This repository contains the code accompanying the paper [1]. We employ Latent Neural Ordinary Differential Equations (LNODEs) to learn the atrial and ventricular pressure-volume temporal dynamics of 400 four-chamber heart electromechanical simulations while spanning 43 physics-based model parameters that describe cell, tissue, whole-heart and cardiovascular system material properties and boundary conditions. We employ the surrogate model based on LNODEs to perform a global sensitivity analysis and robust parameter estimation with inverse uncertainty quantification for different test cases ('TLV', 'Tatria', 'Tventricles', 'Tall').

1. Install a conda environment containing all the required packages:

conda create -n envcardioEM-4CH python=3.8 numpy=1.21.5 matplotlib=3.5.1 pandas=1.3.4 scipy=1.7.3 mpi4py=3.1
conda activate envcardioEM-4CH
conda install tensorflow
conda install -c anaconda scikit-learn
pip install SALib
pip install --upgrade "jax[cpu]"
pip install numpyro
pip install diffrax

2. Clone this repository by typing:

git clone https://github.com/MatteoSalvador/cardioEM-4CH.git

3. Remember to activate the conda environment envcardioEM-4CH by typing conda activate envcardioEM-4CH (in case it is not already active from the installation procedure at point 1).

4. Unzip dataset.zip, which contains a tensor with all the 3D-0D numerical simulations (simulations.pkl), and a table with all the physics-based model parameters (parameters.npy).

5. LNODEs training: execute the Python script train.py with proper settings, such as the total number of Adam (num_iters_ADAM) and BFGS (num_iters_BFGS) epochs, neurons, layers and states.

6. Global sensitivity analysis: execute 4 different Python scripts sequentially (SA_salib_1_params.py, SA_salib_2_execution.py, SA_salib_3_analysis.py, SA_salib_4_postprocessing.py) with proper settings, such as the number of samples (nsamples), input (inputfile) and output (outputfile) files, required during the different stages.

7. Parameter estimation with uncertainty quantification: execute the Python script run_PE_ANN.py with a certain test case ('TLV', 'Tatria', 'Tventricles', 'Tall') and proper settings, such as the folder containing the trained LNODEs (ANN_folder) and the number of trials (n_trials).

• Matteo Salvador ([email protected])
• Francesco Regazzoni ([email protected])

[1] M. Salvador, M. Strocchi, F. Regazzoni, C. Augustin, L. Dede', S. A. Niederer, A. Quarteroni. Whole-heart electromechanical simulations using Latent Neural Ordinary Differential Equations. npj Digital Medicine (2024).

[2] M. Salvador, F. Regazzoni, L. Dede', A. Quarteroni. Fast and robust parameter estimation with uncertainty quantification for the cardiac function. Computer Methods and Programs in Biomedicine (2023).

[3] F. Regazzoni, M. Salvador, L. Dede', A. Quarteroni. A Machine Learning method for real-time numerical simulations of cardiac electromechanics. Computer Methods in Applied Mechanics and Engineering (2022).