Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response

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.

Validating MRI-Derived Myocardial Stiffness Estimates Using In Vitro Synthetic Heart Models

Impaired cardiac filling in response to increased passive myocardial stiffness contributes to the pathophysiology of heart failure. By leveraging cardiac MRI data and ventricular pressure measurements, we can estimate in vivo passive myocardial stiffness using personalized inverse finite element models. While it is well-known that this approach is subject to uncertainties, only few studies quantify the accuracy of these stiffness estimates. This lack of validation is, at least in part, due to the absence of ground truth in vivo passive myocardial stiffness values. Here, using 3D printing, we created soft, homogenous, isotropic, hyperelastic heart phantoms of varying geometry and stiffness and simulate diastolic filling by incorporating the phantoms into an MRI-compatible left ventricular inflation system. We estimate phantom stiffness from MRI and pressure data using inverse finite element analyses based on a Neo-Hookean model. We demonstrate that our identified softest and stiffest values of 215.7 and 512.3 kPa agree well with the ground truth of 226.2 and 526.4 kPa. Overall, our estimated stiffnesses revealed a good agreement with the ground truth ([Formula: see text] error) across all models. Our results suggest that MRI-driven computational constitutive modeling can accurately estimate synthetic heart material stiffnesses in the range of 200-500 kPa.

Robust and efficient fixed-point algorithm for the inverse elastostatic problem to identify myocardial passive material parameters and the unloaded reference configuration

Image-based computational models of the heart represent a powerful tool to shed new light on the mechanisms underlying physiological and pathological conditions in cardiac function and to improve diagnosis and therapy planning. However, in order to enable the clinical translation of such models, it is crucial to develop personalized models that are able to reproduce the physiological reality of a given patient. There have been numerous contributions in experimental and computational biomechanics to characterize the passive behavior of the myocardium. However, most of these studies suffer from severe limitations and are not applicable to high-resolution geometries. In this work, we present a novel methodology to perform an automated identification of in vivo properties of passive cardiac biomechanics. The highly-efficient algorithm fits material parameters against the shape of a patient-specific approximation of the end-diastolic pressure-volume relation (EDPVR). Simultaneously, an unloaded reference configuration is generated, where a novel line search strategy to improve convergence and robustness is implemented. Only clinical image data or previously generated meshes at one time point during diastole and one measured data point of the EDPVR are required as an input. The proposed method can be straightforwardly coupled to existing finite element (FE) software packages and is applicable to different constitutive laws and FE formulations. Sensitivity analysis demonstrates that the algorithm is robust with respect to initial input parameters.

Sensitivity analysis and inverse uncertainty quantification for the left ventricular passive mechanics

Personalized computational cardiac models are considered to be a unique and powerful tool in modern cardiology, integrating the knowledge of physiology, pathology and fundamental laws of mechanics in one framework. They have the potential to improve risk prediction in cardiac patients and assist in the development of new treatments. However, in order to use these models for clinical decision support, it is important that both the impact of model parameter perturbations on the predicted quantities of interest as well as the uncertainty of parameter estimation are properly quantified, where the first task is a priori in nature (meaning independent of any specific clinical data), while the second task is carried out a posteriori (meaning after specific clinical data have been obtained). The present study addresses these challenges for a widely used constitutive law of passive myocardium (the Holzapfel-Ogden model), using global sensitivity analysis (SA) to address the first challenge, and inverse-uncertainty quantification (I-UQ) for the second challenge. The SA is carried out on a range of different input parameters to a left ventricle (LV) model, making use of computationally efficient Gaussian process (GP) surrogate models in place of the numerical forward simulator. The results of the SA are then used to inform a low-order reparametrization of the constitutive law for passive myocardium under consideration. The quality of this parameterization in the context of an inverse problem having observed noisy experimental data is then quantified with an I-UQ study, which again makes use of GP surrogate models. The I-UQ is carried out in a Bayesian manner using Markov Chain Monte Carlo, which allows for full uncertainty quantification of the material parameter estimates. Our study reveals insights into the relation between SA and I-UQ, elucidates the dependence of parameter sensitivity and estimation uncertainty on external factors, like LV cavity pressure, and sheds new light on cardio-mechanic model formulation, with particular focus on the Holzapfel-Ogden myocardial model.

