thomgrand / geasi_grid_demo

A simple implementation demo of the GEASI algorithm. Note that this demo is limited to the isotropic eikonal equation on structured grids.

GEASI Demo

This repository contains a minimal working example of our algorithm presented in GEASI: Geodesic-based Earliest Activation Sites Identification in cardiac models. The main steps of the algorithm are similar, but to offer a concise and short code some simplifying assumptions were made (see Limitations).

The method is demonstrated in GEASI_Grid.ipynb, whereas utils.py contains some utility functions to keep the code short. The presented problem is similar to the square-domain example of the paper with slightly varying results due to the deviations from the original algorithm.

Installation

The notebook can be viewed in github or using jupyter-notebook/-lab. To execute the code, some packages need to be installed to compute the eikonal equation .

Tested on Ubuntu 18.04. To run, switch to the repository directory and execute the following commands in a fresh python environment (e.g. virtual-env, or anaconda).

pip install -r requirements.txt
jupyter-notebook

Limitations

While this repository is meant as a demonstration of the algorithm, several simplifications were made to make the code as short and concise as possible in contrast to the original paper:

• Projection directly from L-BFGS-B method instead of the Moreau-envelope
• A structured grid is used instead of a true FEM mesh
• \nabla \phi is estimated using a finite difference scheme
• Only the isotropic eikonal equation is considered, i.e. |\nabla \phi| = 1/v and solved using skfmm
• We assume quadrilateral basis functions inside the grid
• Contains neither the topological gradient, nor the ECG extension
• No handling of coalescing EASs

Acknowledgements

If this work (or a part of it) helps you in your research, please consider acknowledging the github repository, or citing our paper.

@article{grandits_geasi_2021,
	title = {{GEASI}: {Geodesic}-based earliest activation sites identification in cardiac models},
	volume = {37},
	issn = {2040-7947},
	shorttitle = {{GEASI}},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.3505},
	doi = {10.1002/cnm.3505},
	language = {en},
	number = {8},
	journal = {International Journal for Numerical Methods in Biomedical Engineering},
	author = {Grandits, Thomas and Effland, Alexander and Pock, Thomas and Krause, Rolf and Plank, Gernot and Pezzuto, Simone},
	year = {2021},
	keywords = {eikonal equation, cardiac model personalization, earliest activation sites, Hamilton–Jacobi formulation, inverse ECG problem, topological gradient}
}