Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio

Integrating finite element analysis and physics-informed neural networks for biomechanical modeling of the human lumbar spine

Background

Comprehending the biomechanical characteristics of the human lumbar spine is crucial for managing and preventing spinal disorders. Precise material properties derived from patient-specific CT scans are essential for simulations to accurately mimic real-life scenarios, which is invaluable in creating effective surgical plans. The integration of Finite Element Analysis (FEA) with Physics-Informed Neural Networks (PINNs) offers significant clinical benefits by automating lumbar spine segmentation and meshing.

Methods

We developed a FEA model of the lumbar spine incorporating detailed anatomical and material properties derived from high-quality CT and MRI scans. The model includes vertebrae and intervertebral discs, segmented and meshed using advanced imaging and computational techniques. PINNs were implemented to integrate physical laws directly into the neural network training process, ensuring that the predictions of material properties adhered to the governing equations of mechanics.

Results

The model achieved an accuracy of 94.30% in predicting material properties such as Young's modulus (14.88 GPa for cortical bone and 1.23 MPa for intervertebral discs), Poisson's ratio (0.25 and 0.47, respectively), bulk modulus (9.87 GPa and 6.56 MPa, respectively), and shear modulus (5.96 GPa and 0.42 MPa, respectively). We developed a lumbar spine FEA model using anatomical and material properties from CT and MRI scans. Vertebrae and discs were segmented and meshed with advanced imaging techniques, while PINNs ensured material predictions followed mechanical laws.

Conclusions

The integration of FEA and PINNs allows for accurate, automated prediction of material properties and mechanical behaviors of the lumbar spine, significantly reducing manual input and enhancing reliability. This approach ensures dependable biomechanical simulations and supports the development of personalized treatment plans and surgical strategies, ultimately improving clinical outcomes for spinal disorders. This method improves surgical planning and outcomes, contributing to better patient care and recovery in spinal disorders.

Learning soft tissue deformation from incremental simulations

BACKGROUND

Surgical planning for orthognathic procedures demands swift and accurate biomechanical modeling of facial soft tissues. Efficient simulations are vital in the clinical pipeline, as surgeons may iterate through multiple plans. Biomechanical simulations typically use the finite element method (FEM). Prior works divide FEM simulations into increments to enhance convergence and accuracy. However, this practice elongates simulation time, thereby impeding clinical integration. To accelerate simulations, deep learning (DL) models have been explored. Yet, previous efforts either perform simulations in a single step or neglect the temporal aspects in incremental simulations.

PURPOSE

This study investigates the use of spatiotemporal incremental modeling for biomechanics simulations of facial soft tissue.

METHODS

We implement the method using a graph neural network. Our method synergizes spatial features with temporal aggregation using DL networks trained on incremental FEM simulations from 17 subjects that underwent orthognathic surgery.

RESULTS

Our proposed spatiotemporal incremental method achieved a mean accuracy of 0.37 mm with a mean computation time of 1.52 s. In comparison, a spatial-only incremental method yielded a mean accuracy of 0.44 mm and a mean computation time of 1.60 s, while a spatial-only single-step method yielded a mean accuracy of 0.41 mm and a mean computation time of 0.05 s.

CONCLUSIONS

Statistical analysis demonstrated that the spatiotemporal incremental method reduced mean errors compared to the spatial-only incremental method, emphasizing the importance of incorporating temporal information in incremental simulations. Overall, we successfully implemented spatiotemporal incremental learning tailored to simulate soft tissue deformation while substantially reducing simulation time compared to FEM.

PINNing cerebral blood flow: analysis of perfusion MRI in infants using physics-informed neural networks

Arterial spin labelling (ASL) magnetic resonance imaging (MRI) enables cerebral perfusion measurement, which is crucial in detecting and managing neurological issues in infants born prematurely or after perinatal complications. However, cerebral blood flow (CBF) estimation in infants using ASL remains challenging due to the complex interplay of network physiology, involving dynamic interactions between cardiac output and cerebral perfusion, as well as issues with parameter uncertainty and data noise. We propose a new spatial uncertainty-based physics-informed neural network (PINN), SUPINN, to estimate CBF and other parameters from infant ASL data. SUPINN employs a multi-branch architecture to concurrently estimate regional and global model parameters across multiple voxels. It computes regional spatial uncertainties to weigh the signal. SUPINN can reliably estimate CBF (relative error - 0.3 ± 71.7 ), bolus arrival time (AT) ( 30.5 ± 257.8 ) , and blood longitudinal relaxation time ( T 1 b ) (-4.4 ± 28.9), surpassing parameter estimates performed using least squares or standard PINNs. Furthermore, SUPINN produces physiologically plausible spatially smooth CBF and AT maps. Our study demonstrates the successful modification of PINNs for accurate multi-parameter perfusion estimation from noisy and limited ASL data in infants. Frameworks like SUPINN have the potential to advance our understanding of the complex cardio-brain network physiology, aiding in the detection and management of diseases. Source code is provided at: https://github.com/cgalaz01/supinn.

