A review of state-of-the-art numerical methods for simulating flow through mechanical heart valves
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paper_title: Turbulent stresses downstream of three mechanical aortic valve prostheses in human beings.

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High levels of turbulent stresses resulting from disturbed blood flow may cause damage to red blood cells and platelets. The purpose of this study was to evaluate the spatial distribution and temporal development of turbulent stresses downstream of three mechanical aortic valve prostheses in human subjects: the St. Jude Medical, the CarboMedics, and the Staff-Edwards silicone rubber ball. Blood velocity measurements were taken at 17 measuring points in the cross-sectional area of the ascending aorta 5 to 6 cm downstream of the aortic anulus with the use of a perivascular pulsed Doppler ultrasound system. Turbulence analysis was done for each of the 17 measuring points by calculating the radial Reynolds normal stresses within 50 msec overlapping time windows during systole. By coordinating the calculated Reynolds normal stress values for each time window and for all measuring points, computerized two-dimensional color-coded mapping of the turbulent stress distribution during systole was done. For the St. Jude Medical valves the highest Reynolds normal stress (27 to 63 N/m 2 ) were found along the central slit near the vessel walls. The temporal development and spatial distribution of Reynolds normal stresses for the CarboMedics valves were quite similar to those of the St. Jude Medical valves with maximum Reynolds normal stress values ranging from 19 to 72 N/m 2 . The typical Reynolds normal stress distribution for the Staff-Edwards silicone rubber ball valves was asymmetric, revealing the highest Reynolds normal stresses (11 to 56 N/m 2 ) at various locations in the annular region between the ball and the vessel wall. The spatial distribution and temporal development of turbulent stresses downstream of the three investigated mechanical aortic valve prostheses correlated well with the superstructure of the valves. The maximum Reynolds normal stresses for the three valve types were in the same order of magnitude with exposure times sufficient to cause sublethal damage to red blood cells and platelets.
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paper_title: In Vitro Hydrodynamic Characteristics Among Three Bileaflet Valves in the Mitral Position

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: The non–fully open phenomenon of the advancing standard medical bileaflet heart valves (the ATS valve) are frequently observed in clinical cases, even though there is no problem with their hemodynamic function. The movement of the leaflets was affected easily by the transvalvular flow because of the unique open pivot design of the ATS valve. In this paper, a comparative in vitro hydrodynamic test was conducted among 3 different types of bileaflet valves, and the effect of different shapes of downstream conduits, which induce different transvalvular flow, on hydrodynamic performance was studied. Three bileaflet valves, the ATS valve, CarboMedics valve (CM), and St. Jude Medical valve (SJM), with an annulus diameter of 29 mm for the mitral position were chosen throughout our experiments. First, pressure drops across the valves under steady flow were measured. Then, the valves were tested at the mitral position with our pneumatically driven pulsatile pump. In this pulsatile flow study, 2 different conduits (straight shape and abrupt enlargement shape) were in turn incorporated at the downstream portion of the mitral valve. A high-speed video camera was employed to observe leaflet movements. In a steady-flow test, the ATS and the SJM produced the same pressure drop, but the CM recorded a higher value. In the pulsatile study, it was observed that the ATS leaflets did not open fully in the mitral position when the downstream conduit with an abrupt enlargement shape was incorporated. However, the CM and the SJM always indicated a fully open movement regardless of the shape of downstream conduits. When the straight downstream conduit was incorporated, the ATS produced a similar pressure drop to that of the SJM, which coincided with the steady test results. When the enlargement conduit was incorporated, however, the ATS presented the lowest pressure drop despite the non–fully open movement. The conduit shape at the valve downstream had a significant influence on the closing volume. These findings indicate that the conduit shape at the valve outlet can affect the hydrodynamic characteristics of bileaflet valves.
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paper_title: Turbulent shear stress measurements in the vicinity of aortic heart valve prostheses

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Abstract A two dimensional laser Doppler anemometer system has been used to measure the turbulent shear fields in the immediate downstream vicinity of a variety of mechanical and bioprosthetic aortic heart valves. The measurements revealed that all the mechanical valves studied, created regions of elevated levels of turbulent shear stress during the major portion of systole. The tissue bioprostheses also created elevated levels of turbulence, but they were confined to narrow regions in the bulk of the flow field. The newer generation of bioprostheses create turbulent shear stresses which are considerably lower than those created by the older generation tissue valve designs. All the aortic valves studied (mechanical and tissue) create turbulent shear stress levels which are capable of causing sub-lethal and/or lethal damage to blood elements.
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paper_title: Vorticity dynamics of a bileaflet mechanical heart valve in an axisymmetric aorta

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We present comprehensive particle image velocimetry measurements and direct numerical simulation (DNS) of physiological, pulsatile flow through a clinical quality bileaflet mechanical heart valve mounted in an idealized axisymmetric aorta geometry with a sudden expansion modeling the aortic sinus region. Instantaneous and ensemble-averaged velocity measurements as well as the associated statistics of leaflet kinematics are reported and analyzed in tandem to elucidate the structure of the velocity and vorticity fields of the ensuing flow-structure interaction. The measurements reveal that during the first half of the acceleration phase, the flow is laminar and repeatable from cycle to cycle. The valve housing shear layer rolls up into the sinus and begins to extract vorticity of opposite sign from the sinus wall. A start-up vortical structure is shed from the leaflets and is advected downstream as the leaflet shear layers become wavy and oscillatory. In the second half of flow acceleration the leaflet shea...
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paper_title: Flow-Induced Platelet Activation in Bileaflet and Monoleaflet Mechanical Heart Valves

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A study was conducted to measure in vitro the procoagulant properties of platelets induced by flow through Carbomedics bileaflet and Bjork-Shiley monoleaflet mechanical heart valves (MHVs). Valves were mounted in a left ventricular assist device, and platelets were circulated through them under pulsatile flow. Platelet activation states (PAS) were measured during circulation using a modified prothrombinase method. Computational fluid dynamics (CFD) simulations of turbulent, transient, and non-Newtonian blood flow patterns generated by the two valve designs were done using the Wilcox k - w turbulence model, and platelet shear-stress histories (the integral of shear-stress exposure with respect to time) through the two MHVs were calculated. PAS measurements indicated that the bileaflet MHV activated platelets at a rate more than twice that observed with the monoleaflet MHV. Turbulent flow patterns were evident in CFD simulations for both valves, and corroborated the PAS observations, showing that, for particles close to the leaflet(s), shear-stress exposure in the bileaflet MHV can be more than four times that in the monoleaflet valve.
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paper_title: Experimental Investigation of the Steady Flow Downstream of the St. Jude Bileaflet Heart Valve: A Comparison Between Laser Doppler Velocimetry and Particle Image Velocimetry Techniques

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AbstractThis study investigates turbulent flow, based on high Reynolds number, downstream of a prosthetic heart valve using both laser Doppler velocimetry (LDV) and particle image velocimetry (PIV). Until now, LDV has been the more commonly used tool in investigating the flow characteristics associated with mechanical heart valves. The LDV technique allows point by point velocity measurements and provides enough statistical information to quantify turbulent structure. The main drawback of this technique is the time consuming nature of the data acquisition process in order to assess an entire flow field area. Another technique now used in fluid dynamics studies is the PIV measurement technique. This technique allows spatial and temporal measurement of the entire flow field. Using this technique, the instantaneous and average velocity flow fields can be investigated for different positions. This paper presents a comparison of PIV two-dimensional measurements to LDV measurements, performed under steady flow conditions, for a measurement plane parallel to the leaflets of a St. Jude Medical (SJM) bileaflet valve. Comparisons of mean velocity obtained by the two techniques are in good agreement except for where there is instability in the flow. For second moment quantities the comparisons were less agreeable. This suggests that the PIV technique has sufficient temporal and spatial resolution to estimate mean velocity depending on the degree of instability in the flow and also provides sufficient images needed to duplicate mean flow but not for higher moment turbulence quantities such as maximum turbulent shear stress. © 2000 Biomedical Engineering Society. ::: PAC00: 8719Uv, 4262Be, 8780-y
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paper_title: Regurgitant flow field characteristics of the St. Jude bileaflet mechanical heart valve under physiologic pulsatile flow using particle image velocimetry.

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The regurgitant flow fields of clinically used mechanical heart valves have been traditionally studied in vitro using flow visualization, ultrasound techniques, and laser Doppler velocimetry under steady and pulsatile flow. Detailed investigation of the forward and regurgitant flow fields of these valves can elucidate a valve's propensity for blood element damage, thrombus formation, or cavitation. Advances in particle image velocimetry (PIV) have allowed its use in the study of the flow fields of prosthetic valves. Unlike other flow field diagnostic systems, recent work using PIV has been able to relate particular regurgitant flow field characteristics of the Bjork-Shiley Monostrut valve to a propensity for cavitation. In this study, the regurgitant flow field of the St. Jude Medical bileaflet mechanical heart valve was assessed using PIV under physiologic pulsatile flow conditions. Data collected at selected time points prior to and after valve closure demonstrated the typical regurgitant jet flow patterns associated with the St. Jude valve, and indicated the formation of a strong regurgitant jet, in the B-datum plane, along with twin vortices near the leaflets. Estimated ensemble-average viscous shear rates suggested little potential for hemolysis when the hinge jets collided. However, the vortex motion near the occluder tips potentially provides a low-pressure environment for cavitation.
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paper_title: Velocity measurements and flow patterns within the hinge region of a Medtronic Parallel bileaflet mechanical valve with clear housing.

