Abstract While the mechanical behaviors of the fibrosaand ventricularis layers of the aortic valve (AV) leaflet areunderstood, little information exists on their mechanicalinteractions mediated by the GAG-rich central spongiosalayer. Parametric simulations of the interlayer interactionsof the AV leaflets in flexure utilized a tri-layered finite ele-ment (FE) model of circumferentially oriented tissue sec-tions to investigate inter-layer sliding hypothesized to occur.Simulation results indicated that the leaflet tissue functionsas a tightly bonded structure when the spongiosa effectivemodulus was at least 25 % that of the fibrosa and ventricu-laris layers. Novel studies that directly measured transmuralstrain in flexure of AV leaflet tissue specimens validatedthese findings. Interestingly, a smooth transmural strain dis-tribution indicated that the layers of the leaflet indeed actas a bonded unit, consistent with our previous observations(Stella and Sacks in J Biomech Eng 129:757–766, 2007)of a large number of transverse collagen fibers intercon-necting the fibrosa and ventricularis layers. Additionally,when the tri-layered FE model was refined to match thetransmural deformations, a layer-specific bimodular materialmodel (resulting in four total moduli) accurately matched thetransmural strain and moment-curvature relations simultane-ously. Collectively, these results provide evidence, contraryto previous assumptions, that the valve layers function as abonded structure in the low-strain flexure deformation mode.
R. M. Buchanan · M. S. SacksCenter for Cardiovascular Simulation, Institute for ComputationalEngineering and Sciences, Department of Biomedical Engineering,The University of Texas at Austin, Austin, USA
M. S. Sacks (B)W.A. “Tex” Moncrief, Jr. Simulation-Based Engineering ScienceChair 1, 201 East 24th St, Stop C0200, POB 5.236, Austin,TX 78712-1229, USAe-mail: [email protected]
Most likely, this results directly from the transverse collagenfibers that bind the layers together to disable physical slidingand maintain layer residual stresses. Further, the spongiosamay function as a general dampening layer while the AVleaflets deforms as a homogenous structure despite its het-erogeneous architecture.
AV Aortic valveAC Flexure direction directed against the natural
curvature of the leafletECM Extracellular matrixFE Finite elementGAG GlycosaminoglycansI Second moment of inertia�κ Change in valve leaflet curvature during flexure
testingM Applied bending momentPG Proteoglycanμ Shear modulusTE Tissue engineeringW Strain energy functionWC Flexure direction directed with the natural
curvature of the leaflet
1 Introduction
Over 5 million Americans currently suffer from heart valvedisease (Go et al. 2013), estimating a prevalence of 2.5 % inthe US population. The AV (Fig. 1a) must perform under aunique mechanically stressed environment for an estimated
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Fig. 1 a The aortic valve b A 1 mm ×1 mm section from the central belly region illustrating the 3D, the tri-layered leaflet structure
3–4 billion cycles throughout an average lifetime. For exam-ple, all valvular tissues must exhibit low flexural stiffnessduring valve opening coupled with high tensile stiffness dur-ing closure (Sacks and Yoganathan 2007). This behavior isattainable through a unique hierarchical structure in whichall elements act in unison to provide seamless transition overthe cardiac cycle. Despite this level of functional efficiency,the onset and progression of valve disease ultimately requiresrepair or full replacement in many patients.
The AV leaflet is a heterogeneous structure composed ofthree distinct layers: the fibrosa, spongiosa, and ventricularis(Fig. 1b) (Thubrikar 1990; Sacks et al. 1998; Schoen andLevy 1999; Yacoub and Cohn 2004; Stephens et al. 2008;Wiltz et al. 2013). Each layer contains varying amounts ofcollagen, glycosaminoglycan (GAG), and elastin (Carrutherset al. 2012). The AV valve has multiple biomechanical prop-erties crucial to enabling proper function (Thubrikar et al.1977, 1979; Missirlis and Chong 1978; Sacks and DavidMerryman 2009). It has been speculated that the layered con-figuration of the leaflet facilitates valves under physiologicalconditions (Sauren et al. 1980; Sacks and David Merryman2009). Yet, while the outer fibrous layers of the AV leaflethave been extensively investigated ( Christie and Stephenson1989; Vesely and Noseworthy 1992; Vesely 1996; Stella andSacks 2007), little information exists on interlayer microme-chanics.
To date, only two studies have investigated the mechanicalbehavior of individual leaflet layers using micro-dissectiontechniques. Vesely et al. observed the extensibility of intacttissue under uniaxial tension to be significantly differentfrom the individual layer responses (Vesely and Noseworthy1992). In the second study, Stella et al. observed measurablydifferent behavior under biaxial loading of the separated lay-ers (Stella and Sacks 2007). The intact tissue response wasintermediate to the separated responses. Interestingly, It has
been assumed previously that the spongiosa layer enablessliding between the fibrosa and ventricularis during open-ing and closing (Mohri et al. 1972; Vesely and Boughner1989; Song et al. 1990; Thubrikar 1990; Talman and Bough-ner 1995), yet little data exists to support this theory. Veselyand Boughner (1989) and Song et al. (1990) claim evidenceof measured sliding between the layers; however, the load-ing conditions used were artificial and not representative ofphysiological flexural conditions. Thus, the degree of bond-ing that exists between the two layers and its effects on AVleaflet bending in vivo remain unclear. Moreover, based onthese findings, it is apparent that the individual layers func-tion quite differently in the intact configuration than sepa-rated and must be evaluated in the intact state.