Comparison of interpolation methods of predominant cardiomyocyte orientation from in vivo and ex vivo cardiac diffusion tensor imaging data

Cardiac electrophysiology and cardiac mechanics both depend on the average cardiomyocyte long-axis orientation. In the realm of personalized medicine, knowledge of the patient-specific changes in cardiac microstructure plays a crucial role. Patient-specific computational modelling has emerged as a tool to better understand disease progression. In vivo cardiac diffusion tensor imaging (cDTI) is a vital tool to non-destructively measure the average cardiomyocyte long-axis orientation in the heart. However, cDTI suffers from long scan times, rendering volumetric, high-resolution acquisitions challenging. Consequently, interpolation techniques are needed to populate bio-mechanical models with patient-specific average cardiomyocyte long-axis orientations. In this work, we compare five interpolation techniques applied to in vivo and ex vivo porcine input data. We compare two tensor interpolation approaches, one rule-based approximation, and two data-driven, low-rank models. We demonstrate the advantage of tensor interpolation techniques, resulting in lower interpolation errors than do low-rank models and rule-based methods adapted to cDTI data. In an ex vivo comparison, we study the influence of three imaging parameters that can be traded off against acquisition time: in-plane resolution, signal to noise ratio, and number of acquired short-axis imaging slices.

Surrogate-based Bayesian calibration of biomechanical models with isotropic material behavior

This work introduces a computational methodology to calibrate material models in biomechanical applications under uncertainty. We adopt a Bayesian approach, which estimates the probability distributions of hyperelastic material parameters, based on force-strain measurements. We approximate the parametric biomechanical model by combining a reduced order representation of the force response with a Polynomial Chaos expansion. The surrogate model allows to employ sampling-intensive Markov chain Monte Carlo methods and provides an efficient way to estimate (generalized) Sobol coefficients. We use a Sobol sensitivity analysis to assess the influence of material parameters and present an iterative procedure to quantify the accuracy of the surrogate model as additional uncertainty during Bayesian updating. The methodology is illustrated with three cases, tensile experiments on heat-induced whey protein gel, indentation experiments for oocytes and a manufactured example. Real experimental data are used for the calibration.

Personalization of biomechanical simulations of the left ventricle by in-vivo cardiac DTI data: Impact of fiber interpolation methods

Simulations of cardiac electrophysiology and mechanics have been reported to be sensitive to the microstructural anisotropy of the myocardium. Consequently, a personalized representation of cardiac microstructure is a crucial component of accurate, personalized cardiac biomechanical models. In-vivo cardiac Diffusion Tensor Imaging (cDTI) is a non-invasive magnetic resonance imaging technique capable of probing the heart's microstructure. Being a rather novel technique, issues such as low resolution, signal-to noise ratio, and spatial coverage are currently limiting factors. We outline four interpolation techniques with varying degrees of data fidelity, different amounts of smoothing strength, and varying representation error to bridge the gap between the sparse in-vivo data and the model, requiring a 3D representation of microstructure across the myocardium. We provide a workflow to incorporate in-vivo myofiber orientation into a left ventricular model and demonstrate that personalized modelling based on fiber orientations from in-vivo cDTI data is feasible. The interpolation error is correlated with a trend in personalized parameters and simulated physiological parameters, strains, and ventricular twist. This trend in simulation results is consistent across material parameter settings and therefore corresponds to a bias introduced by the interpolation method. This study suggests that using a tensor interpolation approach to personalize microstructure with in-vivo cDTI data, reduces the fiber uncertainty and thereby the bias in the simulation results.

Space-time shape uncertainties in the forward and inverse problem of electrocardiography

In electrocardiography, the "classic" inverse problem is the reconstruction of electric potentials at a surface enclosing the heart from remote recordings at the body surface and an accurate description of the anatomy. The latter being affected by noise and obtained with limited resolution due to clinical constraints, a possibly large uncertainty may be perpetuated in the inverse reconstruction. The purpose of this work is to study the effect of shape uncertainty on the forward and the inverse problem of electrocardiography. To this aim, the problem is first recast into a boundary integral formulation and then discretised with a collocation method to achieve high convergence rates and a fast time to solution. The shape uncertainty of the domain is represented by a random deformation field defined on a reference configuration. We propose a periodic-in-time covariance kernel for the random field and approximate the Karhunen-Loève expansion using low-rank techniques for fast sampling. The space-time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi-Monte Carlo method. We present several numerical experiments on a simplified but physiologically grounded two-dimensional geometry to illustrate the validity of the approach. The tested parametric dimension ranged from 100 up to 600. For the forward problem, the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi-Monte Carlo method perform as expected, except for the total variation regularisation, where convergence is limited by lack of regularity. We finally investigate an H1/2 regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.

