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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Biomech. Author manuscript; available in PMC 2010 August 25.
Published in final edited form as:
PMCID: PMC2725213

Stress-Strain Behavior of Mitral Valve Leaflets in the Beating Ovine Heart


Excised anterior mitral leaflets exhibit anisotropic, nonlinear material behavior with pre-transitional stiffness ranging from 0.06-0.09 N/mm2 and post-transitional stiffness from 2-9 N/mm2. We used inverse finite element (FE) analysis to test, for the first time, whether the anterior mitral leaflet (AML), in vivo, exhibits similar non-linear behavior during isovolumic relaxation (IVR). Miniature radiopaque markers were sewn to the mitral annulus, AML, and papillary muscles in 8 sheep. 4-D marker coordinates were obtained using biplane videofluoroscopic imaging during three consecutive cardiac cycles. A FE model of the AML was developed using marker coordinates at the end of isovolumic relaxation (when pressure difference across the valve is approximately zero), as the reference state. AML displacements were simulated during IVR using measured left ventricular and atrial pressures. AML elastic moduli in the radial and circumferential directions were obtained for each heartbeat by inverse FEA, minimizing the difference between simulated and measured displacements. Stress-strain curves for each beat were obtained from the FE model at incrementally-increasing transmitral pressure intervals during IVR. Linear regression of 24 individual stress-strain curves (8 hearts, 3 beats each) yielded a mean (±SD) linear correlation coefficient (r2) of 0.994±0.003 for the circumferential direction and 0.995±0.003 for the radial direction. Thus, unlike isolated leaflets, the AML, in vivo, operates linearly over a physiologic range of pressures in the closed mitral valve.

Keywords: mitral valve, finite element analysis, material properties, anisotropy, elastic modulus


The mitral valve has a dual role during left ventricular (LV) systole in the beating heart. The mitral leaflets, the key components of the valve, must maintain appropriate relative positions and geometry during systole to: (1) seal tightly to prevent regurgitant backflow from the LV into the left atrium; and (2) provide an appropriately-shaped portion of the LV outflow tract. The material properties of the leaflets are of critical importance to both of these tasks, as the varying demands on the heart are met with a wide range of LV pressures and volumes.

The material properties of excised mitral leaflets have been well-characterized (Kunzelman and Cochran, 1992; May-Newman and Yin, 1995; May-Newman and Yin, 1998; Chen, McCulloch et al., 2004; Chen, Yin et al., 2004; He, Sacks et al, 2003; Sacks, He et al, 2002; Sacks, Enomoto et al., 2006; He and Ritchie et al, 2005). In uniaxial studies of excised leaflets, Kunzelman and Cochran (1992) measured highly non-linear stress-strain behavior with distinct pre- and post-transitional regions. In biaxial studies of excised anterior leaflets, May-Newman and Yin (1995) found anisotropic, non-linear material properties with pre-transitional stiffness values ranging from 0.06-0.09 N/mm2 and post-transitional stiffness values from 2-9 N/mm2. Recent mitral valve models (Kunzelman, Einstein et al., 2007; Prot, Skallerud et al., 2007; Votta, Caiani et al., 2008) have incorporated these leaflet data into hyperelastic finite element analyses (FEA). Sacks and He (2002) used a left heart simulator and graphite markers to study the in-vitro surface strains in the porcine anterior mitral valve leaflets and showed a non-linear relationship between transmitral pressures and leaflet areal strains. The first attempt at quantifying in-vivo leaflet strains by Sacks and Enomoto (2006) using a sonmicrometry transducer array showed the same non-linear relationship between pressure and leaflet strains in ovine anterior mitral valve leaflets. Sacks and Enomoto (2006) computed only in vivo leaflet strains and did not determine the in vivo leaflet elastic moduli or show the relationship between leaflet stresses and leaflet strains for the beating heart.

Recently, we used inverse FEA to obtain the anisotropic elastic moduli of anterior mitral valve leaflets during isovolumic relaxation (IVR) in the beating ovine heart (Krishnamurthy, Ennis et al., 2008). As a first approximation, we assumed a linear relationship between stress and strain during IVR. The present study was undertaken to the test the validity of this assumption.


