Image-based deep learning of finite element simulations for fast surrogate biomechanical organ deformations

Abstract Introduction: We apply deep learning to emulate Finite Element simulations by exploiting neural networks as universal function approximators. A novel processing pipeline automates the incorporation of three-dimensional biomechanical simulation data into an image-based deep learning task where spatially resolved deformations are predicted using pixelwise regression. We show that a model trained in this way can quickly reproduce simulation results with reasonable preliminary performance. Methods: We trained a customized U-Net (32 input features) to predict displacement models based on the input force vectors. The pipeline was applied to a basic indentation process for an idealized reference geometry (puck) and a 3D liver model from the LiTS dataset. The generated 3D simulation data is normalized and projected onto 2D images using the midpoint circle algorithm such that x, y, and z coordinates are mapped to pixel values of RGB image channels. This way, 1500 simulations of the reference geometry and 500 simulations of the liver deformation with force-controlled, randomly initialized load distributions were performed. Simulations took about 2-5 minutes of computation time with FE models of 6000 and 10000 elements. The neural network was trained on the simulation database using pixelwise regression. Results: We successfully trained a model on the pixelwise regression of simulation results. The model optimization reached an f1- score of 0.69 and 0.61 for the puck and the liver. The displacement images were reprojected and compared to the 3D ground truth. The trained models reached an average surface distance of 3.1 ± 1.2 and 4.2 ± 2.2, as well as 4.3 ± 0.4 and 6.4 ± 0.2 for the Hausdorff distances. Conclusion: The model performance is still impractical for clinical decisions. Still, with mean computation times of under 250 ms, the approach's potential with a more extensive database and a more sophisticated deep learning architecture, e.g., GNN, is recognizable.