Non-invasive Inference of Thrombus Material Properties with Physics-Informed Neural Networks

Super-resolution left ventricular flow and pressure mapping by Navier-Stokes-informed neural networks

Intraventricular vector flow mapping (VFM) is an increasingly adopted echocardiographic technique that derives time-resolved two-dimensional flow maps in the left ventricle (LV) from color-Doppler sequences. Current VFM models rely on kinematic constraints arising from planar flow incompressibility. However, these models are not informed by crucial information about flow physics; most notably the forces within the fluid and the resulting accelerations. This limitation has rendered VFM unable to combine information from different time frames in an acquisition sequence or derive fluctuating pressure maps. In this study, we leveraged recent advances in artificial intelligence (AI) to develop AI-VFM, a vector flow mapping modality that uses physics-informed neural networks (PINNs) encoding mass conservation and momentum balance inside the LV, and no-slip boundary conditions at the LV endocardium. AI-VFM recovers the flow and pressure fields in the LV from standard echocardiographic scans. It performs phase unwrapping and recovers flow data in areas without input color-Doppler data. AI-VFM also recovers complete flow maps at time points without color-Doppler input data, producing super-resolution flow maps. We validate AI-VFM using physiological simulated LV data and show that informing the PINNs with momentum balance is essential for achieving temporal super-resolution and significantly increases the accuracy of AI-VFM compared to informing the PINNs only with mass conservation. AI-VFM is solely informed by each patient's flow physics; it does not utilize explicit smoothness constraints or incorporate data from other patients or flow models. AI-VFM takes 15 min to run in off-the-shelf graphics processing units and its underlying PINN framework could be extended to map other flow-associated metrics such as blood residence time or the concentration of coagulation species.

SELFNet: Denoising Shear Wave Elastography Using Spatial-temporal Fourier Feature Networks

OBJECTIVE

Ultrasound-based shear wave elastography offers estimation of tissue stiffness through analysis of the propagation of a shear wave induced by a stimulus. Displacement or velocity fields during the process can contain noise as a result of the limited number of acquisitions. With advances in physics-informed deep learning, neural networks can approximate a physics field by minimizing the residuals of governing physics equations.

METHODS

In this research, we introduce a shear wave elastography Fourier feature network (SELFNet) using spatial-temporal random Fourier features within a physics-informed neural network framework to estimate and denoise particle displacement signals. The network uses a sparse mapping to increase robustness and incorporates the governing equations for regularization while simultaneously learning the mapping of the shear modulus. The method was evaluated in datasets from tissue-mimicking phantom of lesions and ex vivo tissue.

RESULTS

The findings indicate that SELFNet is capable of smoothing out the noise in phantom lesions with different stiffness and sizes, outperforming a reference Gaussian filtering method by 17% in relative ℓ2 error, 45% in reconstruction root-mean-square error. Furthermore, the ablation study suggested that SELFNet can prevent over-fitting through the Fourier feature mapping module. An ex vivo study confirmed its applicability to different types of tissue.

CONCLUSION

The implementation of SELFNet shows promise for shear wave elastography with limited acquisitions. In this context, subject to successful translation, it has the potential to be extended to clinical applications, such as the diagnosis of cancer or liver disease.

Rapid estimation of left ventricular contractility with a physics-informed neural network inverse modeling approach

