Quasi-static image-based immersed boundary-finite element model of left ventricle under diastolic loading

Modelling coronary flow and myocardial perfusion by integrating a structured-tree coronary flow model and a hyperelastic left ventricle model

BACKGROUND AND OBJECTIVE

There is an increasing demand to establish integrated computational models that facilitate the exploration of coronary circulation in physiological and pathological contexts, particularly concerning interactions between coronary flow dynamics and myocardial motion. The field of cardiology has also demonstrated a trend toward personalised medicine, where these integrated models can be instrumental in integrating patient-specific data to improve therapeutic outcomes. Notably, incorporating a structured-tree model into such integrated models is currently absent in the literature, which presents a promising prospect. Thus, the goal here is to develop a novel computational framework that combines a 1D structured-tree model of coronary flow in human coronary vasculature with a 3D left ventricle model utilising a hyperelastic constitutive law, enabling the physiologically accurate simulation of coronary flow dynamics.

METHODS

We adopted detailed geometric information from previous studies of both coronary vasculature and left ventricle to construct the coronary flow model and the left ventricle model. The structured-tree model for coronary flow was expanded to encompass the effect of time-varying intramyocardial pressure on intramyocardial blood vessels. Simultaneously, the left ventricle model served as a robust foundation for the calculation of intramyocardial pressure and subsequent quantitative evaluation of myocardial perfusion. A one-way coupling framework between the two models was established to enable the evaluation and examination of coronary flow dynamics and myocardial perfusion.

RESULTS

Our predicted coronary flow waveforms aligned well with published experimental data. Our model precisely captured the phasic pattern of coronary flow, including impeded or even reversed flow during systole. Moreover, our assessment of coronary flow, considering both globally and regionally averaged intramyocardial pressure, demonstrated that elevated intramyocardial pressure corresponds to increased impeding effects on coronary flow. Furthermore, myocardial blood flow simulated from our model was comparable with MRI perfusion data at rest, showcasing the capability of our model to predict myocardial perfusion.

CONCLUSIONS

The integrated model introduced in this study presents a novel approach to achieving physiologically accurate simulations of coronary flow and myocardial perfusion. It holds promise for its clinical applicability in diagnosing insufficient myocardial perfusion.

Effect of pulmonary regurgitation on cardiac functions based on a human bi-ventricle model

BACKGROUND AND OBJECTIVE

Assessing the severity of pulmonary regurgitation (PR) and identifying optimal clinically relevant indicators for its treatment is crucial, yet standards for quantifying PR remain unclear in clinical practice. Computational modelling of the heart is in the process of providing valuable insights and information for cardiovascular physiology research. However, the advancements of finite element computational models have not been widely applied to simulate cardiac outputs in patients with PR. Furthermore, a computational model that incorporates both the left ventricle (LV) and right ventricle (RV) can be valuable in assessing the relationship between left and right ventricular morphometry and septal motion in PR patients. To enhance our understanding of the effect of PR on cardiac functions and mechanical behaviour, we developed a human bi-ventricle model to simulate five cases with varying degrees of PR severity.

METHODS

This bi-ventricle model was built using a patient-specific geometry and a widely used myofibre architecture. The myocardial material properties were described by a hyperelastic passive constitutive law and a modified time-varying elastance active tension model. To simulate realistic cardiac functions and the dysfunction of the pulmonary valve in PR disease cases, open-loop lumped parameter models representing systemic and pulmonary circulatory systems were designed.

RESULTS

In the baseline case, pressures in the aorta and main pulmonary artery and ejection fractions of both the LV and RV were within normal physiological ranges reported in the literature. The end-diastolic volume (EDV) of the RV under varying degrees of PR was comparable to the reported cardiac magnetic resonance imaging data. Moreover, RV dilation and interventricular septum motion from the baseline to the PR cases were clearly observed through the long-axis and short-axis views of the bi-ventricle geometry. The RV EDV in the severe PR case increased by 50.3% compared to the baseline case, while the LV EDV decreased by 18.1%. The motion of the interventricular septum was consistent with the literature. Furthermore, ejection fractions of both the LV and RV decreased as PR became severe, with LV ejection fraction decreasing from 60.5% at baseline to 56.3% in the severe case and RV ejection fraction decreasing from 51.8% to 46.8%. Additionally, the average myofibre stress of the RV wall at end-diastole significantly increased due to PR, from 2.7±12.1 kPa at baseline to 10.9±26.5 kPa in the severe case. The average myofibre stress of the LV wall at end-diastole increased from 3.7±18.1 kPa to 4.3±20.3 kPa.

