Models Of Cardiac Mechanics Research Articles

Post-infarct evolution of ventricular and myocardial function

Adverse ventricular remodeling following acute myocardial infarction (MI) may induce ventricular dilation, fibrosis, and loss of global contractile function, possibly resulting in heart failure (HF). Understanding the relation between the time-dependent changes in material properties of the myocardium and the contractile function of the heart may further our understanding of the development of HF post-MI and guide the development of novel therapies. A finite element model of cardiac mechanics was used to model MI in a thick-walled truncated ellipsoidal geometry. Infarct core and border zone comprised 9.6 and 8.1% of the LV wall volume, respectively. Acute MI was modeled by inhibiting active stress generation. Chronic MI was modeled by the additional effect of infarct material stiffening, wall thinning and fiber reorientation. In acute MI, stroke work decreased by 25%. In the infarct core, fiber stress was reduced but fiber strain was increased, depending on the degree of infarct stiffening. Fiber work density was equal to zero. Healthy tissue adjacent to the infarct showed decreased work density depending on the degree of infarct stiffness and the orientation of the myofibers with respect to the infarct region. Thinning of the wall partially restored this loss in work density while the effects of fiber reorientation were minimal. We found that the relative loss in pump function in the infarcted heart exceeds the relative loss in healthy myocardial tissue due to impaired mechanical function in healthy tissue adjacent to the infarct. Infarct stiffening, wall thinning and fiber reorientation did not affect pump function but did affect the distribution of work density in tissue adjacent to the infarct.

Investigating the reference domain influence in personalised models of cardiac mechanics : Effect of unloaded geometry on cardiac biomechanics.

A major concern in personalised models of heart mechanics is the unknown zero-pressure domain, a prerequisite for accurately predicting cardiac biomechanics. As the reference configuration cannot be captured by clinical data, studies often employ in-vivo frames which are unlikely to correspond to unloaded geometries. Alternatively, zero-pressure domain is approximated through inverse methodologies, which, however, entail assumptions pertaining to boundary conditions and material parameters. Both approaches are likely to introduce biases in estimated biomechanical properties; nevertheless, quantification of these effects is unattainable without ground-truth data. In this work, we assess the unloaded state influence on model-derived biomechanics, by employing an in-silico modelling framework relying on experimental data on porcine hearts. In-vivo images are used for model personalisation, while in-situ experiments provide a reliable approximation of the reference domain, creating a unique opportunity for a validation study. Personalised whole-cycle cardiac models are developed which employ different reference domains (image-derived, inversely estimated) and are compared against ground-truth model outcomes. Simulations are conducted with varying boundary conditions, to investigate the effect of data-derived constraints on model accuracy. Attention is given to modelling the influence of the ribcage on the epicardium, due to its close proximity to the heart in the porcine anatomy. Our results find merit in both approaches for dealing with the unknown reference domain, but also demonstrate differences in estimated biomechanical quantities such as material parameters, strains and stresses. Notably, they highlight the importance of a boundary condition accounting for the constraining influence of the ribcage, in forward and inverse biomechanical models.

An oscillation-free fully staggered algorithm for velocity-dependent active models of cardiac mechanics

In this paper we address an unresolved problem in the numerical modeling of cardiac electromechanics, that is the onset of numerical oscillations due to the dependence of force generation models on the fibers shortening velocity. A way to avoid numerical oscillations is to use monolithic schemes for the solution of the coupled problem of active–passive mechanics. However, staggered strategies, which foresee the sequential solution of the models of force generation and of tissue mechanics, are preferable, due to their reduced computational cost and low implementation effort. In this paper we propose a cure for this issue, by introducing, with respect to the standard staggered scheme, a numerically consistent stabilization term. This term is derived in virtue of the identification of the cause of instability in the mismatch between macroscopic and microscopic strains, inconsistently expressed in Lagrangian and Eulerian coordinates, respectively. By considering a model problem of active mechanics we prove that the proposed scheme is unconditionally absolutely stable (i.e. it is stable for any time step size), yet within a fully staggered framework. As such, the new scheme removes the non-physical oscillations, as we prove by applying it to three force generation models, namely the Niederer–Hunter–Smith model, the model by Land and coworkers, and the mean-field force generation model that we have recently proposed.