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.

A 3D personalized cardiac myocyte aggregate orientation model using MRI data-driven low-rank basis functions

Cardiac myocyte aggregate orientation has a strong impact on cardiac electrophysiology and mechanics. Studying the link between structural characteristics, strain, and stresses over the cardiac cycle and cardiac function requires a full volumetric representation of the microstructure. In this work, we exploit the structural similarity across hearts to extract a low-rank representation of predominant myocyte orientation in the left ventricle from high-resolution magnetic resonance ex-vivo cardiac diffusion tensor imaging (cDTI) in porcine hearts. We compared two reduction methods, Proper Generalized Decomposition combined with Singular Value Decomposition and Proper Orthogonal Decomposition. We demonstrate the existence of a general set of basis functions of aggregated myocyte orientation which defines a data-driven, personalizable, parametric model featuring higher flexibility than existing atlas and rule-based approaches. A more detailed representation of microstructure matching the available patient data can improve the accuracy of personalized computational models. Additionally, we approximate the myocyte orientation of one ex-vivo human heart and demonstrate the feasibility of transferring the basis functions to humans.

Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

Efficient uncertainty quantification algorithms are key to understand the propagation of uncertainty-from uncertain input parameters to uncertain output quantities-in high resolution mathematical models of brain physiology. Advanced Monte Carlo methods such as quasi Monte Carlo (QMC) and multilevel Monte Carlo (MLMC) have the potential to dramatically improve upon standard Monte Carlo (MC) methods, but their applicability and performance in biomedical applications is underexplored. In this paper, we design and apply QMC and MLMC methods to quantify uncertainty in a convection-diffusion model of tracer transport within the brain. We show that QMC outperforms standard MC simulations when the number of random inputs is small. MLMC considerably outperforms both QMC and standard MC methods and should therefore be preferred for brain transport models.

Uncertainty in model-based treatment decision support: Applied to aortic valve stenosis

Patient outcome in trans-aortic valve implantation (TAVI) therapy partly relies on a patient's haemodynamic properties that cannot be determined from current diagnostic methods alone. In this study, we predict changes in haemodynamic parameters (as a part of patient outcome) after valve replacement treatment in aortic stenosis patients. A framework to incorporate uncertainty in patient-specific model predictions for decision support is presented. A 0D lumped parameter model including the left ventricle, a stenotic valve and systemic circulatory system has been developed, based on models published earlier. The unscented Kalman filter (UKF) is used to optimize model input parameters to fit measured data pre-intervention. After optimization, the valve treatment is simulated by significantly reducing valve resistance. Uncertain model parameters are then propagated using a polynomial chaos expansion approach. To test the proposed framework, three in silico test cases are developed with clinically feasible measurements. Quality and availability of simulated measured patient data are decreased in each case. The UKF approach is compared to a Monte Carlo Markov Chain (MCMC) approach, a well-known approach in modelling predictions with uncertainty. Both methods show increased confidence intervals as measurement quality decreases. By considering three in silico test-cases we were able to show that the proposed framework is able to incorporate optimization uncertainty in model predictions and is faster and the MCMC approach, although it is more sensitive to noise in flow measurements. To conclude, this work shows that the proposed framework is ready to be applied to real patient data.

