IOP PUBLISHING
PHYSIOLOGICAL MEASUREMENT
Physiol. Meas. 28 (2007) S45–S55
doi:10.1088/0967-3334/28/7/S04
Numerical modelling errors in electrical impedance tomography Hamid Dehghani1 and Manuchehr Soleimani2 1 2
School of Physics, University of Exeter, UK William Lee Innovation Centre, School of Materials, University of Manchester, UK
E-mail:
[email protected] and
[email protected]
Received 12 February 2007, accepted for publication 4 April 2007 Published 26 June 2007 Online at stacks.iop.org/PM/28/S45 Abstract Electrical impedance tomography (EIT) is a non-invasive technique that aims to reconstruct images of internal impedance values of a volume of interest, based on measurements taken on the external boundary. Since most reconstruction algorithms rely on model-based approximations, it is important to ensure numerical accuracy for the model being used. This work demonstrates and highlights the importance of accurate modelling in terms of model discretization (meshing) and shows that although the predicted boundary data from a forward model may be within an accepted error, the calculated internal field, which is often used for image reconstruction, may contain errors, based on the mesh quality that will result in image artefacts. Keywords: electrical impedance tomography, model-based image reconstruction, forward model, finite-element method, mesh resolution (Some figures in this article are in colour only in the electronic version)
1. Introduction Electrical impedance tomography (EIT) is a non-invasive imaging technique that aims to reconstruct images of internal electrical property (conductivity, permittivity and permeability in some high-frequency non-medical applications) distributions from a set of finite and incomplete electrical measurements obtained on the periphery of a domain. EIT has been used to study a variety of clinical problems such as lung ventilation (Newell et al 1993, Metherall et al 1996, Noble et al 1999), cardiac volume changes (Hoetink et al 2001), gastric emptying (Erol et al 1996), head imaging (Bagshaw et al 2003) and breast cancer detection and characterization (Wang et al 2001, Cherepenin et al 2002, Kerner et al 2002). 0967-3334/07/070045+11$30.00 © 2007 IOP Publishing Ltd Printed in the UK
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In EIT, measurements over a region of interest are obtained from a set of electrodes by applying currents and measuring the resulting voltages or alternatively by applying voltages and measuring the induced currents. Depending on the application, various current or voltage driving schemes have been used for electrode excitation, including stimulation of adjacent and opposite pairs or trigonometric spatial patterns (Isaacson 1986, Boone et al 1997). Once boundary measurements are acquired, estimates of the electrical property distributions in tissue can be determined through the appropriate model-based matching of the data (Lionheart 2004, Dehghani et al 2005). The majority of modelling and image reconstruction studies have involved model-based image reconstruction, whereby the measured boundary data are simulated and predicted through a numerical model-based estimation, most commonly using finite-element models (Kerner et al 2002, Molinari et al 2002a, 2002b, Bagshaw et al 2003, Soni et al 2003, Lionheart 2004, Dehghani et al 2005). Although a large number of studies have been done to improve both the experimental setup and instrument development and the use of various reconstruction algorithms (Kolehmainen et al 1997, Borsic et al 2002, Lionheart 2004), little work has been performed to investigate the accuracy of the actual numerical models used. Specifically, although advancements have been made in realistic three-dimensional (3D) modelling of volumes of interest, little work has been done to investigate the errors introduced when the resolution of the mesh discretization is too low to ensure numerical accuracy and stability. Although work has been published that looks at adaptive meshing and mesh refinement in EIT (Johnson and MacLeod 1994, Molinari et al 2001, 2002b), to date, no work has been published that quantifies errors which are primarily due to mesh resolution. Given a discretization of a domain, it is possible to generate simulated boundary data, as well as the internal field due to applied current or voltages. However, the accuracy of such simulations depends largely on the level of discretization, or mesh resolution. Errors in the calculated internal fields, will ultimately lead to errors in reconstructed images of internal electrical properties, and therefore it is essential to quantify such errors as they will lead to image artefacts. This work aims to highlight such problems and introduce methods that can be used to evaluate and quantify modelling inaccuracies associated with numerical approximation to the forward model in EIT. It is of specific interest to be able to establish, that although boundary data may be predicated with some accuracy, the forward model used itself may contain errors that will produce inaccuracies within the reconstructed images in EIT. Specifically, it will be shown that poor spatial resolution in the discretization of the forward model will produce inaccuracies that will present themselves with different magnitude at different locations within a model, depending on the volume being imaged. 2. Theory As defined previously (Vauhkonen 1997, Dehghani et al 2005), under low-frequency assumptions, the full Maxwell’s equations can be simplified to the complex-valued Laplace equation: ∇ · σ ∗ ∇∗ = 0