Historically, the term “tissue engineering” is attributed toY.C. Fung ( Woo and Seguchi 1989). The term underscoredthe importance of “the application of principles and meth-ods of engineering and life sciences toward a fundamentalunderstanding of structure-function relationships in normaland pathologic mammalian tissues and the development ofbiological substitutes to restore, maintain, or improve tissuefunction.” Thus, it is imperative that fundamental structure-function understanding guides the reproduction of native tis-sue if it is to emulate its native counterpart successfully.Clearly, the complex nature of valve biomechanical behav-ior and function (Sacks and David Merryman 2009) cannotbe duplicated with simple homogenous biomaterials. Con-sequently, to develop replacement valvular tissues, we mustmore fully understand the fundamental micromechanics ofthe tissue in both healthy and diseased states (Butler et al.2000).
In the present study, we conducted an integrated simulation-experimental investigation utilizing flexural deformations ofintact AV leaflets as a means to probe interlayer interactions.In addition to being a natural choice to study interlayer micro-
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mechanics, flexure is a major deformation mode of the car-diac cycle (Iyengar et al. 2001; Sacks and Yoganathan 2007)and has been extensively used for valve tissue mechanicalstudies (Gloeckner et al. 1998, 1999; Engelmayr et al. 2005;Merryman et al. 2006; Mirnajafi et al. 2006). Parametricsimulations of interlayer interactions were first conductedusing a tri-layered leaflet tissue finite element (FE) modelto simulate interlayer sliding hypothesized to occur. To vali-date these findings and further refine the model, experimentalstudies on porcine AV leaflet tissue were conducted to exam-ine bi-directional flexural response, and relative interlayermovement using actual transmural strain responses.
2 Methods
2.1 Overview
The flexural deformation mode not only represents a majordeformation mode of heart valve leaflets, but also allowsdirect examination of individual layer responses in tensionand compression. It should be noted that AV leaflets experi-ence complex bidirectional flexure in vivo (Thubrikar et al.1980, 1986). To simplify the problem, we focused on bend-ing in the circumferential direction only, as it is the majorcurvature change in leaflets (Sugimoto and Sacks 2013).A circumferentially oriented rectangular leaflet tissue stripconfiguration, located below the Nodulus of Arantius (Fig.2a), was used for simulation since the central belly regionis structurally most representative of the leaflet (Billiar andSacks 2000). First, an initial model was used to conduct aparametric study of the deformation through the thicknessof the leaflet at varying degrees of layer connectivity. Theresults of this model were verified with novel experimentalflexure studies that quantified the variations in transmuralstrain. Based on these results, a refined flexural model wasthen developed to simultaneously match the moment-changein curvature (M–�κ , where �κ = κ − κ0 and κ0 the initialcurvature) relationship and transmural deformation of the AVin both bending directions. The end result was a clearer pic-ture of interlayer mechanical interactions of the low-strainbehavior that occurs in flexure.
2.2 Initial simulations of AV leaflet tissue flexure
A finite element (FE) model was developed to simulate AVleaflet tissue in flexure using the software package COMSOLMultiphysics v4.3 (COMSOL, Burlington, MA). A rectan-gular model geometry was used, set to 14 mm in length (cir-cumferential direction) by 3 mm in width (radial) by 0.4 mmthick. The tissue layer geometry was derived on previoushistological data (Carruthers et al. 2012), which showed thatthe fibrosa represented 45 % of the volume, the spongiosa
30 %, and ventricularis 25 %. Boundary conditions simulat-ing three-point bending were avoided to ensure no point-loading effects would occur in the center of tissue, and end-loading conditions were used instead. The boundary condi-tions thus consisted of pins at both ends of the model tissue,and a horizontal load was applied at one pin in the X1 direc-tion, causing the model tissue to undergo transverse deflec-tion (Fig. 2b). The mesh consisted of 3,612 8-node brickelements, and shape functions for pressure (linear) were setto one order lower than displacement (quadratic) to avoidlocking. The geometry was modeled as symmetric along theX3 axis to decrease computing time (Fig. 2b).
To establish both the material model and to obtain an ini-tial set of material parameters, we began with previouslypublished AV flexure data for inactivated tissues (Merrymanet al. 2006; Sacks and David Merryman 2009). The follow-ing Ogden model, assuming incompressibility, was chosenas it provides an additional level of flexibility compared witha neo-Hookean model
W =N=1∑
p=1
μp
αp
(λ1
αp + λαp2 + λ
−αp1 λ
−αp2 − 3
)(1)
where α is a constant, μ the shear modulus, and λ1 and λ2
are the principle stretches. When α = 2 and N = 1, Eq. 1will simplify to the standard neo-Hookean form. The classicanalytical solution of an incompressible isotropic beam underflexure (Rivlin 1949) was used to establish an initial materialmodel for a homogenized single layer beam from Eq. 1. Thiswas accomplished by matching the nonlinear shape of theM–�κ relationship of the leaflet bending data to the Ogdenmodel analytical solution from a straight to circular beam(Fig. 3).
