IMPROVEMENT OF THE BIOMEDICAL IMAGE RECONSTRUCTION METHODOLOGY BASED ON IMPEDANCE TOMOGRAPHY

Transcription

1 VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ ÚSTAV TEORETICKÉ A EXPERIMENTÁLNÍ ELEKTROTECHNIKY Ing. Ksenia Kořínková IMPROVEMENT OF THE BIOMEDICAL IMAGE RECONSTRUCTION METHODOLOGY BASED ON IMPEDANCE TOMOGRAPHY VYLEPŠENÍ METODIKY REKONSTRUKCE BIOMEDICÍNSKÝCH OBRAZŮ ZALOŽENÉ NA IMPEDANČNÍ TOMOGRAFII ZKRÁCENÁ VERZE DISERTAČNÍ PRÁCE ABRIDGED DOCTORAL THESIS Obor: Školitel: Teoretická elektrotechnika prof. Ing. Jarmila Dědková, CSc. Rok obhajoby: 2016

2 Abstract The present theoretical thesis discusses the improvement and related research of algorithms for the imaging of the internal structure of conductive objects, biological tissues and organs in particular, via electrical impedance tomography (EIT). Within the thesis, the theoretical framework of EIT is formulated, together with a survey of approaches towards implementing the given technique. More concretely, in this context, algorithms for the solution of the inverse problem are proposed and researched; these algorithms ensure effective reconstruction of the spatial distribution of the electrical properties of the examined object and facilitate the imaging of such properties. The main idea of the algorithm improvement based on the deterministic approach lies in introducing additional techniques, namely, the level set or the fuzzy filter methods. Furthermore, a procedure for the 2-D reconstruction of conductivity distribution utilizing one component of the measured magnetic field, or the z- component of magnetic flux density, is presented. Numerical models for imaging the admittivity (or conductivity) distribution of a biological tissue were created to facilitate the implementation and testing of the algorithms. The results obtained via the basic application of the improved image reconstruction algorithms are discussed and compared. Key words Electrical impedance tomography, inverse problem, image reconstruction algorithm, Tikhonov regularization method, level set method, fuzzy filter Abstrakt Disertační práce, jež má teoretický charakter, je zaměřena na vylepšení a výzkum algoritmů pro zobrazování vnitřní struktury vodivých objektů, hlavně biologických tkání a orgánů pomocí elektrické impedanční tomografie (EIT). V práci je formulován teoretický rámec EIT. Dále jsou prezentovány a porovnány algoritmy pro řešení inverzní úlohy, které zajišťují efektivní rekonstrukci prostorového rozložení elektrických vlastností ve zkoumaném objektu a jejích zobrazení. Hlavní myšlenka vylepšeného algoritmu, který je založen na deterministickém přístupu, spočívá v zavedení dodatečných technik: level set a nebo fuzzy filtru. Kromě toho, je ukázána metoda 2-D rekonstrukce rozložení konduktivity z jediného komponentu magnetického pole a to konkrétně z-tové složky magnetického toku. Byly vytvořeny numerické modely biologické tkáně s určitým rozložení admitivity (nebo konduktivity) pro otestování těchto algoritmů. Výsledky získané z rekonstrukcí pomocí vylepšených algoritmů jsou ukázány a porovnány. Klíčová slova Elektrická impedanční tomografie, inverzní problém, algoritmus pro rekonstrukci obrazu, Tikhonovova regularizační metoda, level set metoda, fuzzy filtr

3 Table of contents 1. INTRODUCTION THE 2-D AND 3-D IMAGING PROBLEMS THE PURPOSE OF THE THESIS AN EIT DEVELOPMENT OVERVIEW The Physical Model of the Solution Domain The Electrode Models The Forward Problem The Finite Element Method The Inverse Problem Solving the Inverse Problem The Improved Image Reconstruction Algorithm The Level Set Method The Fuzzy Filter THE MODELS SIMULATED THE RESULTS OF THE SIMULATIONS The 2-D Simulations The Influence of the Regularization Parameter and the Number of Unknown Values The Influence of Conductivity Changes on the Voltage in the Surface Electrodes Reconstructing the Conductivity Distribution Reconstructing the Admittivity Distribution The 3-D Simulations The Conductivity Distribution Imaging The Influence of Noise IMAGE RECONSTRUCTION BASED ON MAGNETIC FIELD DATA The Theoretical Background The Numerical Simulation CONCLUSION REFERENCES AUTHOR S PUBLICATIONS... 44

4 1. Introduction At present, a large number of different sophisticated numerical methods are available for the image reconstruction of biological tissues in the human body. Modern techniques, along with the necessity to secure exact, detailed, and fast spatial localization of inhomogeneities with minimal auxiliary research, are required to provide for the possibility of supervising the tissue status dynamics on a long-term basis. One of such approaches is Electrical Impedance Tomography (EIT). Electrical impedance tomography is a method to ensure safe, non-invasive image reconstruction of the internal structure of an object via the related spatial distribution of admittivity. EIT is a fast image reconstruction method. The results of the EIT-based reconstruction of spatial admittivity distribution can be represented by both twodimensional and three-dimensional images. The technique in itself is inexpensive, easy to operate, and it offers the possibility of working virtually in real time. The main disadvantage of this approach lies in the relatively low spatial resolution and instability of the imaging process; yet there also exist other drawbacks of major importance, for example, the necessity to secure good contact of the electrodes with the surface of the object. In general terms, the EIT-based image reconstruction is a nonlinear, ill-posed inverse problem, and this fact constitutes the cause of most of the above restrictions. As described above, the principle of EIT-based image reconstruction exploits the evaluation of the physical properties of the examined object, namely, the assessment of the distribution of the admittivity inside the object. The internal admittivity distribution is recalculated from the voltages measured on the electrodes attached to the surface of the object by injecting small amounts of electric current. Each local admittivity change inside the object then causes electric potential changes on the surface of this object. The performed admittivity (electrical conductivity and relative permittivity) tests of tissues showed a difference in the admittivities of a normal and a fatigued tissue; the same result also applies to a healthy and a pathologic tissue. This information about tissue admittivity changes is of major importance for clinical diagnostic procedures, namely, the monitoring of pulmonary and cardiac functions [1], [2], [3], [4], the detection of brain activity [5], [6], and gastric imaging [7], [8]; the discussed data find application also in the therapeutic treatment of clinical problems such as breast cancer [9], [10] and brain function disorders [11], [12], [13]. In addition to the above-presented medical uses, electrical impedance tomography can be also beneficial in geology [14], the chemical industry [15], molecular medicine [16]. The EIT technique is a tool favored in defectoscopy [17], [18]. 2. The 2-D and 3-D Imaging Problems A considerable part of all 2-D image reconstruction algorithms are based on the assumption that the electric current flowing through an examined object is mostly confined to the measurement plane and that its remaining portion, distributing the offplane measurement, is virtually negligible. However, the current in an object tends to spread out in all three dimensions, and, therefore, the off-plane internal structure of the object exerts significant influence on the reconstructed images

