Deep learning-based reduced order models in cardiac electrophysiology

Data-driven reduced order surrogate modeling for coronary in-stent restenosis

BACKGROUND

The intricate process of coronary in-stent restenosis (ISR) involves the interplay between different mediators, including platelet-derived growth factor, transforming growth factor-β, extracellular matrix, smooth muscle cells, endothelial cells, and drug elution from the stent. Modeling such complex multiphysics phenomena demands extensive computational resources and time.

METHODS

This paper proposes a novel non-intrusive data-driven reduced order modeling approach for the underlying multiphysics time-dependent parametrized problem. In the offline phase, a 3D convolutional autoencoder, comprising an encoder and decoder, is trained to achieve dimensionality reduction. The encoder condenses the full-order solution into a lower-dimensional latent space, while the decoder facilitates the reconstruction of the full solution from the latent space. To deal with the 5D input datasets (3D geometry + time series + multiple output channels), two ingredients are explored. The first approach incorporates time as an additional parameter and applies 3D convolution on individual time steps, encoding a distinct latent variable for each parameter instance within each time step. The second approach reshapes the 3D geometry into a 2D plane along a less interactive axis and stacks all time steps in the third direction for each parameter instance. This rearrangement generates a larger and complete dataset for one parameter instance, resulting in a singular latent variable across the entire discrete time-series. In both approaches, the multiple outputs are considered automatically in the convolutions. Moreover, Gaussian process regression is applied to establish correlations between the latent variable and the input parameter.

RESULTS

The constitutive model reveals a significant acceleration in neointimal growth between 30-60 days post percutaneous coronary intervention (PCI). The surrogate models applying both approaches exhibit high accuracy in pointwise error, with the first approach showcasing smaller errors across the entire evaluation period for all outputs. The parameter study on drug dosage against ISR rates provides noteworthy insights of neointimal growth, where the nonlinear dependence of ISR rates on the peak drug flux exhibits intriguing periodic patterns. Applying the trained model, the rate of ISR is effectively evaluated, and the optimal parameter range for drug dosage is identified.

CONCLUSION

The demonstrated non-intrusive reduced order surrogate model proves to be a powerful tool for predicting ISR outcomes. Moreover, the proposed method lays the foundation for real-time simulations and optimization of PCI parameters.

Branched Latent Neural Maps

We introduce Branched Latent Neural Maps (BLNMs) to learn finite dimensional input-output maps encoding complex physical processes. A BLNM is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNMs leverage latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent in-distribution generalization properties with small training datasets and short training times on a single processor. Indeed, their in-distribution generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections, in place of a fully-connected structure, significantly reduce the number of tunable parameters. We show the capabilities of BLNMs in a challenging test case involving biophysically detailed electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a 1D Purkinje network for fast conduction and a 3D heart-torso geometry. Specifically, we trained BLNMs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale, organ-level and electrical dyssynchrony. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNM, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The resulting mean square error is on the order of 10 - 4 on an independent test dataset comprised of 50 additional electrophysiology simulations. In the online phase, the BLNM allows for 5000x faster real-time simulations of cardiac electrophysiology on a single core standard computer and can be employed to solve inverse problems via global optimization in a few seconds of computational time. This paper provides a novel computational tool to build reliable and efficient reduced-order models for digital twinning in engineering applications. The Julia implementation is publicly available under MIT License at https://github.com/StanfordCBCL/BLNM.jl.

