CFD simulation of flow through heart: a perspective review

The heart is an organ which pumps blood around the body by contraction of muscular wall. There is a coupled system in the heart containing the motion of wall and the motion of blood fluid; both motions must be computed simultaneously, which make biological computational fluid dynamics (CFD) difficult. The wall of the heart is not rigid and hence proper boundary conditions are essential for CFD modelling. Fluid-wall interaction is very important for real CFD modelling. There are many assumptions for CFD simulation of the heart that make it far from a real model. A realistic fluid-structure interaction modelling the structure by the finite element method and the fluid flow by CFD use more realistic coupling algorithms. This type of method is very powerful to solve the complex properties of the cardiac structure and the sensitive interaction of fluid and structure. The final goal of heart modelling is to simulate the total heart function by integrating cardiac anatomy, electrical activation, mechanics, metabolism and fluid mechanics together, as in the computational framework.     

Effect of bundle branch block on cardiac output: a whole heart simulation study

The heart is an electrically controlled fluid pump which operates by mechanical contraction. Whole heart modelling is a computationally daunting task which must incorporate several subsystems: mechanical, electrical, and fluidic. Numerous feedback mechanisms on many levels, and operating at different scales, exist to finely control behaviour. Understanding these interactions is necessary to understand heart operation, as well as pathologies and therapies. A review of the components in such a model is given. The authors then present a framework for their electro-mechano-fluidic whole heart model based on cable methods. The model incorporates atria and ventricles, and has functioning valves with papillary muscles. The effect of altered propagation due to left and right bundle branch block on cardiac output is examined using the cable-based model. Results are compared to clinically observed phenomena. Good agreement was obtained, but tighter coupling of mechanical and electrical events is needed to fully account for behaviour. Cable-based models offer an alternative to continuum models.     

An object-oriented modelling framework for the arterial wall

An object-oriented modelling framework for the arterial wall is presented. The novelty of the framework is the possibility to generate customisable artery models, taking advantage of imaging technology. In our knowledge, this is the first object-oriented modelling framework for the arterial wall. Existing models do not allow close structural mapping with arterial microstructure as in the object-oriented framework. In the implemented model, passive behaviour of the arterial wall was considered and the tunica adventitia was the objective system. As verification, a model of an arterial segment was generated. In order to simulate its deformation, a matrix structural mechanics simulator was implemented. Two simulations were conducted, one for an axial loading test and other for a pressure–volume test. Each simulation began with a sensitivity analysis in order to determinate the best parameter combination and to compare the results with analogue controls. In both cases, the simulated results closely reproduced qualitatively and quantitatively the analogue control plots.     

Fast parameter calibration of a cardiac electromechanical model from medical images based on the unscented transform

Patient-specific cardiac modelling can help in understanding pathophysiology and predict therapy planning. However, it requires to personalize the model geometry, kinematics, electrophysiology and mechanics. Calibration aims at providing proper initial values of parameters before performing the personalization stage which involves solving an inverse problem. We propose a fast automatic calibration method of the mechanical parameters of a complete electromechanical model of the heart based on a sensitivity analysis and the Unscented Transform algorithm. A new implementation of the complete Bestel–Clement–Sorine (BCS) cardiac model is also proposed, in a modular and efficient framework. A complete sensitivity analysis is performed that reveals which observations on the volume evolution are significant to characterize the global behaviour of the myocardium. We show that the calibration method gives satisfying results by optimizing up to 5 parameters of the BCS model in only one iteration. This method was evaluated synthetically as well as on 7 volunteers with a mean relative error from the real data of 10 %. This calibration is designed to replace manual parameter estimation as well as initialization steps that precede automatic personalization algorithms based on images.     

A three-dimensional computational analysis of fluid-structure interaction in the aortic valve

Numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle. Moreover, the fluid-structure interaction in the aortic valve system is most frequently ignored in numerical modelling. The effect of this interaction on the valve's behaviour during systolic functioning is investigated. The large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood-valve interaction modelling. This has impeded numerical analyses of valves operating under physiological conditions. A numerical method, known as the Lagrange multiplier based fictitious domain method, is used to describe the large leaflet motion within the computational fluid domain. This method is applied to a three-dimensional finite element model of a stented aortic valve. The model provides both the mechanical behaviour of the valve and the blood flow through it. Results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion.     

Discrete Element Framework for Modelling Extracellular Matrix,Deformable Cells and Subcellular Components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an ‘agent’, meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory.     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.     

Mathematical modelling of the cellular mechanics of plants

The complex mechanical behaviour of plant tissues reflects the complexity of their structure and material properties. Modelling has been widely used in studies of how cell walls, single cells and tissue respond to loading, both externally applied loading and loads on the cell wall resulting from changes in the pressure within fluid-filled cells. This paper reviews what approaches have been taken to modelling and simulation of cell wall, cell and tissue mechanics, and to what extent models have been successful in predicting mechanical behaviour. Advances in understanding of cell wall ultrastructure and the control of cell growth present opportunities for modelling to clarify how growth-related mechanical properties arise from wall polymeric structure and biochemistry.     