All animals received humane care in compliance with the “Principles of Laboratory Animal Care” formulated by the National Society for Medical Research and also in compliance with the “Guide for the Care and Use of Laboratory Animals” prepared by the National Academy of Sciences and published by the National Institutes of Health (U.S. Department of Health and Human Services, NIH Publication 85-23, Revised 1985). This study was approved by the Stanford Medical Center Laboratory Research Animal Preview Committee, which is accredited by the Association for Assessment and Accreditation of Laboratory Animal Care International, and conducted according to Stanford University policy.

Surgical Preparation

Eight adult, Dorsett-hybrid, male sheep (56±8kg) were premedicated with ketamine (25 mg/kg intramuscularly) for venous and arterial catheter placement and monitoring. Anesthesia was induced and maintained with inhalational isoflurane (1-2.5%) and supplemental oxygen. Through a left thoracotomy, 13 miniature tantalum radiopaque markers were implanted in the left ventricle (LV) subepicardial wall silhouetting the LV chamber (Figure 1A). Via a left atriotomy with cardiopulmonary bypass and antegrade cardioplegic arrest, a total of 35 radiopaque tantalum markers were sewn to the following sites: 16 on the atrial aspect of the anterior mitral leaflet (AML) [7 on the AML edge (#1-7, Figure 1B); 9 on the leaflet belly (#8-16, Figure 1B], 16 around the mitral annulus (Figure 1A), 1 on the central edge of the middle scallop of the posterior mitral leaflet (PML, Figure 1A), and 2 on the anterolateral and posteromedial papillary muscle tips (APM, PPM, Figure 1A). A micromanometer pressure transducer (PA4.5-X6, Konigsberg instruments, Inc., Pasadena, CA, USA) was placed in the LV chamber through the left atrium (LA) and exteriorized.

Marker Schematic

Data Acquisition

Immediately after the operation, the animals were transferred to the catheterization laboratory and studied in the right lateral decubitus position with the chest open. Two micromanometer-tipped pressure transducers (model MPC-500; Millar Instruments, Houston, TX, USA) were calibrated and inserted into the LV and ascending aorta via a carotid artery catheter, respectively. A Konigsberg pressure transducer was calibrated against the two Millar pressure transducers while all transducers were in the LV, then pulled back into the LA to record left atrial pressure (LAP). Simultaneous biplane videofluoroscopic images (60 Hz, Philips Medical Systems, Pleasanton, CA, USA), ECG, LV pressure (LVP), aortic pressure and LAP were recorded during a hemodynamically stable interval with the heart in normal sinus rhythm and ventilation transiently arrested at end-expiration. At the completion of each study biplane images of a 3D helical phantom of known dimensions spanning the heart space were recorded. The 2D coordinates of each marker in each projection image were digitized frame-by-frame, using semi-automated image processing and digitization software developed in our laboratory (Niczyporuk and Miller, 1991). Data from the two views were merged using the 3D helical phantom image data and software, described previously, and used to yield the 3D marker coordinates (Daughters, Sanders et al., 1989). The accuracy of the 3D reconstructions from biplane videograms of length measurements was previously shown to be 0.1 ± 0.3 mm (Daughters, Sanders et al., 1989).

Hemodynamics and Cardiac Cycle Timing

Three consecutive beats in sinus rhythm were selected for analysis from each heart. For each beat, end-systole (ES) was defined as the frame containing the minimum second derivative of LVP with respect to time during IVR. Negative dP/dtmax was computed as the maximum time derivative of LVP during IVR. The onset of isovolumic relaxation (IVR1, Figure 2) was defined at ES and the end of isovolumic relaxation (IVR2, Figure 2) as the frame immediately before mitral valve opening, defined as the earliest increase (above the systolic variation) in the separation between the central anterior and posterior leaflet edge markers.

