Dynamic modelling of prosthetic chorded mitral valves using the immersed boundary method

Comparison of Immersed Boundary Simulations of Heart Valve Hemodynamics Against In Vitro 4D Flow MRI Data

The immersed boundary (IB) method is a mathematical framework for fluid-structure interaction problems (FSI) that was originally developed to simulate flows around heart valves. Direct comparison of FSI simulations around heart valves against experimental data is challenging, however, due to the difficulty of performing robust and effective simulations, the complications of modeling a specific physical experiment, and the need to acquire experimental data that is directly comparable to simulation data. Such comparators are a necessary precursor for further formal validation studies of FSI simulations involving heart valves. In this work, we performed physical experiments of flow through a pulmonary valve in an in vitro pulse duplicator, and measured the corresponding velocity field using 4D flow MRI (4-dimensional flow magnetic resonance imaging). We constructed a computer model of this pulmonary artery setup, including modeling valve geometry and material properties via a technique called design-based elasticity, and simulated flow through it with the IB method. The simulated flow fields showed excellent qualitative agreement with experiments, excellent agreement on integral metrics, and reasonable relative error in the entire flow domain and on slices of interest. These results illustrate how to construct a computational model of a physical experiment for use as a comparator.

Some Effects of Different Constitutive Laws on FSI Simulation for the Mitral Valve

In this paper, three different constitutive laws for mitral leaflets and two laws for chordae tendineae are selected to study their effects on mitral valve dynamics with fluid-structure interaction. We first fit these three mitral leaflet constitutive laws and two chordae tendineae laws with experimental data. The fluid-structure interaction is implemented in an immersed boundary framework with finite element extension for solid, that is the hybrid immersed boundary/finite element(IB/FE) method. We specifically compare the fluid-structure results of different constitutive laws since fluid-structure interaction is the physiological loading environment. This allows us to look at the peak jet velocity, the closure regurgitation volume, and the orifice area. Our numerical results show that different constitutive laws can affect mitral valve dynamics, such as the transvalvular flow rate, closure regurgitation and the orifice area, while the differences in fiber strain and stress are insignificant because all leaflet constitutive laws are fitted to the same set of experimental data. In addition, when an exponential constitutive law of chordae tendineae is used, a lower closure regurgitation flow is observed compared to that of a linear material model. In conclusion, combining numerical dynamic simulations and static experimental tests, we are able to identify suitable constitutive laws for dynamic behaviour of mitral leaflets and chordae under physiological conditions.

Complete Unsteady One-Dimensional Model of the Net Aortic Pressure Drop

Background:

A large amount of engineering and medical research has been devoted to the assessment of aortic valve stenosis severity in the past decades. The net transvalvular pressure drop has been recognized as one of the parameters that better reflect stenosis effects on left ventricle overload, and its adoption in clinical assessment of stenosis has been proposed. Flow unsteadiness has been shown to have a non-negligible impact on the net drop; however, a simple formulation for net drop calculation that includes not only flow pulsatility but also the effects of valve dynamics is still lacking.

Objective:

The present contribution is hence aimed at developing a complete unsteady one-dimensional model of the net aortic transvalvular pressure drop that just requires non-invasive data to be implemented.

Methods:

Transvalvular flow is described as a jet of incompressible viscous fluid through a circular orifice placed in a concentric rigid circular tube. The classical one-dimensional mass and total head conservation equations are applied. The effective orifice area and transvalvular flow rate are assumed to vary with time throughout the ejection period.

Results:

The model is found to capture pressure drop oscillations occurring when the valve opens/closes and/or leaflets flutter, thanks to the inclusion of valve dynamics effects. The model is also proposed as a numerical tool for the calculation of the instantaneous effective orifice area once net pressure drop and flow rate are known.

Conclusion:

The model may contribute to the improvement of non-invasive aortic stenosis assessment.

