Predicting left ventricular contractile function via Gaussian process emulation in aortic-banded rats

Whole-heart electromechanical simulations using Latent Neural Ordinary Differential Equations

Cardiac digital twins provide a physics and physiology informed framework to deliver personalized medicine. However, high-fidelity multi-scale cardiac models remain a barrier to adoption due to their extensive computational costs. Artificial Intelligence-based methods can make the creation of fast and accurate whole-heart digital twins feasible. We use Latent Neural Ordinary Differential Equations (LNODEs) to learn the pressure-volume dynamics of a heart failure patient. Our surrogate model is trained from 400 simulations while accounting for 43 parameters describing cell-to-organ cardiac electromechanics and cardiovascular hemodynamics. LNODEs provide a compact representation of the 3D-0D model in a latent space by means of an Artificial Neural Network that retains only 3 hidden layers with 13 neurons per layer and allows for numerical simulations of cardiac function on a single processor. We employ LNODEs to perform global sensitivity analysis and parameter estimation with uncertainty quantification in 3 hours of computations, still on a single processor.

Physics-informed neural networks as surrogate models of hydrodynamic simulators

In response to growing concerns surrounding the relationship between climate change and escalating flood risk, there is an increasing urgency to develop precise and rapid flood prediction models. Although high-resolution flood simulations have made notable advancements, they remain computationally expensive, underscoring the need for efficient machine learning surrogate models. As a result of sparse empirical observation and expensive data collection, there is a growing need for the models to perform effectively in 'small-data' contexts, a characteristic typical of many scientific problems. This research combines the latest developments in surrogate modelling and physics-informed machine learning to propose a novel Physics-Informed Neural Network-based surrogate model for hydrodynamic simulators governed by Shallow Water Equations. The proposed method incorporates physics-based prior information into the neural network structure by encoding the conservation of mass into the model without relying on calculating continuous derivatives in the loss function. The method is demonstrated for a high-resolution inland flood simulation model and a large-scale regional tidal model. The proposed method outperforms the existing state-of-the-art data-driven approaches by up to 25 %. This research demonstrates the benefits and robustness of physics-informed approaches in surrogate modelling for flood and hydroclimatic modelling problems.

Developing Cardiac Digital Twins at Scale: Insights from Personalised Myocardial Conduction Velocity

Large-cohort studies using cardiovascular imaging and diagnostic datasets have assessed cardiac anatomy, function, and outcomes, but typically do not reveal underlying biological mechanisms. Cardiac digital twins (CDTs) provide personalized physics- and physiology-constrained in-silico representations, enabling inference of multi-scale properties tied to these mechanisms. We constructed 3464 anatomically-accurate CDTs using cardiac magnetic resonance images from UK biobank and personalised their myocardial conduction velocities (CVs) from electrocardiograms (ECG), through an automated framework. We found well-known sex-specific differences in QRS duration were fully explained by myocardial anatomy, as CV remained consistent across sexes. Conversely, significant associations of CV with ageing and increased BMI suggest myocardial tissue remodelling. Novel associations were observed with left ventricular ejection fraction and mental-health phenotypes, through a phenome-wide association study, and CV was also linked with adverse clinical outcomes. Our study highlights the utility of population-based CDTs in assessing intersubject variability and uncovering strong links with mental health.

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.

Digital twinning of cardiac electrophysiology for congenital heart disease

In recent years, blending mechanistic knowledge with machine learning has had a major impact in digital healthcare. In this work, we introduce a computational pipeline to build certified digital replicas of cardiac electrophysiology in pediatric patients with congenital heart disease. We construct the patient-specific geometry by means of semi-automatic segmentation and meshing tools. We generate a dataset of electrophysiology simulations covering cell-to-organ level model parameters and utilizing rigorous mathematical models based on differential equations. We previously proposed Branched Latent Neural Maps (BLNMs) as an accurate and efficient means to recapitulate complex physical processes in a neural network. Here, we employ BLNMs to encode the parametrized temporal dynamics of in silico 12-lead electrocardiograms (ECGs). BLNMs act as a geometry-specific surrogate model of cardiac function for fast and robust parameter estimation to match clinical ECGs in pediatric patients. Identifiability and trustworthiness of calibrated model parameters are assessed by sensitivity analysis and uncertainty quantification.