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

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

Efficient estimation of load-free left ventricular geometry and passive myocardial properties using principal component analysis.

Models of cardiac mechanics require a well-defined reference geometry from which deformations and hence myocardial strain and stress can be calculated. In the in vivo beating heart, the load-free (LF) geometry generally cannot be measured directly, since, in many cases, there is no stage at which the lumen pressures and contractile state are all zero. Therefore, there is a need for an efficient method to estimate the LF geometry, which is essential for an accurate mechanical simulation of left ventricular (LV) mechanics, and for estimations of passive and contractile constitutive parameters of the heart muscle. In this paper, we present a novel method for estimating both the LF geometry and the passive stiffness of the myocardium. A linear combination of principal components from a population of diastolic displacements is used to construct the LF geometry. For each estimate of the LF geometry and tissue stiffness, LV inflation is simulated, and the model predictions are compared with surface data at multiple stages during passive diastolic filling. The feasibility of this method was demonstrated using synthetically deformation data that were generated using LV models derived from clinical magnetic resonance image data, and the identifiability of the LF geometry and passive stiffness parameters were analysed using Hessian metrics. Applications of this method to clinical data would improve the accuracy of constitutive parameter estimation and allow a better simulation of LV wall strains and stresses.

In vivo estimation of elastic heterogeneity in an infarcted human heart

In myocardial infarction, muscle tissue of the heart is damaged as a result of ceased or severely impaired blood flow. Survivors have an increased risk of further complications, possibly leading to heart failure. Material properties play an important role in determining post-infarction outcome. Due to spatial variation in scarring, material properties can be expected to vary throughout the tissue of a heart after an infarction. In this study we propose a data assimilation technique that can efficiently estimate heterogeneous elastic material properties in a personalized model of cardiac mechanics. The proposed data assimilation is tested on a clinical dataset consisting of regional left ventricular strains and in vivo pressures during atrial systole from a human with a myocardial infarction. Good matches to regional strains are obtained, and simulated equi-biaxial tests are carried out to demonstrate regional heterogeneities in stress–strain relationships. A synthetic data test shows a good match of estimated versus ground truth material parameter fields in the presence of no to low levels of noise. This study is the first to apply adjoint-based data assimilation to the important problem of estimating cardiac elastic heterogeneities in 3-D from medical images.

Mathematical modeling of left ventricle hypertrophy and dilatation in response to volume overload in heart failure: a coupled renal‐cardiac model

It is well‐established that the heart remodels in response to changes in mechanical loads, and mathematical modeling of cardiac function and remodeling is an advanced field. However, volume regulation is a critical component in the pathogenesis, progression, and management of congestive heart failure, and few existing models account for the interaction between cardiac function and volume regulation by the kidney and neurohumoral signaling. A primary failure in cardiac function is compounded by volume retention as the heart and kidney work together to try to maintain organ perfusion. Eccentric hypertrophy occurs in response to volume overload, as increases in ventricular filling pressure elevate stress and stretch experienced in myocytes during diastole is believed to cause the series replication of sarcomeres and chamber enlargement. In this study, we utilized a coupled model of cardiac mechanics [1] and renal function, volume regulation, and the renin‐angiotensin‐aldosterone system[2]. We previously expanded the model to include mechanisms of cardiac hypertrophy and wall thickening in response to pressure overload – a self‐limiting process. We have now also incorporated a mechanism for left ventricle dilatation in response to volume overload, a progressive process. Specifically, myocyte length is modeled as a function of end‐diastolic myocyte stress, a relationship governed by a time constant. The model reflects the expected behavior of heart failure patients: reduced ejection fraction and increased left ventricle end diastolic volume, stress, and pressure. After inducing “heart failure” by reducing cardiac contractility, we used temporal pressure‐volume data from heart failure patients in the SOLVD study to ensure that the model accurately reflects the structural and hemodynamic changes characteristic of heart failure. We also used published data to establish an empirical relationship between left ventricle End Diastolic Stress and n‐Type Brain Natriuretic Peptide (NT‐pro BNP). We then validated the model by simulating and showing agreement with changes in the pressure‐volume loop in response to enalapril treatment in SOLVD, and NT‐proBNP at baseline and after treatment with valsartan in the VALHEFT[4] trial. This model improve our understanding of the hemodynamic interactions between renal and cardiac function and volume homeostasis in disease, and is a step toward in silico evaluation of new therapeutic mechanisms.Support or Funding InformationThis work was funded by collaborative grant from Pfizer Worldwide Research & Development and Merck & Co., Inc.This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

