Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Models

TL;DR

The capability of PINNs to detect the presence, location and severity of scar tissue is demonstrated, which is beneficial to develop personalised simulation models for disease diagnosis, especially for cardiac applications.

Abstract

The development of biophysical models for clinical applications is rapidly advancing in the research community, thanks to their predictive nature and their ability to assist the interpretation of clinical data. However, high-resolution and accurate multi-physics computational models are computationally expensive and their personalisation involves fine calibration of a large number of parameters, which may be space-dependent, challenging their clinical translation. In this work, we propose a new approach which relies on the combination of physics-informed neural networks (PINNs) with three-dimensional soft tissue nonlinear biomechanical models, capable of reconstructing displacement fields and estimating heterogeneous patient-specific biophysical properties. The proposed learning algorithm encodes information from a limited amount of displacement and, in some cases, strain data, that can be routinely acquired in the clinical setting, and combines it with the physics of the problem, represented by a mathematical model based on partial differential equations, to regularise the problem and improve its convergence properties. Several benchmarks are presented to show the accuracy and robustness of the proposed method and its great potential to enable the robust and effective identification of patient-specific, heterogeneous physical properties, s.a. tissue stiffness properties. In particular, we demonstrate the capability of the PINN to detect the presence, location and severity of scar tissue, which is beneficial to develop personalised simulation models for disease diagnosis, especially for cardiac applications.

Figures (29)

• Figure 1: Schematic of PINNs. Left: A standard fully-connected neural network parameterised by biases and weights $\mathbf{w}$ to approximate a function $\mathbf{u}(\mathbf{x},\boldsymbol{\mu})$. The set of model parameters to estimate is given by $\boldsymbol{\mu}$. Centre: automatic differentiation (AD) is performed to efficiently compute the derivatives involved in the differential operator $\mathbf{\mathcal{L}}(\mathbf{u})$ and the boundary operator $\mathbf{\mathcal{B}}(\mathbf{u})$ on random points. The loss function is computed, composed by the data mismatch on given observation points and the PDE and BC residuals. Minimising the loss with respect to the network parameters $\mathbf{w}$ and the solution parameter $\boldsymbol{\mu}$ produces the PINN $\textrm{NN}_\mathbf{u}$.
• Figure 2: Problem geometry as described in \ref{['sec:soft_tissue_appl']}.
• Figure 3: Isotropic test case of \ref{['sec:iso_hom']}, FEM displacement magnitude in $\mm$. The rest configuration is superposed in shaded grey to the deformed configuration.
• Figure 4: Isotropic material: relative error on the PINN estimation of the passive stiffness $\mu$ considering noise‐free data or data corrupted by Gaussian white noise with different LD. Result of five training processes; the solid lines depict the geometric mean, whereas the shaded region is the area spanned by the trajectories.
• Figure 5: Transverse-isotropic test case of \ref{['sec:TI_hom']}, FEM displacement magnitude in $\mm$. Left: constant fibre orientation along $x-$axis. Right: varying fibre orientation ($\ang{0}$-$\ang{24}$ with respect to the $x$--axis in the $x$--$y$ plane). The rest configuration is superposed in shaded grey to the deformed configuration.