CONCLUSIONS

This study established a foundation for the computational modelling of PR. The simulated results showed that severe PR leads to reduced cardiac outputs in both the LV and RV, clearly observable septum motion, and a significant increase in the average myofibre stress in the RV wall. These findings demonstrate the potential of the model for further exploration of PR.

Increased Blood Residence Time as Markers of High-Risk Patent Foramen Ovale

Previous investigations have suggested that patients with patent foramen ovale (PFO) often have an atrial dysfunction, like to that observed in patients with atrial fibrillation (AF) which may concur to an increased risk of cryptogenic stroke. The aim of the study is to compare the atrial resident time (Rt) of PFO patients to those with sinus rhythm (SR) and AF using patient-specific 3D computational fluid dynamics (CFD) analysis. Models of left atrium (LA) hemodynamics were obtained from time-resolved CT scans and transthoracic echocardiography (TTE). Enrolled patients were divided into three groups: 30 healthy subjects with SR, 30 with PFO, and 30 with AF without PFO. Blood stasis was evaluated by determining the blood residence time (Rt) distribution in the LA and left atrial appendage (LAA). Overall, 90 patients (mean age 47.4 ± 7.5 years, 51 males) were included into the analysis. PFO patients exhibit higher mean Rt values compared to healthy subjects (2.65 ± 0.2 vs 1.5 ± 0.2 s). Conversely, AF patients presented higher Rt when compared to PFO patients (2.9 ± 0.3 vs 2.3 ± 0.2 s). Moreover, PFO patients presenting cerebral lesions at magnetic resonance imaging have a higher Rt compared to those without (2.9 ± 0.3 vs 2.3 ± 0.2 s, respectively, p < 0.001). PFO patients have a higher degree of atrial Rt compared to healthy subjects similar to that observed in AF patients. The higher mean LA Rt values offer an insight into the pathophysiological mechanism linking PFO with cryptogenic stroke and might be a marker of high-risk PFOs.

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.

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.

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.

A poroelastic immersed finite element framework for modelling cardiac perfusion and fluid-structure interaction

Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid-structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre-reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three-phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid. We benchmark our approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry considered in the prior work by Chapelle et al (2010). This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton. With this framework, we also simulate the poroelastic dynamics of a three-dimensional left ventricle, in which the myocardium is described by the Holzapfel-Ogden law. Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously. We find that the poroelastic LV behaves differently from the hyperelastic LV model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well-known wall-stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.

Fluid-structure interaction in a fully coupled three-dimensional mitral-atrium-pulmonary model

This paper aims to investigate detailed mechanical interactions between the pulmonary haemodynamics and left heart function in pathophysiological situations (e.g. atrial fibrillation and acute mitral regurgitation). This is achieved by developing a complex computational framework for a coupled pulmonary circulation, left atrium and mitral valve model. The left atrium and mitral valve are modelled with physiologically realistic three-dimensional geometries, fibre-reinforced hyperelastic materials and fluid–structure interaction, and the pulmonary vessels are modelled as one-dimensional network ended with structured trees, with specified vessel geometries and wall material properties. This new coupled model reveals some interesting results which could be of diagnostic values. For example, the wave propagation through the pulmonary vasculature can lead to different arrival times for the second systolic flow wave (S2 wave) among the pulmonary veins, forming vortex rings inside the left atrium. In the case of acute mitral regurgitation, the left atrium experiences an increased energy dissipation and pressure elevation. The pulmonary veins can experience increased wave intensities, reversal flow during systole and increased early-diastolic flow wave (D wave), which in turn causes an additional flow wave across the mitral valve (L wave), as well as a reversal flow at the left atrial appendage orifice. In the case of atrial fibrillation, we show that the loss of active contraction is associated with a slower flow inside the left atrial appendage and disappearances of the late-diastole atrial reversal wave (AR wave) and the first systolic wave (S1 wave) in pulmonary veins. The haemodynamic changes along the pulmonary vessel trees on different scales from microscopic vessels to the main pulmonary artery can all be captured in this model. The work promises a potential in quantifying disease progression and medical treatments of various pulmonary diseases such as the pulmonary hypertension due to a left heart dysfunction.