5 In several research reports, [19], [20], [21], [22], [23], the effect of 2-D image reconstruction performed by using the data obtained from an object with threedimensional conductivity and resistivity distribution was investigated. The proposed results indicated that the conductivity variations usually depend on all three coordinates, and hence the voltage changes throughout the object as well as the distribution of the current in the given object cannot be confined to 2-D. Consequently, taking into account this information, the electrical impedance imaging of a 3-D object cannot be formed from a set of independent 2-D images. A full 3-D image is reconstructable from the data collected over the whole surface of the object. For many problems, the reconstruction of a three-dimensional image is regarded as important. In addition to providing more detailed information about an examined object, 3-D imaging can be also capable of improving detection changes in the object without the need to carry out multiple 2-D measurements. However, the actual 3-D imaging is associated with a series of difficulties. 3-D EIT assumes the reconstruction of a 3-D object with a complex geometrical shape, which may also be subject to variation due to certain processes (for example, the thorax geometry changes caused by respiration). Thus, the 3-D reconstruction process involves a significantly larger number of unknowns than the 2-D problem. Also, a major difficulty rests in the application of many individual electrodes positioned on the surface of the object in the form of a two-dimensional matrix. Another drawback is the high data collection rate required for the monitoring of the fast-changing processes in the object (this is especially important for monitoring the physiological rates of change within medical imaging). By further extension, 3-D imaging assumes the necessity to work with large datasets. All of these aspects then result in complications within the tomography process and introduce the need to use more complex computational resources with a high performance in 3-D. Much of the research into 3-D EIT is focused on the development of 3-D image reconstruction algorithms or on the adaptation and improvement of the existing 2-D algorithms. 3. The Purpose of the Thesis The thesis discusses the research into and enhancement of algorithms for electrical impedance tomography based on the internal structure imaging of the admittivity of diverse objects, biological tissues and organs in particular. The main purpose of the study is to provide an efficient, stable, and reliable algorithm for 2-D and 3-D image reconstruction. The achievement of the given goal will be made possible through several steps, including the development of a 2-D and a 3-D numerical models for imaging the conductivity and admittivity distribution of a biological tissue. These models will be used for the implementation and testing of an improved image reconstruction algorithm developed on the basis of the knowledge and applications presented within previous studies in this field. The algorithm, utilizing the deterministic approach, must ensure the stability of the image reconstruction process and secure the necessary resolution and imaging speed. The related central objectives of the thesis consist in testing the improved algorithm s stability, examining the accuracy of the obtained results, and analyzing the - 5 -

6 influence exerted by the regularization parameter and the numbers of unknown values on the time and stability of the reconstruction process. The algorithm, after its functionality in 2-D conductivity distribution imaging has been tested, is to be implemented within the imaging of admittivity distribution. Moreover, the algorithm will be adapted for a 3-D numerical model; then, its efficiency (the resolution, stability, and time of the image reconstruction process) will be estimated. The influence of noise in the measured data on the image reconstruction results will be analyzed. The thesis also undertakes the task of developing an algorithm for image conductivity reconstruction on the basis of one component of the measured magnetic field. The image reconstruction results obtained using the proposed algorithm will be compared with those acquired through the use of both the above-mentioned improved algorithm and the conventional EIT algorithm. 4. An EIT Development Overview The principle of image reconstruction using electrical impedance tomography consists in evaluating the spatial distribution of the admittivity (or impedivity) inside an examined object. The internal admittivity distribution is calculated from the measured voltages on the electrodes attached to the surface of the object via injecting a small amount of electric current. The amplitude of the injected current is within the range of 1 10 ma, and its frequency corresponds to khz (in the case of medical imaging applications). In mathematical terms, EIT image reconstruction is a nonlinear, ill-posed inverse problem. Most existing approaches to solving this problem explicitly or implicitly assume the solution of the forward problem of EIT. Such algorithms are reduced to a comparison of the measured boundary voltages (real data) with the boundary voltages calculated from the known admittivity distribution inside the object and the known injected current pattern, i.e., the data defined as a result of the forward problem solution. Thereby, the accuracy and rate of the imaging process directly depend on the accuracy and rate of the forward problem solution. The theoretical background of EIT is characterized in detail within [24], [25], [26], [27] The Physical Model of the Solution Domain The electromagnetic field within the examined object Ω (or the solution domain) is governed by Maxwell s equations. After the transformation of these equations, taking into account quasistatic approximations [25], [27], the equation describing the behavior of the field at any point p of the object, excepting its boundary Ω, has the form div ( z grad u) 0, p, (4.1) where u is the electric potential inside Ω, and z denotes the admittivity inside Ω, which can be expressed in terms of the conductivity and permittivity as z j ; (4.2) 0 r - 6 -

7 here, σ is the conductivity of the medium inside Ω, ω is the angular frequency (ω = 2πf, where f is the frequency), ε 0 denotes the dielectric constant of vacuum, ε r is the relative permittivity of the medium, and j denotes the imaginary unity. The behavior of this field on the boundary of the object is represented by a boundary condition which can be written in the form u z J, p, (4.3) n where n is the outward unit normal to Ω, and J is the current density on Ω. In equation (4.3) the presence of the surface electrodes is not considered. The influence of the electrodes on the examined object is accounted for by imposing the appropriate boundary conditions. A mathematical description of the electrode interaction with the object is determined by the electrode model The Electrode Models Currently, the most accurate electrode model used in EIT is the CEM. In this model, the discreteness of the electrodes, their shunting effect, and the contact impedance are considered. The precision with which the electrode model predicts the measured voltages corresponds to the precision characterizing the measurement of the data [28], [29], [30]. The CEM generally constitutes the more preferred option for the reconstruction of accurate images from in-vivo data. The CEM is mathematically described by equation (4.1) and the following boundary conditions: u u zl z Ul, pal, l 1,..., L, (4.4) n al u z ds n l I, p a, l 1,..., L, l l (4.5) u z 0, n p \ L a l l1, (4.6) L l1 L l1 I 0, (4.7) l U 0, (4.8) l where U l is the voltage measured on the l-th electrode, the coefficient z l is the effective contact impedance between the l-th electrode and the surface of the examined object, a l denotes the l-th electrode, L is the number of electrodes, I l is the electrical current injected to the l-th electrode, and S l is the area of the l-th electrode The Forward Problem The solution of the complete electrode model embodies the solution of the forward problem of electrical impedance tomography. The forward problem is formulated as the determination of the potential distribution of the electric field under - 7 -

8 the condition that the configuration of the currents passing through the surface at the contact points of the electrodes and the admittivity distribution inside the examined object are known. The solution of this problem is unique and stable. The forward problem for the real situation of EIT, especially for cases where biological objects are examined, constitutes the solution of partial differential equations with the complex geometry of the highly heterogeneous examined object and non-trivial boundary conditions. Thus, in view of the information presented in [27], [31], the FEM is a feasible method for solving the forward problem The Finite Element Method As mentioned earlier, the use of the FEM to solve the forward problem involves the discretization of both the solution domain and the admittivity inside this domain. Thereby, the basis for solving the forward problem is a finite element model (FE model) consisting of N U nodes, N E elements, and L electrodes positioned on its surface. The set of other necessary source data comprises the admittivity values of each finite element z (e) ; at that, in accordance with the FEM, the admittivity is constant within the finite element. As a result of the forward problem solution, it is vital to determine the voltage values on each electrode of the FE model. To solve the forward problem based on the CEM, the so-called variational equation proposed in [28] and the theory of finite elements presented in [32], [33] are utilized. A detailed description of this approach is provided in [25], with the solution of the problem reduced to solving a system of linear equations that can be written in the matrix form as [ Z] [ U ] [ I]. (4.9) In this equation, U Σ is an unknown vector consisting of values of the electrical potentials u (e) in the nodes of the FE model and values of the voltages U (l) on the electrodes of the model: ( e) u [ U ] ( l), (4.10) U where u (e) = [u (e) 1, u (e) 2,, u (e) NU ] T and U (l) = [U (l) 1, U (l) 2,, U (l) L 1 ] T. The vector I represents the injected electrical current: 0 I ] ( ), (4.11) I [ l where 0 = [0,,0] T (it is a N U 1 zero vector); I (l) = [I 1 (l),, I j (l),, I L 1 (l) ] T ; and the elements of this vector are defined as I ( l) j L l1 I (b ), (4.12) l where I l is the value of the current injected through the l-th electrode, and b j is an L 1 vector describing the injected current pattern of the j-th measurement. Z then denotes the (N U + L 1) (N U + L 1) block matrix of the form j l K M [ Z] T M P. (4.13) - 8 -

