A deep learning approach to estimate stress distribution: a fast and accurate surrogate of finite-element analysis

Abstract

Structural finite-element analysis (FEA) has been widely used to study the biomechanics of human tissues and organs, as well as tissue-medical device interactions, and treatment strategies. However, patient-specific FEA models usually require complex procedures to set up and long computing times to obtain final simulation results, preventing prompt feedback to clinicians in time-sensitive clinical applications. In this study, by using machine learning techniques, we developed a deep learning (DL) model to directly estimate the stress distributions of the aorta. The DL model was designed and trained to take the input of FEA and directly output the aortic wall stress distributions, bypassing the FEA calculation process. The trained DL model is capable of predicting the stress distributions with average errors of 0.492% and 0.891% in the Von Mises stress distribution and peak Von Mises stress, respectively. This study marks, to our knowledge, the first study that demonstrates the feasibility and great potential of using the DL technique as a fast and accurate surrogate of FEA for stress analysis.

Keywords: deep learning; finite-element analysis; neural network; stress analysis.

Figure 1.

The current workflow versus a machine learning-based solution for patient-specific computational modelling for potential clinical applications. This study is focused on developing the ML–FE surrogate to replace FEA including model set-up and simulation.

Figure 2.

A representative aorta mesh, which is topologically equivalent to a rectangle.

Figure 3.

The overall data flow of the deep learning model which takes an aorta shape as the input and outputs the wall stress distribution.

Figure 4.

The neural network for shape encoding. () is equal to the node-i position of shape X minus the node-i position of the mean shape . The link (i.e. weight) between and is the first component of ; the link between and is the last component of ; and so on. In this application, M = 3 and each has 15 000 components.

Figure 5.

The neural network for mapping the shape code to the stress code. In this study, M = 3 and N = 64.

Figure 6.

The bidirectional neural network for stress decoding (from left to right) and encoding (from right to left). It has 25 subnetworks, corresponding to the 25 regions of the aorta. The aorta mesh is equivalent to a rectangular mesh that is divided into 25 regions.

Figure 7.

Estimated stress distributions for Patient 1. Units are in KPa. Each stress distribution is mapped into a two-dimensional region (figure 2). The first and the second columns are the stress distributions from finite-element simulation. The middle column shows the stress ranges. The fourth and fifth columns are the stress distributions estimated by the DL model.

Figure 8.

Estimated stress distributions for Patient 2. Units are in KPa. Each stress distribution is mapped into a two-dimensional region (figure 2). The first and the second columns are the stress distributions from finite-element simulation. The middle column shows the stress ranges. The fourth and fifth columns are the stress distributions estimated by the DL model.