# Electrical impedance tomography imaging using *a priori* ultrasound data

- Manuchehr Soleimani
^{1}Email author

**5**:8

**DOI: **10.1186/1475-925X-5-8

© Soleimani; licensee BioMed Central Ltd. 2006

**Received: **21 October 2005

**Accepted: **06 February 2006

**Published: **06 February 2006

## Abstract

### Background

Different imaging systems (e.g. electrical, magnetic, and ultrasound) rely on a wide variety of physical properties, and the datasets obtained from such systems provide only partial information about the unknown true state. One approach is to choose complementary imaging systems, and to combine the information to achieve a better representation.

### Methods

This paper discusses the combination of ultrasound and electrical impedance tomography (EIT) information. Ultrasound reflection signals are good at locating sharp acoustic density changes associated with the boundaries of objects. Some boundaries, however, may be indeterminable due to masking from intermediate boundaries or because they are outside the ultrasound beam. Conversely, the EIT data contains relatively low-quality information, but it includes the whole region enclosed by the electrodes.

### Results

Results are shown from a narrowband level-set method applied to 2D and 3D EIT incorporating limited angle ultrasound time of flight data.

### Conclusion

The EIT reconstruction is shown to be faster and more accurate using the additional edge information from both one and four transducer ultrasound systems.

## Background

Electrical impedance tomography (EIT) seeks to image electrical conductivity distribution of an object by measuring the impedance data between electrodes attached to the outer surface of the body [1]. In this paper we are developing an EIT imaging technique combined with a priori ultrasound data. Our proposed method is looking to a localised change in conductivity of part of the imaging area. The application of EIT envisaged in this paper, as motivation, is monitoring of cryosurgery. Cryosurgery is a minimally invasive way of destroying the undesired tissues by freezing them down to between -20 to -80 degree C [2, 3]. Feasibility of EIT for Cryosurgery monitoring has been studied in [4, 5]. In this application a probe is inserted into cancerous tissue. An ice ball forms around the probe destroying the surrounding tissue. It is very important to monitor the location, size and shape of the ice ball, especially if the area to be destroyed is delicate and contains critical tissue. This monitoring is usually achieved using magnetic resonance imaging (MRI) or CT fluoroscopy [6, 7]. Both EIT [5] and ultrasound imaging [8] approaches offer cheaper systems, but individually do not provide sufficient detail to be of practical use.

A standard single transducer ultrasound system would typically only see one face of the ice ball. In contrast, a 3D EIT system would sense the whole volume but the boundaries of the ice ball would be difficult to accurately determine. Full medical ultrasound systems use scanning transducers or phased arrays to produce 2D or 3D images of the body. These systems are expensive and will still only show one face of the ice ball. A simpler and much cheaper system uses a few individual ultrasound transducers placed around the object. This method can estimate the edge position in a few places but cannot produce a full image. This information, however, could be used to improve the reconstruction method of an additional modality, such as EIT. This paper looks to introduce the fusion approach by proposing a narrowband level-set algorithm, which focuses the EIT reconstruction using the ultrasound data.

The level-set method was initially introduced for tracking propagating boundaries, but can also be used to locate static boundaries such as the interface between an inclusion and background [9–11]. There are alternative approaches for the shape reconstruction in EIT, including monotonicty method [11, 12] geometrical based shape recovery [13–15]. Compare to geometrical modelling of the shape recovery [13–15], the level set method has an advantage that it handles multiple objects in an automatic fashion. Monotonicity method is fast and nonlinear, but provides only partial classification of the shape identification problem.

In this paper the finite element method (FEM) has been used to solve the forward problem of EIT. This is implemented in Matlab using codes from EIDORS2D and EIDORS3D [16, 17]. A triangular mesh was used for the FEM model of the forward problem in 2D and a tetrahedral mesh was used in 3D. Using level set method allows us to use a dense mesh for the inverse problem (by applying a narrowband level set the size of the inverse to be solved remains small), consequently large scale forward problem needs to be solved. In order to improve the computational time of the forward solvers especially in 3D, we modified the EIDORS3D by applying algebraic multigrid method (AMG) [18] as a preconditioner for the conjugate gradient as the linear solver from the FEM model. Using AMG scheme improved the speed of the forward problem dramatically; so many forward problems could be solved as needed in level set method. In our modified version of EIDROS3D we developed a nonlinear EIT reconstruction and also minimized the computational time for calculating the Jacobian matrix, so it can be update many times.