Cell to whole organ global sensitivity analysis on a four-chamber heart electromechanics model using Gaussian processes emulators

Cardiac pump function arises from a series of highly orchestrated events across multiple scales. Computational electromechanics can encode these events in physics-constrained models. However, the large number of parameters in these models has made the systematic study of the link between cellular, tissue, and organ scale parameters to whole heart physiology challenging. A patient-specific anatomical heart model, or digital twin, was created. Cellular ionic dynamics and contraction were simulated with the Courtemanche-Land and the ToR-ORd-Land models for the atria and the ventricles, respectively. Whole heart contraction was coupled with the circulatory system, simulated with CircAdapt, while accounting for the effect of the pericardium on cardiac motion. The four-chamber electromechanics framework resulted in 117 parameters of interest. The model was broken into five hierarchical sub-models: tissue electrophysiology, ToR-ORd-Land model, Courtemanche-Land model, passive mechanics and CircAdapt. For each sub-model, we trained Gaussian processes emulators (GPEs) that were then used to perform a global sensitivity analysis (GSA) to retain parameters explaining 90% of the total sensitivity for subsequent analysis. We identified 45 out of 117 parameters that were important for whole heart function. We performed a GSA over these 45 parameters and identified the systemic and pulmonary peripheral resistance as being critical parameters for a wide range of volumetric and hemodynamic cardiac indexes across all four chambers. We have shown that GPEs provide a robust method for mapping between cellular properties and clinical measurements. This could be applied to identify parameters that can be calibrated in patient-specific models or digital twins, and to link cellular function to clinical indexes.

Fast and robust parameter estimation with uncertainty quantification for the cardiac function

BACKGROUND AND OBJECTIVES

Parameter estimation and uncertainty quantification are crucial in computational cardiology, as they enable the construction of digital twins that faithfully replicate the behavior of physical patients. Many model parameters regarding cardiac electromechanics and cardiovascular hemodynamics need to be robustly fitted by starting from a few, possibly non-invasive, noisy observations. Moreover, short execution times and a small amount of computational resources are required for the effective clinical translation.

METHODS

In the framework of Bayesian statistics, we combine Maximum a Posteriori estimation and Hamiltonian Monte Carlo to find an approximation of model parameters and their posterior distributions. Fast simulations and minimal memory requirements are achieved by using an accurate and geometry-specific Artificial Neural Network surrogate model for the cardiac function, matrix-free methods, automatic differentiation and automatic vectorization. Furthermore, we account for the surrogate modeling error and measurement error.

RESULTS

We perform three different in silico test cases, ranging from the ventricular function to the entire cardiocirculatory system, involving whole-heart mechanics, arterial and venous hemodynamics. By employing a single central processing unit on a standard laptop, we attain highly accurate estimations for all model parameters in short computational times. Furthermore, we obtain posterior distributions that contain the true values inside the 90% credibility regions.

CONCLUSIONS

Many model parameters regarding the entire cardiovascular system can be fastly and robustly identified with minimal hardware requirements. This can be achieved when a small amount of non-invasive data is available and when high levels of signal-to-noise ratio are present in the quantities of interest. With these features, our approach meets the requirements for clinical exploitation, while being compliant with Green Computing practices.

Gaussian process emulation of spatio-temporal outputs of a 2D inland flood model

The computational limitations of complex numerical models have led to adoption of statistical emulators across a variety of problems in science and engineering disciplines to circumvent the high computational costs associated with numerical simulations. In flood modelling, many hydraulic and hydrodynamic numerical models, especially when operating at high spatiotemporal resolutions, have prohibitively high computational costs for tasks requiring the instantaneous generation of very large numbers of simulation results. This study examines the appropriateness and robustness of Gaussian Process (GP) models to emulate the results from a hydraulic inundation model. The developed GPs produce real-time predictions based on the simulation output from LISFLOOD-FP numerical model. An efficient dimensionality reduction scheme is developed to tackle the high dimensionality of the output space and is combined with the GPs to investigate the predictive performance of the proposed emulator for estimation of the inundation depth. The developed GP-based framework is capable of robust and straightforward quantification of the uncertainty associated with the predictions, without requiring additional model evaluations and simulations. Further, this study explores the computational advantages of using a GP-based emulator over alternative methodologies such as neural networks, by undertaking a comparative analysis. For the case study data presented in this paper, the GP model was found to accurately reproduce water depths and inundation extent by classification and produce computational speedups of approximately 10,000 times compared with the original simulator, and 80 times for a neural network-based emulator.

