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Compensation of Modelling Errors Due to Unknown Domain Boundary in Electrical Impedance Tomography Antti Nissinen, Ville Petteri Kolehmainen*, and Jari P. Kaipio
Abstract—Electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors. This instability is especially severe when absolute imaging is considered. With clinical measurements, accurate knowledge about the body shape is usually not available, and therefore an approximate model domain has to be used in the computational model. It has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect. In this paper, we adapt the so-called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape. This approach has previously been shown to be applicable to a variety of modeling errors, such as coarse discretization in the numerical approximation of the forward model and domain truncation. We evaluate the approach with a simulated example of thorax imaging, and also with experimental data from a laboratory setting, with absolute imaging considered in both cases. We show that the related modeling errors can be efficiently compensated for by the approximation error approach. We also show that recovery from simultaneous discretization related errors is feasible, allowing the use of computationally efficient reduced order models. Index Terms—Bayesian inversion, electrical impedance tomography, inverse problem, modelling errors, reduced order model.
I. INTRODUCTION
I
N electrical impedance tomography (EIT), electrodes are attached on the boundary of a body and currents are injected into the body through these electrodes. The voltages on all electrodes are measured and the conductivity of the body is reconstructed based on the measured voltages and known currents. The biomedical applications of EIT include detection of the breast cancer [4], [50], imaging of the brain function [44], and monitoring of the lung ventilation [12], [13], [47]. For reviews of EIT, see [5], [8], and [24]. Manuscript received January 08, 2010; revised June 17, 2010; accepted August 29, 2010. Date of publication September 13, 2010; date of current version February 02, 2011. This work was supported by the Academy of Finland under Project 119270, Project 140731, and Project 213476, Finnish Centre of Excellence in Inverse Problems Research 2006-2011. Asterisk indicates corresponding author. A. Nissinen is with the Department of Physics and Mathematics, University of Eastern Finland, FIN-70211 Kuopio, Finland. *V. P. Kolehmainen is with the Department of Physics and Mathematics, University of Eastern Finland, FIN-70211 Kuopio, Finland (e-mail:
[email protected]). J. P. Kaipio is with the Department of Physics and Mathematics, University of Eastern Finland, FIN-70211 Kuopio, Finland and also with the Department of Mathematics, University of Auckland, Auckland 1142, New Zealand. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMI.2010.2073716
The reconstruction of the conductivity is a nonlinear inverse boundary value problem which is highly unstable with respect to measurement and modeling errors. The effect of the measurement errors can be reduced by using an accurate measurement system and by careful modeling of the statistics of the measurement error. Modeling errors, on the other hand, are often related, for example, to discretization of the forward model, truncation of the computational domain and unknown boundary data. Furthermore, a common modeling error in medical EIT is related to inaccurate knowledge on the shape of the target body. Most of the available reconstruction methods assume that the boundary of the target body is known. As an example, consider EIT measurements of pulmonary function from the surface of the thorax. In principle, the shape of the patient’s thorax could be obtained from other imaging modalities such as computerized tomography (CT) or magnetic resonance imaging (MRI). However, such information is often not available and therefore the reconstruction has to be computed using an approximate model domain. Furthermore, the shape of the thorax varies in time due to breathing and is also dependent on the orientation of the patient. Therefore, the body shape, or domain geometry in general, would be inaccurately known even at the best. The use of an incorrect model domain has been shown to produce severe errors in the reconstructed conductivity images, see [14], [22], [26], and [49]. The traditional way to circumvent the problem of inaccurately known body shape has been to use difference imaging. The objective in difference imaging is to reconstruct the change in the conductivity between two measurement times (or two frequencies) using a first-order linear perturbation model [3]. The approach is highly approximative since the actual nonlinear forward mapping is approximated by a linear one, but it is fast since the related Jacobian mappings are precomputed (at some guessed conductivity), and iterations are not possible in the first place. Furthermore, the choice of reconstructing conductivity differences based on differences of measurements is known to reduce the effect of inaccurately known geometry to an extent. In spite of the difference imaging modality being able to suppress some of the effects of model uncertainties, it has been shown that the breathing artefacts are still present in the reconstructions [1]. Furthermore, in high contrast cases such as accumulation of well conducting liquid (haematothorax) or poorly conducting air (pneumothorax) in the lungs, the linear approximation used in difference imaging may be insufficient for the
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detection of the clinically relevant conditions in the lungs [16]. Also, the detection of pneumothorax becomes difficult using difference imaging if the change in the lungs has occurred prior to the measurements [10]. In addition to difference imaging, only a few approaches have been proposed for compensating the effect of inaccurately known boundary shape in EIT. Simultaneous reconstruction of the conductivity and electrode movement have been proposed for difference imaging in [11] and [40]. These approaches are based on a linearized perturbation model and have been evaluated only for relatively small movements of the boundary between the measurement states in difference imaging. Recently, it has been demonstrated that the so called D-bar method, which is a direct method based on a constructive uniqueness proof for 2-D EIT [35], is quite robust against domain modeling errors [34]. The method proposed in [27] and [29] eliminates the errors caused by inaccurately known body shape in 2-D EIT by using the theory of Teichmuller mappings. The method in [27] and [29] is based on the result that there is a unique anisotropic conductivity in the model domain that corresponds to the (noiseless) boundary data on the boundary of the