2.3 Interlayer bonding simulation
To simulate the effects of various levels of interlayer bond-ing, the following parametric simulation was performed. Thefibrosa and ventricularis moduli were set to 45 kPa basedon the prediction of the initial flexure simulation describedabove. Next, bending simulations were performed up to cur-vature changes of 0.1, 0.2 and 0.3 mm−1 with values for thespongiosa shear modulus μS varied. A value of 1 Pa repre-sented relatively unbonded fibrosa and ventricularis layers;then μS was assigned increasing values (μS = 0.1−10 kPa)to simulate increasing levels of bonded states. Lastly, thespongiosa was assigned the same layer properties as thefibrosa and ventricularis to represent a perfectly bonded state.The deformation gradient tensor was derived from the ref-erence (X1, X2) and deformed (x1, x2) coordinates of theelement nodes using Fi j = ∂xi
∂ X j. The resulting deformation
gradient tensor F was decomposed into stretch and rotationtensors, U and R, respectively. The transmural variation in
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816 R. M. Buchanan, M. S. Sacks
X1
X2
Transmural measurement
region 14mm3mm
Bending bar marker
Bendingbar
Tissue sample
(a) (b)
(c)(d)
Fig. 2 a A schematic showing the orientation of the tissue strip andleaflet, and the location of the transmural strain measurement. b A freebody diagram of tissue mounted in the testing device with a force, P,applied to its free end. The orientation of the X1 and X2 coordinate sys-tem specifies the X1 axis coincident with the circumferential direction
and the X2 axis to reflect the thickness of the leaflet. c Markers usedby the macro-imaging system shown on the edge of the tissue strip.d Transmural strain images obtained using micro-camera, showing thereference and deformed tissue states
total axial stretch, Λ1 =√
U211 + U2
12, with respect to thethickness of the tissue was used to determine the overalleffects of bonding on transmural deformations. Normalizedthickness was defined with the ventricularis at the origin andthe fibrosa at unity.
A custom device performed flexure tests on AV leaflets,described in detail in Lam (2004). Briefly, two optical sys-tems simultaneously collect flexural rigidity and transmuraldeformation data. A macro-imaging system tracks markerson one edge of the sample. From these markers the curva-ture was calculated as well as the applied load via a markeron the bending bar transverse linkage. On the opposite edgeof the sample a micro-imaging system employs micron-sizedink marks to determine transmural strain via post- processingtechniques. A specially designed tank and tissue holder main-tains placement and physiological conditions of the samplebeing tested.
Native porcine aortic roots were removed from heartsobtained from an abattoir within three hours of sacrifice.Leaflets were dissected from the root along their attach-
Fig. 3 Comparison of the M–�κ data (Merryman et al. 2006) withthe analytical solution of the one-term Ogden model using various α
parameter values. Note that the experimental data was normalized tothe maximum achieved moment at a �k of 0.28 mm−1. A value for α
of 2.0 approximately captured the shape of the M–�κ
ment points, and cut into a circumferentially oriented rec-tangular strip of tissue for testing using a fresh razor bladeto ensure smooth flat surfaces in the reference state (Fig.2a). On average, specimens were 318 ± 46 microns thick,
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Aortic valve flexure 817
13.7 ± 2.1 mm long, and 2.8 ± 0.2 mm wide (Lam 2004).Five small red ink markers were created on the side of thetissue farthest from the free edge for curvature calculationusing the macro-imaging system (Fig. 2c). Airbrushing theedge of the tissue opposite to the red macro markers usinga Badger Sotar airbrush (Badger Air-Brush Co., FranklinPark, IL) and black India ink created microscopic mark-ers for tracking with the micro-imaging system (Fig. 2d).A stainless steel sleeve was glued to one end of the tissueusing cyanoacrylate and then slipped onto a steel dowel toallow for free rotation and restricted translation of the tis-sue. The tank-tissue holder assembly was mounted to a bi-directional (X1 axis movement) stage fitted with two preci-sion linear actuators (model MM-4M-EX80, National Aper-ture Inc., Salem NH). A small amount of cyanoacrylate glueattached a second stainless steel sleeve to the loading end ofthe tissue; this sleeve then slipped over a vertically mounted316 V stainless steel bending bar of known stiffness. Thestiffness of the bar determined the amount of force pro-duced as a function of the bar’s displacement, eliminatingthe necessity of load cells. Actuation of the bi-directionalstage in the X1-direction moved the entire tank and sample,causing the transverse linkage to bend the bending bar. Inthis design, both positive and negative changes in curvaturecould be tracked, allowing a single-run experiment in boththe with-curvature (WC) and against-curvature (AC) testingdirections.