9 The elements of the matrices K, M, P are definable by the following formulas: ( e) ( e) ( e) 1 ( l) ( l) k ij z grad Ni grad N j d Ni N j dsl, i, j = 1,..., N U, (4.14) z L l1 l Sl where N i (e) and N j (e) are the FEM basis functions for the i-th and the j-th nodes of the finite element, respectively. In turn, N i (l) and N j (l) are also the FEM basis functions for these nodes provided that they participate in the formation of the surface electrode as mentioned above. We have: L 1 ( l) mij ( b j ) l Ni dsl, i = 1,..., N U, j = 1,..., L 1, z p l1 ij l L Sl (4.15) 1 ( i ) l ( j ) l dsl, z b b i, j = 1,..., L 1. (4.16) l1 l Sl The voltages U FEM on the electrodes are calculated as [ U FEM (l) ] [ c] [ U ], (4.17) where c is an L (L 1) matrix representing the measurement voltage pattern The Inverse Problem As pointed out earlier, the solution of the forward problem of EIT constitutes the basis for solving its inverse problem, which is defined as the reconstruction of the unknown admittivity distribution inside the examined object, based on the voltage measurement data on its surface. Mathematically, this problem can be represented as one of determining the absolute minimum of a suitable objective function. In contrast to the forward problem, the inverse problem does not necessarily have unique and stable solutions. For real cases of EIT, image reconstruction constitutes the imaging of an object exhibiting a complex geometry and internal structure and comprising subregions with a high admittivity gradient. In this thesis, the widely adopted deterministic approach, the Gauss-Newton method, is used to solve the inverse problem Solving the Inverse Problem The principle of solving the problem of image reconstruction lies in minimizing the applicable objective function Ψ(z). One of the most common approaches is the minimization of the squared norm of the difference between the measured boundary voltages and the calculated boundary voltages, the so-called least squares method [34],: 1 2 min zψ ( z) min z UM UFEM, (4.18) 2 where U M is the vector of the measured voltages on the surface electrodes. For the solution of equation (4.18), the Gauss-Newton method is applied. A detailed presentation of the proposed approach is described in [25]

10 However, due to the ill-posed nature of the non-linear inverse problem, the described approach does not provide for the stability of the reconstruction process. The solution stability is reached by introducing a regularization technique, namely, the Tikhonov Regularization method (TRM). The idea of the TRM lies in introducing an additional term, the so-called regularization term, into the minimization of the objective function min zψ ( z) min z UM UFEM Rz, (4.19) 2 2 where α is the regularization parameter and R is the regularization matrix connecting the admittivity values of adjacent elements of the FE model [35]. As a result of the actual solution, the admittivity distribution satisfying the above-mentioned condition of minimization is determined as z ( J ( J T T 1 T T i1 zi i Ji R R) i ( UM UFEM ) R i Rz ). (4.20) where i is the i-th iteration and J is the Jacobian representing the sensitivity of the measured voltage on the electrodes to the admittivity change in each element of the reconstructed image. This sensitivity matrix is defined as U J, (4.21) z where U FEM U and N I is the number of current patterns. Detailed information on the computation of the Jacobian is presented in [36], [25]. This regularization technique ensures fast convergence of the inverse problem and relatively good quality of the reconstructed images. However, admittivity oscillation is often observed in the imaging results, especially in the immediate vicinity of the subregion with an abrupt admittivity change [27]. The stability of the presented TRM algorithm is slightly sensitive to the setting of the initial value of the desired admittivity vector, but it also significantly depends on both the number of unknown values, or the degrees of freedom (DOFs), and the optimal choice of the regularization parameter providing balance between the accuracy and stability of the solution [A3]. In order to obtain a stable solution with the required higher accuracy of the imaging results, it is necessary to introduce an auxiliary (or additional) technique into the reconstruction process The Improved Image Reconstruction Algorithm The aim of the improved image reconstruction technique consists in finding a reliable way to detect regional changes of the biological tissue admittivity within the human body. As a result, it is necessary to identify several aspects, namely, the location and geometry of the regions with changed admittivity and the values of such altered admittivities. This algorithm must eliminate the disadvantages of the existing algorithm, meaning that it must be stable, accurate, and not excessively time-consuming within the reconstruction process. This improved algorithm is shown in Fig

11 Inverse problem Forward problem Finite element method Least squares method + Gauss-Newton method Tikhonov regularization method Auxiliary method TRM algorithm Determination of the voltages on the surface electrodes at a given injected current pattern and admittivity distribution Comparison of the calculated voltages with the real data Improvement of the reconstruction process stability A rough estimate of the regions with different admittivity in the FE model Limitation of regions with inhomogeneity: selection of parts of the model with different admittivity and its immediate surroundings TRM algorithm Determination of the resulting admittivity distribution, namely, the admittivity values inside the limited regions Fig A diagram of the improved reconstruction algorithm The Level Set Method One of the auxiliary techniques is the level set method (LSM). This method enables us to identify regions with different parameter values that can be represented in the form of the electrical property of a given material. Thus, the purpose of the LSM technique in the above-presented algorithm is to determine the interface between regions having different unknown admittivity values. The distribution of unknown admittivity can be described, in terms of the level set function F depending on the position of the point p with respect to the boundary Γ between the regions with different values of admittivity z int and z ext, as zint for { p : F( p) 0} z ( p), : { p : F( p) 0}, zext for { p : F( p) 0} p. (4.22) In turn, the level set function satisfies the partial differential equation corresponding to the Hamilton-Jacobi equation [37]: F n grad F 0, (4.23) t where t represents the time and ν n is the velocity function of the evolving contours (ν) in their outward normal direction, defined as

12 grad F n ν. (4.24) grad F The theoretical background of the LSM is described in detail within [38]. During the iteration process based on minimizing the objective function Ψ(z), the boundary is searched in accordance with the requirement that the z(p) minimize the function Ψ(z). This iteration process can be described generally as follows: 1) Initialize the parameters: set the regularization parameters α, the initial values of admittivity z 0, and the objective function Ψ 0 ; 2) Set the constraints for the admittivity values; 3) Perform the iteration procedure based on the TRM algorithm in accordance with equation (4.20) while the Ψ(z) function is decreasing; 4) Determine the interfaces between the relevant regions with different admittivities, in accordance with equations (4.22) and (4.23); 5) Reduce the number of elements with an unknown admittivity value according to the constraints mentioned in 2); 6) Apply the TRM algorithm in accordance with equation (4.20) inside the limited regions while Ψ(z) is decreasing The Fuzzy Filter Several alternative techniques can be considered for implementation into the image reconstruction processes. One of the feasible methods is based on fuzzy logic. In the improved image reconstruction algorithm, the aim of the fuzzy approach, representing the fuzzy filter (FF), is to identify the geometry and location of the regions with different values of admittivity. The FF can be expressed by the following formula: ( e ) e for { e : min( z ) z max( z )}, e = 1,, N E, (4.25) d d where e represents the finite element, and N E denotes the number of elements adjacent to Ω d, representing a region with different values of admittivity z d. The introduction of the proposed filter into the iteration process predetermines the possible geometry of the regions with different admittivities and their location inside the examined domain, further enabling us to determine the values of these admittivities. During the iteration process based on minimizing the objective function Ψ(z), the geometry and location of these regions are searched in accordance with the requirement that the z(p) minimize the function Ψ(z). The iteration process can be described by the following five steps: 1) Initialize the parameters: set the regularization parameters α, the initial values of admittivity z 0, and the objective function Ψ 0 ; 2) Perform the iteration procedure based on the TRM algorithm in accordance with equation (4.20) while Ψ(z) is decreasing; 3) Determine the regions with different admittivities in accordance with equation (4.25); 4) Apply the TRM algorithm in accordance with equation (4.20) inside the limited regions while Ψ(z) is decreasing; 5) If Ψ(z) > , go to 3). d