Neuromechanical wave resonance in jellyfish swimming

For organisms to have robust locomotion, their neuromuscular organization must adapt to constantly changing environments. In jellyfish, swimming robustness emerges when marginal pacemakers fire action potentials throughout the bell's motor nerve net, which signals the musculature to contract. The speed of the muscle activation wave is dictated by the passage times of the action potentials. However, passive elastic material properties also influence the emergent kinematics, with time scales independent of neuromuscular organization. In this multimodal study, we examine the interplay between these two time scales during turning. A three-dimensional computational fluid-structure interaction model of a jellyfish was developed to determine the resulting emergent kinematics, using bidirectional muscular activation waves to actuate the bell rim. Activation wave speeds near the material wave speed yielded successful turns, with a 76-fold difference in turning rate between the best and worst performers. Hyperextension of the margin occurred only at activation wave speeds near the material wave speed, suggesting resonance. This hyperextension resulted in a 34-fold asymmetry in the circulation of the vortex ring between the inside and outside of the turn. Experimental recording of the activation speed confirmed that jellyfish actuate within this range, and flow visualization using particle image velocimetry validated the corresponding fluid dynamics of the numerical model. This suggests that neuromechanical wave resonance plays an important role in the robustness of an organism's locomotory system and presents an undiscovered constraint on the evolution of flexible organisms. Understanding these dynamics is essential for developing actuators in soft body robotics and bioengineered pumps.

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.

Stabilization approaches for the hyperelastic immersed boundary method for problems of large-deformation incompressible elasticity

The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses Lagrangian coordinates to describe the structural deformations, stresses, and resultant forces. Integral transforms with Dirac delta function kernels connect the Eulerian and Lagrangian frames. The fluid and the structure are both typically treated as incompressible materials. Upon discretization, however, the incompressibility of the structure is only maintained approximately. To obtain an immersed method for incompressible hyperelastic structures that is robust under large structural deformations, we introduce a volumetric energy in the solid region that stabilizes the formulation and improves the accuracy of the numerical scheme. This formulation augments the discrete Lagrange multiplier for the incompressibility constraint, thereby improving the original method's accuracy. This volumetric energy is incorporated by decomposing the strain energy into isochoric and dilatational components, as in standard solid mechanics formulations of nearly incompressible elasticity. We study the performance of the stabilized method using several quasi-static solid mechanics benchmarks, a dynamic fluid-structure interaction benchmark, and a detailed three-dimensional model of esophageal transport. The accuracy achieved by the stabilized immersed formulation is comparable to that of a stabilized finite element method for incompressible elasticity using similar numbers of structural degrees of freedom.

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.

Analysis of a coupled fluid-structure interaction model of the left atrium and mitral valve

We present a coupled left atrium-mitral valve model based on computed tomography scans with fibre-reinforced hyperelastic materials. Fluid-structure interaction is realised by using an immersed boundary-finite element framework. Effects of pathological conditions, eg, mitral valve regurgitation and atrial fibrillation, and geometric and structural variations, namely, uniform vs non-uniform atrial wall thickness and rule-based vs atlas-based fibre architectures, on the system are investigated. We show that in the case of atrial fibrillation, pulmonary venous flow reversal at late diastole disappears, and the filling waves at the left atrial appendage orifice during systole have reduced magnitude. In the case of mitral regurgitation, a higher atrial pressure and disturbed flows are seen, especially during systole, when a large regurgitant jet can be found with the suppressed pulmonary venous flow. We also show that both the rule-based and atlas-based fibre defining methods lead to similar flow fields and atrial wall deformations. However, the changes in wall thickness from non-uniform to uniform tend to underestimate the atrial deformation. Using a uniform but thickened wall also lowers the overall strain level. The flow velocity within the left atrial appendage, which is important in terms of appendage thrombosis, increases with the thickness of the left atrial wall. Energy analysis shows that the kinetic and dissipation energies of the flow within the left atrium are altered differently by atrial fibrillation and mitral valve regurgitation, providing a useful indication of the atrial performance in pathological situations.

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.

Simulation of action potential propagation based on the ghost structure method

In this paper, a ghost structure (GS) method is proposed to simulate the monodomain model in irregular computational domains using finite difference without regenerating body-fitted grids. In order to verify the validity of the GS method, it is first used to solve the Fitzhugh-Nagumo monodomain model in rectangular and circular regions at different states (the stationary and moving states). Then, the GS method is used to simulate the propagation of the action potential (AP) in transverse and longitudinal sections of a healthy human heart, and with left bundle branch block (LBBB). Finally, we analyze the AP and calcium concentration under healthy and LBBB conditions. Our numerical results show that the GS method can accurately simulate AP propagation with different computational domains either stationary or moving, and we also find that LBBB will cause the left ventricle to contract later than the right ventricle, which in turn affects synchronized contraction of the two ventricles.

