Physics-Informed Deep Learning for Quasi-Static Elasticity Imaging: Mechanical Characterization of Heterogeneous Domains in Two and Three Dimensions

Abstract

Resolving the spatial distribution of mechanical properties in tissues has major applications in clinical diagnosis, materials characterization, and biomechanics. Elasticity imaging is a technique that reconstructs the hidden mechanical properties of tissues and materials. As a subtype of this imaging technique, quasi-static elasticity imaging is based on acquiring images before and after an induced quasi-static motion, estimating the displacement field from these two states using image correlation, and deducing the spatial distribution of mechanical properties that would produce this deformation. In the past three decades, several mathematical frameworks have been developed to solve this inverse problem. However, several challenges remain in terms of reconstruction accuracy, convergence, and applicability of these methods. More recently, physics-informed deep learning models have emerged as a promising approach to tackle this problem. In this thesis, we introduce physics-informed neural network (PINN) models to solve inverse problems in 2D and 3D linear elasticity and discuss their advantages over conventional methods. We also address specific challenges and propose methods for advancing the accuracy, efficiency, and applicability of these models.In the first study, we present the development of physics-informed neural network models to solve the full inverse elasticity problem in two dimensions. Our model, which consisted of two fully-connected networks, simultaneously discovered the spatial distribution of isotropic linear elasticity parameters, Young's modulus (E) and Poisson’s ratio (ν), using strain data, normal stress boundary conditions, and the governing physics. The inversion model demonstrated accurate localization of embedded inclusions and captured complex mechanical properties and tissue interfaces. Validation with experimental and simulated data showcased the potential of the developed parameter estimation PINN in clinical and material characterization scenarios. Building on the advancements of the first study, our second study introduces a new neural network backbone, El-UNet, based on encoder-decoder convolutional neural networks with skip connections (UNet). El-UNet outperformed fully-connected physics-informed neural networks in accurately estimating two-dimensional (2D) unknown parameters and stress distributions in equal timeframes. We further enhanced the accuracy of reconstructions by incorporating a self-adaptive spatial loss weighting approach. In this study, we also analyzed the benefits and limitations of the proposed method, providing insights for future advancements in solving high-dimensional inverse elasticity problems. Our final study extended the scope of quasi-static elasticity imaging to the reconstruction of three-dimensional (3D) spatial distributions of material parameters for isotropic and transversely isotropic materials. Leveraging the 3D El-UNet, we identified 3D heterogeneous material parameter distributions using strain data and surface-normal stress boundary conditions. We showed the efficacy of El-UNet in estimating material properties for linear isotropic and transversely isotropic materials on finite-element simulation data. This thesis advances the field of elasticity imaging by introducing inversion models leveraging physics-informed deep learning. The major contributions include the development of physics-informed neural network models for the inverse elasticity problem, the enhancement of accuracy and efficiency through El-UNet, and the reconstruction of 3D spatial distributions of material parameters for isotropic and transversely isotropic materials. These advancements expand the understanding and applicability of elasticity imaging techniques and provide stepping stones for future research in more complex material models and potential clinical translation.