Efficient approximation of cardiac mechanics through reduced-order modeling with deep learning-based operator approximation

Reducing the computational time required by high-fidelity, full-order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio-temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high-performance computing architectures. In addition, cardiac models usually depend on a set of input parameters that are calibrated in order to explore multiple virtual scenarios. To compute reliable solutions at a greatly reduced computational cost, we rely on a reduced basis method empowered with a new deep learning-based operator approximation, which we refer to as Deep-HyROMnet technique. Our strategy combines a projection-based POD-Galerkin method with deep neural networks for the approximation of (reduced) nonlinear operators, overcoming the typical computational bottleneck associated with standard hyper-reduction techniques employed in reduced-order models (ROMs) for nonlinear parametrized systems. This method can provide extremely accurate approximations to parametrized cardiac mechanics problems, such as in the case of the complete cardiac cycle in a patient-specific left ventricle geometry. In this respect, a 3D model for tissue mechanics is coupled with a 0D model for external blood circulation; active force generation is provided through an adjustable parameter-dependent surrogate model as input to the tissue 3D model. The proposed strategy is shown to outperform classical projection-based ROMs, in terms of orders of magnitude of computational speed-up, and to return accurate pressure-volume loops in both physiological and pathological cases. Finally, an application to a forward uncertainty quantification analysis, unaffordable if relying on a FOM, is considered, involving output quantities of interest such as, for example, the ejection fraction or the maximal rate of change in pressure in the left ventricle.

Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches

Micro-electro-mechanical-systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from full-order representations, we apply deep learning techniques to generate accurate, efficient, and real-time reduced order models to be used for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches, and gyroscopes, as well as displaying intricate dynamical evolutions such as internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows the extraction of the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.

Meshless Electrophysiological Modeling of Cardiac Resynchronization Therapy—Benchmark Analysis with Finite-Element Methods in Experimental Data

Computational models of cardiac electrophysiology are promising tools for reducing the rates of non-response patients suitable for cardiac resynchronization therapy (CRT) by optimizing electrode placement. The majority of computational models in the literature are mesh-based, primarily using the finite element method (FEM). The generation of patient-specific cardiac meshes has traditionally been a tedious task requiring manual intervention and hindering the modeling of a large number of cases. Meshless models can be a valid alternative due to their mesh quality independence. The organization of challenges such as the CRT-EPiggy19, providing unique experimental data as open access, enables benchmarking analysis of different cardiac computational modeling solutions with quantitative metrics. We present a benchmark analysis of a meshless-based method with finite-element methods for the prediction of cardiac electrical patterns in CRT, based on a subset of the CRT-EPiggy19 dataset. A data assimilation strategy was designed to personalize the most relevant parameters of the electrophysiological simulations and identify the optimal CRT lead configuration. The simulation results obtained with the meshless model were equivalent to FEM, with the most relevant aspect for accurate CRT predictions being the parameter personalization strategy (e.g., regional conduction velocity distribution, including the Purkinje system and CRT lead distribution).

A Review of Healthy and Fibrotic Myocardium Microstructure Modeling and Corresponding Intracardiac Electrograms

Computational simulations of cardiac electrophysiology provide detailed information on the depolarization phenomena at different spatial and temporal scales. With the development of new hardware and software, in silico experiments have gained more importance in cardiac electrophysiology research. For plane waves in healthy tissue, in vivo and in silico electrograms at the surface of the tissue demonstrate symmetric morphology and high peak-to-peak amplitude. Simulations provided insight into the factors that alter the morphology and amplitude of the electrograms. The situation is more complex in remodeled tissue with fibrotic infiltrations. Clinically, different changes including fractionation of the signal, extended duration and reduced amplitude have been described. In silico, numerous approaches have been proposed to represent the pathological changes on different spatial and functional scales. Different modeling approaches can reproduce distinct subsets of the clinically observed electrogram phenomena. This review provides an overview of how different modeling approaches to incorporate fibrotic and structural remodeling affect the electrogram and highlights open challenges to be addressed in future research.