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.

Fully coupled fluid-electro-mechanical model of the human heart for supercomputers

In this work, we present a fully coupled fluid-electro-mechanical model of a 50th percentile human heart. The model is implemented on Alya, the BSC multi-physics parallel code, capable of running efficiently in supercomputers. Blood in the cardiac cavities is modeled by the incompressible Navier-Stokes equations and an arbitrary Lagrangian-Eulerian (ALE) scheme. Electrophysiology is modeled with a monodomain scheme and the O'Hara-Rudy cell model. Solid mechanics is modeled with a total Lagrangian formulation for discrete strains using the Holzapfel-Ogden cardiac tissue material model. The three problems are simultaneously and bidirectionally coupled through an electromechanical feedback and a fluid-structure interaction scheme. In this paper, we present the scheme in detail and propose it as a computational cardiac workbench.

On the chordae structure and dynamic behaviour of the mitral valve

We develop a fluid-structure interaction (FSI) model of the mitral valve (MV) that uses an anatomically and physiologically realistic description of the MV leaflets and chordae tendineae. Three different chordae models-complex, 'pseudo-fibre' and simplified chordae-are compared to determine how different chordae representations affect the dynamics of the MV. The leaflets and chordae are modelled as fibre-reinforced hyperelastic materials, and FSI is modelled using an immersed boundary-finite element method. The MV model is first verified under static boundary conditions against the commercial finite element software ABAQUS and then used to simulate MV dynamics under physiological pressure conditions. Interesting flow patterns and vortex formulation are observed in all three cases. To quantify the highly complex system behaviour resulting from FSI, an energy budget analysis of the coupled MV FSI model is performed. Results show that the complex and pseudo-fibre chordae models yield good valve closure during systole but that the simplified chordae model leads to poorer leaflet coaptation and an unrealistic bulge in the anterior leaflet belly. An energy budget analysis shows that the MV models with complex and pseudo-fibre chordae have similar energy distribution patterns but the MV model with the simplified chordae consumes more energy, especially during valve closing and opening. We find that the complex chordae and pseudo-fibre chordae have similar impact on the overall MV function but that the simplified chordae representation is less accurate. Because a pseudo-fibre chordal structure is easier to construct and less computationally intensive, it may be a good candidate for modelling MV dynamics or interaction between the MV and heart in patient-specific applications.

High Spatial Resolution Multi-Organ Finite Element Modeling of Ventricular-Arterial Coupling

While it has long been recognized that bi-directional interaction between the heart and the vasculature plays a critical role in the proper functioning of the cardiovascular system, a comprehensive study of this interaction has largely been hampered by a lack of modeling framework capable of simultaneously accommodating high-resolution models of the heart and vasculature. Here, we address this issue and present a computational modeling framework that couples finite element (FE) models of the left ventricle (LV) and aorta to elucidate ventricular-arterial coupling in the systemic circulation. We show in a baseline simulation that the framework predictions of (1) LV pressure-volume loop, (2) aorta pressure-diameter relationship, (3) pressure-waveforms of the aorta, LV, and left atrium (LA) over the cardiac cycle are consistent with the physiological measurements found in healthy human. To develop insights of ventricular-arterial interactions, the framework was then used to simulate how alterations in the geometrical or, material parameter(s) of the aorta affect the LV and vice versa. We show that changing the geometry and microstructure of the aorta model in the framework led to changes in the functional behaviors of both LV and aorta that are consistent with experimental observations. On the other hand, changing contractility and passive stiffness of the LV model in the framework also produced changes in both the LV and aorta functional behaviors that are consistent with physiology principles.

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.

Hybrid finite difference/finite element immersed boundary method

The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach uses a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach.