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BACKGROUND AND AIMS OF THE STUDY: During recent clinical trials the Medtronic Parallel bileaflet mechanical heart valve was found to have an unacceptable number of valves with thrombus formation when implanted in the mitral position. Thrombi were observed in the hinge region and also in the upstream portion of the valve housing in the vicinity of the hinge. It was hypothesized that the flow conditions inside the hinge may have contributed to the thrombus formation. METHODS: In order to investigate the flow structures within the hinge, laser Doppler anemometry (LDA) measurements were conducted in both steady and pulsatile flow at approximately 70 predetermined sites within the hinge region of a 27 mm Medtronic Parallel mitral valve with transparent housing. The pulsatile flow velocity measurements were animated in time using a graphical software package to visualize the hinge flow field throughout the cardiac cycle. RESULTS: The LDA measurements revealed that mean forward flow velocities through the hinge region were on the order of 0.10-0.20 m/s. In the inflow channel, a large vortical structure was present during diastole. Upon valve closure, peak reverse velocity reached 3 m/s close to the housing wall in the inflow channel. This area also experienced high turbulent shear stresses (> 6000 dynes/cm2) during the leakage flow phase. A disturbed, vortical flow was again present in the inflow channel after valve closure, while slightly above the leaflet peg and relief the flow was essentially stagnant. The high turbulent stresses near the top of the inflow channel, combined with a persistent vortex, implicate the inflow channel of the hinge as a likely region of thrombus formation. CONCLUSIONS: This experimental investigation revealed zones of flow stagnation in the inflow region of the hinge throughout the cardiac cycle and elevated turbulent shear stress levels in the inflow region during the leakage flow phase. These fluid mechanic phenomena are most likely a direct result of the complex geometry of the hinge of this valve. Although the LDA measurements were conducted at only a limited number of sites within the hinge, these results suggest that the hinge design can significantly affect the washout capacity and thrombogenic potential of the Medtronic Parallel bileaflet mechanical heart valve. The use of LDA within the confines of the hinge region of a mechanical heart valve is a new application, made possible by recent advances in manufacturing technologies and a proprietary process developed by Medtronic that allowed the production of a transparent valve housing. Together, these modalities represent a new method by which future valve designs can be assessed before clinical trials are initiated.
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paper_title: An Analysis of Turbulent Shear Stresses in Leakage Flow Through a Bileaflet Mechanical Prostheses

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In this work, estimates of turbulence were made from pulsatile flow laser Doppler velocimetry measurements using traditional phase averaging and averaging after the removal of cyclic variation. These estimates were compared with estimates obtained from steady leakage flow LDV measurements and an analytical method. The results of these studies indicate that leakage jets which are free and planar in shape may be more unstable than other leakage jets, and that cyclic variation does not cause a gross overestimation of the Reynolds stresses at large distances from the leakage jet orifice.
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paper_title: STEADY FLOW DYNAMICS OF PROSTHETIC AORTIC HEART VALVES : A COMPARATIVE EVALUATION WITH PIV TECHNIQUES

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Abstract Particle Image Velocimetry (PIV), capable of providing full-field measurement of velocities and flow stresses, has become an invaluable tool in studying flow behaviour in prosthetic heart valves. This method was used to evaluate the performances of four prosthetic heart valves; a porcine bioprostheses, a caged ball valve, and two single leaflet tilting disc valves with different opening angles. Flow visualization techniques, combined with velocity vector fields and Reynolds stresses mappings in the aortic root obtained from PIV, and pressure measurements were used to give an overall picture of the flow field of the prosthetic heart valves under steady flow conditions. The porcine bioprostheses exhibited the highest pressure loss and Reynolds stresses of all the valves tested. This was mainly due to the reduction in orifice area caused by the valve mounting ring and the valve stents. For the tilting disc valves, a larger opening angle resulted in a smoother flow profile, and thus lower Reynolds stresses and pressure drops. The St. Vincent valve exhibited the lowest pressure drop and Reynolds stresses.
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paper_title: Fluid mechanics of heart valves.

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Valvular heart disease is a life-threatening disease that afflicts millions of people worldwide and leads to approximately 250,000 valve repairs and/or replacements each year. Malfunction of a native valve impairs its efficient fluid mechanic/hemodynamic performance. Artificial heart valves have been used since 1960 to replace diseased native valves and have saved millions of lives. Unfortunately, despite four decades of use, these devices are less than ideal and lead to many complications. Many of these complications/problems are directly related to the fluid mechanics associated with the various mechanical and bioprosthetic valve designs. This review focuses on the state-of-the-art experimental and computational fluid mechanics of native and prosthetic heart valves in current clinical use. The fluid dynamic performance characteristics of caged-ball, tilting-disc, bileaflet mechanical valves and porcine and pericardial stented and nonstented bioprostheic valves are reviewed. Other issues related to heart valve performance, such as biomaterials, solid mechanics, tissue mechanics, and durability, are not addressed in this review.
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paper_title: Characterization of Hemodynamic Forces Induced by Mechanical Heart Valves: Reynolds vs. Viscous Stresses

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Bileaflet mechanical heart valves (BMHV) are widely used to replace diseased heart valves. Implantation of BMHV, however, has been linked with major complications, which are generally considered to be caused by mechanically induced damage of blood cells resulting from the non-physiological hemodynamics environment induced by BMHV, including regions of recirculating flow and elevated Reynolds (turbulence) shear stress levels. In this article, we analyze the results of 2D high-resolution velocity measurements and full 3D numerical simulation for pulsatile flow through a BMHV mounted in a model axisymmetric aorta to investigate the mechanical environment experienced by blood elements under physiologic conditions. We show that the so-called Reynolds shear stresses neither directly contribute to the mechanical load on blood cells nor is a proper measurement of the mechanical load experienced by blood cells. We also show that the overall levels of the viscous stresses, which comprise the actual flow environment experienced by cells, are apparently too low to induce damage to red blood cells, but could potentially damage platelets. The maximum instantaneous viscous shear stress observed throughout a cardiac cycle is <15 N/m(2). Our analysis is restricted to the flow downstream of the valve leaflets and thus does not address other areas within the BMHV where potentially hemodynamically hazardous levels of viscous stresses could still occur (such as in the hinge gaps and leakage jets).
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paper_title: Turbulent stress measurements downstream of three bileaflet heart valve designs in pigs

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OBJECTIVE ::: Mechanical heart valves can cause thromboembolic complications, possibly due to abnormal flow patterns that produce turbulence downstream of the valve. The objective of this study was to investigate whether three different bileaflet valve designs would exhibit clinically relevant differences in downstream turbulent stresses. ::: ::: ::: METHODS ::: Three bileaflet mechanical heart valves (Medtronic Advantage), CarboMedics Orbis Universal and St. Jude Medical Standard) were implanted into 19 female 90 kg pigs. Blood velocity was measured during open chest conditions in the cross sectional area downstream of the valves with 10 MHz ultrasonic probes connected to a modified Alfred Pulsed Doppler equipment. As a measure of turbulence, Reynolds normal stress (RNS) was calculated at three different cardiac output ranges (3-4, 4.5-5.5, 6-7 L/min). ::: ::: ::: RESULTS ::: Data from 12 animals were obtained. RNS correlated with increasing cardiac outputs. The highest instantaneous RNS observed in these experiments was 47 N/m2, and the mean RNS taken spatially over the cross sectional area of the aorta during systole was between 3 N/m2 and 15 N/m2. In none of the cardiac output ranges RNS values exceeded the lower critical limit for erythrocyte or thrombocyte damage for any of the valve designs. ::: ::: ::: CONCLUSIONS ::: Reynolds normal stress values were below 100 N/m2 for all three valve designs and the difference in design was not reflected in generation of turbulence. Hence, it is unlikely that any of the valve designs causes flow induced damage to platelets or erythrocytes.
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paper_title: An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries

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We present an immersed interface method for the incompressible Navier-Stokes equations capable of handling both rigid and flexible boundaries. The immersed boundaries are represented by a number of Lagrangian control points. In order to ensure that the no-slip condition on the rigid boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep. For flexible boundaries, the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions. The position of the flexible boundary is updated implicitly using a quasi-Newton method (BFGS) within each timestep. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate.
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paper_title: A distributed Lagrange multiplier/fictitious domain method for particulate flows

