A Design-Based Model of the Aortic Valve for Fluid-Structure Interaction
Alexander D. Kaiser, Rohan Shad, William Hiesinger, Alison L. Marsden
AA Design-Based Model of the Aortic Valvefor Fluid-Structure Interaction
Alexander D. Kaiser , , , Rohan Shad , , William Hiesinger , , Alison L. Marsden , , , Institute for Computational and Mathematical Engineering, Stanford University; Department of Pediatrics (Cardiology), Stanford University; Stanford Cardiovascular Institute; Department of Cardiothoracic Surgery, Stanford University; Department of Bioengineering, Stanford University
October 7, 2020
Abstract
This paper presents a new method for modeling the mechanics of the aortic valve, and simulatesits interaction with blood. As much as possible, the model construction is based on first principles,but such that the model is consistent with experimental observations. We require that tension in theleaflets must support a pressure, then derive a system of partial differential equations governing itsmechanical equilibrium. The solution to these differential equations is referred to as the predicted loadedconfiguration; it includes the loaded leaflet geometry, fiber orientations and tensions needed to supportthe prescribed load. From this configuration, we derive a reference configuration and constitutive law.In fluid-structure interaction simulations with the immersed boundary method, the model seals reliablyunder physiological pressures, and opens freely over multiple cardiac cycles. Further, model closureis robust to extreme hypo- and hypertensive pressures. Then, exploiting the unique features of thismodel construction, we conduct experiments on reference configurations, constitutive laws, and grossmorphology. These experiments suggest the following conclusions, which are directly applicable to thedesign of prosthetic aortic valves. (i) The loaded geometry, tensions and tangent moduli primarilydetermine model function. (ii) Alterations to the reference configuration have little effect if the predictedloaded configuration is identical. (iii) The leaflets must have sufficiently nonlinear material response tofunction over a variety of pressures. (iv) Valve performance is highly sensitive to free edge length andleaflet height. For future use, our aortic valve modeling framework offers flexibility in patient-specificmodels of cardiac flow.
The aortic valve is one of four valves in the human heart. It lies between the left ventricle, the main pumpingchamber of the left heart, and the aorta, the central artery through which oxygenated blood leaves the heart.The valve serves to prevent backflow of blood during diastole, the filling phase of the left ventricle, in whichblood enters the chamber through the mitral valve, and opens in systole as the heart beats. In a typicalanatomy, the valve is composed of three thin, flexible leaflets. A highly oriented system of collagen fibersprovides the primary mechanical stiffness of the leaflets, and leads to highly anisotropic material properties.The leaflets are anchored to a non-planar ring called the aortic annulus. During forward flow, the leafletsbend to create an approximately circular orifice at the inlet of the aorta. During closure, the free edges ofthe leaflets coapt, and the orifice is roughly trisected by the lines of contact.In this paper, we present a new method for modeling the mechanics of the aortic valve and simulate itsinteraction with blood. We use nearly first-principles techniques, yet tune the results to empirical knowledgefrom experiments on the gross morphology, kinematics and material properties of valve leaflets. This frame-work allows us the flexibility to adapt the model to a range of patient specific anatomies, without relianceon data that is not typically available in a clinical setting. We assume that the valve leaflets must support apressure, and then derive the valve geometry and material properties from the resulting differential equations.The solution of these differential equations is referred to as the predicted loaded configuration ; this includes a r X i v : . [ q - b i o . T O ] O c t the loaded geometry, its fiber orientations and the tensions required to support such a load. The formulationin this work allows the leaflets to be tuned for a given gross morphology, to fit on a known annular geometry,and directly provides information about the material properties in both the fiber and cross-fiber directions.Since the geometry and material properties are derived and tuned, rather than measured and assigned, werefer to this as a design-based approach to elasticity.A central challenge in fluid-structure interaction (FSI) studies of heart valves is to simulate the multiplecardiac cycles under physiological pressures, and achieve behavior that qualitatively matches that of a realvalve. The model should open freely, allowing a jet of forward flow during the ejection phase, then closeunder back pressure, sealing without leak or regurgitation, and finally open again and repeat the cycle. Tosimulate closure is especially challenging, as leak must be prevented only by the balance of elastic forces inthe leaflets and fluid forces [27].The first goal of this paper is to demonstrate the that our methods achieve the above central challenge.We perform FSI simulations with the immersed boundary (IB) method and show that the model seals underphysiological pressures and opens freely over multiple beats. Further, its closure is robust to pressuresmuch lower and higher than physiological pressures. The second goal is to study the effect of differingreference configurations, constitutive laws and gross morphology for aortic valve tissue. Using the design-based model generation scheme, the reference configuration and constitutive law can be modified whilemaintaining an identical predicted loaded geometry and tension. We exploit this to alter the rest lengthsassociated with given loaded lengths, and study a number of constitutive laws – each of which are equivalentin the predicted loaded configuration – to determine the pressure ranges for which each are effective. Thesenumerical experiments suggest functional explanations for observed native valve material properties andgeometry. Further, observed ranges for good model valve function in turn suggest optimal ranges for aorticvalve prosthetics.Our techniques are directly adapted from prior modeling methods for the mitral valve [24, 25], for whichthere are similar goals but a very different anatomy, as well as prior models of the aortic valve [39]. In theiroriginal study, Peskin and McQueen used the simplification that single fiber family bears all of the load;they treated the cross-fiber direction as not bearing any load and did not include its potential deformation.The entire loaded geometry of the leaflets was determined by the free edge, so the leaflet geometry could notbe forced to conform to a general annular geometry and the gross morphology of the leaflets could not bereadily tuned. Subsequent work using this predicted loaded configuration was effective in FSI simulationsat physiological pressures, but required bending rigidity that to coapt properly [17, 15]. More recently, invitro experiments revealed that the radial, or cross-fiber direction undergoes large strains [57], and materialtests suggests that the radial direction exerts significant stress under such strains [36]. Thus, we seek aformulation with more flexibility in the emergent geometry and that includes radial tension and strain.In the current approach, the fiber structure emerges from the solution of a differential equation, andtherefore can be considered a new “rule based” method for assigning fiber structure. In one such technique,a modified Laplace equation was solved to interpolate the fiber orientation from known points at the boundaryof the leaflets [21]. The modified Laplace equation was selected because its solutions are smooth and readilycomputable with standard numerical techniques. In contrast, the differential equation that we solve ismotivated directly by the valve’s function, and its solution provides simultaneous information about thevalves geometry, fiber orientation, and material properties.There are a variety of other methods for FSI and aortic valve modeling. Griffith and collaborators use theIB method with a finite element formulation on the structure and achieve flow rates comparable to clinicalmeasurements [21]. Their geometry is based on a simple analytic shape from [13], which is manually modifiedto match scan data. Hsu et al. use the immersogeometric framework, which uses non-conforming meshes as inthe IB method, and manually designed geometry with computer aided design (CAD) software [22]. Fictitiousdomain methods are also non-conforming, but require contact forces [3, 50]. Mao et al. use smooth particlehydrodynamics and geometry measured from a CT scan [33]. Marom et al. use a two way coupling approachthat requires contact forces and use a geometry based on a simple analytic shape, tuned to gross morphologyfrom echocardiography [34]. Arbitrary Lagrangian-Eulerian (ALE) schemes, which maintain non-overlappingdomains and a sharp interface between them, appear to have limited success in simulating heart valves. Onestudy with ALE required the addition of a fictitious surface to close the valve orifice once near-contact wasdetected, thus largely prescribing closure rather than having it emerge from the system’s dynamics [52]. Bavoet al. compared ALE methods with IB methods, and did not succeed in simulating an entire cardiac cyclewith ALE because of remeshing problems, and they required complex and nonphysical handling of contact[5]. While some studies focus solely on solid mechanics, studies comparing solids-only simulations withFSI simulations suggest that simulating FSI is essential to study the full dynamics of the valve, revealingqualitative and quantitative differences [30, 34, 5, 22]. Questions related to heart valve prosthetics thatrequire FSI to simulate are reviewed in [59]. See Le et al. [31] for additional review and discussion. Ofresults reported in these studies, methods using non-conforming, IB approaches appear to work best, withmultiple studies reporting realistic behavior though at least one cardiac cycle. Most model geometries areconstructed from a simple analytic shape or a scan, however, analytic shapes are not anatomical, and scandata at sufficient resolution may not be available to construct a good patient-specific valve geometry. Incontrast, our method will allow us to customize model valves to fit in patient-specific model geometries infuture work without the need for individual measurements on valve anatomy or material properties that aretypically not available clinically.The remainder of this paper is organized as follows. In Section 2 we review literature on aortic valveanatomy and physiology. In Section 3, we derive a system of partial differential equations, the solution ofwhich gives the predicted loaded configuration. Using this configuration, we derive a constitutive law forthe valve. Then, we outline methods for FSI including the model test chamber and boundary conditions.In Section 4, we present the model geometry, the emergent heterogeneous constitutive law, and results ofFSI simulations. We then change the reference configuration, the constitutive law, and the model geometryand report results on each. We discuss these results in Section 5, study limitations in Section 6, and finallyconclusions in Section 7.Source code for the project is freely available at github.com/alexkaiser/heart_valves . We wish to construct a model that is consistent with real valves in four categories: the gross morphology ofthe valve, its fiber structure, its material properties, and its loaded strain. Figure 1 reviews the anatomy ofthe aortic valve. The expected properties that we use as targets are:
Gross morphology:
For a given radius r , the loaded leaflet height is approximately 1.4 r , the loaded freeedge length of each leaflet is 2.48 r , and the height of the annulus is 1 .
