Solving the forward problem in electrical impedance tomography for ...

INSTITUTE OF PHYSICS PUBLISHING

PHYSIOLOGICAL MEASUREMENT

Physiol. Meas. 22 (2001) 55–64

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PII: S0967-3334(01)19901-6

Solving the forward problem in electrical impedance tomography for the human head using IDEAS (integrated design engineering analysis software), a finite element modelling tool R H Bayford1 , A Gibson2 , A Tizzard1 , T Tidswell2 and D S Holder2 1 Middlesex University, Archway Campus, Furnival Building, Highgate Hill, London N19 3UA, UK 2 Department of Clinical Neurophysiology, Middlesex Hospital, Cleveland Street, London W1T 3AA, UK

Received 6 November 2000 Abstract If electrical impedance tomography is to be used as a clinical tool, the image reconstruction algorithms must yield accurate images of impedance changes. One of the keys to producing an accurate reconstructed image is the inclusion of prior information regarding the physical geometry of the object. To achieve this, many researchers have created tools for solving the forward problem by means of finite element methods (FEMs). These tools are limited, allowing only a set number of meshes to be produced from the geometric information of the object. There is a clear need for geometrical accurate FEM models to improve the quality of the reconstructed images. We present a commercial tool called IDEAS, which can be used to create FEM meshes for these models. The application of this tool is demonstrated by using segmented data from the human head to model impedance changes inside the head. Keywords: finite element method (FEM), forward problem, EIT (Some figures in this article are in colour only in the electronic version; see www.iop.org)

1. Introduction Electrical impedance tomography (EIT) is a recently developed imaging method, which enables the internal impedance of an object to be imaged with an array of external electrodes. It has been under active development for about two decades. It has substantial potential as a non-invasive technique for tissue characterization in biological material or medicine, as it is safe, inexpensive, rapid and the equipment is small and portable. Although its accuracy has been demonstrated in research studies, it has not yet been established as a routine imaging technique in biology, physiology or medicine. This is mainly because it has relatively poor spatial resolution. There are several factors which influence the quality of EIT images: these include electrode impedance and instrumentation errors. The overriding limiting factor is the reconstruction algorithm. 0967-3334/01/010055+10$30.00

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One focus of the research work in our group is the development of reconstruction algorithms that produce images of impedance changes inside the human head (Gibson et al 1999). To obtain these reconstructed images of changes in resistance we utilize a sensitivity approach (Murai and Kagawa 1985, Metherall et al 1996, Gibson 2000). The mathematical model usually used in EIT is based on the assumption that the material in the body is ohmic and isotropic; it also assumes that the potential φ(x, y, z) for a domain can be governed by Laplace’s equation: ∇ · (σ ∇φ) = 0

(1)

with boundary conditions σ

∂φ =J ∂η

φ=V

(2) (3) ∂ ∂η

is where σ is the conductivity, J is the current density, V is the voltage measurement and the normal derivative to the surface. This assumes that the electrodes are simple point sources. As the images we are aiming to produce are images of changes in impedance, the errors due to electrode models are minimized (Barber and Seager 1987). (Incorporation of other models can be achieved with IDEAS but was not addressed in this paper.) The aim of the reconstruction algorithm is to determine the conductivity distribution from the measurements made at the surface of the domain. For voltage measurements made at the surface of the head the reconstruction problem can be defined as V = Aσ

(4)

where A is known as the sensitivity matrix (Gibson 2000) which relates voltages to image pixels. The sensitivity matrix can be considered as the Jacobian of the forward mapping (matrix of partial derivatives). The inverse problem, to find σ given A and V , becomes σ = A−1 V where A−1 is the inverse of A. A can be found analytically, by assuming the head to be a homogeneous sphere, but this does not use accurate geometrical information regarding the shape of the human head which will introduce errors in the solution. At present we are using such an analytical model to find a solution for A. The aim of this research is to obtain an accurate solution to the forward problem by incorporating the shape and the structure of the human head, and hence obtain an improved estimation of the values of the elements in the sensitivity matrix A. The reconstruction algorithm involves a solution to the forward and inverse problems. The forward problem can be solved in two ways. The first is analytical, for which a number of solutions are available (Pidcock et al 1995a,b); the second is by the use of numerical methods like FEM or boundary element method (BEM) (Vadasz 1995). The analytical methods are preferable as computation time is reduced, but these have been of limited practical value because solutions only exist for a small number of idealized geometries. FEMs have become one of the most commonly used methods of finding solutions to the forward problem. Geometrical information regarding the shape of the object can be included. Many tools are available for finding solutions to the forward problem, which are not specifically designed for EIT. These include TOAST (Arridge and Schweiger 1993) and BEASY (Becker 1992). One of these tools is known as IDEAS (integrated design analysis software). This is a mechanical computer-aided engineering (MCAD) package, which was developed by Structure Dynamics Research Corporation (SDRC). IDEAS contains pre- and post-processing tools for generating 2D and 3D meshes, and visualizing the solution to the forward problem. The meshes can be generated automatically from hybrid surface and solid geometry, which can