A mechanobiological model of endothelial cell migration and proliferation

How angiogenesis is regulated by local environmental cues is still not fully understood despite its importance to many regenerative events. Although mechanics is known to influence angiogenesis, the specific cellular mechanisms influenced by mechanical loading are poorly understood. This study adopts a lattice-based modelling approach to simulate endothelial cell (EC) migration and proliferation in order to explore how mechanical stretch regulates their behaviour. The approach enables the explicit modelling of ECs and, in particular, their migration/proliferation (specifically, rate and directionality) in response to such mechanical cues. The model was first used to simulate previously reported experiments of EC migration and proliferation in an unloaded environment. Next, three potential effects (increased cell migration, increased cell proliferation and biased cellular migration) of mechanical stretch on EC behaviour were simulated using the model and the observed changes in cell population characteristics were compared to experimental findings. Combinations of these three potential drivers were also investigated. The model demonstrates that only by incorporating all three changes in cellular physiology (increased EC migration, increased EC proliferation and biased EC migration in the direction perpendicular to the applied strain) in response to dynamic loading, it is possible to successfully predict experimental findings. This provides support for the underlying model hypotheses for how mechanics regulates EC behaviour.     

Fluid mechanics of the zebrafish embryonic heart trabeculation

Embryonic heart development is a mechanosensitive process, where specific fluid forces are needed for the correct development, and abnormal mechanical stimuli can lead to malformations. It is thus important to understand the nature of embryonic heart fluid forces. However, the fluid dynamical behaviour close to the embryonic endocardial surface is very sensitive to the geometry and motion dynamics of fine-scale cardiac trabecular surface structures. Here, we conducted image-based computational fluid dynamics (CFD) simulations to quantify the fluid mechanics associated with the zebrafish embryonic heart trabeculae. To capture trabecular geometric and motion details, we used a fish line that expresses fluorescence at the endocardial cell membrane, and high resolution 3D confocal microscopy. Our endocardial wall shear stress (WSS) results were found to exceed those reported in existing literature, which were estimated using myocardial rather than endocardial boundaries. By conducting simulations of single intra-trabecular spaces under varied scenarios, where the translational or deformational motions (caused by contraction) were removed, we found that a squeeze flow effect was responsible for most of the WSS magnitude in the intra-trabecular spaces, rather than the shear interaction with the flow in the main ventricular chamber. We found that trabecular structures were responsible for the high spatial variability of the magnitude and oscillatory nature of WSS, and for reducing the endocardial deformational burden. We further found cells attached to the endocardium within the intra-trabecular spaces, which were likely embryonic hemogenic cells, whose presence increased endocardial WSS. Overall, our results suggested that a complex multi-component consideration of both anatomic features and motion dynamics were needed to quantify the trabeculated embryonic heart fluid mechanics.     

A cell based modelling framework for skeletal tissue engineering applications

In this study, a cell based lattice free modelling framework is proposed to study cell aggregate behaviour in bone tissue engineering applications. The model encompasses cell-to-cell and cell–environment interactions such as adhesion, repulsion and drag forces. Oxygen, nutrients, waste products, growth factors and inhibitors are explicitly represented in the model influencing cellular behaviour. Furthermore, a model for cell metabolism is incorporated representing the basic enzymic reactions of glycolysis and the Krebs cycle. Various types of cell death such as necrosis, apoptosis and anoikis are implemented. Finally, an explicit model of the cell cycle controls the proliferation process, taking into account the presence or absence of various metabolites, sufficient space and mechanical stress. Several examples are presented demonstrating the potential of the modelling framework. The behaviour of a synchronised cell aggregate under ideal circumstances is simulated, clearly showing the different stages of the cell cycle and the resulting growth of the aggregate. Also the difference in aggregate development under ideal (normoxic) and hypoxic conditions is simulated, showing hypoxia induced necrosis mainly in the centre of the aggregate grown under hypoxic conditions. The next step in this research will be the application of this modelling framework to specific experimental set-ups for bone tissue engineering applications.     

Mechanical characterisation of the human thoracic descending aorta: experiments and modelling

This work presents experiments and modelling aimed at characterising the passive mechanical behaviour of the human thoracic descending aorta. To this end, uniaxial tension and pressurisation tests on healthy samples corresponding to newborn, young and adult arteries are performed. Then, the tensile measurements are used to calibrate the material parameters of the Holzapfel constitutive model. This model is found to adequately adjust the material behaviour in a wide deformation range; in particular, it captures the progressive stiffness increase and the anisotropy due to the stretching of the collagen fibres. Finally, the assessment of these material parameters in the modelling of the pressurisation test is addressed. The implication of this study is the possibility to predict the mechanical response of the human thoracic descending aorta under generalised loading states like those that can occur in physiological conditions and/or in medical device applications.     