4r [54]. The leaflet thickness isapproximately 0.044 cm [43].
Fiber structure:
Fibers run circumferentially, from commissure to commissure [47].
Material properties:
Material response is highly nonlinear, with stiffening at lower strains in the cir-cumferential direction than in the radial direction [36]. Tangent stiffness in the circumferential andradial directions are of order 10 and 10 dynes/cm respectively, at sufficiently high strains such thatcollagen fibers are fully recruited [40]. Strain:
Fully-loaded strains in the circumferential and radial directions are approximately 0.15 and 0.54,respectively [57].A short review of experimental literature associated with these four categories follows.
Gross morphology:
Swanson and Clark provide a representative description of the gross morphologyof human aortic valve leaflets in a loaded configuration [54]. They claim that the radius at which the leafletattaches is constant though the height of the annulus. For a radius r , the height to the commissure is 1.42 r ,the free edge length of each leaflet is 2.48 r , the leaflet height in the radial direction is 1.4 r . Sch¨afers et al.report similar mean leaflet heights of 2.0 cm with a mean radius of 1.38 cm [49]. A study on human cadaverhearts reported a mean leaflet thickness of 0.044 cm [43]. Fiber structure:
Aortic leaflet material properties are highly nonlinear and anisotropic, attributed tooriented, wavy collagen fibers that are recruited, or straightened, under load and then provide additionalstiffness [48]. These fibers run from commissure to commissure, or circumferentially, and fiber bundles canbe seen with even moderate magnification (Figure 1). These bundles are hierarchical, meaning that fiberbundles visible reflect alignment of smaller scale collagen fibers, fibrils and molecules [16, 51, 1]. The radialdirection is referred to as the cross-fiber direction.
The mechanical properties of porcine aortic valve tissues 329 - - t he or y - e xpe r i me nt I 0 20 Lo
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09 -- - t he or y - e e xpe r r me nt < 08- , E f t J I (0 - Hsl
Fig. 3. (a) H(r) as defined by (3) and calculated from measure- ment data. (b) Experimental and theoretical relaxation curves over 100s. (c)Zoom plot of the first 1.4s of Fig. 3(b). Using the approximation (4) for G(m), the value of K resulted from (2) and (6) -G(m) K= _ . 191 ctc;(co) . , MATERIALS AND METHODS Physiological t&es of strain In order to obtain relevant information from the experiments, the load and deformation values should be in the physiological range. As the specimen length was the controlled variable in our experiments, we shall mainly discuss maximal strain values as de- termined under physiological circumstances.
In in civo experiments on dogs, Brewer et al. (1977) studied the length variations in the leaflet lunula along its free edge during the cardiac cycle. Using the length in diastole at 10.7 kPa pressure difference across the valve as a reference, they observed a strain of about 0.02 from the beginning to the end of diastole. Between the end of diastole and peak systole a shortening of the same order of magnitude was found. Missirlis and Chong (1978) investigated local strains in porcine valve leaflets when pressurizing (16 kPa) an entire aortic root in cirro. In the load bearing leaflet portion (i.e. the portion of the leaflet surface that is not making contact with adjacent leaflets when the valve is closed) they observed strains ranging from 0.05 to 0.1 in the circumferential direction and from 0.1 to more than 1.0 in the radial direction (for the definitions of the directions see Fig. 4). From in ciuo experiments on dogs, Thubrikar et al. (1980) concluded that the leaflet iength of the Ioad bearing portion in the circumferep- tial direction decreased by about 10 ‘X from diastole to systole. Pressure differences across the valve ranged from about I3 kPa to 24 kPa. In presenting typical stress-strain curves for human aortic and sinus tissues, Missirlis (1973) considered the
AORTA LEi T VENTRI CLE
Fig. 4. Exposure of the aortic valve in the closed configur- ation after dissection of one leaflet and the corresponding sinus wall. The position and orientation ofthe specimens used for the experiments are denoted by the dashed lines. (1.2) circumferential and radial leaflet strips, respectively; (3,4) axial and circumferential sinus strips, respectively; (5,6) axial and circumferential aortic strips, respectively. h l cfr F R A C T A L F I BE R A RCH I T E C T UR E O F A O R T I C VA L VE L EA F L E T S H F i g . . C o ll agen f i be r s t ha t s uppo r t ao r t i c l ea f l e t s . Lea f - l e t ha s been m oun t ed f l a t, w h i c h i ne v i t ab l y d i s t o r t s i t s geo m e t r y . H o l e s a r e a r t i f a c t s o f m oun t i ng p r o c e ss . F r ee edge s o f l ea f l e t a r e a t l e ft, and c o mm i ss u r e s a r e a t t op and bo tt o m . N o t e c ab l e li k e qua li t y o f c o ll agen f i be r s and t ha t m o s t c ab l e s l ead t o c o mm i ss u r a l po i n t s . [ R ep r i n t ed b y pe r m i ss i on f r o m A . A . H . J . S au r en ( R e f. , p . ) .] o f c a l c u l u s , t h i s e x p r e ss i on f o r t he ne t f i be r f o r c e a c t i ng on a pa t c h c an be r e w r i tt en a s a doub l e i n t eg r a l T he p r e ss u r e f o r c e a l s o t a k e s t he f o r m o f a doub l e i n t eg r a l , and t he s e doub l e i n t eg r a l s m a y be c o m b i ned t o ob t a i n t he t o t a l f o r c e on t he pa t c h , w h i c h m u s t be z e r o i f t he l ea f l e t i s i n equ ili b r i u m B e c au s e t he li m i t s o f i n t eg r a t i on a r e a r b i t r a r y , t he i n t eg r and m u s t be z e r o . T h i s y i e l d s t he equa t i on o f equ ili b r i u m T h i s equ ili b r i u m equa t i on ha s s e v e r a l i mm ed i a t e c on s e - quen c e s , w h i c h a r e de r i v ed i n APPE ND I X A and s u mm a - r i z ed a s f o ll o w s . I > T = T ( v ) , i ndependen t o f U . T he t en s i on i s c on s t an t on ea c h f i be r . ) T he f i be r s a r e geode s i cs on t he v a l v e l ea f l e t s u r f a c e . ) T he c u r v e s u = c on s t an t and t he c u r v e s u = c on s t an t f o r m an o r t hogo - na l ne t. T o r edu c e E q . t o a pu r e l y geo m e t r i c c ond i t i on , w e m a k e t he c hange o f v a r i ab l e s g i v en b y d V = [ T ( u ) l p ,, l du . W he r ea s u w a s an a r b i t r a r y l abe l a ss i gned t o ea c h f i be r , V ha s t he ph ys i c a l m ean i ng t ha t equa l i n t e r v a l s o f V c a rr y equa l a m oun t s o f f o r c e . V ha s un i t s o f a r ea , and pod V = T ( u ) du i s p r e c i s e l y t he f o r c e t r an s m i tt ed b y t he r a d i a l circumferential Figure 1: Anatomy of the aortic valve. Left: Closed aortic valve from above. Center: Schematic in adissected view from the side, illustration reprinted from [48] with permission from Elsevier. Labeled arethe free edge (f), the leaflet height (l), the annular radius (r), the annular height (h) and a commissurepoint where leaflets meet (c). Right: Excised aortic valve leaflet stained for collagen, reprinted from [46].Tissue-scale collagen fiber bundles are visible with a heavily circumferential orientation.