Forward problem in electrical impedance tomography

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Figure 1. Example of segmented MRI data.

be created using a highly intuitive GUI (graphical user interface). It is a full feature-based modelling system, which means that the creation history as well as the actual geometry is stored so that modifications are easily implemented. There is also associativity between the geometry and the finite element mesh created so that the mesh can be readily updated to reflect any modification made to the geometry. The purpose of this work was to investigate the use of IDEAS to solve the forward problem. The IDEAS tool is equipped with a range of packages, which allow manipulation of the FEM mesh. It also includes geometric modelling software, which was used to import segmented MRI data to build the geometric model of the human head. 2. Method Although IDEAS is not specifically set up to deal with the electrical conductivity problem, it does however contain a thermal modelling simulation package which can be used to solve the piece-wise constant conductivity equation. In the thermal case we solve the heat equation: ∂U = c2 ∇ 2 U ∂t

(5)

and K (6) σρ where U (x, y, z, t) is temperature and t is time. K is the thermal conductivity of the body and in ordinary physical circumstances is a constant. σ is the constant of the specific heat of the material of the body and ρ is the density of the material. U is equivalent to the potential and c the conductivity in the forward problem. Initially our aim is to use IDEAS to produce geometrically accurate meshes to calculate the sensitivity matrix A. In this paper we show the procedure for constructing the geometry of the human head and the mesh generation. One of the specific difficulties with this research is the problem of the skull, which has an impedance 15 times greater than the brain in adults (Geddes and Baker 1967). To obtain accurate reconstructed images it is important that information c2 =

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Figure 2. B-splined MRO data.

regarding the geometry and conductivity of the human head is included in the forward problem. To achieve this we import segmented slices from human MRI scans (figure 1). Segmentation subdivides the image into its constituent parts or objects. The basic aim of this process is the identification of feature boundaries inside the human head by detection of discontinuities. The segmentation process should be automatic, but as illustrated in the image presented in figure 1, anomalies can occur in the feature extraction. These anomalies occur owning to the lack of clearly defined edges, segmenting these structures remains a challenging task. A number of researchers are addressing the problem of automatic segmentation by using prior information. Methods such as deformable models or active contours and shapes (Cootes et al 1994) can capture statistical information about the shape of the structure. Researchers are developing new segmentation methods (Dawant et al 1999) and in principle these methods should produce robust automatic segmentation algorithms for MRI data which could be used to create individual head models for each patient requiring EIT imaging. We manually segmented the MRI slices to avoid any problems with automatic segmentation. The images of the segmented MRI data were imported to IDEAS using DXF format. The segmented images contain too many data points to directly import into IDEAS. To overcome this problem each slice is fitted with a set of B-spline curves which approximate the contours of the boundaries between regions. Each B-spline is fitted interactivity. An example of the fitted spline is shown in figure 2. The slices are then lofted using normalized uniform rational B-spline (NURB) patches to produce a solid model. This is illustrated in figure 3. This process was repeated for the brain and the skull. A shell model was then constructed from the solid models, which consisted of brain, cerebro-spinal fluid (CSF), skull and scalp. The CSF layer is difficult to accurately model even with manual segmentation as it is only 1% (Bayford et al 1996) of the total head diameter. To overcome this problem we created a thin shell which approximated the layer. It was important that the CSF was modelled as a uniform layer. The conductivity of this layer (1.4 m, Geddes and Baker 1967) is greater than the brain (5.8 m) and could act as a shunt for the injected current to flow around the brain. The shell model can be meshed automatically with the IDEAS simulation tool. The tool offers mapped meshes, these are structured meshes with the disadvantage that holes can not be included in the geometry or free mesh which can include holes. The mesh density is controlled by the element length, which is defined by the user. The user can also measure and


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Figure 3. Lofting using normalized uniform rational B-spline (NURB).

examine statistics for the deformity of the elements. A value of 0 to 1 is given which expresses the amount of deformity, where 0 is highly deformed. Other means of establishing element deformities are also provided. Stretch is a measure of the radius of the largest circle which will fit in the triangle of an element face divided by the longest side of that triangle; this value is then normalized as a comparison to the ideal equilateral triangle. Values well below 1.0 may therefore present problems in the final solution (Silvester and Ferrari 1996). Included angles can also be evaluated to ensure that very few or no elements have corner angles below a specified limit. Elements with corner angles which are too small will create inaccuracies in the solution. It must be pointed out, however, that to automatically generate a mesh with no element distortion is virtually impossible with geometry as complex as that generated for the head. IDEAS offers a range of element types for both two- and three-dimensional modelling; these include quadrilateral, triangle, brick, wedge and tetrahedron. We chose tetrahedral elements; this allowed us to deal with the complete structure in the head. The head model required meshing a number of times to minimize the deformity of the elements.