Physics-based computer models based on numerical solutions of the governing equations generally cannot make rapid predictions, which in turn limits their applications in the clinic. To address this issue, we developed a physics-informed neural network (PINN) model that encodes the physics of a closed-loop blood circulation system embedding a left ventricle (LV). The PINN model is trained to satisfy a system of ordinary differential equations (ODEs) associated with a lumped parameter description of the circulatory system. The model predictions have a maximum error of less than 5% when compared to those obtained by solving the ODEs numerically. An inverse modeling approach using the PINN model is also developed to rapidly estimate model parameters (in ∼ 3 min) from single-beat LV pressure and volume waveforms. Using synthetic LV pressure and volume waveforms generated by the PINN model with different model parameter values, we show that the inverse modeling approach can recover the corresponding ground truth values for LV contractility indexed by the end-systolic elastance Ees with a 1% error, which suggests that this parameter is unique. The estimated Ees is about 58% to 284% higher for the data associated with dobutamine compared to those without, which implies that this approach can be used to estimate LV contractility using single-beat measurements. The PINN inverse modeling can potentially be used in the clinic to simultaneously estimate LV contractility and other physiological parameters from single-beat measurements.

Review of Machine Learning Techniques in Soft Tissue Biomechanics and Biomaterials

BACKGROUND AND OBJECTIVE

Advanced material models and material characterization of soft biological tissues play an essential role in pre-surgical planning for vascular surgeries and transcatheter interventions. Recent advances in heart valve engineering, medical device and patch design are built upon these models. Furthermore, understanding vascular growth and remodeling in native and tissue-engineered vascular biomaterials, as well as designing and testing drugs on soft tissue, are crucial aspects of predictive regenerative medicine. Traditional nonlinear optimization methods and finite element (FE) simulations have served as biomaterial characterization tools combined with soft tissue mechanics and tensile testing for decades. However, results obtained through nonlinear optimization methods are reliable only to a certain extent due to mathematical limitations, and FE simulations may require substantial computing time and resources, which might not be justified for patient-specific simulations. To a significant extent, machine learning (ML) techniques have gained increasing prominence in the field of soft tissue mechanics in recent years, offering notable advantages over conventional methods. This review article presents an in-depth examination of emerging ML algorithms utilized for estimating the mechanical characteristics of soft biological tissues and biomaterials. These algorithms are employed to analyze crucial properties such as stress-strain curves and pressure-volume loops. The focus of the review is on applications in cardiovascular engineering, and the fundamental mathematical basis of each approach is also discussed.

METHODS

The review effort employed two strategies. First, the recent studies of major research groups actively engaged in cardiovascular soft tissue mechanics are compiled, and research papers utilizing ML and deep learning (DL) techniques were included in our review. The second strategy involved a standard keyword search across major databases. This approach provided 11 relevant ML articles, meticulously selected from reputable sources including ScienceDirect, Springer, PubMed, and Google Scholar. The selection process involved using specific keywords such as "machine learning" or "deep learning" in conjunction with "soft biological tissues", "cardiovascular", "patient-specific," "strain energy", "vascular" or "biomaterials". Initially, a total of 25 articles were selected. However, 14 of these articles were excluded as they did not align with the criteria of focusing on biomaterials specifically employed for soft tissue repair and regeneration. As a result, the remaining 11 articles were categorized based on the ML techniques employed and the training data utilized.

RESULTS

ML techniques utilized for assessing the mechanical characteristics of soft biological tissues and biomaterials are broadly classified into two categories: standard ML algorithms and physics-informed ML algorithms. The standard ML models are then organized based on their tasks, being grouped into Regression and Classification subcategories. Within these categories, studies employ various supervised learning models, including support vector machines (SVMs), bagged decision trees (BDTs), artificial neural networks (ANNs) or deep neural networks (DNNs), and convolutional neural networks (CNNs). Additionally, the utilization of unsupervised learning approaches, such as autoencoders incorporating principal component analysis (PCA) and/or low-rank approximation (LRA), is based on the specific characteristics of the training data. The training data predominantly consists of three types: experimental mechanical data, including uniaxial or biaxial stress-strain data; synthetic mechanical data generated through non-linear fitting and/or FE simulations; and image data such as 3D second harmonic generation (SHG) images or computed tomography (CT) images. The evaluation of performance for physics-informed ML models primarily relies on the coefficient of determinationR 2 . In contrast, various metrics and error measures are utilized to assess the performance of standard ML models. Furthermore, our review includes an extensive examination of prevalent biomaterial models that can serve as physical laws for physics-informed ML models.