For the bending data, the following information wasrecorded for the duration of the experiment: coordinates ofred macro markers, displacement of the bending bar, and thecurrent curvature of the tissue. Analysis of the bending datawas detailed by (Engelmayr et al. 2003) and summarizedhere. A custom-written Mathcad (PTC, Needham, MA) pro-gram computed curvature, change in curvature, and momentat the middle five markers for each time point of the experi-ment. The resulting marker positions were fit to a fourth-orderpolynomial so that the curvature of the tissue could be deter-mined. The initial curvature, κ0, was recorded by the controlsystem and subtracted from consequent curvature measure-ments to obtain the change in curvature �κ = κ − κ0. 18specimens were tested to a curvature change up to 0.3 mm−1
based on recent in vitro measurements (Sugimoto and Sacks2013). For clarity, results are reported for a change in curva-ture of 0.2 mm−1 and the full transmural results are reportedin Appendix 2. The moment was determined by using theposition of the central marker and the displacement of thebending bar (Eq. 2, Fig. 2b)
M = P y, (2)
where M is the applied moment, y is the deflection fromthe horizontal axis, and P is the axial force. The displace-ment of the bending bar was tracked in real-time since theapplied axial force, P , was a function of the displacement.
The deflection, y (Fig. 2b) was computed using the spatial ycoordinate of the central tissue marker and the y coordinateof the horizontal axis drawn from post to post. The resultingdata is reported as the averaged M/I versus �κ responseof the 18 specimens to normalize with respect to specimengeometry.
To determine transmural deformation, captured imagesfrom the micro-imaging system that tracked the markers onthe edge of the tissue were analyzed (Fig. 2d). A telecentriclens was used to avoid loss of accuracy from taking mea-surements from surface bending, resulting in surfaces notorthogonal to the imaging system, on the strain measure-ments. From the macro-level images, LabVIEW software(National Instruments, Austin, TX) identified and numberedmarkers and stored their centroidal coordinates. For the microimages used to determine the transmural stretch, an image-based particle tracking strain-mapping method, detailed inthe Appendix 1, computed the local deformation gradient Fover the imaged region. Briefly, the reference and deformedstates determined the displacement field and were fitted toobtain F. Rigid body deformation was removed using polardecomposition. The particle tracking method was shown tomeasure stretch as small as 1.001 with 0.05 % net accuracy(Appendix 1) (Lam 2004). The location of the neutral axiswas determined by plotting Λ1 against the thickness of thetissue to determine the location where Λ1 = 1.
2.5 Validation of experimental/simulation method
Validation studies were carried out with 1 mm thick siliconerubber sheets cut into rectangular strips of the same dimen-sions as the experimental studies described above, 14 mm inlength by 3 mm in width. The shear modulus of the stripswas determined using tensile testing with a MTS Tytron 250Mechanical Test System (MTS Systems, Minneapolis, MN)and in flexure using the same custom flex device. The esti-mated shear moduli values obtained from the three methodswere compared to ensure accuracy of both the computationaland experimental methods.
3 Results
3.1 Initial tissue material model
The resulting M–�κ relation from the analytical solutionwas normalized to the maximum achieved moment at a cur-vature of 0.28 mm−1, the maximum observed experimentallymeasured value. Although there was some variation withapplied moment, we determined that a value for α of ∼2.0captured the shape of the M–�κ (Fig. 3, r2 = 0.93). It isimportant to note that the analytical solution utilized a singlelayer and simplified boundary conditions (Rivlin 1949) and
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Fig. 4 Transmural stretch Λ1 simulation results plotted against thenormalized leaflet thickness. For measurable differences to occur intransmural stretch between the fibrosa and ventricularis, simulationsindicate the spongiosa must possess a shear modulus less than 1 kPa. TheWC bending direction simulation results are shown for sake of clarity,although the same simulation results were observed for the AC bendingdirection. The cuvature change reported is 0.2 mm−1 since the reportedresults demonstrated independence of curvature change (Appendix 2)
was thus only used to set the value of the α parameter. Basedon these results, we returned to the isotropic incompressiblehyperelastic neo-Hookean material model to simulate the AVtissue in flexure
W = μL
2(I1 − 3) − p(I3 − 1) (3)
where I1 and I3 are the first and third invariant of the leftCauchy–Green deformation tensor C = FT F, respectively,p is the Lagrange multiplier to enforce incompressibility,and L indicates the layer (L = F, S, or V ). With this modela value of μL = 45 kPa fit the data well (0.974) and wasused for the fibrosa and ventricularis layers in the unimodularsimulations.
3.2 Parametric interlayer bonding model study
As expected, the simulated transmural Λ1 distributiondemonstrated substantial sliding between the fibrosa andventricularis layers at extremely low spongiosa layer mod-uli values (Fig. 4). As the spongiosa modulus more closelymatched that of the outer layers (i.e., layers becoming morebonded), the transmural variation in Λ1 became nearly linear.Both bending directions showed an expected shift in the neu-tral axis (NA) location toward the fibrosa layer since it wasassigned a modulus stiffer than the ventricularis based on pre-vious studies. From these basic simulations, we determinedthat in order for the spongiosa to exhibit measurable differ-ences in transmural stretch from the fibrosa and ventricularis,μs must be less than 1 kPa. This estimated threshold is inde-pendent of imposed curvature as discussed in Appendix 2.
