Parameter estimation in a Holzapfel-Ogden law for healthy myocardium

Using LDDMM and a kinematic cardiac growth model to quantify growth and remodelling in rat hearts under PAH

Pulmonary arterial hypertension (PAH) is a rapidly progressive and fatal disease, with right ventricular failure being the primary cause of death in patients with PAH. This study aims to determine the mechanical stimuli that may initiate heart growth and remodelling (G&R). To achieve this, two bi-ventricular models were constructed: one for a control rat heart and another for a rat heart with PAH. The growth of the diseased heart was estimated by warping it to the control heart using an improved large deformation diffeomorphic metric mapping (LDDMM) framework. Correlation analysis was then performed between mechanical cues (stress and strain) and growth tensors, which revealed that principal strains may serve as a triggering stimulus for myocardial growth and remodelling under PAH. The growth tensors, estimated from in vivo images, could explain 84.3% of the observed geometrical changes in the diseased heart with PAH by using a kinematic cardiac growth model. Our approach has the potential to quantify G&R using sparse in vivo images and to provide insights into the underlying mechanism of triggering right heart failure from a biomechanical perspective.

Application and reduction of a nonlinear hyperelastic wall model capturing ex vivo relationships between fluid pressure, area, and wall thickness in normal and hypertensive murine left pulmonary arteries

Pulmonary hypertension is a cardiovascular disorder manifested by elevated mean arterial blood pressure (>20 mmHg) together with vessel wall stiffening and thickening due to alterations in collagen, elastin, and smooth muscle cells. Hypoxia-induced (type 3) pulmonary hypertension can be studied in animals exposed to a low oxygen environment for prolonged time periods leading to biomechanical alterations in vessel wall structure. This study introduces a novel approach to formulating a reduced order nonlinear elastic structural wall model for a large pulmonary artery. The model relating blood pressure and area is calibrated using ex vivo measurements of vessel diameter and wall thickness changes, under controlled pressure conditions, in left pulmonary arteries isolated from control and hypertensive mice. A two-layer, hyperelastic, and anisotropic model incorporating residual stresses is formulated using the Holzapfel-Gasser-Ogden model. Complex relations predicting vessel area and wall thickness with increasing blood pressure are derived and calibrated using the data. Sensitivity analysis, parameter estimation, subset selection, and physical plausibility arguments are used to systematically reduce the 16-parameter model to one in which a much smaller subset of identifiable parameters is estimated via solution of an inverse problem. Our final reduced one layer model includes a single set of three elastic moduli. Estimated ranges of these parameters demonstrate that nonlinear stiffening is dominated by elastin in the control animals and by collagen in the hypertensive animals. The pressure-area relation developed in this novel manner has potential impact on one-dimensional fluids network models of vessel wall remodeling in the presence of cardiovascular disease.

Simulating impaired left ventricular-arterial coupling in aging and disease: a systematic review

Aortic stenosis, hypertension, and left ventricular hypertrophy often coexist in the elderly, causing a detrimental mismatch in coupling between the heart and vasculature known as ventricular-vascular (VA) coupling. Impaired left VA coupling, a critical aspect of cardiovascular dysfunction in aging and disease, poses significant challenges for optimal cardiovascular performance. This systematic review aims to assess the impact of simulating and studying this coupling through computational models. By conducting a comprehensive analysis of 34 relevant articles obtained from esteemed databases such as Web of Science, Scopus, and PubMed until July 14, 2022, we explore various modeling techniques and simulation approaches employed to unravel the complex mechanisms underlying this impairment. Our review highlights the essential role of computational models in providing detailed insights beyond clinical observations, enabling a deeper understanding of the cardiovascular system. By elucidating the existing models of the heart (3D, 2D, and 0D), cardiac valves, and blood vessels (3D, 1D, and 0D), as well as discussing mechanical boundary conditions, model parameterization and validation, coupling approaches, computer resources and diverse applications, we establish a comprehensive overview of the field. The descriptions as well as the pros and cons on the choices of different dimensionality in heart, valve, and circulation are provided. Crucially, we emphasize the significance of evaluating heart-vessel interaction in pathological conditions and propose future research directions, such as the development of fully coupled personalized multidimensional models, integration of deep learning techniques, and comprehensive assessment of confounding effects on biomarkers.

An image-driven micromechanical approach to characterize multiscale remodeling in infarcted myocardium

Myocardial infarction (MI) is accompanied by the formation of a fibrotic scar in the left ventricle (LV) and initiates significant alterations in the architecture and constituents of the LV free wall (LVFW). Previous studies have shown that LV adaptation is highly individual, indicating that the identification of remodeling mechanisms post-MI demands a fully subject-specific approach that can integrate a host of structural alterations at the fiber-level to changes in bulk biomechanical adaptation at the tissue-level. We present an image-driven micromechanical approach to characterize remodeling, assimilating new biaxial mechanical data, histological studies, and digital image correlation data within an in-silico framework to elucidate the fiber-level remodeling mechanisms that drive tissue-level adaptation for each subject. We found that a progressively diffused collagen fiber structure combined with similarly disorganized myofiber architecture in the healthy region leads to the loss of LVFW anisotropy post-MI, offering an important tissue-level hallmark for LV maladaptation. In contrast, our results suggest that reductions in collagen undulation are an adaptive mechanism competing against LVFW thinning. Additionally, we show that the inclusion of subject-specific geometry when modeling myocardial tissue is essential for accurate prediction of tissue kinematics. Our approach serves as an essential step toward identifying fiber-level remodeling indices that govern the transition of MI to systolic heart failure. These indices complement the traditional, organ-level measures of LV anatomy and function that often fall short of early prognostication of heart failure in MI. In addition, our approach offers an integrated methodology to advance the design of personalized interventions, such as hydrogel injection, to reinforce and suppress native adaptive and maladaptive mechanisms, respectively, to prevent the transition of MI to heart failure. STATEMENT OF SIGNIFICANCE: Biomechanical and architectural adaptation of the LVFW remains a central, yet overlooked, remodeling process post-MI. Our study indicates the biomechanical adaptation of the LVFW post-MI is highly individual and driven by altered fiber network architecture and collective changes in collagen fiber content, undulation, and stiffness. Our findings demonstrate the possibility of using cardiac strains to infer such fiber-level remodeling events through in-silico modeling, paving the way for in-vivo characterization of multiscale biomechanical indices in humans. Such indices will complement the traditional, organ-level measures of LV anatomy and function that often fall short of early prognostication of heart failure in MI.

Advancing clinical translation of cardiac biomechanics models: a comprehensive review, applications and future pathways

Cardiac mechanics models are developed to represent a high level of detail, including refined anatomies, accurate cell mechanics models, and platforms to link microscale physiology to whole-organ function. However, cardiac biomechanics models still have limited clinical translation. In this review, we provide a picture of cardiac mechanics models, focusing on their clinical translation. We review the main experimental and clinical data used in cardiac models, as well as the steps followed in the literature to generate anatomical meshes ready for simulations. We describe the main models in active and passive mechanics and the different lumped parameter models to represent the circulatory system. Lastly, we provide a summary of the state-of-the-art in terms of ventricular, atrial, and four-chamber cardiac biomechanics models. We discuss the steps that may facilitate clinical translation of the biomechanics models we describe. A well-established software to simulate cardiac biomechanics is lacking, with all available platforms involving different levels of documentation, learning curves, accessibility, and cost. Furthermore, there is no regulatory framework that clearly outlines the verification and validation requirements a model has to satisfy in order to be reliably used in applications. Finally, better integration with increasingly rich clinical and/or experimental datasets as well as machine learning techniques to reduce computational costs might increase model reliability at feasible resources. Cardiac biomechanics models provide excellent opportunities to be integrated into clinical workflows, but more refinement and careful validation against clinical data are needed to improve their credibility. In addition, in each context of use, model complexity must be balanced with the associated high computational cost of running these models.