CONCLUSION

ML models offer an accurate, fast, and reliable approach for evaluating the mechanical characteristics of diseased soft tissue segments and selecting optimal biomaterials for time-critical soft tissue surgeries. Among the various ML models examined in this review, physics-informed neural network models exhibit the capability to forecast the mechanical response of soft biological tissues accurately, even with limited training samples. These models achieve highR 2 values ranging from 0.90 to 1.00. This is particularly significant considering the challenges associated with obtaining a large number of living tissue samples for experimental purposes, which can be time-consuming and impractical. Additionally, the review not only discusses the advantages identified in the current literature but also sheds light on the limitations and offers insights into future perspectives.

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.

Super-resolution Left Ventricular Flow and Pressure Mapping by Navier-Stokes-Informed Neural Networks

Intraventricular vector flow mapping (VFM) is a growingly adopted echocardiographic modality that derives time-resolved two-dimensional flow maps in the left ventricle (LV) from color-Doppler sequences. Current VFM models rely on kinematic constraints arising from planar flow incompressibility. However, these models are not informed by crucial information about flow physics; most notably the pressure and shear forces within the fluid and the resulting accelerations. This limitation has rendered VFM unable to combine information from different time frames in an acquisition sequence or derive fluctuating pressure maps. In this study, we leveraged recent advances in artificial intelligence (AI) to develop AI-VFM, a vector flow mapping modality that uses physics-informed neural networks (PINNs) encoding mass conservation and momentum balance inside the LV, and no-slip boundary conditions at the LV endocardium. AI-VFM recovers the flow and pressure fields in the LV from standard echocardiographic scans. It performs phase unwrapping and recovers flow data in areas without input color-Doppler data. AI-VFM also recovers complete flow maps at time points without color-Doppler input data, producing super-resolution flow maps. We show that informing the PINNs with momentum balance is essential to achieving temporal super-resolution and significantly increases the accuracy of AI-VFM compared to informing the PINNs only with mass conservation. AI-VFM is solely informed by each patient's flow physics; it does not utilize explicit smoothness constraints or incorporate data from other patients or flow models. AI-VFM takes 15 minutes to run in off-the-shelf graphics processing units and its underlying PINN framework could be extended to map other flow-associated metrics like blood residence time or the concentration of coagulation species.

Accelerated simulation methodologies for computational vascular flow modelling

Vascular flow modelling can improve our understanding of vascular pathologies and aid in developing safe and effective medical devices. Vascular flow models typically involve solving the nonlinear Navier-Stokes equations in complex anatomies and using physiological boundary conditions, often presenting a multi-physics and multi-scale computational problem to be solved. This leads to highly complex and expensive models that require excessive computational time. This review explores accelerated simulation methodologies, specifically focusing on computational vascular flow modelling. We review reduced order modelling (ROM) techniques like zero-/one-dimensional and modal decomposition-based ROMs and machine learning (ML) methods including ML-augmented ROMs, ML-based ROMs and physics-informed ML models. We discuss the applicability of each method to vascular flow acceleration and the effectiveness of the method in addressing domain-specific challenges. When available, we provide statistics on accuracy and speed-up factors for various applications related to vascular flow simulation acceleration. Our findings indicate that each type of model has strengths and limitations depending on the context. To accelerate real-world vascular flow problems, we propose future research on developing multi-scale acceleration methods capable of handling the significant geometric variability inherent to such problems.

