A symmetry - based algorithm to determine the presence of the internal structure heterogeneity of the of a biological object

. This article proposes a two-level approach for solving the problems of electrical impedance tomography of biological objects, which consists in successively applying the methods of "rough" and "fine" reconstruction to the potentials measured on the surface of the object. To perform a "rough" reconstruction procedure, it is proposed to use a method for determining the presence of heterogeneity in the internal structure of a biological object based on the symmetry algorithm. Thus, consistent localization and identification of the boundaries of the heterogeneity of the internal structure of the biological object is carried out. It is shown that when analyzing data for an arbitrary biological object, the symmetry characteristic of the type of objects under study can be taken into account, and deviations from such a characteristic symmetry will indicate deviations from the norm in the structure of a biological object.


Introduction
When solving the problems of electrical impedance tomography (EIT) [1][2][3][4][5][6][7][8], much attention is paid to the development of effective algorithms for the reconstruction and visualization of the conductivities of the internal structures of biological objects (BO) [9]. However, in this case, it is necessary to take into account all the complexity and specifics of BO, which make it difficult to use a large number of developed algorithms [10]. Therefore, there is a need to create new reconstruction methods and algorithms for processing measurement information. In this regard, it is promising to use a two-level approach in the reconstruction of the internal conduction field of a biological CP, which consists in the sequence of applying the methods of "rough" and "exact" reconstruction to the measured data. For a "rough" reconstruction, a method is proposed for determining the presence of inhomogeneities in the internal structure of an object based on a symmetry algorithm. This is due to the fact that the objects studied by the EIT method can have different types of symmetry, which, if taken into account when constructing an image from the obtained data, allows obtaining additional information about the internal structure of the object. In addition, the application of symmetry principles makes it possible to reduce the dimension of the problem and optimize the choice of the zeroth approximation. Thus, by mathematical calculations, an approximate area is determined in which the heterogeneity is located. Further, using the methods of "exact" reconstruction, further calculations are carried out.

Results
Electrical impedance tomography is one of the developing methods of non-invasive imaging [11][12][13][14][15]. To solve the problems of reconstruction and visualization of the internal structure of the BO, new algorithms are being developed [16][17][18] based on the EIT method. A generalized block diagram of the EIT device is shown in Figure 1. The device is an information-measuring system consisting of a low-amplitude highfrequency current source, a recording unit, a personal computer, and an electrode system, which is located on the BO surface. The essence of the EIT method is as follows. Current I is injected into the BO by means of current-carrying electrodes and potentials φi are simultaneously recorded, where i is the number of the electrode, i = 1, 2, ... , NE. Next, the array φi is transferred to a personal computer for processing, reconstruction and visualization of the internal structures of the BR based on the assessment of their conductivities. Figure 1 schematically shows a BO in the form of a disk with conductivity Ω1, which has in its structure an inhomogeneity with conductivity Ω2. The electrode system is located on the BO surface (for non-invasive EIT). Number of electrodes -NE. The device is controlled by a personal computer. An inhomogeneity is considered to be a homogeneous inclusion in the BR, the conductivity of which differs from the conductivity of the BR itself by a factor of k, with Ω1 ≠ Ω2. The arrows indicate the direction of movement of the current source. To determine the presence of inhomogeneity Ω2 in BO, a method based on the symmetry algorithm of the measured values φi is proposed.
The principle of operation of the method is based on a step-by-step comparison of the magnitude of the potentials φi, where i is the number of the electrode NE, i = 1... NE for different P profiles.
The algorithm is as follows.
1) The conditional axis of symmetry S is set, which divides the investigated BO into two equal halves. The conditional axis S passes through diametrically located electrodes.
2) The values of φi are estimated for each P profile.
3) The conditional axis of symmetry S moves to the next pair of diametrically located electrodes.
4) Item 2 is being implemented. The described actions of paragraphs 1 -3 are repeated for all P profiles. Estimation of φi values for each profile P consists in calculation of stresses U and formation of leads O. The true axis of symmetry of an object for a particular profile P can be determined based on the following. In the absence of inhomogeneity in the structure of the BO (Figure 1, a), the dependence of the voltage U, which is obtained from the measured potentials φi, on the corresponding lead O for all profiles P will have the form shown in Figure 3. When the inhomogeneity Ω2 is introduced into the BO (Figure 1,b), the values of φi, change. Thus, the symmetry is broken. The U(O) plot is stretched and reshaped. Moreover, if the conditional axis S is a real axis of symmetry, then there will be different potentials on the surface of the object, but at the same time symmetry with respect to S will be observed, that is, φ_i/φ_iS =a, where φiS -are the potentials of the electrodes equidistant from the S axis. For example, in Figure 1, these are pairs φ1 and φ7, φ2 and φ6, etc. In the absence of inhomogeneity.
If the conditional axis S is not an axis of symmetry, then there will be different potentials φi on the surface of the object, but at the same time, symmetry with respect to S will not be observed, since a will be largely different from 1.
Thus, with a comprehensive assessment of all measured φi for the "round disk with a circular inhomogeneity" system with an unknown defect (inhomogeneity) position, the presence of the indicated type of symmetry of the measurement results will allow us to detect the BO symmetry plane, which can be used to determine the presence of an inhomogeneity in the BR, the approximate area BO, in which it is present, as well as when constructing a zero approximation.
When analyzing data for arbitrary BO (objects), the symmetry characteristic of the type of objects under study can be taken into account, and significant deviations from such a characteristic symmetry will indicate deviations from the norm in the structure of the BP (sample).
This method can be used for the development and practical application of the algorithm for "rough" reconstruction of the BW conductivity, which is an integral part of the proposed two-level reconstruction and visualization approach for the developed EIT device. A generalized scheme for applying the method is shown in Figure 4. The essence of the approach is as follows. To the measured data (for example, φi), a method is applied to determine the presence of inhomogeneities in the internal structure of the object based on the symmetry algorithm. Further, using the methods of "exact" reconstruction, further calculations are carried out to refine the boundaries of heterogeneity, metalocation, etc. The obtained results of information processing by the methods of "rough" and "exact" reconstruction are processed by the comparison unit. In this case, additional a priori information about BO is used, which makes it possible to increase the reliability of the results. To formalize a priori information, a mathematical model of the BO and its internal structures is constructed. Based on the processing and analysis of multi-parametric information, the final decision is made.
At present, the authors, together with other researchers, have developed and are studying software products for the reconstruction and visualization of the conductivity of the internal structures of the BO. Algorithms of computer programs are aimed at the state registration of the results of intellectual activity.
The EIT information-measuring system for implementing the proposed method and approach is implemented on the basis of National Instruments technology and the LabView graphical programming language [19] using medical reusable electrodes and patient cables [20].

Conclusions
A method is proposed for detecting the presence of inhomogeneity in the BR by processing and analyzing the potentials measured on the BR surface, which consists in step-by-step processing of the measurement results and step-by-step comparison with on the symmetry E3S Web of Conferences 389, 01052 (2023) https://doi.org/10.1051/e3sconf/202338901052 UESF-2023 determination algorithm. The method can be used for preliminary detection of the fact of the presence of heterogeneity in BO when applying a two-level reconstruction approach for EIT. The supposed benefit of accounting for symmetry is to optimize the choice of the zero approximation for solving the inverse problem, and it may also be possible to compile libraries of symmetries characteristic of the studied BOs in the normal (healthy) state and, at least, to determine the presence of pathology in the event of a mismatch of the obtained symmetry with "normal".