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Keywords: Electrical Impedance Tomography, Finite Elements, Mesh Warping. 1. INTRODUCTION. The Finite Element Model (FEM) has become a popular ...
Reconstruction Algorithms

EFFECTS OF WARPING FINITE ELEMENT MESHES FOR THE FORWARD MODEL OF THE HEAD IN EIT Andrew Tizzarda, Richard H Bayfordb, Lior Horeshc, Rebecca Yerworthc, David S. Holderc a

b

Middlesex University, Trent Park, Enfield, London N14 4XS, UK; Middlesex University, Archway Campus, Furnival Building, Highgate, London N19 3UA, UK; c Department of Medical Physics and Bioengineering, and Clinical Neurophysiology, UCL, London, UK

ABSTRACT: The use of realistic FE head models in EIT of brain function confers significant advantages, but image quality is critically dependent on mesh geometrical integrity. Generation of accurate models for individual subjects is time-consuming, so warping an existing idealised mesh to closely approximate specific patient geometry is being investigated. The head was modelled as an ellipsoid to obtain simulated boundary voltages calculated with both a homogeneous mesh of 11820 elements and a shelled mesh of 13985 elements. Images were reconstructed with a linear, truncated SVD algorithm, using spheres linearly warped to the same dimensions as the ellipsoid, with 29435 and 29832 elements for homogeneous and shelled models respectively. Decrease of mesh quality and localisation errors were acceptable in both cases, thus leading towards the conclusion that this method could be uniquely useful for EIT imaging in conditions like acute stroke where it may not be practicable to obtain an individual MRI and mesh Keywords: Electrical Impedance Tomography, Finite Elements, Mesh Warping

1. INTRODUCTION The Finite Element Model (FEM) has become a popular numerical tool in the investigation and application of EIT and a number of researchers have applied the technique to generally simple geometrical models in EIT and EEG [1, 2, 3]. In most cases the meshing of the domain has been carried out using semi-automatic or manual methods with solvers generally written in-house. Some investigations have also taken place to establish methodologies for generating accurate FE meshes of the adult human head [4, 5]. These investigations used I-DEAS (Integrated Design Engineering Analysis Software, www.eds.com); the same software package used for mesh generation in this investigation. A number of systems for acquiring patient-specific geometry for mesh generation do exist, for example: CURRY [6] produces FE meshes, ASA (www.ant-software.nl) and BESA (www.besa.de) generate Boundary Element (BE) meshes. These systems rely on the existence of MRI images, which are segmented to provide the underlying geometry for the final meshes. The need for MRI data can be prohibitive for time-critical clinical use. Therefore, a methodology has yet to emerge that can automatically and rapidly generate complex FE meshes of realistic patient-specific geometry for solving the forward model. One hypothesis upon which this work is based is that if a library of geometrically sound FE meshes exist then any one model can be selected as a close match to a patient and subsequently warped to fit a limited number of registration points or dimensions measured from that patient. Gibson et al [7] have proposed a method of warping an existing head-shaped surface to fit a set of points on an arbitrary head. This method, however, requires subsequent meshing to generate a homogeneous mesh which increases the time taken to produce the forward model. It is now generally accepted that the integrity of the FE model and its underlying geometry has significant and measurable effects on the quality of reconstructed images [8, 9]. However, it is still not fully understood what bearing the deviation of complex geometry from a specific patient’s head shape has on the localisation of conductivity changes in the images. This paper presents initial experimentation and results towards construction of this understanding. Simulated data from conductivity changes at various positions in meshed ellipsoids are used to reconstruct images using FE meshes of spheres that have been warped to the same dimensions as the original meshed ellipsoids.

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2. METHODOLOGY 2.1. Meshing The FE meshes used in this investigation have been generated using I-DEAS as it provides a number of benefits in controlling mesh integrity and quality [4, 5]. The software utilises an advancing front algorithm to generate tetrahedral elements from a solid model. The main benefit of the algorithm is that it allows the user to specify a growth rate factor for interior elements of 1.0 thereby ensuring that the sizes of the elements within the interior of the domain are similar to those at the surface boundaries. This is important for head modelling, as the mesh density within the brain compartment needs to be as fine as that in the outer regions such as the skull. This is because current density within the brain is significantly attenuated owing to the influence of the low conductivity of the skull. I-DEAS can export the meshes as Universal Files; these are text-based files that can easily be parsed to extract nodal positions (vertices) and elemental connectivity (simplices) as well as the location of the elements within any of the considered regions, namely scalp, skull, cerebro-spinal fluid (CSF) and brain. A software package with a graphical user interface is being developed for this and subsequent work that can import the universal files, display the mesh and generate the relevant matrices for forward solution and reconstruction. This software package, written in C++, has been given the working title GraphEIT.