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Abstract A new Lagrange-multiplier based fictitious-domain method is presented for the direct numerical simulation of viscous incompressible flow with suspended solid particles. The method uses a finite-element discretization in space and an operator-splitting technique for discretization in time. The linearly constrained quadratic minimization problems which arise from this splitting are solved using conjugate-gradient algorithms. A key feature of the method is that the fluid–particle motion is treated implicitly via a combined weak formulation in which the mutual forces cancel—explicit calculation of the hydrodynamic forces and torques on the particles is not required. The fluid flow equations are enforced inside, as well as outside, the particle boundaries. The flow inside, and on, each particle boundary is constrained to be a rigid-body motion using a distributed Lagrange multiplier. This multiplier represents the additional body force per unit volume needed to maintain the rigid-body motion inside the particle boundary, and is analogous to the pressure in incompressible fluid flow, whose gradient is the force required to maintain the constraint of incompressibility. The method is validated using the sedimentation of two circular particles in a two-dimensional channel as the test problem, and is then applied to the sedimentation of 504 circular particles in a closed two-dimensional box. The resulting suspension is fairly dense, and the computation could not be carried out without an effective strategy for preventing particles from penetrating each other or the solid outer walls; in the method described herein, this is achieved by activating a repelling force on close approach, such as might occur as a consequence of roughness elements on the particle. The development of physically based mathematical methods for avoiding particle–particle and particle–wall penetration is a new problem posed by the direct simulation of fluidized suspensions. The simulation starts with the particles packed densely at the top of the sedimentation column. In the course of their fall to the bottom of the box, a fingering motion of the particles, which are heavier than the surrounding fluid, develops in a way reminiscent of the familiar dynamics associated with the Rayleigh–Taylor instability of heavy fluid above light. We also present here the results of a three-dimensional simulation of the sedimentation of two spherical particles. The simulation reproduces the familiar dynamics of drafting, kissing and tumbling to side-by-side motion with the line between centers across the flow at Reynolds numbers in the hundreds.
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paper_title: A three-dimensional computational method for blood flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid

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Abstract This paper describes the numerical solution of the 3-dimensional equations of motion of a viscous incompressible fluid that contains an immersed system of elastic fibers. Implementation details such as vectorization and the efficient use of external memory are discussed. The method is applied to the damped vibrations of a fiber-wound toroidal tube, and empirical evidence of convergence is presented.
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paper_title: Flow patterns around heart valves: A numerical method

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Abstract The subject of this paper is the flow of a viscous incompressible fluid in a region containing immersed boundaries which move with the fluid and exert forces on the fluid. An example of such a boundary is the flexible leaflet of a human heart valve. It is the main achievement of the present paper that a method for solving the Navier-Stokes equations on a rectangular domain can now be applied to a problem involving this type of immersed boundary. This is accomplished by replacing the boundary by a field of force which is defined on the mesh points of the rectangular domain and which is calculated from the configuration of the boundary. In order to link the representations of the boundary and fluid, since boundary points and mesh points need not coincide, a semi-discrete analog of the δ function is introduced. Because the boundary forces are of order h −1 , and because they are sensitive to small changes in boundary configuration, they tend to produce numerical instability. This difficulty is overcome by an implicit method for calculating the boundary forces, a method which takes into account the displacements that will be produced by the boundary forces themselves. The numerical scheme is applied to the two-dimensional simulation of flow around the natural mitral valve.
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paper_title: Three-Dimensional Fluid-Structure Interaction Simulation of Bileaflet Mechanical Heart Valve Flow Dynamics

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The wall shear stress induced by the leaflet motion during the valve-closing phase has been implicated with thrombus initiation with prosthetic valves. Detailed flow dynamic analysis in the vicinity of the leaflets and the housing during the valve-closure phase is of interest in understanding this relationship. A three-dimensional unsteady flow analysis past bileaflet valve prosthesis in the mitral position is presented incorporating a fluid-structure interaction algorithm for leaflet motion during the valve-closing phase. Arbitrary Lagrangian-Eulerian method is employed for incorporating the leaflet motion. The forces exerted by the fluid on the leaflets are computed and applied to the leaflet equation of motion to predict the leaflet position. Relatively large velocities are computed in the valve clearance region between the valve housing and the leaflet edge with the resulting relatively large wall shear stresses at the leaflet edge during the impact-rebound duration. Negative pressure transients are computed on the surface of the leaflets on the atrial side of the valve, with larger magnitudes at the leaflet edge during the closing and rebound as well. Vortical flow development is observed on the inflow (atrial) side during the valve impact-rebound phase in a location central to the leaflet and away from the clearance region where cavitation bubbles have been visualized in previously reported experimental studies.
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paper_title: A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids

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In the present note a general reconstruction algorithm for simulating incompressible flows with complex immersed boundaries on Cartesian grids is presented. In the proposed method an arbitrary three-dimensional solid surface immersed in the fluid is discretized using an unstructured, triangular mesh, and all the Cartesian grid nodes near the interface are identified. Then, the solution at these nodes is reconstructed via linear interpolation along the local normal to the body, in a way that the desired boundary conditions for both pressure and velocity fields are enforced. The overall accuracy of the resulting solver is second-order, as it is demonstrated in two test cases involving laminar flow past a sphere.
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paper_title: Numerical simulation of 3D fluid-structure interaction flow using an immersed object method with overlapping grids

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The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151-72] is extended for 3D unsteady flow simulation with fluid-structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier-Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17-20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on ''Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425-36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V"0 can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood-leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.
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paper_title: An immersed boundary method for complex incompressible flows

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An immersed boundary method for time-dependent, three-dimensional, incompressible flows is presented in this paper. The incompressible Navier-Stokes equations are discretized using a low-diffusion flux splitting method for the inviscid fluxes and second-order central-differences for the viscous components. Higher-order accuracy achieved by using weighted essentially non-oscillatory (WENO) or total variation diminishing (TVD) schemes. An implicit method based on artificial compressibility and dual-time stepping is used for time advancement. The immersed boundary surfaces are defined as clouds of points, which may be structured or unstructured. Immersed-boundary objects are rendered as level sets in the computational domain, and concepts from computational geometry are used to classify points as being outside, near, or inside the immersed boundary. The velocity field near an immersed surface is determined from separate interpolations of the components tangent and normal to the surface. The tangential velocity near the surface is constructed as a power-law function of the local wall normal distance. Appropriate choices of the power law enable the method to approximate the energizing effects of a turbulent boundary layer for higher Reynolds number flows. Five different flow problems (flow over a circular cylinder, an in-line oscillating cylinder, a NACA0012 airfoil, a sphere, and a stationary mannequin) are simulated using the present immersed boundary method, and the predictions show good agreement with previous computational and experimental results. Finally, the flow induced by realistic human walking motion is simulated as an example of a problem involving multiple moving immersed objects.
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paper_title: A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies

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A numerical method is developed for solving the 3D, unsteady, incompressible Navier-Stokes equations in Cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics. The governing equations are discretized on a hybrid staggered/non-staggered grid layout using second-order accurate finite-difference formulas. The discrete equations are integrated in time via a second-order accurate dual-time-stepping, artificial compressibility iteration scheme. Unstructured, triangular meshes are employed to discretize complex immersed boundaries. The nodes of the surface mesh constitute a set of Lagrangian control points used to track the motion of the flexible body. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at Cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. Grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity, which show that the method is second-order accurate. The method is validated by applying it to calculate flow in a Cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing. The ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish-like body and flow past an anatomically realistic planktonic copepod performing an escape-like maneuver. euver.
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paper_title: Vorticity dynamics of a bileaflet mechanical heart valve in an axisymmetric aorta

paper_content:

We present comprehensive particle image velocimetry measurements and direct numerical simulation (DNS) of physiological, pulsatile flow through a clinical quality bileaflet mechanical heart valve mounted in an idealized axisymmetric aorta geometry with a sudden expansion modeling the aortic sinus region. Instantaneous and ensemble-averaged velocity measurements as well as the associated statistics of leaflet kinematics are reported and analyzed in tandem to elucidate the structure of the velocity and vorticity fields of the ensuing flow-structure interaction. The measurements reveal that during the first half of the acceleration phase, the flow is laminar and repeatable from cycle to cycle. The valve housing shear layer rolls up into the sinus and begins to extract vorticity of opposite sign from the sinus wall. A start-up vortical structure is shed from the leaflets and is advected downstream as the leaflet shear layers become wavy and oscillatory. In the second half of flow acceleration the leaflet shea...
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paper_title: Collagen fibers reduce stresses and stabilize motion of aortic valve leaflets during systole.

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The effect of collagen fibers on the mechanics and hemodynamics of a trileaflet aortic valve contained in a rigid aortic root is investigated in a numerical analysis of the systolic phase. Collagen fibers are known to reduce stresses in the leaflets during diastole, but their role during systole has not been investigated in detail yet. It is demonstrated that also during systole these fibers substantially reduce stresses in the leaflets and provide smoother opening and closing. Compared to isotropic leaflets, collagen reinforcement reduces the fluttering motion of the leaflets. Due to the exponential stress-strain behavior of collagen, the fibers have little influence on the initial phase of the valve opening, which occurs at low strains, and therefore have little impact on the transvalvular pressure drop.
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paper_title: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

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The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.
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paper_title: Two-dimensional fluid-structure interaction simulation of bileaflet mechanical heart valve flow dynamics.