Identifying Heterogeneous Micromechanical Properties of Biological Tissues via Physics-Informed Neural Networks

The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying full-field heterogeneous elastic properties of soft materials using traditional engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in data-driven models for learning full-field mechanical responses, such as displacement and strain, from experimental or synthetic data. However, research studies on inferring full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, a physics-informed machine learning approach is proposed to identify the elasticity map in nonlinear, large deformation hyperelastic materials. This study reports the prediction accuracies and computational efficiency of physics-informed neural networks (PINNs) in inferring the heterogeneous elasticity maps across materials with structural complexity that closely resemble real tissue microstructure, such as brain, tricuspid valve, and breast cancer tissues. Further, the improved architecture is applied to three hyperelastic constitutive models: Neo-Hookean, Mooney Rivlin, and Gent. The improved network architecture consistently produces accurate estimations of heterogeneous elasticity maps, even when there is up to 10% noise present in the training data.

A theoretical framework for predicting the heterogeneous stiffness map of brain white matter tissue

Finding the stiffness map of biological tissues is of great importance in evaluating their healthy or pathological conditions. However, due to the heterogeneity and anisotropy of biological fibrous tissues, this task presents challenges and significant uncertainty when characterized only by single-mode loading experiments. In this study, we propose a new theoretical framework to map the stiffness landscape of fibrous tissues, specifically focusing on brain white matter tissue. Initially, a finite element (FE) model of the fibrous tissue was subjected to six loading cases, and their corresponding stress-strain curves were characterized. By employing multiobjective optimization, the material constants of an equivalent anisotropic material model were inversely extracted to best fit all six loading modes simultaneously. Subsequently, large-scale FE simulations were conducted, incorporating various fiber volume fractions and orientations, to train a convolutional neural network capable of predicting the equivalent anisotropic material properties solely based on the fibrous architecture of any given tissue. The proposed method, leveraging brain fiber tractography, was applied to a localized volume of white matter, demonstrating its effectiveness in precisely mapping the anisotropic behavior of fibrous tissue. In the long-term, the proposed method may find applications in traumatic brain injury, brain folding studies, and neurodegenerative diseases, where accurately capturing the material behavior of the tissue is crucial for simulations and experiments.

Discovering 3D hidden elasticity in isotropic and transversely isotropic materials with physics-informed UNets

Three-dimensional variation in structural components or fiber alignments results in complex mechanical property distribution in tissues and biomaterials. In this paper, we use a physics-informed UNet-based neural network model (El-UNet) to discover the three-dimensional (3D) internal composition and space-dependent material properties of heterogeneous isotropic and transversely isotropic materials without a priori knowledge of the composition. We then show the capabilities of El-UNet by validating against data obtained from finite-element simulations of two soft tissues, namely, brain tissue and articular cartilage, under various loading conditions. We first simulated compressive loading of 3D brain tissue comprising of distinct white matter and gray matter mechanical properties undergoing small strains with isotropic linear elastic behavior, where El-UNet reached mean absolute relative errors under 1.5 % for elastic modulus and Poisson's ratio estimations across the 3D volume. We showed that the 3D solution achieved by El-UNet was superior to relative stiffness mapping by inverse of axial strain and two-dimensional plane stress/plane strain approximations. Additionally, we simulated a transversely isotropic articular cartilage with known fiber orientations undergoing compressive loading, and accurately estimated the spatial distribution of all five material parameters, with mean absolute relative errors under 5 %. Our work demonstrates the application of the computationally efficient physics-informed El-UNet in 3D elasticity imaging and provides methods for translation to experimental 3D characterization of soft tissues and other materials. The proposed El-UNet offers a powerful tool for both in vitro and ex vivo tissue analysis, with potential extensions to in vivo diagnostics. STATEMENT OF SIGNIFICANCE: Elasticity imaging is a technique that reconstructs mechanical properties of tissue using deformation and force measurements. Given the complexity of this reconstruction, most existing methods have mostly focused on 2D problems. Our work is the first implementation of physics-informed UNets to reconstruct three-dimensional material parameter distributions for isotropic and transversely isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using synthetic data generated using finite element models of biological tissues with high bio-fidelity-the brain and articular cartilage. Our method can be implemented in elasticity imaging scenarios for in vitro and ex vivo mechanical characterization of biomaterials and biological tissues, with potential extensions to in vivo diagnostics.

Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networks

The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying full-field heterogeneous elastic properties of soft materials using traditional engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in using data-driven models to learn full-field mechanical responses such as displacement and strain from experimental or synthetic data. However, research studies on inferring full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, we propose a physics-informed machine learning approach to identify the elasticity map in nonlinear, large deformation hyperelastic materials. We evaluate the prediction accuracies and computational efficiency of physics-informed neural networks (PINNs) by inferring the heterogeneous elasticity maps across three materials with structural complexity that closely resemble real tissue patterns, such as brain tissue and tricuspid valve tissue. We further applied our improved architecture to three additional examples of breast cancer tissue and extended our analysis to three hyperelastic constitutive models: Neo-Hookean, Mooney Rivlin, and Gent. Our selected network architecture consistently produced highly accurate estimations of heterogeneous elasticity maps, even when there was up to 10% noise present in the training data.

Identifying constitutive parameters for complex hyperelastic materials using physics-informed neural networks

Identifying constitutive parameters in engineering and biological materials, particularly those with intricate geometries and mechanical behaviors, remains a longstanding challenge. The recent advent of physics-informed neural networks (PINNs) offers promising solutions, but current frameworks are often limited to basic constitutive laws and encounter practical constraints when combined with experimental data. In this paper, we introduce a robust PINN-based framework designed to identify material parameters for soft materials, specifically those exhibiting complex constitutive behaviors, under large deformation in plane stress conditions. Distinctively, our model emphasizes training PINNs with multi-modal synthetic experimental datasets consisting of full-field deformation and loading history, ensuring algorithm robustness even with noisy data. Our results reveal that the PINNs framework can accurately identify constitutive parameters of the incompressible Arruda-Boyce model for samples with intricate geometries, maintaining an error below 5%, even with an experimental noise level of 5%. We believe our framework provides a robust modulus identification approach for complex solids, especially for those with geometrical and constitutive complexity.

Elastic parameter identification of three-dimensional soft tissue based on deep neural network

In the field of virtual surgery and deformation simulation, the identification of elastic parameters of human soft tissues is a critical technology that directly affects the accuracy of deformation simulation. Current research on soft tissue deformation simulation predominantly assumes that the elasticity of tissues is fixed and already known, leading to the difficulty in populating with the elasticity measured or identified from specific tissues of real patients. Existing elasticity modeling efforts struggle to be implemented on irregularly structured soft tissues, failing to adapt to clinical surgical practices. Therefore, this paper proposes a new method for identifying human soft tissue elastic parameters based on the finite element method and the deep neural network, UNet. This method requires only the full-field displacement data of soft tissues under external loads to predict their elastic distribution. The performance and validity of the algorithm are assessed using test data and clinical data from rhinoplasty surgeries. Experiments demonstrate that the method proposed in this paper can achieve an accuracy of over 99% in predicting elastic parameters. Clinical data validation shows that the predicted elastic distribution can reduce the error in finite element deformation simulations by more than 80% at the maximum compared to the error with traditional uniform elastic parameters, effectively enhancing the computational accuracy in virtual surgery simulations and soft tissue deformation modeling.

Multiscale elasticity mapping of biological samples in 3D at optical resolution

The mechanical properties of biological tissues have emerged as an integral determinant of tissue function in health and disease. Nonetheless, characterizing the elasticity of biological samples in 3D and at high resolution remains challenging. Here, we present a µElastography platform: a scalable elastography system that maps the elastic properties of tissues from cellular to organ scales. The platform leverages the use of a biocompatible, thermo-responsive hydrogel to deliver compressive stress to a biological sample and track its resulting deformation. By surrounding the specimen with a reference hydrogel of known Young's modulus, we are able to map the absolute values of elastic properties in biological samples. We validate the experimental and computational components of the platform using a hydrogel phantom and verify the system's ability to detect internal mechanical heterogeneities. We then apply the platform to map the elasticity of multicellular spheroids and the murine lymph node. With these applications, we demonstrate the platform's ability to map tissue elasticity at internal planes of interest, as well as capture mechanical heterogeneities neglected by most macroscale characterization techniques. The µElastography platform, designed to be implementable in any biology lab with access to 3D microscopy (e.g., confocal, multiphoton, or optical coherence microscopy), will provide the capability to characterize the mechanical properties of biological samples to labs across the large community of biological sciences by eliminating the need of specialized instruments such as atomic force microscopy. STATEMENT OF SIGNIFICANCE: Understanding the elasticity of biological tissues is of great importance, but characterizing these properties typically requires highly specialized equipment. Utilizing stimulus-responsive hydrogels, we present a scalable, hydrogel-based elastography method that uses readily available reagents and imaging modalities to generate resolved maps of internal elasticity within biomaterials and biological samples at optical resolution. This new approach is capable of detecting internal stiffness heterogeneities within the 3D bulk of samples and is highly scalable across both imaging modalities and biological length scales. Thus, it will have significant impact on the measurement capabilities of labs studying engineered biomaterials, mechanobiology, disease progression, and tissue engineering and development.