An updated Lagrangian constrained mixture model of pathological cardiac growth and remodelling

Progressive left ventricular (LV) growth and remodelling (G&R) is often induced by volume and pressure overload, characterized by structural and functional adaptation through myocyte hypertrophy and extracellular matrix remodelling, which are dynamically regulated by biomechanical factors, inflammation, neurohormonal pathways, etc. When prolonged, it can eventually lead to irreversible heart failure. In this study, we have developed a new framework for modelling pathological cardiac G&R based on constrained mixture theory using an updated reference configuration, which is triggered by altered biomechanical factors to restore biomechanical homeostasis. Eccentric and concentric growth, and their combination have been explored in a patient-specific human LV model under volume and pressure overload. Eccentric growth is triggered by overstretching of myofibres due to volume overload, i.e. mitral regurgitation, whilst concentric growth is driven by excessive contractile stress due to pressure overload, i.e. aortic stenosis. Different biological constituent's adaptations under pathological conditions are integrated together, which are the ground matrix, myofibres and collagen network. We have shown that this constrained mixture-motivated G&R model can capture different phenotypes of maladaptive LV G&R, such as chamber dilation and wall thinning under volume overload, wall thickening under pressure overload, and more complex patterns under both pressure and volume overload. We have further demonstrated how collagen G&R would affect LV structural and functional adaption by providing mechanistic insight on anti-fibrotic interventions. This updated Lagrangian constrained mixture based myocardial G&R model has the potential to understand the turnover processes of myocytes and collagen due to altered local mechanical stimuli in heart diseases, and in providing mechanistic links between biomechanical factors and biological adaption at both the organ and cellular levels. Once calibrated with patient data, it can be used for assessing heart failure risk and designing optimal treatment therapies. STATEMENT OF SIGNIFICANCE: Computational modelling of cardiac G&R has shown high promise to provide insight into heart disease management when mechanistic understandings are quantified between biomechanical factors and underlying cellular adaptation processes. The kinematic growth theory has been dominantly used to phenomenologically describe the biological G&R process but neglecting underlying cellular mechanisms. We have developed a constrained mixture based G&R model with updated reference by taking into account different mechanobiological processes in the ground matrix, myocytes and collagen fibres. This G&R model can serve as a basis for developing more advanced myocardial G&R models further informed by patient data to assess heart failure risk, predict disease progression, select the optimal treatment by hypothesis testing, and eventually towards a truly precision cardiology using in-silico models.

A Micro-anatomical Model of the Infarcted Left Ventricle Border Zone to Study the Influence of Collagen Undulation

Myocardial infarction (MI) results in cardiac myocyte death and often initiates the formation of a fibrotic scar in the myocardium surrounded by a border zone. Myocyte loss and collagen-rich scar tissue heavily influence the biomechanical behavior of the myocardium which could lead to various cardiac diseases such as systolic heart failure and arrhythmias. Knowledge of how myocyte and collagen micro-architecture changes affect the passive mechanical behavior of the border zone remains limited. Computational modeling provides us with an invaluable tool to identify and study the mechanisms driving the biomechanical remodeling of the myocardium post-MI. We utilized a rodent model of MI and an image-based approach to characterize the three-dimensional (3-D) myocyte and collagen micro-architecture at various timepoints post-MI. Left ventricular free wall (LVFW) samples were obtained from infarcted hearts at 1-week and 4-week post-MI (n = 1 each). Samples were labeled using immunoassays to identify the extracellular matrix (ECM) and myocytes. 3-D reconstructions of the infarct border zone were developed from confocal imaging and meshed to develop high-fidelity micro-anatomically accurate finite element models. We performed a parametric study using these models to investigate the influence of collagen undulation on the passive micromechanical behavior of the myocardium under a diastolic load. Our results suggest that although parametric increases in collagen undulation elevate the strain amount experienced by the ECM in both early- and late-stage MI, the sensitivity of myocytes to such increases is reduced from early to late-stage MI. Our 3-D micro-anatomical modeling holds promise in identifying mechanisms of border zone maladaptation post-MI.

Image-based estimation of the left ventricular cavity volume using deep learning and Gaussian process with cardio-mechanical applications

In this investigation, an image-based method has been developed to estimate the volume of the left ventricular cavity using cardiac magnetic resonance (CMR) imaging data. Deep learning and Gaussian processes have been applied to bring the estimations closer to the cavity volumes manually extracted. CMR data from 339 patients and healthy volunteers have been used to train a stepwise regression model that can estimate the volume of the left ventricular cavity at the beginning and end of diastole. We have decreased the root mean square error (RMSE) of cavity volume estimation approximately from 13 to 8 ml compared to the common practice in the literature. Considering the RMSE of manual measurements is approximately 4 ml on the same dataset, 8 ml of error is notable for a fully automated estimation method, which needs no supervision or user-hours once it has been trained. Additionally, to demonstrate a clinically relevant application of automatically estimated volumes, we inferred the passive material properties of the myocardium given the volume estimates using a well-validated cardiac model. These material properties can be further used for patient treatment planning and diagnosis.

Computational analysis of ventricular mechanics in hypertrophic cardiomyopathy patients

Hypertrophic cardiomyopathy (HCM) is a genetic heart disease that is associated with many pathological features, such as a reduction in global longitudinal strain (GLS), myofiber disarray and hypertrophy. The effects of these features on left ventricle (LV) function are, however, not clear in two phenotypes of HCM, namely, obstructive and non-obstructive. To address this issue, we developed patient-specific computational models of the LV using clinical measurements from 2 female HCM patients and a control subject. Left ventricular mechanics was described using an active stress formulation and myofiber disarray was described using a structural tensor in the constitutive models. Unloaded LV configuration for each subject was first determined from their respective end-diastole LV geometries segmented from the cardiac magnetic resonance images, and an empirical single-beat estimation of the end-diastolic pressure volume relationship. The LV was then connected to a closed-loop circulatory model and calibrated using the clinically measured LV pressure and volume waveforms, peak GLS and blood pressure. Without consideration of myofiber disarray, peak myofiber tension was found to be lowest in the obstructive HCM subject (60 kPa), followed by the non-obstructive subject (242 kPa) and the control subject (375 kPa). With increasing myofiber disarray, we found that peak tension has to increase in the HCM models to match the clinical measurements. In the obstructive HCM patient, however, peak tension was still depressed (cf. normal subject) at the largest degree of myofiber disarray found in the clinic. The computational modeling workflow proposed here can be used in future studies with more HCM patient data.

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.