13 5. The Models Simulated The numerical simulation results of the proposed algorithm are obtained using three variants of FE models. These models of the electrical properties distribution (the conductivity or admittivity distribution for direct or alternating current simulations, respectively) are applied to simulate the measured voltages on the surface electrodes (U M ). The first model represents a horizontal slice of the human head (Fig. 5.1); the parameters of this 2-D FE model are described in Tab Fig A 2-D FE model of the human head with the arrangement of the biological tissues. Tab The parameters of the models. Model Head model Thorax model Forearm model (Fig. 5.1) (Fig. 5.2) (Fig. 5.3) GEOMETRICAL PARAMETERS Length, x [m] Width, y [m] Height, z [m] PARAMETERS OF THE FE MODEL Type of finite element Triangle Triangle Triangular prism Number of nodes Number of planes 5 Number of elements, (CS 1) /M 2) ) / 2360 / / 1060 Note: 1) CS denotes the number of elements in the cross-section, the so-called plane of the FE model (this parameter is specified for the 3-D FE model). 2) M denotes the number of elements in the FE model

14 The human head model is a simplified model consisting of four homogeneous isotropic subregions: the scalp, skull, and grey (GrM) and white matters (WM). The conductivity values of these biological tissues are presented in Tab Tab The conductivity values of the biological tissues (for direct current measurements). 1 Tissue name Scalp Skull Grey matter White matter Lung (inflated) Heart Body tissue (muscle) Muscle (transverse) Bone Another FE model, shown in Fig. 5.2, is a horizontal slice of the human thorax; the related parameters are presented in Tab This is a 2-D simplified model with three homogeneous isotropic subregions: the lungs, heart, and body tissue (BT). The values of these regional conductivities for the case of direct current simulations are shown in Tab In Tab. 5.3 and Tab. 5.4, the admittivity values of the subregions for simulations with alternating current are presented. Fig A 2-D FE model of the human thorax with the arrangement of the biological tissues. Tissue name Tab The conductivity and relative permittivity values of the biological tissues for the frequencies of 1 khz and 10 khz. 2 Conductivity, σ [S/m] 1 khz 10 khz Relative permittivity, ε r Conductivity, σ [S/m] Relative permittivity, ε r Lung (inflated) Heart Body tissue (muscle) The conductivity values appearing in this table were adopted from [27], [50]. These values are the average values of conductivity presented in the related literature. 2 The admittivity values indicated in this table were adopted from [58]

15 Tissue name Tab The conductivity and relative permittivity values of the biological tissues for the frequencies of 50 khz and 100 khz. 3 Conductivity, σ [S/m] 50 khz 100 khz Relative permittivity, ε r Conductivity, σ [S/m] Relative permittivity, ε r Lung (inflated) Heart Body tissue (muscle) The third model for the simulations is a 3-D finite element representation of the human forearm (Fig. 5.3), whose parameters are described in Tab This FE model embodies a simplified model consisting of bone and muscle homogeneous isotropic subregions. The conductivity values of these subregions are presented in Tab Fig A 3-D FE model of the human forearm with the arrangement of the biological tissues. 6. The Results of the Simulations In this thesis, the improved algorithm proposed in Section is considered for both the 2-D and the 3-D image reconstruction options. At the initial stage, the 2-D simulation results are obtained for the procedure using direct current. In this case, the imaginary part of the above expression (4.2) is neglected, and the conductivity distribution is reconstructed. Then, the given algorithm is implemented for the alternating current simulation (the admittivity distribution imaging). In turn, the 3-D image reconstruction is considered for the direct current simulation The 2-D Simulations In the 2-D image reconstruction, the human thorax (Fig. 5.2) and head (Fig. 5.1) models described in detail within Chapter 5 were used. For the first model, the measured voltages are simulated using 16 surface electrodes (Fig. 6.1(a)). The number of surface electrodes for the head model is 20 (Fig. 6.1(b)). In all the 2-D simulations, trigonometric current patterns are employed for the current injection. In this case, the injected current is given by the following trigonometric function [25]: 3 The admittivity values presented in this table were adopted from [58]

16 I k l cos( kl), sin(( k L/2) l ), l 1,..., L, l 1,..., L, k 1,..., L/2, (6.1) k L/2 1,..., L 1 where ζ l = 2πl/L. This current injection method produces the most homogeneous sensitivity throughout the model in 2-D [39]. (a) (b) Fig The configuration of the surface electrodes for the 2-D human thorax (a) and head (b) models The Influence of the Regularization Parameter and the Number of Unknown Values One of the parameters of the improved algorithm influencing the convergence of the image reconstruction process is the regularization parameter in the TRM algorithm (it is especially important in the first application of the algorithm). The optimal value of this parameter provides the balance between the accuracy and stability of the solution. An analysis targeting the sensitivity of the proposed algorithm to different values of the regularization parameter was performed to determine the optimal value of the parameter. Generally, this analysis is carried out for the case of detecting internal organs via the 2-D simplified model of the human thorax (Chapter 5). The initial value of the conductivity σ 0 corresponds to the conductivity value of the body tissue (Tab. 5.2). The values of the regularization parameters α are assumed in the range of between and 9. During the image reconstruction process, we observe mainly the development of the objective function value, but the reconstruction error defined by expression (6.2) is also included: N U i1 ( ) Err 100 %, (6.2) NU 2 ( ) i1 rec i orig i orig where σ rec and σ orig are the reconstructed and the desired conductivity distributions, respectively. The sensitivity analysis includes two cases. In the first one (1 constraint), a simple constraint is introduced into the iteration process: the value of the reconstructed i

17 conductivity has to be positive or equal to zero. In the second case (2 constraints), an additional constraint is taken into account. During the iteration process, the reconstructed conductivity value can be equal to the conductivity of the body tissue, heart, or lungs. If the reconstructed conductivity is equal to one of these values, then it can be removed from the vector of unknown conductivities, and thus the number of unknown values can be reduced. The results of the sensitivity analysis for the image reconstruction process without and with reduction of the DOFs number after the first application of the TRM algorithm are shown within the graphs presented in Fig The reconstruction process is stable if the values of the regularization parameters remain in the ranges of to for the first case and to 1 for the second one (Fig. 6.3(a)). This result is confirmed by the graph of the conductivity reconstruction error shown in Fig. 6.3(b). For these regularization parameter ranges, the average values of error correspond to 32.64% and 31.57% in the first and the second cases, respectively (Tab. 6.1). The case with reduction of the DOFs number demonstrates the necessity of a smaller number of iterations for almost all values of the regularization parameter from the range (Fig. 6.3(c)). Moreover, in the second case, the number of the DOFs is reduced by 35% on average for this range (Fig. 6.3(d)). These facts further enable us to reduce the time spent on image reconstruction. In Fig. 6.2(a), the conductivity distribution obtained after the first application of the TRM algorithm for the case without reduction of the DOFs number is shown; Fig. 6.2(b) then presents the reconstruction result for the case with reduction of the DOFs number. A simple comparison of the two images indicates that the latter result is clearly more accurate. The number of iterations needed for this reconstruction process is 6, in contrast to the 26 iterations for the first case. In the second case, the number of the DOFs is reduced from 208 to 138 elements. (a) (b) Fig The conductivity distributions after the first application of the TRM algorithm for the cases without (a) and with reduction of the DOFs (b): α = The iteration process defined by equation (4.20) can yield better convergence under the condition that the value of the regularization parameter is dynamically changed during the procedure. The change of the regularization parameter is carried out according to the expression i d, (6.3) 1 i where i is the i-th iteration and dα denotes the value of the dynamic change of the regularization parameter. The results of the sensitivity analysis after the first application of the TRM algorithm for the two above-discussed cases, under the condition of the dynamic change of the regularization parameter, are summarized in Tab