Using an iterative method, with an update formula for the level-set function, the interface between two materials can be recovered. The level set is known to generate good interface results [9, 10] but the computational cost is high, as it will need many iterations (hundreds of nonlinear steps). The number of iterations is a function of the initial guess for the shape as well as the complexity of the objects to be reconstructed. In this article we are using a priori ultrasound data to improve our initial guess for the object location, by identifying some points at the boundary of the object.

## Method

In EIT, the differences in the electrical properties, i.e. conductivity distribution inside the object; is used to generate a tomographic image. EIT is used in both medical and industrial applications. The advantage of such a technique over other imaging modalities is such that, it provides a non-invasive ("non-destructive" in an industrial terminology) method and requires no ionizing radiation. Furthermore, EIT is a relative low cost and simple functional technique. Moreover, a portable measurement system could also be designed for it. The most important drawback of EIT is its poor image resolution, which is often restricted by the number of electrodes used for data acquisition. Data acquisition is typically made by applying an electrical current to the object using a set of electrodes, and measuring the developed voltage between other electrodes.

EIT is suggested to be a good technique for the cryosurgery monitoring. In this article we are reformulating the EIT image reconstruction problem to an interface reconstruction problem between frozen and normal tissue. That would allow us to extract the information we need in more efficient way. Fortunately, this is an application of EIT that the conductivity contrast of the region to be imaged is high and a localised problem as the cryogen probe can tell us where the approximate location of the frozen area is. Here we are attempting to use some additional information about the interface by using ultrasound data.

### Ultrasound transducers

The ultrasound method is a single pulse-echo measurement [19]. The travel time from excitation of the transducer to receiving the reflected signal is recorded. The velocity of the ultrasound wave within the fluid is dependent on its compressibility and density. The distance travelled is calculated from the *time of flight (TOF)* of the ultrasound wave from transducer to the object and back again, and the known velocity of sound in the liquid. Ultrasound technique is based on measuring the TOF of ultrasonic waves arriving at the boundary of the region of interest, which here is the ice ball. Ultrasonic transducers are evenly spaced around the circumference of a body.

In normal medical ultrasound the tissue temperature is approximately constant throughout the body and the velocity is close to that of pure water. However in the cryogenic freezing application there will be noticeable temperature gradient. Since the velocity dependence of water is approximately 3 m/s per degree C, this variation in temperature will have a noticeable effect. It would be possible to model this temperature gradient and variation to calculate the velocity profile and then estimate the ice ball position. However to model the temperature distribution the ice ball position must be known, which leads to an iterative calculation. Another method would be to perform ultrasound measurements across the tissue from known positions on the skin surface. This would aid in the estimation of the velocity and temperature gradients.

### Level set method

The level-set technique is chosen to describe changing shapes since this method is able to easily model topological changes of the boundaries. In the shape reconstruction approach, it is assumed that the approximate values of background parameters and parameters inside the inclusions are known, but that the number, topology and shapes of the inclusions are unknown and have to be recovered from the data. Compared to the more typical pixel/voxel-based reconstruction schemes, the shape reconstruction approach has the advantage that the prior information about the high contrast of the inclusions is incorporated explicitly in the modelling of the problem. In a pixel/ voxel-based reconstruction scheme the approximate locations of the unknown inclusions are found during the early iterations, but it typically takes a large number of additional iterations to achieve accurate information concerning the precise shapes of these objects.

where *F* is the speed function and Φ is the boundary at time t.

We do not use equation (1) to describe the front propagation. Instead we have implemented a narrowband level set method [10]. In narrowband level set method we use an iterative optimisation technique and in each iteration the inverse problem has been solved on the interface between two-phase materials [10].

