Keywords: topology optimization, electrical impedance tomography.

Electrical Impedance Tomography (EIT) is a recent monitoring technique that allows us to obtain images of a transversal plane in any section of human body. In this technique, images are related to the corresponding value of electrical conductivity inside of human body section. Images are generated from voltage values which are obtained by applying a sequence of low amplitude electrical currents on electrodes positioned around of a transversal section of human body. The EIT is based on an inverse problem, in which given the electrical potentials measured on electrodes, this technique tries to find the conductivity distribution inside a transversal section of the body. Technology of EIT is safer and cheaper than other tomography techniques and although it has poor resolution, it has good potential for clinical applications [1]. Moreover, an EIT device is short and portable, which allows its installation close to a bedridden patient. In addition, the patient is not exposed to any type of radiation in EIT.

In this work, a technique called the Topology Optimization Method (TOM) is applied to image reconstruction using the EIT. The TOM is an iterative method that finds systematically a material distribution inside of a design domain, satisfying an objective function requirement and specified constraints [2]. The problem of applying the topology optimization method to obtain an image of body section consists of finding a material distribution (related to conductivity distribution) in the body section domain that minimizes the difference between electrical potential measured on electrodes and electrical potential calculated by using a computational model. In TOM, the design domain (section of body) is divided in a finite number of elements, where each element can vary from a specified material to another material in a continuous way, according a material model called Density Method [2]. In addition, an electrode model, proposed by Hua [3], has been implemented to represent the contact impedance of interface electrode-skin.

The solution of the topology optimization problem is obtained through an iterative computational algorithm that combines the Finite Element Method (FEM) and an optimization algorithm called Sequential Linear Programming (SLP). The SLP solves a non-linear optimization problem considering it as a sequence of linear sub-problems, which can be solved with Linear Programming. Furthermore, the SLP uses gradients of objective function and constraints, related to design variables, for obtaining the linear sub-problems. Mathematical formulation of these gradients can be obtained analytically by using the adjoint method [4].

The optimization algorithm that applies the TOM to obtain image in EIT is implemented in software, which is programmed using C language. This software was applied to image reconstruction of some 2D examples using numerical and experimental data. In this work, 32 electrode elements are uniformly positioned along the boundary of the design domain and their electrode parameter (, where is the contact impedance of electrode) is optimized by the software. Moreover, electrical current is applied through these electrodes following the adjacent and diametral pattern of electrical excitation and the performance was compared. In this case, the adjacent pattern yields better results than the diametral pattern, since by using this pattern absolute conductivity values were found in an average of 93% of expected value. The results also show that by using numerical data the software is able to obtain, in few iterations and with a certain level of precision, the contact impedance values of interface electrodes-skin and the absolute conductivity values of two materials inside of the domain, even if an artificial noise is introduced. Furthermore, by using experimental data the software is able to detect the presence of two materials inside of phantom, however the precision to identify correct position of the glass object must be improved. In this case, these electrode parameters will be obtained by optimizing them together with conductivity values of elements close to the boundary of the domain. In this way, it is believed that the portion of the domain could absorb predominant errors of electrode parameter estimation.

The computational algorithm of the TOM, which is studied in this work, could be seized for obtaining images of the lungs through an EIT device. Applied to EIT image reconstruction, the TOM allows us to include some constraints in the problem of image reconstruction limiting the solution space during tomography examination and avoiding images without clinical meaning. Moreover, it allows us to work with known areas inside of the body section (bone, heart, etc).

1

M. Cheney, D. Isaacson and J.C. Newell, "Electrical Impedance Tomography", SIAM review, 41, 1, 85-101, 1999. doi:10.1137/S0036144598333613

2

M.P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods and Applications", Springer-Verlag, NY, 2003.

3

P. Hua, E.J. Woo, J.G. Webster and W.J. Tompkins, "Finite Element Modelling of Electrode-Skin Contact Impedance in Electrical Impedance Tomography", IEEE Trans. on Biomed. Eng., vol. 40, 335-343, 1993. doi:10.1109/10.222326

4

R.T, Haftka, Z. Gürdal and M.P. Kamat, "Element of Structural Optimization", Kluwer Academic Publishers, Boston, 1996.

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