Abstract

A comprehensive image-based computational modelling pipeline is required for high-fidelity patient-specific cardiac simulations. However, traditional simulation methods are a limitation in these approaches due to their prohibitively slow speeds. We developed a physics-based training scheme using differentiable finite elements to compute the residual force vector of the governing PDE , which is then minimized to find the optimal network parameters. We used neural networks for their representation power, and finite elements for defining the problem domain, specifying the boundary conditions, and performing numerical integrations. We incorporated spatially varying fiber structures into a prolate spheroidal model of the left ventricle. A Fung-type material model including active contraction was used. We developed two versions of our model, one was trained on a reduced basis of the solution space, and one was trained on the full solution space. The models were trained against two pressure-volume loops and validated on a third loop (Fig. 1). We validated our implementation against conventional FEM simulation using FEniCS. While the reduced order model was trained faster than the full-order model, we achieved mean and standard deviation of the nodal error between the NNFE solution and the FE solution with 10 -3 cm, with both models, where the characteristic length was 1 cm (Fig. 2a). The NNFE model predicted each solution within 0.6 ms whereas the FE models took up to 500 ms for each state. The NNFE method can be simultaneously trained over the entire range of physiological boundary conditions. The trained NNFE can predict stress–strain responses for any physiological boundary condition without retraining.