MCNN-DIC: a mechanical constraints-based digital image correlation by a neural network approach

Digital image correlation (DIC) is a widely used photomechanical method for measuring surface deformation of materials. Practical engineering applications of DIC often encounter challenges such as discontinuous deformation fields, noise interference, and difficulties in measuring boundary deformations. To address these challenges, a new, to the best of our knowledge, DIC method called MCNN-DIC is proposed in this study by incorporating mechanical constraints using neural network technology. The proposed method applied compatibility equation constraints to the measured deformation field through a semi-supervised learning approach, thus making it more physical. The effectiveness of the proposed MCNN-DIC method was demonstrated through simulated experiments and real deformation fields of nuclear graphite material. The results show that the MCNN-DIC method achieves higher accuracy in measuring non-uniform deformation fields than a traditional mechanical constraints-based DIC and can rapidly measure deformation fields without requiring extensive pre-training of the neural network.

The Potential of Deep Learning to Advance Clinical Applications of Computational Biomechanics

When combined with patient information provided by advanced imaging techniques, computational biomechanics can provide detailed patient-specific information about stresses and strains acting on tissues that can be useful in diagnosing and assessing treatments for diseases and injuries. This approach is most advanced in cardiovascular applications but can be applied to other tissues. The challenges for advancing computational biomechanics for real-time patient diagnostics and treatment include errors and missing information in the patient data, the large computational requirements for the numerical solutions to multiscale biomechanical equations, and the uncertainty over boundary conditions and constitutive relations. This review summarizes current efforts to use deep learning to address these challenges and integrate large data sets and computational methods to enable real-time clinical information. Examples are drawn from cardiovascular fluid mechanics, soft-tissue mechanics, and bone biomechanics. The application of deep-learning convolutional neural networks can reduce the time taken to complete image segmentation, and meshing and solution of finite element models, as well as improving the accuracy of inlet and outlet conditions. Such advances are likely to facilitate the adoption of these models to aid in the assessment of the severity of cardiovascular disease and the development of new surgical treatments.

Machine Learning and the Conundrum of Stroke Risk Prediction

Stroke is a leading cause of death worldwide. With escalating healthcare costs, early non-invasive stroke risk stratification is vital. The current paradigm of stroke risk assessment and mitigation is focused on clinical risk factors and comorbidities. Standard algorithms predict risk using regression-based statistical associations, which, while useful and easy to use, have moderate predictive accuracy. This review summarises recent efforts to deploy machine learning (ML) to predict stroke risk and enrich the understanding of the mechanisms underlying stroke. The surveyed body of literature includes studies comparing ML algorithms with conventional statistical models for predicting cardiovascular disease and, in particular, different stroke subtypes. Another avenue of research explored is ML as a means of enriching multiscale computational modelling, which holds great promise for revealing thrombogenesis mechanisms. Overall, ML offers a new approach to stroke risk stratification that accounts for subtle physiologic variants between patients, potentially leading to more reliable and personalised predictions than standard regression-based statistical associations.

Recent developments in modeling, imaging, and monitoring of cardiovascular diseases using machine learning

Cardiovascular diseases are the leading cause of mortality, morbidity, and hospitalization around the world. Recent technological advances have facilitated analyzing, visualizing, and monitoring cardiovascular diseases using emerging computational fluid dynamics, blood flow imaging, and wearable sensing technologies. Yet, computational cost, limited spatiotemporal resolution, and obstacles for thorough data analysis have hindered the utility of such techniques to curb cardiovascular diseases. We herein discuss how leveraging machine learning techniques, and in particular deep learning methods, could overcome these limitations and offer promise for translation. We discuss the remarkable capacity of recently developed machine learning techniques to accelerate flow modeling, enhance the resolution while reduce the noise and scanning time of current blood flow imaging techniques, and accurate detection of cardiovascular diseases using a plethora of data collected by wearable sensors.

