Electrical impedance tomography with compensation for electrode positioning variations

Transcription

1 Phys. Med. Biol. 43 (1998) Printed in the UK PII: S (98) Electrical impedance tomography with compensation for electrode positioning variations B H Blott, G J Daniell and S Meeson Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK Received 19 August 1997, in final form 12 January 1998 Abstract. Ideally electrical impedance tomography (EIT) should not be oversensitive to electrode positions, but this conflicts with efforts to produce high-resolution images. Two procedures are presented that balance reducing the sensitivity to electrode position errors with generating practicable EIT images. The first provides a criterion based on electrode sensitivity for regularizing the reconstruction through spectral expansion. The main consequences of this are that smoother images are produced and the number of artefacts and their magnitude are generally reduced. The second modification uses the recorded data to compensate for electrode movements that have occurred after the reference data were measured. Image smoothness is used as the criterion for the readjustment. Computer simulation tests have shown that this modification produces improved image fidelity. 1. Introduction Over the last 15 years electrical impedance tomography (EIT) has been established as a new clinical imaging modality which is economical enough to be allocated to specific patients for long-term monitoring. It poses no known hazards to the patient and it is capable of producing real-time images. However, a series of electrodes must be applied to the patient for data measurement. These electrodes are difficult to position accurately, particularly for patients in intensive care beds. Changes in body position and electrode replacements are also sources of error in long-term in vivo EIT measurements (Lozano et al 1995). This suggests that the electrode spacing will change during data collection, leading to errors in the reconstructed images. Electrode movements during short-term clinical investigations can also introduce image artefacts or contribute to the overall measured change in conductivity. An example of this is chest wall expansion during cardio-pulmonary investigations (Adler et al 1996). Whatever the source of error, either incorrect placement of the electrodes or movement of the electrodes after the reference data set has been recorded, the effect on the images is the introduction of extraneous fine detail and other artefacts. In this paper we present a systematic study of solutions to these problems. Barber and Brown (1988) reported that their approach to reconstructing images of the relative change in resistivity was successful at producing images with a low sensitivity to uneven electrode spacing. Later Adler et al (1996) confirmed that provided the initial reference and the measured data set are recorded with the electrodes in the same positions, only small errors are introduced by non-uniform electrode placement. Ider et al (1992) proposed a modification of the data measurement protocol to avoid electrode position errors, but this requires extra patient manoeuvres and is rather restrictive. Ideally the reconstruction /98/ $19.50 c 1998 IOP Publishing Ltd 1731

2 1732 B H Blott et al process needs to have low sensitivity to the electrode positions, but this must be at the expense of reducing image resolution. Two different procedures are described below. The first reduces the sensitivity of the reconstruction to the electrode positions, and provides a rational criterion for regularizing the reconstruction process. The second procedure aims to compensate for electrode movements that occur during data measurement, by utilizing the data measured in vivo to estimate modifications to the reconstruction process. The effectiveness of these procedures is examined using simulated data sets. 2. A framework for image reconstruction For small changes in conductivity the linear approximation applies. In order to solve the forward problem, the continuous conductivity distribution is divided into a number of finite elements or pixels. The vector of all the voltage measurements v is related to variations in pixel resistivity ρ by the Jacobian matrix J = v i / ρ j : v = J ρ. (1) To obtain an image of the change in resistivity a pseudo-inverse R of the Jacobian matrix J is required so we can write ρ = R v. (2) If R is a good reconstruction matrix leading to a reconstructed resistivity distribution ρ = R v observed then the voltages predicted by the distribution, v predicted = J(R v observed ), should ideally agree with the observed values. So the product JR should equal I, the identity matrix. One method of constructing a pseudo-inverse is by using the singular value decomposition (SVD) of J (Golub and Reinsch 1970). It is possible to write J = UΣV T with U and V orthogonal matrices and Σ diagonal, and possible reconstruction matrices can be written R = VWU T. The resulting image can be regarded as a superposition of basis images, given by the columns of the matrix V weighted with the coefficients WU T v. If JR = I then the elements of W and Σ are related by W i = 1 i. In fact the very small values of i result in unacceptable images unless a filter is introduced to attenuate the contributions of these basis images to the reconstruction. This necessarily means that some misfit to the data must be tolerated. We investigate whether a judicious choice of R within this framework can produce an image that is less sensitive to electrode misplacements. 3. Reconstruction methods with reduced sensitivity to electrode positions 3.1. Theoretical formulation To minimize the sensitivity of the reconstruction to the electrode positions the images reconstructed from the voltages for two slightly different electrode configurations should be as similar as possible. If the voltages measured from the first electrode configuration are v 1 = J 1 ρ and the voltages measured for the second electrode position are v 2 = J 2 ρ then R v 1 and R v 2 should be as similar as possible. Therefore ideally the relationship RδJ = RJ 1 RJ 2 = 0 (3) is desirable for all movements of the electrodes. This is impossible but we can reduce the average value of this difference.