18 DOFs Number of iteration Error, [%] Objective function, Ψ(σ) 1 constraint 1E-04 2 constraints 1E-06 1E-08 1E-07 1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 The regularization parameter, α (a) 1E+04 1E+02 1E+00 1E-02 1 constraint 2 constraints E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 The regularization parameter, α 0 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 The regularization parameter, α (b) 1 constraint 2 constraints (c) constraint 90 2 constraints 40 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 The regularization parameter, α (d) Fig The results of the sensitivity analysis for constant regularization parameters after the first application of the TRM algorithm: the objective function (a), the conductivity reconstruction error (b), the number of iterations (c), and the number of unknown values (d)

19 Tab The results of the sensitivity analysis for the dynamic change of the regularization parameters after the first application of the TRM algorithm. Regularization parameter 1), α Objective function 2), Ψ(σ) Conductivity reconstruction error, Errσ [%] Number of iterations dα 1 constraint 2 constraints 1 constraint 2 constraints 1 constraint 2 constraints 1 constraint 2 constraints min max min max min + max + min + max + min + max + average min + max + average min + max + min + max * Note: * The value of the dynamic change of the regularization parameter corresponding to 1.0 represents the sensitivity analysis for the constant regularization parameters. 2) The values of the minimized objective function. + max/min means maximum/minimum value of the corresponding parameter within the specified range of regularization parameter

20 Wider ranges were obtained for the dynamic change of the regularization parameters than for the constant regularization parameters in both the first and the second cases. In the case with reduction of the DOFs number, the range of the regularization parameter increases only slightly for the dynamic change values of less than or equal to 0.4 and remains stable for all other values (dα 0.4). In turn, in the case with one constraint, changes of the lower limit of the range are also observed within values of the same order. For these ranges, the average values of the conductivity reconstruction error do not exceed 29% and 26.4% in the first and the second cases, respectively (Tab. 6.1). According to the overall trend (Tab. 6.1), it is obvious that the number of iterations required for image reconstruction increases with the increasing value of the dynamic change in both cases. In addition, for a small dynamic change value (dα < 0.4), the number of iterations in the second case corresponds to or is only slightly less than the number of iterations for the first case. With the increasing value of the dynamic change, this trend varies: the case with reduction of the DOFs number demonstrates the necessity of a larger number of iterations for almost all regularization parameter values from the ranges. The dynamic change of the regularization parameters exerts only a minor influence on the reduction of the DOFs number in the second case. For this sensitivity analysis, the number of the DOFs is reduced on average by 33-35% for various dynamic change values. The results presented in Tab. 6.1 show that the dynamic change of the initial value of the regularization parameter (α 0 ) during the iteration process enables us to obtain a more stable solution with a higher accuracy of the reconstruction results. The selection of the optimal values of the regularization parameter and its dynamic change is directly dependent on the actual values of the conductivities and their distribution inside the examined object. As a rule, the correct selection of the regularization parameter results from the combination of a formal approach and intuitive search. In turn, the selection of the optimal value of dα is defined by the balance between the accuracy of the solution and the speed of the image reconstruction process: according to the results presented in Tab. 6.1, the accuracy and stability of the imaging process rise with the increasing dynamic change, but at the same time the number of iterations needed for this process is increased. Based on the considerable number of numerical simulations performed, for the reconstruction of the conductivity distribution under the condition of the dynamic change of the regularization parameter and reduction of the DOFs number during the iteration process (1 st case), the following optimal parameters were selected: α 0 = and dα = 0.7. These parameters enable us to obtain the minimum value of the objective function at the optimal number of iterations. The reconstruction of the conductivity distribution via the improved algorithm using the LSM for the selected optimal parameters is represented in Fig In Fig. 6.5, the results of the reconstruction process performed via the same algorithm with a constant optimal value of the regularization parameters (α 0 = ) and without reduction of the DOFs number (2 nd case) are presented. The parameters and results of these imaging processes are summarized in Tab The presented results show that using the optimal value of the regularization parameters and the related dynamic change in combination with the actual reduction of the DOFs number has a significant influence on the improvement of the stability and accuracy of the image reconstruction process. The results of the sensitivity analysis were published in [A1], [A2], [A3]

21 (a) (b) (c) (d) Fig The original conductivity distribution of the thorax tissues (a); the conductivity distribution after the first application of the TRM algorithm (b), after applying the LSM (c), and after the second application of the TRM algorithm (d) for α 0 = , dα = 0.7. (a) (b) (c) (d) Fig The original conductivity distribution of the thorax tissues (a); the conductivity distribution after the first application of the TRM algorithm (b), after applying the LSM (c), and after the second application of the TRM algorithm (d) for α 0 = , dα =

22 Tab The results of the image reconstruction processes for the optimal values of the regularization parameters. Case 1 st case 2 nd case PARAMETERS OF THE IMAGE RECONSTRUCTION PROCESS Initial value of conductivity, σ 0 [S/m] ) Reduction of the DOFs number yes no Initial value of the regularization parameter, α Dynamic change value, dα RESULTS Objective function, Ψ(σ): TRM 1 /TRM / / # Number of iterations: TRM 1 /TRM 2 47 / / 8 Reconstruction error, Err σ [%]: TRM 1 /LSM/TRM / / / / Number of DOFs + : TRM 1 /LSM 149 / / 165 Time*, [s]: TRM 1 /LSM/TRM / / / / Note: + The number of the DOFs for the FE model of the human thorax is equal to 208. * These results were obtained on 64-bit Windows 7 with a 2.4 GHz Intel dual-core i5 processor. 1) The initial value of conductivity corresponds to the conductivity value of the body tissue. # The value of the objective function after the second application of the TRM algorithm is greater than the value after the first application of the TRM algorithm; this is due to determining the conductivity values inside the limited regions. The TRM 1 /LSM/TRM 2 denote the result after the first application of the TRM algorithm/the LSM/the second application of the TRM algorithm The Influence of Conductivity Changes on the Voltage in the Surface Electrodes The knowledge of the distribution of electrical properties, such as electrical conductivity or admittivity, inside the human body can provide useful information about the state of the relevant biological tissues, namely, changes in the heterogeneous structures of the tissue or in its physiological states and functions. The examples below demonstrate the influence of local changes in the conductivity of head tissues on the voltage measured on the surface electrodes. The voltage distribution on the surface electrodes in the examined healthy tissues of the human head model (Chapter 5) is shown in Fig Fig The voltage distribution on the surface electrodes in the healthy tissues of the human head model