true domain, and has minimal anisotropy. The numerical implementation of the method finds the minimally anisotropic conductivity by minimization of a regularized least squares functional and displays the determinant of the anisotropic conductivity as a deformed image of the original isotropic conductivity. The extension of the method to 3-D EIT was considered in [28]. In this paper, we propose the reduction of the reconstruction errors caused by inaccurately known body shape in EIT by using the Bayesian approximation error approach, which was originally proposed in [23] and [25]. The key idea in the approximation error approach is, loosely speaking, to represent not only the measurement error, but also the effects of the computational model errors and uncertainties as an auxiliary additive noise process in the observation model. The realization of the modeling error is obviously unknown since its value depends on the actual unknown conductivity and body shape. However, the statistics of the related approximation error can be estimated over the prior distribution models of the conductivity and the parameterization of the body shape. The approximation error statistics are then used in the reconstruction process to compensate for the inaccurately known body shape. The approximation error approach was originally applied for discretization error in EIT with numerical examples in [23]. For this reason, the term “approximation error” is commonly used also where “modelling error” might be a more appropriate term. The approach was verified with real EIT data in [36], where the approach was employed for the compensation of discretization errors and the errors caused by inaccurately known height of the air–liquid surface in an industrial mixing tank. The application of the approximation error approach for the discretization errors and the truncation of the computational domain was studied in [33], and for the linearization error in [39]. In [37] the approach was evaluated for the compensation of errors caused by coarse discretization, domain truncation and unknown contact impedances with real EIT data.
In addition to EIT, the approximation error approach has also been applied to other inverse problems and other types of (modelling) errors: model reduction, domain truncation and unknown anisotropy structures in optical diffusion tomography were treated in [2], [18], [19], and [30]. Missing boundary data in the case of image processing was considered in [6]. In [43], again related to optical tomography, an approximative physical model (diffusion model instead of the radiative transfer model) was used for the forward problem. In [31], an unknown uninteresting distributed parameter (scattering coefficient) was treated with the approximation error approach. The extension and application of the modelling error approach to time-dependent inverse problems was considered in [20], [21], and [48]. In this paper, we apply the approximation error approach to the compensation of the modeling error that is caused by unknown body shape in EIT. We consider the absolute imaging problem only. The approach is evaluated with a simulated 2-D example of thorax imaging, and a 3-D example with experimental data from a thorax shaped measurement tank. The results show that the reconstruction errors caused by the inaccurately known body shape can be reduced significantly by employing the approximation error approach. We also demonstrate that the simultaneous treatment of the modeling errors caused by unknown body shape and coarse discretization of the forward model is feasible, allowing the use of computationally efficient reduced order models, in the case of this paper, sparse finite element meshes. The rest of this paper is organized as follows. In Section II, a brief review of the EIT observation model, the Bayesian formulation of the EIT problem and the approximation error approach is given. The practical construction of the approximation error method for the particular problem of unknown boundary shape is explained in Section III. The proposed approach is evaluated using simulated 2-D EIT data in Section IV and with experimental EIT data in Section V. Discussion of technical topics and conclusions are given in Sections VI and VII, respectively. II. ELECTRICAL IMPEDANCE TOMOGRAPHY A. Forward Model and Notation Let , , 3, denote the measurement domain and let denote a parameterization of the domain boundary . In EIT, a set of contact electrodes are attached on the boundary . Using the electrodes, electric currents are injected into the body and corresponding voltages are measured using the same electrodes. We model these measurements with the complete electrode model [9], [41]
(1) (2) (3)
(4)
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where , is the potential distribution inside , is , is the conductivity, the outward unit normal vector at and is the contact impedance between the object and the electrode . The currents satisfy the charge conservation law (5) and a ground level for the voltages can be fixed by (6) The numerical approximation of the forward model (1)–(6) is usually based on the finite element (FEM) approximation. In the FEM approximation, the domain is divided into disjoint elements joined at vertex nodes. The potential and electrode potentials satisfying the variational form [41] of (1)–(6) are approximated as (7)
(8) where the functions are the nodal basis functions of the finite are chosen such that conelement mesh and vectors dition (6) holds. The parameter denotes the size of the largest element in the mesh and defines the discretization level. The theory of elliptic operators guarantees that [41]
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dent with the unknown conductivity. This leads to measurement model (11) where is the vector of the measured voltages, is the conductivity vector, and is a Gaussian distributed and covariance matrix measurement noise with mean . of the target body is not known Note that if the boundary accurately ( is incorrect) or the FEM discretization is too coarse ( is too large), the error in the FEM approximation may become significant compared to the measurement error . Together with the fact that the reconstruction of the conductivity is an ill-posed problem, this modeling error can easily lead to significant artefacts in the reconstructed conductivity. B. Bayesian Approach for EIT Inverse Problem In this section, the Bayesian framework for the EIT inverse problem is reviewed briefly. For more details of the Bayesian framework for inverse problems in general, see [7], [23], [24], [42], and the approximation error approach in particular, for example [2], [23], [25], [31]. In the Bayesian framework, all unknowns and measurements are considered as random variables and the uncertainty related to their values is encoded in their probability distribution models. The joint probability density of conductivity , can be boundary parametrization , and the measurements written as (12)
