Reconstruction of 4D Mitral Regurgitation Hemodynamics from Sparse Planar Data using Deep Operator Networks with Test-Time Adaptation

Jakob Marcel Hoffmann; Yosuke Hasegawa; Alexander Stroh

physics.med-phAug 31, 2026

Reconstruction of 4D Mitral Regurgitation Hemodynamics from Sparse Planar Data using Deep Operator Networks with Test-Time Adaptation

Authors: Jakob Marcel Hoffmann, Yosuke Hasegawa, Alexander Stroh

Organizations: Institute of Fluid Mechanics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany · Center for Research on Innovative Simulation Software, Institute of Industrial Science, The University of Tokyo, Japan

Abstract

Quantifying mitral regurgitation severity remains limited by the assumptions of clinical flow convergence methods, while high-fidelity simulation and volumetric velocimetry are too slow for routine use. We investigate whether a learned solution operator can reconstruct transient three-dimensional transvalvular hemodynamics from the sparse observation an in-vitro experiment actually provides: a single planar velocity slice and two boundary pressure traces. A Deep Operator Network is pretrained on an experimentally benchmarked URANS database spanning eleven mitral regurgitation orifice phantoms, learning a mapping from a masked two-component planar velocity snapshot to the surrounding volumetric field, and is subsequently adapted to unseen target cases by fine-tuning on their sparse measurements. Adaptation reliably corrects the flow topology within the supervised plane, reorienting a strongly eccentric jet that the pretrained operator predicts as straight, and yields full-field predictions in minutes rather than the days required by the underlying simulations. Its influence decays sharply with distance from that plane, however: measured against phase-resolved particle image velocimetry, the reconstruction error rises from 24.6% at 2mm to 52.6% at 6mm, and the resulting mismatch between corrected and uncorrected layers degrades physical consistency. Single-plane supervision thus constrains the observed plane far more effectively than the surrounding volume, which we identify as the principal obstacle to coherent 4D reconstruction from sparse planar data.

Figures & tables

Figure 1: Hemodynamic simulator geometry. (a) CAD cutaway of the chambers, (b) meshed internal fluid domain, (c) mesh slices, (d) magnified Pinhole-L orifice mesh, (e) MROP shapes with height and width markers. LV = Left Ventricle, LA = Left Atrium.

Figure 2: Side-view of rigid orifice plate geometries used in CFD. (a) Height

h

and plate wall thickness

d

for Slot-L. The origin of Cartesian coordinates is defined as this center of the flat opening for all MROPs. (b) Offset

\Delta x

caused by spherical plate wall of Slot-L-Bent.

Shape	height $h$ in mm	width $w$ in mm	area in mm 2
Drop-S	9.7	4.3	27.1
Drop-M	13.7	6.5	52.1
Drop-L	19.8	9.0	108.4
Drop-XL	27.6	13.0	215.5
Slot-S	11.1	3.3	27.0
Slot-M	14.0	4.5	44.8

Table 1: Mitral Regurgitation Orifice Phantom (MROP) sizes. S, M, L, and XL designate Small, Medium, Large, and Extra-Large. Modified to include Drop-XL and EccJet. [ 23 ] For the eccentric case, these values define the ellipse formed on the flat orifice plate.

Figure 3: Cardiac piston pump displacement

\Delta V

(blue) and volume flow rate

\dot{V}

(green) for one Pinhole-L cycle.

Figure 4: Regurgitation volume (RVol) of MROPs measured via ultrasound (US) by three physicians and PIV with uncertainty confidence intervals. CFD results (in green) are added to the data from Leister et al. [ 23 ]

Figure 5: Schematic of the DeepONet architecture for 3D transient flow reconstruction. The branch net receives the constrained 2D2C velocity snapshot in a flattened format. The same snapshots are additionally trained on for Test-Time Adaptation (TTA). The current snapshot’s time-step is the trunk net input $t$ , alongside the spatial inference coordinates. Inputs and outputs are non-dimensional (notation $\square^{*}$ omitted for legibility).

Figure 6: Velocity magnitude distribution at $z^{*}=0$ and $t^{*}=0.25$ for the Drop-XL orifice. (a) Reference CFD solution, (b) DeepONet prediction. (c) Signed, non-linearly displayed and (d) absolute error magnitude with streamline comparison.

Figure 7: Velocity magnitude distribution at $y^{*}=0$ and $t^{*}=0.5$ for the Slot-L-Bent orifice. (a) Reference CFD solution, (b) DeepONet prediction. (c) Signed, non-linearly displayed and (d) absolute error magnitude with streamline comparison.