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.

Non-invasive Model-Based Assessment of Passive Left-Ventricular Myocardial Stiffness in Healthy Subjects and in Patients with Non-ischemic Dilated Cardiomyopathy

Patient-specific modelling has emerged as a tool for studying heart function, demonstrating the potential to provide non-invasive estimates of tissue passive stiffness. However, reliable use of model-derived stiffness requires sufficient model accuracy and unique estimation of model parameters. In this paper we present personalised models of cardiac mechanics, focusing on improving model accuracy, while ensuring unique parametrisation. The influence of principal model uncertainties on accuracy and parameter identifiability was systematically assessed in a group of patients with dilated cardiomyopathy (n=3) and healthy volunteers (n=5). For all cases, we examined three circumferentially symmetric fibre distributions and two epicardial boundary conditions. Our results demonstrated the ability of data-derived boundary conditions to improve model accuracy and highlighted the influence of the assumed fibre distribution on both model fidelity and stiffness estimates. The model personalisation pipeline—based strictly on non-invasive data—produced unique parameter estimates and satisfactory model errors for all cases, supporting the selected model assumptions. The thorough analysis performed enabled the comparison of passive parameters between volunteers and dilated cardiomyopathy patients, illustrating elevated stiffness in diseased hearts.

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.

Analysis of passive cardiac constitutive laws for parameter estimation using 3D tagged MRI.

An unresolved issue in patient-specific models of cardiac mechanics is the choice of an appropriate constitutive law, able to accurately capture the passive behavior of the myocardium, while still having uniquely identifiable parameters tunable from available clinical data. In this paper, we aim to facilitate this choice by examining the practical identifiability and model fidelity of constitutive laws often used in cardiac mechanics. Our analysis focuses on the use of novel 3D tagged MRI, providing detailed displacement information in three dimensions. The practical identifiability of each law is examined by generating synthetic 3D tags from in silico simulations, allowing mapping of the objective function landscape over parameter space and comparison of minimizing parameter values with original ground truth values. Model fidelity was tested by comparing these laws with the more complex transversely isotropic Guccione law, by characterizing their passive end-diastolic pressure–volume relation behavior, as well as by considering the in vivo case of a healthy volunteer. These results show that a reduced form of the Holzapfel–Ogden law provides the best balance between identifiability and model fidelity across the tests considered.