Fig. 5 The M–�κ response of a native porcine AV leaflet in the ACand WC directions that has been bent to a �κ of 0.2 mm−1 in eitherdirection. Simulations using a unimodular (dashed line) and bimod-ular material model (solid line) are superimposed over experimentaldata. The bimodular model represents the bidirectional bending behav-ior slightly better than the unimodular material
3.3 Transmural strain experimental results
The M–�κ response of AV tissue strips was observed tobe slightly nonlinear in both bending directions (Fig. 5). Itshould be noted that our previous work assumed a linearM–�κ relationship to simplify parameter estimation (Sacksand David Merryman 2009). The current approach was moreaccurate since the moduli were estimated directly from theM–�κ data (Fig. 5), including accounting for the observedsmall nonlinearity. Interestingly, as in our three-point bend-ing study (Sacks and David Merryman 2009), no signifi-cant directional differences were found in the overall flexuralresponse (Fig. 5).
However, we noted that the transmural strain measure-ments varied with bending direction. AV results revealedspecimens flexed in the AC direction exhibited a maximumΛ1 stretch that increased from 1.016 to 1.026, with a respec-tive increase in the change of curvature from 0.1 to 0.3 mm−1.The minimum Λ1 stretches decreased, indicating an increasein strain, with increasing curvature with typical values pro-gressing from 0.984 to 0.969. The average neutral planelocation for all specimens showed a shift toward the fibrosain the AC direction, but was not statistically significant(Table 2). The NA results indicated a shift toward the stifferfibrosa layer (Table 1).
3.4 Refined leaflet model
The motivation for refining the initial model was the observeddifferences between the WC and AC bending directions in thetransmural Λ1 distributions (Fig. 6). That is, the M–�κ andtransmural strain responses could not be simultaneously fit in
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Aortic valve flexure 819
Table 1 Silicone material validation study, showing E (MPa)
Specimen Model Flexure Tensile
1 5.973 5.805 5.551
2 6.501 6.558 6.557
3 6.593 6.677 6.333
4 6.749 7.591 6.403
Average 6.454 6.658 6.211
SEM 0.168 0.366 0.225
both bending directions (Fig. 6c, d). This suggests an asym-metry in the tissue layer tensile-compressive responses. Thus,to capture the bidirectional M–�κ and transmural responseof the leaflet observed in flexure, a refined FE model wascreated using a bimodular material model. Further, since theparametric interlayer bonding study and experimental val-idation indicated the leaflet tissue is a functionally bondedunit, we developed the following refined model by absorbingthe spongiosa equally into the two outer layers, resulting ina bilayer model. Geometry and initial curvature of the modelwas matched to the AV experimental specimens. The follow-
ing bimodular incompressible neo-Hookean material modelwas implemented
W ± = μ±L
2(I1 − 3) − p(I3 − 1) (4)
which has four total moduli, μ±F and μ±
F , with the sub-scripts indicating the fibrosa and ventricularis layer and thesuperscripts indicating in tension (+) or compression (−),respectively.
As in the first model, we conducted simulations in bothflexural directions and simultaneously matched to the exper-imental data. Selected nodal positions on the edge of thespecimen, representing tissue marker positions used in exper-imental configuration, were used to compute a correspondingchange in curvature for each applied moment. The momentwas plotted with respect to curvature change taken from thepoint of maximum curvature at the center of the specimen.The experimental M–�κ data was averaged and fit to the FEmodel. The fitting procedure was carried out in a two-stepprocess. First, the interrelationship of the moduli for eachbending direction was determined by matching the location
Fig. 6 Experimental transmural Λ1 results plotted against the normal-ized thickness for a, c WC and b, d AC samples at �κ = 0.2 mm−1,along with the corresponding unimodular (dashed line) and bimodu-lar (solid line) simulation results. a, b Represent strain matched in theWC direction, showing the resulting discrepancy in the AC fit using
a unimodular model. Similarly, c, d represent strain matched in theAC direction, resulting in a poor WC fit using the unimodular model.Although the M–�κ behavior is captured with unimodular model, thetransmural deformation response can only be captured in both the WCand AC directions using the bimodular material model
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Table 2 Predicted andmeasured neutral axis locations �κ(mm−1) Against curvature (AC) With curvature (WC)
of the NA to experimental data for both directions (Table 2).Secondly, the layer moduli values were determined by fittingthe M–�κ with the experimental data using a minimizationof the mean squared error (MSE) (Table 3). The MSE wasdefined as 1
n (y− y)2, where n is number of experimental datapoints and y and y are the simulation and experiment derivedM–�κ relation, respectively. It was crucial to perform thefitting procedure in this way to ensure both the transmuraldeformation and the M–�κ aligned with the experimentalmeasurements. The magnitude values reported are within±0.5 kPa, and the computed Λ1 through the thickness of thetissue is within the range experimental error (±0.0141) ofthe experimental measurements (Fig. 6).
An aspect of our approach is accounting for distortion ofa beam of rectangular cross section undergoing large strains,which is a well-known effect (Timoshenko 1953). This dis-tortion causes out-of-plane warping, and was observed exper-imentally in the AV tissue strips. In the present experimen-tal setup, out-of-plane warping (in the X3 direction) wasestimated to affect the transmural stretch measurements by∼0.02 in the net axial stretch between the center of the speci-men and the edge (Fig. 7). Appendix 2 further investigates theeffects of specimen geometry on the out-of-plane warping.Warping also affected the measured location of the neutralaxis, making it appear almost unchanged with each bendingdirection when taken from the edge of the specimen (Fig. 7b,c). This edge warping affected approximately 40 % of the tis-sue volume (20 % from each edge), leaving the interior 60 %of the tissue largely unaffected. To account for this warp-ing effect, we utilized the edge deformations to match theexperimental data, as well as reported the predicted interiordeformation responses that represent bulk tissue behavior aswould occur in vivo.