Lvp, Lap and Time-Steps

To study the piecewise stress-strain behavior of the anterior mitral leaflet during IVR, for each beat, five frames were selected to span the IVR pressure range, defining four approximately 15 mmHg LVP increments associated with four successive time-intervals (Δt1-Δt4) from IVR2 to IVR1 (Figure 2). For the 24 beats analyzed, for time-step Δt1 the group mean (±SD) transmitral pressure gradient (LVP-LAP) ranged from 0 (IVR2) to 14±1 mmHg; for time-step Δt2 from 14±1 to 30±5 mmHg; for time-step Δt3 from 30±5 to 49±6 mmHg; and for time-step Δt4 from 49±6 to 63±6 mmHg.

Inverse Finite Element Analysis

The inverse finite element analysis methodology to determine the material properties of the anterior mitral valve leaflet has been described in a previous publication (Krishnamurthy, Ennis et al., 2008), thus will only be outlined here.

Finite Element Model

A finite element model of the anterior MV leaflet was developed for each individual time-step (Δt1, Δt2 ,Δt3, Δt4) and for each beat using Hypermesh version 8.0 SR 1 (Altair Hyperworks; Troy, Michigan) to construct the geometry and meshing of the leaflet and Optistruct version 8.0 SR 1 (Altair; Troy, Michigan) as the solver. Thus 96 individual finite element models (8 hearts, 3 beats/heart, 4 time-steps/beat) were analyzed for this study.

For each beat, the geometry of the anterior leaflet was initially defined by the leaflet marker positions (Figure 1) at IVR2 (assumed as the minimum-stress reference state). The x, y, z coordinate values for each of the leaflet and annular marker positions at IVR2 were entered as points in Hypermesh. Five cubic splines were generated through (see Figure 1B): a) Markers 17-1-2-3-4-5-6-7-23; b) Markers 18-8-9-10-11-12-22; c) Markers 19-13-14-15-21; d) Markers 19-16-21; and e) Markers 19-20-21. These splines were used to generate a bicubic leaflet surface.

For the purpose of defining the MV leaflet material properties for Δt1, a coordinate system was defined with origin at the center of the 16 markers defining the saddle-shaped annulus (Levine, Handschumacher et al., 1989) at IVR2. A line from the origin to marker #20 (the “saddlehorn”) was defined as the leaflet radial axis (R, Figure 1). The leaflet circumferential axis (C, Figure 1) was defined normal to R and in the plane containing R and the posterior commissural marker (#23, Figure 1).

A homogeneous leaflet was assumed, with an orthotropic linear elastic material model (MAT8 in Hypermesh). The bicubic surface fit of the MV leaflet was then meshed with plane-stress quadrilateral shell elements. A typical anterior leaflet was meshed with 2200 elements yielding an element size of 0.004 cm2.

The strut chordae were defined as structures undergoing pure tension (MAT1 in Hypermesh). A previously published ex vivo modulus (elastic modulus = 20 N/mm2; cross sectional area = 0.008 cm2 (Kunzelman and Cochran, 1992)) was used for the strut chordae. Tension-only bar elements (PBARL in Hypermesh) were defined as radiating from the papillary muscle tip marker points (APM & PPM, Figure 1A) to leaflet belly insertion positions (Figure 3) defined from anatomical photographs.

Fea Model

The boundary conditions were then enforced on the finite element model. The measured transmitral pressure gradient (LVP-LAP) for the first time-step (Δt1) was applied to the anterior mitral leaflet. The displacements of the annular markers (#17-23, Figure 1B), anterior leaflet edge markers (#1-7, Figure 1B) and papillary tip markers (APM & PPM, Figure 1A) were defined using actual marker data at the end of Δt1.

The Hypermesh finite element model (Figure 3) was then solved for the enforced boundary conditions using Optistruct to obtain the simulated displacements of the leaflet belly markers (#8-16, Figure 1B). This initial run assumed nominal anterior leaflet material properties obtained from previous ex vivo studies (Kunzelman and Cochran, 1992).