Fast left ventricle tracking using localized anatomical affine optical flow

In daily clinical cardiology practice, left ventricle (LV) global and regional function assessment is crucial for disease diagnosis, therapy selection, and patient follow-up. Currently, this is still a time-consuming task, spending valuable human resources. In this work, a novel fast methodology for automatic LV tracking is proposed based on localized anatomically constrained affine optical flow. This novel method can be combined to previously proposed segmentation frameworks or manually delineated surfaces at an initial frame to obtain fully delineated datasets and, thus, assess both global and regional myocardial function. Its feasibility and accuracy were investigated in 3 distinct public databases, namely in realistically simulated 3D ultrasound, clinical 3D echocardiography, and clinical cine cardiac magnetic resonance images. The method showed accurate tracking results in all databases, proving its applicability and accuracy for myocardial function assessment. Moreover, when combined to previous state-of-the-art segmentation frameworks, it outperformed previous tracking strategies in both 3D ultrasound and cardiac magnetic resonance data, automatically computing relevant cardiac indices with smaller biases and narrower limits of agreement compared to reference indices. Simultaneously, the proposed localized tracking method showed to be suitable for online processing, even for 3D motion assessment. Importantly, although here evaluated for LV tracking only, this novel methodology is applicable for tracking of other target structures with minimal adaptations.

Estimating prognosis in patients with acute myocardial infarction using personalized computational heart models

Biomechanical computational models have potential prognostic utility in patients after an acute ST-segment-elevation myocardial infarction (STEMI). In a proof-of-concept study, we defined two groups (1) an acute STEMI group (n = 6, 83% male, age 54 ± 12 years) complicated by left ventricular (LV) systolic dysfunction; (2) an age- and sex- matched hyper-control group (n = 6, 83% male, age 46 ± 14 years), no prior history of cardiovascular disease and normal systolic blood pressure (SBP < 130 mmHg). Cardiac MRI was performed in the patients (2 days & 6 months post-STEMI) and the volunteers, and biomechanical heart models were synthesized for each subject. The candidate parameters included normalized active tension (AT norm) and active tension at the resting sarcomere length (T req, reflecting required contractility). Myocardial contractility was inversely determined from personalized heart models by matching CMR-imaged LV dynamics. Compared with controls, patients with recent STEMI exhibited increased LV wall active tension when normalized by SBP. We observed a linear relationship between T req 2 days post-MI and global longitudinal strain 6 months later (r = 0.86; p = 0.03). T req may be associated with changes in LV function in the longer term in STEMI patients complicated by LV dysfunction. Further studies seem warranted.

An Immersed Boundary method with divergence-free velocity interpolation and force spreading

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid-structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at leastC 1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

A continuum mechanics-based musculo-mechanical model for esophageal transport

In this work, we extend our previous esophageal transport model using an immersed boundary (IB) method with discrete fiber-based structural model, to one using a continuum mechanics-based model that is approximated based on finite elements (IB-FE). To deal with the leakage of flow when the Lagrangian mesh becomes coarser than the fluid mesh, we employ adaptive interaction quadrature points to deal with Lagrangian-Eulerian interaction equations based on a previous work (Griffith and Luo [1]). In particular, we introduce a new anisotropic adaptive interaction quadrature rule. The new rule permits us to vary the interaction quadrature points not only at each time-step and element but also at different orientations per element. This helps to avoid the leakage issue without sacrificing the computational efficiency and accuracy in dealing with the interaction equations. For the material model, we extend our previous fiber-based model to a continuum-based model. We present formulations for general fiber-reinforced material models in the IB-FE framework. The new material model can handle non-linear elasticity and fiber-matrix interactions, and thus permits us to consider more realistic material behavior of biological tissues. To validate our method, we first study a case in which a three-dimensional short tube is dilated. Results on the pressure-displacement relationship and the stress distribution matches very well with those obtained from the implicit FE method. We remark that in our IB-FE case, the three-dimensional tube undergoes a very large deformation and the Lagrangian mesh-size becomes about 6 times of Eulerian mesh-size in the circumferential orientation. To validate the performance of the method in handling fiber-matrix material models, we perform a second study on dilating a long fiber-reinforced tube. Errors are small when we compare numerical solutions with analytical solutions. The technique is then applied to the problem of esophageal transport. We use two fiber-reinforced models for the esophageal tissue: a bi-linear model and an exponential model. We present three cases on esophageal transport that differ in the material model and the muscle fiber architecture. The overall transport features are consistent with those observed from the previous model. We remark that the continuum-based model can handle more realistic and complicated material behavior. This is demonstrated in our third case where a spatially varying fiber architecture is included based on experimental study. We find that this unique muscle fiber architecture could generate a so-called pressure transition zone, which is a luminal pressure pattern that is of clinical interest. This suggests an important role of muscle fiber architecture in esophageal transport.