A displacement-based finite element formulation for incompressible and nearly-incompressible cardiac mechanics

The Lagrange Multiplier (LM) and penalty methods are commonly used to enforce incompressibility and compressibility in models of cardiac mechanics. In this paper we show how both formulations may be equivalently thought of as a weakly penalized system derived from the statically condensed Perturbed Lagrangian formulation, which may be directly discretized maintaining the simplicity of penalty formulations with the convergence characteristics of LM techniques. A modified Shamanskii–Newton–Raphson scheme is introduced to enhance the nonlinear convergence of the weakly penalized system and, exploiting its equivalence, modifications are developed for the penalty form. Focusing on accuracy, we proceed to study the convergence behavior of these approaches using different interpolation schemes for both a simple test problem and more complex models of cardiac mechanics. Our results illustrate the well-known influence of locking phenomena on the penalty approach (particularly for lower order schemes) and its effect on accuracy for whole-cycle mechanics. Additionally, we verify that direct discretization of the weakly penalized form produces similar convergence behavior to mixed formulations while avoiding the use of an additional variable. Combining a simple structure which allows the solution of computationally challenging problems with good convergence characteristics, the weakly penalized form provides an accurate and efficient alternative to incompressibility and compressibility in cardiac mechanics.

Computational model for estimating the short- and long-term cardiac response to arteriovenous fistula creation for hemodialysis

Creation of an arteriovenous fistula (AVF) for hemodialysis may result in cardiac failure due to dramatic increases in cardiac output. To investigate the quantitative relations between AVF flow, changes in cardiac output, myocardial stress and strain and resulting left ventricular adaptation, a computational model is developed. The model combines a one-dimensional pulse wave propagation model of the arterial network with a zero-dimensional one-fiber model of cardiac mechanics and includes adaptation rules to capture the effect of the baro-reflex and long-term structural remodelling of the left ventricle. Using generic vascular and cardiac parameters based on literature, simulations are done that illustrate the model's ability to quantitatively reproduce the clinically observed increase in brachial flow and cardiac output as well as occurence of eccentric hypertrophy. Patient-specific clinical data is needed to investigate the value of the computational model for personalized predictions.

Determinants of left ventricular shear strain

Mathematical models of cardiac mechanics can potentially be used to relate abnormal cardiac deformation, as measured noninvasively by ultrasound strain rate imaging or magnetic resonance tagging (MRT), to the underlying pathology. However, with current models, the correct prediction of wall shear strain has proven to be difficult, even for the normal healthy heart. Discrepancies between simulated and measured strains have been attributed to 1) inadequate modeling of passive tissue behavior, 2) neglecting active stress development perpendicular to the myofiber direction, or 3) neglecting crossover of myofibers in between subendocardial and subepicardial layers. In this study, we used a finite-element model of left ventricular (LV) mechanics to investigate the sensitivity of midwall circumferential-radial shear strain (E(cr)) to settings of parameters determining passive shear stiffness, cross-fiber active stress development, and transmural crossover of myofibers. Simulated time courses of midwall LV E(cr) were compared with time courses obtained in three healthy volunteers using MRT. E(cr) as measured in the volunteers during the cardiac cycle was characterized by an amplitude of approximately 0.1. In the simulations, a realistic amplitude of the E(cr) signal could be obtained by tuning either of the three model components mentioned above. However, a realistic time course of E(cr), with virtually no change of E(cr) during isovolumic contraction and a correct base-to-apex gradient of E(cr) during ejection, could only be obtained by including transmural crossover of myofibers. Thus, accounting for this crossover seems to be essential for a realistic model of LV wall mechanics.

Reconstruction of myocardial tissue motion and strain fields from displacement-encoded MR imaging

A quantitative analysis of myocardial mechanics is fundamental to understanding cardiac function, diagnosis of heart disease, and assessment of therapeutic intervention. Displacement encoding with stimulated-echo (DENSE) magnetic resonance imaging (MRI) technique was developed to track the three-dimensional (3D) displacement vector of discrete material grid points in the myocardial tissue. Despite the wealth of information gained from DENSE images, the current software only provides two-dimensional in-plane deformation. The objective of this study is to introduce a postprocessing method to reconstruct and visualize continuous dynamic 3D displacement and strain fields in the ventricular wall from DENSE data. An anatomically accurate hexagonal finite-element model of the left ventricle (LV) is reconstructed by fitting a prolate spheroidal primitive to contour points of the epi- and endocardial surfaces. The continuous displacement field in the model is described mathematically based on the discrete DENSE vectors using a minimization method with smoothness regularization. Based on the displacement, heart motion and myocardial stretch (or strain) are analyzed. Illustratory computations were conducted with DENSE data of three infarcted and one normal sheep ventricles. The full 3D results show stronger overall axial shortening, wall thickening, and twisting of the normal LV compared with the infarcted hearts. Local myocardial stretches show a dyskinetic LV in the apical region, dilation of apex in systole, and a compensatory increase in strain in the healthy basal region as a compensatory mechanism. We conclude that the proposed postprocessing method significantly extends the utility of DENSE MRI, which may provide a patient-specific 3D model of cardiac mechanics.