23 Fig. 6.7 shows the generated numerical models of the head with three different sizes of the present tissue defect, namely, a blood clot. The conductivity of the inhomogeneous subregion representing this defect is S/m [40]. The inhomogeneities are located inside the white matter region of the right hemisphere, and their size is given by the number of the finite elements: 1 element, 24 elements, and 120 elements in Fig. 6.7(a), (b), and (c), respectively. The differences between the voltage values on the surface electrodes in the healthy tissues of the human head model and the values obtained for the models with three different sizes of the blood clot are presented in Fig This graph demonstrates that the voltage difference value is directly proportional to the size of the defect. The difference was reached with the electrodes placed on the right-hand side of the head model; this corresponds to the location of this defect. (a) (b) (c) Fig The arrangement of the human head tissues with the defect (a blood clot) depending on the size: 1 finite element (a), 24 finite elements (b), 120 finite elements (c). Fig The voltage difference on the surface electrodes for the human head tissues with three different sizes of the blood clot. Additional related models with a pathological tissue such as a blood clot are shown in Fig The pathological tissue regions are located inside the white matter region of the right hemisphere. The inhomogeneities in Fig. 6.9 exhibit the same size, 50 elements; however, they have different positions. The corresponding voltage differences on the surface electrodes are shown in Fig These graphs show that the approximate positions of the inhomogeneities can be estimated under the condition that the difference between the voltage distribution of the healthy and diseased tissues is known; thus, the number of elements with unknown conductivity can be reduced

24 (a) (b) (c) (d) Fig The arrangement of the human head tissues with the defect (a blood clot) depending on the location: anterior (a), posterior (b), left (c), and right (d). The size of the defect in each model is 50 finite elements. (a) (b) Fig The voltage difference on the surface electrodes for the human head tissues with four different locations of the blood clot: anterior and posterior (a), left and right (b). It is obvious from the performed simulations that the voltage distribution depends on the size and position of the inhomogeneities. The considerable number of numerical tests completed shows that the difference between the voltage distributions of the healthy and diseased tissues enables us to determine the approximate positions of the inhomogeneities and to estimate their sizes. Thus, the number of elements with an unknown conductivity can be reduced. The results are used to significantly simplify the image reconstruction process and to reduce the time spent on materializing the solution. These results were published in an international journal with an impact factor [A8] Reconstructing the Conductivity Distribution The two variants of the improved algorithm, namely, the TRM algorithm in combination with the LSM (TRM 1 LSM TRM 2 ), and the second one is the TRM algorithm with the FF (TRM 1 FF TRM 2 ), are utilized for the reconstruction of the conductivity distribution in the case of blood clot detection. This algorithm is applied under the condition that the arrangement and conductivity values of the healthy tissues are known. The inhomogeneity is located inside the white matter region of the right hemisphere; the size of the defect corresponds to 24 finite elements. The conductivity value of the inhomogeneous subregion representing the blood clot is 0.3 S/m [40]. The original conductivity distribution with this type of tissue defect is indicated in

25 Fig. 6.11(a). For the given reconstruction processes, the following parameters were specified: α 0 = and dα = 0.3. The reconstruction of the conductivity distribution by applying the TRM 1 LSM TRM 2 can be seen in Fig. 6.11(b) and (c). The final conductivity distribution obtained via this improved algorithm is identical with the original distribution (Fig. 6.11(a)). The imaging results for the TRM 1 FF TRM 2 are demonstrated in Fig. 6.11(b) and (e). For this algorithm, after the reconstruction process (after the second use of the TRM algorithm), the same conductivity distribution as the original ones was obtained. (a) (b) (c) (d) (e) Fig The original/final conductivity distribution for the case of blood clot detection (a); the conductivity distribution after the first application of the TRM algorithm (b), after applying the LSM (c), and with the FF (e). The size of the defect in the model is 24 finite elements. The objective functions during the blood clot detection (d). Tab The results of the image reconstruction processes for the case of blood clot detection. Algorithm TRM algorithm 1) TRM 1 LSM TRM 2 TRM 1 FF TRM 2 Objective function, Ψ(σ) Number of iterations Reconstruction error +, Err σ [%] Time*, [s] (0.06 # ) (0.04 # ) Note: + The reconstruction error is defined according to the above equation (6.2). * These results were obtained on 64-bit Windows 7 with a 2.4 GHz Intel dual-core i5 processor. 1) The TRM algorithm is described in Section # The time of the corresponding auxiliary techniques

26 The behavior of the objective functions for each modification of the improved algorithm is presented in Fig. 6.11(d). All the reconstructions were successful, and each iteration process was finished when the objective function equalled to the value presented in Tab Other related cases of blood clot and tumor detection inside the human head were published in [A7] and the international journal [A9]. The simulation results obtained using this improved algorithm for the detection of inhomogeneities inside the human lungs and heart were presented in [A1], [A5]. The considerable number of the conducted numerical tests resulted in the conclusion that the TRM reconstruction algorithm combined with the LSM or the FF outperforms the TRM-based algorithm only by providing better accuracy and stability of the reconstruction process. In the detection of inhomogeneities inside the examined object, the objective function of the improved approach minimizes to a value in the order of for the optimal values of the regularization parameter and its dynamic change, in contrast to the conventional TRM algorithm, whose minimized objective function value is not less than In turn, the reconstruction error of the proposed algorithm has a value in the order of %, and the error of the TRM algorithm is more than 5% Reconstructing the Admittivity Distribution The information carried by the imaginary part of admittivity can contribute to the actual image reconstruction. Thus, the improved algorithm, tested for the conductivity reconstruction, was adapted for the imaging of the admittivity distribution. The imaging process was materialized using the 2-D FE model of the human thorax described in detail within Chapter 5. The simulations were carried out with alternating current at the frequencies of 1, 10, 50, and 100 khz. For all the simulations, the following parameters were specified: α 0 = and dα = 0.9. The initial admittivity value corresponds to the value of the body tissue. The distribution of the conductivity and relative permittivity after the application of the proposed algorithm, namely, its modification using the LSM, can be seen in Fig The results of these reconstruction processes are summarized in Tab Other related imaging results of the internal admittivity distribution of the human thorax tissues were published in [A14]. These completed simulations demonstrate that the stability and accuracy of the admittivity distribution imaging process can be secured by introducing an auxiliary technique. According to the data in Tab. 6.4, the reconstruction error of the conductivity and relative permittivity distribution is more than 2 and 1.4 times less, respectively, when the improved algorithm is used instead of the conventional TRM algorithm. In addition, the use of the admittivity values of the biological tissues in the internal structure imaging of an examined object at various frequencies has significant influence on the increase of the stability and accuracy of this reconstruction process. The reconstruction error values in the conductivity and relative permittivity are about 6.7% and 8.4%, respectively. These results correspond to the case without reduction of the DOFs number. In turn, the reconstruction error of the conductivity distribution of the thorax tissues, using direct current for this case, is equal to 23.32% and 4.42% with reduction of the DOFs number (these imaging results are presented in Section 6.1.1)

27 (a) (b) (c) (d) (e) (f) Relative permittivity, εr Relative permittivity, εr Relative permittivity, εr Relative permittivity, εr (g) (h) Fig The resulting conductivity and relative permittivity distribution of the thorax tissues after the application of the improved algorithm for frequencies: 1 khz (a) and (b), 10 khz (c) and (d), 50 khz (e) and (f), and 100 khz (g) and (h), respectively

