Domain Discretisation

The forward problem can be solved analytically on very simple geometries (Isaacson, 1986). In order to solve it on non-symmetric and complicated realistic domains, the domains need to be discretised. Several numerical methods exist to solve the forward problem on discretised domains. The finite difference and finite volume methods usually require regular grids (Liszka and Orkisz, 1980; Eymard et al., 2000). While these methods are computationally efficient, the representation of curved structures is complicated by the regularity requirement (Lionheart et al., 2004). A more powerful, yet less intuitive, method is the finite element method (FEM, Brenner and Scott, 1994), which is the most commonly used in EIT. It does not require any regularity of the discretisation (thefinite element mesh), and instead of evaluating local derivatives (finite difference) or fluxes (finite volume) it solves aweak formulation, which already includes the boundary conditions. As the number of elements in the mesh increases, the FEM solution approaches the real solution of the partial differential equation. On homogeneous domains, the boundary element method (BEM) is a viable alternative to FEM (Gençer and Tanzer, 1999). BEM only requires discretised surfaces and uses an analytical expression of the Green function between surfaces of different characteristics. For complicated geometries (such as the human head), the presence of many

different tissues diminishes the computational advantage of BEM over FEM (Lionheart et al., 2004). node 1 node 2 node 3 (4) (5) (6) (a) node 1 node 2 node 3 (5) (6) (8) node 4 (7) (b) 1 2 4 3 (c) 1 2 3 4 6 7 8 5 (d)

Figure 2.8: Most common finite element shapes- 2D finite elements are usually(a)triangles or(b)quadrilateral elements. The most common 3D finite elements are(c)tetrahedra and(d)hexahedra. For linear shape functions, only the vertices (corners) of the elements are considered as nodes. For quadratic shape functions, additionally the midpoints of the edges are used as nodes (numbering in brackets). More nodes on the edges and inside the elements are added, as the order of the shape functions is increased.

The most common finite elements in 2D are triangles and quadrangles, and in 3D tetrahedra and hexahedra (figure 2.8). While it is possible to mix different element shapes in one mesh, it is not commonly done. Methods for subdividing a domain into a mesh are numerous, but the aim of all methods is the same: to generate a mesh with non-overlapping elements filling the whole domain. Elements should have as few small angles as possible, since this aids accuracy and

convergence of the forward solutions (Shewchuk, 1997). Current meshing packages use different meshing strategies. While Netgen uses an advancing front algorithm (Schöberl, 1997), Cgal (The CGAL Project, 2015) first generates a surface Delaunay triangulation at the object surface and internal structural surfaces (Boissonnat and Oudot, 2005), followed by Delaunay refinement of the volumes between surfaces (Shewchuk, 1997). The fundamental principle of Delaunay triangulation and refinement is to generate triangles and tetrahedra, that do not contain vertices in the interior of their circumcircle or circumsphere (Shewchuk, 1997). In 2D, the Delaunay triangulation can be found by connecting points with neighbouring Voronoi polygons (Ho-Le, 1988).

After the initial meshing, the mesh usually has to be smoothed, in order to remove badly shaped elements. Cgal for instance includes four different mesh optimisation algorithms, which iteratively run through the mesh, moving nodes and splitting badly shaped elements. TheLloyd optimisationminimises a global energy defined as theL1-norm of the error when the square

coordinate functionx2is interpolated on the mesh domain. The energy gradient is set to zero

by moving the vertices. Analogously,optimal Delaunay triangulationminimises global energy by updating the Delaunay triangulation instead of the vertices. Jointly with these two global optimisation methods, Cgal uses two local optimisers to remove the worst elements (perturb andexude).

Figure 2.9: Thorax model created from a 2D outline- Based on a 2D outline of the thorax and lungs segmented from an MRI scan, this 3D head model was generated with Netgen directly in Eidors (Grychtol et al., 2012).

Figure 2.10: Head model based on manual MRI segmentation - This head model consisting of smooth curves (called NURBS) was manually created from MRI datasets (Tizzard et al., 2005).

For most EIT applications, meshes with relatively simple geometries (e.g. cylinders) are used, which can be created with Netgen directly in Eidors (Grychtol and Adler, 2013). Three- dimensional thorax models based on two-dimensional outlines segmented from MRI scans have been created with the same method (Grychtol et al., 2012). The first meshes used for head EIT were either homogeneous spheres or concentric spheres with different conductivities

(Bagshaw et al., 2003). Since the observation, that more accurate head models (figure 2.10; Tizzard et al., 2005) result in better images (Bagshaw et al., 2003), there was an increased interest in generating more physiologically realistic head meshes. Vonach et al. (2012) presented a new, semi-automated workflow for creating head meshes based on either a CT or MRI scan. In a first step, brain, cerebrospinal fluid (CSF), skull and scalp were segmented from the available scan. For MRI scans, the open-source software BrainSuite was used for segmentation, while for CT scans an expectation maximisation algorithm was used to fit prior knowledge of brain, CSF and scalp shapes into the segmented skull with the registration tool NiftyReg (Modat et al., 2010). The created segmentations were then meshed as surfaces with the MeshLab software, and a tetrahedral mesh was generated from the surface meshes with the CUBIT meshing software. When comparing the thickness of the skull in the resulting meshes (figure 2.11), it becomes apparent that CT and MRI scans should be used conjointly to obtain more accurate tissue representations.

Figure 2.11: Meshes generated from either CT or MRI scan- Meshes created by Vonach et al. (2012) by either morphing a brain template model into a skull segmented from a CT scan or by estimating the skull shape from an MRI scan.