Global Sensitivity Analysis of Four Chamber Heart Hemodynamics Using Surrogate Models

Computational Fluid Dynamics (CFD) is used to assist in designing artificial valves and planning procedures, focusing on local flow features. However, assessing the impact on overall cardiovascular function or predicting longer-term outcomes may requires more comprehensive whole heart CFD models. Fitting such models to patient data requires numerous computationally expensive simulations, and depends on specific clinical measurements to constrain model parameters, hampering clinical adoption. Surrogate models can help to accelerate the fitting process while accounting for the added uncertainty. We create a validated patient-specific four-chamber heart CFD model based on the Navier-Stokes-Brinkman (NSB) equations and test Gaussian Process Emulators (GPEs) as a surrogate model for performing a variance-based global sensitivity analysis (GSA). GSA identified preload as the dominant driver of flow in both the right and left side of the heart, respectively. Left-right differences were seen in terms of vascular outflow resistances, with pulmonary artery resistance having a much larger impact on flow than aortic resistance. Our results suggest that GPEs can be used to identify parameters in personalized whole heart CFD models, and highlight the importance of accurate preload measurements.

Quantitative mapping of force-pCa curves to whole heart contraction and relaxation

Abstract

The force–pCa (F–pCa) curve is used to characterize steady‐state contractile properties of cardiac muscle cells in different physiological, pathological and pharmacological conditions. This provides a reduced preparation in which to isolate sarcomere mechanisms. However, it is unclear how changes in the F–pCa curve impact emergent whole‐heart mechanics quantitatively. We study the link between sarcomere and whole‐heart function using a multiscale mathematical model of rat biventricular mechanics that describes sarcomere, tissue, anatomy, preload and afterload properties quantitatively. We first map individual cell‐level changes in sarcomere‐regulating parameters to organ‐level changes in the left ventricular function described by pressure–volume loop characteristics (e.g. end‐diastolic and end‐systolic volumes, ejection fraction and isovolumetric relaxation time). We next map changes in the sarcomere‐regulating parameters to changes in the F–pCa curve. We demonstrate that a change in the F–pCa curve can be caused by multiple different changes in sarcomere properties. We demonstrate that changes in sarcomere properties cause non‐linear and, importantly, non‐monotonic changes in left ventricular function. As a result, a change in sarcomere properties yielding changes in the F–pCa curve that improve contractility does not guarantee an improvement in whole‐heart function. Likewise, a desired change in whole‐heart function (i.e. ejection fraction or relaxation time) is not caused by a unique shift in the F–pCa curve. Changes in the F–pCa curve alone cannot be used to predict the impact of a compound on whole‐heart function.

Key points

The force–pCa (F–pCa) curve is used to assess myofilament calcium sensitivity after pharmacological modulation and to infer pharmacological effects on whole‐heart function.

We demonstrate that there is a non‐unique mapping from changes in F–pCa curves to changes in left ventricular (LV) function.

The effect of changes in F–pCa on LV function depend on the state of the heart and could be different for different pathological conditions.

Screening of compounds to impact whole‐heart function by F–pCa should be combined with active tension and calcium transient measurements to predict better how changes in muscle function will impact whole‐heart physiology.

An Integrated Workflow for Building Digital Twins of Cardiac Electromechanics—A Multi-Fidelity Approach for Personalising Active Mechanics

Personalised computer models of cardiac function, referred to as cardiac digital twins, are envisioned to play an important role in clinical precision therapies of cardiovascular diseases. A major obstacle hampering clinical translation involves the significant computational costs involved in the personalisation of biophysically detailed mechanistic models that require the identification of high-dimensional parameter vectors. An important aspect to identify in electromechanics (EM) models are active mechanics parameters that govern cardiac contraction and relaxation. In this study, we present a novel, fully automated, and efficient approach for personalising biophysically detailed active mechanics models using a two-step multi-fidelity solution. In the first step, active mechanical behaviour in a given 3D EM model is represented by a purely phenomenological, low-fidelity model, which is personalised at the organ scale by calibration to clinical cavity pressure data. Then, in the second step, median traces of nodal cellular active stress, intracellular calcium concentration, and fibre stretch are generated and utilised to personalise the desired high-fidelity model at the cellular scale using a 0D model of cardiac EM. Our novel approach was tested on a cohort of seven human left ventricular (LV) EM models, created from patients treated for aortic coarctation (CoA). Goodness of fit, computational cost, and robustness of the algorithm against uncertainty in the clinical data and variations of initial guesses were evaluated. We demonstrate that our multi-fidelity approach facilitates the personalisation of a biophysically detailed active stress model within only a few (2 to 4) expensive 3D organ-scale simulations—a computational effort compatible with clinical model applications.

