Using position-based dynamics to simulate deformation in aortic valve replacement procedure

Model setup

We use semi-automatic 3D reconstruction based on patient-specific CT or MRI data for generating computational models for the aorta. The aorta model is represented as a surface mesh and consists of a set of vertices, edges, and triangles. The aorta model does not include the model of the diseased aortic valve.

The triangular surface mesh for the Magna Ease replacement valve (Edwards Lifesciences, USA) in its open state was reverse-engineered using micro computed tomography, cf. Figure 1A. A sample with 21 mm diameter was scanned for reference. Since the valves are available in different sizes from the manufacturer, we scale the model uniformly to the desired size. The usage of other models of replacement valves is possible as well.

Figure 1:

(a) Detail view of Magna Ease valve model. Both edge loops in the annulus region are depicted. The two spheres indicate the size of the bounding box in which the deformation will be calculated. (b) View of center-line f(s) and aligned valve. (c) Placement of the valve along the center-line inside aorta model.

To prepare the implantation process, the replacement valve can be placed (manually or automatically) along the path of the aorta center-line, cf. Figure 1B and C. For a manual placement, the center-line is interpreted as a parameterized curve f:R↦R3. The interpolation parameter s∈[0,1] is used for positioning the valve at position f(s)∈R3, the tangent, normal and bi-normal are referred to as t(s), n(s) and b(s)=t(s)×n(s) (TNB frame). Automatic positioning is possible using a local minimum of the center-line radius. Using the local coordinate system induced by the TNB frame of the center-line and the local coordinate system of the valve (its main axis is oriented according to the opening, i.e. the main flow direction through the valve, its origin is located in the center of gravity), we use basis transformation to align the model to the curve and translate it to position f(s). In addition, we allow changes to the orientation of the valve. In particular, the valve can be rotated around the center-line tangent t(s) and can be tilted relative to t(s). Translation of the valve’s origin in the plane defined by n(s) and b(s) at position f(s) is possible as well. The whole process can be automated by choosing the angles and offset in a random fashion. After this, the geometry is setup for simulation of deformation due to valve placement.

Valve replacement simulation

Since the general simulation approach follows the one presented in [3] in the context of mitral valves, we briefly summarize the changes for applying the method to simulate deformation for AVR. In general, depending on the mesh size, the method can be executed at interactive frame rates on current PC hardware which enables us to reduce the time spent for creating the deformed meshes for CFD. This allows for a larger set of models with varying valve angulation to be analyzed. Processing and simulation are implemented using the MeVisLab platform [4].

Using the approach in [3], the aorta shape is approximated by the triangular mesh created in section 2.1 and can be deformed. The valve model created in section 2.1 acts as a rigid body. The initial state is referred to as t=0 and the simulation creates a deformed result at t>0.

The reconstructed mesh of the aorta is augmented with a set of constraints modeling a simplified material for the aortic wall. These constraints relate multiple vertices to each other. The vertices are assigned mass, position and velocity. The extended position-based dynamics (XPBD, [5]) approach taken in [3] uses linearly elastic distance constraints, bending constraints as well as area conservation constraints to model the mitral valve material. The same modeling is used here for the vessel wall, cf. distance constraints shown as edges in Figure 2A–D. Contact between wall and valve is modeled using collision constraints. External forces can be defined for deformation, e.g. a pressure force can drive the aorta model to narrow or widen its diameter.

Figure 2:

(a) Detail view of Magna Ease valve placed inside of aorta model. (b) View of ray connections inside aorta model. The connections reach from the vertices of the non-manifold edges radially outward towards the wall. (c) Deformation of aorta model (inside view). The gap between wall and valve has been closed, the connections are not visible anymore. Both structures meet at the diameter of the replacement valve. (d) Deformation of aorta model (outside view). Compared to the initial state, the aortic wall now meets with the outside diameter of the replacement valve.

To allow for building a surface mesh that serves as a basis for CFD volume mesh generation, the surface mesh of the valve does not form a closed surface, instead it features two free edge loops where the connections to the aorta are placed. This is referred to as the annulus side. The surface is not connected at the annulus side to create the right topology for domain creation, cf. Figure 1A. The two free edge loops in this area can be connected to the outer aortic wall. Here, we use distance constraints with zero rest length between the vertices of the non-manifold boundary edges of the valve mesh and the aorta to simulate sewing. The distance constraints are created by casting rays radially outward from each of the boundary vertices, cf. Figure 2B. The direction is found by forming the cross product of t(s) (TNB frame of the curve and valve are aligned) and the direction of the adjacent edge. The total number of rays is equal to the number of vertices in both edge loops. There are two alternatives for the further processing of the mesh. Either we apply nearest neighbor search for the definition of the distance constraints (remember these connect two vertices), or we subdivide the triangles, create new vertices, and perform mesh post-processing to ensure consistent topology.

In the first case, the closest vertex of the aorta mesh to the hit position of the ray is connected with the boundary vertex and forms a new distance constraint. This implies that multiple boundary vertices can be connected to the same vertex of the aorta mesh, depending on the ratio of boundary vertices to vertices next to the projection of the boundary onto the aorta mesh. For the deformation this is not an important factor, but for domain creation. Since we want to achieve the correct topology for CFD domain creation, this will not result in the correct topology. By applying constructive solid geometry operations, in particular a Boolean union operation, on both meshes afterwards, we can ensure correct topology.

The second alternative uses triangle subdivision to create new vertices. In this case, we have to refine the triangle hit by the ray and subdivide the hit triangle into three new triangles. The fourth new vertex is positioned inside the hit triangle at the hit position. To clean up the mesh topology after refinement, we apply series of post-processing operations such as edge contraction or edge flipping to ensure correct topology, cf. Figure 2D.

In cases where the diameter of the valve implant is larger than that of the aorta at the chosen center-line position and orientation, or the shape of the aortic wall is not convex and, therefore, no placement of the valve that lies fully inside of the aorta mesh can be found, the size of the implant can be scaled to the final size, beginning at a diameter that is known to fit into the aorta. By expanding the rigid body, the collision constraints ensure a simultaneous expansion of the aorta mesh. To limit the computational effort further, a bounding box with an appropriate size can be placed around f(s), cf. spheres shown in Figure 1A. The bounding box vertices are shown as yellow spheres, e.g., in Figure 2A.