where is the solution of the variational formulation of is approximated in a basis (1)–(6). The conductivity (9) typically corresponding to a separate finite element type mesh, where are the nodal basis functions. In the following, we in (9) with the coefficient vector identify the conductivity . By these choices, the numerical forward solution for each current injection is obtained by solving a system of equations. For further details on the FEM approximation of the complete electrode model and the particular implementation used in this paper, see for example [24], [45]. In the following, we use notation: (10) for the FEM based forward solution corresponding to single EIT experiment, that is, the vector contains computed voltages for all the current patterns in the chosen measurement paradigm. The dependence of the forward model on the domain is expressed by the parameterization of the boundary and the subindex denotes the discretization level parameter in (7). The measurement noise in EIT experiments is commonly modeled as Gaussian additive noise which is mutually indepen-
where is the likelihood model and the probability denis the prior model of and . The posterior density, sity which is given by the Bayes formula (13) is the complete probabilistic model of the EIT problem and represents the uncertainty in the unknowns given the measurements. In conventional approaches to EIT, the domain boundary is assumed to be known. Let denote an inaccurate model of the (actual) measurement domain , and let be a parameteriza. In the sequel, the tilde refers to the tion for the boundary models that are to be used in the inversion. In the Bayesian formulation, all variables that are known, such as measurements, or are treated as fixed parameters, appear as conditioning vari, instead of in (13), we ables. Thus, if we fix actually consider (14) Formally, the uncertainty in the primary interesting unknown is obtained by marginalization (integrating) over in (13)
(15)
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The posterior uncertainty of that is predicted by (14) is usually significantly overoptimistic when compared to the actual uncertainty given by (15). With this we mean that the density (14) tends to be significantly narrower than (15), and also most of the mass of the density (14) may be located differently than the mass of (15). In particular, any point estimates, such as the maximum a posteriori estimate (see below), are bound to be highly misleading. It is important to note that generally with any . Unfortunately, the integral in (15) does not generally have an analytical solution and can be computed only with the often excessively resource demanding Markov chain Monte Carlo approach, see for example [23]. For this reason, approximations are usually needed to be considered in applications with limited computational resources. Assuming that the conductivity and the boundary shape are mutually independent with the additive noise in (11), it follows that the likelihood can be written as
where
(16) is the probability density of the noise . In the folwhere can be modelled as mutually inlowing, we assume that dependent so that . Moreover, let the prior equipped with model be the Gaussian distribution the positivity constraint, so that we can write
where is the prior mean and the prior covariance when all elements of are nonnegative, matrix and and zero otherwise. Then, the posterior density of given both the measurements and the boundary shape becomes
(17) Note that the distribution (17) represents the posterior uncertainty in only if is an adequately accurate approximation for and, in particular, if the that is used as a fixed parameter in (17), corresponds to the actual boundary. In the Bayesian approach, the solution is analyzed and visualized by computing point and spread estimates for the posterior model (17), see for example [23]. In practical problems with restricted computational resources, the most commonly computed point estimate is the maximum a posteriori (MAP) estimate, which in the case of (17) leads to the following minimization problem
(18)
and
are Cholesky factors such that
The minimization problem (18) can be solved, for example, by the Gauss-Newton algorithm [38]. We refer to the solution of (18) as MAP with conventional error model (MAP-CEM). Note that when (18) is computed, may or may not correspond to the actual boundary. C. Approximation Error Approach In this section, we explain how the modelling error caused by using the model domain instead of the actual (unknown) domain in the computational (forward) model can be embedded in the likelihood model. Let (19) denote a (sufficiently) accurate model between the unknowns and measurements. Here the parameters of the boundary and discretization level parameter are such that the error in the FEM approximation is smaller than the measurement error. The conductivity is a parametrization in the actual and is dense enough in the above sense. As explained above, in practical clinical measurements one usually lacks accurate knowledge of the shape of the body and therefore the reconstruction is carried out using an approximate model domain . Furthermore, for reasons related to the computation time and resource, there is often pressure to keep the discretization level of the forward model relatively coarse. In such a case, the accurate model (19) is traditionally replaced by the approximate measurement model (20) where the discretization level parameter and are the of the model domain, and one parameters of the boundary hopes that the approximation in (20) is a feasible one. The model is the model that is to be used in the inversion, that is, the discretization level and the boundary are fixed. We refer to in (20) as the target model. the model The relation of the representation of the conductivities in (19) , where and (20) is of the form (21) is a mapping that models the deformation of domain to . Obviously, the true deformation between the measurement domain and model domain is not known, and one has to choose a model for the deformation. In the numerical examples considered in this study is chosen such that the angle and relative distance (between the center of the domain and the boundary) of a coordinate point is preserved. Although this simple deformation model seems to work well with the test cases we have considered, we note that other transformation models may be used as well. More advanced choices for the tranformation model can be sought for from the literature of image registration, see e.g.,
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[15]. The deformation of the conductivity can be represented by a linear transformation
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The computation of the MAP estimate from the posterior model (25) amounts to solving the minimization problem