Imaging and biophysical modelling of thrombogenic mechanisms in atrial fibrillation and stroke

Atrial fibrillation (AF) underlies almost one third of all ischaemic strokes, with the left atrial appendage (LAA) identified as the primary thromboembolic source. Current stroke risk stratification approaches, such as the CHA₂DS₂-VASc score, rely mostly on clinical comorbidities, rather than thrombogenic mechanisms such as blood stasis, hypercoagulability and endothelial dysfunction-known as Virchow's triad. While detection of AF-related thrombi is possible using established cardiac imaging techniques, such as transoesophageal echocardiography, there is a growing need to reliably assess AF-patient thrombogenicity prior to thrombus formation. Over the past decade, cardiac imaging and image-based biophysical modelling have emerged as powerful tools for reproducing the mechanisms of thrombogenesis. Clinical imaging modalities such as cardiac computed tomography, magnetic resonance and echocardiographic techniques can measure blood flow velocities and identify LA fibrosis (an indicator of endothelial dysfunction), but imaging remains limited in its ability to assess blood coagulation dynamics. In-silico cardiac modelling tools-such as computational fluid dynamics for blood flow, reaction-diffusion-convection equations to mimic the coagulation cascade, and surrogate flow metrics associated with endothelial damage-have grown in prevalence and advanced mechanistic understanding of thrombogenesis. However, neither technique alone can fully elucidate thrombogenicity in AF. In future, combining cardiac imaging with in-silico modelling and integrating machine learning approaches for rapid results directly from imaging data will require development under a rigorous framework of verification and clinical validation, but may pave the way towards enhanced personalised stroke risk stratification in the growing population of AF patients. This Review will focus on the significant progress in these fields.

Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems

Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for time-dependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess the accuracy and efficiency, including static and time-dependent problems. We also demonstrate the feasibility of coupling of a continuum model (finite element methods, FEM) with a neural operator, serving as a surrogate of a particle system (Smoothed Particle Hydrodynamics, SPH), for predicting mechanical responses of anisotropic and hyperelastic materials. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.

G2Φnet: Relating genotype and biomechanical phenotype of tissues with deep learning

Many genetic mutations adversely affect the structure and function of load-bearing soft tissues, with clinical sequelae often responsible for disability or death. Parallel advances in genetics and histomechanical characterization provide significant insight into these conditions, but there remains a pressing need to integrate such information. We present a novel genotype-to-biomechanical phenotype neural network (G2Φnet) for characterizing and classifying biomechanical properties of soft tissues, which serve as important functional readouts of tissue health or disease. We illustrate the utility of our approach by inferring the nonlinear, genotype-dependent constitutive behavior of the aorta for four mouse models involving defects or deficiencies in extracellular constituents. We show that G2Φnet can infer the biomechanical response while simultaneously ascribing the associated genotype by utilizing limited, noisy, and unstructured experimental data. More broadly, G2Φnet provides a powerful method and a paradigm shift for correlating genotype and biomechanical phenotype quantitatively, promising a better understanding of their interplay in biological tissues.

Investigating molecular transport in the human brain from MRI with physics-informed neural networks

In recent years, a plethora of methods combining neural networks and partial differential equations have been developed. A widely known example are physics-informed neural networks, which solve problems involving partial differential equations by training a neural network. We apply physics-informed neural networks and the finite element method to estimate the diffusion coefficient governing the long term spread of molecules in the human brain from magnetic resonance images. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small for accurate parameter recovery. To achieve this, we tune the weights and the norms used in the loss function and use residual based adaptive refinement of training points. We find that the diffusion coefficient estimated from magnetic resonance images with physics-informed neural networks becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented here are an important first step towards solving inverse problems on cohorts of patients in a semi-automated fashion with physics-informed neural networks.

