Smoothed finite element methods in simulation of active contraction of myocardial tissue samples.

In modelling and simulation of cardiac mechanics, tetrahedral meshes are often used due to the easy availability of efficient meshing algorithms. This is beneficial in particular when complex geometries such as cardiac structures are considered. The gold standard in simulating the cardiac cycle is to solve the mechanical balance equations with the finite element method (FEM). However, using linear shape functions in the FEM in combination with nearly-incompressible material models is known to produce overly stiff approximations, whereas higher order elements are computationally more expensive. To overcome these problems, smoothed finite element methods (S-FEMs) have been proposed by Liu and co-workers. So far, S-FEMs in 3D have been utilised only in simulations of passive mechanics. In the present work, different S-FEMs are for the first time used for simulation of an active cardiac contraction on three-dimensional myocardial tissue samples. Further, node-based S-FEM (NS-FEM), face-based S-FEM (FS-FEM) and selective FS/NS-FEM are for the first time implemented as user subroutine in the commercial software Abaqus. Our results confirm that all S-FEMs perform softer than linear FEM and volumetric locking is reduced. The FS/NS-FEM produces solutions with the relative error in maximum displacement and rotation being less than 5% with respect to the reference solution obtained by the quadratic FEM for all considered mesh sizes, although linear shape functions are used. We therefore conclude that in particular FS/NS-FEM is an efficient and accurate numerical method in the simulation of an active cardiac muscle contraction.

Transmural fibre orientations based on Laplace-Dirichlet-Rule-Based-Methods and their influence on human heart simulations.

It is well known that the orthotropic tissue structure decisively influences the mechanical and electrical properties of the heart. Numerous approaches to compute the orthotropic tissue structure in computational heart models have been developed in the past decades. In this study, we investigate to what extent different Laplace-Dirichlet-Rule-Based-Methods (LDRBMs) influence the local orthotropic tissue structure and thus the electromechanical behaviour of the subsequent cardiac simulation. In detail, we are utilising three Laplace-Dirichlet-Rule-Based-Methods and compare: (i) the local myofibre orientation; (ii) important global characteristics (ejection fraction, peak pressure, apex shortening, myocardial volume reduction, fractional wall thickening); (iii) local characteristics (active fibre stress, fibre strain). We observe that the orthotropic tissue structures for the three LDRBMs show significant differences in the local myofibre orientation. The global characteristics myocardial volume reduction and peak pressure are rather insensitive to a change in local myofibre orientation, while the ejection fraction is moderately influenced by the different LDRBMs. Moreover, the apical shortening and fractional wall thickening exhibit a sensitive behaviour to a change in the local myofibre orientation. The highest sensitivity can be observed for the local characteristics.

Effects of PTH glandular and external dosing patterns on bone cell activity using a two-state receptor model-Implications for bone disease progression and treatment.

Temporal aspects of ligand specificity have been shown to play a significant role in the case of pulsatile hormone secretion, as exemplified by parathyroid hormone (PTH) binding to its receptor (PTH1R), a G-protein-coupled receptor expressed on surfaces of osteoblasts and osteocytes. The latter binding reaction regulates intracellular signalling and subsequently modulates skeletal homeostasis via bone remodelling. PTH glandular secretion patterns dictate bone cellular activity. In healthy humans, 70% of PTH is secreted in a tonic fashion, whereas 30% is secreted in low-amplitude and high-frequency bursts occurring every 10-20 min, superimposed on the tonic secretion. Changes in the PTH secretion patterns have been associated with various bone diseases. In this paper, we analyse PTH glandular secretion patterns for healthy and pathological states and their link to bone cellular responsiveness (αR). We utilise a two-state receptor ligand binding model of PTH to PTH1R together with a cellular activity function which is able to distinguish various aspects of the stimulation signal including peak dose, time of ligand exposure, and exposure period. Formulating and solving several constrained optimisation problems, we investigate the potential of pharmacological manipulation of the diseased glandular secretion and via clinical approved external PTH injections to restore healthy bone cellular responsiveness. Based on the mean experimentally reported data, our simulation results indicate cellular responsiveness in healthy subjects is sensitive to the tonic baseline stimulus and it is 28% of the computed maximum responsiveness. Simulation results for pathological cases of glucocorticoid-induced osteoporosis, hyperparathyroidism, initial and steady state hypocalcemia clamp tests indicate αR values significantly larger than the healthy baseline (1.7, 2.2, 4.9 and 1.9-times, respectively). Manipulation of the pulsatile glandular secretion pattern, while keeping the mean PTH concentration constant, allowed restoration of healthy baseline values from these catabolic bone diseases. Conversely, PTH glandular diseases that led to maximum bone cellular responsiveness below the healthy baseline value can't be restored to baseline via glandular manipulation. However, external PTH injections allowed restoration of these latter cases.

Support Pressure Acting on the Epicardial Surface of a Rat Left Ventricle-A Computational Study.