Material properties:
Material testing of aortic leaflet tissue reveals that the circumferential (fiber)direction is stiffer, and has a more nonlinear response at smaller strains compared to the radial direction.May-Newman et al. measured nonlinear material properties in both the fiber and radial directions [36].The curves appear approximately exponential with a monotonically increasing slope; there is no clear linearregion after a certain strain. Pham et al. report a tangent modulus of 9 . · dynes/cm circumferentially,and 2 . · dynes/cm radially for an approximately 4:1 ratio of circumferential to radial tangent modulus[40]. Clark found an affine region of material response begins at strains of 0.13 circumferentially and 0.24strain radially, and the fully recruited tangent modulus to be 5 . · dynes/cm circumferentially, and1 . · dynes/cm radially, for an approximately 4:1 ratio of circumferential to radial stiffness [10]. Saurenet al. reported an approximately 20:1 ratio of circumferential to radial stiffness at maximum [48]. Literatureon the shear response appears to be limited, but one modeling study found that shear response had littleinfluence on the closing kinematics of the valve [20]. Kinematics and expected strains:
To consistently define strain, let E denotes engineering strain E = ( L − R ) /R where L denotes the current length and R denotes reference length. Yap et al. testedporcine leaflets in vitro, and found that the principle axes of strain aligned with the radial and circumferentialdirections of the leaflets, and found a near-constant circumferential strain of 0.15 and a radial strain of 0.54during diastole [57]. Additionally, there may be strain at all times in the cardiac cycle or even at restdue to prestrain, which may substantially change the values of strains that are reported in studies [41].One study found prestrain though the cardiac cycle relative to an excised state fixed in glutaraldehyde [2].Gluteraldehyde fixation, however, can markedly change the stress-strain response of the leaflet tissue [7, 6].However, we show in Section 4.4 that the model functions well under a variety of strains assigned to thepredicted loaded configuration. The following assumptions form an idealized summary of the anatomy and function of the loaded aorticvalve, and we will use them directly to construct the model valve.1. The valve is composed of three leaflets, each of which is anchored to the aortic annulus.2. Fibers run from commissure to commissure on each leaflet. Curves in the cross-fiber direction run fromthe annulus to the free edge of each leaflet.3. Each of the leaflets can exert tension in the circumferential, or fiber, direction and the radial, orcross-fiber, direction. Shear tension is assumed to be identically zero.4. Tension in the leaflets supports a uniform pressure load, creating a static mechanical equilibrium inwhich all forces balance.We expect that analysis of the closed configuration in equilibrium to be a good predictor of the loadedconfiguration when interacting with fluid. This is because the inertio-elastic timescale, an estimate of theduration of time it takes the leaflets to deform when pressurized, is much shorter than the expected durationof valve closure. Previously [25], we estimated this timescale to be r (cid:112) ρ/η where r = 1 .
25 cm denotes theradius of the annulus, ρ = 1 g/cm is the density of the leaflets and η = 10 dynes/cm , which is the lowestorder of magnitude for the fully-recruited tangent modulus that we found in the literature. This gives aninertio-elastic timescale of 4 . · − s. Since the valve is closed for approximately 0.5 s per cardiac cycle,analysis based on the closed configuration should be a good predictor of the dynamics in general.We represent the leaflet surfaces as an unknown parametric surface in three dimensional space, X ( u, v ) : Ω ⊂ R → R . (1)The surface X has units of length, or cm, and the parameters u, v are taken to be dimensionless. Curves onwhich u varies and v is constant run circumferentially, conform to and thus represent the fibers. Curves onwhich v varies and u is constant run radially, in the cross-fiber directions. The unit tangents to these twofamilies of curves determine the local directions at which the leaflet exerts force in the fiber and cross-fiberdirections, and are defined as X u | X u | and X v | X v | , (2)respectively. These directions are not required to be orthogonal.Now, consider the mechanical equilibrium on an arbitrary patch of leaflet specified by [ u , v ] × [ u , v ].The pressure p acts normal to the leaflet across the entire patch. Let S denote circumferential tension and T denote radial tension. Circumferential tension acts on the curves specified by v = v and v = v , and radialtension acts on the curves specified by u = u and u = u . Figure 2 shows a free body diagram of theseforces. Summing these forces gives the integral form of the mechanical equilibrium, or0 = (cid:90) v v (cid:90) u u p ( X u ( u, v ) × X v ( u, v )) dudv (3)+ (cid:90) v v (cid:18) S ( u , v ) X u ( u , v ) | X u ( u , v ) | − S ( u , v ) X u ( u , v ) | X u ( u , v ) | (cid:19) dv + (cid:90) u u (cid:18) T ( u, v ) X v ( u, v ) | X v ( u, v ) | − T ( u, v ) X v ( u, v ) | X v ( u, v ) | (cid:19) du. Tension forces apply on the boundary of the patch, and so appear under single integrals, whereas pressureforce appears in an area integral. Next, apply the fundamental theorem of calculus to convert each of thesingle integrals to double integrals. Then, swap the order of integration formally as needed to obtain0 = (cid:90) v v (cid:90) u u (cid:18) p ( X u × X v ) + ∂∂u (cid:18) S X u | X u | (cid:19) + ∂∂v (cid:18) T X v | X v | (cid:19) (cid:19) dudv. (4) v u S X u | X u | dv S X u | X u | dv T X v | X v | du T X v | X v | du p ( X u ⇥ X v ) dudv Figure 2: Free body diagram of forces on the leaflets. Each of the tension forces is evaluated on one edge ofthe patch; the pressure force acts on the entire patch. Tensions on opposite sides of the patch are not equaland do not cancel, but we omit arguments for visual clarity.Since the patch is arbitrary, the integrand must be identically zero and the integrals can be dropped. Thisgives a partial differential equation for equilibrium of the leaflets0 = p ( X u × X v ) + ∂∂u (cid:18) S X u | X u | (cid:19) + ∂∂v (cid:18) T X v | X v | (cid:19) . (5)Note that changes in the tension, that is, curvature in the fiber and cross-fiber directions, as well as localvariation in the magnitude of tension, directly balance the pressure force.Equation (5) has three components and five unknowns, the leaflet configuration X and the tensions S and T , so additional information is required to close this equation. We wish to work directly with the loadedconfiguration, since the loaded configuration determines whether the valve will seal. Further, we do notimmediately possess a realistic reference configuration, nor do we wish to use a simple analytic shape, sinceit is not anatomical. Thus, we specify a tension law that does not rely on a reference configuration. Thesimplest such law would be to prescribe constant tension, but this is not effective, as there are no parametersto tune to alter the gross morphology of the leaflets. Further, the radial tension T is likely to be nonconstant,because at the free edge it must be supported by curvature in circumferential tension in a single fiber at thefree edge only, whereas radial tension near the annulus can be supported by curvature in a number of fibersin the leaflet. Additionally, there is nothing to prevent fibers from nearly colliding during nonlinear iterationsto solve the discretized equations equations of equilibrium. Adjacent points become close and evaluationof finite differences becomes ill-conditioned, and the nonlinear solvers fail to converge. (Discussion of thediscretization and numerical methods follows.)To close equation (5), we prescribe the maximum tension but allow the value of tension to vary belowthat prescribed value. We therefore define S and T as S ( u, v ) = α (cid:18) −
11 + | X u | /a (cid:19) , T ( u, v ) = β (cid:18) −
