«Incorporating apriori anatomical information into image reconstruction in electrical impedance tomography ...»
Incorporating apriori anatomical information into image
reconstruction in electrical impedance tomography
H Dehghani, D C Barber and I BasarabHorwath*
School of Engineering, Sheffield Hallam University, City Campus, Pond
Street, Sheffield, S1 1WB, UK
Department of Medical Physics and Clinical Engineering, Sheffield
University and Health Authority, Royal Hallamshire Hospital, Sheffield, S10 2JF, UK *Corresponding author Abstract. Image reconstruction in electrical impedance tomography using the sensitivity theorem is generally based on the assumption that the initial conductivity distribution of the body being imaged is uniform. The technique of image reconstruction using this method is described and reconstructed images are presented. Improvements in image quality and accuracy are demonstrated when accurate apriori ‘anatomical’ information, in the form of a model of the distribution of conductivity within the region to be imaged, is used to construct the sensitivity matrix. In practice correct apriori information is not available, for example, the conductivity values within the various anatomical regions will not be known. An iterative algorithm is presented which allows the conductivity parameters of the apriori model to be determined during reconstruction.
1. Introduction The methodology of twodimensional (2D) electrical impedance tomography (EIT) has been described in detail elsewhere (Barber and Brown 1984, 1990, Brown and Seagar 1987). Electrodes are positioned with equal spacing around the body to be imaged thus defining a plane through the object. Voltage profiles are collected for all drive and receive electrodepair combinations and images are reconstructed as though the data were from a 2D object. In this work 16 electrodes have been placed around a circular object. Current is driven into the object through two adjacent (drive) electrodes. For each of the 16 driveelectrode pairs, 13 voltage differences between the remaining adjacent noncurrent carrying (receive) electrodes are recorded. In practice objects are threedimensional (3D); current cannot be confined to one plane. However, the reconstruction algorithm usually assumes that the object is 2D. This paper will only deal with proper 2D reconstruction.
Previous investigations of such an approach includes the incorporation of a priori information into the Sheffield filtered backprojection image reconstruction algorithm, which showed some success (Avis et al 1995). Also, Zadehkoochak et al (1991) presented a reconstruction algorithm based on the inversion of the sensitivity matrix associated with a nonuniform conductivity distribution using the singular value decomposition (SVD) method. They did not present any reconstructed images. Zadehkoochak et al (1993) have also investigated the use of apriori information associated with imaging the thorax and reported artefacts in the resulting images. The use of a universal model (a standard model of internal conductivity distribution within the human thorax) has been speculated by Zadehkoochak et al (1993) to be invalid; however, it will been shown that providing the assumed anatomical model is closely matched to the patient's anatomy an improvement in the reconstructed image is achieved.
The conductivity of a region, discretised into small areas or elements, can be written as a column matrix or vector. When this conductivity changes from a reference conductivity distribution σ ref (where each element has units of 1 conductivity (Sm )) to a conductivity σ dat (where each element has units of 1 Sm ) it represents a change in conductivity ∆ σ = σ dat σ ref, where ∆ σ is a vector of the same size as both σ dat and σ ref. Images of this change in internal impedance distribution can be reconstructed from the resulting change in differential boundary voltages measured in volts (∆ v = vdat vref) from a uniform conductivity distribution, where vdat , vref and ∆ v are column vectors, usually of a different size from the conductivity vectors; the actual size depending on the number of independent differential boundary voltage measurements. These images of conductivity change can be obtained using a relationship described by a sensitivity matrix (S) which is derived using a
theorem by Geselowitz (1971). This relation is given by:
∆v = S∆σ (1) where S is a sensitivity matrix (usually nonsquare), ∆ σ is a vector containing the changes in conductivity and ∆ v is a vector containing the corresponding changes in voltage profiles. S relates the small conductivity change within each individual pixel of the image to the corresponding differential boundary voltage change for each and every pixel and each and every drivereceive electrode combination. The relationship between ∆ v and ∆ σ is often assumed to be linear (Barber and Brown 1990) and thus the elements of S are independent of conductivity. S is usually calculated by assuming that the initial conductivity distribution is uniform, i.e. all elements within a region to be imaged have the same conductivity value (Barber and Brown 1990) and then calculating the resulting boundary voltage changes due to small perturbations in the conductivity values for each and every element. The sensitivity matrix derived in this manner is, in this paper, called the uniform sensitivity matrix, denoted by Sunif. However, the relationship between the conductivity changes and the corresponding differential boundary voltages is not linear and virtually no interrogated region is uniform. The current distribution and therefore the voltage distribution within a body depends on the conductivity distribution within the body and as a consequence this is also true of the sensitivity matrix. For example, a small change in conductivity next to a physically large conductive area will have a smaller effect on the boundary voltages than the same small change in conductivity next to a relatively small area. Thus the coefficients of the true sensitivity matrix vary with the conductivity distribution and this matrix is referred to in this paper as the ideal sensitivity matrix, Sideal. In general therefore, although the sensitivity matrix S changes with the conductivity distribution and equation (1) is therefore nonlinear, it has been argued that for small changes in conductivity this nonlinearity can be ignored (Barber and Brown, 1990). Shaw et al (1993) have also reported a similar result, namely that for a small region, the linear approximation is valid for the conductivity changes of up to 300% of the background.