Bayesian optimisation for efficient parameter inference in a cardiac mechanics model of the left ventricle

We consider parameter inference in cardio-mechanic models of the left ventricle, in particular the one based on the Holtzapfel-Ogden (HO) constitutive law, using clinical in vivo data. The equations underlying these models do not admit closed form solutions and hence need to be solved numerically. These numerical procedures are computationally expensive making computational run times associated with numerical optimisation or sampling excessive for the uptake of the models in the clinical practice. To address this issue, we adopt the framework of Bayesian optimisation (BO), which is an efficient statistical technique of global optimisation. BO seeks the optimum of an unknown black-box function by sequentially training a statistical surrogate-model and using it to select the next query point by leveraging the associated exploration-exploitation trade-off. To guarantee that the estimates based on the in vivo data are realistic also for high-pressures, unobservable in vivo, we include a penalty term based on a previously published empirical law developed using ex vivo data. Two case studies based on real data demonstrate that the proposed BO procedure outperforms the state-of-the-art inference algorithm for the HO constitutive law.

Estimation of left ventricular parameters based on deep learning method

Estimating material properties of personalized human left ventricular (LV) modelling is a central problem in biomechanical studies. In this work we use deep learning (DL) method to evaluating the passive myocardial mechanical properties inversely. In the first part of the paper, we establish a standardized geometric model of the LV. The geometric model parameters are optimized based on 27 different healthy volunteers. In the second part, we use statistical methods and Latin hypercube sampling (LHS) to obtain the geometric parameters data. The LV myocardium is described using a structure-based orthotropic Holzapfel-Ogden constitutive law. The LV diastolic pressure-volume (PV) curves are calculated by numerical simulation. Tn the third part, we establish the multiple neural networks to pblackict PV curve parameters. Then, instead of using constrained optimization problems to solve constitutive parameters, DL was used to establish the nonlinear mapping relationship of geometric parameters, PV curve parameters and constitutive parameters. The results show that the deep learning method can greatly improve the computational efficiency of numerical simulation and increase the possibility of its application in rapid feedback of clinical data.

A machine learning model to estimate myocardial stiffness from EDPVR

In-vivo estimation of mechanical properties of the myocardium is essential for patient-specific diagnosis and prognosis of cardiac disease involving myocardial remodeling, including myocardial infarction and heart failure with preserved ejection fraction. Current approaches use time-consuming finite-element (FE) inverse methods that involve reconstructing and meshing the heart geometry, imposing measured loading, and conducting computationally expensive iterative FE simulations. In this paper, we propose a machine learning (ML) model that feasibly and accurately predicts passive myocardial properties directly from select geometric, architectural, and hemodynamic measures, thus bypassing exhaustive steps commonly required in cardiac FE inverse problems. Geometric and fiber-orientation features were chosen to be readily obtainable from standard cardiac imaging protocols. The end-diastolic pressure-volume relationship (EDPVR), which can be obtained using a single-point pressure-volume measurement, was used as a hemodynamic (loading) feature. A comprehensive ML training dataset in the geometry-architecture-loading space was generated, including a wide variety of partially synthesized rodent heart geometry and myofiber helicity possibilities, and a broad range of EDPVRs obtained using forward FE simulations. Latin hypercube sampling was used to create 2500 examples for training, validation, and testing. A multi-layer feed-forward neural network (MFNN) was used as a deep learning agent to train the ML model. The model showed excellent performance in predicting stiffness parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_f$$\end{document}af and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document}bf associated with fiber direction (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{a_f}=99.471\%$$\end{document}Raf2=99.471% and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{b_f}=92.837\%$$\end{document}Rbf2=92.837%). After conducting permutation feature importance analysis, the ML performance further improved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document}bf (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2_{b_f}=96.240\%$$\end{document}Rbf2=96.240%), and the left ventricular volume and endocardial area were found to be the most critical geometric features for accurate predictions. The ML model predictions were evaluated further in two cases: (i) rat-specific stiffness data measured using ex-vivo mechanical testing, and (ii) patient-specific estimation using FE inverse modeling. Excellent agreements with ML predictions were found for both cases. The trained ML model offers a feasible technology to estimate patient-specific myocardial properties, thus, bridging the gap between EDPVR, as a confounded organ-level metric for tissue stiffness, and intrinsic tissue-level properties. These properties provide incremental information relative to traditional organ-level indices for cardiac function, improving the clinical assessment and prognosis of cardiac diseases.

Effects of dispersed fibres in myocardial mechanics, Part II: active response

This work accompanies the first part of our study "effects of dispersed fibres in myocardial mechanics: Part I passive response" with a focus on myocardial active contraction. Existing studies have suggested that myofibre architecture plays an important role in myocardial active contraction. Following the first part of our study, we firstly study how the general fibre architecture affects ventricular pump function by varying the mean myofibre rotation angles, and then the impact of fibre dispersion along the myofibre direction on myocardial contraction in a left ventricle model. Dispersed active stress is described by a generalised structure tensor method for its computational efficiency. Our results show that both the myofibre rotation angle and its dispersion can significantly affect cardiac pump function by redistributing active tension circumferentially and longitudinally. For example, larger myofibre rotation angle and higher active tension along the sheet-normal direction can lead to much reduced end-systolic volume and higher longitudinal shortening, and thus a larger ejection fraction. In summary, these two studies together have demonstrated that it is necessary and essential to include realistic fibre structures (both fibre rotation angle and fibre dispersion) in personalised cardiac modelling for accurate myocardial dynamics prediction.

Effects of dispersed fibres in myocardial mechanics, Part I: passive response

It is widely acknowledged that an imbalanced biomechanical environment can have significant effects on myocardial pathology, leading to adverse remodelling of cardiac function if it persists. Accurate stress prediction essentially depends on the strain energy function which should have competent descriptive and predictive capabilities. Previous studies have focused on myofibre dispersion, but not on fibres along other directions. In this study, we will investigate how fibre dispersion affects myocardial biomechanical behaviours by taking into account both the myofibre dispersion and the sheet fibre dispersion, with a focus on the sheet fibre dispersion. Fibre dispersion is incorporated into a widely-used myocardial strain energy function using the discrete fibre bundle approach. We first study how different dispersion affects the descriptive capability of the strain energy function when fitting to ex vivo experimental data, and then the predictive capability in a human left ventricle during diastole. Our results show that the chosen strain energy function can achieve the best goodness-of-fit to the experimental data by including both fibre dispersion. Furthermore, noticeable differences in stress can be found in the LV model. Our results may suggest that it is necessary to include both dispersion for myofibres and the sheet fibres for the improved descriptive capability to the ex vivo experimental data and potentially more accurate stress prediction in cardiac mechanics.

A clinically applicable strategy to estimate the in vivo distribution of mechanical material properties of the right ventricular wall

A clinically applicable approach to estimate the in vivo mechanical material properties of the heart wall is presented. This optimization-based inverse estimation approach applies a shape-based objective functional combined with rigid body registration and incremental parameterization of heterogeneity to use standard clinical imaging data along with simplified representations of cardiac function to provide consistent and physically meaningful solution estimates. The capability of the inverse estimation algorithm is evaluated through application to two clinically obtained human datasets to estimate the passive elastic mechanical properties of the heart wall, with an emphasis on the right ventricle. One dataset corresponded to a subject with normal heart function, while the other corresponded to a subject with severe pulmonary hypertension, and therefore expected to have a substantially stiffer right ventricle. Patient-specific pressure-driven bi-ventricle finite element analysis was used as the forward model and the endocardial surface of the right ventricle was used as the target data for the inverse problem. By using the right ventricle alone as the target of the inverse problem the relative sensitivity of the objective function to the right ventricle properties is increased. The method was able to identify material properties to accurately match the corresponding shape of the simplified forward model to the clinically obtained target data, and the properties obtained for the example cases are consistent with the clinical expectation for the right ventricle. Additionally, the material property estimates indicate significant heterogeneity in the heart wall for both subjects, and more so for the subject with pulmonary hypertension.