11 + | X v | /b (cid:19) . (6)Here, α denotes the maximum tension in the fiber direction, β denotes the maximum tension in the cross-fiber direction. The parameters a, b are tunable free parameters with units of length that can be adjustedto control the fiber spacing and gross morphology of the loaded configuration of the valve. Note that theseparameters are not required to be constants, and indeed using a nonconstant value of a was necessary toachieve anatomical gross morphology of the leaflets.Substituting Equation (6) into Equation (5) gives the final form of the equilibrium equations0 = p ( X u × X v ) + ∂∂u (cid:18) α (cid:18) −
11 + | X u | /a (cid:19) X u | X u | (cid:19) + ∂∂v (cid:18) β (cid:18) −
11 + | X v | /b (cid:19) X v | X v | (cid:19) . (7)Equation (7) is discretized using a centered finite difference scheme. Let X j,k denote an arbitrary pointinternal to the leaflets, and we refer to the connection between vertices in the discretized model as links oredges. The nonlinear system of equations associated with this point is given by0 = p (cid:18) ( X j +1 ,k − X j − ,k )2∆ u × ( X j,k +1 − X j,k − )2∆ v (cid:19) (8)+ α ∆ u (cid:18) − (cid:30)(cid:18) | X j +1 ,k − X j,k | a (∆ u ) (cid:19)(cid:19) X j +1 ,k − X j,k | X j +1 ,k − X j,k |− α ∆ u (cid:18) − (cid:30)(cid:18) | X j,k − X j − ,k | a (∆ u ) (cid:19)(cid:19) X j,k − X j − ,k | X j,k − X j − ,k | + β ∆ v (cid:18) − (cid:30)(cid:18) | X j,k +1 − X j,k | b (∆ v ) (cid:19)(cid:19) X j,k +1 − X j,k | X j,k +1 − X j,k |− β ∆ v (cid:18) − (cid:30)(cid:18) | X j,k − X j,k − | b (∆ v ) (cid:19)(cid:19) X j,k − X j,k − | X j,k − X j,k − | . The nonlinear system is solved using Newton’s method with line search. To compute the Jacobian of equation(8), each term was differentiated analytically, then the entire sparse matrix is constructed using the analyticforms at each step of the Newton’s iteration.The equations are solved on the domain Ω = [0 , × [0 , / v = 0 represents the annulus;its location is prescribed as a Dirichlet boundary condition. The radius r = 1 .
25 cm [8], and the annularheight is 1 .
4r [54]. The curve v = 1 / u = 0 , / , / .
9r to 1 . u = 1 is identified with the curve u = 0 by periodicity. The geometryof the aortic annulus is shown in Figure 3.Figure 3: Two views of the three-dimensional aortic annulus. This curve is prescribed as a Dirichlet bound-ary condition at the bottom of the leaflets and the commissures when solving the equations of equilibrium (8).The position of the free edge emerges from the solution. There zero radial tension (homogeneous Neumann)boundary conditions, are prescribed.We seek to match the loaded geometry provided in [54]. The loaded leaflet height at the center of theleaflet is then targeted to be 1 . r = 1 .
75 cm. They report the free edge length as 2 . r = 3 . . r + 0 . . . p = 60 mmHg, slightly below the nominal diastolic pressuredifference of 80 mmHg across the fully-loaded valve. Each of the tunable parameters are selected by trialand error, to match achieve the desired gross morphology in the predicted loaded configuration. We select α, β and b to be constants, and we tune a to vary in a linear manner from the annulus to the free edge.Values are shown in Table 1.Type Max tension Max tension Dec tension Dec tensionname dynes name cmCircumferential α . · a β . · b a varies linearly from the annulus to the free edge.Note that this phrasing does not include any notion of contact, and we allow the leaflets to interpenetrateon solving equation (8). We allow this because the overlapping creates a bit of extra length on the free edgesallows the leaflets to obtain good coaptation and seal, or more informally mash together, when running FSIsimulations (in which the leaflets are not allowed to interpenetrate). Solving the equations is simpler withoutthis additional force, and when simulated with fluid, the IB method prevents further interpenetration withoutspecifying additional contact forces [32]. Further, not allowing contact sets the material properties such thatthe leaflet could bear a pressure load. In in vitro experiments, we occasionally see one leaflet begin to closeand appear to support a pressure on its own and form a bowl-like shape immediately prior to any contact[12], so rather than specifically account for different strains or loading conditions locally in the coaptationregion, we load the entire leaflet uniformly. The solutions to equations (7), as shown in Section 4.1, specify the loaded geometry of the valve, and thetensions required to support such a load. For each link in the discretized model, this then gives a single pointon the tension/strain curve. In this section, we apply this information to set a reference configuration andconstitutive law for the valve. We do this in such a way that using the new constitutive law, the equilibriumequations (5) are still satisfied, but with the constitutive law defining tensions, rather than the formulas inequation (6).First, we prescribe uniform strain to the loaded configuration. Let E c denote the circumferential strain,and E r denote the radial strain, L the length of any link, and R its associated rest length. Yap et al. foundthat the aortic valve achieves a nearly constant strain of E c = 0 . , E r = 0 . . (9)in the belly region when fully loaded [57]. We prescribe these values in each direction, then solve E = L − RR (10)for the rest length R in each link in the model.Next, we assign each link a constitutive law. We base the shape (but not the local stiffness) of theconstitutive law on the experimental results of [36]. Their strip biaxial tests on specimen P35 in their Figure4 appear to be approximately exponential, and are taken to be representative curves. Thus, we assume thatthe tension/strain relationship is exponential through the origin for positive strains, and zero for compressivestrains, or τ ( E ) = (cid:40) κ ( e λE −
1) : E ≥
00 :
E < , (11)The exponential rate of the curves associated with data in the circumferential and radial directions wereestimated using a nonlinear least squares fit using Matlab [35] and take values λ c = 57 .
46 and λ r = 22 . κ scale the constitutive law. The value of κ is set individually for each link inthe model to achieve the value of tension in the solution of the equilibrium equations (8) at the prescribedstrains E c and E r . Let S j +1 / ,k denote the circumferential tension between X j,k and X j +1 ,k . We solve S j +1 / ,k = κ j +1 / ,kc ( e λ c E c −
1) (12)for the circumferential membrane stiffness coefficients κ j +1 / ,kc . Similarly, if T j,k +1 / denotes the radialtension between X j,k and X j,k +1 , we solve T j,k +1 / = κ j,k +1 / r ( e λ r E r −
1) (13)for the radial membrane stiffness coefficient κ j,k +1 / r associated with the current link.Using this constitutive law, we solve an addition equilibrium problem where the pressure is set to zero, p = 0 mmHg. A Dirichlet boundary condition is prescribed at the free edge to ensure that the solution isfully open and without self-intersections. We then expand the leaflet membrane in the normal direction tothe open configuration to obtain a model with anatomical thickness. Three total layers are placed 0 . .
044 cm, as measured in [43]. This is to mitigate the “grid alignedartifact” that may appear in IB simulations when a discontinuous pressure is supported by an infinitely thinmembrane [25, 24]. The stiffness in the circumferential and radial direction of each layer is set to one thirdof the membrane stiffness calculated above. Linear springs in the normal direction keep the layers adjacent.This completes the constitutive law for the leaflets, and this configuration is then used as an initialcondition for FSI simulations. This constitutive is fiber-based, in that all forces are determined based on anetwork of linear and nonlinear springs, and the network is structured into curves that represent biologicalfiber bundles in the circumferential direction and material response in the radial direction. This is similar toa mass-spring model, except that the structure is assumed to be neutrally buoyant and mass is thus handledas fluid density by the IB method without additional mass at structure nodes. In a comparative study ofthe aortic valve, mass-spring models had similar deformations to those of hyperelastic finite element baseddiscretization, and were approximately ten times faster [20].