Physics-informed UNets for discovering hidden elasticity in heterogeneous materials

Soft biological tissues often have complex mechanical properties due to variation in structural components. In this paper, we develop a novel UNet-based neural network model for inversion in elasticity (El-UNet) to infer the spatial distributions of mechanical parameters from strain maps as input images, normal stress boundary conditions, and domain physics information. We show superior performance - both in terms of accuracy and computational cost - by El-UNet compared to fully-connected physics-informed neural networks in estimating unknown parameters and stress distributions for isotropic linear elasticity. We characterize different variations of El-UNet and propose a self-adaptive spatial loss weighting approach. To validate our inversion models, we performed various finite-element simulations of isotropic domains with heterogenous distributions of material parameters to generate synthetic data. El-UNet is faster and more accurate than the fully-connected physics-informed implementation in resolving the distribution of unknown fields. Among the tested models, the self-adaptive spatially weighted models had the most accurate reconstructions in equal computation times. The learned spatial weighting distribution visibly corresponded to regions that the unweighted models were resolving inaccurately. Our work demonstrates a computationally efficient inversion algorithm for elasticity imaging using convolutional neural networks and presents a potential fast framework for three-dimensional inverse elasticity problems that have proven unachievable through previously proposed methods.

Mechanical characterization of human skin-A non-invasive digital twin approach using vibration-response integrated with numerical methods

This paper proposes an innovative approach to identify elastic material properties and mass density of soft tissues based on interpreting their mechanical vibration response, externally excited by a mechanical indenter or acoustic waves. A vibration test is performed on soft sheets to measure their response to a continuous range of excitation frequencies. The frequency responses are collected with a pair of high-speed cameras in conjunction with 3-D digital image correlation (DIC). Two cases are considered, including suspended/fully-free rectangular neoprene sheets as artificial tissue cutout samples and continuous layered human skin vibrations. An efficient theoretical model is developed to analytically simulate the free vibrations of the neoprene artificial sheet samples as well as the continuous layered human skins. The high accuracy and validity of the presented analytical simulations are demonstrated through comparison with the DIC measurements and the conducted frequency tests, as well as a number of finite element (FE) modeling. The developed analytical approach is implemented into a numerical algorithm to perform an inverse calculation of the soft sheets' elastic properties using the imported experimental vibration results and the predicted system's mass via the system equivalent reduction/expansion process (SEREP) method. It is shown that the proposed frequency-dependent inverse approach is capable of rapidly predicting the material properties of the tested samples with high accuracy.

Physics-Informed Neural Networks for Tissue Elasticity Reconstruction in Magnetic Resonance Elastography

Magnetic resonance elastography (MRE) is a medical imaging modality that non-invasively quantifies tissue stiffness (elasticity) and is commonly used for diagnosing liver fibrosis. Constructing an elasticity map of tissue requires solving an inverse problem involving a partial differential equation (PDE). Current numerical techniques to solve the inverse problem are noise-sensitive and require explicit specification of physical relationships. In this work, we apply physics-informed neural networks to solve the inverse problem of tissue elasticity reconstruction. Our method does not rely on numerical differentiation and can be extended to learn relevant correlations from anatomical images while respecting physical constraints. We evaluate our approach on simulated data and in vivo data from a cohort of patients with non-alcoholic fatty liver disease (NAFLD). Compared to numerical baselines, our method is more robust to noise and more accurate on realistic data, and its performance is further enhanced by incorporating anatomical information.

Physics-Informed Deep-Learning For Elasticity: Forward, Inverse, and Mixed Problems

Elastography is a medical imaging technique used to measure the elasticity of tissues by comparing ultrasound signals before and after a light compression. The lateral resolution of ultrasound is much inferior to the axial resolution. Current elastography methods generally require both axial and lateral displacement components, making them less effective for clinical applications. Additionally, these methods often rely on the assumption of material incompressibility, which can lead to inaccurate elasticity reconstruction as no materials are truly incompressible. To address these challenges, a new physics-informed deep-learning method for elastography is proposed. This new method integrates a displacement network and an elasticity network to reconstruct the Young's modulus field of a heterogeneous object based on only a measured axial displacement field. It also allows for the removal of the assumption of material incompressibility, enabling the reconstruction of both Young's modulus and Poisson's ratio fields simultaneously. The authors demonstrate that using multiple measurements can mitigate the potential error introduced by the "eggshell" effect, in which the presence of stiff material prevents the generation of strain in soft material. These improvements make this new method a valuable tool for a wide range of applications in medical imaging, materials characterization, and beyond.