The conductivity at each point *r* can be described in terms of the level-set function depending on the position of the point *r* with respect to the boundary ∂*D* of the inclusion D as

and

∂*D* = {*r*:Φ(*r*) = 0}. (3)

Here σ_{int} and σ_{
ext
}are the conductivity of the inclusion and the background respectively. The describing level-set function is a function form *R*^{3} → *R* for three-dimensional case, and its value is zero on the boundary, it has a negative sign inside and a positive sign outside of the boundary.

The inverse boundary value problem is to find the boundary ∂*D* (which in turn describes a conductivity distribution) that minimizes the mismatch between the measured and fitted voltage data. The mismatch function, Ψ(∂D), is defined as

Ψ(∂*D*) = ||*V*
_{
m
}- *V*(∂*D*(σ))|| + *G*(∂*D*(σ)) (4)

*V*

_{ m }are the measured voltages, and

*V*(∂

*D*(σ)) are the voltages calculated from the conductivity distribution, σ, derived from the corresponding boundary ∂

*D*and

*G*. is a regularisation term applied to the interface. In this paper the regularisation matrix is the identity matrix. The derivative of the voltage to a change in interface was derive [9] and [10, 11] and has been used in this study. Regularised Gauss-Newton update formula [11] has been used to reconstruct the interface. Small values of the mismatch between measured and simulated voltage indicates good locations for the boundary where as large values indicate poor estimation of the boundary. In particular, the boundary which makes this as small as possible is used, along with the inclusion and background resistivities, to give the level-set reconstruction. In practice this minimum cannot be found in a closed form and so a numerical procedure is required. To perform the minimization the following algorithm [10] is used

- 1.
Start with an initial guess for the shape of the inclusion, which is an initial level-set function, in our case a circle located in the centre.

- 2.
Define the interface and narrow band; the narrow band is an area that includes pixels sharing points with the interface.

- 3.
Solve the forward problem and calculate the Jacobian with respect to the boundary.

- 4.
Update the level-set function and calculate a new interface boundary and narrow band.

- 5.
Check the misfit in the data and if the error is small enough then stop.

- 6.
If the misfit is not small go to Step 2

## Results

Using a single ultrasound transducer a point, (0.3, 0, 0.4) m, on the boundary of the inclusion is identified and used to help start the level-set algorithm. The initial guess is of a spherical object with diameter 0.07 m centred at (0.3, 0, 0.33) m. The convergence improves (see Figure 8 left) and in a few iterations a very good estimate of the true object can be reconstructed.

Further improvement, both in terms of the speed of the convergence and the accuracy of the reconstructed shapes, is expected with multiple ultrasound transducers. To investigate this, ultrasound probes help to provide the positions of four points in object boundary. A spherical object passing through these four points is implemented as the initial guess. The convergence of the level-set method is faster as expected (Figure 8 right), however the shape reconstruction accuracy does not improve significantly over the single transducer results. This is likely to be due to the coarseness of the level-set mesh. The smallest size of the ice ball that can be reconstructed depends on the accuracy of the measurement system and computational model. In [18] for brain cryosurgery, we have shown that the voltage differences between frozen and normal tissues in mv are proportional to the volume of ice ball in cm^{3}. On the other hand the choice of mesh density and size of narrowband has an impact in accuracy of the interface.

## Conclusion

- 1.
Generate a pixel based EIT image (using a linear image reconstruction).

- 2.
Evaluate an approximated speed profile for the ultrasound based on conductivity profile (approximate it to temperature gradient) generated using linear EIT image and consider the speed with different organs based on EIT image.

- 3.
Calculate some points at the boundary of the inclusion (here is frozen tissue) using ultrasound data.

- 4.
Applying narrowband level set method as a more accurate interface reconstruction technique using EIT data and a priori ultrasound data.

With above strategy, we have an EIT image but with less accurate interface information and a localised conductivity interface reconstruction with more detailed information about the interface between frozen and normal tissues. Combination of full ultrasound tomography with consideration of different speed of ultrasound in different organs and different temperature is our main aim for future study.

## Declarations

## Authors’ Affiliations

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## Copyright

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