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BACKGROUND AND AIM OF THE STUDY: Mechanical heart valve implantation requires long-term anticoagulation because of thromboembolic complications. Recent studies have indicated that the relatively high wall shear stresses and negative pressure transients developed during the valve closing phase may be dominant factors inducing thrombus initiation. The study aim was a two-dimensional (2D) functional simulation of flow past bileaflet heart valve prosthesis during the closing phase, incorporating the fluid-structure interaction analysis to induce motion of the leaflets. METHODS: The fluid-structure interaction model used was based on unsteady 2D Navier-Stokes equations with the arbitrary Lagrangian-Eulerian method for moving boundaries, coupled with the dynamic equation for leaflet motion. Parametric analysis of the effect of valve size, leaflet density, and the coefficient of resilience at the instant of impact of the leaflet with the housing were also performed. RESULTS: Comparing the predicted motion of the leaflet with previous experimental results validated the simulation. The results showed the presence of negative pressure transients near the inflow side of the leaflet at the instant of valve closure, and the negative pressure transients were augmented during the leaflet rebound process. Relatively high velocities and wall shear stresses, detrimental to the formed elements in blood were present in the clearance region between the leaflet and valve housing at the instant of valve closure. CONCLUSION: The simulation can be potentially applied to analyze the effects of valve geometry and dimensions, and the effect of leaflet material on the flow dynamics past the valve prosthesis during the opening and closing phases for design improvements in minimizing problems associated with thromboembolic complications.
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paper_title: A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries

paper_content:

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.
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paper_title: IMMERSED BOUNDARY METHODS

paper_content:

The term “immersed boundary method” was first used in reference to a method developed by Peskin (1972) to simulate cardiac mechanics and associated blood flow. The distinguishing feature of this method was that the entire simulation was carried out on a Cartesian grid, which did not conform to the geometry of the heart, and a novel procedure was formulated for imposing the effect of the immersed boundary (IB) on the flow. Since Peskin introduced this method, numerous modifications and refinements have been proposed and a number of variants of this approach now exist. In addition, there is another class of methods, usually referred to as “Cartesian grid methods,” which were originally developed for simulating inviscid flows with complex embedded solid boundaries on Cartesian grids (Berger & Aftosmis 1998, Clarke et al. 1986, Zeeuw & Powell 1991). These methods have been extended to simulate unsteady viscous flows (Udaykumar et al. 1996, Ye et al. 1999) and thus have capabilities similar to those of IB methods. In this review, we use the term immersed boundary (IB) method to encompass all such methods that simulate viscous flows with immersed (or embedded) boundaries on grids that do not conform to the shape of these boundaries. Furthermore, this review focuses mainly on IB methods for flows with immersed solid boundaries. Application of these and related methods to problems with liquid-liquid and liquid-gas boundaries was covered in previous reviews by Anderson et al. (1998) and Scardovelli & Zaleski (1999). Consider the simulation of flow past a solid body shown in Figure 1a. The conventional approach to this would employ structured or unstructured grids that conform to the body. Generating these grids proceeds in two sequential steps. First, a surface grid covering the boundaries b is generated. This is then used as a boundary condition to generate a grid in the volume f occupied by the fluid. If a finite-difference method is employed on a structured grid, then the differential form of the governing equations is transformed to a curvilinear coordinate system aligned with the grid lines (Ferziger & Peric 1996). Because the grid conforms to the surface of the body, the transformed equations can then be discretized in the
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paper_title: A ghost-cell immersed boundary method for flow in complex geometry

paper_content:

An efficient ghost-cell immersed boundary method (GCIBM) for simulating turbulent flows in complex geometries is presented. A boundary condition is enforced through a ghost cell method. The reconstruction procedure allows systematic development of numerical schemes for treating the immersed boundary while preserving the overall second-order accuracy of the base solver. Both Dirichlet and Neumann boundary conditions can be treated. The current ghost cell treatment is both suitable for staggered and non-staggered Cartesian grids. The accuracy of the current method is validated using flow past a circular cylinder and large eddy simulation of turbulent flow over a wavy surface. Numerical results are compared with experimental data and boundary-fitted grid results. The method is further extended to an existing ocean model (MITGCM) to simulate geophysical flow over a three-dimensional bump. The method is easily implemented as evidenced by our use of several existing codes.
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paper_title: The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources

paper_content:

The authors develop finite difference methods for elliptic equations of the form \[ \nabla \cdot (\beta (x)\nabla u(x)) + \kappa (x)u(x) = f(x)\] in a region $\Omega $ in one or two space dimension...
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paper_title: A three-dimensional computational analysis of fluid–structure interaction in the aortic valve

paper_content:

Abstract 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.
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paper_title: Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations

paper_content:

A second-order accurate, highly efficient method is developed for simulating unsteady three-dimensional incompressible flows in complex geometries. This is achieved by using boundary body forces that allow the imposition of the boundary conditions on a given surface not coinciding with the computational grid. The governing equations, therefore, can be discretized and solved on a regular mesh thus retaining the advantages and the efficiency of the standard solution procedures. Two different forcings are tested showing that while the quality of the results is essentially the same in both cases, the efficiency of the calculation strongly depends on the particular expression. A major issue is the interpolation of the forcing over the grid that determines the accuracy of the scheme; this ranges from zeroth-order for the most commonly used interpolations up to second-order for an ad hoc velocity interpolation. The present scheme has been used to simulate several flows whose results have been validated by experiments and other results available in the literature. Finally in the last example we show the flow inside an IC piston/cylinder assembly at high Reynolds number; to our knowledge this is the first example in which the immersed boundary technique is applied to a full three-dimensional complex flow with moving boundaries and with a Reynolds number high enough to require a subgrid-scale turbulence model.
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paper_title: A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve.

paper_content:

The importance of the aortic root compliance in the aortic valve performance has most frequently been ignored in computational valve modeling, although it has a significant contribution to the functionality of the valve. Aortic root aneurysm or (calcific) stiffening severely affects the aortic valve behavior and, consequently, the cardiovascular regulation. The compromised mechanical and hemodynamical performance of the valve are difficult to study both 'in vivo' and 'in vitro'. Computational analysis of the valve enables a study on system responses that are difficult to obtain otherwise. In this paper a numerical model of a fiber-reinforced stentless aortic valve is presented. In the computational evaluation of its clinical functioning the interaction of the valve with the blood is essential. Hence, the blood-tissue interaction is incorporated in the model using a combined fictitious domain/arbitrary Lagrange-Euler formulation, which is integrated within the Galerkin finite element method. The model can serve as a diagnostic tool for clinical purposes and as a design tool for improving existing valve prostheses or developing new concepts. Structural mechanical and fluid dynamical aspects are analyzed during the systolic course of the cardiac cycle. Results show that aortic root compliance largely influences the valve opening and closing configurations. Stresses in the delicate parts of the leaflets are substantially reduced if fiber-reinforcement is applied and the aortic root is able to expand.
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paper_title: A two-dimensional fluid–structure interaction model of the aortic value

paper_content:

Failure of synthetic heart valves is usually caused by tearing and calci"cation of the lea#ets. Lea#et "ber-reinforcement increases the durability of these values by unloading the delicate parts of the lea#ets, maintaining their physiological functioning. The interaction of the valve with the surrounding #uid is essential when analyzing its functioning. However, the large di!erences in material properties of #uid and structure and the "nite motion of the lea#ets complicate blood}valve interaction modeling. This has, so far, obstructed numerical analyses of valves operating under physiological conditions. A two-dimensional #uid}structure interaction model is presented, which allows the Reynolds number to be within the physiological range, using a "ctitious domain method based on Lagrange multipliers to couple the two phases. The extension to the three-dimensional case is straightforward. The model has been validated experimentally using laser Doppler anemometry for measuring the #uid #ow and digitized high-speed video recordings to visualize the lea#et motion in corresponding geometries. Results show that both the #uid and lea#et behaviour are well predicted for di!erent lea#et thicknesses. ( 2000 Elsevier Science Ltd. All rights reserved.
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paper_title: An adaptive, formally second order accurate version of the immersed boundary method

paper_content:

Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509–534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75–105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.
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paper_title: Computation of Solid-Liquid Phase Fronts in the Sharp Interface Limit on Fixed Grids

paper_content:

A finite-difference formulation is applied to track solid?liquid boundaries on a fixed underlying grid. The interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked. The imposition of boundary conditions exactly on a sharp interface that passes through the Cartesian grid is performed using simple stencil readjustments in the vicinity of the interface. Attention is paid to formulating difference schemes that are globally second-order accurate in x and t. Error analysis and grid refinement studies are performed for test problems involving the diffusion and convection?diffusion equations, and for stable solidification problems. Issues concerned with stability and change of phase of grid points in the evolution of solid?liquid phase fronts are also addressed. It is demonstrated that the field calculation is second-order accurate while the position of the phase front is calculated to first-order accuracy. Furthermore, the accuracy estimates hold for the cases where there is a property jump across the interface. Unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work. The results indicate the need to begin an effort to benchmark computations of instability phenomena.
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paper_title: Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations

paper_content:

A Cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three-dimensional immersed boundaries. The embedded boundaries are represented using level-sets and treated in a sharp manner without the use of source terms to represent boundary effects. The narrow-band distance function field in the level-set boundary representation facilitates implementation of the finite-difference flow solver. The resulting algorithm is implemented in a straightforward manner in three-dimensions and retains global second-order accuracy. The accuracy of the finite-difference scheme is established and shown to be comparable to finite-volume schemes that are considerably more difficult to implement. Moving boundaries are handled naturally. The pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries. Benchmarking of the method is performed against available numerical as well as experimental results.
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paper_title: Analysis and Stabilization of Fluid-Structure Interaction Algorithm for Rigid-Body Motion

paper_content:

Fluid-structure interaction computations in geometries where different chambers are almost completely separated from each other by a movable rigid body but connected through very small gaps can encounter stability problems when a standard explicit coupling procedure is used for the coupling of the fluid flow and the movement of the rigid body. An example of such kind of flows is the opening and closing of valves, when the valve motion is driven by the flow. A stability analysis is performed for the coupling procedure of the movement of a cylinder in a cylindrical tube, filled with fluid, Between the moving cylinder and the tube, a small gap is present, so that two chambers are formed. It is shown that a standard explicit coupling procedure or an implicit coupling procedure with explicit coupling in the subiterations steps can lead to unstable motion depending on the size of the gaps, the density of the rigid body, and the density of the fluid. It is proven that a reduction of the time-step size cannot stabilize the coupling procedure. An implicit coupling procedure with implicit coupling in the subiterations has to be used. An illustration is given on how such a coupling procedure can be implemented in a commercial computational fluid dynamics (CFD) software package. The CFD package FLUENT (Fluent, Inc.) is used. As an application, the opening and the closing of a prosthetic aortic valve is computed.
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paper_title: Numerical simulation of the dynamics of a bileaflet prosthetic heart valve using a fluid-structure interaction approach

paper_content:

Abstract The main purpose of this study is to reproduce in silico the dynamics of a bileaflet mechanical heart valve (MHV; St Jude Hemodynamic Plus, 27 mm characteristic size) by means of a fully implicit fluid–structure interaction (FSI) method, and experimentally validate the results using an ultrafast cinematographic technique. The computational model was constructed to realistically reproduce the boundary condition (72 beats per minute (bpm), cardiac output 4.5 l/min) and the geometry of the experimental setup, including the valve housing and the hinge configuration. The simulation was carried out coupling a commercial computational fluid dynamics (CFD) package based on finite-volume method with user-defined code for solving the structural domain, and exploiting the parallel performance of the whole numerical setup. Outputs are leaflets excursion from opening to closure and the fluid dynamics through the valve. Results put in evidence a favorable comparison between the computed and the experimental data: the model captures the main features of the leaflet motion during the systole. The use of parallel computing drastically limited the computational costs, showing a linear scaling on 16 processors (despite the massive use of user-defined subroutines to manage the FSI process). The favorable agreement obtained between in vitro and in silico results of the leaflet displacements confirms the consistency of the numerical method used, and candidates the application of FSI models to become a major tool to optimize the MHV design and eventually provides useful information to surgeons.
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paper_title: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

paper_content:

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.
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paper_title: High-Resolution Fluid–Structure Interaction Simulations of Flow Through a Bi-Leaflet Mechanical Heart Valve in an Anatomic Aorta

paper_content:

We have performed high-resolution fluid-structure interaction simulations of physiologic pulsatile flow through a bi-leaflet mechanical heart valve (BMHV) in an anatomically realistic aorta. The results are compared with numerical simulations of the flow through an identical BMHV implanted in a straight aorta. The comparisons show that although some of the salient features of the flow remain the same, the aorta geometry can have a major effect on both the flow patterns and the motion of the valve leaflets. For the studied configuration, for instance, the BMHV leaflets in the anatomic aorta open much faster and undergo a greater rebound during closing than the same valve in the straight axisymmetric aorta. Even though the characteristic triple-jet structure does emerge downstream of the leaflets for both cases, for the anatomic case the leaflet jets spread laterally and diffuse much faster than in the straight aorta due to the aortic curvature and complex shape of the anatomic sinus. Consequently the leaflet shear layers in the anatomic case remain laminar and organized for a larger portion of the accelerating phase as compared to the shear layers in the straight aorta, which begin to undergo laminar instabilities well before peak systole is reached. For both cases, however, the flow undergoes a very similar explosive transition to the small-scale, turbulent-like state just prior to reaching peak systole. The local maximum shear stress is used as a metric to characterize the mechanical environment experienced by blood cells. Pockets of high local maximum shear are found to be significantly more widespread in the anatomic aorta than in the straight aorta throughout the cardiac cycle. Pockets of high local maximum shear were located near the leaflets and in the aortic arc region. This work clearly demonstrates the importance of the aortic geometry on the flow phenomena in a BMHV and demonstrates the potential of our computational method to carry out image-based patient-specific simulations for clinically relevant studies of heart valve hemodynamics.
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paper_title: Comparison of the Hemodynamic and Thrombogenic Performance of Two Bileaflet Mechanical Heart Valves Using a CFD/FSI Model

paper_content:

The hemodynamic and the thrombogenic performance of two commercially available bileaflet mechanical heart valves (MHVs)--the ATS Open Pivot Valve (ATS) and the St. Jude Regent Valve (SJM), was compared using a state of the art computational fluid dynamics-fluid structure interaction (CFD-FSI) methodology. A transient simulation of the ATS and SJM valves was conducted in a three-dimensional model geometry of a straight conduit with sudden expansion distal the valves, including the valve housing and detailed hinge geometry. An aortic flow waveform (60 beats/min, cardiac output 4 l/min) was applied at the inlet. The FSI formulation utilized a fully implicit coupling procedure using a separate solver for the fluid problem (FLUENT) and for the structural problem. Valve leaflet excursion and pressure differences were calculated, as well as shear stress on the leaflets and accumulated shear stress on particles released during both forward and backward flow phases through the open and closed valve, respectively. In contrast to the SJM, the ATS valve opened to less than maximal opening angle. Nevertheless, maximal and mean pressure gradients and velocity patterns through the valve orifices were comparable. Platelet stress accumulation during forward flow indicated that no platelets experienced a stress accumulation higher than 35 dyne x s/cm2, the threshold for platelet activation (Hellums criterion). However, during the regurgitation flow phase, 0.81% of the platelets in the SJM valve experienced a stress accumulation higher than 35 dyne x s/cm2, compared with 0.63% for the ATS valve. The numerical results indicate that the designs of the ATS and SJM valves, which differ mostly in their hinge mechanism, lead to different potential for platelet activation, especially during the regurgitation phase. This numerical methodology can be used to assess the effects of design parameters on the flow induced thrombogenic potential of blood recirculating devices.
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paper_title: Partitioned analysis of coupled mechanical systems

paper_content:

Abstract This is a tutorial article that reviews the use of partitioned analysis procedures for the analysis of coupled dynamical systems. Attention is focused on the computational simulation of systems in which a structure is a major component. Important applications in that class are provided by thermomechanics, fluid–structure interaction and control–structure interaction. In the partitioned solution approach, systems are spatially decomposed into partitions. This decomposition is driven by physical or computational considerations. The solution is separately advanced in time over each partition. Interaction effects are accounted for by transmission and synchronization of coupled state variables. Recent developments in the use of this approach for multilevel decomposition aimed at massively parallel computation are discussed.
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paper_title: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

paper_content:

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.
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paper_title: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

paper_content:

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.
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paper_title: Numerical Simulation of Flow in Mechanical Heart Valves: Grid Resolution and the Assumption of Flow Symmetry

paper_content:

A numerical method is developed for simulating unsteady, 3-D, laminar flow through a bileaflet mechanical heart valve with the leaflets fixed. The method employs a dual-timestepping artificial-compressibility approach together with overset (Chimera) grids and is second-order accurate in space and time. Calculations are carried out for the full 3-D valve geometry under steady inflow conditions on meshes with a total number of nodes ranging from 4310 5 to 1.6310 6 . The computed results show that downstream of the leaflets the flow is dominated by two pairs of counter-rotating vortices, which originate on either side of the central orifice in the aortic sinus and rotate such that the common flow of each pair is directed away from the aortic wall. These vortices intensify with Reynolds number, and at a Reynolds number of approximately 1200 their complex interaction leads to the onset of unsteady flow and the break of symmetry with respect to both geometric planes of symmetry. Our results show the highly 3-D structure of the flow; question the validity of computationally expedient assumptions of flow symmetry; and demonstrate the need for highly resolved, fully 3-D simulations if computational fluid dynamics is to accurately predict the flow in prosthetic mechanical heart valves. @DOI: 10.1115/1.1614817#
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paper_title: Unsteady Effects on the Flow Across Tilting Disk Valves

paper_content:

The present study simulates numerically the ﬂow across two-dimensional tilting disk mod-els of mechanical heart valves. The time-dependent Navier-Stokes equations are solved toassess the importance of unsteady effects in the fully open position of the valve. Flowcases with steady or physiological inﬂow conditions and with ﬁxed or moving valves aresolved. The simulations lead into mixed conclusions. It is obvious that steady inﬂow casesthat account for vortex shedding only cannot model realistic physiological cases. In caseswith imposed physiological inﬂow, the details of the ﬂow ﬁeld for ﬁxed and moving valvesmight differ in the fully open position as well, although the gross features are quitesimilar. The ﬁxed valve case consistently results in safe estimations of several criticalquantities such as the axial force, the maximal shear stress on the valve, or the transval-vular pressure drop. Thus, ﬁxed valve simulations can provide useful information for thedesign of prosthetic heart valves, as long as the properties in the fully open position onlyare sought. @DOI: 10.1115/1.1427696#
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paper_title: Numerical Simulations of Fluid-Structure Interaction Problems in Biological Flows

paper_content:

University of Minnesota Ph.D. dissertation. June, 2008. Major: Mechanical Engineering. Advisor: Fotis Sotiropoulos. 1 computer file (PDF); xvi, 273 pages.
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paper_title: Two-Dimensional Dynamic Simulation of Platelet Activation During Mechanical Heart Valve Closure

paper_content:

A major drawback in the operation of mechanical heart valve prostheses is thrombus formation in the near valve region. Detailed flow analysis in this region during the valve closure phase is of interest in understanding the relationship between shear stress and platelet activation. A fixed-grid Cartesian mesh flow solver is used to simulate the blood flow through a bi-leaflet mechanical valve employing a two-dimensional geometry of the leaflet with a pivot point representing the hinge region. A local mesh refinement algorithm allows efficient and fast flow computations with mesh adaptation based on the gradients of the flow field in the leaflet-housing gap at the instant of valve closure. Leaflet motion is calculated dynamically based on the fluid forces acting on it employing a fluid-structure interaction algorithm. Platelets are modeled and tracked as point particles by a Lagrangian particle tracking method which incorporates the hemodynamic forces on the particles. A platelet activation model is included to predict regions which are prone to platelet activation. Closure time of the leaflet is validated against experimental studies. Results show that the orientation of the jet flow through the gap between the housing and the leaflet causes the boundary layer from the valve housing to be drawn in by the shear layer separating from the leaflet. The interaction between the separating shear layers is seen to cause a region of intensely rotating flow with high shear stress and high residence time of particles leading to high likelihood of platelet activation in that region.
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paper_title: Three-Dimensional Fluid-Structure Interaction Simulation of Bileaflet Mechanical Heart Valve Flow Dynamics

paper_content:

The wall shear stress induced by the leaflet motion during the valve-closing phase has been implicated with thrombus initiation with prosthetic valves. Detailed flow dynamic analysis in the vicinity of the leaflets and the housing during the valve-closure phase is of interest in understanding this relationship. A three-dimensional unsteady flow analysis past bileaflet valve prosthesis in the mitral position is presented incorporating a fluid-structure interaction algorithm for leaflet motion during the valve-closing phase. Arbitrary Lagrangian-Eulerian method is employed for incorporating the leaflet motion. The forces exerted by the fluid on the leaflets are computed and applied to the leaflet equation of motion to predict the leaflet position. Relatively large velocities are computed in the valve clearance region between the valve housing and the leaflet edge with the resulting relatively large wall shear stresses at the leaflet edge during the impact-rebound duration. Negative pressure transients are computed on the surface of the leaflets on the atrial side of the valve, with larger magnitudes at the leaflet edge during the closing and rebound as well. Vortical flow development is observed on the inflow (atrial) side during the valve impact-rebound phase in a location central to the leaflet and away from the clearance region where cavitation bubbles have been visualized in previously reported experimental studies.
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paper_title: Numerical simulation of 3D fluid-structure interaction flow using an immersed object method with overlapping grids

paper_content:

The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151-72] is extended for 3D unsteady flow simulation with fluid-structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier-Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17-20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on ''Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425-36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V"0 can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood-leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.
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paper_title: Vorticity dynamics of a bileaflet mechanical heart valve in an axisymmetric aorta

paper_content:

We present comprehensive particle image velocimetry measurements and direct numerical simulation (DNS) of physiological, pulsatile flow through a clinical quality bileaflet mechanical heart valve mounted in an idealized axisymmetric aorta geometry with a sudden expansion modeling the aortic sinus region. Instantaneous and ensemble-averaged velocity measurements as well as the associated statistics of leaflet kinematics are reported and analyzed in tandem to elucidate the structure of the velocity and vorticity fields of the ensuing flow-structure interaction. The measurements reveal that during the first half of the acceleration phase, the flow is laminar and repeatable from cycle to cycle. The valve housing shear layer rolls up into the sinus and begins to extract vorticity of opposite sign from the sinus wall. A start-up vortical structure is shed from the leaflets and is advected downstream as the leaflet shear layers become wavy and oscillatory. In the second half of flow acceleration the leaflet shea...
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paper_title: Numerical simulation of the dynamics of a bileaflet prosthetic heart valve using a fluid-structure interaction approach

paper_content:

Abstract The main purpose of this study is to reproduce in silico the dynamics of a bileaflet mechanical heart valve (MHV; St Jude Hemodynamic Plus, 27 mm characteristic size) by means of a fully implicit fluid–structure interaction (FSI) method, and experimentally validate the results using an ultrafast cinematographic technique. The computational model was constructed to realistically reproduce the boundary condition (72 beats per minute (bpm), cardiac output 4.5 l/min) and the geometry of the experimental setup, including the valve housing and the hinge configuration. The simulation was carried out coupling a commercial computational fluid dynamics (CFD) package based on finite-volume method with user-defined code for solving the structural domain, and exploiting the parallel performance of the whole numerical setup. Outputs are leaflets excursion from opening to closure and the fluid dynamics through the valve. Results put in evidence a favorable comparison between the computed and the experimental data: the model captures the main features of the leaflet motion during the systole. The use of parallel computing drastically limited the computational costs, showing a linear scaling on 16 processors (despite the massive use of user-defined subroutines to manage the FSI process). The favorable agreement obtained between in vitro and in silico results of the leaflet displacements confirms the consistency of the numerical method used, and candidates the application of FSI models to become a major tool to optimize the MHV design and eventually provides useful information to surgeons.
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paper_title: Collagen fibers reduce stresses and stabilize motion of aortic valve leaflets during systole.

paper_content:

The effect of collagen fibers on the mechanics and hemodynamics of a trileaflet aortic valve contained in a rigid aortic root is investigated in a numerical analysis of the systolic phase. Collagen fibers are known to reduce stresses in the leaflets during diastole, but their role during systole has not been investigated in detail yet. It is demonstrated that also during systole these fibers substantially reduce stresses in the leaflets and provide smoother opening and closing. Compared to isotropic leaflets, collagen reinforcement reduces the fluttering motion of the leaflets. Due to the exponential stress-strain behavior of collagen, the fibers have little influence on the initial phase of the valve opening, which occurs at low strains, and therefore have little impact on the transvalvular pressure drop.
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paper_title: Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies

paper_content:

The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.
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paper_title: Two-dimensional fluid-structure interaction simulation of bileaflet mechanical heart valve flow dynamics.

paper_content:

BACKGROUND AND AIM OF THE STUDY: Mechanical heart valve implantation requires long-term anticoagulation because of thromboembolic complications. Recent studies have indicated that the relatively high wall shear stresses and negative pressure transients developed during the valve closing phase may be dominant factors inducing thrombus initiation. The study aim was a two-dimensional (2D) functional simulation of flow past bileaflet heart valve prosthesis during the closing phase, incorporating the fluid-structure interaction analysis to induce motion of the leaflets. METHODS: The fluid-structure interaction model used was based on unsteady 2D Navier-Stokes equations with the arbitrary Lagrangian-Eulerian method for moving boundaries, coupled with the dynamic equation for leaflet motion. Parametric analysis of the effect of valve size, leaflet density, and the coefficient of resilience at the instant of impact of the leaflet with the housing were also performed. RESULTS: Comparing the predicted motion of the leaflet with previous experimental results validated the simulation. The results showed the presence of negative pressure transients near the inflow side of the leaflet at the instant of valve closure, and the negative pressure transients were augmented during the leaflet rebound process. Relatively high velocities and wall shear stresses, detrimental to the formed elements in blood were present in the clearance region between the leaflet and valve housing at the instant of valve closure. CONCLUSION: The simulation can be potentially applied to analyze the effects of valve geometry and dimensions, and the effect of leaflet material on the flow dynamics past the valve prosthesis during the opening and closing phases for design improvements in minimizing problems associated with thromboembolic complications.
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paper_title: High-Resolution Fluid–Structure Interaction Simulations of Flow Through a Bi-Leaflet Mechanical Heart Valve in an Anatomic Aorta

paper_content:

