Biomedical Engineering Reference
In-Depth Information
walls provide a viable option for understanding the complex nature of blood flow and
arterial wall mechanics and for obtaining those relevant quantities. These numeri-
cal computations require meshes describing the patient-specific three-dimensional
cardiovascular geometry.
The quality of the meshes is of great importance since it impacts both on the ac-
curacy and the efficiency of the numerical method [5, 40]. For example, it is well
known that for finite element computations, the discretization error in the finite ele-
ment solution increases when the angles of the mesh elements become too large [4],
and that the condition number of the finite element matrix increases with small an-
gles [16] which hinders the numerical convergence.
Two important elements have to be taken into account in order to generate a high
quality tetrahedral mesh for cardiovascular simulations from medical images: (i) the
quality of the triangular surface mesh, (ii) the capability to generate boundary layers
meshes.
Most of the current imaging techniques allow to extract only the inner wall of
the arteries, called also lumen surface. The outcome of the segmentation procedure
is then a triangulation of the surface. These triangulations are however not suited
for subsequent numerical simulations since they are generally oversampled and of
very low quality (with poorly shaped and distorted triangles). It is then desirable
to modify the initial surface mesh to generate a new one with nearly equilateral
triangles of given triangle density (e.g. density based on the vessel radius). There
exists mainly two approaches for surface remeshing: mesh adaptation strategies [6,
23, 41] and meshing techniques that rely on a suitable surface parametrization [7,
27]. The mesh adaptation strategies belong to the direct meshing methods and use
local mesh modifications in order both to improve the quality of the input surface
mesh and to adapt the mesh to a given mesh size criterion. The parametrization
techniques belong to the indirect meshing approach. The initial 3D surface mesh
is first parametrized onto a 2D planar surface mesh; the initial surface can then be
remeshed using any 2D mesh generation procedure with any given mesh size field by
subsequently mapping the new mesh back to the original surface. In this paper we
first propose an efficient approach based on parametrization for recovering a high
quality surface mesh from a low quality input triangulation. The parametrization
technique is based on discrete finite element conformal maps [29, 34] and the density
of the new mesh can be for example adapted to the vessel radius or the discrete mean
curvature (for an example of a curvature adapted mesh of an aneurysm, see [35]).
The proposed quality surface remeshing algorithm is of high importance for sub-
sequent three-dimensional blood flow simulations. Indeed, the lumen surface trian-
gulation is most of the time taken as input for the tetrahedral mesh generator (e.g.
Delaunay, Frontal), which retains the remeshed surface as the boundary of the result-
ing tetrahedral mesh. Hence if the surface mesh contains low quality triangles with
small angles, the resulting tetrahedral mesh might contain some degenerate tetrahe-
dra with small volumes and small dihedral angles. Those degenerate triangles may
lead to large interpolation errors, and have a negative effect on the convergence rate
of the solution procedure. The worst impact results in an unresolvable system of
equations.