(1)
where ∗ is the complex-valued electric potential and σ ∗ is the complex conductivity of the medium (σ ∗ = σ − iωε0 εr , for ω is the frequency, ε0 and εr are the free space and relative permittivity). In this work, we use the complete electrode model, which takes into account
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both the shunting effect of the electrodes and the contact impedance between the electrodes and tissue: ∂∗ ∗ + z l σ ∗ = Vl∗ , dn ∂∗ dS = Il∗ , σ∗ dn el σ∗
∂∗ = 0, dn
x ∈ el , l = 1, 2, . . . , L
(2)
x ∈ el , l = 1, 2, . . . , L
(3)
x ∈ ∂ ∪Ll el
(4)
where x is a point within the domain, zl is the effective contact impedance between the lth voltage, I∗ is the electrode and the tissue, n is the outward normal, V∗ is the complex-valued L complex-valued current and el denotes the electrode l. x ∈ ∂ ∪l el indicates a point on the boundary not under the electrodes. 2.1. Finite-element implementation The finite-element discretization of a domain of interest can be obtained by subdividing the domain into D elements joined at N vertex nodes. In finite-element formalism, (r) at spatial by a piecewise continuous polynomial function N point r is happroximated h u (r) ∈ , where is a finite-dimensional subspace spanned by basis h (r) = i i i functions {ui (r); i = 1, . . . , N } chosen to have limited support. The problem of solving for h then conveniently becomes a sparse matrix inversion for which many methods are available (Press et al 1992). In this work, we use a bi-conjugate gradient stabilized solver with a tolerance level of 1 × 10−12. Equation (1) in the finite-element model (FEM) framework can be expressed as a system of linear algebraic equations: (K(σ ∗ ) + z−1 F )∗ = 0
(5) ∗
where z is the effective contact impedance and the matrices K(σ ) and F have entries given by Kij = σ ∗ (r)∇ui (r).∇uj (r) dn r (6)
Fij =
ui (r)uj (r) dn−1 r
(7)
∂∈l
where δ ∈ l is the boundary under each electrode. 3. Methods In order to evaluate the effect of mesh resolution on numerical modelling accuracy, a total of seven, three-dimensional (3D) meshes were generated using NETGEN (Schoberl), figure 1. The meshes were constructed such that they approximated a symmetrically ideal elliptic breast shape with a dimension of 100 mm in length (z direction) and a diameter of 102 mm at the plane of measurements. Care was taken to model the exact size and shape of 16 electrodes at a single plane, placed midway along the length of the breast model (z = −50 mm). Each mesh, whose specific resolution details are given in table 1, contained 16 circular electrodes of diameter of 5 mm and for each specific mesh, each electrode has an equal and exact number of nodes on the external boundary (19 nodes), to reduce error bias
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(a)
(b)
(c)
Figure 1. Plot of the breast shaped (a) high resolution mesh (mesh 1) used as reference, (b) medium resolution mesh (mesh 4), and (c) low resolution mesh (mesh 7). Table 1. Mesh identification and resolution in terms of minimum, maximum and average element volume, together with the associated error for the calculated boundary data.
Mesh
Number of nodes
Number of elements
Minimum element volume (m3)
Maximum element volume (m3)
Average element volume (m3)
Boundary data error (equation (8))
1 2 3 4 5 6 7
59 228 55 820 39 115 13 537 4 841 4 270 3 796
288 353 273 337 186 656 61 870 21 039 18 824 16 981
1.31 × 10−10 1.31 × 10−10 1.34 × 10−10 1.28 × 10−10 1.27 × 10−10 1.46 × 10−10 1.27 × 10−10
1.47 × 10−8 1.59 × 10−8 1.72 × 10−8 9.27 × 10−8 6.98 × 10−7 9.00 × 10−7 2.70 × 10−6
2.6 × 10−9 2.8 × 10−9 4.0 × 10−9 1.2 × 10−8 3.6 × 10−8 3.9 × 10−8 4.4 × 10−8
— 0.06 × 10−3 0.28 × 10−3 0.39 × 10−3 0.42 × 10−3 0.51 × 10−3 0.61 × 10−3
between the meshes. The models all assumed bulk homogenous electrical properties of σ = 0.47 S m−1 and εr = 12 312, which have been previously measured in a study of 23 women
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with negative clinical findings (Poplack et al 2004). All of the data presented in this work were confined to an excitation frequency of 125 kHz. Boundary data were calculated assuming a ‘current’ mode, whereby a set of current patterns are applied at each electrode simultaneously and measuring the resulting voltages at the same electrodes. In this work, a set of 15 sinusoidal current patterns (maximum amplitude of 1 mA) distributed circumferentially are applied, to give rise to 240 boundary voltage measurement. For each set of calculated boundary data, the error between each mesh was calculated, as compared to the highest resolution mesh (mesh 1): error =
y − yref 2 yref 2
(8)