Physically-based structural modeling of a typical regenerative tissue analog bridges material macroscale continuum and cellular microscale discreteness and elucidates the hierarchical characteristics of cell-matrix interaction

This paper presents a comprehensive physically-based structural modelling for the passive and active biomechanical processes in a typical engineered tissue - namely, cell-compacted collagen gel. First, it introduces a sinusoidal curve analog for quantifying the mechanical response of the collagen fibrils and a probability distribution function of the characteristic crimp ratio for taking into account the fibrillar geometric entropic effect. The constitutive framework based on these structural characteristics precisely reproduces the nonlinearity, the viscoelasticity, and fairly captures the Poisson effect exhibiting in the macroscale tensile tests; which, therefore, substantially validates the structural modelling for the analysis of the cell-gel interaction during collagen gel compaction. Second, a deterministic molecular clutch model specific to the interaction between the cell pseudopodium and the collagen network is developed, which emphasizes the dependence of traction force on clutch number altering with the retrograde flow velocity, actin polymeric velocity, and the deformation of the stretched fibril. The modelling reveals the hierarchical features of cellular substrate sensing, i.e. a biphasic traction force response to substrate elasticity begins at the level of individual fibrils and develops into the second biphasic sensing by means of the fibrillar number integration at the whole-cell level. Singular in crossing the realms of continuum and discrete mechanics, the methodologies developed in this study for modelling the filamentous materials and cell-fibril interaction deliver deep insight into the temporospatially dynamic 3D cell-matrix interaction, and are able to bridge the cellular microscale and material macroscale in the exploration of related topics in mechanobiology.

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.

Neural network-based left ventricle geometry prediction from CMR images with application in biomechanics

Combining biomechanical modelling of left ventricular (LV) function and dysfunction with cardiac magnetic resonance (CMR) imaging has the potential to improve the prognosis of patient-specific cardiovascular disease risks. Biomechanical studies of LV function in three dimensions usually rely on a computerized representation of the LV geometry based on finite element discretization, which is essential for numerically simulating in vivo cardiac dynamics. Detailed knowledge of the LV geometry is also relevant for various other clinical applications, such as assessing the LV cavity volume and wall thickness. Accurately and automatically reconstructing personalized LV geometries from conventional CMR images with minimal manual intervention is still a challenging task, which is a pre-requisite for any subsequent automated biomechanical analysis. We propose a deep learning-based automatic pipeline for predicting the three-dimensional LV geometry directly from routinely-available CMR cine images, without the need to manually annotate the ventricular wall. Our framework takes advantage of a low-dimensional representation of the high-dimensional LV geometry based on principal component analysis. We analyze how the inference of myocardial passive stiffness is affected by using our automatically generated LV geometries instead of manually generated ones. These insights will inform the development of statistical emulators of LV dynamics to avoid computationally expensive biomechanical simulations. Our proposed framework enables accurate LV geometry reconstruction, outperforming previous approaches by delivering a reconstruction error 50% lower than reported in the literature. We further demonstrate that for a nonlinear cardiac mechanics model, using our reconstructed LV geometries instead of manually extracted ones only moderately affects the inference of passive myocardial stiffness described by an anisotropic hyperelastic constitutive law. The developed methodological framework has the potential to make an important step towards personalized medicine by eliminating the need for time consuming and costly manual operations. In addition, our method automatically maps the CMR scan into a low-dimensional representation of the LV geometry, which constitutes an important stepping stone towards the development of an LV geometry-heterogeneous emulator.

The Comparison of Different Constitutive Laws and Fiber Architectures for the Aortic Valve on Fluid-Structure Interaction Simulation

Built on the hybrid immersed boundary/finite element (IB/FE) method, fluid-structure interaction (FSI) simulations of aortic valve (AV) dynamics are performed with three different constitutive laws and two different fiber architectures for the AV leaflets. An idealized AV model is used and mounted in a straight tube, and a three-element Windkessel model is further attached to the aorta. After obtaining ex vivo biaxial tensile testing of porcine AV leaflets, we first determine the constitutive parameters of the selected three constitutive laws by matching the analytical stretch-stress relations derived from constitutive laws to the experimentally measured data. Both the average error and relevant R-squared value reveal that the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions could be more suitable for characterizing the mechanical behaviors of the AV leaflets. We then thoroughly compare the simulation results from both structural mechanics and hemodynamics. Compared to the other two constitutive laws, the anisotropic non-linear constitutive law with exponential terms for both the fiber and cross-fiber directions shows the larger leaflet displacements at the opened state, the largest forward jet flow, the smaller regurgitant flow. We further analyze hemodynamic parameters of the six different cases, including the regurgitant fraction, the mean transvalvular pressure gradient, the effective orifice area, and the energy loss of the left ventricle. We find that the fiber architecture with body-fitted orientation shows better dynamic behaviors in the leaflets, especially with the constitutive law using exponential terms for both the fiber and cross-fiber directions. In conclusion, both constitutive laws and fiber architectures can affect AV dynamics. Our results further suggest that the strain energy function with exponential terms for both the fiber and cross-fiber directions could be more suitable for describing the AV leaflet mechanical behaviors. Future experimental studies are needed to identify competent constitutive laws for the AV leaflets and their associated fiber orientations with controlled experiments. Although limitations exist in the present AV model, our results provide important information for selecting appropriate constitutive laws and fiber architectures when modeling AV dynamics.

Finite-element based optimization of left ventricular passive stiffness in normal volunteers and patients after myocardial infarction: Utility of an inverse deformation gradient calculation of regional diastolic strain

INTRODUCTION

Left ventricular (LV) diastolic dysfunction (DD) is common after myocardial infarction (MI). Whereas current clinical assessment of DD relies on indirect markers including LV filling, finite element (FE) -based computational modeling directly measures regional diastolic stiffness. We hypothesized that an inverse deformation gradient (DG) method calculation of diastolic strain (IDGDS) allows the FE model-based calculation of regional diastolic stiffness (material parameters; MP) in post-MI patients with DD.

METHODS

Cardiac magnetic resonance (CMR) with tags (CSPAMM) and late gadolinium enhancement (LGE) was performed in 10 patients with post-MI DD and 10 healthy volunteers. The 3-dimensional (3D) LV DG from end-diastole (ED) to early diastolic filling (EDF; DG_ED→EDF) was calculated from CSPAMM. Diastolic strain was calculated from DG_EDF→ED by inverting the DG_ED→EDF. FE models were created with MI and non-MI (remote; RM) regions determined by LGE. Guccione MPs C, and exponential fiber, b_f, and transverse, b_t , terms were optimized with IDGDS strain.