Fluid-structure interaction of a pulsatile flow with an aortic valve model: A combined experimental and numerical study

The complex fluid-structure interaction problem associated with the flow of blood through a heart valve with flexible leaflets is investigated both experimentally and numerically. In the experimental test rig, a pulse duplicator generates a pulsatile flow through a biomimetic rigid aortic root where a model of aortic valve with polymer flexible leaflets is implanted. High-speed recordings of the leaflets motion and particle image velocimetry measurements were performed together to investigate the valve kinematics and the dynamics of the flow. Large eddy simulations of the same configuration, based on a variant of the immersed boundary method, are also presented. A massively parallel unstructured finite-volume flow solver is coupled with a finite-element solid mechanics solver to predict the fluid-structure interaction between the unsteady flow and the valve. Detailed analysis of the dynamics of opening and closure of the valve are conducted, showing a good quantitative agreement between the experiment and the simulation regarding the global behavior, in spite of some differences regarding the individual dynamics of the valve leaflets. A multicycle analysis (over more than 20 cycles) enables to characterize the generation of turbulence downstream of the valve, showing similar flow features between the experiment and the simulation. The flow transitions to turbulence after peak systole, when the flow starts to decelerate. Fluctuations are observed in the wake of the valve, with maximum amplitude observed at the commissure side of the aorta. Overall, a very promising experiment-vs-simulation comparison is shown, demonstrating the potential of the numerical method.

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.

INFLUENCE OF LEAFLET’S MATRIX STIFFNESS AND FIBER ORIENTATION ON THE OPENING DYNAMICS OF A PROSTHETIC TRILEAFLET HEART VALVE

Biological valves are employed for aortic valve substitution since a long time but there is a growing effort toward the development of new engineered tissues, in which the complex mechanical response of native leaflets is replicated using composite materials consisting of a soft matrix with embedded reinforcing fibers. The main goal of the present study is to investigate the influence that variations on fiber orientation and matrix stiffness may have on valve dynamics. To this aim, a fluid–structure interaction (FSI) model of a trileaflet valve was implemented in which the opening phase was simulated and leaflet matrix stiffness and fiber orientation were varied in the framework of an anisotropic hyperelastic strain energy function. Results show that both parameters may affect significantly transvalvular pressure gradient and effective orifice area (EOA). For the opening phase of the valve examined, less favorable flow conditions were found when preferred fiber orientation is circumferential, due to lower maximum EOA achievable. Such configuration in combination with stiffer matrix may result in significant degradation of valve performances. Overall fiber orientation can potentially be taylored to optimize valve dynamics, provided also structural aspects that may be prominent in the closure phase, are considered.

Modelling and simulation of flow and agglomeration in deep veins valves using discrete multi physics

The hemodynamics in flexible deep veins valves is modelled by means of discrete multi-physics and an agglomeration algorithm is implemented to account for blood accrual in the flow. Computer simulations of a number of valves typologies are carried out. The results show that the rigidity and the length of the valve leaflets play a crucial role on both mechanical stress and stagnation in the flow. Rigid and short membranes may be inefficient in preventing blood reflux, but reduce the volume of stagnant blood potentially lowering the chances of thrombosis. Additionally, we also show that in venous valves, cell agglomeration is driven by stagnation rather than mechanical stress.

E, Cerrolaza M. ANALYSIS OF BLOOD FLOW PASSING THROUGH AORTIC AND MITRAL VALVES USING A COMPUTATIONAL MODEL OF CONCENTRATED PARAMETERS

Blood flow has been extensively studied because of its close relationship with cardiovascular disease. Heart valves blood flow analysis is particularly complex due to the high mobility of its leaflets, a fact that has stimulated the development of computational models aimed to its better understanding. For studying heart valves blood flow, we developed a mathematical model derived from clinical observations based on echocardiographic images, which describe valve leaflets motion and its influence on blood flow. This work presents a concentrated-parameters-based model of heart valves blood flow that takes into consideration five main factors affecting such a flow in the mitral and aortic valves. This model considers factors that are related to blood fluid and valve leaflets characteristics. Considering the main factors involved, it was found that blood flow exhibit an abnormal behavior in response to small variations (less than 10%) in blood pressure gradient or in leaflets stiffness. Likewise, after changing the roughness of the leaflets, the impact is smaller, only slightly affecting blood flow behavior with changes beyond 30%. Moreover, it was observed that the influence of fluid vortices originated behind the valves can be disregarded and the kinetic energy induced by them is almost negligible.