We have performed high-resolution fluid-structure interaction simulations of physiologic pulsatile flow through a bi-leaflet mechanical heart valve (BMHV) in an anatomically realistic aorta. The results are compared with numerical simulations of the flow through an identical BMHV implanted in a straight aorta. The comparisons show that although some of the salient features of the flow remain the same, the aorta geometry can have a major effect on both the flow patterns and the motion of the valve leaflets. For the studied configuration, for instance, the BMHV leaflets in the anatomic aorta open much faster and undergo a greater rebound during closing than the same valve in the straight axisymmetric aorta. Even though the characteristic triple-jet structure does emerge downstream of the leaflets for both cases, for the anatomic case the leaflet jets spread laterally and diffuse much faster than in the straight aorta due to the aortic curvature and complex shape of the anatomic sinus. Consequently the leaflet shear layers in the anatomic case remain laminar and organized for a larger portion of the accelerating phase as compared to the shear layers in the straight aorta, which begin to undergo laminar instabilities well before peak systole is reached. For both cases, however, the flow undergoes a very similar explosive transition to the small-scale, turbulent-like state just prior to reaching peak systole. The local maximum shear stress is used as a metric to characterize the mechanical environment experienced by blood cells. Pockets of high local maximum shear are found to be significantly more widespread in the anatomic aorta than in the straight aorta throughout the cardiac cycle. Pockets of high local maximum shear were located near the leaflets and in the aortic arc region. This work clearly demonstrates the importance of the aortic geometry on the flow phenomena in a BMHV and demonstrates the potential of our computational method to carry out image-based patient-specific simulations for clinically relevant studies of heart valve hemodynamics.
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paper_title: Flow-driven opening of a valvular leaflet

paper_content:

The understanding of valvular opening is a central issue in cardiac flows, whose analysis is often prohibited by the unavailability of ( in vivo ) data about tissue properties. Asymptotic or approximate representations of fluid–structure interaction are thus sought. The dynamics of an accelerated stream, in a two-dimensional channel initially closed by a rigid inertialess movable leaflet, is studied as a simple model problem aimed at demonstrating the main phenomena contributing to the fluid–structure interaction. The problem is solved by the coupled numerical solution of equations for the flow and solid. The results show that the leaflet initially opens in a no-shedding regime, driven by fluid mass conservation and a predictable dynamics. Then the leaflet motion jumps, after the saturation of a very rapid intermediate vortex-shedding phase, to the asymptotic slower regime with a stable self-similar wake structure.
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paper_title: A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries

paper_content:

A novel numerical method is developed that integrates boundary-conforming grids with a sharp interface, immersed boundary methodology. The method is intended for simulating internal flows containing complex, moving immersed boundaries such as those encountered in several cardiovascular applications. The background domain (e.g. the empty aorta) is discretized efficiently with a curvilinear boundary-fitted mesh while the complex moving immersed boundary (say a prosthetic heart valve) is treated with the sharp-interface, hybrid Cartesian/immersed-boundary approach of Gilmanov and Sotiropoulos [A. Gilmanov, F. Sotiropoulos, A hybrid cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics 207 (2005) 457-492.]. To facilitate the implementation of this novel modeling paradigm in complex flow simulations, an accurate and efficient numerical method is developed for solving the unsteady, incompressible Navier-Stokes equations in generalized curvilinear coordinates. The method employs a novel, fully-curvilinear staggered grid discretization approach, which does not require either the explicit evaluation of the Christoffel symbols or the discretization of all three momentum equations at cell interfaces as done in previous formulations. The equations are integrated in time using an efficient, second-order accurate fractional step methodology coupled with a Jacobian-free, Newton-Krylov solver for the momentum equations and a GMRES solver enhanced with multigrid as preconditioner for the Poisson equation. Several numerical experiments are carried out on fine computational meshes to demonstrate the accuracy and efficiency of the proposed method for standard benchmark problems as well as for unsteady, pulsatile flow through a curved, pipe bend. To demonstrate the ability of the method to simulate flows with complex, moving immersed boundaries we apply it to calculate pulsatile, physiological flow through a mechanical, bileaflet heart valve mounted in a model straight aorta with an anatomical-like triple sinus.
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paper_title: Flow in a Mechanical Bileaflet Heart Valve at Laminar and Near-Peak Systole Flow Rates: CFD Simulations and Experiments

paper_content:

Time-accurate, fully 3D numerical simulations and particle image velocity laboratory experiments are carried out for flow through a fully open bileaflet mechanical heart valve under steady (nonpulsatile) inflow conditions. Flows at two different Reynolds numbers, one in the laminar regime and the other turbulent (near-peak systole flow rate), are investigated. A direct numerical simulation is carried out for the laminar flow case while the turbulent flow is investigated with two different unsteady statistical turbulence modeling approaches, unsteady Reynolds-averaged Navier-Stokes (URANS) and detached-eddy simulation (DES) approach. For both the laminar and turbulent cases the computed mean velocity profiles are in good overall agreement with the measurements. For the turbulent simulations, however, the comparisons with the measurements demonstrate clearly the superiority of the DES approach and underscore its potential as a powerful modeling tool of cardiovascular flows at physiological conditions. The study reveals numerous previously unknown features of the flow.
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paper_title: An Analysis of Turbulent Shear Stresses in Leakage Flow Through a Bileaflet Mechanical Prostheses

paper_content:

In this work, estimates of turbulence were made from pulsatile flow laser Doppler velocimetry measurements using traditional phase averaging and averaging after the removal of cyclic variation. These estimates were compared with estimates obtained from steady leakage flow LDV measurements and an analytical method. The results of these studies indicate that leakage jets which are free and planar in shape may be more unstable than other leakage jets, and that cyclic variation does not cause a gross overestimation of the Reynolds stresses at large distances from the leakage jet orifice.
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paper_title: A three-dimensional computational analysis of fluid–structure interaction in the aortic valve

paper_content:

Abstract 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.
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paper_title: A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve.

paper_content:

The importance of the aortic root compliance in the aortic valve performance has most frequently been ignored in computational valve modeling, although it has a significant contribution to the functionality of the valve. Aortic root aneurysm or (calcific) stiffening severely affects the aortic valve behavior and, consequently, the cardiovascular regulation. The compromised mechanical and hemodynamical performance of the valve are difficult to study both 'in vivo' and 'in vitro'. Computational analysis of the valve enables a study on system responses that are difficult to obtain otherwise. In this paper a numerical model of a fiber-reinforced stentless aortic valve is presented. In the computational evaluation of its clinical functioning the interaction of the valve with the blood is essential. Hence, the blood-tissue interaction is incorporated in the model using a combined fictitious domain/arbitrary Lagrange-Euler formulation, which is integrated within the Galerkin finite element method. The model can serve as a diagnostic tool for clinical purposes and as a design tool for improving existing valve prostheses or developing new concepts. Structural mechanical and fluid dynamical aspects are analyzed during the systolic course of the cardiac cycle. Results show that aortic root compliance largely influences the valve opening and closing configurations. Stresses in the delicate parts of the leaflets are substantially reduced if fiber-reinforcement is applied and the aortic root is able to expand.
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paper_title: Vortex Shedding as a Mechanism for Free Emboli Formation in Mechanical Heart Valves

paper_content:

The high incidence of thromboembolic complications of mechanical heart valves (MHV) limits their success as permanent implants. The thrombogenicity of all MHV is primarily due to platelet activation by contact with foreign surfaces and by nonphysiological flow patterns. The latter include elevated flow stresses and regions of recirculation of blood that are induced by valve design characteristics. A numerical simulation of unsteady turbulent flow through a bileaflet MHV was conducted, using the Wilcox k-omega turbulence model for internal low-Reynolds-number flows, and compared to quantitative flow visualization performed in a pulse duplicator system using Digital Particle Image Velocimetry (DPIV). The wake of the valve leaflet during the deceleration phase revealed an intricate pattern of interacting shed vortices. Particle paths showed that platelets that were exposed to the highest flow stresses around the leaflets were entrapped within the shed vortices. Potentially activated, such platelets may tend to aggregate and form free emboli. Once formed, such free emboli would be convected downstream by the shed vortices, increasing the risk of systemic emboli.
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paper_title: Asymmetric opening of a simple bileaflet valve.

paper_content:

The opening motion of a bileaflet valve driven by a pulsed stream is studied in the limiting condition of rigid massless leaflets closing a two-dimensional channel. This simple arrangement allows us to write simple dynamical equations for the solid that are solved numerically with the Navier-Stokes equation for the fluid. The analysis is focused on the influence of asymmetry on the coupled fluid-valve dynamics when parameters are taken with references to cardiac valves. Results show that the wake generation from the leaflet's trailing edge can be partly inhibited; the primary vortex downstream does not occur in an intermediate range of asymmetry. The potential emergence of such a phenomenon in realistic cases would present implications in development of diagnostic schemes.
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paper_title: Numerical Analysis of Three-Dimensional Björk–Shiley Valvular Flow in an Aorta

paper_content:

Laminar vortical flow around a fully opened Bjork-Shiley valve in an aorta is obtained by solving the three-dimensional incompressible Navier-Stokes equations. Used is a non iterative implicit finite-element Navier-Stokes code developed by the authors, which makes use of the well-known finite difference algorithm PISO. The code utilizes segregated formulation and efficient iterative matrix solvers such as PCGS and ICCG. Computational results show that the three-dimensional vortical flow is recirculating with large shear in the sinus region of the valve chamber. Passing through the valve, the flow is split into major upper and lower jet flows. The spiral vortices generated by the disk are advected in the wake and attenuated rapidly downstream by diffusion. It is shown also that the shear stress becomes maximum near the leading edge of the disk valve.
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paper_title: Fluid mechanics of heart valves.

paper_content:

Valvular heart disease is a life-threatening disease that afflicts millions of people worldwide and leads to approximately 250,000 valve repairs and/or replacements each year. Malfunction of a native valve impairs its efficient fluid mechanic/hemodynamic performance. Artificial heart valves have been used since 1960 to replace diseased native valves and have saved millions of lives. Unfortunately, despite four decades of use, these devices are less than ideal and lead to many complications. Many of these complications/problems are directly related to the fluid mechanics associated with the various mechanical and bioprosthetic valve designs. This review focuses on the state-of-the-art experimental and computational fluid mechanics of native and prosthetic heart valves in current clinical use. The fluid dynamic performance characteristics of caged-ball, tilting-disc, bileaflet mechanical valves and porcine and pericardial stented and nonstented bioprostheic valves are reviewed. Other issues related to heart valve performance, such as biomaterials, solid mechanics, tissue mechanics, and durability, are not addressed in this review.
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paper_title: A two-dimensional fluid–structure interaction model of the aortic value

paper_content:

Failure of synthetic heart valves is usually caused by tearing and calci"cation of the lea#ets. Lea#et "ber-reinforcement increases the durability of these values by unloading the delicate parts of the lea#ets, maintaining their physiological functioning. The interaction of the valve with the surrounding #uid is essential when analyzing its functioning. However, the large di!erences in material properties of #uid and structure and the "nite motion of the lea#ets complicate blood}valve interaction modeling. This has, so far, obstructed numerical analyses of valves operating under physiological conditions. A two-dimensional #uid}structure interaction model is presented, which allows the Reynolds number to be within the physiological range, using a "ctitious domain method based on Lagrange multipliers to couple the two phases. The extension to the three-dimensional case is straightforward. The model has been validated experimentally using laser Doppler anemometry for measuring the #uid #ow and digitized high-speed video recordings to visualize the lea#et motion in corresponding geometries. Results show that both the #uid and lea#et behaviour are well predicted for di!erent lea#et thicknesses. ( 2000 Elsevier Science Ltd. All rights reserved.
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paper_title: Computational Approach for Probing the Flow Through Artificial Heart Devices

paper_content:

Computational fluid dynamics (CFD) has become an indispensable part of aerospace research and design. The solution procedure for incompressible Navier-Stokes equations can be used for biofluid mechanics research. The computational approach provides detailed knowledge of the flowfield complementary to that obtained by experimental measurements. This paper illustrates the extension of CFD techniques to artificial heart flow simulation. Unsteady incompressible Navier-Stokes equations written in three-dimensional generalized curvilinear coordinates are solved iteratively at each physical time step until the incompressibility condition is satisfied. The solution method is based on the pseudocompressibility approach. It uses an implicit upwind-differencing scheme together with the Gauss-Seidel line-relaxation method. The efficiency and robustness of the time-accurate formulation of the numerical algorithm are tested by computing the flow through model geometries. A channel flow with a moving indentation is computed and validated by experimental measurements and other numerical solutions. In order to handle the geometric complexity and the moving boundary problems, a zonal method and an overlapped grid embedding scheme are employed, respectively. Steady-state solutions for the flow through a tilting-disk heart valve are compared with experimental measurements. Good agreement is obtained. Aided by experimental data, the flow through an entire Penn State artificial heart model is computed.
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paper_title: Characterization of Hemodynamic Forces Induced by Mechanical Heart Valves: Reynolds vs. Viscous Stresses

paper_content:

Bileaflet mechanical heart valves (BMHV) are widely used to replace diseased heart valves. Implantation of BMHV, however, has been linked with major complications, which are generally considered to be caused by mechanically induced damage of blood cells resulting from the non-physiological hemodynamics environment induced by BMHV, including regions of recirculating flow and elevated Reynolds (turbulence) shear stress levels. In this article, we analyze the results of 2D high-resolution velocity measurements and full 3D numerical simulation for pulsatile flow through a BMHV mounted in a model axisymmetric aorta to investigate the mechanical environment experienced by blood elements under physiologic conditions. We show that the so-called Reynolds shear stresses neither directly contribute to the mechanical load on blood cells nor is a proper measurement of the mechanical load experienced by blood cells. We also show that the overall levels of the viscous stresses, which comprise the actual flow environment experienced by cells, are apparently too low to induce damage to red blood cells, but could potentially damage platelets. The maximum instantaneous viscous shear stress observed throughout a cardiac cycle is <15 N/m(2). Our analysis is restricted to the flow downstream of the valve leaflets and thus does not address other areas within the BMHV where potentially hemodynamically hazardous levels of viscous stresses could still occur (such as in the hinge gaps and leakage jets).
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paper_title: Numerical Simulations of Fluid-Structure Interaction Problems in Biological Flows

paper_content:

University of Minnesota Ph.D. dissertation. June, 2008. Major: Mechanical Engineering. Advisor: Fotis Sotiropoulos. 1 computer file (PDF); xvi, 273 pages.
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paper_title: High-Resolution Fluid–Structure Interaction Simulations of Flow Through a Bi-Leaflet Mechanical Heart Valve in an Anatomic Aorta

paper_content:

We have performed high-resolution fluid-structure interaction simulations of physiologic pulsatile flow through a bi-leaflet mechanical heart valve (BMHV) in an anatomically realistic aorta. The results are compared with numerical simulations of the flow through an identical BMHV implanted in a straight aorta. The comparisons show that although some of the salient features of the flow remain the same, the aorta geometry can have a major effect on both the flow patterns and the motion of the valve leaflets. For the studied configuration, for instance, the BMHV leaflets in the anatomic aorta open much faster and undergo a greater rebound during closing than the same valve in the straight axisymmetric aorta. Even though the characteristic triple-jet structure does emerge downstream of the leaflets for both cases, for the anatomic case the leaflet jets spread laterally and diffuse much faster than in the straight aorta due to the aortic curvature and complex shape of the anatomic sinus. Consequently the leaflet shear layers in the anatomic case remain laminar and organized for a larger portion of the accelerating phase as compared to the shear layers in the straight aorta, which begin to undergo laminar instabilities well before peak systole is reached. For both cases, however, the flow undergoes a very similar explosive transition to the small-scale, turbulent-like state just prior to reaching peak systole. The local maximum shear stress is used as a metric to characterize the mechanical environment experienced by blood cells. Pockets of high local maximum shear are found to be significantly more widespread in the anatomic aorta than in the straight aorta throughout the cardiac cycle. Pockets of high local maximum shear were located near the leaflets and in the aortic arc region. This work clearly demonstrates the importance of the aortic geometry on the flow phenomena in a BMHV and demonstrates the potential of our computational method to carry out image-based patient-specific simulations for clinically relevant studies of heart valve hemodynamics.
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Title: A Review of State-of-the-Art Numerical Methods for Simulating Flow through Mechanical Heart Valves

Section 1: Introduction
Description 1: Introduce the need for heart valve surgeries, complications with bileaflet mechanical heart valves (BMHV), and the significance of numerical simulations in understanding BMHV hemodynamics.

Section 2: Governing Equations and Numerical Methods
Description 2: Describe the fluid dynamics equations governing BMHV flows and review numerical methods for handling fluid-structure interaction problems.

Section 3: Leaflet Equations
Description 3: Discuss the equations governing the motion of BMHV leaflets and their numerical implementation.

Section 4: Boundary Conditions and Coupling for FSI Problems
Description 4: Detail the boundary conditions and methodologies for coupling fluid and structure dynamics in FSI problems involving BMHV.

Section 5: Stability of the FSI Coupling for MHV Simulations
Description 5: Analyze the stability and computational considerations for different FSI coupling methods used in BMHV simulations.

Section 6: Recent Simulations and Insights into the Flow Physics
Description 6: Summarize recent simulation studies, their findings, and insights gained into the flow physics of BMHV.

Section 7: Future Outlook
Description 7: Provide a future perspective on the challenges and potential developments in numerical simulation methods for BMHV.