Figure 8: Temporal evolution of NRMSE for the three velocity components over the cardiac cycle. The vertical line marks the moment of flow reversal, indicating the onset of the diastolic jet forming in the LV. Includes pre-trained baseline and fine-tuned hybrid results for validation cases and select training cases. Horizontal dashed lines indicate the respective aggregate values computed over the entire period.

Figure 9: Mitral volume flow and integrated regurgitation volume (RVol, horizontal lines) for select MROPs. Comparison of the CFD reference (solid) with baseline (dashed) and fine-tuned hybrid (dash-dotted) DeepONet predictions: (a) Pinhole-M and Slot-M training cases (baseline only), (b) Drop-XL and (c) Slot-L-Bent validation cases.

Figure 10: DeepONet predictions for mitral volume flow and integrated regurgitation volume (RVol, horizontal lines) driven by experimental PIV input. PIV-derived reference RVol (dotted) for Pinhole-L and Slot-L available from Fig. 4 and corresponding paper. [ 23 ] Dots mark the discrete phase positions captured by the phase-averaged PIV.

Panel of Figure 11 — Figure 11: DeepONet velocity magnitude predictions for the Pinhole-L orifice based on experimental PIV input. (a) $x$ - $y$ -plane at $z^{*}=0$ and $t^{*}=0.26$ , (b) $x$ - $z$ -plane at $y^{*}=0$ and $t^{*}=0.087$ , (c) $x$ - $y$ -plane at $z^{*}=0$ and $t^{*}=0.527$ .

Figure 12: URANS residuals aggregated over the cardiac cycle. Blue bars denote results driven by synthetic CFD input, including the aggregate of all pre-training cases and the two fine-tuned validation cases. Orange bars represent the fine-tuned experimental cases driven by real-world PIV data.

Figure 13: (a) Reference PIV velocity magnitude at $z^{*}=-0.02$ and $t^{*}=0.067$ for the EccJet orifice. (b) DeepONet prediction. (c) Absolute error magnitude with streamline comparison.

NRMSE in %	$z^{*}=-0.02$	$z^{*}=-0.04$	$z^{*}=-0.06$
$t^{*}=0.040$	28.2	62.0	96.3
$t^{*}=0.053$	26.5	52.9	72.1
$t^{*}=0.067$	24.6	43.1	52.6
$t^{*}=0.080$	23.9	41.6	49.8
$t^{*}=0.093$	21.0	35.1	43.9

Table 2: NRMSE values for the velocity magnitude prediction compared against PIV measurements for the EccJet case.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

	Volumetric Size		Surface Size
Region	in %	in mm	in %	in mm
Main (LV & LA)	100	3.00	50	1.50
Reservoir	200	6.00	50	3.00
Pump	200	6.00	100	6.00
Aorta	50	1.50	50	0.75
Orifice Plate	-	-	25	0.75

Appendix

Table 3: Mesh parameters. The surface size settings simultaneously describe the prism layer thickness and the target and minimum surface sizes of the polyhedral mesher. The lower half of listed regions denote subsections found in the main (LV & LA) domain part.

Mesh	in mm	in %	in mm	in %	in mm	Count
	Base Size	Aorta Sphere		Orifice Cylinder		Cell
1	5.0	50	2.500	25	1.250	145,678
2 1 1 1 Selected baseline configuration. Other settings match with Table 3 .	3.0	50	1.500	25	0.750	427,557
3	2.0	75	1.500	25	0.500	1,042,883
4	1.5	75	1.125	25	0.375	1,993,642

Appendix

Table 4: Configurations evaluated during the mesh independence study for the Pinhole-L geometry. Relative refinement percentages are shown alongside their resulting absolute cell sizes.

Figure 14: Mass flow through Pinhole-L MROP (dashed) and maximum regurgitation jet velocity magnitude (solid) for the first systole of the 4 studied meshes. Experimental velocity values (black dotted) included from PIV.

Figure 15: Pinhole-L velocity magnitude snapshot for (a) PIV and (b) CFD at $t=0.085\,\mathrm{s}$ and $z=0\,\mathrm{mm}$ . Pump BC used in CFD and resulting maximum velocity magnitudes through the cardiac cycle shown in (c) with the vertical line marking the phase of (a, b).

Figure 16: (a) Total aorta flow rate and (b) ventricle and (c) atrium pressure in the experiment (dashed) and CFD (solid). The three sizes of Slot MROPs are offset by $0.25\,\mathrm{s}$ for legibility. Two unit scalings shown each.