RESULTS

3D circumferential and longitudinal diastolic strain (E_cc;E_ll) calculated using IDGDS in CSPAMM obtained in volunteers and MI patients were [Formula: see text]  = 0.27 ± 0.01, [Formula: see text]  = 0.24 ± 0.03 and [Formula: see text]  = 0.21 ± 0.02, and [Formula: see text]  = 0.15 ± 0.02, respectively. MPs in the volunteer group were C_H = 0.013 [0.001, 0.235] kPa, [Formula: see text]  = 20.280 ± 4.994, and [Formula: see text]  = 7.460 ± 2.171 and C_RM = 0.0105 [0.010, 0.011] kPa, [Formula: see text]  = 50.786 ± 13.511 (p = 0.0846), and [Formula: see text]  = 17.355 ± 2.743 (p = 0.0208) in the remote myocardium of post-MI patients.

CONCLUSION

Diastolic strain, calculated from CSPAMM with IDGDS, enables calculation of FE model-based regional diastolic material parameters. Transverse stiffness of the remote myocardium, , may be a valuable new metric for determination of DD in patients after MI.

Passive mechanical properties in healthy and infarcted rat left ventricle characterised via a mixture model

During the cardiac cycle, electrical excitation is coupled with mechanical response of the myocardium. Besides the active contraction, passive mechanics plays an important role, and its behaviour differs in healthy and diseased hearts as well as among different animal species. The aim of this study is the characterisation of passive mechanical properties in healthy and infarcted rat myocardium by means of mechanical testing and subsequent parameter fitting. Elasticity assessments via uniaxial extension tests are performed on healthy and infarcted tissue samples from left ventricular rat myocardium. In order to fully characterise the orthotropic cardiac tissue, our experimental data are combined with other previously published tests in rats - shear tests on healthy myocardium and equibiaxial tests on infarcted tissue. In a first step, we calibrate the Holzapfel-Ogden strain energy function in the healthy case. Sa far, this orthotropic constitutive law for the passive myocardium has been fitted to experimental data in several species, however there is a lack of an appropriate parameter set for the rat. With our determined parameters, a finite element simulation of the end-diastolic filling is performed. In a second step, we propose a model for the infarcted tissue. It is represented as a mixture of intact myocardium and a transversely isotropic scar structure. In our mechanical experiments, the tissue after myocardial infarction shows significantly stiffer behaviour than in the healthy case, and the stiffness correlates with the amount of fibrosis. A similar relationship is observed in the computational simulation of the end-diastolic filling. We conclude that our new proposed material model can capture the behaviour of two kinds of tissues - healthy and infarcted rat myocardium, and its calibration with the fitted parameters represents the experimental data well.

Comparisons of simulation results between passive and active fluid structure interaction models for left ventricle in hypertrophic obstructive cardiomyopathy

BACKGROUND

Patient-specific active fluid-structure interactions (FSI) model is a useful approach to non-invasively investigate the hemodynamics in the heart. However, it takes a lot of effort to obtain the proper external force boundary conditions for active models, which heavily restrained the time-sensitive clinical applications of active computational models.

METHODS

The simulation results of 12 passive FSI models based on 6 patients' pre-operative and post-operative CT images were compared with corresponding active models to investigate the differences in hemodynamics and cardiac mechanics between these models.

RESULTS

In comparing the passive and active models, it was found that there was no significant difference in pressure difference and shear stress on mitral valve leaflet (MVL) at the pre-SAM time point, but a significant difference was found in wall stress on the inner boundary of left ventricle (endocardium). It was also found that pressure difference on the coapted MVL and the shear stress on MVL were significantly decreased after successful surgery in both active and passive models.

CONCLUSION

Our results suggested that the passive models may provide good approximated hemodynamic results at 5% RR interval, which is crucial for analyzing the initiation of systolic anterior motion (SAM). Comparing to active models, the passive models decrease the complexity of the modeling construction and the difficulty of convergence significantly. These findings suggest that, with proper boundary conditions and sufficient clinical data, the passive computational model may be a good substitution model for the active model to perform hemodynamic analysis of the initiation of SAM.

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.

Apparent growth tensor of left ventricular post myocardial infarction - In human first natural history study

An outstanding challenge in modelling biomechanics after myocardial infarction (MI) is to estimate the so-called growth tensor. Since it is impossible to track pure growth induced geometry change from in vivo magnetic resonance images alone, in this work, we propose a way of estimating a surrogate or apparent growth tensor of the human left ventricle using cine magnetic resonance (CMR) and late gadolinium enhanced (LGE) images of 16 patients following acute MI. The apparent growth tensor is evaluated at four time-points following myocardial reperfusion: 4-12 h (baseline), 3 days, 10 days and 7 months. We have identified three different growth patterns classified as the Dilation, No-Change and Shrinkage groups defined by the left ventricle end-diastole cavity volume change from baseline. We study the- trends in both the infarct and remote regions. Importantly, although the No-Change group has little change in the ventricular cavity volume, significant remodelling changes are seen within the myocardial wall, both in the infarct and remote regions. Through statistical analysis, we show that the growth tensor invariants can be used as effective biomarkers for adverse and favourable remodelling of the heart from 10 days onwards post-MI with statistically significant changes over time, in contrast to most of the routine clinical indices. We believe this is the first time that the apparent growth tensor has been estimated from in vivo CMR images post-MI. Our study not only provides much-needed information for understanding growth and remodelling in the human heart following acute MI, but also identifies novel biomarker for assessing heart disease progression.

Identification of tumor nodule in soft tissue: An inverse finite-element framework based on mechanical characterization

Identification and characterization of nodules in soft tissue, including their size, shape, and location, provide a basis for tumor identification. This study proposes an inverse finite-element (FE) based computational framework, for characterizing the size of examined tissue sample and detecting the presence of embedded tumor nodules using instrumented palpation, without a priori anatomical knowledge. The inverse analysis was applied to a model system, the human prostate, and was based on the reaction forces which can be obtained by trans-rectal mechanical probing and those from an equivalent FE model, which was optimized iteratively, by minimizing an error function between the two cases, toward the target solution. The tumor nodule can be identified through its influence on the stress state of the prostate. The effectiveness of the proposed method was further verified using a realistic prostate model reconstructed from magnetic resonance (MR) images. The results show the proposed framework to be capable of characterizing the key geometrical indices of the prostate and identifying the presence of cancerous nodules. Therefore, it has potential, when combined with instrumented palpation, for primary diagnosis of prostate cancer, and, potentially, solid tumors in other types of soft tissue.

Mechanical and morphometric study of mitral valve chordae tendineae and related papillary muscle

The mitral valve (MV) apparatus is a complex mechanical structure including annulus, valve leaflets, papillary muscles (PMs) and connected chordae tendineae. Chordae anchor to the papillary muscles to help the valve open and close properly during one cardiac cycle. It is of paramount importance to understand the functional, mechanical, and microstructural properties of mitral valve chordae and connecting PMs. In particular, little is known about the biomechanical properties of the anterior and posterior papillary muscle and corresponding chords. In this work, we performed uniaxial and biaxial tensile tests on the anterolateral (APM) and posteromedial papillary muscle (PPM), and their respective corresponding chordae tendineae, chordae^APM and chordae^PPM, in porcine hearts. Histology was carried out to link the microstructure and macro-mechanical behavior of the chordae and PMs. Our results demonstrate that chordae^PPM are less in number, but significantly longer and stiffer than chordae^APM. These different biomechanical properties may be partially explained by the higher collagen core ratio and larger collagen fibril density of chordae^PPM_. No significant mechanical or microstructural differences were observed along the circumferential and longitudinal directions of APM and PPM samples. Data measured on chordae and PMs were further fitted with the Ogden and reduced Holzapfel - Ogden strain energy functions, respectively. This study presents the first comparative anatomical, mechanical, and structural dataset of porcine mitral valve chordae and related PMs. Results indicate that a PM based classification of chordae will need to be considered in the analysis of the MV function or planning a surgical treatment, which will also help developing more precise computational models of MV.