Modelling mitral valvular dynamics-current trend and future directions

Dysfunction of mitral valve causes morbidity and premature mortality and remains a leading medical problem worldwide. Computational modelling aims to understand the biomechanics of human mitral valve and could lead to the development of new treatment, prevention and diagnosis of mitral valve diseases. Compared with the aortic valve, the mitral valve has been much less studied owing to its highly complex structure and strong interaction with the blood flow and the ventricles. However, the interest in mitral valve modelling is growing, and the sophistication level is increasing with the advanced development of computational technology and imaging tools. This review summarises the state-of-the-art modelling of the mitral valve, including static and dynamics models, models with fluid-structure interaction, and models with the left ventricle interaction. Challenges and future directions are also discussed.

Image-based immersed boundary model of the aortic root

Each year, approximately 300,000 heart valve repair or replacement procedures are performed worldwide, including approximately 70,000 aortic valve replacement surgeries in the United States alone. Computational platforms for simulating cardiovascular devices such as prosthetic heart valves promise to improve device design and assist in treatment planning, including patient-specific device selection. This paper describes progress in constructing anatomically and physiologically realistic immersed boundary (IB) models of the dynamics of the aortic root and ascending aorta. This work builds on earlier IB models of fluid-structure interaction (FSI) in the aortic root, which previously achieved realistic hemodynamics over multiple cardiac cycles, but which also were limited to simplified aortic geometries and idealized descriptions of the biomechanics of the aortic valve cusps. By contrast, the model described herein uses an anatomical geometry reconstructed from patient-specific computed tomography angiography (CTA) data, and employs a description of the elasticity of the aortic valve leaflets based on a fiber-reinforced constitutive model fit to experimental tensile test data. The resulting model generates physiological pressures in both systole and diastole, and yields realistic cardiac output and stroke volume at physiological Reynolds numbers. Contact between the valve leaflets during diastole is handled automatically by the IB method, yielding a fully competent valve model that supports a physiological diastolic pressure load without regurgitation. Numerical tests show that the model is able to resolve the leaflet biomechanics in diastole and early systole at practical grid spacings. The model is also used to examine differences in the mechanics and fluid dynamics yielded by fresh valve leaflets and glutaraldehyde-fixed leaflets similar to those used in bioprosthetic heart valves. Although there are large differences in the leaflet deformations during diastole, the differences in the open configurations of the valve models are relatively small, and nearly identical hemodynamics are obtained in all cases considered.

Patient-specific modeling of left ventricular electromechanics as a driver for haemodynamic analysis

Aims

Models of blood flow in the left ventricle (LV) and aorta are an important tool for analysing the interplay between LV deformation and flow patterns. Typically, image-based kinematic models describing endocardial motion are used as an input to blood flow simulations. While such models are suitable for analysing the hemodynamic status quo, they are limited in predicting the response to interventions that alter afterload conditions. Mechano-fluidic models using biophysically detailed electromechanical (EM) models have the potential to overcome this limitation, but are more costly to build and compute. We report our recent advancements in developing an automated workflow for the creation of such CFD ready kinematic models to serve as drivers of blood flow simulations.

Methods and results

EM models of the LV and aortic root were created for four pediatric patients treated for either aortic coarctation or aortic valve disease. Using MRI, ECG and invasive pressure recordings, anatomy as well as electrophysiological, mechanical and circulatory model components were personalized.

Results

The implemented modeling pipeline was highly automated and allowed model construction and execution of simulations of a patient’s heartbeat within 1 day. All models reproduced clinical data with acceptable accuracy.

Conclusion

Using the developed modeling workflow, the use of EM LV models as driver of fluid flow simulations is becoming feasible. While EM models are costly to construct, they constitute an important and nontrivial step towards fully coupled electro-mechano-fluidic (EMF) models and show promise as a tool for predicting the response to interventions which affect afterload conditions.

A coupled mitral valve-left ventricle model with fluid-structure interaction

Understanding the interaction between the valves and walls of the heart is important in assessing and subsequently treating heart dysfunction. This study presents an integrated model of the mitral valve (MV) coupled to the left ventricle (LV), with the geometry derived from in vivo clinical magnetic resonance images. Numerical simulations using this coupled MV-LV model are developed using an immersed boundary/finite element method. The model incorporates detailed valvular features, left ventricular contraction, nonlinear soft tissue mechanics, and fluid-mediated interactions between the MV and LV wall. We use the model to simulate cardiac function from diastole to systole. Numerically predicted LV pump function agrees well with in vivo data of the imaged healthy volunteer, including the peak aortic flow rate, the systolic ejection duration, and the LV ejection fraction. In vivo MV dynamics are qualitatively captured. We further demonstrate that the diastolic filling pressure increases significantly with impaired myocardial active relaxation to maintain a normal cardiac output. This is consistent with clinical observations. The coupled model has the potential to advance our fundamental knowledge of mechanisms underlying MV-LV interaction, and help in risk stratification and optimisation of therapies for heart diseases.

