RECONSTRUCTION PROCESS

5.1

In t r o d u c t io n

5.1.1

The need for a realistic model of the head

The work described in Chapter 4 showed that reliable impedance changes can be measured during human evoked responses, but images reconstructed from these measurements could not localise the changes reliably. It was unclear whether errors in the reconstruction process or complex physiological changes were primarily responsible for the localisation errors, given that the reconstruction algorithm used was able to reconstruct images accurately from data acquired in a head-shaped saline-filled tank.

This suggests that assuming the head to be a sphere was not the main source o f error in the images. Even so, the inclusion o f a real head model in the reconstruction process could still have a number o f advantages. From the point o f view o f reconstruction, the forward model would more closely match the anatomy o f the head, allowing the inversion o f the sensitivity matrix to be more heavily regularised, which in turn would improve the image resolution. In addition, the imaging system used for the study in Chapter 4 was slow, allowing only 258 measurements to be taken for each image. If a faster BIT system was available, more data could be acquired and the advantage o f an accurate forward model would be increased.

Additionally, an accurate numerical model o f the head would be useful for test purposes. It would be possible to simulate test data which could be used for testing new, improved reconstruction techniques. Furthermore, the effect o f the anatomy o f the head could be examined in detail, including, for example, the effect o f variability in the thickness and resistivity o f the skull, both between sites in an individual head and between individuals. Another potentially significant effect is that o f holes in the skull, such as those through which the optic nerves pass, which has been shown to distort significantly the localisation o f the BEG (van den Broek 1997).

The approach taken in this work was to use the finite element method to solve the forward problem numerically for a realistic head-shaped mesh. The principles o f the finite element method, its advantages over other approaches and its implementation in this work are discussed below.

5.1.2

The Finite Element Method

5.1.2.1 F inite a n d B oundary elem ent m ethods

To solve the forward problem, the most commonly used numerical methods are the finite element method (FEM) and the boundary element method (BEM). In FEM, the entire volume o f the object is divided into elements and the potential distribution throughout the object is approximated by polynomial functions within each element. In BEM, the potential is found only on the internal and external boundaries o f the object. Additional processing is required to obtain the potential at an arbitrary point in the object. BEM is often used in preference to FEM as the computational demands are considerably less.

The head is a naturally layered structure which would appear to be suited to BEM. However, all regions within a BEM model must be isotropic, piecewise homogeneous and continuous. The skull, as well as other tissues in the head, follows none o f these constraints: it is anisotropic, its conductivity varies from place to place and, possibly most significantly, there are gaps such as those through which the optic nerves pass. These discontinuities could be significant and should be included in any realistic model o f current flow in the head. If these effects are to be modelled accurately, the FEM must be used (Pruis et al. 1993) and it was therefore decided that the FEM would be used in this work.

This section continues with an outline o f FEM, concentrating on those aspects that are o f particular relevance to this work, and defining terms that are used later in italics. For a more complete description, see Tong and Rossettos (1977), Burnett (1987), Silvester and Ferrari (1996) orR ao(1999).

5 .1 .2 .2 M esh g en eration

The first step in FEM is to define the region in the model, a process known as segmentation.

Where the model is taken from a MR image o f the head, the boundaries may not be clear, in which case segmentation is not a trivial problem and may require manual intervention. The segmented model is then divided into a mesh. This is a grid o f connected elements which are usually triangles in 2D, or tetrahedrons in 3D, although quadrilateral and hexahedral elements have also been used. The comers o f the elements are known as nodes. A good quality mesh accurately represents the geometry o f the model, and has elements with similar sizes and shapes. The aspect ratio is often used as a measure o f mesh quality. It is given by the length

good quality mesh can be difficult. Three o f the more commonly used meshing algorithms are discussed below (Owen 1998):

• Octree:

The model is divided into a coarse grid o f square or cubic cells which are then repeatedly divided until the required resolution is obtained. The resolution is preferentially increased at boundaries by subdividing cells which intersect the boundary until it is resolved. The conditioning o f the mesh often needs to be improved, for example by prohibiting sudden jumps in the size o f adjacent cells, or by “jiggling” the mesh by moving all the vertices randomly. Finally, tetrahedral elements are generated from the cells to give a finite element mesh.

