Fig 1.
Schematic representation of the EP problem to address.
In particular, we aim at reconstructing the activation/repolarization times of the cardiac tissue given the initial stimulus applied for 1 ms.
Fig 2.
Schematic architecture of Fourier Neural Operators (FNO).
Fig 3.
Kernel Operator Learning (KOL) diagram.
Starting from the input function a, A collects observations of the function at different collocation points through ϕ. Then, the vector-valued f† processes observations of the input into observations of the output . Finally, the reconstruction operator is applied to determine the output function
.
Fig 4.
Grids adopted for the numerical simulations: (a) 2D grid (structured, elements, physical area 1 cm × 1 cm = 1 cm2), (b) 3D slab (structured,
elements, physical volume 3.84 cm × 3.84 cm × 0.64 cm = 9.44 cm3.
The figure displays the coarser elements grid used to save the dataset employed to train the operator learning models.) and (c) 3D unstructured ventricle (about 35k nodes, physical volume 100 cm3). For all the geometries, we have considered Q1 elements (regular squares/cubes for structured 2D and 3D cases, hexaedral elements for the 3D unstructured case).
Table 1.
Parameters of the high-fidelity solvers used for the 2D, 3D, and 3D unstructured cases.
Here, h is the element diameter, dt the time step, the stimulation duration,
the membrane capacitance per unit surface area,
,
,
the conductivities along the principal directions, and
the amplitude of the applied stimulus current density and
is the simulated physical time. The 2D case has been solved using MATLAB’s direct solver, while the 3D cases have been solved using the Conjugate Gradient (CG) solver provided by the PETSc library and Hypre BoomerAMG as preconditioner [4,59].
Fig 5.
Comparison of FNO (A) and KOL (B) activation time predictions for the 2D case (acti with 2000 samples).
Colorbars indicate time in milliseconds (ms).
Fig 6.
Comparison of FNO (A) and KOL (B) repolarization time predictions for the 2D case (repo with 2000 samples).
Colorbars indicate time in milliseconds (ms).
Table 2.
Performance comparison of FNO (reduce-on-plateau learning rate policy) and KOL (iq4 kernel) methods on 2D datasets.
Table 3.
Performance comparison of FNO and KOL methods on 3D datasets.
Fig 7.
FNO box plot (A) and histogram (B) for the 3D dataset acti 2000 relative to the best model trained.
For the training set, 3% of the data have a relative L2 error greater than 4%, while for the test set, 3.5% of the data exceed this threshold.
Fig 8.
KOL box plot (A) and histogram (B) for 3D dataset acti 2000 relative to the best model trained.
Training results are not shown since we achieve machine precision. For the test set, 1.5% of the data have a relative L2 error greater than 10%.
Fig 9.
FNO loss plot for the 3D case (acti 2000) dataset).
Mean train and test loss of three different randomly initialized models (dashed line). Standard deviation over the three models for each epoch is also reported (light shadow).
Fig 10.
Example of FNO activation time prediction for the 3D case (acti with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 11.
Example of FNO repolarization time prediction for the 3D case (repo with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 12.
Example of KOL activation time prediction for the 3D case (acti with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 13.
Example of KOL repolarization time prediction for the 3D case (repo with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 14.
Example of FNO activation time prediction for the 3D heterogeneous case (acti with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 15.
Example of FNO repolarization time prediction for the 3D heterogeneous case (repo with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 16.
Example of KOL activation time prediction for the 3D heterogeneous case (acti with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Fig 17.
Example of KOL repolarization time prediction for the 3D heterogeneous case (repo with 2000 samples).
The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom). Colorbars indicate time in milliseconds (ms).
Table 4.
Performance comparison of FNO and KOL on 3D unstructured datasets (cfr. Table D in S1 File for the performance comparison of FNO architectures with different layers and widths).
Fig 18.
Example of FNO predictions for the 3D unstructured case: (A) Activation times (acti 2000), (B) Repolarization times (acti 2000). Colorbars indicate time in milliseconds (ms).
Fig 19.
Example of KOL predictions for the 3D unstructured case: (A) Activation times (acti 2000), (B) Repolarization times (acti 2000).
Colorbars indicate time in milliseconds (ms).