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.

Structural analysis of a stented pericardial heart valve with leaflets mounted externally

Our aim was to understand the structural and functional behaviour of a pericardial heart valve with biological leaflets attached externally to a stent. To our knowledge, there is little if any literature concerning these kinds of bioprosthetic heart valves, while there is more concerning bioprosthetic heart valves with leaflets mounted internally. We studied the problem using a finite element approach considering leaflets and stent interaction, the influence of leaflet anisotropy and stent stiffness, by comparing quasi-static and dynamic loadings. Although we considered the problem to be symmetric and fluid-structure interaction was not implemented, we believe that our results could be a solid basis for valve optimization. We found regions of high stress concentration at the commissure near the stent tip and at the base of the leaflet cusp. The structural behaviour in the first region was complex, while the stress in the second region acted radially because of high bending. Although leaflet tissue anisotropy and stent stiffness exerted a significant influence on the structural and functional behaviours, they had a contrasting effect on leaflet stress state, coaptation and valve opening. Therefore, a good optimization should take into account both structural and functional requirements when tuning tissue properties and stent stiffness.

Evaluation of a transient, simultaneous, arbitrary Lagrange-Euler based multi-physics method for simulating the mitral heart valve

A transient multi-physics model of the mitral heart valve has been developed, which allows simultaneous calculation of fluid flow and structural deformation. A recently developed contact method has been applied to enable simulation of systole (the stage when blood pressure is elevated within the heart to pump blood to the body). The geometry was simplified to represent the mitral valve within the heart walls in two dimensions. Only the mitral valve undergoes deformation. A moving arbitrary Lagrange-Euler mesh is used to allow true fluid-structure interaction (FSI). The FSI model requires blood flow to induce valve closure by inducing strains in the region of 10-20%. Model predictions were found to be consistent with existing literature and will undergo further development.

Biomechanical assessment of surgical repair of the mitral valve

Repair of the mitral valve is defined (loosely) as a procedure that alters the valve structure, without replacement, enabling the natural valve itself to continue to perform under the physical conditions to which it is exposed. As the mitral valve is driven by flow and pressure, it should be feasible to analyse and assess its function, failure and repair as a mechanical system. This article reviews the current state of mechanical evaluation of surgical repairs of the failed mitral valve of the heart. This review describes the anatomy and physiology of the mitral valve, followed by the failure of the mitral valve from a mechanical point of view. The surgical methods used to repair failed valves are introduced, while the use of engineering analysis to aid understanding of mitral valve repair is also reviewed. Finally, a section on recommendations for development and future uses of engineering techniques to surgical repair are presented.

A personalized mitral valve closure simulator

We describe a method for performing modeling and simulation to predict the closure of the Mitral Valve (MV) using patient specific anatomical information derived from 3D Transesophageal Echocardiography (3D TEE). The ability to predict the MV closure behavior is an important step along the way of developing personalized simulation tools that would allow a surgeon to perform preoperative planning and decide between various MV repair options. While the MV is an important use case because of its relevance and prevalence among reconstructive cardiac interventions, the study described here can provide a blueprint to perform pre-operative planning for other cardiac surgical interventions. The method reported here exploits the Saint Venant-Kirchhoff elasticity model that is tuned to match empirical observations of the MV strainstress behavior. Using intraoperative 3D TEE data, the proposed simulator was evaluated over 10 test cases and resulted in mean prediction absolute error values of 1.81 mm.