A mathematical model for active contraction in healthy and failing myocytes and left ventricles

Cardiovascular disease is one of the leading causes of death worldwide, in particular myocardial dysfunction, which may lead to heart failure eventually. Understanding the electro-mechanics of the heart will help in developing more effective clinical treatments. In this paper, we present a multi-scale electro-mechanics model of the left ventricle (LV). The Holzapfel-Ogden constitutive law was used to describe the passive myocardial response in tissue level, a modified Grandi-Pasqualini-Bers model was adopted to model calcium dynamics in individual myocytes, and the active tension was described using the Niederer-Hunter-Smith myofilament model. We first studied the electro-mechanics coupling in a single myocyte in the healthy and diseased left ventricle, and then the single cell model was embedded in a dynamic LV model to investigate the compensation mechanism of LV pump function due to myocardial dysfunction caused by abnormality in cellular calcium dynamics. The multi-scale LV model was solved using an in-house developed hybrid immersed boundary method with finite element extension. The predictions of the healthy LV model agreed well with the clinical measurements and other studies, and likewise, the results in the failing states were also consistent with clinical observations. In particular, we found that a low level of intracellular Ca2+ transient in myocytes can result in LV pump function failure even with increased myocardial contractility, decreased systolic blood pressure, and increased diastolic filling pressure, even though they will increase LV stroke volume. Our work suggested that treatments targeted at increased contractility and lowering the systolic blood pressure alone are not sufficient in preventing LV pump dysfunction, restoring a balanced physiological Ca2+ handling mechanism is necessary.

Immersed boundary-finite element model of fluid-structure interaction in the aortic root

It has long been recognized that aortic root elasticity helps to ensure efficient aortic valve closure, but our understanding of the functional importance of the elasticity and geometry of the aortic root continues to e-volve as increasingly detailed in vivo imaging data become available. Herein, we describe a fluid-structure interaction model of the aortic root, including the aortic valve leaflets, the sinsuses of Valsalva, the aortic annulus, and the sinotubular junction, that employs a version of Peskin's immersed boundary (IB) method with a finite element (FE) description of the structural elasticity. As in earlier work, we use a fiber-based model of the valve leaflets, but this study extends earlier IB models of the aortic root by employing an incompressible hyperelastic model of the mechanics of the sinuses and ascending aorta using a constitutive law fit to experimental data from human aortic root tissue. In vivo pressure loading is accounted for by a backward displacement method that determines the unloaded configurations of the root model. Our model yields realistic cardiac output at physiological pressures, with low transvalvular pressure differences during forward flow, minimal regurgitation during valve closure, and realistic pressure loads when the valve is closed during diastole. Further, results from high-resolution computations indicate that although the detailed leaflet and root kinematics show some grid sensitivity, our IB model of the aortic root nonetheless produces essentially grid-converged flow rates and pressures at practical grid spacings for the high-Reynolds number flows of the aortic root. These results thereby clarify minimum grid resolutions required by such models when used as stand-alone models of the aortic valve as well as when used to provide models of the outflow valves in models of left ventricular fluid dynamics.

Study of cardiovascular function using a coupled left ventricle and systemic circulation model

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV–SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i.e. the SA model feeds back the pressure as a boundary condition to the LV model, and the aortic flow rate from the LV model is used as the input for the SA model. The governing equations of the coupled system are solved using a combined immersed-boundary finite-element (IB/FE) method and a Lax–Wendroff scheme. A baseline case using physiological measurements of healthy subjects, and four exemplar cases based on different physiological and pathological scenarios are studied using the LV–SA model. The results of the baseline case agree well with published experimental data. The four exemplar cases predict varied pathological responses of the cardiovascular system, which are also supported by clinical observations. The new model can be used to gain insight into cardio-arterial interactions across a range of clinical applications.

Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour

Models of cardiac mechanics are increasingly used to investigate cardiac physiology. These models are characterized by a high level of complexity, including the particular anisotropic material properties of biological tissue and the actively contracting material. A large number of independent simulation codes have been developed, but a consistent way of verifying the accuracy and replicability of simulations is lacking. To aid in the verification of current and future cardiac mechanics solvers, this study provides three benchmark problems for cardiac mechanics. These benchmark problems test the ability to accurately simulate pressure-type forces that depend on the deformed objects geometry, anisotropic and spatially varying material properties similar to those seen in the left ventricle and active contractile forces. The benchmark was solved by 11 different groups to generate consensus solutions, with typical differences in higher-resolution solutions at approximately 0.5%, and consistent results between linear, quadratic and cubic finite elements as well as different approaches to simulating incompressible materials. Online tools and solutions are made available to allow these tests to be effectively used in verification of future cardiac mechanics software.

Emerging Trends in Heart Valve Engineering: Part IV. Computational Modeling and Experimental Studies

In this final portion of an extensive review of heart valve engineering, we focus on the computational methods and experimental studies related to heart valves. The discussion begins with a thorough review of computational modeling and the governing equations of fluid and structural interaction. We then move onto multiscale and disease specific modeling. Finally, advanced methods related to in vitro testing of the heart valves are reviewed. This section of the review series is intended to illustrate application of computational methods and experimental studies and their interrelation for studying heart valves.

Parameter estimation in a Holzapfel-Ogden law for healthy myocardium

A central problem in biomechanical studies of personalized human left ventricular (LV) modelling is to estimate material properties from in vivo clinical measurements. In this work we evaluate the passive myocardial mechanical properties inversely from the in vivo LV chamber pressure-volume and strain data. The LV myocardium is described using a structure-based orthotropic Holzapfel-Ogden constitutive law with eight parameters. In the first part of the paper we demonstrate how to use a multi-step non-linear least-squares optimization procedure to inversely estimate the parameters from the pressure-volume and strain data obtained from a synthetic LV model in diastole. In the second part, we show that to apply this procedure to clinical situations with limited in vivo data, additional constraints are required in the optimization procedure. Our study, based on three different healthy volunteers, demonstrates that the parameters of the Holzapfel-Ogden law could be extracted from pressure-volume and strain data with a suitable multi-step optimization procedure. Although the uniqueness of the solution cannot be addressed using our approaches, the material response is shown to be robustly determined.

A finite strain nonlinear human mitral valve model with fluid-structure interaction

A computational human mitral valve (MV) model under physiological pressure loading is developed using a hybrid finite element immersed boundary method, which incorporates experimentally-based constitutive laws in a three-dimensional fluid-structure interaction framework. A transversely isotropic material constitutive model is used to characterize the mechanical behaviour of the MV tissue based on recent mechanical tests of healthy human mitral leaflets. Our results show good agreement, in terms of the flow rate and the closing and opening configurations, with measurements from in vivo magnetic resonance images. The stresses in the anterior leaflet are found to be higher than those in the posterior leaflet and are concentrated around the annulus trigons and the belly of the leaflet. The results also show that the chordae play an important role in providing a secondary orifice for the flow when the valve opens. Although there are some discrepancies to be overcome in future work, our simulations show that the developed computational model is promising in mimicking the in vivo MV dynamics and providing important information that are not obtainable by in vivo measurements.

Dynamic finite-strain modelling of the human left ventricle in health and disease using an immersed boundary-finite element method

Detailed models of the biomechanics of the heart are important both for developing improved interventions for patients with heart disease and also for patient risk stratification and treatment planning. For instance, stress distributions in the heart affect cardiac remodelling, but such distributions are not presently accessible in patients. Biomechanical models of the heart offer detailed three-dimensional deformation, stress and strain fields that can supplement conventional clinical data. In this work, we introduce dynamic computational models of the human left ventricle (LV) that are derived from clinical imaging data obtained from a healthy subject and from a patient with a myocardial infarction (MI). Both models incorporate a detailed invariant-based orthotropic description of the passive elasticity of the ventricular myocardium along with a detailed biophysical model of active tension generation in the ventricular muscle. These constitutive models are employed within a dynamic simulation framework that accounts for the inertia of the ventricular muscle and the blood that is based on an immersed boundary (IB) method with a finite element description of the structural mechanics. The geometry of the models is based on data obtained non-invasively by cardiac magnetic resonance (CMR). CMR imaging data are also used to estimate the parameters of the passive and active constitutive models, which are determined so that the simulated end-diastolic and end-systolic volumes agree with the corresponding volumes determined from the CMR imaging studies. Using these models, we simulate LV dynamics from enddiastole to end-systole. The results of our simulations are shown to be in good agreement with subject-specific CMR-derived strain measurements and also with earlier clinical studies on human LV strain distributions.