3. Results Each compartment of the model was meshed independently. Figure 4 shows the scalp, skull, CSF and brain with 23 219, 63 982, 47 140 and 21 574 elements respectively. The fully assembled model contains over 155 000 elements. The resultant mesh had a very small percentage of its elements with a stretch less than 0.3, the mean being a little less than 0.7. This equated to a similarly small percentage of elements with minimum included face triangle angles of less than 15 degrees, the mean being a little less than 40 degrees. The statistics for the stretch and minimum element angle are shown in figure 5. An example of the solution for half the head model with a single source and sink is shown in figure 6. A current of ±1 mA was injected on diametrically opposed points on the head model to simulate a polar drive. This gives a greater current density in the brain region of the human head compared to current injected using an adjacent pair of electrodes (Bayford et al 1996). Current was injected on the boundary using a single node at the front and back

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Figure 4. Brain meshed, CSF meshed, skull meshed and scalp meshed.

of the head. This required approximately 2 minutes to solve on an Intel Pentium III 500 MHz with 128 MB RAM. The PC used Microsoft Windows NT 4 (Service Pack 6) as the operating system. The maximum time mesh each compartment of the head required was approximately 30 s. 4. Discussion We have investigated other meshes generated for the human head from packages for electroencephalography (EEG) and magnetoencephalography (EMG) source location. These included: CURRY (Buchner et al 1997), ASA (www.ant-software.nl) and BESA (www.besa.de). Only CURRY has a FEM capability at present; the others are based on BEMs. CURRY offers similar quality meshes to IDEAS, but it is limited to a mesh element size of


Figure 5. Statistics for stretch (left) and minimum element angle (right).

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0

(a)

(b)

Figure 6. Initial solutions for the head model using IDEAS. A current of ±1 mA was injected in the front and back of the head. (a) A continuous plot of the magnitude of the current. (b) A vector plot showing the direction of the current flow.

1.2 mm. We have also been provided with a mesh from the University of Utah (Weinstein et al 1995). This mesh has 59 379 nodes and 327 015 elements. We are in the process of evaluating this mesh. Creating meshes for complex structure of the human head is difficult. No one tool appears to exist that can automatically segment MRI or computer tomography (CT) images then provided suitable meshes for EIT. However, IDEAS provide a route for generating the FEM meshes from segmented data. The elements need to be small enough to follow the contours of the region between the scalp, skull and CSF. The human head can be modelled as a set of thin shells. The accuracy of the results will depend on the number of elements in the thin shells. In this model the CSF which is the thinnest layer is approximately two elements thick. A key problem when meshing any thin shell is the deformation of elements. If the angles of the elements are too acute then the solution to the forward problem will have large errors in it. To simplify the model of the skull we have removed the lower jaw; we plan to improve the geometrical accuracy of this structure in our next model. A file can be generated from the meshed head model, which contains details of the elements and nodes in text format known as a universal file format. The file can then be used to generate the sensitivity matrix A. At present we are using TOAST to create the sensitivity matrix. The desired mesh quality for the forward problem should have elements with ideal equilateral triangle, and hence a stretch factor of 1. At present, to our knowledge, the limits on the deformity of the mesh are not known. We have achieved a minimum stretch factor of 0.008 with a mean value of 0.68. This may be improved by using a different type of mesh element. The accuracy of the forward model may not need to be the same as the accuracy required for the sensitivity matrix. We hope to investigate the limits of deformity and accuracy in future research. The advantage of using a commercial package like IDEAS as a forward solver is its adaptability. Difficult regions can be remeshed to improve the accuracy of the results and the geometry can easy be changed to accommodate holes to model eyes sockets. These could have a major effect of the current distribution in the brain and effect the location of impedance


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changes indicated by the reconstruction algorithm. The geometry and conductivity of the object can be easily exported to other FEM solvers allowing comparison. The next stage of this research is the incorporation of the new meshes into our EIT reconstruction algorithms. We are also planning to produce a mesh for neonates, which should, in principle, be easier to model as the skull will not have formed (Gibson et al 2000). We have made initial investigations into using IDEAS to solve the inverse problem by solving a single source and sink as shown in the results (figure 6). It may also be possible to use IDEAS to solve the inverse problem. We have not fully addressed this issue in the research presented here, as the focus of the research was the use of IDEAS to generate improved head meshes. IDEAS is a commercial package; this limits our access to the source code to investigate modifications that may be required to adapt it fully to the EIT problem. We hope to explore this in future research. We are aiming to offer the meshes generated using IDEAS to other researchers through the EIDORS (electrical impedance and diffuse optical reconstruction software) project (Vauhkonen et al 2000). We believe that incorporating this new FEM model into the reconstruction algorithm will improve the quality of the impedance images obtained from the head.

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Vauhkonen M, Lionheart W R B, Heikkinen L M, Vauhkonen P J and Kaipio J P 2000 A Matlab Toolbox for the EIDORS project to reconstruct two and three dimensional EIT images 2nd EPSRC Engineering Network Meeting, Biomedical Applications Weinstein D M, Johnson C R and Schmidt J A 1995 Effects of adaptive refinements on the inverse EEG solution. Experimental and numerical methods for solving ill-posed inverse problems Proc. SPIE 2570 2–11