Pacemaking function of two simplified cell models

Simplified nonlinear models of biological cells are widely used in computational electrophysiology. The models reproduce qualitatively many of the characteristics of various organs, such as the heart, brain, and intestine. In contrast to complex cellular ion-channel models, the simplified models usually contain a small number of variables and parameters, which facilitates nonlinear analysis and reduces computational load. In this paper, we consider pacemaking variants of the Aliev-Panfilov and Corrado two-variable excitable cell models. We conducted a numerical simulation study of these models and investigated the main nonlinear dynamic features of both isolated cells and 1D coupled pacemaker-excitable systems. Simulations of the 2D sinoatrial node and 3D intestine tissue as application examples of combined pacemaker-excitable systems demonstrated results similar to obtained previously. The uniform formulation for the conventional excitable cell models and proposed pacemaker models allows a convenient and easy implementation for the construction of personalized physiological models, inverse tissue modeling, and development of real-time simulation systems for various organs that contain both pacemaker and excitable cells.

Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification

Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.

EP-PINNs: Cardiac Electrophysiology Characterisation Using Physics-Informed Neural Networks

Accurately inferring underlying electrophysiological (EP) tissue properties from action potential recordings is expected to be clinically useful in the diagnosis and treatment of arrhythmias such as atrial fibrillation. It is, however, notoriously difficult to perform. We present EP-PINNs (Physics Informed Neural Networks), a novel tool for accurate action potential simulation and EP parameter estimation from sparse amounts of EP data. We demonstrate, using 1D and 2D in silico data, how EP-PINNs are able to reconstruct the spatio-temporal evolution of action potentials, whilst predicting parameters related to action potential duration (APD), excitability and diffusion coefficients. EP-PINNs are additionally able to identify heterogeneities in EP properties, making them potentially useful for the detection of fibrosis and other localised pathology linked to arrhythmias. Finally, we show EP-PINNs effectiveness on biological in vitro preparations, by characterising the effect of anti-arrhythmic drugs on APD using optical mapping data. EP-PINNs are a promising clinical tool for the characterisation and potential treatment guidance of arrhythmias.

POD-Enhanced Deep Learning-Based Reduced Order Models for the Real-Time Simulation of Cardiac Electrophysiology in the Left Atrium

The numerical simulation of multiple scenarios easily becomes computationally prohibitive for cardiac electrophysiology (EP) problems if relying on usual high-fidelity, full order models (FOMs). Likewise, the use of traditional reduced order models (ROMs) for parametrized PDEs to speed up the solution of the aforementioned problems can be problematic. This is primarily due to the strong variability characterizing the solution set and to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To enhance ROM efficiency, we proposed a new generation of non-intrusive, nonlinear ROMs, based on deep learning (DL) algorithms, such as convolutional, feedforward, and autoencoder neural networks. In the proposed DL-ROM, both the nonlinear solution manifold and the nonlinear reduced dynamics used to model the system evolution on that manifold can be learnt in a non-intrusive way thanks to DL algorithms trained on a set of FOM snapshots. DL-ROMs were shown to be able to accurately capture complex front propagation processes, both in physiological and pathological cardiac EP, very rapidly once neural networks were trained, however, at the expense of huge training costs. In this study, we show that performing a prior dimensionality reduction on FOM snapshots through randomized proper orthogonal decomposition (POD) enables to speed up training times and to decrease networks complexity. Accuracy and efficiency of this strategy, which we refer to as POD-DL-ROM, are assessed in the context of cardiac EP on an idealized left atrium (LA) geometry and considering snapshots arising from a NURBS (non-uniform rational B-splines)-based isogeometric analysis (IGA) discretization. Once the ROMs have been trained, POD-DL-ROMs can efficiently solve both physiological and pathological cardiac EP problems, for any new scenario, in real-time, even in extremely challenging contexts such as those featuring circuit re-entries, that are among the factors triggering cardiac arrhythmias.

