Fluid-Structure Interactions of the Mitral Valve and Left Heart: Comprehensive Strategies, Past, Present and Future

Abstract

The remodeling that occurs after a posterolateral myocardial infarction can alter mitral valve function by creating conformational abnormalities in the mitral annulus and in the posteromedial papillary muscle, leading to mitral regurgitation (MR). It is generally assumed that this remodeling is caused by a volume load and is mediated by an increase in diastolic wall stress. Thus, mitral regurgitation can be both the cause and effect of an abnormal cardiac stress environment. Computational modeling of ischemic MR and its surgical correction is attractive because it enables an examination of whether a given intervention addresses the correction of regurgitation (fluid-flow) at the cost of abnormal tissue stress. This is significant because the negative effects of an increased wall stress due to the intervention will only be evident over time. However, a meaningful fluid-structure interaction model of the left heart is not trivial; it requires a careful characterization of the in-vivo cardiac geometry, tissue parameterization though inverse analysis, a robust coupled solver that handles collapsing Lagrangian interfaces, automatic grid-generation algorithms that are capable of accurately discretizing the cardiac geometry, innovations in image analysis, competent and efficient constitutive models and an understanding of the spatial organization of tissue microstructure. In this manuscript, we profile our work toward a comprehensive fluid-structure interaction model of the left heart by reviewing our early work, presenting our current work and laying out our future work in four broad categories: data collection, geometry, fluid-structure interaction and validation.

Figure 1

Axisymmetric geometry of the fluid-coupled mitral valve model. The fluid-domain was modeled as an Eulerian box. Pressure curves were applied to the atrial and ventricular surfaces. The model was axisymmetric; the half indicated by the light blue shading was not modeled.

Figure 2

Streamlines at four time points during the simulation: A) the beginning of the simulation before diastolic flow has reversed; B) the end of the simulation after coaptation and as the atrial flow begins to settle; C & D the moments preceding and immediately following coaptation, respectively. Note: chordae have been omitted from the figure for clarity. Panels C & D are cross-sections at the symmetrical midline shown in Figure 1.

Figure 3

Predictions of the computational model compared to experimental in-vitro (A & B) and animal (D, E & F) data: (A) transvalvular pressure, (B) trans-mitral flow. Solid line - reference. Dotted line - computational model. The predicted closing flow shows excellent agreement with in vitro data over the simulation window [50]. Simulated intraventricular S1 (C) and measured thoracic S1 (D). Linear relationship between dP/dt max and S1 RMS (open squares). Linear relationship between dP and S1 RMS (open circles) (E). The increased dP/dt has a more profound effect on acoustic power. (F) Predicted anterolateral papillary muscle force (solid line with squares) and ventricular pressure (broken line).

Figure 4

Semi-automatic multi-material segmentation of ex-vivo sheep heart. A) volumetric visualization of whole heart. B) left and right ventricular blood masses in whole heart transparency. C) whole heart transparency. D & E) cut-away sections though the long-axis of the heart, showing both atrioventricular valves (tricuspid and mitral on the left and right hand sides respectively). F) blood with long-axis cut-away. G & H) details of tricuspid and mitral valves.

Figure 5

Automatic mapping of 3D cellular information. Data is the direction vector of the myofibers in a sheep heart, as revealed by diffusion tensor MRI. A) the direction and magnitude of the vector in a voxelated MRI dataset. B) The overlap between the unstructured grid (grey) and the voxel grid (green). Note that the algorithm is very tolerant of small geometric mismatching. C) The vector field mapped onto the unstructured grid.

Figure 6

3D cardiac strain analysis from in-vivo tagged MR images. Endocardial and epicardial contours as well as segmented tag-lines were traced from (A) short and (B) long-axis MR images to create (C) a 3D geometry. (D) Each short axis slice was divided into 12 sectors and a 4D B-spline-based motion tracking technique was applied to the tag-line (dotted lines) deformations in order to calculate the Lagrangian Green’s strains in cylindrical coordinates. For each sector of each short axis slice, longitudinal, radial, (E) circumferential and shear strains throughout systole were determined.