(22) where is a matrix that interpolates the nodal conductivity [see (9)] in into a nodal conductivity in according to the deformation . In the approximation error approach, instead of writing the approximation (20), the accurate measurement model (19) is written in the form
(23) represents the modelling error due to the diswhere . cretization and incorrect boundary, and we denote Being a function of random variables, is a random variable as well as the marginal density and the joint density can be computed in principle, but in most cases these do not have analytical expressions. Furthermore, in what follows, , since we are to use the variable in the inversion and . we consider the joint distribution The objective in the modelling error approach is to derive for the posa computationally efficient approximation based on the measurement model (23). terior density Since and were modelled as mutually independent, and the only term that depends on the random variable in (23) is , the posterior model corresponding to (23) can be written as (24)
see [31] for details. A complication is that the likelihood in (24) does not in general have an analytic expression. To obtain , a computationally feasible and efficient approximation we make the Gaussian approximation for the joint distribution . This is the core of the most common implementation of the approximation error approach, in particular when computational efficiency is sought. Then, we obtain the Gaussian approximation for the likelihood in (24), and the approximation for the posterior model becomes
(25) where (26) (27) and where and . Notice that in the usual situation in which the measurement errors and conductivity are mutually independent, that is, , we in (25)–(27). have
(28) where the Cholesky factor . Thus, the MAP estimation problem with the modelling error approach is formally similar to the MAP estimation (18) with the conventional noise model, and therefore the functional (28) can be minimized with the same algorithms as the MAP with conventional noise model (18). We refer to the MAP estimate (28) as MAP with the approximation error model (MAP-AEM). While it is clear that and are not independent, it has turned out in several applications that a feasible approximation is oband . With this further aptained by setting , we have proximation, and the earlier assumption (29) in (26) and (27). This approximation is called the enhanced error model, see [23] and [25]. Note that in the case of nonlinear forward models, the mean and the covariances , , and in (26) and (27) need to be estimated based on Monte Carlo simulations. However, this task can be done offline and needs to be done only once for a given measurement setup, and for the expected range of uncertainties, in this paper, the range of boundary shapes and the specified prior model for the conductivity. III. ESTIMATES AND CONSTRUCTION OF THE APPROXIMATION ERROR STATISTICS In Sections IV and V, we consider simulated and real data, respectively. In both cases, the prior model is constructed similarly, and the approximation error statistics is computed likewise. In this section, we discuss these constructions. A. Estimates to be Computed The model domain that is to be used in the inversion, is circular with diameter . For both test cases, the following reconstructions were computed. 1) MAP with conventional noise model (18) (MAP-CEM) in the correct domain using fine discretization (forward model ). This reconstruction serves as reference reconstruction with conventional measurement error model when no domain modeling or discretization errors are present. 2) MAP-CEM in the model domain using fine discretiza), representing conventional recontion (model struction in presence of domain modelling errors. 3) MAP-CEM in the model domain using coarse discretization (model ), representing conventional reconstruction in the presence of combined domain modeling and discretization errors. 4) MAP with the approximation error approach (28) (MAPusing fine discretization AEM) in the model domain ). (model
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5) MAP-AEM in the model domain using coarse discretiza). tion (model The MAP-AEM estimates (4–5) were computed using the enhanced error model with the statistics of the modeling error approximated as in (29). The minimization problems (18) and (28) were solved with the Gauss-Newton algorithm using a line search algorithm [38]. The consumed CPU times were recorded in all cases. B. Prior Model As the prior model , we used a proper Gaussian smoothness prior, constructed similarly as in [2], [23], and [30]. In this paper, the conductivity is considered in the form
where is a spatially inhomogeneous conductivity with is a spatially homogeneous conductivity zero mean, and , of non-zero mean. For the latter, we can write . Setting where is a vector of ones and the basis for conductivity, we have the coordinates , , and set . We model and as mutually independent, that is, with respect to the prior model, the background conductivity is mutually independent with the inhomogeneities in the conductivity. (arbitrary For the homogeneous background, we set When constructing the prior covariance units) and , we set the variance of elements (diagonal elements of ) , and the correlation length to 7 cm. The correlation to length expresses roughly our prior estimate about the expected size of the inhomogeneities in the medium. This also means that in the model for , any two elements that correspond to spatial locations that are further away from each other than 7 cm, are (approximately) mutually independent. and Thus, we have the prior covariance
This prior model is a proper distribution, in that the covariance exists in the first place. This model also allows (restricted) variation for the background, see Fig. 1. Traditional smoothness prior models are improper, that is, the variances are infinite, and samples cannot be drawn from such distributions. The approximation error approach, on the other hand, is based on computing the statistics of over the prior distribution. This is not possible with a prior of unbounded variances. is needed only for drawing samples of The prior model the chest shapes for the estimation of the approximation error statistics. The samples of chest shapes can in practice be obtained either as random draws from the prior or by using real geometries from atlases of anatomical images. In this study, we employ the latter approach. That is, we do not construct the explicitly at all—a natural and representative set of model samples is instead obtained from a set of chest CT images of different individuals in the population. We note that if one would , the construction of a Gaussian wish to construct the model is straightforward based on the sample approximation for CT ensemble.