The present computational study investigates the effects of an epicardial support pressure mimicking a heart support system without direct blood contact. We chose restrictive cardiomyopathy as a model for a diseased heart. By changing one parameter representing the amount of fibrosis, this model allows us to investigate the impairment in a diseased left ventricle, both during diastole and systole. The aim of the study is to determine the temporal course and value of the support pressure that leads to a normalization of the cardiac parameters in diseased hearts. These are quantified via the end-diastolic pressure, end-diastolic volume, end-systolic volume, and ejection fraction. First, the amount of fibrosis is increased to model diseased hearts at different stages. Second, we determine the difference in the left ventricular pressure between a healthy and diseased heart during a cardiac cycle and apply for the epicardial support as the respective pressure difference. Third, an epicardial support pressure is applied in form of a piecewise constant step function. The support is provided only during diastole, only during systole, or during both phases. Finally, the support pressure is adjusted to reach the corresponding parameters in a healthy rat. Parameter normalization is not possible to achieve with solely diastolic or solely systolic support; for the modeled case with 50% fibrosis, the ejection fraction can be increased by 5% with purely diastolic support and 14% with purely systolic support. However, the ejection fraction reaches the value of the modeled healthy left ventricle (65.6%) using a combination of diastolic and systolic support. The end-diastolic pressure of 13.5 mmHg cannot be decreased with purely systolic support. However, the end-diastolic pressure reaches the value of the modeled healthy left ventricle (7.5 mmHg) with diastolic support as well as with the combination of the diastolic and systolic support. The resulting negative diastolic support pressure is -4.5 mmHg, and the positive systolic support pressure is 90 mmHg. We, thereby, conclude that ventricular support during both diastole and systole is beneficial for normalizing the left ventricular ejection fraction and the end-diastolic pressure, and thus it is a potentially interesting therapy for cardiac insufficiency.

Comparison of stress and stress-strain approaches for the active contraction in a rat cardiac cycle model.

In the last decades, different strategies to model the active electromechanically coupled behaviour of the cardiac tissue were proposed in order to simulate electromechanics of the heart under healthy and pathological conditions. The main objective of this work is to compare two approaches for modelling the active contraction during the electromechanically coupled rat cardiac cycle -- the stress and the stress-strain approach. Firstly, a cylindrical benchmark is considered and secondly, for a generic model of a rat left ventricle, a simulation including the Windkessel model, excitation via Purkinje fibre network and mechano-electrical feedback is performed. The model is calibrated with experimental data for rats, partly from own measurements via cardiac ultrasound, partly from the literature. Further, possibilities to reach higher ejection fractions are discussed and considered for an exemplary rat left ventricle. Within each approach, we observe regionally different active stresses and fibre stretches. Moreover, the transmural active stress and fibre stretch distribution is influenced by the pressure load on the endocardial surface. The active stress approach is not sensitive to the fibre stretch and transmurally varying fibre stretch in the left ventricular domain is observed. The active stress-strain approach leads to transmurally more homogeneous fibre stretch at the end-systolic state.

A Transmural Path Model Improves the Definition of the Orthotropic Tissue Structure in Heart Simulations.

In the past decades, the structure of the heart, human as well as other species, has been explored in a detailed way, e.g., via histological studies or diffusion tensor magnetic resonance imaging. Nevertheless, the assignment of the characteristic orthotropic structure in a patient-specific finite element model remains a challenging task. Various types of rule-based models, which define the local fiber and sheet orientation depending on the transmural depth, have been developed. However, the correct assessment of the transmural depth is not trivial. Its accuracy has a substantial influence on the overall mechanical and electrical properties in rule-based models. The main purpose of this study is the development of a finite element-based approach to accurately determine the transmural depth on a general unstructured grid. Instead of directly using the solution of the Laplace problem as the transmural depth, we make use of a well-established model for the assessment of the transmural thickness. It is based on two hyperbolic first-order partial differential equations for the definition of a transmural path, whereby the transmural thickness is defined as the arc length of this path. Subsequently, the transmural depth is determined based on the position on the transmural path. Originally, the partial differential equations were solved via finite differences on structured grids. In order to circumvent the need of two grids and mapping between the structured (to determine the transmural depth) and unstructured (electromechanical heart simulation) grids, we solve the equations directly on the same unstructured tetrahedral mesh. We propose a finite-element-based discontinuous Galerkin approach. Based on the accurate transmural depth, we assign the local material orientation of the orthotropic tissue structure in a usual fashion. We show that this approach leads to a more accurate definition of the transmural depth. Furthermore, for the left ventricle, we propose functions for the transmural fiber and sheet orientation by fitting them to literature-based diffusion tensor magnetic resonance imaging data. The proposed functions provide a distinct improvement compared to existing rules from the literature.

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

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

Towards the simulation of active cardiac mechanics using a smoothed finite element method.

In the last decades, various computational models have been developed to simulate cardiac electromechanics. The most common numerical tool is the finite element method (FEM). However, this method crucially depends on the mesh quality. For complex geometries such as cardiac structures, it is convenient to use tetrahedral discretisations which can be generated automatically. On the other hand, such automatic meshing with tetrahedrons together with large deformations often lead to elements distortion and volumetric locking. To overcome these difficulties, different smoothed finite element methods (S-FEMs) have been proposed in the recent years. They are known to be volumetric locking free, less sensitive to mesh distortion and so far have been used e.g. in simulation of passive cardiac mechanics. In this work, we extend for the first time node-based S-FEM (NS-FEM) towards active cardiac mechanics. Firstly, the sensitivity to mesh distortion is tested and compared to that of FEM. Secondly, an active contraction in circumferentially aligned fibre direction is modelled in the healthy and the infarcted case. We show, that the proposed method is more robust with respect to mesh distortion and computationally more efficient than standard FEM. Being furthermore free of volumetric locking problems makes S-FEM a promising alternative in modelling of active cardiac mechanics, respectively electromechanics.