Uncertainty quantification and sensitivity analysis of left ventricular function during the full cardiac cycle

Patient-specific computer simulations can be a powerful tool in clinical applications, helping in diagnostics and the development of new treatments. However, its practical use depends on the reliability of the models. The construction of cardiac simulations involves several steps with inherent uncertainties, including model parameters, the generation of personalized geometry and fibre orientation assignment, which are semi-manual processes subject to errors. Thus, it is important to quantify how these uncertainties impact model predictions. The present work performs uncertainty quantification and sensitivity analyses to assess the variability in important quantities of interest (QoI). Clinical quantities are analysed in terms of overall variability and to identify which parameters are the major contributors. The analyses are performed for simulations of the left ventricle function during the entire cardiac cycle. Uncertainties are incorporated in several model parameters, including regional wall thickness, fibre orientation, passive material parameters, active stress and the circulatory model. The results show that the QoI are very sensitive to active stress, wall thickness and fibre direction, where ejection fraction and ventricular torsion are the most impacted outputs. Thus, to improve the precision of models of cardiac mechanics, new methods should be considered to decrease uncertainties associated with geometrical reconstruction, estimation of active stress and of fibre orientation. This article is part of the theme issue 'Uncertainty quantification in cardiac and cardiovascular modelling and simulation'.

Effect of myofibre architecture on ventricular pump function by using a neonatal porcine heart model: from DT-MRI to rule-based methods

Myofibre architecture is one of the essential components when constructing personalized cardiac models. In this study, we develop a neonatal porcine bi-ventricle model with three different myofibre architectures for the left ventricle (LV). The most realistic one is derived from ex vivo diffusion tensor magnetic resonance imaging, and other two simplifications are based on rule-based methods (RBM): one is regionally dependent by dividing the LV into 17 segments, each with different myofibre angles, and the other is more simplified by assigning a set of myofibre angles across the whole ventricle. Results from different myofibre architectures are compared in terms of cardiac pump function. We show that the model with the most realistic myofibre architecture can produce larger cardiac output, higher ejection fraction and larger apical twist compared with those of the rule-based models under the same pre/after-loads. Our results also reveal that when the cross-fibre contraction is included, the active stress seems to play a dual role: its sheet-normal component enhances the ventricular contraction while its sheet component does the opposite. We further show that by including non-symmetric fibre dispersion using a general structural tensor, even the most simplified rule-based myofibre model can achieve similar pump function as the most realistic one, and cross-fibre contraction components can be determined from this non-symmetric dispersion approach. Thus, our study highlights the importance of including myofibre dispersion in cardiac modelling if RBM are used, especially in personalized models.

Using machine learning to characterize heart failure across the scales

Heart failure is a progressive chronic condition in which the heart undergoes detrimental changes in structure and function across multiple scales in time and space. Multiscale models of cardiac growth can provide a patient-specific window into the progression of heart failure and guide personalized treatment planning. Yet, the predictive potential of cardiac growth models remains poorly understood. Here, we quantify predictive power of a stretch-driven growth model using a chronic porcine heart failure model, subject-specific multiscale simulation, and machine learning techniques. We combine hierarchical modeling, Bayesian inference, and Gaussian process regression to quantify the uncertainty of our experimental measurements during an 8-week long study of volume overload in six pigs. We then propagate the experimental uncertainties from the organ scale through our computational growth model and quantify the agreement between experimentally measured and computationally predicted alterations on the cellular scale. Our study suggests that stretch is the major stimulus for myocyte lengthening and demonstrates that a stretch-driven growth model alone can explain [Formula: see text] of the observed changes in myocyte morphology. We anticipate that our approach will allow us to design, calibrate, and validate a new generation of multiscale cardiac growth models to explore the interplay of various subcellular-, cellular-, and organ-level contributors to heart failure. Using machine learning in heart failure research has the potential to combine information from different sources, subjects, and scales to provide a more holistic picture of the failing heart and point toward new treatment strategies.

A Contemporary Look at Biomechanical Models of Myocardium

Understanding and predicting the mechanical behavior of myocardium under healthy and pathophysiological conditions are vital to developing novel cardiac therapies and promoting personalized interventions. Within the past 30 years, various constitutive models have been proposed for the passive mechanical behavior of myocardium. These models cover a broad range of mathematical forms, microstructural observations, and specific test conditions to which they are fitted. We present a critical review of these models, covering both phenomenological and structural approaches, and their relations to the underlying structure and function of myocardium. We further explore the experimental and numerical techniques used to identify the model parameters. Next, we provide a brief overview of continuum-level electromechanical models of myocardium, with a focus on the methods used to integrate the active and passive components of myocardial behavior. We conclude by pointing to future directions in the areas of optimal form as well as new approaches for constitutive modeling of myocardium.