Coupling of a 3D Finite Element Model of Cardiac Ventricular Mechanics to Lumped Systems Models of the Systemic and Pulmonic Circulation

In this study we present a novel, robust method to couple finite element (FE) models of cardiac mechanics to systems models of the circulation (CIRC), independent of cardiac phase. For each time step through a cardiac cycle, left and right ventricular pressures were calculated using ventricular compliances from the FE and CIRC models. These pressures served as boundary conditions in the FE and CIRC models. In succeeding steps, pressures were updated to minimize cavity volume error (FE minus CIRC volume) using Newton iterations. Coupling was achieved when a predefined criterion for the volume error was satisfied. Initial conditions for the multi-scale model were obtained by replacing the FE model with a varying elastance model, which takes into account direct ventricular interactions. Applying the coupling, a novel multi-scale model of the canine cardiovascular system was developed. Global hemodynamics and regional mechanics were calculated for multiple beats in two separate simulations with a left ventricular ischemic region and pulmonary artery constriction, respectively. After the interventions, global hemodynamics changed due to direct and indirect ventricular interactions, in agreement with previously published experimental results. The coupling method allows for simulations of multiple cardiac cycles for normal and pathophysiology, encompassing levels from cell to system.

Assessment of Cardiac Performance with Magnetic Resonance Imaging

A quantitative assessment of regional cardiac performance is required for the diagnosis of disease, evaluation of severity and the quantification of treatment effect. MRI allows the noninvasive quantification of motion and deformation in the heart, including the precise assessment of all components of deformation in all regions of the heart throughout the cardiac cycle. In recent years, these imaging protocols have become standardized in both the research and clinical settings. However, adoption in the routine clinical environment has been hindered by the complex and time-consuming nature of the image post-processing. Model-based image analysis procedures provide a powerful mechanism for the fast, accurate assessment of cardiac MRI data and lend themselves to biophysical analysis and standardized functional mapping proce- dures. This paper reviews the current state of the art in MRI assessment of cardiac performance with an emphasis on mathematical modeling analysis procedures. Firstly, fast and accurate evaluation of mass and volume is discussed using interactive 4D modeling techniques. Analysis of tissue function, strain and strain rate is then reviewed. Mathematical models of regional tissue function and wall motion allow registration between cases and across groups, enabling quantifi- cation of multidimensional patterns of wall motion between disease and treatment groups. Finally, information on myo- cardial tissue kinematics can be incorporated into biophysical models of cardiac mechanics and used to gain an under- standing of how physiological tissue parameters such as contractility, ventricular compliance and electrical activation combine to effect whole heart function.

Towards model-based analysis of cardiac MR tagging data: Relation between left ventricular shear strain and myofiber orientation