Fig. 1. Four 2-D samples f ; g for the construction of the approximation error model. The sample domains f g are from an ensemble of chest CT images of different individuals and the conductivities are drawn from the ). The unit of the conductivity is arbitrary. The color scale deprior model ( notes the common minimum to maximum scale of the conductivity values in the samples.
C. Estimation of Modeling Error Statistics In cases in which the measurement model is linear and the prior model is Gaussian, the approximation error statistics can be computed analytically, see [23]. The statistics is, however, typically estimated by Monte Carlo simulation. It is notable that in this case, a closed form representation for the prior is not needed when the approximation error statistics is computed. The Monte Carlo approach thus facilitates the use of, for example, anatomical atlases, such as below. In this paper, the approximation error statistics was computed over the prior model that was used in the inversion. In general, this is not required: the approx, imation error statistics can be computed over any model is used in the inversion. and a Gaussian approximation for For the Monte Carlo simulation, we generate a set of draws from the prior models and . In both the simdraws were used ulation and the real data cases, only and . to approximate For the draws for the domains and the boundary shapes, CT images of different individuals were segmented. This reand sulted in the ensemble of domains the corresponding parametric representations for the boundaries . The domains were scaled so that the horizontal diameter is equivalent to the diameter of the circular model domain . For the generation of the FEM meshes that are needed in the computation of the accurate measurements, a Fourier parameterization of the form (30) where are the sine and cosine functions and the angular variable, was constructed for each of the segmented and scaled . Using the parameterized boundaries, a FEM boundaries mesh was automatically created using the MatLab and Comsol for the mesh generators for each of the sample domains computation of forward solutions. Note that once the approximation error statistics is computed, none of these models are
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Fig. 3. Left: the actual domain is shown as gray patch. The domain was obtained by segmenting a CT image of the thorax. The circular model domain
~ is shown as bold solid line. The locations of the 16 electrodes attached on the boundary are shown with thick line segments. Right: arrangement of lungs and heart in the actual domain. Fig. 2. The actual thorax domain in the simulation case is shown as a gray patch. The boundaries of ten sample domains that were used in the construction of the approximation error model are also shown.
needed any more. Fig. 2 shows ten of the sample boundaries . in the sample domains The samples were drawn from the smoothness prior model which was constructed as in Section III-B. To simplify the sampling was defined on a FEM mesh in rectanprocess, the model . To gular domain that encloses all the sample domains , a random sample was then drawn from create the sample in the rectangular domain and interpolated into nodal values in the sample domain , see (9). Fig. 1 shows four on the CT based sample domains . The samples samples were then used for the computation of the accurate for each of the sample forward solution domains. and , To compute the target models to model dothe conductivity samples were mapped from main by
where is a matrix that interpolates nodal conductivity from to according to the deformation , see (21) and (22). As explained in Section III-C, in this study we employ a simple deformation model that preserves angle and relative distance (between the center of the domain and the boundary) of a coordinate point. for conductivities of the The construction of the matrix be the coordinate form (9) is implemented as follows; let vector of the th node of the FEM mesh in the model domain . We first find the point such that and then which encloses point find the index of the FEM element in . Once this has been done, the indices of nonzero elements for are the indices of the nodes of the element the th row of enclosing and the values are the values of respective nodal evaluated at point . basis functions Given the accurate and target forward solutions, the samples of the approximation error were obtained as
for the case of pure domain modeling errors and
for the combined domain modeling and discretization errors. The second-order statistics of the modeling errors were then estimated for both cases as
where tion
, and we write the Gaussian approxima.
IV. TEST CASE WITH SIMULATED DATA AND 2-D FEM MODEL A. Target Conductivity and Simulation Parameters The actual thorax domain is shown in Fig. 3. The thorax domain as well as the subdomains for the lungs and the heart were obtained from a segmented CT reconstruction of the human used to compute the approximathorax. The ensemble tion error statistics did not include this particular CT sample. cm and is The horizontal diameter of the domain was also used as the diameter of the model domain, see Fig. 3. Sixteen equally spaced measurement electrodes were mod. The locations of the electrodes are elled on the boundary indicated in Fig. 3, as well as the locations of the lungs and is shown the heart. The actual conductivity distribution in Fig. 4. The target conductivities were set to 1.2, 2, and 3.6 (arbitrary units) for the lungs, background and heart, respectively. In the simulation of the EIT measurements, sixteen adjacent current patterns were used and the voltages were measured between adjacent electrode pairs, leading to 256 voltage . The simulated measurements measurements, that is, , were computed with 2-D FEM approximation using mesh see Table I. Gaussian mutually independent noise was added to the simulated measurements. The added noise was zero mean and the standard deviation of the noise was 1% of the computed simulated voltages corresponding to the signal to noise . ratio SNR B. Reconstructions Using Simulated Data The reconstructions of type (1–5) described in Section III-A are shown in Fig. 4. In addition to the different reconstructions,
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Fig. 4. The actual conductivity and MAP estimates with simulated data. forward mesh.