The IB method is a framework for the modeling and simulation of FSI [38]. The method uses two differentframes of reference, a lab-based or Eulerian reference frame for the fluid, and a material or Lagrangianframe for the structure. Let x denote a physical location in the fluid domain, and t denote time. Let thefield u ( x , t ) represent fluid velocity and p ( x , t ) represent pressure. These fields are defined with respect tothe Eulerian frame, and accept arguments that are fixed with respect to spatial location. The field f ( x , t )denotes a body force exerted by the structure onto the fluid. This is one distinctive feature of the IB method:the fluid interacts with the structure through such a body force. The parameters ρ and µ represent densityand dynamic viscosity, respectively. Let s denote material points on the structure. (We previously used u, v to denote material points on the leaflets, but change to s here to avoid confusion with fluid velocity.) Let X ( s , t ) denote the position of the structure associated with material point s at time t . Let F ( s , t ) d s denotethe force exerted by the structure onto the fluid associated with patch d s . These two frames are coupled viaconvolutions with the Dirac delta function in a manner shown below.0The governing equations of the IB method are ρ (cid:18) ∂ u ( x , t ) ∂t + u ( x , t ) · ∇ u ( x , t ) (cid:19) = −∇ p ( x , t ) + µ ∆ u ( x , t ) + f ( x , t ) (14) ∇ · u ( x , t ) = 0 (15) F ( · , t ) = S ( X ( · , t )) (16) ∂ X ( s , t ) ∂t = u ( X ( s , t ) , t ) (17)= (cid:90) u ( x , t ) δ ( x − X ( s , t )) d xf ( x , t ) = (cid:90) F ( s , t ) δ ( x − X ( s , t )) d s . (18)Equations (14), (15) are the Navier Stokes equations describing the dynamics of a viscous, incompressiblefluid. Equation (14) represents momentum conservation, and equation (15) represents volume conservationor incompressibility. Equation (16) represents a mapping from the configuration of the structure to the forcethat the structure exerts on the fluid. This includes the nonlinear constitutive law of the leaflets, and anyother forces prescribed, including those that keep the leaflet mounted in place on its edges. The omittedargument indicates that the mapping S takes the entire configuration of the structure X as an argument,and produces the entire function F as output. Equations (17) and (18) are interaction equations that couplethe two frames. Equation (17) is the equation of velocity interpolation, and says that structure moves withvelocity equal to the local fluid velocity. Equation (18) is the equation of force spreading that computes theEulerian frame force from the Lagrangian frame force. When discretized, the delta function in equations(17) and (18) is replaced with a regularized delta function; we use the 5-point delta function derived in [4].The equations were solved with the software library IBAMR using a staggered grid discretization [18, 19].Simulations were run on Stanford University’s Sherlock cluster, with 48 Intel Xeon Gold 5118 cores with a2.30GHz clock speed.The fluid domain is a taken to be a box of dimensions [ − L, L ] × [ − L, L ] × [ − L +1 , L +1] where L = 2 . × ×
192 points for a spatial resolution∆ x = 0 . t = 5 · − s, which was the largest valuethat we found to be stable. The density is ρ = 1 g/cm , and the viscosity is µ = 0 .
04 Poise. No slip wallsare placed on the sides of the domain. The domain is taken to be periodic in the flow direction, and to hideperiodic effects, we add a mathematical flow straightener to the bottom 0 . d = − η ( u, v, w − ¯ w ), where ¯ w denotes the mean of the z-component of velocity. Thiscreates a friction-like force that approximately enforces zero flow in the x - and y -directions and enforces themean in the z -direction. Overlap and interpenetration of the leaflets is prevented by the IB method withoutany specifically added contact forces [32].A pressure difference across the valve drives the simulations. This is prescribed as a uniform body forcein the z direction, which is equivalent to prescribing the pressure difference directly via a change of variables.The value of the aortic pressure is determined by the dynamics of a Windkessel or rcr (resistor capacitorresistor) lumped parameter network. The aortic pressure is governed by the ordinary differential equations1 R p ( P ao − P wk ) = Q ao (19) C dP wk dt + 1 R d P wk = Q ao , (20)where P ao is the prescribed aortic pressure, P wk is the pressure in the Windkessel, Q ao is flow throughthe aortic valve, R p is proximal resistance, R d is the distal resistance and C is the capacitance [28, 17].The flow target, Q mean is set to 6.1 L/min, slightly higher than the nominal mean flow of 5.6 L/min, toaccount for some extra resistance from the flow straightener. We target a pressure of 120 mmHg in systolefor approximately 40% of the cardiac cycle and 80 mmHg in diastole for approximately 60% of the cardiac1 Periodic in flow directionPartition to hold valveDriving body force d ( z ) = p/H Flow straightener in x, y directions,enforce mean in z direction s = ( u, v, w w )No-slip walls Figure 4: Simulation setup. The model valve is mounted in a cylindrical scaffold, and attached to a flatpartition. Sides are treated with no-slip boundary conditions. The domain is periodic in the flow direction.Ventricular pressure is prescribed, aortic pressure is determined by a lumped parameter network, and thepressure difference is prescribed as a body force. An additional force that acts as a flow straightener in the x, y directions and approximately enforces the mean in the z direction is applied on a thin slab at the bottomof the domain.cycle. The mean pressure is set to be be a weighted average of the systolic and diastolic pressures, giving P mean = 96 mmHg. The total resistance is then computed as R total = P mean /Q mean . The ratio of theproximal to distal resistors is taken to be 0 .
064 [29]. This gives proximal resistance R p = 76 .
81 s dynescm − and distal resistance R d = 1182 .
10 s dynes cm − . To tune the capacitance, we select an early diastolicpressure of P = 100 mmHg, since the pressure drops from systolic pressure in closure, an end diastolicpressure of P = 80 mmHg, and the predicted time as t as the time in diastole. Assuming perfect closureduring diastole so Q ao = 0, equation (20) has an exact exponential solution. We substitute these values intothe exact solution to obtain C = − tR d log( P /P ) = 0 . dynes − . (21)The ventricular pressure is prescribed following the experimental measurements shown in [58]. The heartrate is set to a nominal value of 75 beats per minute, or 0.8 s. All simulations are run for three cardiaccycles, which is sufficient to test multiple cycles, then stopped due to high overall computational time.The structure mesh is targeted to approximately twice as fine as the fluid resolution. The annulus hasthree-dimensional length 11 .
28 cm. Since N is required to be a power of two on each leaflet, this correspondsto N = 384 points around the annulus. Note that the precise length of links in the leaflets at any time aredetermined as solutions to equations (14)-spreading and change according to the dynamics of system. Aflat partition is added to the plane z = 0 outside of the annulus. The leaflets are mounted to a cylindricalscaffold of height πr/ .
31 cm that serves to cover holes between the partition and the annulus. Thecylinder and partition are approximately rigid, and held to a fixed position using target points . For a point X and its desired position X target , this is a force f = − k ( X − X target ), representing a linear spring of zerorest length. The predicted loaded configuration of the valve is shown in Figure 5. This configuration arises as the solutionto the static equations of equilibrium (8), and predicts the geometry and tension required to support the2prescribed pressure load. Since we allow the leaflets to interpenetrate and do not include contact forces atthis stage, the leaflets cross near the free edge. The right panel shows only one leaflet to clarify the othertwo views. The emergent tension in the leaflets is shown in Figure 6. The values of tension that emerge areheterogeneous and much lower in the radial direction than the circumferential direction. More meaningfulthan the local tensions is the predicted tangent modulus that results, see Section 3.2.Figure 5: Predicted loaded configuration of the aortic valve. From left to right, the valve is viewed fromabove, at an angle and showing one leaflet only. Every eighth contour in the fiber and cross-fiber directionsin the computational mesh is shown in black for visual clarity. At this model-construction stage, we allowthe leaflets to interpenetrate, as depicted on the left and center panel, so each leaflet bears pressure onits entirety. This configuration is used to generate the reference configuration and constitutive law for thevalve. An open configuration without interpenetration will be prescribed as an initial condition to the FSIsimulations. circumferential tension radial tension dynes · Figure 6: Emergent tension in one aortic valve leaflet in the predicted loaded configuration showing thecircumferential, fiber direction (left) and radial, cross-fiber direction (right). The tension is much larger inthe circumferential direction than the radial.