The resulting simulations indicated that Λ1 in the cen-ter of the specimen (representing the bulk of the tissue)demonstrated significant changes in NA location with bend-ing direction (Fig. 8). In the WC bending direction, the NA
shifted toward the ventricularis approximately 0.35 of thenormalized thickness. Interestingly, this indicates the wholefibrosa is under compression and the tensile load is carriedentirely by the ventricularis (Fig. 8). This shift in the NAtoward the ventricularis was reflected in the estimated mod-uli (Table 3) indicating greater tensile stiffness in the ven-tricularis than compressive stiffness in the fibrosa (a ratio ofapproximately 2:1). In the AC bending direction the NA shiftsto approximately 0.79 of the normalized thickness, indicatingthe fibrosa was much stiffer under tension than compression(Fig. 8). Again, this shift was reflected in the estimated mod-uli values, indicating greater tensile stiffness of the fibrosacompared with compressive stiffness of both the fibrosa andventricularis (ratio of approximately 4:1).
3.5 Model validation
Silicone rubber test specimens demonstrated good agreementwith tensile test shear moduli measurements. Specifically,the model estimated 6.454 ± 0.168 MPa, the experimentalflex measurement was 6.658 ± 0.366, and the tensile testsmeasuring 6.211 ± 0.225 MPa (Table 1).
4 Discussion
4.1 Overview
The present study investigated the interlayer micromechanicsof the AV leaflet undergoing flexure using an integrated sim-ulation/experimental approach. While the modeling effortwas rather straightforward, we noted the need for exam-ining both the macro-level flexural behavior (M–�κ) andthe micro-level transmural deformation, integrated into a 3Dbeam model, to accurately capture AV leaflet layer interac-tions. Our major findings were that (1) the AV leaflet layersfunction in flexure as a perfectly bonding unit, and (2) the
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Aortic valve flexure 821
F V
Location of optical strain measurement
X3
X2
X3
X2
0.08
0.040.06
0.020 - 0.02- 0.04- 0.06- 0.08
E11
(b)
(c)
(a)
Fig. 7 a Transmural Green–Lagrange strain distribution (E11) fromthe simulated flexure model demonstrating a disruption at the edgesdue to standard warping that occurs during bending. b A significantshift (2 % stretch) in axial deformation is observed by plotting Λ1 ver-
sus normalized thickness in both the WC and c AC bending directions.It is important to note the associated shift in the neutral access locationwhen observing deformation at the edge or in the unaffected centralregion of the tissue
Fig. 8 A schematicsummarizing the changes in NAlocation experienced by the bulktissue for both bendingdirections. In the WC bendingdirection, the NA shift towardthe ventricularis approximately0.35 of the normalizedthickness, demonstrating thewhole fibrosa undercompression (−) and themajority of the ventricularisunder tension (+). In the ACbending direction the NA shiftsto approximately 0.79 of thenormalized thickness. Theseresults indicate that both theventricularis and fibrosa aremuch stiffer under tension (+)than compression
AC
WC
Neutral Axis0.79+ _
_
_
+_
F: 60%
V: 40%
Neutral Axis0.35
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822 R. M. Buchanan, M. S. Sacks
spongiosa layer did not have measurably different mechan-ical properties compared with the fibrosa and ventricularis.This last point was primarily evidenced by the smooth trans-mural stretch distributions (Fig. 4). Thus, in the low-strainenvironment, the spongiosa may be described mechanicallyas contiguous extension of the two outer layers.
4.2 Bimodular material model for leaflet tissue in flexure
It is important to emphasize here that a standard neo-Hookeanmaterial, single layer model was only used in the first part ofthe study to set the basic model form and a range for the mate-rial model parameters. Interestingly in the refined model, abimodular material model for the AV leaflet tissue in flexurewas identified, similar to approaches used for other fibroustissues and part of a larger sub-class of bimodular materials(Curnier et al. 1995; Ateshian 2007). We also determinedthat both the fibrosa and ventricularis have greater stiffnessin tension than in compression (a ratio of approximately 4:1,Table 3). The fibrosa was found to be consistently stiffer thanthe ventricularis, approximately 2:1 in both tension and com-pression (Table 3). The bimodular behavior was not entirelysurprising based on the unique arrangement of ECM compo-nents throughout the three layers. Moreover, the transmuraldeformation of the AV leaflet tissue indicated that the NAshifted closer to the ventricularis in the WC direction at allcurvature levels. This implied that when the ventricularis wasunder mainly tensile loading in the circumferential directionand the fibrosa was fully under compression, the ventricu-laris provided most of the stiffness (Fig. 8). When the valveflexed in the opposite direction (AC), the NA shifted closerto the fibrosa, leaving the entire ventricularis and one third ofthe fibrosa under compressive loading in the circumferentialdirection, and the remaining two thirds of the fibrosa undertension, providing most of the stiffness (Fig. 8). Structurally,this behavior was likely due to the greater concentration of thecomparatively stiff type I collagen within the fibrosa and thegreater concentration of the more compliant elastin found inthe ventricularis. Previous studies have shown that the elastinin the ventricularis forms a honeycomb network around thecollagen fibers, allowing the fibers to stretch and return totheir initial state (Scott and Vesely 1995; Vesely 1998). Thecollagen fibers in the fibrosa do not possess an extensiveor organized elastin network and are bound together moretightly (Schoen and Levy 1999). Additionally, proteoglycansintimately bind to collagen fibers and may provide signifi-cant reinforcement to the fibrosa in doing so, compared withthe collagen-depleted ventricularis.