• Delauney

The Delauney criterion provides a rigorous way to connect a set o f nodes to give a mesh o f elements and is widely used in 2D and 3D meshing. Nodes are first inserted throughout the model, either as the vertices o f a regular grid, or using some more sophisticated method. The nodes are then connected to give elements, with the constraint that no node falls within a sphere which circumscribes an element. For example, in 3D, each tetrahedral element has four nodes which can be used to define a circumsphere. If such a sphere is dravm for every element, and no node falls within a circumsphere, the Delauney criterion has been met. The solution is unique and gives elements with the best possible aspect ratios, given the initial set o f nodes - although if the locations o f the nodes are not ideal, even the optimal mesh could be poor (George 1999).

• Advancing Front

In the advancing fron t method, the surface o f the object is meshed as a 2D triangular grid. Each triangle provides the base for a tetrahedral element which is completed using a fourth, optimally placed, node inside the object. This first layer o f tetrahedra is used to build more tetrahedra, and this process is continued until the entire model is meshed. Constraints are used to ensure that the elements do not overlap and that the sizes o f the elements remain similar. There is no guarantee that this method will always provide a solution (George 1999).

5.1.2.3 In terpolation m odel

The next step is to approximate the solution to the entire model as a series o f functions which are fitted within each element. In EIT, these interpolation functions (sometimes called basis functions) represent the potential distribution in each element. At the simplest level, the potential might be assumed to be constant within each element, giving a piecewise constant approximation to the entire model. In practice, the interpolation functions are usually either linear or quadratic. A linear interpolation function applied to a 3D tetrahedral mesh, for example, models the potential throughout each element as a constant offset, plus linear changes in %, y and z, giving 4 degrees o f freedom for each element. A better fit, particularly where the potential changes rapidly (such as near the electrodes in EIT), is provided by a

quadratic interpolation function. In this case, nodes are added to the midpoints o f each element face, to give a quadratic element which has 10 degrees o f freedom. This can provide a better fit to the potential than simply increasing the number o f linear elements, for the same computation time.

5.1 .2 .4 M esh m odification

It is often necessary to improve a mesh after it has been generated (Owen 1998). Three techniques are commonly used (Molinari et al. 2000): h-refinement, where existing elements are subdivided, p-refinement, where the order o f the interpolation function is increased, and r- refinement, where the positions o f the existing nodes are modified.

5.1.2.5 A ssem b ly o f elem ent a n d g lo b a l m atrices

Each element can be described in terms o f a matrix K and a vector F, known by analogy with structural engineering as the stiffness matrix and the force vector (equation 5.1). This equation is then solved to find the displacement o f each node when a force is applied. If there are n nodes in each element, then (j) and F are vectors o f length n and K is « x «. The techniques used to find K and F are reviewed in any good text on FEM, e.g. Tong and Rossettos (1977) or Burnett (1987).

K(zJ=F (5.1)

In EIT, equation 5.1 is used to find the potential, at each node when a current is applied to a body with a given conductivity. Then the interpolation functions give an estimate o f the potential, and K and F depend on the interpolation functions, and the conductivity and applied

Equation 5.1 is found for each element, but every node is shared between many elements. To solve the model, a global stiffness matrix and global force vector are assembled which contain all the element matrices and vectors, linked so that nodes which are shared between elements are identified. This gives an equation which is identical to equation 5.1, except

K

F

K

\ s m x m and (j) and

F

5 .1 .2 .6 Solution o f finite elem ent equations

The final stage is to solve the global equations by inverting

K

5.1.3

Alternative implementations of FEM

5.1.3.1 C hoice o f F E M softw are f o r this p r o je c t

There are many programs available for FEM, with varying degrees o f complexity, functionality and expense. Owen (1998) gave a thorough survey o f programs for mesh generation. The authors o f over 100 programs were surveyed and 81 replied, o f which around half had a facility for 3D meshing.

To determine whether the inclusion o f an accurate numerical model o f the head improved the performance o f EIT images, it was not necessary to have a flexible meshing program. Only a single, generic head model was required. If it was found that the image quality was improved, individual meshes could be generated for each patient at a later date.

The finite element solver itself would have to be used as part o f the EIT reconstruction algorithm developed in Chapter 3. For example, it should be able to deal with current injection on an electrode, and not require a new model to be created and solved each time the current injection pattern is changed. It was also important that the solution could be exported in a format that could be read into the reconstruction algorithm. Finally, it would be useful, but not vital, if the program was open-source. This would allow it to be modified, and reduce the need for support.