Analysis of Cardiac Amyloidosis Progression Using Model-Based Markers

Deposition of amyloid in the heart can lead to cardiac dilation and impair its pumping ability. This ultimately leads to heart failure with worsening symptoms of breathlessness and fatigue due to the progressive loss of elasticity of the myocardium. Biomarkers linked to the clinical deterioration can be crucial in developing effective treatments. However, to date the progression of cardiac amyloidosis is poorly characterized. There is an urgent need to identify key predictors for disease progression and cardiac tissue function. In this proof of concept study, we estimate a group of new markers based on mathematical models of the left ventricle derived from routine clinical magnetic resonance imaging and follow-up scans from the National Amyloidosis Center at the Royal Free in London. Using mechanical modeling and statistical classification, we show that it is possible to predict disease progression. Our predictions agree with clinical assessments in a double-blind test in six out of the seven sample cases studied. Importantly, we find that multiple factors need to be used in the classification, which includes mechanical, geometrical and shape features. No single marker can yield reliable prediction given the complexity of the growth and remodeling process of diseased hearts undergoing high-dimensional shape changes. Our approach is promising in terms of clinical translation but the results presented should be interpreted with caution due to the small sample size.

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.

An orthotropic electro-viscoelastic model for the heart with stress-assisted diffusion

We propose and analyse the properties of a new class of models for the electromechanics of cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law, for both active stress and active strain formulations of active mechanics, coupled with a four-variable phenomenological model for human cardiac cell electrophysiology, which produces an accurate description of the action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms, and the activation model assumes a simplified stretch-calcium interaction generating active tension or active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.

Insights into the passive mechanical behavior of left ventricular myocardium using a robust constitutive model based on full 3D kinematics

Myocardium possesses a hierarchical structure that results in complex three-dimensional (3D) mechanical behavior, forming a critical component of ventricular function in health and disease. A wide range of constitutive model forms have been proposed for myocardium since the first planar biaxial studies were performed by Demer and Yin (J. Physiol. 339 (1), 1983). While there have been extensive studies since, none have been based on full 3D kinematic data, nor have they utilized optimal experimental design to estimate constitutive parameters, which may limit their predictive capability. Herein we have applied our novel 3D numerical-experimental methodology (Avazmohammadi et al., Biomechanics Model. Mechanobiol. 2018) to explore the applicability of an orthotropic constitutive model for passive ventricular myocardium (Holzapfel and Ogden, Philos. Trans. R. Soc. Lond.: Math. Phys. Eng. Sci. 367, 2009) by integrating 3D optimal loading paths, spatially varying material structure, and inverse modeling techniques. Our findings indicated that the initial model form was not successful in reproducing all optimal loading paths, due to previously unreported coupling behaviors via shearing of myofibers and extracellular collagen fibers in the myocardium. This observation necessitated extension of the constitutive model by adding two additional terms based on the I8(C) pseudo-invariant in the fiber-normal and sheet-normal directions. The modified model accurately reproduced all optimal loading paths and exhibited improved predictive capabilities. These unique results suggest that more complete constitutive models are required to fully capture the full 3D biomechanical response of left ventricular myocardium. The present approach is thus crucial for improved understanding and performance in cardiac modeling in healthy, diseased, and treatment scenarios.

Interaction of the Mechano-Electrical Feedback With Passive Mechanical Models on a 3D Rat Left Ventricle: A Computational Study

In this paper, we are investigating the interaction between different passive material models and the mechano-electrical feedback (MEF) in cardiac modeling. Various types of passive mechanical laws (nearly incompressible/compressible, polynomial/exponential-type, transversally isotropic/orthotropic material models) are integrated in a fully coupled electromechanical model in order to study their specific influence on the overall MEF behavior. Our computational model is based on a three-dimensional (3D) geometry of a healthy rat left ventricle reconstructed from magnetic resonance imaging (MRI). The electromechanically coupled problem is solved using a fully implicit finite element-based approach. The effects of different passive material models on the MEF are studied with the help of numerical examples. It turns out that there is a significant difference between the behavior of the MEF for compressible and incompressible material models. Numerical results for the incompressible models exhibit that a change in the electrophysiology can be observed such that the transmembrane potential (TP) is unable to reach the resting state in the repolarization phase, and this leads to non-zero relaxation deformations. The most significant and strongest effects of the MEF on the rat cardiac muscle response are observed for the exponential passive material law.

Fast parameter inference in a biomechanical model of the left ventricle by using statistical emulation

A central problem in biomechanical studies of personalized human left ventricular modelling is estimating the material properties and biophysical parameters from in vivo clinical measurements in a timeframe that is suitable for use within a clinic. Understanding these properties can provide insight into heart function or dysfunction and help to inform personalized medicine. However, finding a solution to the differential equations which mathematically describe the kinematics and dynamics of the myocardium through numerical integration can be computationally expensive. To circumvent this issue, we use the concept of emulation to infer the myocardial properties of a healthy volunteer in a viable clinical timeframe by using in vivo magnetic resonance image data. Emulation methods avoid computationally expensive simulations from the left ventricular model by replacing the biomechanical model, which is defined in terms of explicit partial differential equations, with a surrogate model inferred from simulations generated before the arrival of a patient, vastly improving computational efficiency at the clinic. We compare and contrast two emulation strategies: emulation of the computational model outputs and emulation of the loss between the observed patient data and the computational model outputs. These strategies are tested with two interpolation methods, as well as two loss functions. The best combination of methods is found by comparing the accuracy of parameter inference on simulated data for each combination. This combination, using the output emulation method, with local Gaussian process interpolation and the Euclidean loss function, provides accurate parameter inference in both simulated and clinical data, with a reduction in the computational cost of about three orders of magnitude compared with numerical integration of the differential equations by using finite element discretization techniques.

Gaussian process emulation to accelerate parameter estimation in a mechanical model of the left ventricle: a critical step towards clinical end-user relevance

In recent years, we have witnessed substantial advances in the mathematical modelling of the biomechanical processes underlying the dynamics of the cardiac soft-tissue. Gao et al. (Gao et al. 2017 J. R. Soc. Interface 14, 20170203 ( doi:10.1098/rsif.2017.0203 )) demonstrated that the parameters underlying the biomechanical model have diagnostic value for prognosticating the risk of myocardial infarction. However, the computational costs of parameter estimation are prohibitive when the goal lies in building real-time clinical decision support systems. This is due to the need to repeatedly solve the mathematical equations numerically using finite-element discretization during an iterative optimization routine. The present article presents a method for accelerating the inference of the constitutive parameters by using statistical emulation with Gaussian processes. We demonstrate how the computational costs can be reduced by about three orders of magnitude, with hardly any loss in accuracy, and we assess various alternative techniques in a comparative evaluation study based on simulated data obtained by solving the left ventricular model with the finite-element method, and real magnetic resonance images data for a human volunteer.