where y is the real part of the calculated boundary data for a given mesh, and yref is the real part of the calculated boundary data for the finest mesh. This work is constrained to analysing the real part of the complex data, but the presented trend is also present in the imaginary part. However, it should be noted that the magnitude of the errors within the imaginary data will increase with increasing frequency. The resulting internal field within each mesh was also calculated and linearly interpolated onto two sets of two-dimensional grids: (a) a uniform grid of 40 000 points (200 × 200) that was confined only to the x and y plane of measurement (at z = −50 mm), and (b) a uniform grid of 40 000 points (200 × 200) that was confined perpendicular to the plane of measurement (at y = 0 mm). The magnitude of the residual error was calculated as residual = | − ref |
(9)
where is the real part of the internal scalar field for a given mesh and ref is the real part of the internal scalar field for the finest mesh. The linear interpolation function was identical in each case and each model to reduce errors. Two 3D meshes were also constructed whereby a single higher conductivity anomaly (σ = 1 S m−1 and εr = 12 312) of radius 10 mm was placed at mid-point between the external boundary and the centre of the imaging plane, centred at x = 25 mm, y = 0 mm and z = −50 mm. The first mesh created, figure 2(a), was such that to ensure a high-density resolution at the interface of the anomaly and the surrounding background, whereas the second mesh created assumed a homogeneous mesh resolution at the interface, figure 2(b). The internal field within each mesh was also calculated and interpolated onto the two sets of two-dimensional grids as defined above and the magnitude of the residual error was also calculated. However, in this case, given equation (9), is the real part of the internal scalar field for the mesh without a fine boundary meshing and ref is the real part of the internal scalar field for the mesh with a fine anomaly boundary. 4. Results The magnitude of error in the boundary data, as calculated by equation (8), is shown in table 1. It is seen that as the mesh resolution is increased, the magnitude of the error in the calculated data is decreased. As a comparison, the error in the boundary data, for the highest resolution mesh with 1% added (randomly distributed) noise was also calculated to be 2.39 × 10−5 and for 2% noise it was calculated as 9.1 × 10−5. The quality of the mesh resolution, in the context of this work, is defined as the volume of the largest element within a given mesh, and it should be noted that smallest element in all meshes correspond to those placed directly under each electrode.
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(a)
(b)
Figure 2. Cross section of 3D meshes used in the presence of a single anomaly. (a) Mesh with fine resolution at the anomaly and background boundary and (b) mesh with equal resolution at the anomaly and background boundary.
In order to evaluate the region within each mesh, which would give rise to numerical inaccuracies, the residual error for a single sinusoidal current pattern was calculated for three meshes at two specific planes of interest and these are shown in figure 3. It is seen that the largest residual errors are seen near the electrodes where that gradient of the applied electric field is the largest. However, with the mesh with the largest element volume, the residual error, deep within the volume, away from the electrodes is also significant. Finally, to evaluate the numerical error due to poor mesh resolution at anomaly and background interfaces, the residual error for a single current pattern was calculated at the two planes of interest and are shown in figure 4. It is seen that although the largest errors are seen near the electrodes, there exists a significant error at the anomaly boundary.
5. Discussions In order to evaluate the numerical errors introduced by meshing resolution in finite-element modelling for image reconstruction in EIT, a set of 3D breast-like meshes have been used. In each case, the meshes were ensured to have identical electrodes modelled as part of the mesh, in terms of geometry and number of nodes on the external boundary. A fine mesh, consisting of 59 228 nodes corresponding to 288 353 linear tetrahedral elements, figure 1(a), was used as the reference mesh, and a number of other meshes of lower resolution but with exact geometry were also used to evaluate errors in the calculated boundary measurements. Mesh quality, for the purpose of this work, is defined as the range of element volumes as well as the ‘average’ element volumes for each mesh. The results, as shown in table 1, show that as the element volume within the model increases, the error in the calculated boundary measurement also increases. Given that the presence of 1% added (randomly distributed) noise within the boundary data calculated from the finest mesh would correspond to a calculated error of 2.39 × 10−5, it is seen that even with a mesh resolution closest to the fine mesh, the
Numerical modelling errors in electrical impedance tomography XY Plane (z = -0.05m)
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XZ Plane (y = -0.05m)
10.0
0.0
Figure 3. Images of residual error associated with each mesh as compared to the fine resolution mesh (mesh 1). Top row corresponds to residual error in mesh 2, middle row to mesh 4 and bottom row to mesh 7. The solid dots around the periphery indicate the position of the electrodes.