3 EIT with compensation for electrode position variation 1733 Let δj (r) be the change in J when electrode r is moved a small fixed distance, then a convenient measure of the average change in RJ is (RδJ (r) ) 2 ij = trace[(rδj (r) ) T (RδJ (r) )] = trace[r R T ] (4) r ij r where = (δj (r) )(δj (r) ) T. (5) r If the resistivity changes in a particular region of the image are required, for instance the lungs in a cross section of the thorax, it might be useful to weight the different pixels in equation (4). We try to minimize (4) by varying R. Ideally one might hope that there was enough freedom in choosing R so that JR could be exactly equal to I and the resulting resistivity distribution would predict an exact fit to the data. Unfortunately this leads to the reconstruction matrix R = J T [JJ T ] 1 (6) independent of. We conclude that all reconstruction matrices that predict an exact fit to the data are equally sensitive to electrode misplacements, using our definition of sensitivity. We must accept that reducing the sensitivity to the electrode positions can only be achieved at the expense of a misfit to the data. Therefore instead of constraining R so strongly we define a measure of the misfit to the data by (JR I) 2 ij = trace[(jr I)T (JR I)]. (7) ij We can now minimize a linear combination of (4) and (7) λ trace[r R T ] + trace[(jr I) T (JR I)]. (8) The constant λ describes the relative importance we wish to attach to fitting the data and reducing sensitivity to electrode displacement. In terms of the singular value decomposition for J this expression becomes particularly simple: λ Wi 2 δ i + (W i i 1) 2 (9) i i where the δ i are the diagonal elements of U T U. The values of W i can then be found by minimizing expression (9), leading to i W i = i 2. (10) + λδ i When i is much larger than λδ i, which is the case for low-resolution basis images, W i 1 i. For the higher-resolution basis images, which are more sensitive to electrode positions, i is less than λδ i and the values of W i are small. Therefore the addition of λδ i in equation (10) controls the magnitudes of the W i preventing small values of i from dominating the reconstruction Evaluation of coefficients W i All the calculations are done using a finite element model (FEM) of the conductivity distribution, with 208 square pixels (shown later in figure 2), leading to an admittance matrix relating the pixel node currents and the pixel node potentials. Initially the 16 electrodes are placed at known positions (nodes) on the perimeter of the distribution and the Jacobian

4 1734 B H Blott et al J computed. To calculate δj (r), electrode r is moved by one pixel, first clockwise then anticlockwise, from its original position. Let J a and J b be the Jacobian matrices computed for these electrode movements respectively. Then the matrix δj (r) is given by δj (r) = J b J a. (11) 2 The eigenvalues i required to evaluate W i using (10) are known since they are generated by the SVD of the initial Jacobian. The values of δ i are derived using equation (5), U obtained from the SVD of J, and the 16 matrices δj (r) corresponding to moving each of the 16 electrodes used for data acquisition. On a coarse grid a movement of one pixel in electrode position is rather large, therefore the values of δj (r) were also calculated for single-node displacements on a finer pixel grid. The aim was to use these results with the faster reconstruction that uses the coarse grid. Unfortunately the orders of the image space eigenvectors for the fine and coarse grids are different, due to the finer grid more closely resembling a circle. It does not therefore appear possible to use these more accurate values of δj (r) except by doing the whole reconstruction on the finer grid. Figure 1. Coefficients (W i ) to be applied to the basis image expansion, designed to minimize the sensitivity of the reconstruction to the electrode spacing. Figure 1 is a plot of the W i values, derived using a pixel grid and with λ = 1. The basis images are ordered according to the size i, that is, broadly speaking, resolution and the values of W i are plotted against basis image number. The general trend is an increase in W i, from an initially low value, to a maximum before falling again. Against this background trend certain values, representing specific basis images, are significantly higher than their close neighbours. These high W i values indicate which basis images are least sensitive to the electrode positions. Emphasizing these images is exactly the behaviour required to minimize the sensitivity of the reconstruction process to the electrode positions. The shape of the plot in figure 1 suggests that there is no need for a regularization filter, as used in the standard reconstruction (Zadehkoochak et al 1991), since the elements of W are designed to preferentially select those basis images that are least sensitive to