Functional and structural differences between skinned and intact muscle preparations

Myofilaments and their associated proteins, which together constitute the sarcomeres, provide the molecular-level basis for contractile function in all muscle types. In intact muscle, sarcomere-level contraction is strongly coupled to other cellular subsystems, in particular the sarcolemmal membrane. Skinned muscle preparations (where the sarcolemma has been removed or permeabilized) are an experimental system designed to probe contractile mechanisms independently of the sarcolemma. Over the last few decades, experiments performed using permeabilized preparations have been invaluable for clarifying the understanding of contractile mechanisms in both skeletal and cardiac muscle. Today, the technique is increasingly harnessed for preclinical and/or pharmacological studies that seek to understand how interventions will impact intact muscle contraction. In this context, intrinsic functional and structural differences between skinned and intact muscle pose a major interpretational challenge. This review first surveys measurements that highlight these differences in terms of the sarcomere structure, passive and active tension generation, and calcium dependence. We then highlight the main practical challenges and caveats faced by experimentalists seeking to emulate the physiological conditions of intact muscle. Gaining an awareness of these complexities is essential for putting experiments in due perspective.

In silico identification of potential calcium dynamics and sarcomere targets for recovering left ventricular function in rat heart failure with preserved ejection fraction

Heart failure with preserved ejection fraction (HFpEF) is a complex disease associated with multiple co-morbidities, where impaired cardiac mechanics are often the end effect. At the cellular level, cardiac mechanics can be pharmacologically manipulated by altering calcium signalling and the sarcomere. However, the link between cellular level modulations and whole organ pump function is incompletely understood. Our goal is to develop and use a multi-scale computational cardiac mechanics model of the obese ZSF1 HFpEF rat to identify important biomechanical mechanisms that underpin impaired cardiac function and to predict how whole-heart mechanical function can be recovered through altering cellular calcium dynamics and/or cellular contraction. The rat heart was modelled using a 3D biventricular biomechanics model. Biomechanics were described by 16 parameters, corresponding to intracellular calcium transient, sarcomere dynamics, cardiac tissue and hemodynamics properties. The model simulated left ventricular (LV) pressure-volume loops that were described by 14 scalar features. We trained a Gaussian process emulator to map the 16 input parameters to each of the 14 outputs. A global sensitivity analysis was performed, and identified calcium dynamics and thin and thick filament kinetics as key determinants of the organ scale pump function. We employed Bayesian history matching to build a model of the ZSF1 rat heart. Next, we recovered the LV function, described by ejection fraction, peak pressure, maximum rate of pressure rise and isovolumetric relaxation time constant. We found that by manipulating calcium, thin and thick filament properties we can recover 34%, 28% and 24% of the LV function in the ZSF1 rat heart, respectively, and 39% if we manipulate all of them together. We demonstrated how a combination of biophysically based models and their derived emulators can be used to identify potential pharmacological targets. We predicted that cardiac function can be best recovered in ZSF1 rats by desensitising the myofilament and reducing the affinity to intracellular calcium concentration and overall prolonging the sarcomere staying in the active force generating state.

We developed a computational model of the ZSF1 rat model of heart failure with preserved ejection fraction. We validated that the model can link simulated pharmacological interventions from cellular to whole heart pump function. Our computational model identified calcium dynamics as the main determinant of left ventricular contractile behaviour. We demonstrated that the highest degree of LV function recovery could be achieved when calcium dynamics is manipulated in conjunction with both thin and thick filament kinetics.

