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Part IV. Computational Modeling and Experimental Studies

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Part IV. Computational Modeling and Experimental Studies Annals of Biomedical Engineering (Ó 2015) DOI: 10.1007/s10439-015-1394-4 Emerging Trends in Heart Valve Engineering: Part IV. Computational Modeling and Experimental Studies 1,2 1,2 1 3 ARASH KHERADVAR, ELLIOTT M. GROVES, AHMAD FALAHATPISHEH, MOHAMMAD K. MOFRAD, S. 1 4 5 6,7 8 HAMED ALAVI, ROBERT TRANQUILLO, LAKSHMI P. DASI, CRAIG A. SIMMONS, K. JANE GRANDE-ALLEN, 9 10 11 12 13,14 CRAIG J. GOERGEN, FRANK BAAIJENS, STEPHEN H. LITTLE, SUNCICA CANIC, and BOYCE GRIFFITH 1Department of Biomedical Engineering, The Edwards Lifesciences Center for Advanced Cardiovascular Technology, University of California, Irvine, 2410 Engineering Hall, Irvine, CA 92697-2730, USA; 2Department of Medicine, Division of Cardiology, University of California, Irvine School of Medicine, Irvine, CA, USA; 3Department of Bioengineering and Mechanical Engineering, University of California, Berkeley, CA, USA; 4Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN, USA; 5Department of Mechanical Engineering, School of Biomedical Engineering, Colorado State University, Fort Collins, CO, USA; 6Department of Mechanical & Industrial Engineering, University of Toronto, Toronto, ON, Canada; 7Institute of Biomaterials & Biomedical Engineering, University of Toronto, Toronto, ON, Canada; 8Department of Bioengineering, Rice University, Houston, TX, USA; 9Weldon School of Biomedical Engineering, Purdue University, West Lafayette, IN, USA; 10Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands; 11Houston Methodist DeBakey Heart & Vascular Center, Houston, TX, USA; 12Department of Mathematics, University of Houston, Houston, TX, USA; 13Department of Mathematics, Center for Interdisciplinary Applied Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA; and 14McAllister Heart Institute, University of North Carolina at Chapel Hill School of Medicine, Chapel Hill, NC, USA (Received 26 April 2015; accepted 14 July 2015) Associate Editor K. A. Athanasiou oversaw the review of this article. Abstract—In this final portion of an extensive review of heart logical conditions. These studies have aided the assess- valve engineering, we focus on the computational methods ment of heart valves, interventional and surgical and experimental studies related to heart valves. The procedures, heart valve design, and general understanding discussion begins with a thorough review of computational modeling and the governing equations of fluid and structural of healthy and abnormal cardiac valve function. interaction. We then move onto multiscale and disease To be viable, computational models must be evidently specific modeling. Finally, advanced methods related to based on an accurate representation of the mechanical in vitro testing of the heart valves are reviewed. This section behavior of the valves’ material and flow passing through of the review series is intended to illustrate application of the valves. Native heart valve leaflet tissue has many computational methods and experimental studies and their interrelation for studying heart valves. important biomechanical attributes contributing to its elastic response to the loads along with a viscous reaction Keywords—Computational modeling, Heart valves, Particle that damps out flow-related vibrations, making its overall image velocimetry, Biaxial testing, Multiscale modeling, response as viscoelastic. Nonetheless, water comprises a Numerical simulation. major fraction of the collagenous tissue by weight and is tightly bound to the fibrous network; hence the heart valve tissue behaves nearly or completely incompressible. INTRODUCTION Moreover, the aligned fibers of the leaflet tissue make the stress–strain response highly anisotropic.14 The waviness Over the past two decades, computational modeling of the fibers also significantly affects the stress–strain and experimental studies have been used as powerful tools response.156 A constitutive model for heart valve tissue to understand and predict the behavior and mechanics of must incorporate important features of the heart valves. native and prosthetic heart valves in normal and patho- Namely it should describe a pseudoelastic, incompress- ible, anisotropic, nonlinear material. Extensive work has been devoted to developing constitutive equations to Address correspondence to Arash Kheradvar, Department of describe the mechanics of the heart valve leaflet tis- Biomedical Engineering, The Edwards Lifesciences Center for 119,144,156,157,159 Advanced Cardiovascular Technology, University of California, sues. Different derivations for the stress– Irvine, 2410 Engineering Hall, Irvine, CA 92697-2730, USA. Elec- strain behavior of the heart valves have been attempted, tronic mail: [email protected] which treat the structure of the native tissue using Ó 2015 Biomedical Engineering Society KHERADVAR et al. different approaches ranging from phenomological valves. Let X R3 denote the computational domain, models that include no information about the structure to with XfðtÞX indicating the physical region occupied unit-cell models that are completely derived from net- by the fluid at time t, XsðtÞX indicating the physical work structure as extensively described by Weinberg and region occupied by the solid, and X ¼ XfðtÞ[XsðtÞ. Kaazempur-Mofrad.156 The fluid–structure interface is CfsðÞ¼t CfðÞ\t CsðÞt , In this final section of a four-part review series, we in which CsðÞt is the boundary of XsðtÞ and CfðÞt is the focus on the various aspects of computational modeling boundary of XfðtÞ. Finally, let ufðx; tÞ be the velocity of and experimental studies related to heart valves. Rang- the fluid at position x 2 XfðtÞ at time t, and let usðx; tÞ ing from pure computational models to in vitro studies, be the velocity of the structure at position x 2 XsðtÞ at modeling is an important component of the armamen- time t. To simplify notation, we shall generally omit tarium of those who study and design heart valves. the superscripts ‘‘f’’ and ‘‘s’’ on the velocity field. In- stead, we identify uðx; tÞ as the velocity of whichever material happens to be located at position x at time t. COMPUTATIONAL MODELING The equations of momentum conservation for the fluid and solid are: Native and prosthetic valve leaflets are thin, flexible Du structures that are passively carried by the blood flow qf ðÞ¼rÁx; t rfðÞx; t in XfðtÞ; ð1Þ while also applying forces to the blood that affect the fluid Dt motion and, thereby, the motion of the leaflets them- Du selves. Static or quasi-static valve models can be used to qs ðÞ¼rÁx; t rsðÞx; t in XsðtÞ; ð2Þ study the mechanics of native and prosthetic valve leaf- Dt lets, and standard computational fluid dynamics (CFD) in which qf and qs are, respectively, the mass densities approaches can be applied to study the flow patterns and of the fluid and solid, rfðÞx; t and rsðÞx; t are the fluid stresses resulting from imposed or measured leaflet (Cauchy) stress tensors of the fluid and the solid, and kinematics.132 Alternatively, predicting the kinematics of DðÁÞ @ðÁÞ Dt ¼ @t þ uxð; t Þ Á rðÁÞ is the material (convective) the valve leaflets, or the dynamic interaction between the time derivative. The fluid and structure may both be valve leaflets and flow, requires a fluid–structure inter- modeled as incompressible, so that action (FSI) approach. Such dynamic models can be used to simulate the conditions that ultimately lead to the rÁuxðÞ¼; t 0: ð3Þ deterioration of the bioprosthetic valve leaflets, and val- Typical coupling conditions for the fluid and idated FSI models could thereby be used to help to structure regions are that of the velocity and traction, develop improved leaflet designs with durable lifetimes and are continuous along CfsðÞt , i.e., that are longer than present bioprosthetic valves. Several computational challenges must be addressed ufðÞ¼x; t usðx; tÞ on CfsðÞt ; ð4Þ when developing FSI models of heart valves. For instance, the valve leaflets experience large displacements rfðÞÁx; t n ¼ rsðÞÁx; t n on CfsðÞt ; ð5Þ and strains during the cardiac cycle. Further, FSI models fs that seek to model realistic loading of the closed valve in which n is a unit normal vector along C ðÞt . must account for contact between the leaflets. Native and Boundary and initial conditions, which we omit here, bioprosthetic valve leaflets generally exhibit a complex, are required to complete the specification of the nonlinear stress response that requires mathematical equations of motion. formulations appropriate for finite-strain structural models. Finally, the high Reynolds number flows char- Modeling the Fluid acteristic of the heart valves, especially the aortic and For the flows in the vicinity of the heart valves, pulmonary valves, generally necessitates the use of very blood is typically described as an incompressible high spatial resolution to resolve the flow features. Newtonian fluid, so that the fluid stress tensor is Despite substantial work over several decades, the devel- ÀÁ opment of computational approaches that address all of rfðÞ¼Àx; t pðÞx; t I þ l ruðÞþrx; t uTðÞx; t ; ð6Þ these challenges remains an area of active investigation. in which l is the (constant) dynamic viscosity of the fluid and pðÞx; t is the pressure. In this case, the Governing Equations dynamics of the fluid in XfðtÞ are modeled by the incompressible Navier–Stokes equations, The same fundamental equations of fluid–solid interaction can be used to describe the dynamics of Du qf ðÞ¼Àrx; t pðÞþx; t lr2uxðÞ; t ; ð7Þ native, bioprosthetic, and even mechanical heart
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