Figure 17: Comparison of (a) PIV and CFD velocity magnitude for Slot-L with (b) bent and (c) flat rigid orifice plate at $t=0.1\,\mathrm{s}$ and $z=0\,\mathrm{mm}.$

Figure 18: Drop-XL velocity magnitude snapshot for (a) PIV and (b) CFD at $t=0.045\,\mathrm{s}$ and $z=0\,\mathrm{mm}$ . Pump BC used in CFD and resulting maximum velocity magnitudes through the cardiac cycle shown in (c) with the vertical line marking the phase of (a, b).

Panel of Figure 19 — Figure 19: DeepONet velocity magnitude predictions for the Drop-XL orifice based on experimental PIV input. (a) $x$ - $y$ -plane at $z^{*}=0$ and $t^{*}=0.193$ , (b) $x$ - $z$ -plane at $y^{*}=0$ and $t^{*}=0.26$ , (c) $x$ - $y$ -plane at $z^{*}=0$ and $t^{*}=0.393$ .

Figure 20: Pump BC flow rate (solid), mitral valve flow rate (dash-dotted), and maximum velocity magnitude (dashed) across Slot-S (red), Slot-M (green), and Slot-L (blue) MROPs in the URANS CFD simulation.

Panel of Figure 21 — Figure 21: DeepONet velocity magnitude predictions for the Slot-L orifice based on experimental PIV input. (a) $x$ - $y$ -plane at $z^{*}=0$ and $t^{*}=0.147$ , (b) $x$ - $z$ -plane at $y^{*}=0$ and $t^{*}=0.087$ .

Figure 22: Velocity magnitude distribution at $y^{*}=0$ and $t^{*}=0.08$ for the Drop-XL orifice. (a) Reference CFD solution, (b) baseline DeepONet prediction before TTA. (c) Signed, non-linearly displayed and (d) absolute error magnitude with streamline comparison.

Figure 23: Velocity magnitude distribution at $y^{*}=0$ and $t^{*}=0.08$ for the Drop-XL orifice. (a) Reference CFD solution, (b) hybrid DeepONet prediction after TTA. (c) Signed, non-linearly displayed and (d) absolute error magnitude with streamline comparison.

4D Reconstruction

4D reconstruction recovers changing 3D scenes, shapes, and motion over time from images, videos, or other sparse measurements.

See all 139 papers

A Physics-Conditioned Neural Operator for Generalization of Atrioventricular Valve Mechanics across Pressure and Tissue Properties
Sep 20, 2026Shawn Koohy, Wensi Wu, Matthew A Jolley +1Surrogate Modeling Physics-Informed Neural Operators
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA. · Cardiovascular Institute, Children’s Hospital of Philadelphia, Philadelphia, PA, USA. · Department of Anesthesiology and Critical Care Medicine, Children’s Hospital of Philadelphia, Philadelphia, PA, USA. +1
Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach
May 22, 2026Qian Zhang, George Em KarniadakisPDE Operator Learning Operator Learning
Division of Applied Mathematics, Brown University
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

Sep 20, 2026J/K move · Enter open · S save

A Physics-Conditioned Neural Operator for Generalization of Atrioventricular Valve Mechanics across Pressure and Tissue Properties

Shawn Koohy, Wensi Wu, Matthew A Jolley, Paris Perdikaris

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA, USA. · Cardiovascular Institute, Children’s Hospital of Philadelphia, Philadelphia, PA, USA. · Department of Anesthesiology and Critical Care Medicine, Children’s Hospital of Philadelphia, Philadelphia, PA, USA.+1

Surrogate Modeling Physics-Informed Neural Operators Neural Operators

Open paper AskPDF arXiv

Mitral and tricuspid regurgitation are the most common regurgitant valvular lesions, yet only a minority of severe cases undergo corrective surgery. Rapid assessment of valve mechanics could enable earlier, more precise intervention, but finite element (FE) analysis is slow to repeat across the many loading and tissue-property values of interest, which for a given valve are not known in advance. We introduce the Physics-Conditioned Neural Operator (PCNO), a transformer-based neural operator predicting leaflet displacement, strain, and stress fields conditioned on systolic blood pressure and tissue properties. PCNO is trained on, and evaluated against, FEBio simulations of functional and regurgitant mitral and tricuspid valves and of three mitral pathologies. With pressure and all material parameters simultaneously outside the training support, displacement error against FE reaches 4.48% and the mean errors of unsupervised geometric measures of valve function stay within 3.5%, indicating a conditioned solution operator over parameter space rather than an interpolator of the training set. Under this shift, PCNO is also more accurate than graph neural network and graph neural operator baselines trained on the same simulations, with the largest margins in stress.

Abstract

4D Reconstruction

Explore similar work

A Physics-Conditioned Neural Operator for Generalization of Atrioventricular Valve Mechanics across Pressure and Tissue Properties

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

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