Fast Posterior Estimation of Cardiac Electrophysiological Model Parameters via Bayesian Active Learning

Probabilistic estimation of cardiac electrophysiological model parameters serves an important step toward model personalization and uncertain quantification. The expensive computation associated with these model simulations, however, makes direct Markov Chain Monte Carlo (MCMC) sampling of the posterior probability density function (pdf) of model parameters computationally intensive. Approximated posterior pdfs resulting from replacing the simulation model with a computationally efficient surrogate, on the other hand, have seen limited accuracy. In this study, we present a Bayesian active learning method to directly approximate the posterior pdf function of cardiac model parameters, in which we intelligently select training points to query the simulation model in order to learn the posterior pdf using a small number of samples. We integrate a generative model into Bayesian active learning to allow approximating posterior pdf of high-dimensional model parameters at the resolution of the cardiac mesh. We further introduce new acquisition functions to focus the selection of training points on better approximating the shape rather than the modes of the posterior pdf of interest. We evaluated the presented method in estimating tissue excitability in a 3D cardiac electrophysiological model in a range of synthetic and real-data experiments. We demonstrated its improved accuracy in approximating the posterior pdf compared to Bayesian active learning using regular acquisition functions, and substantially reduced computational cost in comparison to existing standard or accelerated MCMC sampling.

Efficient Ventricular Parameter Estimation Using AI-Surrogate Models

The onset and progression of pathological heart conditions, such as cardiomyopathy or heart failure, affect its mechanical behaviour due to the remodelling of the myocardial tissues to preserve its functional response. Identification of the constitutive properties of heart tissues could provide useful biomarkers to diagnose and assess the progression of disease. We have previously demonstrated the utility of efficient AI-surrogate models to simulate passive cardiac mechanics. Here, we propose the use of this surrogate model for the identification of myocardial mechanical properties and intra-ventricular pressure by solving an inverse problem with two novel AI-based approaches. Our analysis concluded that: (i) both approaches were robust toward Gaussian noise when the ventricle data for multiple loading conditions were combined; and (ii) estimates of one and two parameters could be obtained in less than 9 and 18 s, respectively. The proposed technique yields a viable option for the translation of cardiac mechanics simulations and biophysical parameter identification methods into the clinic to improve the diagnosis and treatment of heart pathologies. In addition, the proposed estimation techniques are general and can be straightforwardly translated to other applications involving different anatomical structures.

Bayesian Calibration of Electrophysiology Models Using Restitution Curve Emulators

Calibration of cardiac electrophysiology models is a fundamental aspect of model personalization for predicting the outcomes of cardiac therapies, simulation testing of device performance for a range of phenotypes, and for fundamental research into cardiac function. Restitution curves provide information on tissue function and can be measured using clinically feasible measurement protocols. We introduce novel "restitution curve emulators" as probabilistic models for performing model exploration, sensitivity analysis, and Bayesian calibration to noisy data. These emulators are built by decomposing restitution curves using principal component analysis and modeling the resulting coordinates with respect to model parameters using Gaussian processes. Restitution curve emulators can be used to study parameter identifiability via sensitivity analysis of restitution curve components and rapid inference of the posterior distribution of model parameters given noisy measurements. Posterior uncertainty about parameters is critical for making predictions from calibrated models, since many parameter settings can be consistent with measured data and yet produce very different model behaviors under conditions not effectively probed by the measurement protocols. Restitution curve emulators are therefore promising probabilistic tools for calibrating electrophysiology models.

Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning

We present a new, computationally efficient framework to perform forward uncertainty quantification (UQ) in cardiac electrophysiology. We consider the monodomain model to describe the electrical activity in the cardiac tissue, coupled with the Aliev-Panfilov model to characterize the ionic activity through the cell membrane. We address a complete forward UQ pipeline, including both: (i) a variance-based global sensitivity analysis for the selection of the most relevant input parameters, and (ii) a way to perform uncertainty propagation to investigate the impact of intra-subject variability on outputs of interest depending on the cardiac potential. Both tasks exploit stochastic sampling techniques, thus implying overwhelming computational costs because of the huge amount of queries to the high-fidelity, full-order computational model obtained by approximating the coupled monodomain/Aliev-Panfilov system through the finite element method. To mitigate this computational burden, we replace the full-order model with computationally inexpensive projection-based reduced-order models (ROMs) aimed at reducing the state-space dimensionality. Resulting approximation errors on the outputs of interest are finally taken into account through artificial neural network (ANN)-based models, enhancing the accuracy of the whole UQ pipeline. Numerical results show that the proposed physics-based ROMs outperform regression-based emulators relying on ANNs built with the same amount of training data, in terms of both numerical accuracy and overall computational efficiency.