From the steps described in Section 3.2, a constitutive law was constructed. We estimate the fully-loadedtangent modulus that emerges from the process as follows. Equation (11) (including the local stiffness3circumferential tangent modulus · dynescm radial tangent modulus · dynescm ratio circumferential/radial tangent modulus radial tangent modulus (rescaled) · dynescm Figure 7: Emergent tangent modulus and ratio of tangent moduli in one aortic valve leaflet. The top rowshows the emergent tangent modulus in the circumferential direction (left) and radial direction (right). Thebottom row shows the ratio of circumferential over the radial tangent moduli (left) and the radial tangentmodulus, but plotted on a smaller scale (right).coefficient κ ) is differentiated with respect to E and evaluated at E c or E r , depending on the direction oflink. This gives the tangent stiffness in units of force, which is divided by an area element to convert to thetangent modulus. The length of the area element is computed as half the distance to each adjacent pointin the opposing direction to the link, and the thickness is 0 .
044 cm [43]. The mean circumferential tangentmodulus is 1 . · dynes/cm , the mean radial tangent modulus is 5 . · dynes/cm , and the ratio ofthe means is approximately 25.The emergent tangent moduli of this constitutive law at the prescribed strains of E r and E c , as well asthe local ratio of the moduli, are shown in Figure 7. Both the radial and circumferential directions have aheterogeneous modulus. The circumferential, fiber direction in the central, belly region of the leaflet has alocal tangent modulus of order 10 dynes/cm . Near the commissures and the annulus, the circumferentialtangent modulus has a locally higher modulus. One experimental study found that vast variety in tissue,including tendon like structures near the commissures, which suggests that there may be heterogeneousthickening or higher elastic modulus there [7]. The radial direction, is an order of magnitude less stiffoverall. When viewed on the same scale as the circumferential tension, the tangent modulus appears lowoverall, and differences in the load radial tangent modulus are barely visible. Viewing on a smaller scalereveals that the radial direction is most stiff near the annulus and much less stiff near the free edge. Thisis consistent with experiments that show less stiffness in the radial direction near the free edge [7]. Nearthe annulus, there are many circumferential fibers in the direction of the free edge to support tension in theradial direction via curvature. Near the free edge, the radial-direction contours end, their tension must be4balanced by curvature of fewer circumferential fibers, and a much lower local tangent modulus results. Using the simulation setup described in Section 3.3, we simulate the model valve with the IB method. Figure8 shows the configuration of the valve and slice views of the velocity field through the cardiac cycle in thethird beat. The top panels show the valve from the side, and a slice view of the z -component of velocity inthe x = 0 plane. The bottom panels at each time point show the valve from above, and a slice view of the z -component of velocity at the z = 0 plane. We discuss the panels from left to right. The first frame showsthe fully closed valve. Its configuration has been nearly static for the previous approximately 0.4s. The flowhas relaxed from previous cycles and is relatively still. Since the IB method regularizes the forces due to thestructure, so from the perspective of the numerical method, this configuration is closed despite small visiblegaps. Then, the valve begins to unload. The free edges are still close together, but the belly of the leafletshas started to rise, and at the center a small opening has begun to form. Next, the valve in the middle ofopening shows transient ripples on the free edge. Forward flow has begun, but mostly fluid that is movingwith the leaflets, rather than flow moving through the open configuration. Then, the valve has fully opened,achieving near-maximum open area, and the forward jet is starting to develop. At peak systolic flow, thevalve is fully open and a strong jet flows forward. Next, the jet continues but at a lower magnitude, whilethe valve remains nearly as open and forward flow has begun to slow. In the penultimate frame, closurebegins. The jet breaks off and the leaflets move towards the center of the orifice. A slight, transient, localflow reversal appears between the leaflets. This is not true leakage, but rather represents fluid that had yetto “clear” the leaflets during the closure. Finally, the valve has just closed and is still vibrating. A slight“puff” of reverse flow is all that remains of the closing transient below the valve.Figure 8: Slice view of the z component of velocity through the cardiac cycle. From left to right, the framesshow the fully closed valve, the initial unloading prior to opening, the valve in middle of opening, the justopen valve, the fully open valve at peak forward flow, the open valve as forward flow begins to slow, thevalve initiating closure, and the just-closed, still-vibrating valve.Pressures in the physiological range and the flow rates that result are shown in Figure 9. The simulationbegins with diastole, and the aortic valve immediately closes. After the initial vibration concludes, flowis approximately zero for about half a second. The ventricular pressure rises during systole, and with aslight lag, the aortic pressure follows. A forward pressure difference is established and the forward flow ratethrough the valve rapidly rises. The pressure difference declines gradually through systole, and rapid ejectioncontinues. Note that the pressure difference during forward flow includes the pressure difference across theflow straightener; the pressure difference across the valve itself is lower.At end systole, the ventricular pressure drops below the aortic pressure and continues to fall and the valvebegins to close. A prominent dicrotic notch appears in the aortic pressure, which emerges from the combineddynamics of the fluid, valve and lumped parameter network. Immediately after, an violent vibration beginsthen is quickly damped out. The aortic pressure, the dynamics of which are governed by equation (20),5 Time (s) -20020406080100120140 P r e ss u r e ( mm H g ) p Ao p LV Time (s) -600-400-2000200400600 F l o w ( m l / s ) , C u m u l a t i v e F l o w ( m l ) FlowCumulative Flow
Figure 9: Driving pressures at physiological values and emergent flows.shows an oscillation at the same time. This vibration in pressure and flow causes the S heart sound orcolloquially the “dub” of “lub-dub.” Next, the flow is approximately zero; the valve is tightly sealed. Thisrepeats over the next two cardiac cycles. Note that the back pressure is over an order of magnitude greaterthan the forward driving pressures. This asymmetry creates demanding conditions for the valve, as it mustsupport a pressure, then open freely under a forward pressure that is much smaller.See also movies M1,M2 , which show this simulation visualized with slice views in slow motion and realtime, respectively, and M3 , which shows pathlines in slow motion.Next, we “stress test” the model to ensure its function over higher and lower driving pressures. First, wesimulate with lower, or hypotensive, systolic aortic and ventricular pressure pressure. We target a pressureof 60/40 mmHg, or half of the original targets. In the lumped parameter network, the total resistance isturned down by half, while the ratio of proximal to distal resistance is kept constant. The capacitance isset using equation (21) with P = 40 mmHg and P = 50 mmHg. This gives R p = 38 .
41 s dynes cm − , R d = 591 .
05 s dynes cm − and C = 0 . dynes − . Ventricular pressure remains constant thoughdiastole, then is halved in systole, as it only needs to rise above the lower aortic systolic pressure that occurswith half resistance. Flows and pressures are shown in Figure 10. Despite lower pressures, the valve is fullcompetent and seals without leak over three cardiac cycles.Next, we test the valve under extreme hypertensive, high-pressure conditions. Total resistance is doubled,the ratio of proximal to distal resistance is maintained, and capacitance is tuned using equation (21) with P = 160 mmHg and P = 200 mmHg. Note that pulse pressure may rise in conditions such as essentialhypertension [9]. This gives R p = 153 .
62 ml − s dynes cm − , R d = 2364 .
20 ml − s dynes cm − and C = 0 . dynes − . Ventricular pressure remains constant though diastole, then is doubled in systole,as it must rise above the higher systemic systolic pressure that occurs with double resistance. Flows andpressures are shown in Figure 11. Despite extreme hypertensive pressures, the valve seals reliably on allbeats.Figure 12 shows the valve in hypotensive, physiological and hypertensive pressures during diastole inthe third cardiac cycle. In all cases, the loaded configurations are visually similar. The lowest pressure hasthe least curvature, which is expected as it is the least loaded, then the model with physiological pressure,then the hypertensive model, which is again expected as it is the most loaded. In the hypotensive case, thematerial model is compliant enough such that with much lower pressure, the valve achieves a similar loadedconfiguration as in the standard cases. In the hypertensive cases, the material, which stiffens more underhigher load, deforms only slightly more than under standard pressures. This suggests that the nonlinearityof the material enables the valve to function effectively over a range of pressures.6 Time (s) -20020406080 P r e ss u r e ( mm H g ) p Ao p LV Time (s) -400-2000200400600 F l o w ( m l / s ) , C u m u l a t i v e F l o w ( m l ) FlowCumulative Flow
Figure 10: Hypotensive driving pressures and emergent flows. Despite much lower loading pressures, thevalve seals competently over multiple beats.