Inverse Finite Element Analysis Algorithm

The Optistruct solver was then interlinked with commercial optimization software, Hyperstudy version 8.0 SR 1 (Altair Hyperworks, Troy, Michigan) to run an inverse finite element analysis to yield the in vivo material properties of the mitral valve during Δt1. In this algorithm, the model-simulated displacements of the nine leaflet belly markers (#8-16, Figure 1B) from the nominal run were compared with the actual measured displacements of these 9 markers during time-step Δt1 to yield a response function defined as the root mean squared (RMS) displacement difference between measured and actual displacements of the nine leaflet belly markers. Hyperstudy then used a parameter identification algorithm, the “Method of Feasible Directions” (Belegundu, Damle et al., 2004), to minimize the response function by repeated iterations of the material properties (Ecirc, Erad) in the finite element model until a global minimum was obtained (Figure 4). Leaflet circumferential-radial shear during IVR proved sufficiently small that Ecirc and Erad values so obtained were unchanged with inclusion or exclusion of this shear in the parameter identification process. The material property values (Ecirc, Erad) obtained at the end of the material identification run with the response function at its global minimum were interpreted as the in vivo material properties of the anterior MV leaflet belly during Δt1. That is, these material property values, when used in the finite element model for the anterior leaflet belly under the enforced pressure and geometric boundary conditions, produced, as closely as possible, the same displacements of the 9 leaflet belly markers as those measured experimentally during time-step Δt1.

Material Parameter Identification

Forward Analysis and Stress-Strain curve

The computed in vivo material properties were then used in the finite element model, to determine the stress and strain in the circumferential and radial directions for Δt1 at marker #14 (as a representative case).

Using marker coordinates at the end of the previous time-step to build the finite element model for the next time-step, a similar inverse finite element analysis was employed to determine the material properties for each time-step during IVR, and forward analysis to determine the stresses and strains were performed for time-steps Δt2, Δt3 and Δt4. The circumferential and radial stresses at marker #14 were plotted against the corresponding circumferential and radial strains at marker #14 for successive time-steps to construct piece-wise circumferential and radial stress-strain curves for each beat.

The linearity of the stress-strain curve for each beat was characterized by the correlation coefficient (r2) associated with a linear regression analysis of each curve.


Table 1 displays the group mean heart rate, -dP/dtmax during IVR, and left ventricular and left atrial pressures for each heart at IVR1 and IVR2. Variations in these parameters are seen from heart-to-heart, showing that the stress-strain curves represent a variety of hemodynamic conditions.


Table 2 gives the values of the circumferential and radial stresses and strains for the four transmitral pressure intervals associated with each of the 3 beats for the 8 hearts studied along with the r2 correlation coefficient values for each curve. Figure 5 displays these data as stress-strain plots, with each panel displaying the circumferential and radial stress-strain relations for each of the three beats from each heart. The group mean (±SD) stress-strain linear correlation coefficient (r2) values were 0.995±0.003 for the circumferential curves and 0.994±0.003 for the radial curves. The beat-to-beat reproducibility of the stress-strain curves in each heart (Figure 5) validates the robustness of this method to study the stress-strain behavior of the anterior mitral leaflet during IVR. Consistent with results from ex vivo testing (May-Newman and Yin, 1995), in vivo radial strains are higher than circumferential strains at all time-steps.

Stress- Strain Curves
Raw Stress and Strain Values


This study introduces a novel methodology using a combination of inverse and forward finite element analysis for the piece wise construction of stress-strain curves of the mitral valve leaflets, in vivo. Sacks and Enomoto (2006) reported, for the first time, the in vivo anterior leaflet strains, but the relationship between in vivo leaflet stresses and in vivo leaflet strains has not been reported thus far. This is the first report of the stress-strain behavior of the anterior mitral valve leaflet in the beating heart. The key finding of the current study is that both circumferential and radial stress-strain curves are linear over a physiologic range of pressures, in the closed mitral valve. This finding validates the material linearity assumption made in our earlier study (Krishnamurthy, Ennis et al., 2008) that determined the in vivo material properties of the anterior mitral leaflet during IVR.