On the AIC-based model reduction for the general Holzapfel-Ogden myocardial constitutive law

Constitutive laws that describe the mechanical responses of cardiac tissue under loading hold the key to accurately model the biomechanical behaviour of the heart. There have been ample choices of phenomenological constitutive laws derived from experiments, some of which are quite sophisticated and include effects of microscopic fibre structures of the myocardium. A typical example is the strain-invariant-based Holzapfel–Ogden 2009 model that is excellently fitted to simple shear tests. It has been widely used and regarded as the state-of-the-art constitutive law for myocardium. However, there has been no analysis to show if it has both adequate descriptive and predictive capabilities for other tissue tests of myocardium. Indeed, such an analysis is important for any constitutive laws for clinically useful computational simulations. In this work, we perform such an analysis using combinations of tissue tests, uniaxial tension, biaxial tension and simple shear from three different sets of myocardial tissue studies. Starting from the general 14-parameter myocardial constitutive law developed by Holzapfel and Ogden, denoted as the general HO model, we show that this model has good descriptive and predictive capabilities for all the experimental tests. However, to reliably determine all 14 parameters of the model from experiments remains a great challenge. Our aim is to reduce the constitutive law using Akaike information criterion, to maintain its mechanical integrity whilst achieving minimal computational cost. A competent constitutive law should have descriptive and predictive capabilities for different tissue tests. By competent, we mean the model has least terms but is still able to describe and predict experimental data. We also investigate the optimal combinations of tissue tests for a given constitutive model. For example, our results show that using one of the reduced HO models, one may need just one shear response (along normal-fibre direction) and one biaxial stretch (ratio of 1 mean fibre : 1 cross-fibre) to satisfactorily describe Sommer et al. human myocardial mechanical properties. Our study suggests that single-state tests (i.e. simple shear or stretching only) are insufficient to determine the myocardium responses. We also found it is important to consider transmural fibre rotations within each myocardial sample of tests during the fitting process. This is done by excluding un-stretched fibres using an “effective fibre ratio”, which depends on the sample size, shape, local myofibre architecture and loading conditions. We conclude that a competent myocardium material model can be obtained from the general HO model using AIC analysis and a suitable combination of tissue tests.

Kinematic boundary conditions substantially impact in silico ventricular function

Computational cardiac mechanical models, individualized to the patient, have the potential to elucidate the fundamentals of cardiac (patho-)physiology, enable non-invasive quantification of clinically significant metrics (eg, stiffness, active contraction, work), and anticipate the potential efficacy of therapeutic cardiovascular intervention. In a clinical setting, however, the available imaging resolution is often limited, which limits cardiac models to focus on the ventricles, without including the atria, valves, and proximal arteries and veins. In such models, the absence of surrounding structures needs to be accounted for by imposing realistic kinematic boundary conditions, which, for prognostic purposes, are preferably generic and thus non-image derived. Unfortunately, the literature on cardiac models shows no consistent approach to kinematically constrain the myocardium. The impact of different approaches (eg, fully constrained base, constrained epi-ring) on the predictive capacity of cardiac mechanical models has not been thoroughly studied. For that reason, this study first gives an overview of current approaches to kinematically constrain (bi) ventricular models. Next, we developed a patient-specific in silico biventricular model that compares well with literature and in vivo recorded strains. Alternative constraints were introduced to assess the influence of commonly used mechanical boundary conditions on both the predicted global functional behavior of the in-silico heart (cavity volumes, stroke volume, ejection fraction) and local strain distributions. Meaningful differences in global functioning were found between different kinematic anchoring strategies, which brought forward the importance of selecting appropriate boundary conditions for biventricular models that, in the near future, may inform clinical intervention. However, whilst statistically significant differences were also found in local strain distributions, these differences were minor and mostly confined to the region close to the applied boundary conditions.

A short history of the development of mathematical models of cardiac mechanics

Cardiac mechanics plays a crucial role in atrial and ventricular function, in the regulation of growth and remodelling, in the progression of disease, and the response to treatment. The spatial scale of the critical mechanisms ranges from nm (molecules) to cm (hearts) with the fastest events occurring in milliseconds (molecular events) and the slowest requiring months (growth and remodelling). Due to its complexity and importance, cardiac mechanics has been studied extensively both experimentally and through mathematical models and simulation. Models of cardiac mechanics evolved from seminal studies in skeletal muscle, and developed into cardiac specific, species specific, human specific and finally patient specific calculations. These models provide a formal framework to link multiple experimental assays recorded over nearly 100 years into a single unified representation of cardiac function. This review first provides a summary of the proteins, physiology and anatomy involved in the generation of cardiac pump function. We then describe the evolution of models of cardiac mechanics starting with the early theoretical frameworks describing the link between sarcomeres and muscle contraction, transitioning through myosin-level models to calcium-driven systems, and ending with whole heart patient-specific models.

Construction and Validation of Subject-Specific Biventricular Finite-Element Models of Healthy and Failing Swine Hearts From High-Resolution DT-MRI

Predictive computational modeling has revolutionized classical engineering disciplines and is in the process of transforming cardiovascular research. This is particularly relevant for investigating emergent therapies for heart failure, which remains a leading cause of death globally. The creation of subject-specific biventricular computational cardiac models has been a long-term endeavor within the biomedical engineering community. Using high resolution (0.3 × 0.3 × 0.8 mm) ex vivo data, we constructed a precise fully subject-specific biventricular finite-element model of healthy and failing swine hearts. Each model includes fully subject-specific geometries, myofiber architecture and, in the case of the failing heart, fibrotic tissue distribution. Passive and active material properties are prescribed using hyperelastic strain energy functions that define a nearly incompressible, orthotropic material capable of contractile function. These materials were calibrated using a sophisticated multistep approach to match orthotropic tri-axial shear data as well as subject-specific hemodynamic ventricular targets for pressure and volume to ensure realistic cardiac function. Each mechanically beating heart is coupled with a lumped-parameter representation of the circulatory system, allowing for a closed-loop definition of cardiovascular flow. The circulatory model incorporates unidirectional fluid exchanges driven by pressure gradients of the model, which in turn are driven by the mechanically beating heart. This creates a computationally meaningful representation of the dynamic beating of the heart coupled with the circulatory system. Each model was calibrated using subject-specific experimental data and compared with independent in vivo strain data obtained from echocardiography. Our methods produced highly detailed representations of swine hearts that function mechanically in a remarkably similar manner to the in vivo subject-specific strains on a global and regional comparison. The degree of subject-specificity included in the models represents a milestone for modeling efforts that captures realism of the whole heart. This study establishes a foundation for future computational studies that can apply these validated methods to advance cardiac mechanics research.