Time (s) P r e ss u r e ( mm H g ) p Ao p LV Time (s) -1000-800-600-400-2000200400600 F l o w ( m l / s ) , C u m u l a t i v e F l o w ( m l ) FlowCumulative Flow
Figure 11: Hypertensive driving pressures and emergent flows. With much higher loads, the valve remainscompletely sealed during diastole over three cardiac cycles.
In this section, we examine variations in the prescribed loaded strain. In experimental studies on thekinematics of the aortic valve, reported values of strain or stretch may vary due to the definition of thereference configuration. The reference configuration may be taken to be the open, in vivo state, attached to7hypotensive physiological pressures hypertensiveFigure 12: Slice view of the z component of velocity with low, physiological and high pressure. Despitevastly different loading conditions, the valve functions well in each case, with slightly less curvature at lowpressure and slightly more at higher pressure, as expected.the aortic root but in a dissected specimen (as in [57]), excised and floating in liquid, crumpled, or stretchedto remove initial buckling. Pre-strain alters the deformation between the reference and loaded state, and candramatically change the reported values of local tangent moduli [41]. Since the strains reported in reportedin [57] are measured from the in situ state, there may still be pre-strain present; perhaps these models wouldbehave differently if pre-strain was added or potentially excess strain was removed.For a given loaded state as the solution to equations (8), we then ask whether the behavior in FSIsimulations is sensitive to the values of strain prescribed. We first prescribe smaller strains while maintainingthe ratio of circumferential to radial strain, corresponding to longer lengths for all links in the model. Thevalues are E c = 0 . , E r = 0 . . (22)We then prescribe a larger strain, corresponding to a shorter resting lengths for all links in the model, withvalues E c = 0 . , E r = 0 . . (23)This change preserves exactly the predicted loaded geometry and the force exerted in the loaded geometry,but alters the reference geometry or the model.This change in strain nearly preserves the tangent modulus, the slope of tension with respect to strain,as well. Consider two values of the fully-loaded strain E and E at which equal tension is achieved. Byconstruction, T ( E ) = T ( E ) or κ ( e λE −
1) = κ ( e λE − , (24)8lower predicted experimentally higher predictedstrain, longer predicted strain strain, shorterreference lengths reference lengthsFigure 13: Slice view of the z component of velocity with low, physiological and high predicted strainin construction of the model. Each has a distinct reference configuration, but identical predicted loadedconfigurations. Despite different reference configurations, the performance during closure of these threemodels is nearly identical.so T (cid:48) ( E ) | E = E T (cid:48) ( E ) | E = E = κ λe λE κ λe λE = ( e λE − e λE ( e λE − e λE = 1 − e − λE − e − λE ≈ . (25)Both the numerator and denominator exponentially approach one away from E = 0. Since the exponentialrate b = 57 .
5, and we consider E to be on the order of .
1, this ratio is very close to one. This implies thatchanging the loaded strain very nearly maintains the tangent modulus of the predicted loaded configuration.Flows during closure in the third cycle are shown in Figure 13. The three models have nearly identicalperformance, including highly similar closed configurations. This suggests that the predicted loaded config-uration is indeed predictive of valve performance, regardless of alterations to the reference configuration.
In this section, we vary the constitutive law in the valve. First, we apply a linear law to show where themodel maintains its performance and where it fails, relative to a model with a more realistic constitutive law.Results with this constitutive law are meant to serve as a “negative example”, and illustrate why nonlinearbehavior is required for robust valve function. We replace equation (11) with the linear law τ ( E ) = κE (26)The value of tensions for strains lower than E c and E r is greater than the standard model, equal at thestrains E c and E r , and lower for strains larger than E c and E r . This creates a lower mean tangent modulus9at the predicted strains E c and E r . The mean circumferential tangent modulus is 1 . · dynes/cm , themean radial tangent modulus is 4 . · dynes/cm , and the ratio of the means is approximately 35. Thesemoduli are approximately one order of magnitude lower than the tangent moduli of the standard model.Results under low, standard and high pressures for the model with linear constitutive law are shownin Figure 14. At standard pressure, the loaded model is fully sealed and appears qualitatively similar tothe standard model. At low pressure, the leaflets fail to deform enough to create a good coaptation, andregurgitation results. At high pressure, the valve seals, but deforms so much that the belly of the leafletprolapses below the lowest point on the annulus. Aortic valve prolapse may lead to regurgitation or requiresurgery [14].Next, we reduce the exponential rates by a factor of one half, representing a less nonlinear material thanthe standard model. The exponential rates are taken to be half of the basic model, as derived from [36]. Thecircumferential rate takes value λ c = 57 . / .
73 and the radial rate takes value λ r = 22 . / . . · dynes/cm , the mean radial tangent modulusis 2 . · dynes/cm , and the ratio of the means is approximately 25. These values are approximately half ofthe tangent modulus computed for the standard configuration. This is because this change in exponential ratealters the tangent modulus, the slope of tension with respect to strain, as follows. Consider two exponentialrates λ and λ at which equal tension is achieved at strain E ∗ . By construction, T ( E ∗ ) = T ( E ∗ ) or κ ( e λ E ∗ −
1) = κ ( e λ E ∗ − , (27)so T (cid:48) ( E ∗ ) T (cid:48) ( E ∗ ) = κ λ e λ E ∗ κ λ e λ E ∗ = λ ( e λ E ∗ − e λ E ∗ λ ( e λ E ∗ − e λ E ∗ = λ (1 − e − λ E ∗ ) λ (1 − e − λ E ∗ ) ≈ λ λ . (28)Thus by taking half the exponential rate, the local tangent moduli approximately half that of the standardmodel.Figure 15 shows results with half exponential rates and approximately half the tangent modulus at thepredicted strain values. At low pressure, the valve develops regurgitation. Similarly to the linear model, thismodel is much stiffer at lower strains than the standard model, and the leaflets do not not coapt well underthis smaller load. At standard pressures, the model also leaks with a central jet. It is not obvious why thismodel works more poorly than the linear model under standard pressure. At high pressure, the valve sealswell. Despite being less stiff at higher strains, the exponential law is able to generate sufficient tension tosupport this load.Last, we double the exponential rates, with values λ c = 57 . · .
91 and λ r = 22 . · . . · and1 . · dynes/cm circumferentially and radially, respectively. Figure 15 shows results at low, standardand high pressures. The valve functions well in all cases, with notably little visible difference in the closedconfigurations in all cases. This model is more compliant at strains below the predicted strain, and stifferstill at higher strains. This suggests that the model functions best when it is compliant enough to easilyobtain the closed configuration, then stiff enough due to nonlinearity around that configuration that it doesnot strain much more. In this section, we test four slightly different geometries under physiological pressures. These are all lesseffective than the standard model geometry. This serves to illustrate how slight changes in gross morphologycan degrade valve performance. Each model is constructed as described in Sections 3.1 and 3.2, and theconstitutive law is emergent from this process and varies slightly from model to model.In the first example, we remove 0.2 cm of length from the free edge of the model in the predicted loadedconfiguration. Note that the removed reference length is less than this value. This valve leaks dramatically,forming a large, central jet of backflow. In the second example, we add 0.2 cm of length from the free edgeof the model, again in the predicted loaded configuration. This seals acceptably, but the extra leaflet causesthe center coaptation region to not form, meaning that there is a lack of vertically aligned leaflet tissuenear the free edge. In the third example, we remove 0.2 cm of height from leaflets in the predicted loaded0hypotensive physiological pressures hypertensiveFigure 14: Results with linear constitutive law under hypotensive pressure (left), physiological pressure(center) and hypertensive pressure (right). This model leaks under low pressure, seals at physiologicalpressure, and prolapses at high pressure.configuration. As with removing length from the free edge, this leaks dramatically. In the last example, theleaflet is 2 mm taller than the standard example in the predicted loaded configuration. This model closeseffectively, but a buckle appears at the free edge due to the extra material. The valve fails to form a flatcoaptation zone, instead coapting only on where the buckling leaflet touches the other leaflets. A normal,healthy aortic valve is expected to have a coaptation height of approximately 0 . r = 0 .