An experimental and computational analysis of primary cilia deflection under fluid flow

In this study we have developed a novel model of the deflection of primary cilia experiencing fluid flow accounting for phenomena not previously considered. Specifically, we developed a large rotation formulation that accounts for rotation at the base of the cilium, the initial shape of the cilium and fluid drag at high deflection angles. We utilised this model to analyse full 3D data-sets of primary cilia deflecting under fluid flow acquired with high-speed confocal microscopy. We found a wide variety of previously unreported bending shapes and behaviours. We also analysed post-flow relaxation patterns. Results from our combined experimental and theoretical approach suggest that the average flexural rigidity of primary cilia might be higher than previously reported (Schwartz et al. 1997, Am J Physiol. 272(1 Pt 2):F132–F138). In addition our findings indicate that the mechanics of primary cilia are richly varied and mechanisms may exist to alter their mechanical behaviour.     

Mapping cardiac surface mechanics with structured light imaging

Cardiovascular disease often manifests as a combination of pathological electrical and structural heart remodeling. The relationship between mechanics and electrophysiology is crucial to our understanding of mechanisms of cardiac arrhythmias and the treatment of cardiac disease. While several technologies exist for describing whole heart electrophysiology, studies of cardiac mechanics are often limited to rhythmic patterns or small sections of tissue. Here, we present a comprehensive system based on ultrafast three-dimensional (3-D) structured light imaging to map surface dynamics of whole heart cardiac motion. Additionally, we introduce a novel nonrigid motion-tracking algorithm based on an isometry-maximizing optimization framework that forms correspondences between consecutive 3-D frames without the use of any fiducial markers. By combining our 3-D imaging system with nonrigid surface registration, we are able to measure cardiac surface mechanics at unprecedented spatial and temporal resolution. In conclusion, we demonstrate accurate cardiac deformation at over 200,000 surface points of a rabbit heart recorded at 200 frames/s and validate our results on highly contrasting heart motions during normal sinus rhythm, ventricular pacing, and ventricular fibrillation.     

Empirical modelling using dummy atoms (EMUDA): an alternative approach for studying “auxetic” structures

Auxetics (materials or structures) are systems with a negative Poisson's ratio, a property that arises from the way various geometric features in the structure (or internal structure in the case of materials) deform when subjected to uniaxial loads. Such systems are normally studied by examining the behaviour of idealised representations of structures, which deform in a controlled fashion (e.g. deforming solely through hinging or stretching). Methods used for the analysis typically involve construction of real physical macro-models and/or derivation of analytical expressions for the mechanical properties. This paper proposes an alternative method for analysing such structures whereby idealised “hinging” or “stretching” structures are constructed within a molecular modelling environment using dummy atoms and examined using standard molecular mechanics techniques. We will show that this methodology of “empirical modelling using dummy atoms” (EMUDA) successfully reproduces the known properties of 2D conventional and auxetic hexagonal honeycombs hence confirming the suitability of this technique for studying auxetic structures.     

Electromechanical models of the ventricles

Computational modeling has traditionally played an important role in dissecting the mechanisms for cardiac dysfunction. Ventricular electromechanical models, likely the most sophisticated virtual organs to date, integrate detailed information across the spatial scales of cardiac electrophysiology and mechanics and are capable of capturing the emergent behavior and the interaction between electrical activation and mechanical contraction of the heart. The goal of this review is to provide an overview of the latest advancements in multiscale electromechanical modeling of the ventricles. We first detail the general framework of multiscale ventricular electromechanical modeling and describe the state of the art in computational techniques and experimental validation approaches. The powerful utility of ventricular electromechanical models in providing a better understanding of cardiac function is then demonstrated by reviewing the latest insights obtained by these models, focusing primarily on the mechanisms by which mechanoelectric coupling contributes to ventricular arrythmogenesis, the relationship between electrical activation and mechanical contraction in the normal heart, and the mechanisms of mechanical dyssynchrony and resynchronization in the failing heart. Computational modeling of cardiac electromechanics will continue to complement basic science research and clinical cardiology and holds promise to become an important clinical tool aiding the diagnosis and treatment of cardiac disease.     

A cell-based framework for modeling cardiac mechanics

Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell–cell and cell–matrix interactions.

     

Passive myocardial mechanical properties: meaning,measurement, models

Passive mechanical tissue properties are major determinants of myocardial contraction and relaxation and, thus, shape cardiac function. Tightly regulated, dynamically adapting throughout life, and affecting a host of cellular functions, passive tissue mechanics also contribute to cardiac dysfunction. Development of treatments and early identification of diseases requires better spatio-temporal characterisation of tissue mechanical properties and their underlying mechanisms. With this understanding, key regulators may be identified, providing pathways with potential to control and limit pathological development. Methodologies and models used to assess and mimic tissue mechanical properties are diverse, and available data are in part mutually contradictory. In this review, we define important concepts useful for characterising passive mechanical tissue properties, and compare a variety of in vitro and in vivo techniques that allow one to assess tissue mechanics. We give definitions of key terms, and summarise insight into determinants of myocardial stiffness in situ. We then provide an overview of common experimental models utilised to assess the role of environmental stiffness and composition, and its effects on cardiac cell and tissue function. Finally, promising future directions are outlined.