Anterior mitral leaflet stiffness during isovolumic contraction (IVC) is much greater than that during isovolumic relaxation (IVR). We have hypothesized that this stiffening is due to transient early systolic force development in the slip of cardiac myocytes in the annular third of the anterior leaflet. Because the atrium is excited before IVC and leaflet myocytes contract for ≤250 ms, this hypothesis predicts that IVC leaflet stiffness will drop to near-IVR values in the latter half of ventricular systole. We tested this prediction using radiopaque markers and inverse finite element analysis of 30 beats in 10 ovine hearts. For each beat, circumferential (Ec) and radial (Er) stiffness was determined during IVC (Δt1), end IVC to midsystole (Δt2), midsystole to IVR onset (Δt3), and IVR (Δt4). Group mean stiffness (Ec ± SD; Er ± SD; in N/mm2) during Δt1 (44 ± 16; 15 ± 4) was 1.6–1.7 times that during Δt4 (28 ± 11; 9 ± 3); Δt2 stiffness (39 ± 15; 14 ± 4) was 1.3–1.5 times that of Δt4, but Δt3 stiffness (32 ± 12; 11 ± 3) was only 1.1–1.2 times that of Δt4. The stiffness drop during Δt3 supports the hypothesis that anterior leaflet stiffening during IVC arises primarily from transient force development in leaflet cardiac myocytes, with stiffness reduced as this leaflet muscle relaxes in the latter half of ventricular systole.
- finite element analysis
- material properties
- elastic modulus
the anterior leaflet of the mitral valve (MV) has been shown to be an active structure rather than a simple passive flap (4, 7). We have previously reported that anterior leaflet stiffness varies actively over the cardiac cycle, with stiffness during isovolumic contraction (IVC) being 1.5–1.7 times greater than during isovolumic relaxation (IVR). In this previous report, we hypothesized that this higher stiffness during IVC was most likely due to transient force development in the slip of cardiac myocytes in the annular third of the anterior leaflet. Because the atrium is excited before IVC and cardiac myocytes contract for 250 ms or less (3), this hypothesis predicts that IVC leaflet stiffness will drop to near-IVR values in the latter half of ventricular systole. As our previous analysis focused on material properties only during IVC and IVR, we did not test this prediction at that time. In the present study we extend this previous analysis, using the same radiopaque marker dataset from the previous study (4), to test the prediction that IVC leaflet stiffness drops to near-IVR values in the latter half of ventricular systole.
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 1996). 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.
Data acquisition methods have been presented in detail in our previous report (4) and thus will only be summarized here. Thirteen radiopaque markers were surgically implanted to silhouette the left ventricular chamber in each heart (ventricular markers, Fig. 1A), one marker at the tip of the anterior (APM) and posterior (PPM) papillary muscle (Fig. 1A), 16 around the mitral annulus (annular markers, Fig. 1A), 16 on the atrial aspect of the anterior MV leaflet [7 on the MV leaflet edge (No. 1–7, Fig. 1B); 9 on the leaflet belly (No. 8–16, Fig. 1B)], and one on the central edge of the middle scallop of the posterior mitral leaflet (Fig. 1A). In the catheterization laboratory, under open-chest conditions, videofluoroscopic images (60 frames/s) of all markers were acquired with the heart in normal sinus rhythm and ventilation transiently arrested at end expiration. Left atrial pressure (LAP), left ventricular pressure (LVP), and aortic pressure were continuously measured by catheter-tip manometers. Marker coordinates from each view were then merged to yield the three-dimensional coordinates of the centroid of each marker in each frame (2, 10).
Three consecutive beats in sinus rhythm were selected for analysis in each of the 10 hearts studied. For each of these beats, the valve closure time point (IVC1, Fig. 2) was defined as the frame when the distance between marker 4 on the anterior leaflet and its counterpart on the central posterior leaflet initially fell to a stable minimum; left ventricular and LAPs at that time were noted (LVP1, LAP1). The end IVC time point (IVC2, Fig. 2) was defined from the aortic pressure deflection, and LVPs and LAPs at that time were noted (LVP2, LAP2). For each beat, end systole was defined as the frame containing the minimum second derivative of LVP with respect to time. Midsystole (Fig. 2) was defined at the frame nearest midway between end diastole and end systole. The frame during IVR when LVP most closely matched LVP2 was defined as IVR1 (Fig. 2). The frame during IVR when LVP most closely matched LVP1 was defined as IVR2 (Fig. 2).
Material property identification using inverse finite element analysis was performed over the four time intervals shown in Fig. 2: Δt1 from IVC1 to IVC2; Δt2 from IVC2 to midsystole; Δt3 from midsystole to IVR1; and Δt4 from IVR1 to IVR2.
Inverse Finite Element Analysis
The inverse finite element analysis methodology to determine the material properties of the anterior MV leaflet has been described in previous publications (5, 6) and thus will only be outlined here. A model validation can be found in Krishnamurthy et al. (5) where experimentally determined leaflet displacements were very accurately predicted [group mean displacement residuals of 0.39 ± 0.17 (SD) mm] when the model used in the present study was applied to data from 51 beats in 17 ovine hearts.