calculated error in boundary measurement corresponds to a larger error than that expected in most experimental instruments (Halter et al 2004). In order to evaluate the source of such errors, the residual in the internal scalar field of each model was calculated with respect to the finest mesh, and these are shown in figure 3. It is seen that the largest error within each mesh is associated with region directly under each electrode, which is the region where the gradient of the applied electric field is the largest. This is not surprising, since in the area of a rapidly changing quantity, the discretization of the region of interest must be such that to ensure numerical accuracy. In finite-element formulation, whereby the quantity of interest within each element is approximated by a basis (shape) function, the residual will only vanish as the size of the finite elements approach zero. In reality, this is not possible, and therefore mesh discretization must be such that to ensure minimum residual error. As the overall mesh resolution is decreased, the residual error away from the electrodes is also shown to increase where the gradient of the field is not large as compared to near the electrodes. This result indicated that it is not only the areas under the
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XZ Plane (y = 0m) 2.0
0.0
Figure 4. Images of residual error in the presence of an anomaly in the imaging plane.
electrodes that must have a fine resolution, but also other regions within the volume must also be adequately meshed to minimize residual error. In the case of anomalies present within the imaging domain, two sets of meshed models were investigated, figure 2, where (a) the boundary between the anomaly and the background is meshed using layers to ensure high resolution meshing and (b) where no additional layers were used and therefore the mesh resolution is comparable to the surrounding area. The residual error between these two meshes were calculated and the cross sections of these are shown in figure 4. It is seen from these results, that although errors directly near the electrodes exist, there are also errors associated at the boundaries of the modelled anomaly. At the interface of a change in conductivity, the rate of change of the induced electric field is also large and the results indicate that these must also be accounted for. The results presented have several implications. It is not always adequate to assume that given a set of boundary data that matches closely that of those obtained from other models, the model used is accurate. This becomes more important for image reconstruction in EIT, since the reconstructed images rely on the internal field calculated (Polydorides and Lionheart 2002, Dehghani et al 2005). The sensitivity matrix (also known as the Jacobian or weight matrix) which relates a change in measured boundary data to changes in the internal impedance changes rely on accurate calculations of the internal fields. Large errors in these calculations will in turn lead to large errors in the reconstructed images by means of image artefacts. Consider, for example, the image in figure 5, which is a cross-section (xz plane) of the normalized difference in the two Jacobians calculated using the two meshes shown in figure 2. This plot demonstrates that although the largest error seen is directly under the electrodes, there is an associated error due to the meshing of interfaces between regions where there is a large change in conductivity. It is worth noting here that in industrial applications of EIT, where the change of conductivity within regions is greater, this error may also be larger. The results, presented within this work, indicate that the most likely areas of reconstructing image artefacts lies at areas of large potential gradient, such as those found directly under the electrodes and those at anomaly and background interfaces. These may be of paramount importance, given that in EIT the sensitivity near the external boundaries is much larger than those deep within the volume, and therefore any associated error within these regions will result in image artefact. Also, given that in EIT one aims to reconstruct the size, shape and
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XZ Plane (y = 0 m) 1%
0
Figure 5. Images of percentage error in the calculated Jacobian for meshes in the presence of an anomaly in the imaging plane.
location of anomalies within the volume of interest, errors in modelling of such boundaries become significant.
6. Conclusions The use of numerical models within EIT has been investigated, specifically concerning the associated inaccuracies with mesh resolution. It is shown that given a set of boundary data, although they can be predicted with some desired accuracy, the accuracy of the calculated internal field may not be high. There are several situations where mesh resolution becomes important, namely in the areas of large field gradients such as those seen directly under the electrodes and those at the interface of anomalies within the volume of interest. This paper has attempted to highlight the importance of accurate modelling and meshing within EIT, and to demonstrate their importance in image reconstruction. In the inverse problem, there are many developed techniques that can be incorporated to allow for uncertainties or errors to be penalized (suppressed) (Lionheart 2004, Arridge et al 2006) but this suppression of errors does not necessarily imply that the model is numerically correct or appropriate to provide the most adequate and accurate information. It is also important to acknowledge that this study has assumed correct a priori information regarding the external boundary and geometry of the domain being investigated. It is well known that inaccurate information regarding the geometry itself can lead to errors within the modelling and image reconstruction algorithm, which has been investigated in other similar imaging modalities (Gibson et al 2003, Xu et al 2005), which have also shown a high sensitivity to errors due to incorrect information about the model geometry. An interesting area of further study, which can be based on the presented result, includes establishing statistical analysis of the discretization error. Such an analysis and method can be used for the development of an enhanced error model (Arridge et al 2006) that allows the use of a low-density mesh for the forward model by tacking into account the associated modelling error.
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Although some work has been done in the use of adaptive meshing to improve accuracy (Johnson and MacLeod 1994, Molinari et al 2001, 2002b), further work is needed in identifying the most suitable method which can also be incorporated within image reconstruction. Of specific interest would be the effect of the unknown conductivity and permittivity on mesh resolution, which will be the subject of future investigation.
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