1
TABLE I 2-D FINITE ELEMENT MESHES FOR THE SIMULATION CASE. IS THE NUMBER OF NODES AND THE NUMBER OF TRIANGULAR ELEMENTS. THE BASIS FUNCTIONS IN THE APPROXIMATION (7) FOR THE POTENTIAL AND IN (9) FOR THE CONDUCTIVITY WERE PIECEWISE LINEAR NODAL BASIS FUNCTIONS
N
N
the corresponding numbers of nodes in the FEM meshes and the CPU times, as well as the relative estimation errors (31) are also shown. The MAP-CEM with the accurate forward model using the correct measurement domain can be taken as a reference estimate. The error in the estimate is due to the ill-posed
(%) are the relative estimation errors (31) and
N
is the number of nodes in the
nature of the EIT problem. We note, however, that the particular prior model that has been used here, has been proven to be well suited for diffuse tomography, see for example [2], [23]. As explained in Section III-B, the prior model is proper, that is, it has finite covariances. Furthermore, by construction, the model can accommodate to an unknown background conductivity. The reconstructions in which the incorrect model domain is used but the approximation error is not accounted for, show intolerable errors. This applies to both the reconstruction with and the approximate model the accurate FEM model . What is evident from the similarity of these two reconstructions, is that the error due to the mismodelling of the domain clearly dominates the errors due to the discretization. In contrast, the corresponding reconstructions in which the approximation errors are accounted for, these anomalies are present in a much lesser degree. Most importantly, this applies also to the one in which the reduced order FEM model is used. The computational time needed for the termination of the GN iteration when using the more accurate model is about three times that needed for the approximate model. Also here, the errors due to the mismodelling of the domain dominate over the FEM discretization errors. In addition to the reconstruction of
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Fig. 5. The measurement phantom and MAP estimates with real data. The images show the central horizontal cross sections from the 3-D reconstructions. the number of nodes in the forward mesh.
the shapes of the organs, the actual conductivity values match the reality quite well also. Although it is clear that the MAP-CEM estimate with the actual geometry is a better estimate than the MAP-AEM with , the computational time with the latter is only about 7% of the former. Of course, the central issue is that the actual domain does not need to be known. To assess the approximation error approach with more eccentric target chests than the one in Fig. 4, we computed the results with exactly the same models and settings using simulated data from different chests. Based on the simulations one can, roughly speaking, say that the modelling error approach (MAP-AEM) gives clear improvement over the conventional error model (MAP-CEM) whenever the target geometry is plausible with respect the sample statistics of the domain shape in that is used for the construction of the the CT ensemble approximation error model. When the degree of eccentricity becomes large (target shape clearly outside the two standard deviation limits of the sample statistics), the usefullness of the estimates becomes questionable.
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N
is
V. TEST CASE USING EXPERIMENTAL DATA AND 3-D FEM MODEL A. Experimental Setup The experimental data was measured from a vertically symmetric measurement tank , see Fig. 5. The horizontal cross sections of the measurement tank and cylindrical model domain were the same as in the simulated test case, see Fig. 3. Sixteen equally spaced stainless steel electrodes were attached of the tank. The height of the tank and the on the boundary model domain was 5 cm. Thus, no model error due to truncation of the computational domain in the vertical direction was present. To construct the phantom, heart and lung shaped inclusions were made of agar and placed in the measurement tank filled cm The inclusions were with saline of conductivity 3.0 constructed using vertically symmetric moulds. The conduccm and tivity of the lung and heart targets were 0.73 cm , respectively. 5.8
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TABLE II 3-D FINITE ELEMENT MESHES FOR THE TEST CASE WITH EXPERIMENTAL IS NUMBER OF NODES AND THE NUMBER OF TETRAHEDRAL DATA. ELEMENTS. THE BASIS FUNCTIONS IN THE APPROXIMATION (7) FOR THE IN (9) FOR THE CONDUCTIVITY WERE PIECEWISE LINEAR POTENTIAL AND NODAL BASIS FUNCTIONS
mean, in particular, that we assume that the background conductivity is between (2.5, 3.5) with approximately 99% probability. The statistics of the modeling errors was again estimated conductivity and boundary samples only, simusing ilarly as explained in Section III-C. The conductivity samples were drawn from the 3-D prior model. The cross-sections of the (vertically symmetric) boundary samples were the same CT based samples that were used in the 2-D case. The contact impedances were estimated before the actual reconstruction by solving least squares problem
The measurements were carried out with the KIT 4 measurement system [32]. Sixteen adjacent current patterns were used and the voltages differences between the adjacent electrodes were used as measurements, leading to 256 voltage measurements. The amplitude of the injected currents was 5 mA with frequency 10 kHz. For the estimation of measurement error statistics, 40 000 realization were measured. The distribution of the measurement noise conformed well with the Gaussian model. The means and the covariances were formed as sample averages for current patterns separately. In KIT 4, each of the different current patterns use partially different circuit boards and have different switch states, which may result in different measurement error statistics for different current patterns. Significant external low frequency sources were not present, and the serial autocorrelation was verified to essentially vanish after the zeroth lag. Thus, the errors in the demodulated voltages could be modelled as mutually independent between current patterns was constructed as a block diagand the overall covariance onal matrix
(32) is a homogeneous conductivity value and where are the electrode contact impedances. The parameters and are fixed to correspond to the model, that is, , and depending on the forward model. Likewise, the mean of the total additive errors and the Cholesky factor depend and . Similar on the estimate, with least squares approaches for estimation of contact impedances have been proposed in [17], [29], and [46]. We note that the inaccurately known contact impedances could alternatively have been included in the approximation error statistics, see [37].