Finite element model.
A finite element model of the anterior MV leaflet was developed for each individual interval (Δt1, Δt2, Δt3, and Δt4) and for each beat using Hypermesh version 8.0 SR 1 (Altair Hyperworks; Troy, MI) to construct the geometry and meshing of the leaflet and Optistruct version 8.0 SR 1 (Altair Hyperworks) as the solver. Thus 120 individual finite element models (10 hearts, 3 beats/heart, and 4 intervals/beat) were analyzed for this study.
For each beat, for interval Δt1, the geometry of the anterior leaflet was initially defined by the leaflet marker positions (Fig. 1) at IVC1. The x-, y-, and z-coordinate values for each of the leaflet and annular marker positions at IVC1 were entered as points in Hypermesh. Five cubic splines were generated through (see Fig. 1B) 1) markers 17-1-2-3-4-5-6-7-23, 2) markers 18-8-9-10-11-12-22, 3) markers 19-13-14-15-21, 4) markers 19-16-21, and 5) 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 the origin at the center of the 16 markers defining the saddle-shaped annulus (9) at IVR2. A line from the origin to marker 20 (the “saddlehorn”) was defined as the leaflet radial axis (R, Fig. 1). The leaflet circumferential axis (C, Fig. 1) was defined normal to R and in the plane containing R and the posterior commissural marker (No. 23, Fig. 1).
Two separate shells (PSHELL in Hypermesh) were used to define the varying thickness of the leaflet regions using data obtained from our histological study of an anterior leaflet from a representative ovine heart. The first shell defined a region radiating from the annulus saddlehorn marker (No. 20) to 75% of the leaflet toward the free edge; this region had thickness values that varied linearly from 1.2 mm at the annulus to 0.7 mm at 75% toward the leaflet free edge. The second shell defined the remaining 25% of the leaflet with a uniform thickness value of 0.2 mm. The measured thickness change from 0.2 to 0.7 mm at the 75% boundary reflects the presence of strut chords that insert at that boundary.
A homogeneous leaflet was assumed, with an orthotropic linear elastic material model (MAT8 in Hypermesh) with elastic moduli in the circumferential direction (Ec) and in the radial direction (Er). Our linear elastic assumption is justified not only by the small pressure gradients (and leaflet displacements) in the present study but also by Krishnamurthy et al. (6) where a linear relationship was observed between stress and strain (linear correlation coefficient, r2 = 0.99) over the physiological range of LVPs in eight ovine hearts. The bicubic surface fit of the MV leaflet was then meshed with plane-stress quadrilateral shell elements. A typical anterior leaflet was meshed with 2,200 elements, yielding an element size of 0.004 cm2. This mesh size yielded a total computational time of 1.5 min for each material parameter identification run, allowing experimental displacements to be matched with <5% error for the final material properties obtained.
Strut chordae were defined as structures undergoing pure tension (MAT1 in Hypermesh) with a previously published ex vivo elastic modulus of 20 N/mm2 and cross-sectional area of 0.008 cm2 (8). Struts were defined as tension-only bar elements (PBARL in Hypermesh) radiating from the papillary muscle tip marker points (APM and PPM, Fig. 1A) to leaflet belly insertion positions defined from anatomical photographs.
The boundary conditions were then enforced on the finite element model. The measured transmitral pressure gradient (LVP-LAP) for the first interval (Δt1) was applied to the anterior mitral leaflet. The displacements of the annular markers (No. 17–23, Fig. 1B), anterior leaflet edge markers (No. 1–7, Fig. 1B), and papillary tip markers (APM and PPM, Fig. 1A) were defined using actual marker data at the end of Δt1.
The Hypermesh finite element model was then solved for the enforced boundary conditions using Optistruct to obtain the simulated displacements of the leaflet belly markers (No. 8–16, Fig. 1B). This initial run assumed nominal anterior leaflet material properties obtained from previous ex vivo studies (8).
Inverse finite element analysis algorithm.
The Optistruct solver was then interlinked with commercial optimization software, Hyperstudy version 8.0 SR 1 (Altair Hyperworks) to run an inverse finite element analysis to yield the in vivo material properties of the MV during Δt1. In this algorithm, the model-simulated displacements of the nine leaflet belly markers (No. 8–16, Fig. 1B) from the nominal run were compared with the actual measured displacements of these nine markers during interval Δt1 to yield a response function defined as the root mean squared 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” (1), to minimize the response function by repeated iterations of the material properties (Ec, Er) in the finite element model until a global response function minimum was obtained. Leaflet circumferential-radial shear during IVR proved sufficiently small that Ec and Er values so obtained were unchanged with the inclusion or exclusion of this shear in the parameter identification process. The material property values (Ec, Er) 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 nine leaflet belly markers as those measured experimentally during interval Δt1.
