Recovery algorithm of electrical conductance distribution in the volume of the object of research in electrical impedance tomography

. An efficient, rigorously substantiated algorithm for solving electrical impedance tomography problems is proposed and tested. The features of the algorithm are the use of a methodology based on solving conditionally correct inverse problems and the presence of a search stage for the optimal current configuration that provides the maximum sensitivity of the problem functionals to the desired parameters, which allows to speed up the iterative process and improve the accuracy of the problem solution. The algorithm can be used in medicine and industry to solve diagnostic problems.


Introduction
At present, electrical impedance tomography (EIT) is widely used in medicine and industry to assess and diagnose the state of an object [1][2][3][4][5]. Currently, EIT is developing rapidly. The research geography covers more than 20 countries [6]. In the field of medicine, electrical impedance tomographs have been developed and put into operation for examining the mammary gland, in gynecology [7,8], for monitoring the respiratory system during artificial lung ventilation [9][10][11], as well as many technical means for bioimpedancemetry [12][13][14][15][16][17][18][19][20]. Electrical impedance tomography can be attributed to the field of medical imaging [21]. This method is distinguished by non-invasiveness, painlessness and safety for both the patient and the attendants, ease of maintenance, relatively low cost, and the possibility of multiple examinations. But, despite the advantages, EIT has a low resolution and sensitivity, a complex mathematical apparatus for reconstruction and visualization, limited areas of application, and lack of universality in application [5,22]. The solution of these problems largely depends on the development of new approaches, methods and devices for EIT, which allow solving existing problems in this area.

Algorithm for solving the inverse problem
In general terms, the essence of the EIT method is as follows. An electric current I of 1 to 5 mA with a frequency f of 10 to 150 kHz is passed through the object under study (bioobject (BO)) . Electric potentials φi., where i = 1, 2, ... , n, are measured on the surface of the object using measuring medical non-invasive electrodes. The function of receiving signals from the electrodes, their processing and transfer to a personal computer (PC) is performed by the BI measurement unit. The reconstruction and visualization of the conductivity of inhomogeneities (H1, H2, ..., Hk, k = 1, 2, ..., q.) of the internal structures of the BO is carried out on a PC. In this work, we develop and test an algorithm for restoring the conductivities of the internal structures of the BO. It is a set of rules for solving a conditionally correct inverse problem for an equation describing an electric field in a conducting medium where -φ ̇ complex electric potential; i=√(-1); ε -the dielectric constant; σ -electrical conductivity; ω -circular frequency, ω=2πf [23].
with boundary conditions on the outer surface of the object: = -on one current electrode; = − -on another current electrode; = 0 -on the rest of the outer surface of the object, where S -electrode area. Let us formulate the inverse EIT problem: to find the distribution σ(x,y,z) and the potential φ(x,y,z) in the volume of the object under study that satisfies equation (1) and boundary conditions (2), (3) and (4). Additional information is available for measurement values of the electric potential on the surface of the object izm ( = 1,2, … , ) with an error izm . Here N is the number of electrodes on the outer surface; i -current configuration number.
Consider the main stages of the algorithm for solving the inverse problem. Stage 1. Setting the initial distribution (0) ( , , ). In a particular case, this is a vector If condition (6) is not met, then we proceed to the next stage. Stage 6. We find the following approximations of the desired values by the gradient method The step ( ) -decrease as it approaches the minimum of ( ) .
Let's move on to stage 2.
We will use four options for the location of current electrodes (current configurations), i = 1,2,3,4 ( Figure 2). We solve direct FEM problems [23] for the equation  We find the potentials on the outer circumference of the region * ( = 1,2,3,4; = 1,2, … , ) for four current configurations (see Figure 4 (a, b, c, d), which in our problem they play the same role that izm in the study of real objects.We assume that the potentials * are determined with an error of * = 0,01. For the functional, we use formula (5). The process termination condition is the fulfillment of inequality (6). Let us introduce the notation: to the parameter σ1 for i. Similarly, we define the sensitivity to other parameters: 2 ( , +1) ; 2 ( , +1) ; ( , +1) ; ( , +1) .
We investigate the dependences of the sensitivities to the desired parameters on the current configurations (Figures 5 -7). The following designations are accepted in the figures: 1, 2, 3, 4 -numbers of current configurations; 1-3 -when calculating the sensitivities, the squares of the potential differences obtained with current configurations 1 and 3 were summed; 2-4 -with current configurations 2 and 4; 1-2-3-4 -with current configurations 1, 2, 3, 4.   Further, when minimizing the functionals by the gradient method of coordinate descent [25], we will use the maximum sensitivity values (that is, the results of calculations for current configuration 1), which will speed up the iterative process and improve the accuracy of the results of solving the problem. Thus, the algorithm described above must be supplemented with the stage of searching for the optimal current configuration, at which the maximum sensitivity of the functionals to the desired parameters takes place.

Conclusion
Thus, an effective, strictly justified algorithm for solving EIT problems has been proposed and tested. The features of the algorithm are the use of a methodology based on solving conditionally correct inverse problems and the presence of a search stage for the optimal current configuration that provides the maximum sensitivity of the problem functionals to the desired parameters, which allows to speed up the iterative process and improve the accuracy of the problem solution. The algorithm can be used in medicine and industry to solve diagnostic problems.