N
N
The standard deviations of the noise were in the range from 0.01% to 2.64% of the mean of the measured voltages and the signal-to-noise ratio (SNR) was 65.52 dB. The structure of (essentially) nonzero elements in the estimates was tridiagonal, implying that the three voltage difference measurements (between adjacent electrode pairs) that involve electrode are correlated. The details of the 3-D FEM meshes used in the different computational models are given in Table II. We note that due to the translational symmetry of the phantom, a 2-D model could have been used. We use, however, a 3-D computational model for both the forward problem and the representation of the conductivity, to be able to assess the related computational times in 3-D. The prior for the 3-D model was constructed similarly as for the 2-D model explained in Section III-B. The correlation length in the prior model was set to 7 cm as in the simulation case. The other prior parameters were set as follows. The prior mean was set to the conductivity of the saline background, that is, cm cm , and cm . The correlation length of 7 cm in a 5 cm tall tank means that (according to the prior model) the conductivities on the bottom and top layers are almost mutually independent. These choices
B. Reconstructions Using Experimental Data The reconstructions of type (1–5) described in Section III-A are shown in Fig. 5, together with a photograph of the measurement phantom. The images that are shown, are the central horizontal cross sections of the reconstructed 3-D conductivity, that is, 2.5 cm from both the top and bottom layers. The characteristics of the reconstructions in Fig. 5 are similar to those with the simulated 2-D case shown in Fig. 4. Again, the MAP-AEM models are able to capture the geometry of the organs well. Also, the actual values of the conductivity distribution match well. VI. DISCUSSION We computed the reconstructions with both the (full Gaussian) approximation error and the enhanced error model approximation, in which one makes the further approximation . The reconstructions were essentially identical. This is definitely not the case in general, see for example [25] for a deconvolution example in which the omission of the above cross-covariance effectively destroyed the approach. In the present case, however, this was not the case. This is good news from the practical point of view, since the estimation of the overall covariance structure such that the positive definiteness of the covariance estimate in (27) is ensured necessitates significantly larger sample from the prior than the approximation (29). For example, in the 2-D simulation example a set of 1000 samples was not sufficient. On the other hand, with the ensamples hanced error model (29), we could do with only. Furthermore, when anatomical atlases are used, such a large number might not be available and further regularization methods would have to be resorted to. Overall, the number of samples needed to capture the second order statistics of the approximation errors, depends heavily on the range of the primary uncertainties, in our case, the range of
NISSINEN et al.: COMPENSATION OF MODELLING ERRORS DUE TO UNKNOWN DOMAIN BOUNDARY IN ELECTRICAL IMPEDANCE TOMOGRAPHY
different boundary shapes and the prior for the conductivity. As a rule, the smaller the range, the more significant (relatively) is the approximation error mean over the covariance. Thus, with small range of uncertainties, a small number of samples is needed since the sample estimate for the mean converges fast. With some problems, even a few samples ( 10) can suffice, see for example [33]. We made the technical approximation in the prior model that the boundary shape and the conductivity distributions were independent. This assumption cannot be sustained in principle since a perturbation of the boundary effects indirectly also the conductivity via the mapping [see (21) and (22)]. However, based on the results this technical approximation seems to be acceptable in practice. In the present study, the heights of the measurement domain and the incorrect model domain were equal in the 3-D example with data from the thorax shaped measurement tank. In such a case the boundary condition (4) holds at the top and bottom surfaces of the model domain and there are no modelling errors caused by truncation of the computation domain in the computational model. When imaging the human chest with EIT the situation becomes slightly more complicated since one typically is bound to use for computational reasons a truncated model domain (e.g., a cylinder domain that has height equal to the height of the lungs) instead of using a sufficiently tall model domain so that the error induced by incorrect zero current boundary condition at the truncation surfaces in the computational model would become negligible. Treatment of domain truncation by the approximation error approach in geophysical applications of EIT was first considered in [33]. In [37] we demonstrated using EIT data from an industrial process tomography application that simultaneous compensation of errors caused by domain truncation, unknown