The same material property identification procedures were then performed for intervals Δt2, Δt3, and Δt4. For interval Δt2, marker coordinates at IVC2 were used as the reference configuration and midsystole as the deformed configuration; for interval Δt3, marker coordinates at midsystole were used as the reference configuration and IVR1 as the deformed configuration; for interval Δt4, marker coordinates at IVR1 were used as the reference configuration and IVR2 as the deformed configuration.
For each heart, excellent beat-to-beat reproducibility was noted, supporting the robustness of the stiffness measuring algorithm as illustrated in Table 1 that shows the transmitral pressure gradient (ΔP, mmHg), Ec and Er for each interval (Δt1–Δt4), and each beat for the 10 hearts studied.
The key finding of this study is the transient behavior of the anterior leaflet stiffness as shown in Table 2 for group mean Ec and Er for each interval (Δt1–Δt4). Also shown are the differences between the stiffness values for each interval and the stiffness values during Δt4, expressed as a percentage of the Δt4 material property values. Stiffness during Δt1 was 1.6–1.7 times the stiffness during Δt4, stiffness during Δt2 was 1.3–1.5 times the stiffness during Δt4, and stiffness during Δt3 fell to 1.1–1.2 times the stiffness during Δt4 (Fig. 3).
The principal finding of this study is that the IVC-IVR stiffness difference is a transient phenomenon that is considerably diminished by midsystole. This phenomenon is consistent with the behavior of the contraction of the slip of atrial-like cardiac myocytes in the proximal part of anterior mitral leaflet that contracts for ∼250 ms after the onset of ventricular systole (3). As can be seen in Fig. 2, 250 ms (15 frames) correspond to roughly half of ventricular systole in ovine hearts.
A diminution of the measured leaflet stiffness by midsystole could result from a change in shape of the leaflet because of the loss of muscle contraction by midsystole. Even though the cardiac muscle is restricted to the annular third of the leaflet, the loss of its contraction could propagate as a shape change throughout the leaflet.
This cardiac muscle twitch at the beginning of the cardiac cycle may be important because it may aid valve closure (12); ablation of this muscle with phenol was shown to delay valve closure by 35 ms in the beating heart. Second, it may help set initial anterior mitral leaflet shape during valve closure/early systole to provide a consistent and efficient left ventricular outflow tract to funnel the blood flowing under the anterior leaflet toward the aortic valve during ejection. A previous study (11) has shown that the leaflet maintains its compound curvature along the central meridian even after cutting of major second order chordae, suggesting the role of active components such as myocytes in maintaining this compound leaflet shape in the beating heart. Third, it will stiffen the valve to help the leaflets withstand the mechanical shock associated with rapid LVP rise during early systole.
This study was performed using a homogeneous finite element model of the anterior mitral leaflet. Although we have developed a heterogeneous finite element model of the anterior leaflet (7), it was not possible to use this heterogeneous model to characterize the material properties during the intervals Δt2 and Δt3 since the small transmitral pressure gradients (and hence small leaflet displacements) during these systolic intervals provided insufficient data for consistent convergence to characterize the additional variables in the more demanding six-parameter heterogeneous model. An attempt to address this issue by assuming a constant ratio between Ec and Er could not be employed because our previous heterogeneous analysis (7) using larger transmitral pressure gradients has shown that the Ec-to-Er ratio is not constant.
The results from two of our previous studies support the attribution of the change in stiffness versus time derived from the homogeneous leaflet model primarily to the change in stiffness of the annular region. First, Krishnamurthy et al. (7), employing a heterogeneous analysis, have shown that the leaflet is, in fact, nearly homogenous with very similar stiffness values obtained from each region: annulus, belly, and edge. Second, Itoh et al. (4), employing a homogeneous leaflet analysis, showed that the β-blocker esmolol, which reduces contractile force generation in myocardial cells (found primarily in the annular third of the leaflet), reduces global IVC stiffness to IVR values. Finally, Krishnamurthy et al. (7), employing the heterogeneous analysis, showed that such esmolol administration almost exclusively reduces annular region stiffness to IVR values. Thus it is highly likely that the stiffness increase (over IVR values) measured globally, that is greatest at IVC and fades away throughout systole, is due to a twitch-like contractile response of the myocardial cells in the annular third of the leaflet. This behavior is much like the contractile activity of the atrial myocardium which also lasts only about halfway through ventricular systole.
We should also note that although the transmitral pressure gradients are not equal for each of the four intervals, this disparity of transmitral pressure gradients will not affect the material properties, since we have previously shown that the stress-strain behavior of the anterior leaflet of the closed MV is nearly linear over the physiological range of pressures (6).
This study was supported by National Heart, Lung, and Blood Institute Grants HL-29589 and HL-67025. G. Krishnamurthy was supported by the Medtronic BioX graduate fellowship, and J. C. Swanson was supported by the Western States Affiliate American Heart Association postdoctoral fellowship.
No conflicts of interest, financial or otherwise, are declared by the author(s).
We gratefully acknowledge the expert technical assistance of Sigurd Hartnett, Maggie Brophy, and George T. Daughters.
- Copyright © 2010 the American Physiological Society