MECHANICAL PROPERTIES OF CHORDAE TENDINEAE OF THE MITRAL HEART VALVE: YOUNG'S MODULUS, STRUCTURAL STIFFNESS, AND EFFECTS OF AGING

Young's modulus and structural stiffness were determined for chordae tendineae of the mitral valve from young (18–26 weeks) and old (over 2 years) porcine hearts. For chordae from the posterior leaflet of the valve, the Young's modulus values were significantly higher (p < 0.05) for the thinner marginal chordae (59 ± 31 MPa young; 88 ± 21 MPa old) than for the thicker basal chordae (31 ± 4 MPa young; 28 ± 9 MPa old). Marginal chordae (both anterior and posterior) had significantly higher (p < 0.05) value for their Young's modulus in old (88 ± 21 MPa anterior and posterior) than in young (62 ± 17 MPa anterior, 59 ± 18 MPa posterior) pig hearts. There was no significant difference in structural stiffness between marginal and basal (anterior and posterior leaflets) or between strut chordae (that are associated with anterior the leaflet only) and marginal and basal chordae. However, the value of structural stiffness of chordae was significantly higher (p < 0.05) for old (2.2 ± 0.2 kN/m) than for young (2.0 ± 0.4 kN/m) chordae. These results show that aging affects the properties of chordae and that all chordae need to be included in finite element models of valve function.

A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana

Mathematical and experimental studies of the flows generated by jellyfish have focused primarily on mechanisms of swimming. More recent work has also considered the fluid dynamics of feeding from currents generated during swimming. Here we capitalize on the benthic lifestyle of the upside-down jellyfish (Cassiopea xamachana) to explore the fluid dynamics of feeding uncoupled from swimming. A two-dimensional mathematical model is developed to capture the fundamental characteristics of the motion of the unique concave bell shape. Given the prominence of the oral arms, this structure is included and modeled as a porous layer that perturbs the flow generated by bell contractions. The immersed boundary method is used to solve the fluid–structure interaction problem. Velocity fields obtained from live organisms using digital particle image velocimetry were used to validate the numerical simulations. Parameter sweeps were used to numerically explore the effects of changes in pulse dynamics and the properties of the oral arms independently. Numerical experiments allow the opportunity to examine physical effects and limits within and beyond the biologically relevant range to develop a better understanding of the system. The presence of the prominent oral arm structures in the field of flow increased the flux of new fluid from along the substrate to the bell. The numerical simulations also showed that the presence of pauses between bell expansion and the next contraction alters the flow of the fluid over the bell and through the oral arms.

Mass-spring model for simulation of heart valve tissue mechanical behavior

Heart valves are functionally complex, making surgical repair difficult. Simulation-based surgical planning could facilitate repair, but current finite element (FE) studies are prohibitively slow for rapid, clinically oriented simulations. Mass-spring (M-S) models are fast but can be inaccurate. We quantify speed and accuracy differences between an anisotropic, nonlinear M-S and an efficient FE membrane model for simulating both biaxial and pressure loading of aortic valve (AV) leaflets. The FE model incurs approximately 10 times the computational cost of the M-S model. For simulated biaxial loading, mean error in normal strains is <1% for both FE and M-S models for equibiaxial loading but increases for non-equibiaxial states for the M-S model (7%). The M-S model was less able to simulate shear behavior, with mean strain error of approximately 80%. For pressurized AV leaflets, the M-S model predicts similar leaflet dimensions to the FE model (within 2.6%), and the coaptation zone is similar between models. The M-S model simulates in-plane behavior of AV leaflets considerably faster than the FE model and with only minor differences in the deformed mesh. While the M-S model does not allow explicit control of shear response, shear does not strongly influence shape of the simulated AV under pressure.