Many cardiac pathologies are reflected in abnormal myocardial deformation, accessible through magnetic resonance tagging (MRT). Interpretation of the MRT data is difficult, since the relation between pathology and deformation is not straightforward. Mathematical models of cardiac mechanics could be used to translate measured abnormalities into the underlying pathology, but, so far, they even fail to correctly simulate myocardial deformation in the healthy heart. In this study we investigated to what extent (1) our previously published three-dimensional finite element model of cardiac mechanics [Kerckhoffs, R.C.P., Bovendeerd, P.H.M., Kotte, J.C.S., Prinzen, F.W., Smits, K., Arts, T., 2003. Homogeneity of cardiac contraction despite physiological asynchrony of depolarization: a model study. Ann. Biomed. Eng. 31, 536-547] can simulate measured cardiac deformation, and (2) discrepancies between strains in model and experiment are related to the choice of the myofiber orientation in the model. To this end, we measured midwall circumferential strain E(cc) and circumferential-radial shear strain E(cr) in three healthy subjects using MRT. E(cc) as computed in the model agreed well with measured E(cc). Computed E(cr) differed significantly from measured E(cr). The time course of E(cr) was found to be very sensitive to the choice of the myofiber orientation, in particular to the choice of the transverse angle. Discrepancies between circumferential-radial shear strain in model and experiment were reduced strongly by increasing the transverse angle in the original model by 25%.

End–diastolic myofiber stress and ejection strain increase with ventricular volume overload

Myocardial stress and strain are considered primary mechanical stimuli for hypertrophic remodeling. Their values and significance in the intact beating heart during chronic overload remain poorly characterized. Left-ventricular (LV) dimensions (echocardiography) and pressure (invasive) were simultaneously recorded in anesthetized dogs at sinus rhythm (SR), acute and 1, 2, 6, 12 weeks of atrioventricular block (AVB), leading to structural, electrical and contractile remodeling. Mechanical load of the myocardium was quantified as myofiber stress (sigma(f)), being force along myofiber orientation per cross-sectional area, and natural myofiber strain (e(f)), being change in natural logarithm of myofiber length (l) divided by its reference length: e(f) = ln(l/l(ref)). Time courses of sigma(f) and e(f) were calculated from LV pressure and dimensions, using a validated mathematical model of cardiac mechanics. End-diastolic sigmaf increased from 2.0 +/- 0.1 kPa at SR to 3.4 +/- 0.3 kPa at acute AVB, remaining elevated for > 6 weeks. Systolic sigma(f) was not affected by AVB. Ejection strain rose instantly upon AVB, reaching a maximum at 2 weeks: 0.24 +/- 0.02 vs. 0.10 +/- 0.01 at SR. The increase of myofiber stroke work (sigma(f)-e(f) loop area) from 3.1 +/- 0.3 at SR to 6.0 +/- 0.5 kJ/m(3)/beat at 1 week AVB was attributed mainly to an increase of strain during ejection. Stroke work and ejection strain remained elevated up to 12 weeks. The rate of LV-mass increase was maximal (2.2 +/- 0.4 g/day) at 1 week AVB. Serial mechanical phenotyping is feasible in the intact anesthetized dog with chronic ventricular overload. Our new approach yields values of mechanical load that are comparable to those found in isolated myocardium by others. In chronic AVB, both end-diastolic myofiber stress and ejection strain are increased. Early increases of both parameters coincide with peak hypertrophic growth, suggesting their important role for mechanotransduction. Peak systolic sigmaf is likely not important for hypertrophy in this model, since it does not change throughout the experiment.

Nonlinear incompressible finite element for simulating loading of cardiac tissue--Part II: Three dimensional formulation for thick ventricular wall segments.

A three dimensional incompressible and geometrically as well as materially nonlinear finite element is formulated for future implementation in models of cardiac mechanics. The stress-strain relations in the finite element are derived from a recently proposed constitutive law which is based on the histological composition of the myocardium. The finite element is formulated for large deformations and considers incompressibility by introducing the hydrostatic pressure as an additional variable. The results of passive loading cases simulated by this element allow to analyze the mechanical properties of ventricular wall segments, the main of which are that the circumferential direction is stiffer than the longitudinal one, that its shear stiffness is considerably lower than its tensile and compressive stiffness and that, due to its mechanically prominent role, the collagenous matrix may affect the myocardial perfusion.