28 Tab The results of the admittivity distribution reconstruction processes for the frequencies of 1, 10, 50, and 100 khz. Algorithm TRM algorithm 1) TRM 1 LSM TRM 2 FREQUENCY: 1 khz Objective function module, Ψ(σ) Number of iterations Reconstruction error +, Err σ /Err εr [%]: / / 5.32 Time*, [s] (0.02 # ) FREQUENCY: 10 khz Objective function module, Ψ(σ) Number of iterations 3 92 Reconstruction error +, Err σ /Err εr [%]: / / Time*, [s] (0.02 # ) FREQUENCY: 50 khz Objective function module, Ψ(σ) Number of iterations Reconstruction error +, Err σ /Err εr [%]: / / 8.53 Time*, [s] (0.02 # ) FREQUENCY: 100 khz Objective function module, Ψ(σ) Number of iterations Reconstruction error +, Err σ /Err εr [%]: / / 8.26 Time*, [s] (0.02 # ) Note: + The reconstruction error is defined according to the above equation (6.2). * These results were obtained on 64-bit Windows 7 with a 2.4 GHz Intel dual-core i5 processor. 1) The TRM algorithm is described in Section # The time of the LSM The 3-D Simulations For the 3-D imaging, the FE model of the human forearm (Fig. 5.3) described in detail within Chapter 5 was used. In this case, the measured voltages are simulated by 32 surface electrodes placed in two planes, namely, 16 electrodes in one plane (Fig. 6.13). The opposite method of voltage data collection (Fig. 6.14) is used for the 3-D simulations. Based on the results presented in [39], this method is not inferior to the technique with current injection as a trigonometric current pattern in the efficiency of the 3-D image reconstruction

29 Fig A configuration of the surface electrodes for the 3-D FE model of the human forearm. Fig The opposite method of voltage data collection [41] The Conductivity Distribution Imaging A series of simulations were materialized to estimate the functional efficiency of the proposed improved algorithm in 3-D. We performed the detection of bones with a fracture in one of them, namely, in the right bone (the third plane of the FE model) inside the human forearm. The size of the fracture in the model is 1 finite element. The conductivity value of the defect corresponds to the zero value of conductivity. The initial conductivity value for both the TRM 1 LSM TRM 2 and the TRM 1 FF TRM 2 is S/m (1.3 σ muscle ). The original conductivity distributions for this detection case are shown in Fig For these two reconstruction processes, the following optimal parameters were specified: α 0 = and dα = 0.7. The results of these reconstruction processes are summarized in Tab Tab The results of the image reconstruction processes for the detection of bones: a fracture inside the forearm. Algorithm TRM algorithm 1) TRM 1 LSM TRM 2 TRM 1 FF TRM 2 Objective function, Ψ(σ) Number of iterations Reconstruction error +, Err σ [%] Time*, [s] (0.106 # ) (0.003 # ) Note: + The reconstruction error is defined according to the above equation (6.2). * These results were obtained on 64-bit Windows 7 with a 2.4 GHz Intel dual-core i5 processor. 1) The TRM algorithm is described in Section 0. # The time of the corresponding auxiliary techniques. The reconstruction error obtained by applying the variants of the improved algorithm is less than 1%; in turn, the imaging error for the conventional TRM algorithm is more than 2.5% These results are confirmed by the values of the minimized objective functions presented in Tab The detection of a bone inside the human forearm model is considered in [A15]. The obtained results enable us to conclude that the proposed algorithm is effective in both 3-D and 2-D imaging. This algorithm provides for efficient reconstruction of the spatial distribution of electrical properties inside the object with the necessary resolution and imaging speed

30 The Influence of Noise The influence of noise on the image reconstruction process by applying the improved algorithm was examined and analyzed for the case of imaging the internal conductivity distribution of the human forearm tissues; thus, in other words, we focused on the detection of forearm bones with a fracture in the right bone. The noise analysis was performed using the 3-D FE model of the human forearm (Chapter 5). The size of the fracture located in the third plane of the FE model is 1 finite element. The conductivity value of the defect corresponds to the zero value of conductivity. The original conductivity distribution of the biological tissues for this case can be seen in Fig Further, the analysis was carried out for ±0.01%, ±0.05%, and ±0.10% of noise added to the measured voltage values. The number of reconstructions executed for each noise range is 100. The parameters and results of this noise analysis are summarized in Tab The resulting conductivity distribution with the reconstruction error corresponding to the median values is shown in Fig for the TRM 1 LSM TRM 2 and in Fig for the TRM 1 FF TRM 2. The noise added to the measured voltages has an effect on the reconstruction obtained with the proposed algorithm. As is obvious from the presented results, for noise in the range of ±0.01%, the reconstructed conductivity distributions are almost comparable to the original. Both modifications of the improved algorithm successfully determine the location and geometry of the forearm bones and fracture in the right bone. The conductivity values of these biological tissues were determined at the reconstruction error of 2.39% for the TRM 1 LSM TRM 2 and 2.25% for the TRM 1 FF TRM 2 (Tab. 6.6). In turn, considering the presence of ±0.05% noise in the measured values, the algorithm enables us to establish the location and geometry of the bones and to detect the approximate location of the fracture. The conductivity values are close to the original ones. The reconstruction errors equal to 3.16% and 3.34% for the variants using the LSM and FF, respectively. In the case of ±0.10% of noise, the proposed algorithm determines the approximate location of the bone in the human forearm model. The related imaging errors are 3.91% for the LSM modification and 4.75% for the FF variant. Furthermore, as can be seen in Tab. 6.6, the sensitivity of the algorithm to error in the initial data decreases with increasing values of the regularization parameter. Similar results of the noise analysis were obtained for the other FE model and published in [A4]

31 Tab The parameters and results of the noise analysis. Algorithm TRM1 LSM TRM2 TRM1 FF TRM2 PARAMETERS OF THE IMAGE RECONSTRUCTION PROCESS Noise value, [%] ±0.01 ±0.05 ±0.10 ±0.01 ±0.05 ±0.10 Initial conductivity value, σ0 [S/m] S/m (1.3 σmuscle) Initial regularization parameter value, α0 Dynamic change value, dα RESULTS* Objective function #, Ψ(σ): TRM1/TRM2 Number of iterations: TRM1/TRM2 Reconstruction error +, Errσ [%]: TRM1/AT/TRM / / / 3.35 / / / / 4.76 / / / / / / / 6.71 / 4.75 Note: * The presented results are the median values of the corresponding parameters (the number of reconstructions for each noise range is 100). # The objective function value after the second application of the TRM algorithm is greater than the value after the first application of the same algorithm. This is due to the determination of the conductivity values inside the limited regions. + The reconstruction error is defined according to the above equation (6.2). The TRM1/AT/TRM2 denote the result after the first application of the TRM algorithm/the corresponding auxiliary technique (AT)/the second application of the TRM algorithm. 3 / / 7.62 / / / 2.89 / / / 4.08 /

32 st 1 plane 2 nd plane rd 3 plane th 4 plane th 5 plane Original ±0.01% Fig The original and the resulting conductivity distribution obtained with the TRM1 LSM TRM2 algorithm for ±0.01%, ±0.05%, and ±0.1% of noise added ±0.05% ±0.10%

33 st 1 plane 2 nd plane rd 3 plane th 4 plane th 5 plane Original ±0.01% Fig The original and the resulting conductivity distribution obtained with the TRM1 FF TRM2 algorithm for ±0.01%, ±0.05%, and ±0.1% of noise added ±0.05% ±0.10%