Figure 7

Circumferential strains predicted from the present FE model are generally in reasonable agreement with the values measured in-vivo from tagged MR images. Slice 1 is the most basal while slice 16 is the most apical. I is the posterior right ventricular insertion, II is the free wall, III is the anterior right ventricular insertion and IV is the septum. The area of largest discrepancy between the measured and predicted circumferential strains is at the insertion points of the right ventricle to the left ventricle since the right ventricle was not included in the model. Overall the minimized mean square error was 4.7, motivating the development of improved methods.

Figure 8

Automatic layered tetrahedral grid generation of the imaging data shown in Figure 4. Panels A & B: feature size field from two points of view. Panels D–I: details of the layered tetrahedral mesh. Shown are the mitral and tricuspid valves, with chordae. Note the size variation of the tetrahedra with respect to the local feature size.

Figure 9

Automatic layered tetrahedral grid generation of micro-CT data of a porcine aortic valve. Panel A: feature size field f[x]. Panel B: f_out[x] field. Note the automatic detection of leaflets in close apposition. Panel C shows the surface adaptation by both f[x] and f_out[x]. Panels D–F: details of the layered tetrahedral mesh. Note the preserved layering throughout. Also notable is the thick muscular shelf below the aortic cusps that are generally omitted in idealizations of the aortic geometry.

Figure 10

Multi-material boundary layer mesh of a human heart, consisting of heart tissue, blood boundary layer and blood domains. Panels A and B show the feature size field on the outer surface of the heart tissue domain and the blood domain, respectively. Panel B shows the feature size field on the outer surface of the blood domain. Panel C shows a cut through the heart muscle tessellated with layered tetrahedra [70], where the orientation of the cut plane is indicated in Panel A. Panel D shows the detail of the layered tetrahedral mesh of the heart tissue. Panels E and F are zoomed-in views on the regions of the cardiac valves, showing the prismatic boundary layer alone. Panels G, H and I show the prismatic boundary layer at about 25% of GLFS of the blood domain, sandwiched between the layered tetrahedra of the tissue and the Delaunay tetrahedra of the blood. Heart model and un-refined surface mesh are courtesy of the NYU Medical Center and Zhang et al. [85, 86].

Figure 11

Mesh quality statistics for the prismatic layers and interior tetrahedra of the human heart mesh generated using layered tetrahedra and face-offsetting. Panel A shows the the scaled aspect ratio for prisms of each prismatic layer (1 is ideal, and a negative value would indicate an inverted prism). Panel D shows the aspect ratio for tetrahedral elements (1 is ideal)

Figure 12

Comparison of complexity between the Delaunay algorithm and the cojugate gradient algorithm.

Figure 13

Sketch and dimensions for the aortic valve validation case.

Figure 14

Interpolated velocity profiles matched to experimental profiles at five time points labeled according to the lower right hand panel. The dashed line represents the experiment and the solid line the numerical results.

Figure 15

Computed opening angle (full line) compared to experimental results (dashed line).

Figure 16

Predicted velocity vector field for the 2D rigid aortic leaflet and sinus.

Figure 17

Re-meshing step preserving the local node density around the flap for two consecutive time steps.

Figure 18

Comparison of two re-meshing techniques as explained in section 3.3.3. In (A) and B the nodal density is kept constant throughout the computation. In (C) and D the nodal density is adapted based on a local error metric.

Figure 19

Number of elements as a function of time for the adaptive re-meshing with an error estimate.

Figure 20

Lagrangian fluid-structure interaction study of a large deformation, solid neo-Hookean leaflet coupled with a dynamic tetrahedral fluid grid: A) problem domain and input parameters, B) input velocity profile and transfered force, c) velocity streamlines at five points in time.