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

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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 hh and plate wall thickness dd for Slot-L. The origin of Cartesian coordinates is defined as this center of the flat opening for all MROPs. (b) Offset Δx\Delta x caused by spherical plate wall of Slot-L-Bent.
Shapeheight hh in mmwidth ww in mmarea in mm 2
Drop-S9.74.327.1
Drop-M13.76.552.1
Drop-L19.89.0108.4
Drop-XL27.613.0215.5
Slot-S11.13.327.0
Slot-M14.04.544.8
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 ΔV\Delta V (blue) and volume flow rate V˙\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 tt , alongside the spatial inference coordinates. Inputs and outputs are non-dimensional (notation □∗\square^{*} omitted for legibility).
Figure 6: Velocity magnitude distribution at z∗=0z^{*}=0 and t∗=0.25t^{*}=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∗=0y^{*}=0 and t∗=0.5t^{*}=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.
Figure 11: DeepONet velocity magnitude predictions for the Pinhole-L orifice based on experimental PIV input. (a) xx - yy -plane at z∗=0z^{*}=0 and t∗=0.26t^{*}=0.26 , (b) xx - zz -plane at y∗=0y^{*}=0 and t∗=0.087t^{*}=0.087 , (c) xx - yy -plane at z∗=0z^{*}=0 and t∗=0.527t^{*}=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∗=−0.02z^{*}=-0.02 and t∗=0.067t^{*}=0.067 for the EccJet orifice. (b) DeepONet prediction. (c) Absolute error magnitude with streamline comparison.
NRMSE in %z∗=−0.02z^{*}=-0.02z∗=−0.04z^{*}=-0.04z∗=−0.06z^{*}=-0.06
t∗=0.040t^{*}=0.04028.262.096.3
t∗=0.053t^{*}=0.05326.552.972.1
t∗=0.067t^{*}=0.06724.643.152.6
t∗=0.080t^{*}=0.08023.941.649.8
t∗=0.093t^{*}=0.09321.035.143.9
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 SizeSurface Size
Regionin %in mmin %in mm
Main (LV & LA)1003.00501.50
Reservoir2006.00503.00
Pump2006.001006.00
Aorta501.50500.75
Orifice Plate--250.75
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.
 Base SizeAorta SphereOrifice CylinderCell
Meshin mmin %in mmin %in mmCount
15.0502.500251.250145,678
2 1 1 1 Selected baseline configuration. Other settings match with Table 3 .3.0501.500250.750427,557
32.0751.500250.5001,042,883
41.5751.125250.3751,993,642
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 st=0.085\,\mathrm{s} and z=0 mmz=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 s0.25\,\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 st=0.1\,\mathrm{s} and z=0 mm.z=0\,\mathrm{mm}.
Figure 18: Drop-XL velocity magnitude snapshot for (a) PIV and (b) CFD at t=0.045 st=0.045\,\mathrm{s} and z=0 mmz=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 19: DeepONet velocity magnitude predictions for the Drop-XL orifice based on experimental PIV input. (a) xx - yy -plane at z∗=0z^{*}=0 and t∗=0.193t^{*}=0.193 , (b) xx - zz -plane at y∗=0y^{*}=0 and t∗=0.26t^{*}=0.26 , (c) xx - yy -plane at z∗=0z^{*}=0 and t∗=0.393t^{*}=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.
Figure 21: DeepONet velocity magnitude predictions for the Slot-L orifice based on experimental PIV input. (a) xx - yy -plane at z∗=0z^{*}=0 and t∗=0.147t^{*}=0.147 , (b) xx - zz -plane at y∗=0y^{*}=0 and t∗=0.087t^{*}=0.087 .
Figure 22: Velocity magnitude distribution at y∗=0y^{*}=0 and t∗=0.08t^{*}=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∗=0y^{*}=0 and t∗=0.08t^{*}=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.
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