contact impedances and reduced discretization by the approximation error approach is feasible. Adding domain truncation error to the present model would be obtained by using truncated model domain in the target model (20). In the present 3-D example the chest shaped measurement domain was vertically symmetric and therefore accurate reprefor the sentation for the shapes of the sample domains construction of the approximation error statistics was available from ensemble of 2-D CT images of different individuals. When applying the approach to imaging of human chest, a more realistic approximation for the error statistics can be obtained if one has access to ensemble of 3-D chest CT (or MRI) images. VII. CONCLUSION In this paper, we considered electrical impedance tomography and the recovery from incorrectly modelled boundary shape. An incorrectly modeled boundary has been known to induce significant errors to the estimates, typically making absolute imaging impossible. Another issue with absolute imaging has been that the errors that are induced by the approximations of the forward problems solvers, should be smaller than the measurement noise. This, in turn, has necessitated the use of sometimes infeasibly heavy computational models. In this paper, we applied the recently proposed approximation error approach for the modelling of the errors induced by the
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unknown boundary shape. The approximation error approach is based on the Bayesian framework for statistics, in which statistical models for all unknowns and uncertainties, or prior models, are constructed. In this approach, the approximate statistics of these errors are computed over the prior models, and the likelihood model is formulated accordingly. We considered both a simulation example as well as a real tank measurement case, both corresponding to a (absolute) thorax imaging problem. For the construction of the prior distribution for the thorax shape, CT images were used. The results show that the approximation error approach is well suited for the recovery from simultaneous uncertainty in boundary shape and the errors caused by using highly approximative forward solvers. Furthermore, the approach can readily use information from anatomical atlases. In biomedical applications of EIT, the boundary shape is seldom known to the accuracy required by absolute imaging. The results of this paper, together with earlier results concerning the handling of the truncation of the computational domain, suggest that absolute EIT imaging might be a clinically relevant possibility. ACKNOWLEDGMENT The authors would like to thank Dr. L. Heikkinen and Dr. T. Savolainen for the construction of the measurement tank and for assistance with the measurements. REFERENCES [1] A. Adler, R. Guardo, and Y. Berthiaume, “Impedance imaging of lung ventilation: Do we need to account for chest expansion?,” IEEE Trans. Biomed. Eng., vol. 43, no. 4, pp. 414–420, 1996. [2] S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inv. Problems, vol. 22, pp. 175–195, 2006. [3] D. C. Barber and B. H. Brown, “Applied potentian tomography,” J. Phys. E: Sci. Instrum., vol. 17, pp. 723–733, 1984. [4] G. Boverman, T.-K. Kao, R. Kulkarni, B. S. Kim, D. Isaacson, G. J. Saulnier, and J. C. Newell, “Robust linearized image reconstruction for multifrequence EIT of the breast,” IEEE Trans. Med. Imag., vol. 27, no. 10, pp. 1439–1448, Oct. 2008. [5] B. H. Brown, “Electrical impedance tomography (EIT): A review,” J. Med. Eng. Technol., vol. 27, no. 3, pp. 97–108, 2003. [6] D. Calvetti, J. P. Kaipio, and E. Somersalo, “Aristotelian prior boundary conditions,” Int. J. Math., vol. 1, pp. 63–81, 2006. [7] D. Calvetti and E. Somersalo, An Introduction to Bayesian Scientific Computing—Ten Lectures on Subjective Computing. New York: Springer, 2007. [8] M. Cheney, D. Isaacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev., vol. 41, pp. 85–101, 1999. [9] K.-S. Cheng, D. Isaacson, J. C. Newell, and D. G. Gisser, “Electrode models for electric current computed tomography,” IEEE Trans. Biomed. Eng., vol. 36, no. 9, pp. 918–924, Sep. 1989. [10] E. L. V. Costa, C. N. Chaves, S. Gomes, M. A. Beraldo, M. S. Volpe, M. R. Tucci, I. A. L. Schettino, S. H. Bohm, C. R. R. Carvalho, H. Tanaka, R. G. Lima, and M. B. A. Amato, “Real-time detection of pneumothorax using electrical impedance tomography,” Crit. Care Med., vol. 36, pp. 1230–1238, 2008. [11] T. Dai, C. Gomez-Laberge, and A. Adler, “Reconstruction of conductivity changes and electrode movements based on EIT temporal sequences,” Physiol. Meas., vol. 29, pp. S77–S88, 2008. [12] I. Frerichs, “Electrical impedance tomography (EIT) in applications related to lung and ventilation: A review of experimental and clinical activities,” Physiol. Meas., vol. 21, pp. R1–R21, 2000. [13] I. Frerichs, J. Hinz, P. Herrmann, G. Weisser, G. Hahn, T. Dudykevych, M. Quintel, and G. Hellige, “Detection of local lung air content by electrical impedance tomography compared with electron beam CT,” J. Appl. Physiol., vol. 93, pp. 660–666, 2002.
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