4.3 Implications for AV micromechanical function
Functionally, the leaflet’s ability to present low flexural stiff-ness during valve opening (WC) and high stiffness during
closure (AC) is only attainable through its distinctive hierar-chical structure in which all elements act in unison to provideseamless transition from the diastole to systole cycle. Thepresent study indicated that, contrary to previous hypothe-ses (Mohri et al. 1972; Vesely and Boughner 1989; Songet al. 1990; Thubrikar 1990; Talman and Boughner 1995),the spongiosa did not allow the fibrosa and ventricularis lay-ers to slide with respect to one another. If this did occur(i.e., when functioning as loosely bonded layers), each layerwould have its own neutral axis (Fig. 4), greatly compli-cating intra-layer deformation patterns. Instead, the leafletsappear to function as a fully bonded unit. These findingsare also consistent with our observations (Stella and Sacks2007) of the presence of a large number of collagen fiberinterconnections between the fibrosa and ventricularis lay-ers. While recent work by Tseng and Grande-Allen (2012)shows intriguing results of elastin presence in the spongiosalayer, the nature of the fibrous interconnections between thelayers at a gross level appears to be dominated by collagenousfibers. However, the exact composition and structure of theseinterconnecting fibers remains unknown. More importantly,the composition of these fibers has no effect on the findingsof the current study, since what is important is that they existand appear to bind the fibrosa and ventricularis layers.
The question remains: what is the mechanical role, ifany, of the spongiosa layer? In a recent study, we notedthat viscoelastic behavior of heart valve tissues only mani-fested themselves in the low-strain region (Eckert et al. 2013).Also, other studies have demonstrated a functionally elasticbehavior for native valvular tissues in tension (Grashow et al.2006a,b; Stella et al. 2007). Yet, it can be speculated that thecentrally located spongiosa layer functions to reduce high-frequency motions of the leaflet during the opening/closingphases. Such dampening-like behavior might be a mecha-nism to reduce hemodynamic energy loss during the open-ing and closing phases. Current information on this aspectof valve function is scant and must be the subject of futureinvestigations.
4.4 Limitations and the need for an integratedexperimental/simulation approach
In actual heart valves, flexure occurs during the open-ing/closing phases and is bidirectional (Iyengar et al. 2001;Sugimoto and Sacks 2013). Since it is not currently possibleto experimentally investigate this behavior directly, we choseinstead to approximate this behavior with a strip model incircumferential bending (Sugimoto and Sacks 2013). Whilenot accounting for bending in both directions, this is also theprimary fiber direction and any measured behavior shouldapproximate the native tissue response in vivo. It should alsobe noted that Vesely and Boughner (1989) investigated theflexural behavior of native AV leaflets and found that the NA
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Aortic valve flexure 823
was close to the fibrosa surface under AC flexure, suggest-ing a very low compressive modulus compared with tensilein agreement with the present study. However, testing wasperformed under unrealistic loading conditions, making itdifficult to correlate with in vivo function. In the presentapproach, an end-loading configuration was used, whichshould be more representative of the in vivo responses ofthe bulk leaflet tissue since it avoided any artificial contactpoints. We also note that the measured transmural deforma-tion data must be matched with the computational model atthe edge of the specimen and then reported from the centralregion in order to represent the bulk tissue behavior accu-rately.
5 Conclusions
In summary, the behavior of AV leaflet tissue as a compositebeam was determined by examining the M–�κ relationshipand the stretch along the edge of the tissue during bending.We conclude that while it has been previously speculatedthat valve layers slide with respect to one another duringvalve opening/closing, our evidence suggests that the leafletlayers function as a single bonded unit. Layers appear to bebonded by transverse collagen fibers that hold residual strainsin place. Thus, despite a heterogeneous structure and dif-ferences in stress distribution throughout the layers, the AVleaflet deforms as a homogenous structure. Furthermore, thevalve tissue in flexure requires a bimodular material model.Further studies are needed to investigate the leaflet microme-chanics in activated cellular conditions that represent healthyand diseased physiological conditions.
Acknowledgments This research was supported by NIH Grants HL-068816, HL-089750, HL-070969, and HL-108330. The authors wouldlike to thank Thanh V. Lam for the development of the flexure-testingdevice and Brett Zubiate for the later improvements made to the trans-mural strain system. Also, recognition goes to Kristen Feaver for hercontribution of the bimodular schematic (Fig. 8).