Deep Learning for Computational Hemodynamics: A Brief Review of Recent Advances

Computational fluid dynamics (CFD) modeling of blood flow plays an important role in better understanding various medical conditions, designing more effective drug delivery systems, and developing novel diagnostic methods and treatments. However, despite significant advances in computational technology and resources, the expensive computational cost of these simulations still hinders their transformation from a research interest to a clinical tool. This bottleneck is even more severe for image-based, patient-specific CFD simulations with realistic boundary conditions and complex computational domains, which make such simulations excessively expensive. To address this issue, deep learning approaches have been recently explored to accelerate computational hemodynamics simulations. In this study, we review recent efforts to integrate deep learning with CFD and discuss the applications of this approach in solving hemodynamics problems, such as blood flow behavior in aorta and cerebral arteries. We also discuss potential future directions in the field. In this review, we suggest that incorporating physiologic understandings and underlying fluid mechanics laws in deep learning models will soon lead to a paradigm shift in the development novel non-invasive computational medical decisions.

Analyses of internal structures and defects in materials using physics-informed neural networks

Characterizing internal structures and defects in materials is a challenging task, often requiring solutions to inverse problems with unknown topology, geometry, material properties, and nonlinear deformation. Here, we present a general framework based on physics-informed neural networks for identifying unknown geometric and material parameters. By using a mesh-free method, we parameterize the geometry of the material using a differentiable and trainable method that can identify multiple structural features. We validate this approach for materials with internal voids/inclusions using constitutive models that encompass the spectrum of linear elasticity, hyperelasticity, and plasticity. We predict the size, shape, and location of the internal void/inclusion as well as the elastic modulus of the inclusion. Our general framework can be applied to other inverse problems in different applications that involve unknown material properties and highly deformable geometries, targeting material characterization, quality assurance, and structural design.

Simulating progressive intramural damage leading to aortic dissection using DeepONet: an operator-regression neural network

Aortic dissection progresses mainly via delamination of the medial layer of the wall. Notwithstanding the complexity of this process, insight has been gleaned by studying in vitro and in silico the progression of dissection driven by quasi-static pressurization of the intramural space by fluid injection, which demonstrates that the differential propensity of dissection along the aorta can be affected by spatial distributions of structurally significant interlamellar struts that connect adjacent elastic lamellae. In particular, diverse histological microstructures may lead to differential mechanical behaviour during dissection, including the pressure-volume relationship of the injected fluid and the displacement field between adjacent lamellae. In this study, we develop a data-driven surrogate model of the delamination process for differential strut distributions using DeepONet, a new operator-regression neural network. This surrogate model is trained to predict the pressure-volume curve of the injected fluid and the damage progression within the wall given a spatial distribution of struts, with in silico data generated using a phase-field finite-element model. The results show that DeepONet can provide accurate predictions for diverse strut distributions, indicating that this composite branch-trunk neural network can effectively extract the underlying functional relationship between distinctive microstructures and their mechanical properties. More broadly, DeepONet can facilitate surrogate model-based analyses to quantify biological variability, improve inverse design and predict mechanical properties based on multi-modality experimental data.

Mesh-Free Surrogate Models for Structural Mechanic FEM Simulation: A Comparative Study of Approaches

The technical world of today fundamentally relies on structural analysis in the form of design and structural mechanic simulations. A traditional and robust simulation method is the physics-based finite element method (FEM) simulation. FEM simulations in structural mechanics are known to be very accurate; however, the higher the desired resolution, the more computational effort is required. Surrogate modeling provides a robust approach to address this drawback. Nonetheless, finding the right surrogate model and its hyperparameters for a specific use case is not a straightforward process. In this paper, we discuss and compare several classes of mesh-free surrogate models based on traditional and thriving machine learning (ML) and deep learning (DL) methods. We show that relatively simple algorithms (such as k-nearest neighbor regression) can be competitive in applications with low geometrical complexity and extrapolation requirements. With respect to tasks exhibiting higher geometric complexity, our results show that recent DL methods at the forefront of literature (such as physics-informed neural networks) are complicated to train and to parameterize and thus, require further research before they can be put to practical use. In contrast, we show that already well-researched DL methods, such as the multi-layer perceptron, are superior with respect to interpolation use cases and can be easily trained with available tools. With our work, we thus present a basis for the selection and practical implementation of surrogate models.