34 7. Image Reconstruction Based on Magnetic Field Data In this chapter, a direct method for reconstructing the internal conductivity is proposed; it is necessary to know only one component of the magnetic flux density data when injecting a current into the examined object through a single pair of surface electrodes. The proposed method reconstructs the density of the projected current, namely, a uniquely determined current from the measured one component of the magnetic flux density outside the given object. Using the relationship between the electric field and the current density, based on Ohm s law, an additional condition can be introduced to the implicit matrix system for the determination of the internal conductivity distribution. The described techniques utilize internal information on the induced magnetic field in addition to the boundary current-voltage measurements to produce images of the internal current density and conductivity distribution. Thus, a new procedure to obtain the distribution without knowing the voltage data on the boundary of the examined object is presented. This technique can be conveniently applied to identify the location of regions with different conductivity values or to identify local changes of these values The Theoretical Background The current density J in a linear medium with the interior electrical conductivity σ can be obtained from the electric field E or the corresponding potential distribution u J E grad u. (7.1) The impedance tomography problem consists in reconstructing the conductivity distribution satisfying the continuity equation divj 0. (7.2) In considering a 2-D numerical model with linear triangular elements, the equation (7.1) is approximated from the nodal values of the potential u i using the linear approximation functions N i (e) on a mesh of linear triangular finite elements: N u u i N i. (7.3) U i1 Then, the current density J (e) on each finite element (e) can be described by ( e) ( e) 3 j1 ( e) j ( e) ( e) j J u grad N, (7.4) where σ (e) (e) is the conductivity value of a finite element, and N j denotes the linear approximation function for the j-th node of the finite element. The magnetic flux density B, inducted by the injected current into the 2-D electrically conductive object and corresponding to J, can be described using the Biot

35 Savart law. If the magnetic flux density B due to the injection current is known from the measurements, an image of the corresponding internal current density distribution J can be obtained from, for example, Ampere s law: J rot B /, (7.5) where μ 0 is the permeability of vacuum. This method naturally requires the knowledge of all the magnetic field components; therefore, in the following text, a simpler procedure will be proposed. The magnetic field B i can be calculated numerically at the general point given by the coordinates [x i, y i, z i ], using the Biot-Savart law and the principle of superposition: 0 B( r) 4 B i 0 4 S N 0 J( r) D( r, r) ds, 3 (7.6) D ( r, r) E j1 J D j D 3 ij ij S j, (7.7) where J j is the current density of the j-th finite element, S j represents the area of the j-th element, and D ij is the distance between the centre of the j-th element with the coordinates [x j, y j, z j ] and the general point [x i, y i, z i ]. The components of the magnetic field can be expressed as B xij J D 0 yj zij 3 4 Dij B zij S ( J j, D B yij J J D 0 xj zij 3 4 Dij D ) 0 xj yij yj xij 3 4 Dij S j. S j, (7.8) It is obvious that, to obtain the N E pair of the J x and J y current density components, we need to know either the same number of the B x and B y components or the double number of the B z component of the magnetic field. Suppose that the 2 N E values of B z are known, with each of them given by B zi N E j1 ( K Dyij J xj K Dxij J yj ), i = 1,, 2 N E. (7.9) The matrix notation of the 2 N E algebraic equations is J x KD KD Bz KD J B x y. (7.10) J y From equation (7.10), we can obtain the source current density distribution in the form 1 J K D B. (7.11) The conductivity reconstruction process lies in minimizing the difference between the current densities J M and J FEM on the elements. The J FEM vector corresponds to the calculated voltage U FEM from the above equation (4.17). The vector can be computed using U FEM and equation (7.4). The J M vector is obtainable from the measured magnetic field value using equation (7.11). Thus, the objective function J () can be written as

36 Ψ J ( σ) JM JFEM Rσ. (7.12) 2 2 The meaning of the parameters α and R is the same as in Section 0. As a result of solving equation (7.12), the iteration procedure is obtained: σ σ T T 1 T T ( J J R R) ( J ( J J ) R Rσ ), (7.13) i1 i Ji Ji Ji M FEM i where i is the i-th iteration and J J is the corresponding Jacobian: J J FEM J. (7.14) σ From the difference of the current density J in the samples without and with inhomogeneity, the conductivity distribution can be obtained using the conditions on the boundary between two media with different conductivities. It is this difference of the current density vectors, as the results show [A6], that enables us to obtain significant information about the influence of inhomogeneity on the magnetic field distribution outside the examined object The Numerical Simulation The results of the numerical simulation performed within the process of detecting a pathological tissue were obtained using a 2-D FE model representing a simplified horizontal slice of the human head described in detail in Chapter 5. Fig. 7.1 demonstrates two different reconstruction results related to the detection of several small brain tumors. A tumor is represented by six finite elements with the conductivity of S/m. The original conductivity distribution for this detection case is shown in Fig. 7.1(b). In Fig. 7.1(a) and (b), we can observe the final conductivity distribution obtained through minimizing the objective functions U and J given by equations (4.19) and (7.12), respectively. The behavior of the objective functions during the reconstruction process is shown in Fig (a) (b) Fig The resulting conductivity distribution for the case of brain tumor detection via the TRM algorithm (a) and the approach based on using the z-component of the magnetic flux density (b). The defect is represented by 6 finite elements. In another example, reduction of the DOFs number was introduced into both the above-mentioned algorithms. The images in Fig. 7.3(a) and (b) present the

37 reconstruction results obtained by minimizing the U and J functions, respectively. The conductivity distribution, reconstructed by means of the approach based on using the z-component of the magnetic flux density, corresponds the original one. The behavior of the object functions U and J during the reconstruction process is similar to the characteristics shown in Fig The time needed to perform these reconstruction processes was significantly reduced in both algorithms: three times in the conventional TRM algorithm and fifteen times in the approach based on the magnetic field data. Fig The behavior of the objective functions in the case of brain tumor detection. (a) (b) Fig The resulting conductivity distribution in the detection of three brain tumors via the TRM algorithm (a), and the approach based on using the z-component of the magnetic flux density (b) with reduction of the DOFs. All the obtained reconstruction results showed that, in contrast to the conventional TRM algorithm, the procedure based on using the z-component of the magnetic flux density steadily and reliably determines the conductivity changes in an imaging slice. The high degree of accuracy was achieved by applying this algorithm, namely, the same conductivity distributions as the original ones were obtained as a result of imaging in all the given cases. The presented results and the outcome of the reconstruction of conductivity changes in the heart and lungs within the related 2-D FE human thorax model were published in an international journal with an impact factor [A13]. Further, similar results were obtained for the case of skull trauma detection [A12]. Another paper, [A10], then demonstrated the efficiency of using the approach based on measuring a magnetic field component to monitor brain tumor evolution

38 In another case of simulating blood clot detection, the defect, whose size is represented by seven finite elements, is located inside the white matter region of the right hemisphere. The original conductivity distribution with the related inhomogeneity is shown in Fig. 7.4(a). The reconstruction results in this process were obtained by applying both the above-mentioned algorithms and the modifications of the improved approach. The simulated results, which demonstrate the reconstruction process using TRM 1 LSM TRM 2, can be seen in Fig. 7.4(a) (c). The imaging results for the TRM 1 FF TRM 2 are demonstrated in Fig. 7.4(b) (e) (a), respectively. The behaviour of the objective functions for each variant of the improved algorithm is presented in Fig. 7.4(d). (a) (b) (c) (d) (e) Fig The original/final conductivity distribution for the case of blood clot detection (a); the conductivity distribution after the first application of the TRM algorithm (b), after applying the LSM (c), and with the FF (e). The size of the defect in the model is 7 finite elements. The objective functions during the blood clot detection procedure (d). The same conductivity distribution as the original one (Fig. 7.4(a)) was obtained through imaging with the algorithm based on the measurement of the z-component of the magnetic flux density. The behavior of the objective functions for this reconstruction process is presented in Fig The relevant results were published in [A11]