Appendix 1
Transmural deformation analysis
The images captured using the micro-imaging system (Fig.2d) were analyzed by locating the markers that had been air-brushed onto the edge of the tissue. A custom program waswritten in LabVIEW (National Instruments, Austin, TX) topost-process the images taken by the micro-imaging systemso that displacement fields could be determined. The mark-ers were identified, numbered, and their areas and centroidcoordinates were determined. This procedure was performedsimultaneously for the reference image and for the deformed
image. The software then displayed both altered imagesconcurrently so that the user of the program could matchmarkers between the reference image and the deformedimage.
From the resulting images, the coordinates of the refer-ence markers were referred to as the (X1, X2) system, andthe deformed coordinates were referred to as the (x1, x2) sys-tem. The displacements, u and v, were calculated from theformer using u = x1 − X1 and v = x2 − X2, respectively.These quantities were then fitted to the surface described byEq. (5).
The surface fit to the u and v coordinates achieved an r2
value of approximately 0.9. A higher order fit could havebeen used resulting in a higher r2, this would produce a roughsurface due to variations in marker location from threshold-ing. The lower order fit maintains a smooth surface, true tothe nature of the sample tested, by not overfitting the curve toall variations in marker location. By evaluating Eq. (6), thedeformation gradient was obtained.
F = H + I =[
∂u∂ X1
∂u∂ X2
∂v∂ X1
∂v∂ X2
]+
[1 00 1
]=
[λ1 k1
k2 λ2
](6)
The F was then decomposed into its stretch and rotation ten-sors, U and R, respectively, Eq. (7). The polar decompositionof the deformation tensor removes rigid body motion effectsinto the rotation tensor, leaving only the stretch deforma-tion information in the stretch tensor. The rigid body rota-tion information in R was calculated to determine the degreeof rotation experienced by the tissue during flexure. Higherlevels of rigid body rotation were determined to be coinci-dent with measurements taken away from the center of thetissue.
F = RU
U2 = FTF
R = FU−1 (7)
The stretch tensor components U11 and U22 correspond tolocal tissue strains in the X1 and X2 directions, respectively.Thus, the location of the neutral axis was determined byplotting U11 against the thickness of the tissue. The depthof the tissue that coincided with the U11 value of unity wasthe corresponding location of the neutral axis. Rotation thatoccurred in the displacement field was characterized by deter-mining the angle of rotation, α, incurred in the deformedsystem from the reference state.
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824 R. M. Buchanan, M. S. Sacks
Appendix 2
Parametric interlayer bonding study supplement
The reported results for the parametric bonding study rep-resented a curvature change of 0.2 mm−1 solely for sake ofclarity, as the same trends were observed at curvature changesof 0.1, 0.2 and 0.3 mm−1 (Fig. 9a–c). These choices of cur-vature change were taken from our in vitro measurements(Sugimoto and Sacks 2013). Not surprisingly, we noted thatthe presence of interlayer sliding (not magnitude) either sim-ulated (Fig. 9) or experimentally (Fig. 10) was not a functionof the level of bending (i.e., �κ), but only of the ratio ofthe ventricularis and fibrosa:spongiosa moduli for the simu-lations. Thus, the estimated μS threshold is independent ofimposed curvature. Greater bending simply created greatersliding.
Parametric out-of-plane warping study supplement
To investigate the effects of leaflet geometry on the sim-ulation findings of out-of-plane warping effects, a para-
Fig. 10 Experimental results obtained from transmural bending testsperformed on native aortic valve tissue. The tissue was bent to threedifferent changes of curvature, 0.1, 0.2, and 0.3 mm−1. As curvatureincreased, the deformation increased as expected, yet no sliding isobserved
metric simulation was performed varying the thickness ofthe leaflets as well as the curvature change. The specimenlength and width remained constant for the simulations.
= 0.1 mm-1 = 0.2 mm-1
= 0.3 mm-1
Fig. 9 Transmural deformation results of parametric bonding simula-tion,Λ1 is plotted against the normalized leaflet thickness for a curvaturechange of 0.1, 0.2, and 0.3 mm−1. A tri-layered rectangular strip rep-resented the AV and shear moduli values ranging from 1.0 Pa to 45 kPawere assigned to the central spongiosa layer to emulate varying degrees
of connectivity between the outside layers and identify the followingrelationship: μF = μV : μS . Results indicate for all curvature levelsthat for measurable sliding to occur between the fibrosa and ventricu-laris, the spongiosa must possess a shear modulus less than 1 kPa
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Aortic valve flexure 825
Fig. 11 The effect of leaflet thickness on the out-of-plane warping esti-mated by the simulation (Fig. 7). The degree of warping, measured bythe absolute change in net axial stretch (Λ1), increases with increas-ing leaflet thickness. Additionally, this relationship is maintained andexaggerated with increasing curvature (0.1, 0.2 and 0.3 mm−1)
Figure 7 demonstrates the significant change in net axialstretch between the center of the specimen and the edge.Therefore, this change in absolute axial stretch (Λ1) wasused as a metric of warping and plotted against the change inspecimen thickness (Fig. 11). Results found that increasingthickness of the specimens exaggerated the degree of warp-ing. Furthermore, as expected, this warping effect increaseswith increasing curvature change.
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