Whole-heart ventricular arrhythmia modeling moving forward: Mechanistic insights and translational applications

Ventricular arrhythmias are the primary cause of sudden cardiac death and one of the leading causes of mortality worldwide. Whole-heart computational modeling offers a unique approach for studying ventricular arrhythmias, offering vast potential for developing both a mechanistic understanding of ventricular arrhythmias and clinical applications for treatment. In this review, the fundamentals of whole-heart ventricular modeling and current methods of personalizing models using clinical data are presented. From this foundation, the authors summarize recent advances in whole-heart ventricular arrhythmia modeling. Efforts in gaining mechanistic insights into ventricular arrhythmias are discussed, in addition to other applications of models such as the assessment of novel therapeutics. The review emphasizes the unique benefits of computational modeling that allow for insights that are not obtainable by contemporary experimental or clinical means. Additionally, the clinical impact of modeling is explored, demonstrating how patient care is influenced by the information gained from ventricular arrhythmia models. The authors conclude with future perspectives about the direction of whole-heart ventricular arrhythmia modeling, outlining how advances in neural network methodologies hold the potential to reduce computational expense and permit for efficient whole-heart modeling.

Real-Time Simulation of Parameter-Dependent Fluid Flows through Deep Learning-Based Reduced Order Models

Simulating fluid flows in different virtual scenarios is of key importance in engineering applications. However, high-fidelity, full-order models relying, e.g., on the finite element method, are unaffordable whenever fluid flows must be simulated in almost real-time. Reduced order models (ROMs) relying, e.g., on proper orthogonal decomposition (POD) provide reliable approximations to parameter-dependent fluid dynamics problems in rapid times. However, they might require expensive hyper-reduction strategies for handling parameterized nonlinear terms, and enriched reduced spaces (or Petrov–Galerkin projections) if a mixed velocity–pressure formulation is considered, possibly hampering the evaluation of reliable solutions in real-time. Dealing with fluid–structure interactions entails even greater difficulties. The proposed deep learning (DL)-based ROMs overcome all these limitations by learning, in a nonintrusive way, both the nonlinear trial manifold and the reduced dynamics. To do so, they rely on deep neural networks, after performing a former dimensionality reduction through POD, enhancing their training times substantially. The resulting POD-DL-ROMs are shown to provide accurate results in almost real-time for the flow around a cylinder benchmark, the fluid–structure interaction between an elastic beam attached to a fixed, rigid block and a laminar incompressible flow, and the blood flow in a cerebral aneurysm.

Polygonal surface processing and mesh generation tools for the numerical simulation of the cardiac function

In order to simulate the cardiac function for a patient-specific geometry, the generation of the computational mesh is crucially important. In practice, the input is typically a set of unprocessed polygonal surfaces coming either from a template geometry or from medical images. These surfaces need ad-hoc processing to be suitable for a volumetric mesh generation. In this work we propose a set of new algorithms and tools aiming to facilitate the mesh generation process. In particular, we focus on different aspects of a cardiac mesh generation pipeline: (1) specific polygonal surface processing for cardiac geometries, like connection of different heart chambers or segmentation outputs; (2) generation of accurate boundary tags; (3) definition of mesh-size functions dependent on relevant geometric quantities; (4) processing and connecting together several volumetric meshes. The new algorithms-implemented in the open-source software vmtk-can be combined with each other allowing the creation of personalized pipelines, that can be optimized for each cardiac geometry or for each aspect of the cardiac function to be modeled. Thanks to these features, the proposed tools can significantly speed-up the mesh generation process for a large range of cardiac applications, from single-chamber single-physics simulations to multi-chambers multi-physics simulations. We detail all the proposed algorithms motivating them in the cardiac context and we highlight their flexibility by showing different examples of cardiac mesh generation pipelines.

Data integration for the numerical simulation of cardiac electrophysiology

The increasing availability of extensive and accurate clinical data is rapidly shaping cardiovascular care by improving the understanding of physiological and pathological mechanisms of the cardiovascular system and opening new frontiers in designing therapies and interventions. In this direction, mathematical and numerical models provide a complementary relevant tool, able not only to reproduce patient-specific clinical indicators but also to predict and explore unseen scenarios. With this goal, clinical data are processed and provided as inputs to the mathematical model, which quantitatively describes the physical processes that occur in the cardiac tissue. In this paper, the process of integration of clinical data and mathematical models is discussed. Some challenges and contributions in the field of cardiac electrophysiology are reported.