2.2. Simulated data Simulated data was generated from FE meshes of an ellipsoid modelled in I-DEAS. The dimensions of the ellipsoid have been chosen to match the aspect ratio of the existing accurate head model used in [9]: 150 mm in the x-direction, 190 mm in the y-direction and 200 mm in the z-direction. Two FE models were created: a homogeneous model consisting of 11820 elements and 2387 nodes and a four-shell model consisting of 13985 elements and 2662 nodes. The four-shell model was partitioned into regions representing brain (5020 elements), CSF (2693 elements), skull (2972 elements) and scalp (3300 elements). The relative geometrical radii of the regions used were for brain 85%, CSF 3%, skull 9% and scalp 3% [10]. The elements within these regions were grouped in I-DEAS prior to exporting the universal file and the import facility in GraphEIT assigns conductivities to the regions. The values used for the conductivities are those adopted by Liston et al [11]; specifically: scalp 0.44 Sm-1, skull 0.018 Sm1 , CSF 1.79 Sm-1 and brain 0.25 Sm-1. The number of elements chosen for each model is such that the time taken for generating simulated data is reasonable (around 100s per forward solve on a Pentium 3 866MHz using EIDORS) whilst ensuring an acceptable level of accuracy of the boundary voltages at the electrodes. For each model, conductivity perturbations were simulated and these were localised at approximately –50 to +50 mm along the x-axis, -70 to +70 mm along the y-axis and –30 to +70 mm along the z–axis in steps of 10 mm for each axis. The radius of the perturbations was set to 5 mm which resulted in one element representing the conductivity change. A total of 37 perturbations were therefore simulated for each ellipsoid model. In the case of the homogeneous ellipsoid the conductivity perturbations represented a 100% change and for the shelled model a value representing that of CSF was used; in the latter case, all perturbations were localised in the brain region. GraphEIT has functionality to define ideal perturbation positions, sizes and conductivities and reports back actual position and size having established the nearest element or elements to the ideal. It can then perform a forward solve for each perturbation using imported electrode position and protocol files. The forward solution is generated by communication with a MATLAB engine and execution of the relevant EIDORS routines [12].

2.3. Mesh warping and the forward model The main emphasis of this work is to determine the effects of warping a Finite Element model so that its underlying geometry matches that from which an EIT data set as taken. Two FE models were constructed for warping both of which were generated from solid spheres of 200mm diameter in I-DEAS. The first was a homogeneous spherical mesh of 29435 elements, 6135 nodes. The second was a shelled mesh of 29832 elements and 5586 nodes (brain 12554 elements, CSF 5079 elements, skull 5825 elements and scalp 6374 elements). A function exists in GraphEIT to scale a complete FE model non-proportionally, i.e using separate scaling factors for each orthogonal direction, and this was used to warp the meshes such that the underlying geometry was the same as the ellipsoids used for generating the simulated data described in §2.2 above. The distribution of elemental volume and element stretch were evaluated before

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and after warping to establish its effects on mesh quality and integrity. Element stretch is a popular means of evaluating mesh quality; it is a measure of how much an element is distorted from an ideal regular tetrahedron (or equilateral triangle in 2D systems). In 3D systems large quantities of elements with stretch values below 0.05 will introduce significant errors in the forward model [13]. For a threedimensional tetrahedral element the stretch value is the normalised ratio of inscribed sphere radius (R) to maximum edge length (Lmax) and is given by: S = 12

(1)

R Lmax

The inscribed radius is the calculated as the volume divided by the sum of the face areas.

2.4. Reconstruction The methodology used to reconstruct data was based on a linearised sensitivity approach. Electrode placement was based on EEG 10/20 positions resulting in a total of 31 electrodes and a protocol used involving bipolar current injection resulting in a total of 258 measurements. The sensitivity matrix was generated using EIDORS. The matrix was inverted using Singular Value Decomposition (SVD), truncated to 70 values to correspond with the schema used by Bagshaw et al [9]. Prior to inversion the sensitivity matrix was row and column normalised. The resulting image matrix was transferred back to GraphEIT and displayed using a slicer-plot showing unsmoothed images of conductivity changes. Facilities for analysing the images include a peak conductivity search and a means of determining the mean centre and size of a region of interest based on conductivity values lying in a given percentage of the peak conductivity. Setting this deviation to 50% allows a full-width at half maximum (FWHM) to be evaluated; an example of one view of the image is shown in fig. 1.

Figure 1: Sample Slicer plot showing peak conductivity change (arbitrary units).

3. RESULTS Results of the changes to mesh parameters and quality are given in the following table along with the localisation errors for perturbations along the x, y and z-axes. Volumes are in mm3 and localisation errors in mm. Element Volume Homogeneous

Range

Mean±SD

Stretch Range

Localisation error (Mean±SD)

Mean±SD

Before 33.20 - 530.37 141.97 ± 56.83 0.344 - 0.986 0.681 ± 0.098 After 23.66 - 377.89 101.16 ± 40.49 0.318 - 0.965 0.653 ± 0.104 Shelled Before After

7.05 - 605.92 139.69 ± 78.32 0.134 - 0.986 0.639 ± 0.151 5.03 - 431.72

99.53 ± 55.80 % change

0.102 - 0.966 0.615 ± 0.150 -4.1

x

y

z

5.0±5.8

1.2±0.8

1.6±1.6

2.4±1.7

2.1±1.9

3.9±2.9

-3.8

For the homogeneous sphere, element volume distribution throughout the region is random, with no specific element size confined to a particular region. Distribution of the volumes for the shelled sphere indicated a greater concentration of smaller elements in the thin shells representing scalp and CSF as well as to some extent in the skull.

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4. CONCLUSIONS AND FURTHER WORK The procedures and the results described above have provided a useful insight into the effects of warping finite elements models of spheres into ellipsoids. The warping of the shapes is far more extreme than that expected if a human head is to be warped to that of a known patient. The mesh quality degradation for both homogeneous and shelled models is minimal and result in meshes with acceptable element quality. Localisation errors are within the accepted ranges for this type of linearised sensitivity matrix reconstruction algorithm. This work leads on to a number of activities for further investigation. Standard FE meshes of the adult human head will require minimal warping to achieve a close geometric match to a specific patient. Under investigation at present are a number of approaches of acquiring key registration points from the heads of a number of subjects. Among these registration points are the actual electrode positions plus additional orientation points defined by nasion and inion and left and right temporal positions. The standard head mesh can be scaled and orientated to these latter positions and the mesh warped to fit the electrode positions. This could result in a final model where electrode positions are accurately defined allowing non-linear reconstruction techniques to be adopted. Non-linear reconstruction relies heavily upon accurate electrode positioning and is necessary for the absolute imaging requirements of stroke victims.

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