A coupled sharp-interface immersed boundary-finite-element method for flow-structure interaction with application to human phonation

A new flow-structure interaction method is presented, which couples a sharp-interface immersed boundary method flow solver with a finite-element method based solid dynamics solver. The coupled method provides robust and high-fidelity solution for complex flow-structure interaction (FSI) problems such as those involving three-dimensional flow and viscoelastic solids. The FSI solver is used to simulate flow-induced vibrations of the vocal folds during phonation. Both two- and three-dimensional models have been examined and qualitative, as well as quantitative comparisons, have been made with established results in order to validate the solver. The solver is used to study the onset of phonation in a two-dimensional laryngeal model and the dynamics of the glottal jet in a three-dimensional model and results from these studies are also presented.

Fluid-Structure Interactions of the Mitral Valve and Left Heart: Comprehensive Strategies, Past, Present and Future

The remodeling that occurs after a posterolateral myocardial infarction can alter mitral valve function by creating conformational abnormalities in the mitral annulus and in the posteromedial papillary muscle, leading to mitral regurgitation (MR). It is generally assumed that this remodeling is caused by a volume load and is mediated by an increase in diastolic wall stress. Thus, mitral regurgitation can be both the cause and effect of an abnormal cardiac stress environment. Computational modeling of ischemic MR and its surgical correction is attractive because it enables an examination of whether a given intervention addresses the correction of regurgitation (fluid-flow) at the cost of abnormal tissue stress. This is significant because the negative effects of an increased wall stress due to the intervention will only be evident over time. However, a meaningful fluid-structure interaction model of the left heart is not trivial; it requires a careful characterization of the in-vivo cardiac geometry, tissue parameterization though inverse analysis, a robust coupled solver that handles collapsing Lagrangian interfaces, automatic grid-generation algorithms that are capable of accurately discretizing the cardiac geometry, innovations in image analysis, competent and efficient constitutive models and an understanding of the spatial organization of tissue microstructure. In this manuscript, we profile our work toward a comprehensive fluid-structure interaction model of the left heart by reviewing our early work, presenting our current work and laying out our future work in four broad categories: data collection, geometry, fluid-structure interaction and validation.

Patient-specific modeling of cardiovascular mechanics

Advances in numerical methods and three-dimensional imaging techniques have enabled the quantification of cardiovascular mechanics in subject-specific anatomic and physiologic models. Patient-specific models are being used to guide cell culture and animal experiments and test hypotheses related to the role of biomechanical factors in vascular diseases. Furthermore, biomechanical models based on noninvasive medical imaging could provide invaluable data on the in vivo service environment where cardiovascular devices are employed and on the effect of the devices on physiologic function. Finally, patient-specific modeling has enabled an entirely new application of cardiovascular mechanics, namely predicting outcomes of alternate therapeutic interventions for individual patients. We review methods to create anatomic and physiologic models, obtain properties, assign boundary conditions, and solve the equations governing blood flow and vessel wall dynamics. Applications of patient-specific models of cardiovascular mechanics are presented, followed by a discussion of the challenges and opportunities that lie ahead.

Mitral valve finite-element modelling from ultrasound data: a pilot study for a new approach to understand mitral function and clinical scenarios

In the current scientific literature, particular attention is dedicated to the study of the mitral valve and to comprehension of the mechanisms that lead to its normal function, as well as those that trigger possible pathological conditions. One of the adopted approaches consists of computational modelling, which allows quantitative analysis of the mechanical behaviour of the valve by means of continuum mechanics theory and numerical techniques. However, none of the currently available models realistically accounts for all of the aspects that characterize the function of the mitral valve. Here, a new computational model of the mitral valve has been developed from in vivo data, as a first step towards the development of patient-specific models for the evaluation of annuloplasty procedures. A structural finite-element model of the mitral valve has been developed to account for all of the main valvular substructures. In particular, it includes the real geometry and the movement of the annulus and papillary muscles, reconstructed from four-dimensional ultrasound data from a healthy human subject, and a realistic description of the complex mechanical properties of mitral tissues. Preliminary simulations allowed mitral valve closure to be realistically mimicked and the role of annulus and papillary muscle dynamics to be quantified.

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.