5 EIT with compensation for electrode position variation 1735 electrode positions and the plot shows how the values are already constrained by the δ term in equation (10). This is certainly an advantage of this new technique, because deciding the point where the regularization filter should truncate the spectral expansion has so far been rather arbitrary in most applications of EIT Computer simulation The finite element net shown in figure 2 was used to represent a uniform resistivity distribution with electrodes equally spaced (not as shown) around the perimeter; the spectral expansion of basis images was developed from the sensitivity Jacobian for this arrangement. Data for the comparison were generated after displacing the electrodes by one net node either way along the periphery, in a random manner. Two data sets were produced, one from a uniform resistivity distribution and the other from the simulated blood distribution shown in figure 2. Figure 2. Finite element net showing the electrodes perturbed by random displacements. The position of the simulated blood pool (1.5 m) in the conductivity distribution (1 m) is also shown. To demonstrate the effectiveness of the new procedure set out above the results are compared with the use of the arbitrary cut-off filter given by Zadehkoochak et al (1991), which we call the standard reconstruction. The resulting images are shown in figure 3 for the standard and modified reconstructions. The top row images are taken with the same set of (perturbed) positions for the electrodes, and show that a flatter region is generated around a smaller peak in the modified reconstruction. The image differences introduced by the electrode perturbations are illustrated in the bottom row of figure 3, for the two reconstruction methods. These demonstrate that fewer and smaller artefacts are generated, particularly in the central regions of the image, by the modified reconstruction. The overall effect is to provide a trade-off between image resolution and insensitivity to electrode positioning.

6 1736 B H Blott et al Figure 3. Contour images of the simulated blood pool generated using the standard and modified spectral expansions (left- and right-hand columns respectively). The top row of images show the effects of the perturbed electrode positions. The bottom row shows the differences between images generated before and after perturbation of the electrode configuration. The contour intervals are 0.03 m starting at 0.45 m but excluding the zero level contour. 4. Compensation for electrode movement during measurement Up to this point the electrode positions have been assumed to remain fixed once they have been attached to the patient. In long-term monitoring the measurement electrodes may become displaced after recording the reference data set, and if this occurs Barber and Brown (1988) and Adler et al (1996) have shown that the resulting image artefacts in EIT are a major problem Theoretical formulation We can compensate for small movements in the electrode positions by using a modified Jacobian matrix. To determine this modification a simple regularized reconstruction algorithm that minimizes the mean square value of the change in resistivity ( ρ) with a constraint on the fit to the measured data is used, that is we construct the minimum of λ ρ T ρ + ( v J 0 ρ) T ( v J 0 ρ) (12) where J 0 represents the Jacobian matrix corresponding to uniformly spaced electrodes and λ is a constant. Note that in contrast to section 3.1 a misfit to the observed data is used rather than a general condition requiring that JR I. The minimization gives ρ = J T 0 ( λi + J0 J T ) 1 0 v. (13)

7 EIT with compensation for electrode position variation 1737 By performing the SVD of J this can be seen to be equivalent to the spectral expansion with 1 W i =. (14) λ + i The expression in equation (13) defines a particular value of ρ T ρ. The effect of electrode movements, and hence the wrong matrix J 0, is to add extraneous detail to the image and probably leads to an increased value of ρ T ρ. To determine if it is possible to reduce this value, J 0 is modified to be J = J 0 + ɛ r δj (r). (15) r The δj (r) are the changes in J 0 when electrode r is moved (defined earlier in section 3.1), and the ɛ r are the fraction of δj (r) required to model small electrode displacements. It is convenient in the following to denote the set of ɛ r by the vector ɛ. It turns out in test simulations that ρ T ρ can be reduced considerably but at the expense of very large values for ɛ r. These are greater than could be justified in any linear approximation and it is unlikely that electrode displacements made during clinical investigations will be greater than a small fraction of the interelectrode separation. A limit must be placed on the magnitudes of ɛ r to prevent the electrode corrections to J 0 being too large. To restrict the magnitudes of ɛ r equation (12) is modified to include a constraint on ɛ T ɛ: λ ρ T ρ + ( v J ρ) T ( v J ρ) µɛt ɛ. (16) The constant µ, which determines the dominance of the ɛ term in equation (16), is selected to limit the maximum displacement to a specified fraction of the interelectrode separation. Equation (16) is minimized with respect to ρ, which again gives equation (13), with J replacing J 0, and with respect to ɛ k which gives ρ T δj (k)t ( v J ρ) + ( v J ρ) T ( δj (k) ρ) + µɛ k = 0 (17) remembering that J depends on ɛ through equation (15). Rearranging equation (17) for ɛ leads to [ ρ T (δj (k)t δj (r) + δj (r)t δj (k) ) ρ + µδ kr ]ɛ r r = ρ T δj (k)t v + v T δj (k) ρ ρ T (δj (k)t J 0 J T 0 δj(k) ) ρ (18) where δ kr is the Kronecker delta. An iterative solution must be sought to solve for ɛ in equation (18). Initially J = J 0 and ɛ = 0 are used to evaluate ρ using equation (13). With J = J 0 and the initial ρ known, ɛ can be calculated from equation (18). The improved J and the updated ρ can be calculated using equation (15) and equation (13) respectively. This procedure is repeated until the relative change in ρ T ρ between iterations is less than 1%. This modified Jacobian is specific to the recorded data set v, since v is used to derive ɛ. This is in contrast to the previous method which derived a modified reconstruction process that was applicable to any data set Computer simulation We used the same model of resistivity distribution and electrode perturbations as in section 3.3 above, except that the electrode displacements were made after generating the