43 cm for a 1.25 cmradius valve, in which the leaflets are flush against each other [54]. We hypothesize that such a buckle mayallowing aortic pressure to push on the buckled tissue, and so separate the leaflets then initiate regurgitation.Slice views of velocity on these three examples are shown in Figure 16.These results suggest that the valve performance is highly sensitive to gross morphology. Too little lengthon the free edge or in leaflet height can cause the models to leak. Any extra height can create buckles andcause a coaptation zone of appropriate length to fail to form.
The results of FSI simulations in Section 4.3 suggest that the model is robust and effective under physiological,hypo- and hypertensive pressures. Achieving such function in a variety of conditions is essential for futurestudies of patient-specific cardiac flow.In Section 4.2, we showed that our model construction produces the fully-loaded mean tangent moduli of1 . · and 5 . · dynes/cm , circumferentially and radially, respectively, for a ratio of approximately25. These values are comparable to experimental results. Recall that Pham et al. found a circumferentialtangent modulus of 9 . · dynes/cm and a radial tangent modulus of 2 . · dynes/cm [40]. Our meancircumferential, fiber-direction tangent modulus is 44% stiffer than their experimental results. Our radialstiffness is 25% as stiff as their experimental results, though their experiments were on older human tissue1hypotensive physiological pressures hypertensiveFigure 15: Results with alternative constitutive laws under hypotensive pressure (left), physiological pressure(center) and hypertensive pressure (right). The top row shows a model with half the experimentally measuredexponential rates in the constitutive law, shows leak at low and standard pressure, but a good seal at highpressure. The bottom row shows a model with half the experimentally measured exponential rates in theconstitutive law, and this model seals well at all tested pressures.22 mm shorter free edge 2 mm extra free edge 2 mm shorter height 2 mm extra heightFigure 16: Slice view of the z component of velocity showing poor results with altered gross morphology.The upper left panel shows a model with 2mm of predicted loaded length removed at the free edge. It leakscatastrophically. The upper right shows a model with 2mm of predicted loaded length added to the freeedge. It seals acceptably, but fails to form a good coaptation surface of positive length. The lower left panelshows a model with 2mm of predicted loaded leaflet height removed. It shows significant regurgitation. Thelower right panel shows a model with 2mm of extra height. While the valve seals acceptably, an unappealingand unphysiological buckle has appeared near the free edge.that may become less compliant with age. Sauren et al. report an stiffness ratio of approximately 20:1,which is very close to our findings [48]. While a pressure load, strains and exponential rates were prescribed,stiffnesses and anisotropy ratios were not. The stiffnesses and ratio of anisotropy that emerges from thisprocess – solving the equations of equilibrium and tuning the constitutive law to match – reproduce thematerial properties and anisotropy ratios that are found in natural aortic valve tissue.Our method produces heterogeneous material properties, we hypothesize that heterogeneity assists ingood aortic valve function. There are “hot spots” of higher circumferential tangent modulus in near thecommissures, and the radial tangent modulus generally decreases moving from the annulus to the free edge.Experiments revealed heterogeneity in material properties and thickness [42, 26], as well as variation in thedistinct histological layers of the leaflet [53]. An experimental study noted the tendinous bunches near thecommissures, and decreasing radial tangent modulus toward the free edge [7]. There is some appearance ofradially-oriented fibers near the annulus, that thin and dissipate before the belly of the leaflet [47]. Radialcollagen fiber bundles were observed on the ventricular side of the leaflet [26, 54], but other potential causesof material nonlinearity in the radial direction have been proposed [7]. This suggests an experimentalstudy: measure heterogeneous material properties, produce a map of fiber orientations and local nonlinearforce response, then study their functional significance. Such a study could validate our predictions ofheterogeneity and would advance understanding of the mechanics of the aortic valve.In Section 4.4, we showed that with a consistent predicted loaded configuration, varying the prescribedstrain and thus the reference configuration had little effect. This suggests that the loaded configuration, itstension and its tangent modulus are the primary determinants of good valve closure. In particular, whether3the reference length is based on an excised valve, an in situ but resting valve, the valve at some point in thecardiac cycle, or includes pre-strain does not appear to be as important having the appropriate tension at ageometry that allows good coaptation.In Section 4.5, we simulated with a linear law, and changed the exponential rates in the constitutivelaw. The linear and half exponential rate functioned poorly, regurgitating at low pressures because they aretoo stiff at lower strains to achieve good coaptation. The linear law prolapsed under high pressure. Theseresults suggest that the leaflets need sufficient nonlinearity to function well over a large range of pressures.This suggests that the aortic valve should operate in two regimes – very compliant from open to closed andbarely loaded, loaded and very stiff once the loaded configuration is achieved to maintain a nearly constantposition.In Section 4.6, we showed that under changes in geometry, however, the models regurgitated with modelsthat are superficially similar. With too little length at the free edge, strong central jets of regurgitationformed. With too much height on the leaflet, the free edge buckled unphysiologically in the coaptationregion. This suggests an extremely precise range of lengths at the free edge, in the leaflet belly and in heightis required for robust valve function.These experiments are directly relevant to the design of prosthetic aortic valves. Our conclusions can besummarized as follows:1. Valve performance during closure is primarily determined by the loaded geometry, the force it exertsin the loaded geometry, and the tangent modulus in the loaded geometry.2. The reference configuration and pre-strain are not central to the valve closure, when the loaded con-figuration remains similar.3. To function over a range or pressures, the leaflets must be have in a sufficiently nonlinear manner.Materials that are linear or not nonlinear enough may function at physiological pressure, but fail athypotensive, hypertensive pressures or both.4. The range of free edge lengths and leaflet heights that function well is narrow. Regurgitation occursin leaflets that are too small and poor coaptation occurs in leaflets that are too large. While many in vitro studies use a box-shaped valve tester, downstream hemodynamics are affected by thegeometry, and testing in a model aortic root or beating heart would be a step forward. Despite testing anextensive variety of constitutive laws, we use a highly-specific, fiber-based constitutive law, in contrast to amore standard, three-dimensional hyperelastic formulation with a volumetric penalty term. Bending rigidityis not included, nor is shear tension, though both are expected to be small. We leave a direct comparisonof such models for future work. Anatomical details such as the nodules of Arantius, which may introducesome bending rigidity, were omitted. Finally, tuning of the models is highly manual; automating the tuningprocess would be a significant step forward.
We began with a near first-principles statement, that the aortic valve leaflets must support a pressure, andderived a corresponding partial differential equation, the solution of which specified the predicted loadedconfiguration of the valve. Parameters were tuned to make the solutions match experimental observationsabout the gross morphology of loaded aortic valve leaflets and a constitute law was created. This thencreates a model suitable for simulation with the IB method. When simulated under physiological pressures,the leaflets coapt well, and the valve seals over multiple cardiac cycles, and allows physiological flow ratesthrough. When simulated under pressures much lower or much higher than normal, the valve performssimilarly. Real aortic valves are robust to a variety of loading pressures, and the model appears to mimicthis property. We then conducted a number of experiments on reference configurations, constitutive lawsand gross morphology.4In conclusions, this method is robust and effective for FSI simulations involving the aortic valve. Havingsuch a model is essential for further studies of healthy and pathological patient-specific cardiac flow. We hopethat this model will be useful for such studies, and that the conclusions are provide insights for designers ofprosthetic aortic valves.
ADK was supported in part by a grant from the National Heart, Lung and Blood Institute (1T32HL098049),Training Program in Mechanisms and Innovation in Vascular Disease at Stanford. ADK and ALM weresupported in part by the National Science Foundation SSI grant
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