The question may arise that because we made use of a linear elastic material model to quantify the material parameters, the overall stress-strain curves during IVR had to be linear. This is not the case. Each time-step analysis was independent. Thus, while each time step yields a modulus based on a linear material model for that time step, the next time-step could yield a completely different modulus. Thus there is no requirement for the combined time-steps to exhibit linear behavior. If the relationship was truly nonlinear, this approach used here should have detected this nonlinearity. Of interest, a prior study (Sacks, Enomoto et al., 2006) also demonstrated linear material behavior of the ovine anterior mitral leaflet in the closed valve, although their results are not directly comparable with the results of the present study as they plotted areal strain against left ventricular pressure,.

Next, we should comment on our choice of a minimum-stress, zero-strain reference state at IVR2 (Figure 2). First, to the best of our knowledge, it is impossible, using any technique currently available, to measure the loading on each side of the leaflet in the open valve. Moreover, leaflet shape in the open valve varies dramatically as it responds to complex blood flow patterns. Thus, we cannot perform a stress-strain analysis of the leaflet during diastole and thereby cannot establish whether a truly stress-free state exists at any time in an open valve. Thus the current inverse finite element analysis methodology is applicable only to the closed mitral valve, when transleaflet pressures can be defined, and cannot be applied to the open mitral valve when transleaflet pressures are undefined and large geometric non-linearities may be present as observed in in vitro studies (He, Sacks et al, 2003; Sacks, He et al, 2002). Second, in order to determine leaflet material properties using inverse FEA, we need to measure transleaflet pressures (both left atrial and left ventricular) over the widest possible range. At IVR2, left atrial and left ventricular pressures are virtually equal (Figure 2), yet the valve is still closed, thus the conditions at IVR2 provide the nearest possible approximation to an unloaded state. Further, the interval between IVR1 and IVR2 provides the full physiological range of pressures encountered by the closed valve, allowing material properties to be estimated over this wide pressure range. Finally, it is possible, but difficult to imagine, that these thin membranous leaflets, have significant residual stress, relative to the stresses encountered when they are supporting left ventricular pressures, although this possibility remains to be determined. For these reasons, then, we think that the choice of IVR2 is appropriate as a minimum-stress minimum-strain reference configuration in FEA studies of mitral valve leaflet dynamics.

It must be noted that the leaflet edge has inconsistent deformations and mechanics. Taking this into account, we have reported stresses and strains only for the leaflet belly (marker 14, Figure 1), as the belly region of the leaflet is homogeneous and has consistent deformations. In the FEA model, we incorporate the actual motion of the leaflet edge as seen in our experiments in order to enforce accurate boundary conditions to the model, but understanding that the edge behavior is inconsistent we report only the stresses and strains at the center of the leaflet.

Finally, it has to be noted that leaflets are known to be heterogeneous with different regions of the leaflet having different material properties. Strut chordal insertions into the leaflet introduce material heterogeneities in the leaflet (Chen, Yin et al., 2004). Scanning acoustic microscopy indicates that human anterior leaflets are stiffer in the fibrous middle layer than atrial and ventricular layers and the entire leaflet is stiffer at the annulus than at the free edge (Jensen, Baandrup et al., 2006). Leaflet homogeneity was a simplification for this initial effort to quantify the in vivo material behavior of the anterior mitral valve leaflet. Developing a heterogeneous finite element model is an ongoing effort in our group.


The authors gratefully acknowledge the expert technical assistance of Sigurd Hartnett, BS, Maggie Brophy, AS, Eleazar Briones, BA, Lauren Davis, BS, and George T. Daughters, MS. This study was supported by National Heart, Lung and Blood Institute Grants HL-29589 and HL-67025. Gaurav Krishnamurthy was supported by the BioX Medtronic Graduate Fellowship, Wolfgang Bothe was supported by the Deutsche Herzstiftung, Frankfurt, Germany, and Julia C. Swanson was supported by the Western States Affiliate AHA Postdoctoral Fellowship.


Conflicts of Interest None of the authors have any conflicts of interest.

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