An in silico framework to analyze the anisotropic shear wave mechanics in cardiac shear wave elastography

Shear wave elastography (SWE) is a potential tool to non-invasively assess cardiac muscle stiffness. This study focused on the effect of the orthotropic material properties and mechanical loading on the performance of cardiac SWE, as it is known that these factors contribute to complex 3D anisotropic shear wave propagation. To investigate the specific impact of these complexities, we constructed a finite element model with an orthotropic material law subjected to different uniaxial stretches to simulate SWE in the stressed cardiac wall. Group and phase speed were analyzed in function of tissue thickness and virtual probe rotation angle. Tissue stretching increased the group and phase speed of the simulated shear wave, especially in the direction of the muscle fiber. As the model provided access to the true fiber orientation and material properties, we assessed the accuracy of two fiber orientation extraction methods based on SWE. We found a higher accuracy (but lower robustness) when extracting fiber orientations based on the location of maximal shear wave speed instead of the angle of the major axis of the ellipsoidal group speed surface. Both methods had a comparable performance for the center region of the cardiac wall, and performed less well towards the edges. Lastly, we also assessed the (theoretical) impact of pathology on shear wave physics and characterization in the model. It was found that SWE was able to detect changes in fiber orientation and material characteristics, potentially associated with cardiac pathologies such as myocardial fibrosis. Furthermore, the model showed clearly altered shear wave patterns for the fibrotic myocardium compared to the healthy myocardium, which forms an initial but promising outcome of this modeling study.

Changes and classification in myocardial contractile function in the left ventricle following acute myocardial infarction

In this research, we hypothesized that novel biomechanical parameters are discriminative in patients following acute ST-segment elevation myocardial infarction (STEMI). To identify these biomechanical biomarkers and bring computational biomechanics ‘closer to the clinic’, we applied state-of-the-art multiphysics cardiac modelling combined with advanced machine learning and multivariate statistical inference to a clinical database of myocardial infarction. We obtained data from 11 STEMI patients (ClinicalTrials.gov NCT01717573) and 27 healthy volunteers, and developed personalized mathematical models for the left ventricle (LV) using an immersed boundary method. Subject-specific constitutive parameters were achieved by matching to clinical measurements. We have shown, for the first time, that compared with healthy controls, patients with STEMI exhibited increased LV wall active tension when normalized by systolic blood pressure, which suggests an increased demand on the contractile reserve of remote functional myocardium. The statistical analysis reveals that the required patient-specific contractility, normalized active tension and the systolic myofilament kinematics have the strongest explanatory power for identifying the myocardial function changes post-MI. We further observed a strong correlation between two biomarkers and the changes in LV ejection fraction at six months from baseline (the required contractility (r = − 0.79, p < 0.01) and the systolic myofilament kinematics (r = 0.70, p = 0.02)). The clinical and prognostic significance of these biomechanical parameters merits further scrutinization.

In vivo estimation of passive biomechanical properties of human myocardium

Identification of in vivo passive biomechanical properties of healthy human myocardium from regular clinical data is essential for subject-specific modelling of left ventricle (LV). In this work, myocardium was defined by Holzapfel-Ogden constitutive law. Therefore, the objectives of the study were (a) to estimate the ranges of the constitutive parameters for healthy human myocardium using non-invasive routine clinical data, and (b) to investigate the effect of geometry, LV end-diastolic pressure (EDP) and fibre orientations on estimated values. In order to avoid invasive measurements and additional scans, LV cavity volume, measured from routine MRI, and empirical pressure-normalised-volume relation (Klotz-curve) were used as clinical data. Finite element modelling, response surface method and genetic algorithm were used to inversely estimate the constitutive parameters. Due to the ill-posed nature of the inverse optimisation problem, the myocardial properties was extracted by identifying the ranges of the parameters, instead of finding unique values. Additional sensitivity studies were carried out to identify the effect of LV EDP, fibre orientation and geometry on estimated parameters. Although uniqueness of the solution cannot be achieved, the normal ranges of the parameters produced similar mechanical responses within the physiological ranges. These information could be used in future computational studies for designing heart failure treatments. Graphical abstract.

In vivo estimation of passive biomechanical properties of human myocardium

Identification of in vivo passive biomechanical properties of healthy human myocardium from regular clinical data is essential for subject-specific modelling of left ventricle (LV). In this work, myocardium was defined by Holzapfel-Ogden constitutive law. Therefore, the objectives of the study were (a) to estimate the ranges of the constitutive parameters for healthy human myocardium using non-invasive routine clinical data, and (b) to investigate the effect of geometry, LV end-diastolic pressure (EDP) and fibre orientations on estimated values. In order to avoid invasive measurements and additional scans, LV cavity volume, measured from routine MRI, and empirical pressure-normalised-volume relation (Klotz-curve) were used as clinical data. Finite element modelling, response surface method and genetic algorithm were used to inversely estimate the constitutive parameters. Due to the ill-posed nature of the inverse optimisation problem, the myocardial properties was extracted by identifying the ranges of the parameters, instead of finding unique values. Additional sensitivity studies were carried out to identify the effect of LV EDP, fibre orientation and geometry on estimated parameters. Although uniqueness of the solution cannot be achieved, the normal ranges of the parameters produced similar mechanical responses within the physiological ranges. These information could be used in future computational studies for designing heart failure treatments.

Advances in computational modelling for personalised medicine after myocardial infarction

Myocardial infarction (MI) is a leading cause of premature morbidity and mortality worldwide. Determining which patients will experience heart failure and sudden cardiac death after an acute MI is notoriously difficult for clinicians. The extent of heart damage after an acute MI is informed by cardiac imaging, typically using echocardiography or sometimes, cardiac magnetic resonance (CMR). These scans provide complex data sets that are only partially exploited by clinicians in daily practice, implying potential for improved risk assessment. Computational modelling of left ventricular (LV) function can bridge the gap towards personalised medicine using cardiac imaging in patients with post-MI. Several novel biomechanical parameters have theoretical prognostic value and may be useful to reflect the biomechanical effects of novel preventive therapy for adverse remodelling post-MI. These parameters include myocardial contractility (regional and global), stiffness and stress. Further, the parameters can be delineated spatially to correspond with infarct pathology and the remote zone. While these parameters hold promise, there are challenges for translating MI modelling into clinical practice, including model uncertainty, validation and verification, as well as time-efficient processing. More research is needed to (1) simplify imaging with CMR in patients with post-MI, while preserving diagnostic accuracy and patient tolerance (2) to assess and validate novel biomechanical parameters against established prognostic biomarkers, such as LV ejection fraction and infarct size. Accessible software packages with minimal user interaction are also needed. Translating benefits to patients will be achieved through a multidisciplinary approach including clinicians, mathematicians, statisticians and industry partners.

High-resolution data assimilation of cardiac mechanics applied to a dyssynchronous ventricle

Computational models of cardiac mechanics, personalized to a patient, offer access to mechanical information above and beyond direct medical imaging. Additionally, such models can be used to optimize and plan therapies in-silico, thereby reducing risks and improving patient outcome. Model personalization has traditionally been achieved by data assimilation, which is the tuning or optimization of model parameters to match patient observations. Current data assimilation procedures for cardiac mechanics are limited in their ability to efficiently handle high-dimensional parameters. This restricts parameter spatial resolution, and thereby the ability of a personalized model to account for heterogeneities that are often present in a diseased or injured heart. In this paper, we address this limitation by proposing an adjoint gradient-based data assimilation method that can efficiently handle high-dimensional parameters. We test this procedure on a synthetic data set and provide a clinical example with a dyssynchronous left ventricle with highly irregular motion. Our results show that the method efficiently handles a high-dimensional optimization parameter and produces an excellent agreement for personalized models to both synthetic and clinical data.