8 1738 B H Blott et al reference data set. Preliminary results obtained with the rather coarse net shown in figure 2 confirmed the extreme sensitivity to electrode drift. The small electrode displacements expected in clinical practice were therefore modelled by selecting values of ɛ r in the modified Jacobian of equation (15), and deriving the consequent change in voltage measurements with equation (1). The images obtained after perturbing the electrodes are shown in the top row of figure 4; the left-hand diagram shows the severe distortion produced by equation (12) containing the unmodified Jacobian J 0. The reconstruction with the modified Jacobian, shown in the top right, clearly shows the improvement in image fidelity, and the restoration of the expected symmetry. Figure 4. Contour images generated using the standard (left) and corrected (right) Jacobians, with electrodes displaced after the reference dataset. The contour plots in the bottom row show the differences between the distorted images in the top row and the image obtained with the standard Jacobian and without electrode displacements. The contour intervals are as in figure 3. The bottom row of figure 4 shows the difference between the top row images and one reconstructed with J 0 from data generated with no electrode perturbations. This demonstrates very clearly that the compensation procedure, although rather time-consuming to implement, can provide improved image quality when electrode movements are suspected. 5. Conclusions A modification has been presented that reduces the sensitivity of EIT reconstruction to the electrode positions, and which represents an improvement over the method of Barber and Brown (1988). The main consequences of this change are that smoother images can be produced, and the number and magnitude of artefacts are generally reduced. A major advantage of the modified spectral expansion is that regularization, to control noise in the images, is automatically provided by the preferential selection of those basis images that are least sensitive to electrode positions. This is an important result for most applications

9 EIT with compensation for electrode position variation 1739 of EIT, since a clear rationale is used to constrain the reconstruction in contrast with the general practice of arbitrarily choosing a regularization filter to control noise in the images. The data-specific procedure is a more computationally intensive modification to the Jacobian matrix, used to compensate for electrode movements that occur between data measurements. Computer simulations suggest that such a modification would be beneficial in clinical practice, since image peaks are better defined when the modified Jacobian is used. Therefore if patient movement occurs during monitoring, this procedure is likely to deliver more credible reconstructed images. In summary the spectral expansion method modified to reduce the sensitivity to non-ideal electrode positions will be advantageous in real-time imaging; and long-term monitoring with EIT will benefit from the ability to use data to reduce the impact of patient movement on electrode spacings. References Adler A, Guardo R and Berthiaume Y 1996 Impedance imaging of lung ventilation: do we need to account for chest expansion? IEEE Trans. Biomed. Eng Barber D C and Brown B H 1988 Errors in reconstruction of resistivity images using a linear reconstruction technique Clin. Phys. Physiol. Meas. 9 (suppl A) Golub G H and Reinsch C 1970 Singular value decomposition and least squares solutions Numer. Math Ider Y Z, Nakiboglu B, Kuzuoglu M and Gencer N G 1992 Determination of the boundary of an object inserted into a water-filled cylinder Clin. Phys. Physiol. Meas. 13 (suppl A) Lozano A, Rosell J and Pallas-Areny R 1995 Errors in prolonged electrical impedance measurements due to electrode repositioning and postural change Physiol. Meas Zadehkoochak M, Blott B H, Hames T K and George R F 1991 Spectral expansion analysis in electrical impedance tomography J. Phys. D: Appl. Phys