PDE-aware deep learning for inverse problems in cardiac electrophysiology: datasets

This repository contains the data accompanying the paper:

PDE-aware deep learning for inverse problems in cardiac electrophysiology, by R. Tenderini, S.Pagani, A. Quarteroni, and S. Deparis. (2022). SIAM Journal on Scientific Computing, 44(3), B605-B639. https://doi.org/10.1137/21M1438529

All data have been generated in-silico, by numerically approximating cardiac electrophysiology during ventricular systole, employing the bidomain model. We refer to the aforementioned paper and to the related appendix for all details regarding both the physical modeling and the choice of the main model parameters.

The dataset is partitioned into two blocks, depending on the number of body surface signals that have been numerically estimated. They can be identified by means of the first part of the related filenames; specifically

1. ECG_*-named files: they contain data relative to numerical simulations in which 12-lead ECG signals have been estimated (see Fig. 6).
2. Vest_*-named files: they contain data relative to numerical simulations in which 158 body surface potentials have been estimated (see Fig. 6).

The following quantities will be referred to in the files content description:

• \(N\): number of snapshots, characterized by different ventricular activation patterns and cardiac electrical conductivities. It equals 400.
• \(M\): number of estimated body surface signals. It equals either 12 or 158, depending on the block.
• \(N_s\): number of FOM degrees of freedom located on the epicardial surface. It equals 5148.
• \(N_t\): number of time instants at which the FOM problem has been solved, equispaced between 0 and 160 ms. It equals 80.
• \(n_s\): number of spatial reduced basis functions.
• \(n_t\): vector of length equal to \(n_s\), storing the number of temporal reduced basis functions tailored to each spatial basis function.
• \(n_{st}\): number of spatio-temporal reduced basis functions. It equals \(\sum\limits_{i=1}^{n_s} n_t^i\), being \(n_t^i\) the \(i\)-th entry of  \(n_t\)

Each of the 2 blocks features the following files, in .csv format:

• *_ BasesTimeDims.csv: dimensionalities of the temporal reduced bases tailored to each spatial reduced basis function. Data are stored in a vector of integers of dimension \(n_s\)
• *_Basis.csv: spatial reduced basis functions for the epicardial potential field. Data are stored in a matrix of dimension \(N_s \times n_s\)
• *_BasisTime.csv: temporal reduced bases functions, each one tailored to one spatial reduced basis function, for the epicardial potential field. Data are stored in a matrix of dimension \(n_s \times N_t\max(n_t)\). So, each row is subdivided into \(n_t^i\) horizontally stacked blocks - being \(i\) the row index - each one being a temporal basis function tailored to the \(i\)-th spatial basis function. If the \(i\)-th spatial basis function is associated with \(\tilde{n_t} < \max(n_t)\) temporal bases functions, the last \((\max(n_t) - \tilde{n_t})N_t\) entries of the \(i\)-th row of the matrix are filled with NaN values.
• *_Sigmas.csv: singular values associated with the spatial reduced bases functions for the epicardial potential field. Data are stored in a vector of dimension \(n_s\)
• *_SigmasTime.csv: singular values associated with the tailored temporal reduced bases functions for the epicardial potential field. Data are stored in a matrix of dimension \(n_s \times \max(n_t)\). If the \(i\)-th spatial basis function is associated with \(\tilde{n_t} < \max(n_t)\) temporal bases functions, the last \(\max(n_t) - \tilde{n_t}\) entries of the \(i\)-th row of the matrix are filled with NaN values.
• *_Signals.csv: numerically estimated body surface signals. Data are stored in a matrix of dimension \(N \times M N_t\); each row features then \(M\) horizontally stacked blocks, each one being the time course of a body surface potential over \(N_t\) equispaced time instants.
• *_SnapshotsReduced.csv: spatio-temporal projection of the \(N\) FOM snapshots of the epicardial potential field. Data are stored in a matrix of dimension \(N \times n_{st}\)