction

Prior-Guided Implicit Neural Representations for Single-Subject Diffusion MRI Super-Resolution