We have determined that the assumption of linearity by Barber and Brown (1990) is violated in conditions where there is a wide range of conductivity values; this is in accordance with findings by Seagar (1983). Given accurate spatial and conductivity information about a region to be imaged, an accurate image of the impedance distribution can be reconstructed from the boundary voltage data. Although this may be selfevident, it shows whether or not the relationship described by equation (1) is valid. It also shows the extent of the blurring of the image due to the algorithm and computational process. Also, it serves as a benchmark for images obtained using less apriori information and as such is the "best" image that can be obtained. Recognising that in general complete anatomical information will not be available, this paper will address the degree of accuracy needed in the apriori information in order to produce a useful reconstructed image. If the method is reasonably robust it may be possible to use anatomical information taken from a database rather than from the subject being imaged, contrary to previous speculations (Zadehkoochak et al (1993)). However it will be shown in the work presented in the paper that providing the model is closely matched to that of the interrogated area an improvement in image quality is achieved. It also will be demonstrated how the present widely used sensitivity algorithm can be combined with an approximate knowledge of the spatial distribution of the tissues to provide a better estimate of the conductivity values.
In order to generate the sensitivity matrix S the area of interest, a 2D circular tank, is split into 1920 triangular and brick elements as shown in figure 1. The elements representing the modelled electrodes are not shown in the reconstructed images and the central elements are not displayed well due to their small size.
The sensitivity matrix S is a matrix of 208 by 1920 coefficients. The 208 rows of the S matrix relate to the 208 different differential voltage readings and the 1920 columns to the sensitivity coefficient for each independent element. There are 13 voltage measurements for each of the sixteen projections, giving 208 voltage readings. Each of the 1920 elements in the model has its conductivity value perturbed from a uniform value and this small change generates 208 differential boundary voltage changes.
The sensitivity coefficient for each element i was calculated from Geselowitz’s lead theory (Geselowitz 1971) and is given by:
S( j,i ) = − ∫ ∇Φ m ⋅ ∇Ψn ⋅ du (2) u where j = j(m,n), Φ m is the potential distribution generated in an object when unit current is passed through the electrode pair m before a change in conductivity (σ ref) and Ψ n is the potential distribution produced if unit current had been injected through electrode pair n after the change in conductivity to σ dat has occurred. j is the drive and receive electrode combination and the integration is over the area of the element u, i.e. multiplication of the dot product with the area of the element u.
The electric fields ∇Φ and ∇Ψ are calculated for each element using an available finite element package. The two vector components of the electric field (Exi and Eyi) are calculated at the centre of each element for every drive electrode combination.
As Φ m and Ψ n are derived for different conductivity distributions equation (1) is nonlinear in terms of conductivity. However, for small changes in conductivity about σ ref, Ψ n can be replaced by the equivalent potential calculated for σ ref and this linearises the problem. For large changes the assumption of linearity is no longer correct: a problem this paper addresses.
The linearised sensitivity matrix Sunif is calculated assuming that the conductivity distribution is uniform before and after a change has occurred and this is the matrix which has been used to date even when the reference conductivity is not uniform (Barber and Brown 1990, Kotre 1989).
The sensitivity coefficient for each element of Sunif was calculated using equation (3), where for the electric field (Exm, Eym) in element i is due to the current injected through electrode pair m and the electric field (Ex n, Eyn) is that produced when the same current is injected through electrode pair n.
The integration is again over the area of the element u.
In principle the calculated sensitivity matrix can now be used to reconstruct an image of the change in conductivity distribution calculated from the boundary potential differences. This relationship can be derived from equation (1) to