Linking statistical shape models and simulated function in the healthy adult human heart

Cardiac anatomy plays a crucial role in determining cardiac function. However, there is a poor understanding of how specific and localised anatomical changes affect different cardiac functional outputs. In this work, we test the hypothesis that in a statistical shape model (SSM), the modes that are most relevant for describing anatomy are also most important for determining the output of cardiac electromechanics simulations. We made patient-specific four-chamber heart meshes (n = 20) from cardiac CT images in asymptomatic subjects and created a SSM from 19 cases. Nine modes captured 90% of the anatomical variation in the SSM. Functional simulation outputs correlated best with modes 2, 3 and 9 on average (R = 0.49 ± 0.17, 0.37 ± 0.23 and 0.34 ± 0.17 respectively). We performed a global sensitivity analysis to identify the different modes responsible for different simulated electrical and mechanical measures of cardiac function. Modes 2 and 9 were the most important for determining simulated left ventricular mechanics and pressure-derived phenotypes. Mode 2 explained 28.56 ± 16.48% and 25.5 ± 20.85, and mode 9 explained 12.1 ± 8.74% and 13.54 ± 16.91% of the variances of mechanics and pressure-derived phenotypes, respectively. Electrophysiological biomarkers were explained by the interaction of 3 ± 1 modes. In the healthy adult human heart, shape modes that explain large portions of anatomical variance do not explain equivalent levels of electromechanical functional variation. As a result, in cardiac models, representing patient anatomy using a limited number of modes of anatomical variation can cause a loss in accuracy of simulated electromechanical function.

Surrogate models based on machine learning methods for parameter estimation of left ventricular myocardium

A long-standing problem at the frontier of biomechanical studies is to develop fast methods capable of estimating material properties from clinical data. In this paper, we have studied three surrogate models based on machine learning (ML) methods for fast parameter estimation of left ventricular (LV) myocardium. We use three ML methods named K-nearest neighbour (KNN), XGBoost and multi-layer perceptron (MLP) to emulate the relationships between pressure and volume strains during the diastolic filling. Firstly, to train the surrogate models, a forward finite-element simulator of LV diastolic filling is used. Then the training data are projected in a low-dimensional parametrized space. Next, three ML models are trained to learn the relationships of pressure-volume and pressure-strain. Finally, an inverse parameter estimation problem is formulated by using those trained surrogate models. Our results show that the three ML models can learn the relationships of pressure-volume and pressure-strain very well, and the parameter inference using the surrogate models can be carried out in minutes. Estimated parameters from both the XGBoost and MLP models have much less uncertainties compared with the KNN model. Our results further suggest that the XGBoost model is better for predicting the LV diastolic dynamics and estimating passive parameters than other two surrogate models. Further studies are warranted to investigate how XGBoost can be used for emulating cardiac pump function in a multi-physics and multi-scale framework.

Deep learning-based reduced order models in cardiac electrophysiology

Predicting the electrical behavior of the heart, from the cellular scale to the tissue level, relies on the numerical approximation of coupled nonlinear dynamical systems. These systems describe the cardiac action potential, that is the polarization/depolarization cycle occurring at every heart beat that models the time evolution of the electrical potential across the cell membrane, as well as a set of ionic variables. Multiple solutions of these systems, corresponding to different model inputs, are required to evaluate outputs of clinical interest, such as activation maps and action potential duration. More importantly, these models feature coherent structures that propagate over time, such as wavefronts. These systems can hardly be reduced to lower dimensional problems by conventional reduced order models (ROMs) such as, e.g., the reduced basis method. This is primarily due to the low regularity of the solution manifold (with respect to the problem parameters), as well as to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To overcome this difficulty, in this paper we propose a new, nonlinear approach relying on deep learning (DL) algorithms—such as deep feedforward neural networks and convolutional autoencoders—to obtain accurate and efficient ROMs, whose dimensionality matches the number of system parameters. We show that the proposed DL-ROM framework can efficiently provide solutions to parametrized electrophysiology problems, thus enabling multi-scenario analysis in pathological cases. We investigate four challenging test cases in cardiac electrophysiology, thus demonstrating that DL-ROM outperforms classical projection-based ROMs.